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abstract: 'We deal with a degenerate model in divergence form describing the dynamics of a population depending on time, on age and on space. We assume that the degeneracy occurs in the interior of the spatial domain and we focus on null controllability. To this aim, first we prove Carleman estimates for the associated adjoint problem, then, via cut off functions, we prove the existence of a null control function localized in the interior of the space domain. We consider two cases: either the control region contains the degeneracy point $x_0$, or the control region is the union of two intervals each of them lying on one side of $x_0$. This paper complement some previous results, concluding the study of the subject.'
address: |
Dipartimento di Matematica\
Università di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari - Italy
author:
- Genni Fragnelli
title: Controllability for a population equation with interior degeneracy
---
*Dedicated to Irena, on the occasion of her 70th birthday,*
*with sincere esteem*
Introduction
============
We consider the following degenerate population model describing the dynamics of a single species: $$\label{1}
\begin{cases}
\displaystyle {\frac{\partial y}{\partial t}+\frac{\partial y}{\partial a}}
- (ky_x)_x+\mu(t, a, x)y =f(t,a,x)\chi_{\omega} & \quad \text{in } Q,\\
y(t, a, 1)=y(t, a, 0)=0 & \quad \text{on }Q_{T,A},\\
y(0, a, x)=y_0(a, x) &\quad \text{in }Q_{A,1},\\
y(t, 0, x)=\int_0^A \beta (a, x)y (t, a, x) da &\quad \text{in } Q_{T,1}
\end{cases}$$ where $Q:=(0,T)\times(0,A)\times(0,1)$, $Q_{T,A} := (0,T)\times (0,A)$, $Q_{A,1}:=(0,A)\times(0,1)$ and $Q_{T,1}:=(0,T)\times(0,1)$. Here $y(t,a,x)$ is the distribution of certain individuals at location $x \in (0,1)$, at time $t\in(0,T)$, where $T$ is fixed, and of age $a\in (0,A)$. $A$ is the maximal age of life, while $\beta$ and $\mu$ are the natural fertility and the natural death rate, respectively. Thus, the formula $\int_0^A \beta y da$ denotes the distribution of newborn individuals at time $t$ and location $x$. The function $k$, which is the dispersion coefficient, depends on the space variable $x$ and we assume that it degenerates in an interior point $x_0$ of the state space. In particular, we say that
\[Ass0\] The function $k$ is [**weakly degenerate (WD)**]{} if there exists $x_0 \in (0,1)$ such that $k(x_0)=0$, $k>0$ on $[0, 1]\setminus \{x_0\}$, $k\in
W^{1,1}(0,1)$ and there exists $M\in (0,1)$ so that $(x-x_0)k'
\le Mk$ a.e. in $[0,1]$.
\[Ass01\] The function $k$ is [**strongly degenerate (SD)**]{} if there exists $x_0 \in (0,1)$ such that $k(x_0)=0$, $k>0$ on $[0, 1]\setminus \{x_0\}$, $k\in
W^{1, \infty}(0,1)$ and there exists $M \in [1,2)$ so that $(x-x_0)k' \le M k$ a.e. in $[0,1]$.
For example, as $k$ one can consider $k(x)=|x-x_0|^\alpha$, $\alpha >0$.
Finally, in the model, $ \chi_\omega$ is the characteristic function of the control region $\omega\subset(0,1)$ which can contain $x_0$ or can be the union of two intervals each of them lying on different sides of the degeneracy point, more precisely: $$\omega= \omega_1 \cup \omega_2$$ where $$\omega_i= (\lambda_i , \beta_i ) \subset (0, 1), i = 1, 2, \text{ and } \beta_1 < x_0 < \lambda_2.$$
It is known that the asymptotic behavior of the solution for the Lotka-McKendrick system depends on the so called net reproduction rate $R_0$: indeed the solution can be exponentially growing if $R_0> 1$, exponentially decaying if $R_0<1$ or tends to the steady state solution if $R_0=1$. Clearly, if the system represents the distribution of a damaging insect population or of a pest population and $R_0>1$, it is very worrying. For this reason, recently great attention is given to null controllability issues. For example in [@he], where models an insect growth, the control corresponds to a removal of individuals by using pesticides. If [*$k$ is a constant or a strictly positive function*]{}, null controllability for is studied, for example, in [@An]. If [*$k$ degenerates at the boundary or at an interior point of the domain*]{} and $y$ is independent of $a$ we refer, for example, to [@acf], [@fm], [@fm1] and to [@fm2], [@fm_hatvani], [@fm_opuscola] if $\mu$ is singular at the same point of $k$. Actually, [@aem] is the first paper where $y$ depends on $t$, $a$ and $x$ and the dispersion coefficient $k$ degenerates. In particular, in [@aem], $k$ degenerates at the boundary of the domain (for example $k(x) = x^\alpha,$ being $x \in (0,1)$ and $\alpha >0$). Using Carleman estimates for the adjoint problem, the authors prove null controllability for under the condition $T\ge A$. The case $T<A$ is considered in [@idriss], [@em], [@f_anona] and [@fJMPA]. In [@em] the problem is always in [*divergence form*]{} and the authors assume that $k$ degenerates only at a point of the boundary; moreover, they use the fixed point technique in which the birth rate $\beta$ must be of class $C^2(Q)$ (necessary requirement in the proof of [@em Proposition 4.2]). A more general result is obtained in [@f_anona] where $\beta$ is only a continuous function, but $k$ can degenerate at both extremal points. In [@idriss] the problem is in [*divergence form*]{} and $k$ degenerates at an interior point and it belongs to $C[0,1]\cap C^1([0,1]\setminus\{x_0\})$. Finally, in [@fJMPA], we studied null controllability for in [*non divergence form*]{} and with a diffusion coefficient degenerating at a one point of the boundary domain or in an interior point. In this paper we study the null controllability for assuming that $k$ degenerates at $x_0 \in (0,1)$ and $T<A$ or $T >A$. We underline that here, contrary to [@idriss], the function $k$ is less regular, the control region $\omega$ not only can contain $x_0$, but can be also the union of two intervals each of them lying on one side of $x_0$ and $T$ can be greater than $A$. Moreover, contrary to [@aem], where $T>A$ and $k$ degenerates at the boundary, here we assume that $T$ can be smaller than $A$ and $k$ degenerates at $x_0 \in (0,1)$. Hence, this paper is the completion of all the previous ones. Moreover, the technique used in Theorem \[CorOb1’\] can be also applied either when $k$ degenerates at the boundary of the domain, completing [@f_anona], or when $k$ is in non divergence form and $k$ degenerates at the boundary or in the interior of the domain, completing [@fJMPA]. Finally, observe that in this paper, as in [@f_anona] or in [@fJMPA], we do not consider the positivity of the solution, even if it is clearly an interesting question to face: this problem is related to the minimum time, i.e. $T$ cannot be too small (see [@zuazua] for related results in non degenerate cases. This topic will be the subject of further investigations.
A final comment on the notation: by $c$ or $C$ we shall denote [*universal*]{} strictly positive constants, which are allowed to vary from line to line.
Well posedness results {#sec3-1}
======================
For the well posedness of the problem, we assume the following hypotheses on the rates $\mu$ and $\beta$ :
\[ratesAss\] The functions $\mu$ and $\beta$ are such that $$\label{3}
\begin{aligned}
&\bullet \beta \in C(\bar Q_{A,1}) \text{ and } \beta \geq0 \text{ in } Q_{A,1}, \\
&\bullet \mu \in C(\bar Q) \text{ and } \mu\geq0\text{ in } Q.
\end{aligned}$$
To prove well possessedness of , we introduce, as in [@fm1], the following Hilbert spaces $$\begin{aligned}
H^1_{k} (0,1):=\Big\{ u \in W^{1,1}_0(0,1) \,:\, \sqrt{k} u' \in L^2(0,1)\Big\}
\end{aligned}$$ and $$H^2_k := \{ u \in H^1_k(0,1) |\,ku_x \in
H^1(0,1)\}.$$ We have, as in [@fm1], that the operator $$\mathcal A_0u:= (ku_{x})_x,\qquad D(\mathcal A_0): = H^2_{k}(0,1)$$ is self–adjoint, nonpositive and generates an analytic contraction semigroup of angle $\pi/2$ on the space $L^2(0,1)$.
As in [@f_anona], setting $ \mathcal A_a u := \ds \frac{\partial u}{\partial a}$, we have that $$\mathcal Au:= \mathcal A_a u -
\mathcal A_0 u,$$ for $$\begin{aligned}
u \in D(\mathcal A) =&\left\{u \in L^2(0,A;D(\mathcal A_0)) : \frac{\partial u}{\partial a} \in L^2(0,A;H^1_k(0,1)), \right. \\&
\left.\quad u(0, x)= \int_0^A \beta(a, x) u(a, x) da\right\},
\end{aligned}$$ generates a strongly continuous semigroup on $L^2(Q_{A,1}):= L^2(0,A; L^2(0,1))$ (see also [@iannelli]). Moreover, the operator $B(t)$ defined as $$B(t) u:= \mu(t,a,x) u,$$ for $u \in D(\mathcal A)$, can be seen as a bounded perturbation of $\mathcal A$ (see, for example, [@acf]); thus also $
(\mathcal A + B(t), D(\mathcal A))
$ generates a strongly continuous semigroup.
Setting $L^2(Q):= L^2(0,T;L^2(Q_{A,1}))$, the following well posedness result holds (see [@f_anona] for the proof):
\[theorem\_existence\] Assume that $k$ is weakly or strongly degenerate at $0$ and/or at $1$. For all $f \in
L^2(Q)$ and $y_0 \in L^2(Q_{A,1})$, the system admits a unique solution $$y \in \mathcal U:= C\big([0,T];
L^2(Q_{A,1}))\big) \cap L^2 \big(0,T;
H^1(0,A; H^1_k(0,1))\big)$$ and $$\label{stimau}
\begin{aligned}
\sup_{t \in [0,T]} \|y(t)\|^2_{L^2(Q_{A,1})} &+\int_0^T\int_0^A\|\sqrt{k}y_x\|^2_{L^2(0,1)}dadt \\
&\le C \|y_0\|^2_{L^2(Q_{A,1})} + C\|f\|^2_{L^2(Q)},
\end{aligned}$$ where $C$ is a positive constant independent of $k, y_0$ and $f$.
In addition, if $f\equiv 0$, then $
y\in C^1\big([0,T];L^2(Q_{A,1})\big).
$
Carleman estimates {#sec-3}
==================
In this section we show degenerate Carleman estimates for the following adjoint system associated to : $$\label{adjoint}
\begin{cases}
\ds \frac{\partial z}{\partial t} + \frac{\partial z}{\partial a}
+(k(x)z_{x})_x-\mu(t, a, x)z =f ,& (t,a,x) \in Q,\\
z(t, a, 0)=z(t, a, 1)=0, & (t,a) \in Q_{T,A},\\
z(t,A,x)=0, & (t,x) \in Q_{T,1}.
\end{cases}$$
On $k$ we make additional assumptions:
\[BAss01\] The function $k$ is [**(WD)**]{} or [**(SD)**]{}. Moreover, if $M > \ds \frac{4}{3}$, then there exists a constant $\theta \in \left(0, M\right]$ such that $$\label{dainfinito_1}
\begin{array}{ll}
x \mapsto \dfrac{k(x)}{|x-x_0|^{\theta}} &
\begin{cases}
& \mbox{ is non increasing on the left of $x=x_0$,}\\
& \mbox{ is non decreasing on the right of $x=x_0$}.
\end{cases}
\end{array}$$ In addition, when $ M >\displaystyle \frac{3}{2}$ the function in is bounded below away from $ 0$ and there exists a constant $\Gamma >0$ such that $$\label{Sigma}
|k'(x)|\leq \Gamma |x-x_0|^{2\theta-3} \mbox{ for a.e. }x\in
[0,1].$$
Now, let us introduce the weight function
$$\label{13}
\varphi(t,a,x):=\Theta(t,a)\psi(x),$$
where $$\label{theta}
\Theta(t,a):=\frac{1}{[t(T-t)]^4a^4}\quad \text{and}\quad
\psi(x) :=
c_1\left[\int_{x_0}^x \frac{y-x_0}{k(y)}dy- c_2\right].$$ The following estimate holds:
\[Cor1\] Assume that Hypothesis $\ref{BAss01}$ is satisfied. Then, there exist two strictly positive constants $C$ and $s_0$ such that every solution $v$ of in $$\mathcal{V}:=L^2\big(Q_{T,A}; H^2_k(0,1)\big) \cap H^1\big(0,T; H^1(0,A;H^1_k(0,1))\big)$$ satisfies, for all $s \ge s_0$, $$\begin{aligned}
&\int_{Q} \left(s\Theta k (v_x)^2 + s^3 \Theta^3
\frac{(x-x_0)^2}{k}v^2\right)e^{2s\varphi}dxdadt\\
&\le C\left(\int_{Q} f^{2}e^{2s\varphi}dxdadt +
sc_1\int_0^T\int_0^A\left[k\Theta e^{2s\varphi}(x-x_0)(v_x)^2
dadt\right]_{x=0}^{x=1}dadt\right)
\end{aligned}$$
Clearly the previous Carleman estimate holds for every function $v$ that satisfies in $(0,T)\times(0,A)\times (B,C)$ as long as $(0,1)$ is substituted by $(B,C)$ and $k$ satisfies Hypothesis \[BAss01\] in $(B,C)$.
#### Proof of Theorem \[Cor1\]
The proof of Theorem \[Cor1\] follows the ideas of the one of [@f_anona Theorem 3.1] or [@fJMPA Theorem 3.6] (for the non divergence case). As in the previous papers, we consider, first of all, the case when $\mu\equiv 0$: for every $s> 0$ consider the function $$w(t,a, x) := e^{s \varphi(t,a, x)}v(t,a, x),$$ where $v$ is any solution of in $\mathcal{V}$, so that also $w\in\mathcal{V}$, since $\varphi<0$. Moreover, $w$ satisfies $$\label{1'}
\begin{cases}
(e^{-s\varphi}w)_t + (e^{-s\varphi}w)_a +(k (e^{-s\varphi}w)_x)_{x} =f(t,a,x), & (t,x) \in
Q,
\\[5pt]
w(0, a, x)= w(T,a, x)= 0, & (a,x) \in Q_{A,1},
\\[5pt]
w(t,A,x)=w(t,0,x)=0, & (t,x) \in Q_{T,1},
\\[5pt]
w(t, a,0)= w(t, a, 1)= 0, & (t,a) \in Q_{T,A}.
\end{cases}$$ and [@f_anona Lemma 3.1] still holds. In particular, setting $$\begin{cases}
L^+_sw :=( kw_{x})_x
- s (\varphi_t+ \varphi_a) w + s^2k \varphi_x^2 w,
\\[5pt]
L^-_sw := w_t + w_a-2sk\varphi_x w_x -
s(k\varphi_{x})_xw,
\end{cases}$$we have
\[lemma1\]\[see [@f_anona Lemma 3.1]\] Assume Hypothesis $\ref{BAss01}$. The following identity holds $$\label{D&BT}
\left.
\begin{aligned}
<L^+_sw,L^-_sw>_{L^2(Q)}
\;&=\;
\frac{s}{2} \int_Q(\varphi_{tt}+\varphi_{aa}) w^2dxdadt \\
&+ s \int_Qk(x) (k(x)
\varphi_x)_{xx} w w_xdxdadt
\\&- 2s^2 \int_Qk \varphi_x \varphi_{tx}w^2dxdadt - 2s^2\int_{Q}k \varphi_x\varphi_{xa}w^2dxdadt\\
&+s
\int_Q(2 k^2\varphi_{xx} + kk'\varphi_x)w_x^2 dxdadt
\\
&+ s^3 \int_Q(2k \varphi_{xx} + k'\varphi_x)k
\varphi^2_x w^2dxdadt \\
&+s\int_{Q}\varphi_{at}w^2 dxdadt.
\end{aligned}\right\}\;\text{\{D.T.\}}$$ $$\nonumber
\hspace{55pt}
\text{\{B.T.\}}\;\left\{
\begin{aligned}
&
\int_{Q_{T,A}}[kw_xw_t]_{0}^{1} dadt+\int_{Q_{T,A}}\big[kw_xw_a\big]_0^{1}dadt\\& -\frac{s}{2}\int_{Q_{A,1}} \left[\varphi_a w^2\right]_0^T dxda.\\& +
\int_{Q_{T,A}}[-s\varphi_x (k(x)w_x)^2 +s^2k(x)\varphi_t \varphi_x w^2\\& -
s^3 k^2\varphi_x^3w^2 ]_{0}^{1}dadt\\
& +
\int_{Q_{T,A}}[-sk(x)(k(x)\varphi_x)_xw w_x]_{0}^{1}dadt\\&+ s^2 \int_{Q_{T,A}}\big[k\varphi_x\varphi_aw^2\big]_0^{1}dadt\\[3pt]&
-\frac{1}{2}\int_{Q_{T,1}} \big[kw_x^2\big]_0^Adxdt
+\frac{1}{2}\int_{Q_{T,1}}\big[ \big(s^2k \varphi_x^2 \\&- s (\varphi_t+\varphi_a) \big)w^2\big]_0^Adxdt.\end{aligned}\right.$$
We underline the fact that in this case all integrals and integrations by parts are justified by the definition of $D(\mathcal A)$ and the choice of $\varphi$, while, if the degeneracy is at the boundary of the domain as in [@f_anona], they were guaranteed by the choice of Dirichlet conditions at $x=0$ or $x=1$, i.e. where the operator is degenerate.
As a consequence of the definition of $\varphi$, one has the next estimate:
\[lemma2\]Assume Hypothesis $\ref{BAss01}$. There exist two strictly positive constants $C$ and $s_0$ such that, for all $s\ge s_0$, all solutions $w$ of satisfy the following estimate $$sC\int_{Q}\Theta k w_x^2 dxdadt
+s^3C\int_{Q}\Theta^3 \frac{(x-x_0)^2}{k}w^2 dxdadt \le \big\{D.T.\big\} .$$
Using the definition of $\varphi$, the distributed terms given in Lemma \[lemma1\] take the form $$\text{\{D.T.\}}=\;\left\{\begin{aligned}
&\frac{s}{2}\int_{Q}(\Theta_{tt}+ \Theta_{aa})\psi w^2 dxdadt
-2s^2c_1\int_{Q}\Theta{\Theta_t}\frac{(x-x_0)^2}{k}w^2 dxdadt\\
&-2s^2c_1\int_{Q}\Theta{\Theta_a}\frac{(x-x_0)^2}{k}w^2 dxdadt\\
&+ sc_1 \int_{Q}\Theta \left(2-\frac{k'}{k}
(x-x_0)\right)k(w_x)^2 dxdadt\\
&+ s^3c_1^3\int_{Q}\Theta^3\left(2-\frac{k'}{k}
(x-x_0)\right)\frac{(x-x_0)^2}{k}w^2dxdadt\\
&+ s \int_{Q}\Theta_{ta}\psi w^2 dxdadt.
\end{aligned}\right.$$ Because of the choice of $\varphi(x)$, one has, as in [@fm1], $$2-\frac{(x-x_0)k'}{k}\ge 2-M \quad \text{a.e. } \; x\in[0,1].$$ Thus, there exists $C>0$ such that, the distributed terms satisfy the estimate $$\label{aaaaa}
\begin{aligned}
\{D.T.\} &\ge\frac{s}{2}\int_{Q}(\Theta_{tt}+ \Theta_{aa})\psi w^2 dxdadt -s^2C\int_{Q}|\Theta\Theta_t|\frac{(x-x_0)^2}{k}w^2
dxdadt\\
&-s^2C\int_{Q}|\Theta\Theta_a|\frac{(x-x_0)^2}{k}w^2
dxdadt\\
&+ s C\int_{Q}\Theta (w_x)^2 dxdadt+ s^3C\int_{Q}\Theta^3\frac{(x-x_0)^2}{k}w^2
dxdadt\\
&+s\int_{Q}\Theta_{ta}\psi w^2 dxdadt.
\end{aligned}$$ By [@fJMPA Lemma 3.5], we conclude that, for $s$ large enough, $$\begin{aligned}
s^2C\int_{Q}(|\Theta\Theta_t|+|\Theta \Theta_a|)\frac{(x-x_0)^2}{k}
w^2 dxdadt&\le C s^2
\int_{Q}\Theta^3\frac{(x-x_0)^2}{k}w^2 dxdadt\\
&\le \frac{C^3}{4}s^3\int_{Q}\Theta^3 \frac{(x-x_0)^2}{k}w^2 dxdadt.
\end{aligned}$$ Again as in [@fm1 Lemma 4.1], we get $$\label{quasfin}
\begin{aligned}
\left| \frac{s}{2}\int_{Q}(\Theta_{tt} + \Theta_{aa})\psi
w^2dxdadt \right|
&\leq sC\int_Q\Theta ^{3/2} w^2 dxdadt
\\&
\le \frac{C}{4}s\int_{Q} \Theta k (w_x)^2
dxdadt
\\&+ \frac{C^3}{4}s^3\int_{Q}\Theta^3
\frac{(x-x_0)^2}{k}w^2 dxdadt.
\end{aligned}$$ Analogously, one has that the last term in , i.e. $s\int_{Q} \Theta_{ta}\psi w^2 dxdadt$ satisfies $$\begin{aligned}
\left|s\int_{Q} \Theta_{ta}\psi w^2 dxdadt\right| &\le \frac{C}{4}s\int_{Q} \Theta k (w_x)^2
dxdadt \\
&+ \frac{C^3}{4}s^3\int_{Q}\Theta^3
\frac{(x-x_0)^2}{k}w^2 dxdadt.
\end{aligned}$$ Summing up, we obtain $$\begin{aligned}
\{D.T.\}&\ge -\frac{C}{4}s\int_{Q} \Theta (w_x)^2 dxdadt -
\frac{C^3}{4}s^3\int_{Q}\Theta^3
\left(\frac{x-x_0}{k}\right)^2w^2 dxdadt \\
& -\frac{C^3}{4}s^3\int_{Q}\Theta^3 \left(\frac{x-x_0}{k}
\right)^2w^2 dxdadt
\\&
+ s C\int_{Q}\Theta (w_x)^2 dxdadt+ s^3C\int_{Q}\Theta^3\left(\frac{x-x_0}{k} \right)^2w^2 dxdadt\\&
-\frac{C}{4}s\int_{Q} \Theta (w_x)^2 dxdadt -
\frac{C^3}{4}s^3\int_{Q}\Theta^3(w_x)^2 dxdadt
\\&
\ge \frac{C}{4}s\int_{Q} \Theta (w_x)^2 dxdadt +\frac{C^3}{4}s^3
\int_{Q}\Theta^3 \left(\frac{x-x_0}{k} \right)^2 w^2 dxdadt.
\end{aligned}$$
Proceeding as in [@f_anona] and in [@fm1], one has for the boundary terms the following lemma:
\[lemma41\] Assume Hypothesis $\ref{BAss01}$. The boundary terms in reduce to $$-sc_1\int_0^T\int_0^A\Theta(t)k\Big[(x-x_0)(w_x)^2\Big]_{x=0}^{x=1}dadt.$$
By Lemmas \[lemma1\]-\[lemma41\], there exist $C>0$ and $s_0>0$ such that all solutions $w$ of satisfy, for all $s \ge s_0$, $$\begin{aligned}
& s\int_{Q}\Theta k w_x^2 dxdadt
+s^3\int_{Q}\Theta^3 \frac{(x-x_0)^2}{k}w^2 dxdadt \\
&\le C\left(\int_{Q} f^2 e^{2s\varphi}dxdadt+
sc_1\int_0^{T}\int_0^A \left[\Theta k(x)
(x-x_0)(w_x)^2\right]_{x=0}^{x=1}dadt \right).
\end{aligned}$$
Hence, if $\mu \equiv 0$, Theorem \[Cor1\] follows recalling the definition of $w$ and the fact that $$L^+_sw + L^-_sw=e^{s\varphi}f,$$
If $\mu \not \equiv 0$, we consider the function $\overline{f}=f+\mu v$. Hence, there are two strictly positive constants $C$ and $s_0$ such that, for all $s\geq s_0$, the following inequality holds $$\label{fati1?}
\begin{aligned}
&\int_{Q} \left(s\Theta k (v_x)^2 + s^3 \Theta^3
\frac{(x-x_0)^2}{k}v^2\right)e^{2s\varphi}dxdadt\\
&\le C\left(\int_{Q} \bar f^{2}e^{2s\varphi}dxdadt +
s\int_0^T\int_0^A\left[k\Theta e^{2s\varphi}(x-x_0)(v_x)^2
dadt\right]_{x=0}^{x=1}dadt\right).
\end{aligned}$$ On the other hand, we have $$\label{4'?}
\begin{aligned}
\int_{Q}\overline{f}^{2}e^{2s\varphi}\,dxdadt
\leq 2\Big(\int_{Q}|f|^{2}e^{2s\varphi}\,dxdadt
+\int_{Q}|\mu|^2|v|^{2}e^{2s\varphi}\,dxdadt\Big).
\end{aligned}$$ Now, setting $\nu:=e^{s\varphi}v$, we obtain $$\label{sopra}
\begin{aligned}
\int_{Q}|\mu|^2|v|^{2}e^{2s\varphi}\,dxdadt&\le \|\mu\|_{\infty}^2\int_0^1
\nu^2 dx \\
&= \|\mu\|_{\infty}^2\int_0^1\left(\frac{k^{1/3}}{|x-x_0|^{2/3}}\nu^2\right)^{3/4}\left(
\frac{|x-x_0|^2}{k} \nu^2\right)^{1/4} \\
&\le C \int_0^1
\frac{k^{1/3}}{|x-x_0|^{2/3}}\nu^2dx +C
\int_0^1 \frac{|x-x_0|^2}{k} \nu^2 dx.
\end{aligned}$$ As in , proceeding as in [@fm1] and applying the Hardy-Poincaré inequality proved in [@fm] to the function $\nu$ with weight $p(x)= |x-x_0|^{4/3}$, if $K \le \displaystyle \frac{4}{3}$, or $p(x) = (k(x)|x-x_0|^4)^{1/3}$, if $K>4/3$, we can prove that
$$\label{hpapplbis1}
\begin{aligned}
\int_0^1 \frac{ k^{1/3}}{|x-x_0|^{2/3}}\nu^2dx &\le C\int_0^1 k (\nu_x)^2 dx
\\
&\le
C\int_{Q} k(x) e^{2s\varphi}v_x^2dxdadt
\\& + Cs^2\int_{Q}\Theta^2 e^{2s\varphi}\frac{(x-x_0)^2}{k} v^2dxdadt.
\end{aligned}$$
In any case, by , and , we have $$\label{fati2?}
\begin{aligned}
\int_{Q} |\bar{f}|^{2}\text{\small$~e^{2s\varphi}$\normalsize}~dxdadt
&\le
2\int_{Q} |f|^{2}\text{\small$~e^{2s\varphi}$\normalsize}~dxdadt
+C\int_{Q}k(x) e^{2s\varphi} v_x^2 dxdadt
\\&+ Cs^2\int_{Q} \Theta^2 e^{2s\varphi}\frac{(x-x_0)^2}{k} v^2dxdadt\\
&\le C\int_{Q} |f|^{2}\text{\small$~e^{2s\varphi}$\normalsize}~dxdadt
+C\int_{Q}\Theta k(x) e^{2s\varphi} v_x^2 dxdadt
\\&+ Cs^2\int_{Q} \Theta^3 e^{2s\varphi}\frac{(x-x_0)^2}{k} v^2dxdadt.
\end{aligned}$$ Substituting in , one can conclude $$\begin{aligned}
& \int_{Q}\left(s \Theta k v_x^2 +s^3\Theta^3\frac{(x-x_0)^2}{k} v^2\right)e^{2s\varphi}dxdadt
\le
C\Big(\int_{Q} |f|^{2}e^{2s\varphi}dxdadt
\\[3pt]&
+ s\int_0^T\int_0^A\left[k\Theta e^{2s\varphi}(x-x_0)(v_x)^2
dadt\right]_{x=0}^{x=1}dadt \Big),
\end{aligned}$$ for all $s$ large enough.\
Observability and controllability {#osservabilita}
=================================
In this section we will prove, as a consequence of the Carleman estimates established in Section 3, observability inequalities for the associated adjoint problem of : $$\label{h=0}
\begin{cases}
\ds \frac{\partial v}{\partial t} + \frac{\partial v}{\partial a}
+(k(x)v_{x})_x-\mu(t, a, x)v +\beta(a,x)v(t,0,x)=0,& (t,x,a) \in Q,
\\[5pt]
v(t,a,0)=v(t,a,1) =0, &(t,a) \in Q_{T,A},\\
v(T,a,x) = v_T(a,x) \in L^2(Q_{A,1}), &(a,x) \in Q_{A,1} \\
v(t,A,x)=0, & (t,x) \in Q_{T,1}.
\end{cases}$$ From now on, we assume that the control set $\omega$ is such that $$\label{omega0}
x_0 \in \omega = (\alpha, \rho) \subset\subset (0,1),$$ or $$\label{omega_new}
\omega = \omega_1 \cup
\omega_2,$$ where $$\label{omega2}
\omega_i=(\lambda_i,\rho_i) \subset (0,1), \, i=1,2, \mbox{ and
$\rho_1 < x_0< \lambda_2$}.$$
\[beta1\] Observe that, if holds, we can find two subintervals $\omega_1=(\lambda_1,\rho_1)\subset \subset(\alpha, x_0),
\omega_2=(\lambda_2,\rho_2) \subset\subset (x_0,\rho)$.
Moreover, on $\beta$ we assume the following assumption:
\[conditionbeta\] Suppose that there exists $\bar a <A$ such that $$\label{conditionbeta1}
\beta(a, x)=0 \; \text{for all $(a, x) \in [0, \bar a]\times [0,1]$}.$$
Observe that Hypothesis \[conditionbeta\] has a biological meaning. Indeed, $\bar a$ is the minimal age in which the female of the population become fertile, thus it is natural that before $\bar a$ there are no newborns. For other comments on Hypothesis \[conditionbeta\] we refer to [@fJMPA].
In order to prove the desired observability inequality for the solution $v$ of we proceed, as usual, using a density argument. To this purpose, we consider, first of all the space $${\mathcal W}:=\Big\{ v\;\text{solution of \eqref{h=0}}\;\big|\;v_T \in
D(\mathcal A^2)\Big\},$$ where $D(\mathcal A^2) = \Big\{u \in
D(\mathcal A) \;\big|\; \mathcal A u \in
D(\mathcal A)\;\Big\}
$. Clearly $D(\mathcal A^2)$ is densely defined in $D({\mathcal A})$ (see, for example, [@b Lemma 7.2]) and hence in $L^2(Q_{A,1})$ and $$\begin{aligned}
{\mathcal W}&= C^1\big([0,T]\:;D(\mathcal A)\big)\\&\subset \mathcal{V}:=L^2\big(Q_{T,A}; H^2_k(0,1)\big) \cap H^1\big(0,T; H^1(0,A;H^1_k(0,1))\big)
\subset \mathcal{U}.
\end{aligned}$$
\[caccio\] Let $\omega'$ and $\omega$ two open subintervals of $(0,1)$ such that $\omega'\subset \subset \omega \subset (0,1)$ and $x_0 \not
\in \bar\omega'$. Let $\psi(t,a,x):=\Theta(t,a)\Psi(x)$, where $$\label{Theta}
\Theta(t, a)= \frac{1}{t^{4}(T-t)^{4}a^{4}}$$ and $
\Psi \in C([0,1],(-\infty,0))\cap
C^1([0,1]\setminus\{x_0\},(-\infty,0))
$ is such that $$\label{stimayx}
|\Psi_x|\leq \frac{c}{\sqrt{k}} \mbox{ in }[0,1]\setminus\{x_0\}.$$ Then, there exist two strictly positive constants $C$ and $s_0$ such that, for all $s \ge s_0$, $$\label{caccioeq}
\begin{aligned}
\int_{0}^T\int_0^A \int _{\omega'} v_x^2e^{2s\psi } dxdadt
\ &\leq \ C\left( \int_{0}^T\int_0^A \int _{\omega} v^2 dxdadt + \int_Q f^2 e^{2s\psi } dxdadt\right),
\end{aligned}$$ for every solution $v$ of .
The proof of the previous proposition is similar to the one given in [@f_anona Proposition 4.2] and [@fm Proposition 4.2], so we omit it.
Moreover, the following non degenerate inequality proved in [@fJMPA] is crucial:
\[nondegenere\]\[see [@fJMPA Theorem 3.2]\] Let $z\in
\mathcal{Z}$ be the solution of , where $f \in L^{2}(Q)$, $k \in C^{1}([0,1])$ is a strictly positive function and $$\mathcal{Z}:=L^2\big(Q_{T,A}; H^2(0,1)\cap H^1_0(0,1)\big) \cap H^1\big(0,T; H^1(0,A;H^1_0(0,1))\big).$$ Then, there exist two strictly positive constants $C$ and $s_0$, such that, for any $s\geq s_0$, $z$ satisfies the estimate $$\label{570'}
\begin{aligned}
&\int_{Q}(s^{3}\phi^{3}z^{2}+s\phi z_{x}^{2})e^{2s\Phi} dxdadt \leq C \int_{Q}f^{2}e^{2s\Phi}dxdadt
\\&-C
s\kappa\int_0^T\int_0^A\left[ke^{2s\Phi}\phi(z_x)^2
\right]_{x=0}^{x=1}dadt.
\end{aligned}$$ Here the functions $\phi$ and $\Phi$ are defined as follows $$\label{571}
\begin{gathered}
\phi(t,a,x)=\Theta(t,a)e^{\kappa\sigma(x)},\\
\Phi(a,t,x)=\Theta(t,a)\Psi(x), \quad
\Psi(x)=e^{\kappa\sigma(x)}-e^{2\kappa\|\sigma\|_{\infty}},
\end{gathered}$$ where $(t,a,x)\in Q$, $\kappa>0$, $\sigma (x) :=\mathfrak{d}\int_x^1\frac{1}{k(t)}dt$, $\fd=\|k'\|_{L^\infty(0,1)}$ and $\Theta$ is given in .
The previous Theorem still holds under the weaker assumption $k \in W^{1, \infty}(0,1)$ without any additional assumption.\
On the other hand, if we require $k \in W^{1,1}(0,1)$ then we have to add the following hypothesis: [*there exist two functions $\fg \in L^1(0,1)$, $\fh \in W^{1,\infty}(0,1)$ and two strictly positive constants $\fg_0$, $\fh_0$ such that $\fg(x) \ge \fg_0$ and $$\label{debole}
-\frac{k'(x)}{2\sqrt{k(x)}}\left(\int_x^1\fg(t) dt + \fh_0 \right)+ \sqrt{k(x)}\fg(x) =\fh(x)\quad \text{for a.e.} \; x \in [0,1].$$*]{}\
In this case, i.e. if $k \in W^{1,1}(0,1)$, the function $\Psi$ in becomes $$\label{Psi_new}
\Psi(x):= - r\left[\int_0^x
\frac{1}{\sqrt{k(t)}} \int_t^1
\fg(s) dsdt + \int_0^x \frac{\fh_0}{\sqrt{k(t)}}dt\right] -\mathfrak{c},$$ where $r$ and $\mathfrak{c}$ are suitable strictly positive functions. For other comments on Theorem \[nondegenere\] we refer to [@fJMPA].
In the following, we will apply Theorem \[nondegenere\] in the intervals $[\lambda_2, 1]$ and $[-\rho_1, \rho_1]$ under these weaker assumptions. In particular, on $k$ we assume:
\[ipogenerale\] The function $k$ satisfies Hypothesis $\ref{BAss01}$. Moreover, if $k \in W^{1,1}(0,1)$, then there exist two functions $\fg \in L^\infty_{\rm loc}([-\rho_1,1]\setminus \{x_0\})$, $\fh \in W^{1,\infty}_{\rm loc}([-\rho_1,1]\setminus \{x_0\}, L^\infty(0,1))$ and two strictly positive constants $\fg_0$, $\fh_0$ such that $\fg(x) \ge \fg_0$ and $$\label{aggiuntivastrana}
-\frac{\tilde k'(x)}{2\sqrt{\tilde k(x)}}\left(\int_x^B\fg(t) dt + \fh_0 \right)+ \sqrt{\tilde k(x)}\fg(x) =\fh(x,B)$$ for a.e. $x \in [-\rho_1,1], B \in [0,1]$ with $x<B<x_0$ or $x_0<x<B$, where $$\label{tildek}
\tilde k(x):= \begin{cases}k(x), & x \in [0,1],\\
k(-x), & x \in [-1,0].
\end{cases}$$
With the aid of Theorems \[Cor1\], \[nondegenere\] and Proposition \[caccio\], we can now show $\omega-$local Carleman estimates for .
\[Cor2\] Assume Hypothesis \[ipogenerale\]. Then, there exist two strictly positive constants $C$ and $s_0$ such that every solution $v$ of in $
\mathcal {V}
$ satisfies, for all $s \ge s_0$, $$\begin{aligned}
\int_{Q}\left(s \Theta k v_x^2
+ s^3\Theta^3\text{\small$\displaystyle \frac{(x-x_0)^2}{k}$\normalsize}
v^2\right)e^{2s\varphi}dxdadt
&\le
C\int_{Q}f^{2}\text{\small$e^{2s\Phi}$\normalsize}~dxdadt\\
&+C \int_0^T \int_0^A\int_ \omega v^2 dx dadt.
\end{aligned}$$
First assume that $\omega$ satisfies and take $w_i$, $i=1,2$, as in Remark \[beta1\]. Now, fix $\bar \lambda_i, \bar \rho_i \in \omega_i=(\lambda_i, \rho_i)$, $i=1,2$, such that $\bar \lambda_ i < \bar \rho_i$ and consider a smooth function $\xi:[0,1]\to[0,1]$ such that $$\xi(x)=\begin{cases} 0&x\in [0,\bar\lambda_1],\\
1 & x\in[\tilde \lambda_1,\tilde \lambda_2],\\
0&x\in [\bar\rho_2,1],
\end{cases}$$ where $\tilde \lambda_i=(\bar \lambda_i+\bar \rho_i)/2$, $i=1,2$. Define $w:= \xi v$, where $v$ is any fixed solution of . Then $w$ satisfies $$%\label{eq}
\begin{cases}
w_t +w_a+(k w_{x})_x- \mu w= \xi f + (k\xi_xv)_x+\xi_xkv_x=:h,&
(t,a, x) \in Q,
\\[5pt]
w(t,a,0)= w(t,a,1)=0, & (t,a) \in Q_{T,A}.
\end{cases}$$ Thus, applying Theorem \[Cor1\], Proposition \[caccio\], and proceeding as in [@f_anona], we have $$\label{add1}
\begin{aligned}
&\int_0^T\int_0^A \int_{\tilde \lambda_1}^{\tilde \lambda_2}\left(s \Theta k v_x^2
+ s^3\Theta^3\text{\small$\displaystyle\frac{(x-x_0)^2}{k}$\normalsize}
v^2\right)e^{2s\varphi}dxdadt \\
&=\int_0^T\int_0^A \int_{\tilde \lambda_1}^{\tilde \lambda_2}\left(s \Theta k w_x^2
+ s^3\Theta^3\text{\small$\displaystyle\frac{x^2}{k}$\normalsize}
w^2\right)e^{2s\varphi}dxdadt\\
&
\le C \left( \int_{Q} f^2e^{2s\varphi} dxdadt+ \int_{0}^T\int_0^A \int _{\omega} v^2 dxdadt \right).
\end{aligned}$$ Now, consider a smooth function $\eta: [0,1] \to [0,1]$ such that $$\eta(x) =\begin{cases} 0& x\in [0,\bar\lambda_2],\\
1& x\in [\tilde\lambda_2, 1],
\end{cases}$$ and define $z:= \eta v$. Then $z$ satisfies $$\label{problemz}
%\label{eq}
\begin{cases}
z_t +z_a+(k z_{x})_x- \mu z= \eta f +(k\eta_xv)_x+\eta_xkv_x=:h,&
(t,a, x) \in Q_{T,A}\times (\lambda_2,1),\\
z(t,a, \lambda_2)= z(t,a, 1)=0, & t \in Q_{T,A},
\end{cases}$$ Clearly the equation satisfied by $z$ is not degenerate, thus applying Theorem \[nondegenere\] and [@fm_opuscola Lemma 4.1] on $(\lambda_2,1)$, one has $$\begin{aligned}
&\int_0^T\int_0^A\int_{\lambda_2}^1(s^{3}\phi^{3}z^{2}+s\phi z_{x}^{2})e^{2s\Phi} dxdadt \leq C \int_0^T\int_0^A\int_{\lambda_2}^1h^{2}e^{2s\Phi}dxdadt
\\
&\le C \left( \int_Qf^{2}e^{2s\Phi}dxdadt+ \int_0^T\int_0^A \int_{\omega}v^2dxdadt\right).
\end{aligned}$$ Hence $$\begin{aligned}
& \int_0^T\int_0^A\int_{\tilde\lambda_2}^1 (s^{3}\phi^{3}v^{2}+s\phi v_{x}^{2})e^{2s\Phi} dxdadt=
\int_0^T\int_0^A\int_{\tilde\lambda_2}^1 (s^{3}\phi^{3}z^{2}+s\phi z_{x}^{2})e^{2s\Phi} dxdadt\\
&\le C \left( \int_Qf^{2}e^{2s\Phi}dxdadt+ \int_0^T\int_0^A \int_{\omega} v^2dxdadt\right),
\end{aligned}$$ for a strictly positive constant $C$. Proceeding, for example, as in [@fm1] one can prove the existence of $\varsigma>0$, such that, for all $(t,a,x)\in [0,T]\times[0,A]\times[\lambda_2,1]$, we have $$\label{stimaphi}
e^{2s\varphi}\leq\varsigma e^{2s\Phi},
\frac{(x-x_0)^2}{k(x)}e^{2s\varphi}\leq\varsigma e^{2s\Phi}.$$ Thus, for a strictly positive constant $C$, $$\label{add2}
\begin{aligned}
&\int_0^T\int_0^A \int_{\tilde\lambda_2}^1 \left(s \Theta k v_x^2
+ s^3\Theta^3\frac{(x-x_0)^2}{k}
v^2\right)e^{2s\varphi}dxdadt \\
&\le C\left( \int_0^T\int_0^A \int_{\tilde\lambda_2}^1 (s^{3}\phi^{3}v^{2}+s\phi v_{x}^{2})e^{2s\Phi} dxdadt\right)\\
&
\le C \left( \int_{Q} f^2 e^{2s\Phi} dxdadt+ \int_{0}^T\int_0^A \int _{\omega} v^2 dxdadt \right).
\end{aligned}$$ Hence, $$\label{stimavec}
\begin{aligned}
&\int_0^T\int_0^A \int_{\tilde\lambda_1}^1 \left(s \Theta k v_x^2
+ s^3\Theta^3\frac{(x-x_0)^2}{k}
v^2\right)e^{2s\varphi}dxdadt \\
&
\le C \left( \int_{Q} f^2 e^{2s\Phi} dxdadt+ \int_{0}^T\int_0^A \int _{\omega} v^2 dxdadt \right).
\end{aligned}$$ To complete the proof it is sufficient to prove a similar inequality for $x\in[0,\tilde\lambda_1]$. To this aim, we use the reflection procedure as in [@fJMPA]; thus we consider the functions $$W(t,a,x):= \begin{cases} v(t,a,x), & x \in [0,1],\\
-v(t,a,-x), & x \in [-1,0],
\end{cases}$$ $$\tilde f(t,a,x):= \begin{cases} f(t,a,x), & x \in [0,1],\\
-f(t,a,-x), & x \in [-1,0],
\end{cases}$$ $$\tilde \mu(t,a,x):= \begin{cases} \mu(t,a,x), & x \in [0,1],\\
\mu(t,a,-x), & x \in [-1,0],
\end{cases}$$ so that $W$ satisfies the problem $$\begin{cases}
W_t +W_a +(\tilde k W_{x})_x - \tilde \mu W= \tilde f, &(t,x) \in Q_{T,A}\times (-1,1),\\
W(t,a,-1)=W(t,a,1) =0, & t \in Q_{T,A},
\end{cases}$$ (by the way, observe that in [@fJMPA] there is a misprint in the definition of $\mu$; it clearly must be defined in this way, otherwise $W$ is not the solution of the associated problem). Now, consider a cut off function $\zeta: [-1,1] \to [0,1]$ such that $$\zeta (x) =\begin{cases} 0 & x\in[-1,-\bar \rho_1],\\
1& x\in [-\tilde\lambda_1, \tilde\lambda_1],\\
0&x\in [\bar \rho_1,1],
\end{cases}$$ and define $Z:=\zeta W$. Then $Z$ satisfies $$\label{eq-Z*}
\begin{cases}
Z_t + Z_a+ (\tilde kZ_{x})_x -\tilde \mu Z=\tilde h, &(t,x) \in Q_{T,A}\times (-\rho_1,\rho_1),\\
Z(t,a,-\rho_1)= Z(t,a,\rho_1)=0, & t \in Q_{T,A},
\end{cases}$$ where $\tilde h=\zeta \tilde f+ (\tilde k\zeta_xW)_x+\zeta_x\tilde kW_x$. Now, applying the analogue of Theorem \[nondegenere\] on $(- \rho_1,
\rho_1)$ in place of $(0,1)$, using the definition of $W$, the fact that $Z_x(t, a,-\rho_1)=Z_x(t,a,
\rho_1)=0$ and since $\zeta$ is supported in $\left[-\bar \rho_1, -\tilde \lambda_1\right] \cup\left[\tilde\lambda_1, \bar \rho_1\right]$, we get $$\begin{aligned}
& \int_0^T\int_0^A\int_{0}^{\tilde \lambda_1} \left(s\Theta k (W_x)^2 + s^3
\Theta^3
\frac{(x-x_0)^2}{k} W^2\right)e^{2s\varphi}dxdadt\\
&= \int_0^T\int_0^A\int_{0}^{\tilde \lambda_1} \left(s\Theta k (Z_x)^2 + s^3
\Theta^3
\frac{(x-x_0)^2}{k} Z^2\right)e^{2s\varphi}dxdadt\\
&\le C \int_0^T\int_0^A\int_{0}^{\rho_1}\left(s\Theta (Z_x)^2 + s^3 \Theta^3
Z^2\right)e^{2s\Phi}dxdadt\\
&\le C \int_0^T\int_0^A\int_{-\rho_1}^{\rho_1}\left(s\Theta (Z_x)^2 + s^3 \Theta^3
Z^2\right)e^{2s\Phi}dxdadt
\end{aligned}$$ $$\begin{aligned}
& \le C
\int_0^T\int_0^A\int_{-\rho_1}^{\rho_1} \tilde h^{2}e^{2s\Phi}dxdadt\le C \int_0^T\int_0^A\int_{-\rho_1}^{\rho_1} \tilde f^{2}e^{2s\Phi}dxdadt \\
&+ C \int_0^T\int_0^A \int_{-\bar \rho_1}^{-\tilde\lambda_1}(
W^2+ (W_x)^2)e^{2s\Phi}dxdadt \\
&+ C\int_0^T\int_0^A \int_{\tilde \lambda_1}^{ \bar \rho_1}(W^2+ (W_x)^2)e^{2s\Phi}dxdadt\\
&\le C \int_0^T\int_0^A\int_{-\rho_1}^{\rho_1} \tilde f^{2}dxdadt + C\int_0^T\int_0^A \int_{-\rho_1}^{- \lambda_1}W^2dxdadt \\
&+C\int_0^T\int_0^A\int_{\lambda_1}^{ \rho_1} W^2dxdadt \\
& \mbox{ (by \cite[Lemma 4.1]{fm_opuscola} and since $\tilde f(t,a,x)= -f(t,a,-x)$, for $x <0$) }\\
& \le C \int_0^T\int_0^A\int_0^1 f^2 dxdadt +C\int_0^T\int_0^A\int_\omega v^2 dxdadt,
\end{aligned}$$ for some strictly positive constants $C$ and $s$ large enough. Here $\Phi$ is related to $ (-\rho_1,\rho_1)$.
Hence, by definitions of $Z$, $W$ and $\zeta$, and using the previous inequality one has $$\label{car101}
\begin{aligned}
&\int_0^T\int_0^A\int_{0}^{\tilde\lambda_1} \left(s\Theta k (v_x)^2 + s^3 \Theta^3
\frac{(x-x_0)^2}{k} v^2\right)e^{2s\varphi}dxdadt\\
&= \int_0^T\int_0^A\int_{0}^{\tilde\lambda_1} \left(s\Theta k(W_x)^2 + s^3 \Theta^3
\frac{(x-x_0)^2}{k}W^2\right)e^{2s\varphi}dxdadt\\
&\le C\left( \int_Q f^{2}dxdadt + \int_0^T\int_0^A \int_{\omega}v^2dxdadt\right).
\end{aligned}$$ Moreover, by and , the conclusion follows.
Nothing changes in the proof if $\omega= \omega_1\cup \omega_2$ and each of these intervals lye on different sides of $x_0$, as the assumption implies.
\[remarkultimo\] Observe that the results of Theorem \[Cor2\] still holds true if we substitute the domain $(0,T)\times (0,A)$ with a general domain $(T_1,T_2)\times (\delta,A)$, provided that $\mu$ and $\beta$ satisfy the required assumptions. In this case, in place of the function $\Theta$ defined in , we have to consider the weight function $$\tilde \Theta(t,a):= \frac{1}{(t-T_1)^4 (T_2-t)^4(a-\delta)^4}.$$
Using the previous local Carleman estimates one can prove the next observability inequalities.
\[Theorem4.4\] Assume Hypotheses $\ref{conditionbeta}$, with $ \bar a<T \le A$, and $\ref{ipogenerale}$. Then, for every $\delta \in (0,A)$, there exists a strictly positive constant $C=C(\delta)$ such that every solution $v$ of in $\mathcal V$ satisfies $$\label{T<A}
\begin{aligned}
&\int_0^A\int_0^1 v^2( T-\bar a,a,x) dxda \le C\int_0^{T} \int_0^\delta \int_0^1v^2(t,a,x) dxdadt\\
&+ C\left( \int_0^{T}\int_0^1 v_T^2(a,x)dxda+ \int_0^T \int_0^A\int_ \omega v^2 dx dadt\right).
\end{aligned}$$ Moreover, if $v_T(a,x)=0$ for all $(a,x) \in (0, T) \times (0,1)$, one has $$\label{T<A1}
\begin{aligned}
\int_0^A\!\!\int_0^1 v^2(T -\bar a,a,x) dxda &\le C\int_0^{T} \int_0^\delta\!\! \int_0^1v^2(t,a,x) dxdadt \\&+C\int_0^T\!\! \int_0^A\!\!\int_ \omega v^2 dx dadt.
\end{aligned}$$
Observe that in [@fJMPA Theorem 4.4], which is the analogue of Theorem \[Theorem4.4\] in the non divergence case, there is a mistake in the statement. Indeed, we assumed $\ds \frac{k'}{\sqrt{k}} \in L^\infty_{\text{loc}}([0,1]\setminus\{x_0\})$, which was a consequence of below (see the remark after $(46)$ in [@fJMPA]); the precise assumption is:\
[*there exist two functions $\fg \in L^\infty_{\rm loc}([-\rho_1,1]\setminus \{x_0\})$, $\fh \in W^{1,\infty}_{\rm loc}([-\rho_1,1]\setminus \{x_0\}, L^\infty(0,1))$ and two strictly positive constants $\fg_0$, $\fh_0$ such that $\fg(x) \ge \fg_0$ and $$\label{aggiuntivastrana1}
\frac{\tilde k'(x)}{2\sqrt{\tilde k(x)}}\left(\int_x^B\fg(t) dt + \fh_0 \right)+ \sqrt{\tilde k(x)}\fg(x) =\fh(x,B)$$ for a.e. $x \in [-\rho_1,1], B \in [0,1]$ with $x<B<x_0$ or $x_0<x<B$, where $\tilde k$ is defined in .* ]{} Indeed, in order to prove [@fJMPA Theorem 4.4], we use [@fJMPA Theorem 4.3] which holds under . On the other hand, the statement of [@fJMPA Corollary 4.1], which is also a consequence of [@fJMPA Theorem 4.4], is correct.
The proof follows the one of [@f_anona Theorem 4.4], but we repeat here in a briefly way for the reader’s convenience underling the differences since in [@f_anona Theorem 4.4] $k$ degenerates at the boundary of the domain, while hereit degenerates in the interior.
As in [@fJMPA], using the method of characteristic lines, one can prove the following implicit formula for $v$ solution of : $$\label{implicitformula}
S(T-t) v_T(T+a-t, \cdot),$$ if $t \ge \tilde T + a$ and $$\label{implicitformula1}
v(t,a, \cdot)=\begin{cases}
S(T-t) v_T(T+a-t, \cdot)\!+\int_a^{T+a-t}S(s-a)\beta(s, \cdot)v(s+t-a, 0, \cdot) ds, &\Gamma\!= \!\bar a \\
\int_a^AS(s-a)\beta(s, \cdot)v(s+t-a, 0, \cdot) ds, & \Gamma\!= \!\Gamma_{A,T},
\end{cases}$$ otherwise. Here $(S(t))_{t \ge0}$ is the semigroup generated by the operator $\mathcal A_0 -\mu Id$ for all $u \in D(\mathcal A_0)$ ($Id$ is the identity operator), $\Gamma_{A,T}:= A -a +t-\tilde T$ and $$\label{Gamma}
\Gamma:= \min \{\bar a, \Gamma_{A,T}\}.$$ In particular, it results $$\label{v(0)}
v(t,0, \cdot):= S(T-t) v_T(T-t, \cdot).$$ Proceeding as in [@f_anona Theorem 4.4], with suitable changes, one has that there exists a positive constant $C$ such that: $$\label{t=01}
\int_{Q_{A,1}} v^2(\tilde T,a,x) dxda \le C\int_{\frac{T}{4}}^{\frac{3T}{4}} \int_{Q_{A,1}} v^2(t,a,x) dxdadt.$$ Take $\delta \in (0, A)$. By the previous inequality, we have $$\label{t=0}
\begin{aligned}
\int_{Q_{A,1}} v^2(\tilde T,a,x) dxda \le C \int_{\frac{T}{4}}^{\frac{3T}{4}} \left(\int_0^\delta + \int_\delta^A \right)\int_0^1 v^2(t,a,x) dxdadt.
\end{aligned}$$ Now, we will estimate the term $\ds\int_{\frac{T}{4}}^{\frac{3T}{4}} \int_\delta^A\int_0^1v^2(t,a,x) dxdadt$. It results that $$\label{terminenuovo11}
\begin{aligned}
\int_0^1v^2dx
&
\le C\left( \int_0^1 k v_x^2 dx +\int_0^1 \frac{(x-x_0)^2}{k} v^2dx\right) ,
\end{aligned}$$ for a strictly positive constant $C.$ Indeed, using the Young’s inequality to the function $v$, we obtain $$\label{nu'}
\begin{aligned}
\int_0^1|v|^{2}\,dx
& \le C
\int_0^1\left(\frac{k^{1/3}}{(x-x_0)^{2/3}}v^2\right)^{3/4}\left(
\frac{(x-x_0)^2}{k}v^2\right)^{1/4}dx \\
&\le C\int_0^1
\frac{k^{1/3}}{(x-x_0)^{2/3}}v^2dx+
C \int_0^1
\frac{(x-x_0)^2}{k} v^2 dx.
\end{aligned}$$ Now, consider the term $$\int_0^1
\frac{k^{1/3}}{(x-x_0)^{2/3}}v^2dx.$$ If $M > \ds \frac{4}{3}$, take the function $\gamma(x) = (k(x)|x-x_0|^4)^{1/3}$. Clearly, $\displaystyle \gamma(x)= k(x)
\left(\frac{(x-x_0)^2}{k(x)}\right)^{2/3}\le C k(x)$ and $\displaystyle \frac{k^{1/3}}{(x-x_0)^{2/3}}=
\frac{\gamma(x)}{(x-x_0)^2}$. Moreover, using Hypothesis \[BAss01\], one has that the function $\displaystyle\frac{\gamma(x)}{|x-x_0|^q} = \left(\frac{k(x)}{|x-x_0|^\theta}\right)^{\frac{1}{3}}$, where $\displaystyle q: =\frac{4+\vartheta}{3}\in(1,2)$, is non increasing on the left of $x=x_0$ and non decreasing on the right of $x=x_0$. Hence, by the Hardy-Poincaré inequality given in [@fm Proposition 2.6], $$\int_0^1
\frac{k^{1/3}}{(x-x_0)^{2/3}}v^2dx = \int_0^1
\frac{\gamma(x)}{(x-x_0)^2} v^2 dx \le C \int_0^1 k v_x^2 dx.$$ Thus, if $M > \ds\frac{4}{3}$, by , holds. Now, assume $M \le \ds\frac{4}{3}$ and introduce the function $p(x) = |x-x_0|^{4/3}$. Obviously, there exists $ q \in
\left(1, \displaystyle\frac{4}{3}\right)$ such that the function $\displaystyle x\mapsto\frac{p(x)}{|x-x_0|^q}$ is nonincreasing on the left of $x=x_0$ and nondecreasing on the right of $x=x_0$. Thus, applying again [@fm Proposition 2.6], one has $$\label{hpapplbis}
\begin{aligned}
\int_0^1 \frac{k^{1/3}}{|x-x_0|^{2/3}}v^2dx & \le \max_{[0,1]}
k^{1/3}\int_0^1 \frac{1}{|x-x_0|^{2/3}}v^2dx \\
&= \max_{ [0,1]} k^{1/3}\int_0^1 \frac{p}{(x-x_0)^2} v^2 dx\\
&\le \max_{[0,1]} k^{1/3}C\int_0^1 p (v_x)^2 dx \\
& = \max_{[0,1]} k^{1/3} C\int_0^1 k \frac
{|x-x_0|^{4/3}}{k} (v_x)^2 dx\\
&\le \max_{ [0,1]}k^{1/3} C \int_0^1 k (v_x)^2 dx,
\end{aligned}$$ Hence, still holds and $$\label{ribo1}
\begin{aligned}
\int_{\frac{T}{4}}^{\frac{3T}{4}} \int_\delta^A\int_0^1v^2(t,a,x) dxdadt &\le C \int_{\frac{T}{4}}^{\frac{3T}{4}} \int_\delta^A\int_0^1\tilde \Theta v_x^2e^{2s\varphi} dxdadt \\
& +C \int_{\frac{T}{4}}^{\frac{3T}{4}} \int_\delta^A\int_0^1 \tilde \Theta^3 \frac{(x-x_0)^2}{k}v^2e^{2s\varphi} dxdadt.
\end{aligned}$$ The rest of the proof follows as in [@f_anona Theorem 4.4], so we omit it.
\[CorOb\] Assume Hypotheses $\ref{conditionbeta}$, with $\bar a=T <A$, and $\ref{ipogenerale}$. Then, for every $\delta \in (0,A)$, there exists a strictly positive constant $C=C(\delta)$ such that every solution $v$ of in $\mathcal V$ satisfies $$\begin{aligned}
\int_0^A\int_0^1 v^2( 0,a,x) dxda &\le C\int_0^T \int_0^\delta \int_0^1v^2(t,a,x) dxdadt\\
&+ C\left( \int_0^T\int_0^1 v_T^2(a,x)dxda+ \int_0^T \int_0^A\int_ \omega v^2 dx dadt\right).
\end{aligned}$$ Moreover, if $v_T(a,x)=0$ for all $(a,x) \in (0, T) \times (0,1)$, one has $$\begin{aligned}
\int_0^A\!\!\int_0^1 v^2(0,a,x) dxda &\le C\left(\int_0^T \int_0^\delta\!\! \int_0^1 v^2(t,a,x) dxdadt +\!\! \int_0^T\!\! \int_0^A\!\!\int_ \omega v^2 dx dadt\right).
\end{aligned}$$
Proceeding as in Theorem \[Theorem4.4\], one can prove the analogous result in the case $T>A$. Indeed, with suitable changes, one can prove again , if $t \ge \tilde T + a$, and , otherwise. In particular, we have again . Thus:
\[Theorem4.4\_new\] Assume Hypotheses $\ref{conditionbeta}$, with $ \bar a<A<T$, and $\ref{ipogenerale}$. Then, for every $\delta \in (0,A)$, there exists a strictly positive constant $C=C(\delta)$ such that every solution $v$ of in $\mathcal V$ satisfies . Moreover, if $v_T(a,x)=0$ for all $(a,x) \in (0, T) \times (0,1)$, one has .
Actually, proceeding as in [@fJMPA] with suitable changes, we can improve the previous results in the following way:
\[CorOb1’\]Assume Hypotheses $\ref{conditionbeta}$ and $\ref{ipogenerale}$. If $T<A$, then, for every $\delta \in (T,A)$, there exists a strictly positive constant $C=C(\delta)$ such that every solution $v$ of in $\mathcal V$ satisfies $$\label{ribo}
\int_0^A\int_0^1 v^2(T-\bar a,a,x) dxda \le
C\left( \int_0^\delta \int_0^1 v_T^2(a,x)dxda+ \int_0^T \int_0^A\int_ \omega v^2 dx dadt\right).$$ If $A<T$, then, for every $\delta \in (\bar a,A)$, there exists a strictly positive constant $C= C(\delta)$ such that every solution $v$ of in $\mathcal V$ satisfies .
If $T<A$ the proof of the previous theorem is analogous to the one of [@fJMPA Theorem 4.6], with suitable changes, so we omit it.\
Now, consider the case $A<T$ and fix $\delta \in (\bar a, A)$. We distinguish between the two cases $\delta < 2\bar a$ and $\delta \ge 2\bar a$.
First of all, consider $\delta < 2\bar a$: as in [@f_anona Theorem 4.4.], we can prove $$\label{bo}
\int_{Q_{A,1}} v^2(T-\bar a,a,x) dxda \le C \int_{Q_{A,1}} v^2(t,a,x) dxda.$$ Then, integrating over $\ds \left[T-\bar a, T+\delta -2\bar a\right]$, we have the following inequality: $$\label{t=041}
\int_{Q_{A,1}} v^2(T-\bar a,a,x) dxda \le C\int_{T-\bar a}^{T+\delta -2\bar a} \left(\int_0^{\delta - \bar a} + \int_{\delta - \bar a}^A \right)\int_0^1 v^2(t,a,x) dxdadt.$$ Using Theorem \[Cor2\], we can prove $$\label{bo1}
\begin{aligned}
\int_{T-\bar a}^{T+\delta -2\bar a}\int_{\delta - \bar a}^A \!\!\int_0^1v^2(t,a,x) dxdadt &\le C\int_0^\delta\int_0^1\!\! v_T^2(a,x)dxda\\&+C \int_0^T \!\!\int_0^A\!\!\int_ \omega v^2 dx dadt.
\end{aligned}$$ Indeed, by applied to $[T-\bar a, +\delta -2\bar a]$ and Theorem \[Cor2\], we have $$\begin{aligned}
&\int_{T-\bar a}^{T+\delta -2\bar a} \int_\delta^A\int_0^1v^2(t,a,x) dxdadt \le C \int_{T-\bar a}^{T+\delta -2\bar a} \int_\delta^A\int_0^1\tilde \Theta v_x^2e^{2s\varphi} dxdadt \\
& +C \int_{T-\bar a}^{T+\delta -2\bar a} \int_\delta^A\int_0^1 \tilde \Theta^3 \frac{(x-x_0)^2}{k}v^2e^{2s\varphi} dxdadt\\
& \le
C\left(\int_{T-\bar a}^{T+\delta -2\bar a} \int_0^A\int_0^1f^2dxdadt+ \int_0^T \int_0^A\int_ \omega v^2 dx dadt\right),
\end{aligned}$$ where, in this case, $f(t,a,x):=-\beta(a,x)v(t,0,x)$. Hence, $$\begin{aligned}
&\int_{T-\bar a}^{T+\delta -2\bar a} \int_\delta^A\int_0^1v^2(t,a,x) dxdadt \le
C\int_{T-\bar a}^{T+\delta -2\bar a} \int_0^A\int_0^1v^2(t,0,x)dxdadt\\&+ C\int_0^T \int_0^A\int_ \omega v^2 dx dadt\\
&\le C \left(\int_{T-\bar a}^{T+\delta -2\bar a} \int_0^1v_T^2(T-t,x)dxdt+ \int_0^T \int_0^A\int_ \omega v^2 dx dadt\right)\\
&=C \left(\int_{-\delta+2\bar a}^{\bar a}\int_0^1v_T^2(a,x)dxda+ \int_0^T \int_0^A\int_ \omega v^2 dx dadt\right)\\
&\le C \left(\int_0^{\delta} \int_0^1v_T^2(a,x)dxda+ \int_0^T \int_0^A\int_ \omega v^2 dx dadt\right).
\end{aligned}$$ Hence follows.
It remains to estimate the following integral: $$\int_{T-\bar a}^{T+\delta -2\bar a}\int_0^{\delta - \bar a}\int_0^1 v^2(t,a,x) dxdadt.$$ Observe that, since $t \le T+\delta -2\bar a$, $t-T+\bar a < \delta- \bar a$, hence $$\label{zero}
\begin{aligned}
\int_{T-\bar a}^{T+\delta -2\bar a}\int_0^{\delta - \bar a}\int_0^1 v^2(t,a,x) dxdadt&= \int_{T-\bar a}^{T+\delta -2\bar a}\int_0^{t-T+\bar a}\int_0^1 v^2(t,a,x) dxdadt\\& + \int_{T-\bar a}^{T+\delta -2\bar a}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 v^2(t,a,x) dxdadt.
\end{aligned}$$ Now, by and by the boundedness of $(S(t))_{t \ge0}$, $$\label{prima}
\begin{aligned}
&\int_{T-\bar a}^{T+\delta -2\bar a}\int_0^{t-T+\bar a}\int_0^1 v^2(t,a,x) dxdadt\\
&=\int_{T-\bar a}^{T+\delta -2\bar a}\int_0^{t-T+\bar a}\int_0^1 (S(T-t)v_T(T+a-t,x))^2dxdadt\\
& \le C\int_{T-\bar a}^{T+\delta -2\bar a}\int_0^{t-T+\bar a}\int_0^1 v_T^2(T+a-t,x)dxdadt\\
&= C\int_{-\delta +2\bar a}^{\bar a}\int_0^{\bar a-z}\int_0^1 v_T^2(z+a,x)dxdadz\\
&= C\int_{-\delta +2\bar a}^{\bar a}\int_z^{\bar a}\int_0^1 v_T^2(\sigma,x)dxd\sigma dz\\
&\le C\int_{-\delta +2\bar a}^{\bar a} \int_0^{\bar a}\int_0^1 v_T^2(\sigma,x)dxd\sigma dz\\
&\le C \int_0^{\bar a}\int_0^1 v_T^2(\sigma,x)dxd\sigma \le C\int_0^\delta\int_0^1 v_T^2(\sigma,x)dxd\sigma
\end{aligned}$$ On the other hand, if $t \ge T-\bar a$ and $a \in (t-T+\bar a, \delta -\bar a)$, it results that $T-t < A-a$, thus $\Gamma = \bar a$ (to this purpose recall that $\delta \in (\bar a, A)$ and $\Gamma$ is defined in ). Hence in we have to consider the first formula, i.e. $$v(t,a, \cdot)=
S(T-t) v_T(T+a-t, \cdot)\!+\int_a^{T+a-t}S(s-a)\beta(s, \cdot)v(s+t-a, 0, \cdot) ds.$$ It follows that, proceeding as in , $$\begin{aligned}
&
\int_{T-\bar a}^{T+\delta -2\bar a}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 v^2(t,a,x) dxdadt\\
&\le C\int_{T-\bar a}^{T+\delta -2\bar a}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 v^2_T(T+a-t,x)dxdadt \\
&+C \int_{T-\bar a}^{T+\delta -2\bar a}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 \left(\int_a^{T+a-t}v^2(s+t-a,0,x)ds \right)dxdadt\\
&= C\int_{-\delta+2 \bar a}^{\bar a}\int_{\bar a-z}^{\delta -\bar a} \int_0^1 v^2_T(z+a,x)dxdadz \\
& +C \int_{T-\bar a}^{T+\delta -2\bar a}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 \left(\int_a^{T+a-t}v_T^2(T-s-t+a,x)ds \right)dxdadt
\\
&=C\int_{-\delta+2 \bar a}^{\bar a}\int_{\bar a}^{z+\delta -\bar a} \int_0^1 v^2_T(\sigma,x)dxd\sigma dz \\
&+C \int_{T-\bar a}^{T+\delta -2\bar a}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 \left(\int_{-a}^{T-a-t}v_T^2(a+z,x)dz \right)dxdadt
\end{aligned}$$ $$\begin{aligned}
& \le C\int_{-\delta+2 \bar a}^{\bar a}\int_{\bar a}^{z+\delta -\bar a} \int_0^1 v^2_T(\sigma,x)dxd\sigma dz \\
&+C \int_{T-\bar a}^{T+\delta -2\bar a}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 \left(\int_{-a}^{T-a-t}v_T^2(a+z,x)dz \right)dxdadt.
\end{aligned}$$ Using the fact that in the first integral $z \in (-\delta+2 \bar a, \bar a)$ and in the second one $t \ge T-\bar a$, one has $z+\delta-\bar a \le \delta$ and $T-t \le \bar a$, respectively, this implies $$\label{seconda}
\begin{aligned}
&\int_{T-\bar a}^{T+\delta -2\bar a}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 v^2(t,a,x) dxdadt\le C\int_{-\delta+2 \bar a}^{\bar a}\int_{\bar a}^{\delta} \int_0^1 v^2_T(\sigma,x)dxd\sigma dz \\
&+C \int_{T-\bar a}^{T+\delta -2\bar a}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 \left(\int_0^{T-t}v_T^2(\sigma, x)d\sigma \right)dxdadt\\
&\le C \int_0^{\delta} \int_0^1 v^2_T(\sigma,x)dxd\sigma + C\int_{T-\bar a}^{T+\delta -2\bar a}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 \left(\int_0^{\bar a}v_T^2(\sigma, x)d\sigma \right)dxdadt\\
&\le C \int_0^{\delta} \int_0^1 v^2_T(\sigma,x)dxd\sigma.
\end{aligned}$$ Hence, by - , follows.
Now, consider the case $\delta \ge 2\bar a$ and, in place of $\ds \left[T-\bar a, T+\delta -2\bar a\right]$, take the interval $\left[T-\bar a, T-\ds\frac{\bar a}{4}\right]$. Hence we have $$\label{t=041new}
\int_{Q_{A,1}} v^2(T-\bar a,a,x) dxda \le C\int_{T-\bar a}^{T-\frac{\bar a}{4}} \left(\int_0^{\delta - \bar a} + \int_{\delta - \bar a}^A \right)\int_0^1 v^2(t,a,x) dxdadt.$$ Proceeding as before, we can prove the analogous of , i.e. $$\label{bo2}
\int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_{\delta - \bar a}^A \!\!\int_0^1v^2(t,a,x) dxdadt \le C\left(\int_0^\delta\int_0^1\!\! v_T^2(a,x)dxda+ \int_0^T \!\!\int_0^A\!\!\int_ \omega v^2 dx dadt\right).$$ It remains to estimate $$\int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_0^{\delta - \bar a}\int_0^1 v^2(t,a,x) dxdadt.$$ Also in this case, since $t \in \left[T-\bar a, T-\ds\frac{\bar a}{4}\right]$, it follows that $t-T+\bar a \le \delta -\bar a$ (recall that we are in the case $\delta \ge 2\bar a$). Proceeding as before, one has $$\begin{aligned}
\int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_0^{\delta - \bar a}\int_0^1 v^2(t,a,x) dxdadt&= \int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_0^{t-T+\bar a}\int_0^1 v^2(t,a,x) dxdadt\\& + \int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 v^2(t,a,x) dxdadt.
\end{aligned}$$ As for and , we have: $$\label{primanew}
\begin{aligned}
& \int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_0^{t-T+\bar a}\int_0^1 v^2(t,a,x) dxdadt\\
&
\le C\int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_0^{t-T+\bar a}\int_0^1 v_T^2(T+a-t,x)dxdadt\\
&= C\int_{\frac{\bar a}{4}}^{\bar a}\int_0^{\bar a-z}\int_0^1 v_T^2(z+a,x)dxdadz= C\int_{\frac{\bar a}{4}}^{\bar a}\int_z^{\bar a}\int_0^1 v_T^2(\sigma,x)dxd\sigma dz\\
&\le C\int_{\frac{\bar a}{4}}^{\bar a} \int_0^{\bar a}\int_0^1 v_T^2(\sigma,x)dxd\sigma dz \le C\int_0^\delta\int_0^1 v_T^2(\sigma,x)dxd\sigma
\end{aligned}$$ and $$\label{secondanew}
\begin{aligned}
&
\int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 v^2(t,a,x) dxdadt\\
&\le C\int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 v^2_T(T+a-t,x)dxdadt \\
&+C \int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 \left(\int_a^{T+a-t}v^2(s+t-a,0,x)ds \right)dxdadt\\
&=C\int_{\frac{\bar a}{4}}^{\bar a}\int_{\bar a}^{z+\delta -\bar a} \int_0^1 v^2_T(\sigma,x)dxd\sigma dz \\
&+C \int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 \left(\int_{-a}^{T-a-t}v_T^2(a+z,x)dz \right)dxdadt\\
& \le C\int_{\frac{\bar a}{4}}^{\bar a}\int_{\bar a}^{\delta} \int_0^1 v^2_T(\sigma,x)dxd\sigma dz \\
&+C \int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 \left(\int_0^{T-t}v_T^2(\sigma,x)d\sigma \right)dxdadt\\
& \le C\int_{0}^{\delta} \int_0^1 v^2_T(\sigma,x)dxd\sigma +C \int_{T-\bar a}^{T-\frac{\bar a}{4}}\int_{t-T+\bar a}^{\delta -\bar a} \int_0^1 \left(\int_0^{\bar a}v_T^2(\sigma,x)d\sigma \right)dxdadt\\
& \le C\int_0^\delta\int_0^1v_T^2(\sigma,x)d\sigma dx.
\end{aligned}$$ By -, follows.
By Theorem \[CorOb1’\] and using a density argument, one can deduce the following observability result:
\[PropOI\] \[obser.\] Assume Hypotheses $\ref{conditionbeta}$ and $\ref{ipogenerale}$. If $T<A$, then, for every $\delta \in (T,A)$, there exists a strictly positive constant $C=C(\delta)$ such that every solution $v\in \mathcal U$ of satisfies $$\label{OI}
\int_0^A\int_0^1 v^2(T-\bar a,a,x) dxda \le
C\left( \int_0^\delta \int_0^1v_T^2(a,x)dxda+ \int_0^T \int_0^A\int_ \omega v^2 dx dadt\right).$$ If $A<T$, then, for every $\delta \in (\bar a,A)$, there exists a strictly positive constant $C= C(\delta)$ such that every solution $v$ of satisfies .\
Here $v_T(a,x)$ is such that $v_T(A,x)=0$ in $(0,1)$.
Observe that in the statements of the analogous results given in [@fJMPA] for the non divergence case there is a misprint. Indeed the constant $C$ depends on $\delta$, as one can deduce by the proofs. The right statement is
[*...for every $\delta \in (T,A),$ there exists $C=C(\delta)$ such that...*]{}
We underline that the results are correct and in the correct way they are used to prove [@fJMPA Theorems 4.7 and 4.8].
As a consequence of Proposition \[PropOI\] one can prove, as in [@f_anona Theorem 4.7], the following null controllability result:
\[ultimo\] Assume Hypotheses $\ref{conditionbeta}$ and $\ref{ipogenerale}$ and take $y_0 \in L^2(Q_{A,1})$. Then for every $\delta \in (T,A)$, if $T<A$, or for every $\delta \in (\bar a,A)$, if $A<T$, there exists a control $f_\delta \in L^2(Q)$ such that the solution $y_\delta$ of satisfies $$\label{chissa}
y_\delta(T,a,x) =0 \quad \text{a.e. } (a,x) \in (\delta, A) \times (0,1).$$ Moreover, there exists $C=C(\delta)>0$ $$\label{stimaf}
\|f_\delta\|_{L^2( Q)} \le C \|y_0\|_{L^2(Q_{A,1})}.$$
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and she is partially supported by the FFABR “Fondo per il finanziamento delle attività base di ricerca” 2017.
[99]{}
B. Ainseba, Y. Echarroudi, L. Maniar [*Null controllability of population dynamics with degenerate diffusion*]{}, Differential Integral Equations (2013), 1397–1410.
F. Alabau-Boussouira, P. Cannarsa, G Fragnelli, [*Carleman estimates for degenerate parabolic operators with applications to null controllability*]{}, J. Evol. Equ. **6** (2006), 161–204.
S. Aniţa, [*Analysis and control of age-dependent population dynamics*]{}, Mathematical Modelling: Theory and Applications **11** (2000), Kluwer Academic Publishers, Dordrecht. V. Barbu, M. Iannelli, M. Martcheva, [*On the controllability of the Lotka-McKendrick model of population dynamics*]{}, J. Math. Anal. Appl. **253** (2001), 142–-165. I. Boutaayamou, Y. Echarroudi, [*Null controllability of a population dynamics with interior degeneracy*]{}, accepted in Journal of Mathematics and Statistical Science.
H. Brezis, [*Functional Analysis, Sobolev Spaces and Partial Differential Equations*]{}, Springer Science+Business Media, LLC 2011. Y. Echarroudi, L. Maniar, [*Null controllability of a model in population dynamics*]{}, Electron. J. Differential Equations **2014** (2014), 1–20.
G. Fragnelli, [*Null controllability for a degenerate population model in divergence form via Carleman estimates*]{}, submitted. G. Fragnelli, [*Carleman estimates and null controllability for a degenerate population model*]{}, Journal de Mathématiques Pures et Appliqués, **115** (2018), 74–-126.
G. Fragnelli, D. Mugnai, [*Carleman estimates and observability inequalities for parabolic equations with interior degeneracy*]{}, Advances in Nonlinear Analysis **2** (2013), 339–378.
G. Fragnelli, D. Mugnai, [*Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations*]{}, Mem. Amer. Math. Soc., **242** (2016), v+84 pp. [*Corrigendum*]{}, to appear.
G. Fragnelli, D. Mugnai, [*Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients*]{}, Adv. Nonlinear Anal., **6** (2017), 61–84.
G. Fragnelli, D. Mugnai, [*Controllability of strongly degenerate parabolic problems with strongly singular potentials*]{}, Electron. J. Qual. Theory Differ. Equ., **50** (2018), 1–11.
G. Fragnelli, D. Mugnai, [*Controllability of degenerate and singular parabolic problems: the double strong case with Neumann boundary conditions*]{}, Opuscula Math. **39** (2019), 207–-225. Y. He, B. Ainseba, [*Exact null controllability of the Lobesia botrana model with diffusion*]{}, J. Math. Anal. Appl. **409** (2014), 530–543. D. Maity, M. Tucsnak and E. Zuazua, [*Controllability and positivity constraints in population dynamics with age structuring and diffusion*]{}, Journal de Mathématiques Pures et Appliqués. In press, 10.1016/j.matpur.2018.12.006.
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abstract: 'In this paper, we consider the tensor completion problem representing the solution in the tensor train (TT) format. It is assumed that tensor is high-dimensional, and tensor values are generated by an unknown smooth function. The assumption allows us to develop an efficient initialization scheme based on Gaussian Process Regression and TT-cross approximation technique. The proposed approach can be used in conjunction with any optimization algorithm that is usually utilized in tensor completion problems. We empirically justify that in this case the reconstruction error improves compared to the tensor completion with random initialization. As an additional benefit, our technique automatically selects rank thanks to using the TT-cross approximation technique.'
bibliography:
- 'references.bib'
---
tensor completion, Gaussian Processes, tensor train, cross-approximation
68W25, 65F99, 60G15
Introduction
============
In this paper we consider the tensor completion problem. We suppose that values of tensor $\mathcal{X}$ are generated by some smooth function, i.e. $$\mathcal{X}_{i_1, \ldots, i_d} = f(x_{i_1}, \ldots, x_{i_d}),$$ where $(x_{i_1}, \ldots, x_{i_d})$ is a point on some multi-dimensional grid and $f(\cdot)$ is some unknown smooth function. However, the tensor values are known only at some small subset of the grid. The task is to complete the tensor, i.e., to reconstruct the tensor values at all points on the grid taking into account the properties of the [*data generating process*]{} $f(\cdot)$.
This problem statement differs from the traditional problem statement, which does not use any assumptions about the function $f(\cdot)$. Knowing some properties of the data generating function provides insights about how the tensor values relate to each other, and this, in turn, allows us to improve the results. In this work we assume that function $f(\cdot)$ is smooth.
There are a lot of practical applications that suit the statement. For example, modeling of physical processes, solutions of differential equations, modeling probability distributions.
In this paper we propose to model the smoothness of the data generating process by using Gaussian Process Regression (GPR). In GPR the assumptions about the function that we approximate are controlled via the kernel function. The GPR model is then used to construct the initial solution to the tensor completion problem In principle, such initialization can improve any other tensor completion technique. It means that using the proposed initialization state-of-the-art results can be obtained employing some simple optimization procedure like Stochastic Gradient Descent.
When the tensor order is high the problem should be solved in some low-rank format because the number of elements of the tensor grows exponentially. The proposed approach is based on the tensor train (TT) format for its computational efficiency and ability to handle large dimensions [@oseledets2010tt].
The contributions of this paper are as follows
- We introduce new initialization algorithm which takes into account the tensor generating process. The proposed algorithm is described in \[sec:initialization\].
- The proposed initialization technique automatically selects the rank of the tensor, the details are given in \[sec:tt\_cross\].
- We conducted empirical evaluations of the proposed approach and compared it with tensor completion techniques without our initialization. The results are given in \[sec:experiments\] and show the superiority of the proposed algorithm.
Tensor completion {#sec:tensor_completion}
=================
The formal problem statement is as follows. Suppose that $\mathcal{Y}$ is a $d$-way tensor, $\mathcal{Y} \in \mathbb{R}^{n_1 \times n_2 \times \cdots \times n_d}$ (by tensor here we mean a multi-dimensional array). Tensor values are known only at some subset of indices $\Omega \subset \{1, \ldots, n_1\} \times \cdots \times \{1, \ldots, n_d\}$. By $P_\Omega$ we denote the projection onto the set $\Omega$, i.e. $$P_{\Omega} \mathcal{X} = \mathcal{Z}, \quad
\mathcal{Z}(i_1, i_2, \ldots, i_d) = \begin{cases}
\mathcal{X}(i_1, i_2, \ldots, i_d) & \mbox{if } (i_1, \ldots, i_d) \in \Omega, \\
0 & \mbox{otherwise}.
\end{cases}$$ We formulate the tensor completion as an optimization problem $$\begin{aligned}
\label{eq:tensor_completion}
\min_{\mathcal{X}} \quad & f(\mathcal{X}) = \|P_{\Omega} \mathcal{X} - P_{\Omega} \mathcal{Y}\|_F^2 \\
\mbox{subject to} \quad & \mathcal{X} \in \mathcal{M}_r =
\{\mathcal{X} \in \mathbb{R}^{n_1 \times \cdots n_d} \; | \; {\rm rank}_{TT}
(\mathcal{X}) = \mathbf{r} \},
\end{aligned}$$ where ${\rm rank}_{TT}(\mathcal{X})$ is a tensor train rank of $\mathcal{X}$ [@oseledets2011tensor], which is a generalization of the matrix rank, and $\|\cdot\|_F$ is the Frobenius norm. A tensor $\mathcal{X}$ is said to be in tensor train format if its elements are represented as $$\mathcal{X}(i_1, \ldots, i_d) = \sum_{j_1, j_2, \ldots j_d}
\mathcal{G}^{(1)}_{1, i_1, j_1}
\mathcal{G}^{(2)}_{j_1, i_2, j_2} \cdots
\mathcal{G}^{(d)}_{j_{d - 1}, i_d, 1},$$ where $\mathcal{G}^{(i)}$ is a three-way tensor core with size $r_{i - 1} \times n_i \times r_{i}$, $r_0 = r_{d} = 1$. Vector $\mathbf{r}_{TT} = (r_0, \ldots, r_d)$ is called TT-rank.
Tensor train format assumes that the full tensor can be approximated by a set of $3$-way core tensors, the total number of elements in core tensors is $\mathcal{O}(dnr^2)$, where $r = \max\limits_{i = 0, \ldots, d}\{r_i\}$, $n = \max\limits_{i = 1, \ldots, d}\{n_i\}$, which is much smaller than $n^d$.
In problem we optimize the objective function straightforwardly with respect to tensor cores $\mathcal{G}^{(1)}, \ldots \mathcal{G}^{(d)}$ while having their sizes fixed. Problem is non-convex, so optimization methods can converge to a local minimum. To get an efficient solution we impose two requirements:
1. Initial tensor $\mathcal{X}_0$ in tensor train format should be as close to the optimum as possible.
2. Availability of an efficient optimization procedure that will be launched from the obtained initial tensor.
These steps are independent, and one can apply any desired algorithm in each of them.
In this work we develop the initialization algorithm that allows obtaining accurate initial tensor for the case when the tensor of interest is generated by some smooth function. The experimental section demonstrates that our initialization can improve the results of any optimization procedure, ensuring our approach to be universal.
Initialization {#sec:initialization}
==============
We consider tensors that are generated by some function, i.e. tensor values are computed as follows $$\mathcal{Y}_{i_1, \ldots, i_d} = f(x_{i_1}, \ldots, x_{i_d}),$$ where $f(\cdot)$ is some unknown smooth function and $(x_{i_1}, \ldots, x_{i_d})^\top \in \mathbb{R}^d$, $i_k = 1, \ldots, n_k$, $n_1, \ldots, n_d$ are tensor sizes. The set of points $\{(x_{i_1}, \ldots x_{i_d}): i_k = 1, \ldots, n_k; k = 1, \ldots, d\}$ is a full factorial Design of Experiments, i.e. a multi-dimensional grid, and we also assume that the grid is uniform.
In this setting the tensor completion can be considered as a regression problem and can be solved by any regression technique that guarantees the smoothness of the solution. However, in the tensor completion problem we are interested in a tensor of values of $f(\cdot)$ at a predefined finite grid of points. The tensor should be in a low-rank format to be able to perform various operations with tensor efficiently (e.g. calculation of the norm of the tensor, dot product and other).
These observations give us the solution — build regression model $\widehat{f}$ using the observed values of the tensor, then use the obtained approximation as a black-box for the TT-cross approximation algorithm [@oseledets2010tt]. The last step results in a tensor $\widehat{\mathcal{X}}$ in TT format, which is a low-rank format and allows efficient computations. The next step (which is optional) is to improve the obtained solution $\widehat{\mathcal{X}}$ by using it as initialization for any other tensor completion technique.
Let us write down the set of observed tensor values into a vector $\mathbf{y}$ and the corresponding indices into a matrix $\mathbf{X}$ (each row is a vector of indices $(i_1, i_2, \ldots, i_d)$). Then the approach for tensor completion (in TT format) can be written as follows
1. Construct initial tensor $\mathcal{X}_0$ in TT format:
1. Apply some regression technique using given data set $(\mathbf{X}, \mathbf{y})$ to construct approximation of the function that generates tensor values.
2. Apply TT-cross method (see \[sec:tt\_cross\], [@oseledets2010tt]) to the constructed approximation to obtain $\mathcal{X}_0$.
2. Apply some tensor completion technique using $\mathcal{X}_0$ as an initial value.
At step $1$(a) the choice of the regression technique affects the result of the initialization, although it can be arbitrary. It is required to choose the regression algorithm such that it will capture the peculiarities of the tensor we would like to restore. In this work we suppose that the tensor generating function is smooth (which is a common situation when modeling physical processes). Therefore, we choose a regression technique that is good at approximating smooth functions. A reasonable choice, in this case, is to use Gaussian Process (GP) Regression [@rasmussen2004gaussian]. GP models is a favorite tool in many engineering applications as they proved to be efficient, especially for problems where it is required to model some smooth function [@belyaev2016gtapprox]. The points $(x_{i_1}, \ldots, x_{i_d})$ are not given, all we know is that at the point with multi-index $(i_1, \ldots, i_d)$ on the grid the function value is equal to $\mathcal{X}_{i_1, \ldots, i_d}$. To make the problem statement reasonable we assume that the indices are connected with the points as follows: $x_{i_k} = a_k i_k + b_k$, where $a_k, b_k \in \mathbb{R}$. So, as an input for the approximation we set $a_k$ and $b_k$ such that $x_{i_k} \in [0, 1]$.
At step $1$(b) we use TT-cross because it allows to efficiently approximate black-box function by a low-rank tensor in TT format. Moreover, this approach can automatically select TT-rank making it more desirable. More details on the technique are given in \[sec:tt\_cross\].
The described approach has the following benefits
1. Initial tensor $\mathcal{X}_0$ which is close to the optimal value in terms of the reconstruction error at observed values. It will push the optimization to faster convergence.
2. Better generalization ability — there are many degrees of freedom: a lot of different tensor train factors can give low reconstruction error at observed positions but can give a large error at other locations. Accurate approximation model will push the initial tensor to be closer to the original tensor in both the observed positions and unobserved ones.
3. TT-cross technique chooses rank automatically, so there is no need to tune the rank of the tensor manually.
The described approach leads to the \[alg:initialization\]. The steps 3 and 4 of the algorithm are described in \[sec:gp\] and \[sec:tt\_cross\] correspondingly.
**Input**: $\mathbf{y}, \Omega$
1. Construct the training set $(\mathbf{X}, \mathbf{y})$ from $\mathbf{y}, \Omega$
2. Rescale inputs $\mathbf{X}$ to $[0, 1]$ interval
3. 4.
Gaussian Process Regression {#sec:gp}
---------------------------
One of the most efficient tools for approximating smooth functions is the Gaussian Process (GP) Regression [@burnaev2016regression]. GP regression is a Bayesian approach where a prior distribution over continuous functions is assumed to be a Gaussian Process, i.e. $$\mathbf{y} \, | \, \mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \, \mathbf{K}_f + \sigma_{noise}^2\mathbf{I}),$$ where $\mathbf{y} = (y_1, y_2, \ldots, y_N)$ is a vector of outputs, $\mathbf{X} = (\mathbf{x}_1^{\top}, \mathbf{x}_2^{\top}, \ldots, \mathbf{x}_N^{\top})^{\top}$ is a matrix of inputs, $\mathbf{x}_i \in \mathbb{R}^d$, $\sigma_{noise}^2$ is a noise variance, $\boldsymbol{\mu} = (\mu(\mathbf{x}_1), \mu(\mathbf{x}_2), \ldots, \mu(\mathbf{x}_N))$ is a mean vector modeled by some function $\mu(\mathbf{x})$, $\mathbf{K}_f = \{ k(\mathbf{x}_i, \mathbf{x}_j) \}_{i, j = 1}^N$ is a covariance matrix for some a priori selected covariance function $k$ and $\mathbf{I}$ is an identity matrix. An example of such function is a squared exponential kernel $$k(\mathbf{x}, \mathbf{x}') = \exp \left (
- \frac{1}{2} \sum_{i=1}^d \left (
\frac{\mathbf{x}^{(i)} - \mathbf{x}'{}^{(i)}}{\sigma_i}
\right )^2
\right ),$$ where $\sigma_i, i = 1, \ldots, d$ are parameters of the kernel (hyperparameters of the GP model). The hyperparameters should be chosen according to the given data set.
Without loss of generality we make the standard assumption of zero-mean data. Now, for a new unseen data point $\mathbf{x}_*$ we have $$\label{eq:gp_posterior}
\hat{f}(\mathbf{x}_*) \sim \mathcal{N}\left (\mu(\mathbf{x}_*), \sigma^2(\mathbf{x}_*) \right ),$$ $$\mu(\mathbf{x}_*) = \mathbf{k}(\mathbf{x}_*)^\top \mathbf{K}_y^{-1}\mathbf{y},$$ $$\sigma^2(\mathbf{x}_*) = k(\mathbf{x}_*, \mathbf{x}_*) - \mathbf{k}(\mathbf{x}_*)^\top \mathbf{K}_y^{-1} \mathbf{k}(\mathbf{x}_*),$$ where $\mathbf{k}(\mathbf{x}_*) = (k(\mathbf{x}_*, \mathbf{x}_1), \ldots, k(\mathbf{x}_*, \mathbf{x}_N))^T$ and $\mathbf{K}_y = \mathbf{K}_f + \sigma_{noise}^2\mathbf{I}$.
Let us denote the vector of hyperparameters $\sigma_i, i=1, \ldots, d, \sigma_f$ and $\sigma_{noise}$ by $\boldsymbol{\theta}$. To choose the hyperparameters of our model we consider the log-likelihood $$\log p(\mathbf{y} \, | \, \mathbf{X}, \boldsymbol{\theta}) =
-\frac12 \mathbf{y}^T \mathbf{K}_y^{-1}\mathbf{y} - \frac12 \log |\mathbf{K}_y| -
\frac{N}{2} \log 2 \pi$$ and optimize it over the hyperparameters [@rasmussen2004gaussian]. The runtime complexity of learning GP regression is $\mathcal{O}(N^3)$ as we need to calculate the inverse of $\mathbf{K}_y$, its determinant and derivatives of the log-likelihood. If the sample size $N$ ($|\Omega|$ in our case) is large, the computational complexity becomes an issue. There are several ways to overcome it. If the data set has a factorial structure (multidimensional grid in a simple case) we can use the algorithm from [@belyaev2015gaussian]. If the structure is factorial with a small number of missing values the method from [@belyaev2016computationally] should be applied. For general unstructured cases, the approximate GP model can be built using, for example, the model described in [@munkhoeva2018quadrature] or use a subsample as a training set.
After tuning of the hyperparameters, we can use the mean of the posterior distribution as a prediction of the model.
Note, that the input points $\mathbf{X}$ in our case is a set of indices of the observed values $\mathbf{y}$. For the GP model we scale each index to $[0, 1]$ interval.
Tensor-Train cross-approximation {#sec:tt_cross}
--------------------------------
To approximate tensor $\widehat{\mathcal{X}}$ generated by $\hat{f}$ we use Tensor-Train cross-approximation. First, let us consider the matrix case. Suppose that we are given a rank-$r$ matrix $\mathbf{A}$ of size $m \times n$. A cross-approximation for the matrix is represented as $$\mathbf{A} = \mathbf{C} \widehat{\mathbf{A}}^{-1} \mathbf{R},$$ where $\mathbf{C} = \mathbf{A}(:, J), \mathbf{R} = \mathbf{A}(I, :)$ are some $r$ columns and rows of the matrix $\mathbf{A}$ and $\widehat{\mathbf{A}} = A(I, J)$ is the submatrix on the intersection of these rows and columns. To construct accurate approximation it is required to find submatrix $\widehat{\mathbf{A}}$ of large volume. It can be done in $\mathcal{O}(nr^2)$ operations [@tyrtyshnikov2000incomplete].
Now for tensor $\widehat{\mathcal{X}} \in \mathbb{R}^{n_1 \times \cdots \times n_d}$ the procedure is the following. At the first step let us consider unfolding $\mathbf{X}_1$ of size $n_1 \times n_2 n_3 \cdot \cdots \cdot n_d$ and rank $r_1$. Using row-column alternating algorithm from [@tyrtyshnikov2000incomplete] we can find $r_1$ linearly independent columns of matrix $\mathbf{X}_1$, these columns form matrix $\mathbf{C}$. After that applying maxvol procedure [@tyrtyshnikov2000incomplete] to the matrix $\mathbf{C}$ we can find set of row indices $I_1 = \big[i_1^{\alpha_1} \big], \alpha_1 = 1, \ldots, r_1$, matrix $\mathbf{R}$ and matrix $\widehat{\mathbf{A}}_1$ that will give the cross-approximation of unfolding $\mathbf{X}_1$: $$\mathbf{X}_1 = \mathbf{C}\widehat{\mathbf{A}}_1^{-1}\mathbf{R}.$$ We set $$\mathbf{G}_1 = \mathbf{C}\widehat{\mathbf{A}}_1^{-1},$$ where $\mathbf{G}_1$ is of size $n_1 \times r_1$. Next, let us form tensor $\mathcal{R}$ from $r_1$ rows of $\mathbf{X}_1$: $$\mathcal{R}(\alpha_1, i_2, \ldots, i_d) = \widehat{\mathcal{X}}(i_1^{\alpha_1}, i_2, \ldots, i_d),$$ and reshape it into a tensor of size $r_1n_2 \times n_3 \times \cdots \times n_d$. Next step is to apply the same procedure to the unfolding $\mathbf{R}_1$ of the tensor $\mathcal{R}$ and obtain the matrices $\mathbf{C}$, $\widehat{\mathbf{A}}_2$ and $$\mathbf{G}_2 = \mathbf{C} \widehat{\mathbf{A}}_2^{-1}$$ of size $r_1n_2 \times r_2$.
Repeating the described procedure $d$ times we will end up with matrices $\mathbf{G}_1, \allowbreak \mathbf{G}_2, \ldots, \mathbf{G}_d$ of sizes $n_1 \times r_1, r_1n_2 \times r_2, \ldots, r_{d-1}n_d \times 1$. Then each matrix can be reshaped to the $3$-way tensor of size $r_{d - 1} \times n_d \times r_d$, $r_0 = r_d = 1$ and can be used as core tensors for TT format. It turns out that such representation is a TT decomposition of the initial tensor $\widehat{\mathcal{X}}$.
The exact ranks $r_1, \ldots, r_d$ are not known to us in general. They can only be estimated from the above (e.g., by the maximum rank of the corresponding unfolding). If the rank is overestimated then the calculation of matrices $\mathbf{G}_i$ becomes unstable operation (because we obtain almost rank-deficient unfolding matrices). However, in [@oseledets2010tt] the authors suggest some simple modification that overcomes this issue. Therefore, we need to estimate the ranks from the above, but the estimate should not be much larger than the real rank. So, the approach is to start from some small rank, construct the tensor in TT format and then apply recompression (see [@oseledets2011tensor]). If there is a rank that is not reduced, then we underestimated that rank and should increase it and repeat the procedure.
Experimental results {#sec:experiments}
====================
In this section we present the results of the application of our approach to $2$ engineering problems.
The experimental setup is the following. We try the following optimization algorithms
1. SGD – stochastic gradient descent [@zhao2018high],
2. Ropt – Riemannian optimization [@steinlechner2016riemannian],
3. TTWopt – weighted tensor train optimization [@zhao2018high],
4. ALS – alternating least squares [@grasedyck2013alternating].
We run each algorithm with random initialization and with the proposed GP-based initialization and then compare results.
The quality of the methods is measured using mean squared error (MSE) $$MSE = \frac{1}{|\Omega_{test}|}\|P_{\Omega_{test}} \widehat{\mathcal{Y}} - P_{\Omega_{test}}\mathcal{Y} \|_F^2,$$ where $\Omega_{test}$ is some set of indices independent from the given observed set of indices $\Omega$, $|\Omega_{test}|$ is a size of the set $\Omega_{test}$ and $\widehat{\mathcal{Y}}$ is an obtained approximation of the actual tensor $\mathcal{Y}$.
The listed above algorithms were compared on two problems: CMOS oscillator model and Cookie problem (see \[sec:cmos\_ring\_oscillator\] and \[sec:cookie\_problem\] correspondingly). In CMOS oscillator problem we run each optimization $10$ times with different random training sets and then calculate the average reconstruction error as well as standard deviation. However, for Cookie problem we performed $10$ runs, and the training set was the same during all runs because it is more computationally intensive and generating several training sets takes more time.
Note, that when we use GP based initialization, the TT-rank $\mathbf{r}_{{\rm TT}}$ of the tensor is selected automatically by the TT-cross algorithm and max value of $\mathbf{r}_{{\rm TT}}$ can be larger than $n$. The optimization algorithms with random initialization do not have a procedure for automatic rank selection, so we ran them with different ranks (from $1$ to $\min_{k}{n_k}$) and then chose the best one.
TTWopt implementation[^1] does not support high-dimensional problems. For higher dimensional problems the authors of TTWopt propose to use SGD. The authors of TTWopt also propose truncated SVD based initialization. The idea is to fill missing values using the mean value of the observed part of the tensor and then to apply truncated SVD to obtain TT cores. However, such approach is only applicable to low-dimensional tensors as it requires to calculate full matrices of large size.
For Ropt and ALS we used publically available MATLAB codes [^2].
Cookie problem {#sec:cookie_problem}
--------------
Let us consider parameter-dependent PDE [@ballani2015hierarchical; @tobler2012low]: $$\begin{aligned}
-{\rm div} (a(x, p) \nabla u(x, p)) &= 1, \quad x \in D = [0, 1]^2, \\
u(x, p) &= 0, \quad x \in \partial D,\end{aligned}$$ where $$a(x, p) = \begin{cases}
p_\mu, \quad \mbox{if } x\in D_{s, t}, \mu = mt + s, \\
1, \quad \mbox{otherwise},
\end{cases}$$ $D_{s, t}$ is a disk of radius $\rho=\frac{1}{4m + 2}$ and $m^2$ is a number of disks which form $m \times m$ grid. This is a heat equation where heat conductivity $a(x, p)$ depends on $x$ (see illustration in \[fig:cookie\_problem\]) and $p$ is an $m^2$-dimensional.
We are interested in average temperature over $D$: $u(p) = \int_{[0, 1]^2} u(x, p){\rm d}x$. If $p$ takes $10$ possible values then there are $\mathbf{10^{m^2}}$ possible values of $u(p)$.
In this work we used the following setup for the Cookie problem: each parameter $p$ lie in the interval $[0.01, 1]$, number of levels for each p is $10$, number of cookies is $m^2 = 9$ and $16$, size of the observed set is $N = 5000$, for the test set we used $10000$ independently generated points.
![Illustration of Cookie problem with $m=3$ ($9$ cookies).[]{data-label="fig:cookie_problem"}](figures/CookieProblem.pdf){width="50.00000%"}
The results of tensor completion are presented in \[tab:results\_cookie\]. One can see that GP based initialization gives lower reconstruction errors both on the training set and test set except for ALS technique. ALS method with the proposed initialization overfits: the error on the training set is close to $0$, whereas the test error is much more significant. The error on the training set is about $10^{-29}$, which means that the training set was approximated with machine precision. It is not surprising if we recall that there are only $5000$ observed values, while the number of free parameters that are used to construct TT is much higher.
CMOS ring oscillator {#sec:cmos_ring_oscillator}
--------------------
Let us consider the CMOS ring oscillator [@zhang2017big]. It is an electric circuit which consists of $7$ stages of CMOS inverters. We are interested in the oscillation frequency of the oscillator. The characteristics of the electric circuit are described by $57$ parameters. Each parameter can take one of $3$ values, so the total size of the tensor is $3^{57} \approx 1.57 \times 10^{27}$. The number of observed values that were used during the experiments is $N = 5000$. For the test set we used $10000$ independently generated points.
The results of the experiments are given in \[tab:results\_cmos\]. The table demonstrates that utilizing GP based initialization improves the results for all algorithms except ALS. ALS, in this case, overfits again: training error is extremely small, whereas the test error is much larger, though it is rather small compared to other techniques and ALS with random initialization.
\[tab:results\_cookie\]
-------- -------------------------------------------- -------- -------------------------------------------- --------
SGD $(1.66 \pm 0.067) \times 10^{-2}$ $1500$ $\mathbf{(2.86 \pm 0.18) \times 10^{-5}}$ $150$
Ropt $(4.13 \pm 2.20) \times 10^{-8}$ $1000$ $\mathbf{(5.48 \pm 1.10) \times 10^{-10}}$ $1000$
TTWopt $(2.73 \pm 0.19) \times 10^{-4}$ $100$ $\mathbf{(9.21 \pm 2.17) \times 10^{-7}}$ $100$
ALS $(1.07 \pm 1.07) \times 10^{-4}$ $100$ $\mathbf{(2.39 \pm 0.60) \times 10^{-30}}$ $100$
SGD $(3.14 \pm 1.08) \times 10^{-2}$ $1500$ $\mathbf{(1.65 \pm 0.13) \times 10^{-5}}$ $150$
Ropt $(1.42 \pm 0.01) \times 10^{-2}$ $1000$ $\mathbf{(3.42 \pm 0.50) \times 10^{-4}}$ $1000$
TTWopt $(1.31 \pm 0.00) \times 10^{-4}$ $100$ $\mathbf{(1.80 \pm 0.16) \times 10^{-6}}$ $100$
ALS $(6.59 \pm 3.30) \times 10^{-5}$ $100$ $\mathbf{(1.33 \pm 0.46) \times 10^{-29}}$ $100$
SGD $(2.06 \pm 2.31) \times 10^{-1}$ — $\mathbf{(9.97 \pm 0.40) \times 10^{-5}}$ —
Ropt $\mathbf{(1.48 \pm 0.90) \times 10^{-7}}$ — $(3.45 \pm 0.0165) \times 10^{-4}$ —
TTWopt $(4.52 \pm 0.50) \times 10^{-4}$ — $\mathbf{(5.27 \pm 0.74) \times 10^{-6}}$ —
ALS $\mathbf{(4.37 \pm 7.73) \times 10^{-2})}$ — $(3.78 \pm 1.08) \times 10^{0}$ —
SGD $(2.40 \pm 2.76) \times 10^{1}$ — $\mathbf{(1.15 \pm 0.05) \times 10^{-4}}$ —
Ropt $(1.47 \pm 0.003 \times 10^{-2}$ — $\mathbf{(5.38 \pm 0.07) \times 10^{-4}}$ —
TTWopt $(2.42 \pm 0.00) \times 10^{-4}$ — $\mathbf{(3.02 \pm 0.17) \times 10^{-5}}$ —
ALS $\mathbf{(3.57 \pm 5.65) \times 10^{-1}}$ — $(1.85 \pm 60.5) \times 10^{0}$ —
-------- -------------------------------------------- -------- -------------------------------------------- --------
: MSE errors for Cookie problem
------ ---------------------------------- --------- -------------------------------------------- ---------
error N iters error N iters
SGD $(7.77 \pm 15.25) \times 10^5$ $1500$ $\mathbf{(3.11 \pm 4.87) \times 10^{-4}}$ $150$
Ropt $(6.22 \pm 0.01) \times 10^3$ $1000$ $\mathbf{(9.50 \pm 4.28) \times 10^{-5}}$ $1000$
ALS $(9.95 \pm 0.26) \times 10^{-2}$ $300$ $\mathbf{(3.57 \pm 0.45) \times 10^{-26}}$ $300$
SGD $(3.45 \pm 9.68) \times 10^8$ — $\mathbf{(4.65 \pm 5.01) \times 10^{-4}}$ —
Ropt $(6.23 \pm 0.0) \times 10^3$ — $\mathbf{(9.68 \pm 4.16) \times 10^{-5}}$ —
ALS $(1.03 \pm 0.01) \times 10^{-1}$ — $\mathbf{(4.09 \pm 3.10) \times 10^{-4}}$ —
------ ---------------------------------- --------- -------------------------------------------- ---------
: MSE errors for CMOS oscillator[]{data-label="tab:results_cmos"}
All in all, the obtained results prove that GP based initialization allows improving the tensor completion results in general. At least it provides better training error. As for the error on the test set one should be more careful as the number of degrees of freedom is large and there exist many solutions that give a small error for the observed values but large errors for other values.
Related works {#sec:related_works}
=============
One set of approaches to tensor completion is based on nuclear norm minimization. The nuclear norm of a matrix is defined as a sum of all singular values of the matrix. This objective function is a convex envelope of the rank function. For a tensor the nuclear norm is defined as a sum of singular values of matricizations of the tensor.
There are efficient off-the-shelf techniques for such types of problems that apply interior-point methods. However, they are second-order methods and scale poorly with the dimensionality of the problem. Special optimization technique was derived for nuclear norm minimization [@gandy2011tensor; @liu2013tensor; @recht2010guaranteed]. More often such techniques are applied to matrices or low-dimensional tensors as their straightforward formulation allows finding the full tensor. It becomes infeasible when we come to high-dimensional problems.
The second type of approaches is based on low-rank tensor decomposition [@acar2011scalable; @chen2013simultaneous; @kressner2014low; @steinlechner2016riemannian; @yuan2017completion]. There are several tensor decompositions, and all these papers derive some optimization procedure for one of them, namely, CP decomposition, Tucker decomposition or TT/MPS decomposition. The simplest technique is the alternating least squares [@grasedyck2015alternating]. It just finds the solution iteratively at each iteration minimizing the objective function w.r.t. one core while other cores are fixed.
Another approach is based on Riemannian optimization that tries to find the optimal solution on the manifold of low-rank tensors of the given structure [@steinlechner2016riemannian]. The same can be done by using Stochastic Gradient Descent [@yuan2017completion]. Riemannian optimization, TTWopt, ALS and its modifications (e.g., ADF, alternating directions fitting [@grasedyck2013alternating]) try to find the TT representation of the actual tensor iteratively. At each iteration it optimizes TT cores such that the resulting tensor approximates well the tensor which coincides with the real tensor at observed indices and with the result of the previous iteration at other indices. All these approaches need to specify rank manually. The authors of [@zhao2015bayesian] apply the Bayesian framework for CP decomposition which allows them to select the rank of the decomposition automatically.
In some papers the objective is modified by introducing special regularizers to suit the problem better [@yokota2016smooth]. For example, in [@chen2013simultaneous; @zhao2015bayesian] to obtain better results for visual data a special prior regularizer was utilized.
Our proposed algorithm is an initialization technique for the tensor completion problems in TT format and can be used with most of the algorithms solving such problems. If the assumptions from \[sec:initialization\] (the tensor values are values of some rather smooth function of tensor indices) are satisfied the initial value will be close to the optimal providing better results. The question of a good initialization is rarely taken into account. In paper [@ko2018fast] a special initialization if proposed for visual data. The idea is to use some crude technique (like bilinear interpolation) to fill missing values and after that apply SVD-based tensor train decomposition. The drawback of the approach is that it can be applied only in case of small-dimensional tensors as we need to fill all missing values. In [@grasedyck2013alternating] they propose special initialization for the Alternating Direction Fitting (ADF) method. This is a general technique for the tensor completion and it does not take into account the assumptions on the data generating function.
Conclusions {#sec:conclusions}
===========
We proposed a new initialization algorithm for high-dimensional tensor completion in TT format. The approach is designed mostly for the cases when some smooth function generates the tensor values. It can be combined with any optimization procedure that is used for tensor completion. Additionally, the TT-rank of the initial tensor is adjusted automatically by the TT-cross method and defines the resulting rank of the tensor. So, the approach provides an automatic rank selection. Our experimental study confirms that the proposed initialization delivers lower reconstruction errors for many of the optimization procedures.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Zheng Zhang, who kindly provided us CMOS ring oscillator data set.
[^1]: <https://github.com/yuanlonghao/T3C_tensor_completion>
[^2]: <https://anchp.epfl.ch/index-html/software/ttemps/>
|
---
author:
- 'Maarten A. Breddels'
- Jovan Veljanoski
bibliography:
- 'vaexpaper.bib'
title: 'Vaex: Big Data exploration in the era of Gaia'
---
Introduction {#sec:intro}
============
Visualization plays an important role in astronomy, and is often used to find and display trends in data in the form of two dimensional scatter plots. The Herzsprung-Russel diagram for example, is one of the most famous scatter plots, which shows the relationship between the temperature and the luminosity of stars. Before the era of computers, these plots were drawn by hand, while now it is customary to use a software package or a library to produce them.
While two dimensional scatter plots may reveal trends or structure in a dataset of relatively small size, they become illegible when the number of samples exceeds $\sim 10^6$: the symbols overlap and ultimately fill up the plot in a uniform colour, obscuring any information such a figure may contain. When a dataset contains more than $10^6$ samples, it is more meaningful to visualize the local density in a two dimensional plot. The density can be determined using a kernel density estimator (KDE) or by a binning technique, equivalent to constructing a histogram in one dimension. The value of the local density can then be translated to a colour using a colourmap, which makes for an informative visualization.
[cc]{}  & \
To illustrate this concept, in Figure \[fig:scatter\_vs\_density\] we show the positions in equatorial coordinates of the stars in the *Gaia* DR1 catalogue [@GaiaDR1cat], which contains over 1 billion entries in total. On the top left panel we show a scatter plot containing only $10^4$ randomly chosen stars. This plot shows some structure, the Galactic disk is clearly seen, and one can also see the Large Magellanic Cloud as an over-density of points at (ra, dec) $\approx (80,-70)$. These structures are largely smeared out and nearly unnoticeable on the right panel where we show a scatter plot with $1\,000\,000$ stars. On the other hand, we get significantly more information if we visualize the data with a density plot. The bottom panel in Figure \[fig:scatter\_vs\_density\] shows a density plot of the entire *Gaia* DR1 catalogue, where one can see in great detail the structure of the disk, the Magellanic Clouds, patterns related to how the satellite scans the sky, and even some dwarf galaxies and globular clusters. All these details are lost when we represent the data with a scatter plot. However, a visualization library cannot stand on its own, and needs additional support for efficient transformation, filtering and storing of the data, as well as efficient algorithms to calculate statistics that form the basis of the visualization.
In this paper we present a new [`Python`]{} library called [`vaex`]{}, which is able to handle extremely large tabular datasets such as astronomical catalogues, N-body simulations or any other regular datasets which can be structured in rows and columns. Fast computations of statistics on regular N-dimensional grids allows analysis and visualizations in the order of a billion rows per second. We use streaming algorithms, memory mapped files and a zero memory copy policy to allow exploration of datasets larger than the Read Access Memory (RAM) of a computer would normally allow, e.g. out-of-core algorithms. [`Vaex`]{} allows arbitrary mathematical transformations using normal [`Python`]{} expressions and [`numpy`]{} functions which are lazily evaluated, meaning that they are only computed when needed, and this is done in small chunks which optimizes the RAM usage. Boolean expressions, which are also lazily evaluated, can be used to explore subsets of the data, which we call selections. [`Vaex`]{} uses a similar DataFrame API as [`Pandas`]{} [@PandasMckinney-proc-scipy-2010], a very popular [`Python`]{} library, which lessens the learning curve and makes its usage more intuitive. Visualization is one of the focus points of [`vaex`]{}, and is done using binned statistics in one dimension (histogram), in two dimensions (2d histograms with colour-mapping) and in three dimensions (using volume rendering). [`Vaex`]{} is split in several packages: `vaex-core` for the computational part, `vaex-viz` for visualization mostly based on matplotlib, `vaex-server` for the optional client-server communication, `vaex-ui` for the Qt based interface, `vaex-jupyter` for interactive visualization in the Jupyter notebook/lab based on `IPyWidgets`, `vaex-hdf5` for [`hdf5`]{}based memory mapped storage and `vaex-astro` for astronomy related selections, transformations and `(col)fits` storage.
Other similar libraries or programs exist, but do not match the performance or capabilities of [`vaex`]{}. [`TOPCAT`]{}[@Topcat2005ASPC], a common tool in astronomy, has support for density maps, but in general is focussed on working on a per row basis, and does not handle $10^9$ objects efficiently. The [`Pandas`]{} library can be used for similar purposes, but its focus is on in memory data structures. The `datashader`[^1] library can handle large volumes of data, focuses mainly on visualization in two dimensions, and lacks tools for exploration of the data. `Dask`[^2] and especially its `DataFrame` library is a good alternative for the computational part of [`vaex`]{} but it is accompanied with a rather steep learning curve.
This paper is structured as follows. In Section \[sec:main\] we begin by laying out the main ideas that form [`vaex`]{}, which we support with the relevant calculations. In Section \[sec:library\], we first present the basis of the library (`vaex-core`), and discuss all other packages that are subsequently built on top of it, such as `vaex-astro` for the astronomy related package, and `vaex-ui` for the Qt based user interface. We summarize our work in Section \[sec:conclusions\].
Note that this paper does not document all the features and options of the the [`vaex`]{}library. It lays out the principle ideas and motivation for creating the software, and presents the main capabilities of [`vaex`]{}. The full documentation can be found at: <https://vaex.io>. [`Vaex`]{}is open source and available under the MIT licence on github at: <https://github.com/maartenbreddels/vaex>.
Main ideas {#sec:main}
==========
In this section, we lay out the main ideas and the motivation for developing [`vaex`]{}. We start by discussing the possibilities and limitations when dealing with large tabular datasets. We then present some calculations to show that it is indeed theoretically possible to process 1 billion samples per second, and reflect back on that with an implementation.
Constraints and possibilities
-----------------------------
In the Introduction we clearly showed how a scatter plot displaying $\sim 10^9$ samples is usually not meaningful due to over-plotting of the symbols. In addition, when one want to process $10^9$ samples in one second on a Intel(R) Core(TM) i7-4770S CPU $3.1$ GHz machine with four cores, only $12.4$ CPU cycles are available per sample. That does not leave room for plotting even one glyph per object, as only a few CPU instructions are available. Furthermore, considering numerical data for two columns of the double precision floating point type, the memory usage is 16 GB $(10^9\times 2 \times 8~\text{bytes} =
16\times 10^9~\text{bytes} \approx 15~\text{GiB})$, which is quite large compared to a maximum bandwidth of $~25.6 \text{GB/s}$ for the same CPU.
Therefore, for the [`vaex`]{}library we only consider streaming or out-of-core algorithms which need one or a few passes over the data, and require few instructions per sample. The computation of statistics such as the mean or higher moments of the data are examples of such algorithms. The computation of a histogram on a regular grid can also be done with only few instructions, enabling us to efficiently visualize large amounts of data in one dimension by means of a histogram, in two dimensions via a density plot, and in three dimensions by the means of volume or isosurface rendering.
Preprocessing the data can lead to additional increase in performance. Given that users often perform various transformations on the data while they are exploring it, such as taking the log of a quantity or the difference between two quantities, we do not consider any preprocessing.
Real performance
----------------
We implemented a simple binning algorithm in [`C`]{}with a Python binding, finding that we can create a $256\times256$ two dimensional histogram from a dataset with [0.6 billion]{}samples in [0.55 seconds]{}, processing [1.1 billion objects/s]{}, which we consider acceptable for interactive visualization. This code uses multi-threading[^3] to achieve this speed, while using $\sim 75-85\%$ (15-17 GB/s) of the maximum memory bandwidth[^4].
N-dimensional statistics
------------------------
Apart from simply counting the number of samples in each bin, one can generalize this idea to calculate other statistics per bin using extra columns. Instead of simply summing up the number of samples that fall into each bin, one can use the same algorithm to perform other computations on a particular column, effectively calculating many statistics on a regular grid in N dimensions, where 0 dimensions implies a scalar. For example, let us consider a dataset that features four columns [$x$, $y$, $v_x$, $v_y$]{}, where the first two represent the position and the last two the corresponding velocity components of a particle or a star. One can construct a two dimensional grid spanned by $x$ and $y$ displaying the mean $v_x$ by first summing up the $v_x$ values and then dividing by the total number of samples that fall into each bin. The same exercise can be repeated to calculate the mean velocity in $y$ direction. Higher order moment can also be calculated, allowing one to compute and visualize vector and tensor quantities in two and three dimensions. The types of statistics available in [`vaex`]{}are listed in Section \[sec:statistics\] and Table \[tab:algo\].
Implementation
--------------
These ideas and algorithms, which are efficiently implemented, form the basis of the [`vaex`]{}library. [`Vaex`]{}exposes them in a simple way, allowing users to perform computations and scientific visualizations of their data with minimal amount of code. The graphical user interface program, from now on referred to as the program, uses the library to directly visualize datasets to users, and allows for interactive exploration. By this we mean the user is not only able to navigate (zoom and pan), but also to make interactive selections (visual queries), which can be viewed in other windows that display a plot created by a different combination of columns from those on which the selection was made on (linked views). [`Vaex`]{}also provides ranking of subspaces[^5], by calculating their mutual information or correlation coefficient in order to find out which subspaces contain more information.
Using a part of the data
------------------------
In some cases, it may be useful to do computations a smaller random subset of of data. This is beneficial for devices that do not have enough storage to keep the whole dataset such as laptops, and will also require less computing power. This is also useful for servers, as we will see in Section \[sec:vaex-server\] in order to handle many requests per second. Instead of drawing a random subset of rows the the full dataset, we store the dataset with the rows in a random order, and than ‘draw’ a random subset of rows (which will be the same every time), by only processing the first N rows. To support this, the library includes the option to covert and export a dataset with the rows in a random order. Note that to shuffle more than $2^{32}\approx4.2\times10^9$ rows, a 64 bit random number generator is needed. For the moment, this is only supported on the Linux operation system.
[`Vaex`]{} {#sec:library}
==========
The ideas of the previous section form the basis of the [`vaex`]{}library. [`Vaex`]{}is a [`Python`]{}package, consisting of pure [`Python`]{}modules as well as a so called extension module, written in [`C`]{}, which contains the fast performing algorithms, such as those for binning the data. The [`vaex`]{}library can be installed using `pip`, or (ana)conda[^6]. Its source code and issue tracker are on-line at <https://github.com/maartenbreddels/vaex>, and the homepage is at <https://vaex.io>.
[`Vaex`]{}is available as one (meta) package which will install all packages in the [`vaex`]{}family. However, if only a few functionalities are needed, only the relevant packages can be installed. For instance in many cases only `vaex-core`, `vaex-hdf5` and `vaex-viz` are needed. One can thus avoid installing `vaex-ui` since it has (Py)Qt as a dependency, which can be more difficult to install on some platforms.
vaex-core
---------
The foundation of all [`vaex`]{}packages is `vaex-core`. This contains the most important part, the Dataset class, which wraps a series of columns ([`numpy`]{}arrays) in an API similar to Pandas’ DataFrames, and gives access to all the operations that can be performed on them, such as calculating statistics on N-dimensional grids or the joining of two tables. On top of that, the Dataset class does bookkeeping to track virtual columns, selections and filtering. Note that in [`vaex`]{}almost no operation makes copies of the data, since we expect the full dataset to be larger than the RAM of typical computer.
### (Lazy) Expressions
In practice, one rarely works only with the columns as they are stored in the table. Within the [`vaex`]{}framework, every statistic is based on a mathematical expression, making it possible to not just plot the logarithm of a quantity for example, but to plot and compute statistics using an arbitrary, often user defined expression. For instance, there is no difference in usage when calculating statistics on existing columns, for example the mean of $x$, or any mathematical operation using existing columns, for example $x+y$, where $x$ and $y$ are two columns of a [`vaex`]{}dataset. The last expression will be calculated on the fly using small chunks of the data in order to minimize memory impact, and optimally make use of the CPU cache. Being able to calculate statistics on an N-dimensional grid for arbitrary expressions is crucial for exploring large datasets, such as the modern astronomical catalogues or outputs of large-scale numerical simulations. For instance taking the logarithm of a column is quite common, as well as calculating vector lengths (e.g. $\sqrt{x^2+y^2+z^2}$). No pre-computations are needed, giving users the complete freedom of what to plot or compute.
Contrary to the common [`Pandas`]{}library, a statement like , where df is a [`Pandas`]{}DataFrame containing the columns `b` and `c`, would be directly computed and will results in additional memory usage equal to that of the columns `b` or `c`. In [`vaex`]{}, the statement , where `ds` is a [`vaex`]{}Dataset, results in an expression which only stores the information of how the computation should be done. The expression will only be calculated when the result of `a` is needed, which if often referred to as a lazy evaluation. For convenience, a [`vaex`]{}Dataset can also hold what we refer to as virtual columns, which is a column that does not refer to a [`numpy`]{}array, but is an expression. This means that many columns can be added to a dataset, without causing additional memory usage, and in many cases causing hardly any performance penalty (when we are not CPU-bound). For instance, a vector length can be added using , which can then be used in subsequent expressions. This minimizes the amount of code that needs to be written and thus leads to less mistakes.
### Just-in-time compilation
Once the result of an expressions is needed, it is evaluated using the [`numpy`]{}library. For complex expressions this can result in the creation of many temporary arrays, which may decrease performance. In these cases, the computation speed of such complex expressions can be improved using just-in-time (JIT) compilation by utilizing the `Pythran` [@Pythran] or `Numba` [@Numba] libraries which optimize the code at runtime. Note that the JIT compilation will not be done automatically, but needs to be manually applied on an expression, e.g. .
[ll]{} Statistic & Description\
count & Counts the number of rows, or non-missing values of an expression.\
sum & Sum of non-missing values of an expression.\
mean & The sample mean of an expression.\
var & The sample variance of an expression, using a non-stable algorithm.\
std & The sample standard deviation of an expression using a non-stable algorithm.\
min & The minimum value of an expression.\
max & The maximum value of an expression.\
minmax & The minimum and maximum value of an expression (faster than min and max seperately).\
covar & The sample covariance between two expressions.\
correlation & The sample correlation coefficient between two expressions, i.e. $\text{cov}[{x,y}] / \left(\sqrt{\text{var}[x]\text{var}[y]} \right)$.\
cov & The full covariance matrix for a list of expressions.\
percentile\_approx &\
\
\
median\_approx & Approximation of the median, based on the percentile statistic.\
mode & Estimates the mode of an expression by calculating the peak of its histogram.\
mutual\_information & Calculates the mutual information for two or more expression, see Section \[sec:mi\] for details.\
nearest & Finds the nearest row to a particular point for given a metric.\
### Selections/Filtering
In many cases one wants to visualize or perform computations on a specific subset of the data. This is implemented by doing so called ‘selections’ on the dataset, which are one or more boolean expressions combined with boolean operators. Such selections can be defined in two ways, via boolean expressions, or via a geometrical (lasso) selection. The boolean expressions have the same freedom as expressions applied on the dataset when computing statistics, and can be a combination of any valid logical expressions supported by [`Python`]{}using one or more (virtual) columns. Examples of such booleans expressions are or , where the ampersand means logical “and”. Although the geometrical lasso selection can be implemented via boolean expressions, it is implemented separately for performance reasons. The lasso selection can be used a in graphical user interfaces to select regions with a mouse or other pointing device, or to efficiently select complex regions in two dimensions such as geographical regions. A dataset can have multiple selections, and statistics or visualizations can be computed for one or more selections at the same time (e.g. ) in a single pass over the data. When selections are created using ‘select’ method , they can be named by passing a string to the name argument, and the result of the selection, which is a boolean array, will be cached in memory leading to a performance increase. If no name is given, it will assume the name ’default’. Thus all selection arguments in [`vaex`]{}can take a boolean expression as argument, a name (referring to a selection made previously with ), or a boolean, where refers to no selection and to the default selection. This is useful for selections that are computationally expensive or selections that are frequently used. In the current implementation, a named selection will consume one byte per row, leading to a memory usage of 1 GB for a dataset containing $10^9$ rows. Note that no copies of the data are being made, only a boolean mask for the selection is created.
Often, a part of the data will not be used at all as part of preprocessing or cleaning up. In this case we want a particular selection to be always applied, without making a copy of the data. We refer to this as filtering, and is similarly done as in [`Pandas`]{}, e.g. ). The filtering feature is implemented in exactly the same way as the selections, except that a filter will always be applied, whereas a selection has to be passed to a calculation explicitly each time.
A history of the expressions that define the selections is also kept, which leads to less memory usage and enables users to undo and redo selections. In addition, the expressions which define a selection can be stored on disk, making the steps that led to a particular selection traceable and reproducible.
### Missing values
It often happens that some samples in a column of a dataset lack an entry. Such missing values are supported using [`numpy`]{}’s masked array type, where a boolean is kept for every row of a column, specifying whether a value is missing or not. For floating point numbers a NaN (Not a Number) values are also interpreted as missing values.
### Units
Optionally, a unit can be assigned to a column. Expressions based on columns which all have units assigned to them will also result is an unit for that expression. For visualization, the units can be used in the labelling of axis, as is done in `vaex-viz` (see the x-axis labelling of the top panel of Figure \[fig:viz1d\] for an example). The use of units fully relies on the `Astropy` [`Python`]{}package [@Astropy2013].
### Statistics on N dimensional grids {#sec:statistics}
One of the main features of [`vaex`]{}is the calculations of statistics on regular N-dimensional grids. The statistics, listed in Table \[tab:algo\] can be computed for zero[^7] or higher dimensional grids, where for each dimension an expression, a bin-count and a range (a minimum and a maximum) must be specified. All these calculations make use of the fast N-dimensional binning algorithm which is explained in more detail in Appendix \[app:algo\]. Each method can take one or more selections as arguments, which will then stream over the data once, or as few times as possible, giving optimal performance especially when the size of the data exceeds that of the RAM.
vaex-hdf5
---------
In order to achieve the performance estimated in Section \[sec:main\], we need to put some constraints on how the data is stored and accessed. If we use the typical unbuffered POSIX read method, assuming all the data from disk is cached in memory, we would still have the overhead of the memory copy, in addition to the system call overhead. Alternatively, if the data is stored in a file format that can be memory mapped, it will enable us to directly use the physical memory of the operating system cache, eliminating unnecessary overheads[^8]. Aside from the memory mapping requirements, we also impose additional constrains on the file format in which the data is to be stored. First, we require the data to be stored in the native format of the CPU (IEEE 754), and preferably in the native byte order (little endian for the x86 CPU family). Our second requirement is that the data needs to be stored in a column based format, meaning that the datum of the next row is in the next memory location. In cases where we only use a few columns, such as for visualization, reading from column based storage is optimal since the reading from disk is sequential, giving maximum read performance.
The well known and flexible file format [`hdf5`]{}has the capabilities to do both column based storage and to store the data in little and in big endian formats. The downside of the [`hdf5`]{}format is that it can store almost anything, and there are no standards for storing meta information or where in the file a table should be stored. To reconcile this issue, we adopted the VOTable as an example [@std:VOTABLE], and also implemented support for Unified Content Descriptors @std:UCD [UCD], units and descriptions for every column, and a description for the tables. Having UCDs and units as part of the column description allows the software to recognize the meaning of the columns, and suggest appropriate transformations for example. The layout of the file is explained in more detail in <https://vaex.io>.
Although [`vaex`]{}can read other formats, such as FITS, ascii or VOTable, these require parsing the file or keeping a copy in memory, which is not ideal for datasets larger than $\gtrsim 100$ MB. For superior performance, users can convert these formats to [`hdf5`]{}using [`vaex`]{}. An intermediate solution is the column based FITS format, which we will discuss in section \[sec:vaex-astro\].
vaex-viz {#sec:vaex-viz}
--------
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Example of one dimensional visualization for the `vaex-viz` package. **Top:** Histogram of [L$_z$]{}, the angular momentum in the $z$ direction. Because the units are specified in the data file for this column, it will be included in the labelling of the x axis by default. **Bottom:** Similar as above, but showing the mean of the energy [E]{}in each bin.[]{data-label="fig:viz1d"}](fig/vaex-viz1d-a "fig:")
![Example of one dimensional visualization for the `vaex-viz` package. **Top:** Histogram of [L$_z$]{}, the angular momentum in the $z$ direction. Because the units are specified in the data file for this column, it will be included in the labelling of the x axis by default. **Bottom:** Similar as above, but showing the mean of the energy [E]{}in each bin.[]{data-label="fig:viz1d"}](fig/vaex-viz1d-b "fig:")
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
---------------------------- -------------------------------
 
 
 
---------------------------- -------------------------------
A core feature of [`vaex`]{}is the visualization based on statistics calculated on N-dimensional grids. The `vaex-viz` package provides visualization utilising the [`matplotlib`]{}library [@MPLHunter:2007].
To showcase the main feature of [`vaex`]{}, we use a random 10% subset of the dataset generated by @Helmi2000MNRAS which will be downloaded on the fly when is executed. This dataset is a simulation of the disruption of 33 satellite galaxies in a Galactic potential. The satellites are almost fully phase-mixed, making them hard to disentangle in configuration space[^9], but they are still separable in the space spanned by the integrals of motion: [E]{}(Energy), [L]{}(total angular momentum)[^10], and the angular momentum around the z axis [L$_z$]{}. Even though this dataset contains only $330\,000$ rows, it serves well to demonstrate what can be done with [`vaex`]{}while being reasonably small in size.
Larger datasets, such as 100% of the @Helmi2000MNRAS dataset, the Gaia DR1 catalogue [@GaiaDR1cat], or over 1 billion taxi trips in New York can be found at <https://vaex.io>.
For one dimensional visualizations, all statistics listed in Table \[tab:algo\] can be plotted as a function of a single parameter or an expression, where the count method will be used to create a histogram. An example is shown in Figure \[fig:viz1d\], where on the top panel we show a regular one dimensional histogram for [L$_z$]{}, while in the bottom panel we visualize the mean energy [E]{}in each bin in log radius. Note that the x-axis in the top panel include a unit by default because these are include in the data file.
For two dimensional visualizations, we can display a two dimensional histogram as an image. An example of this is shown in the top left panel of Figure \[fig:viz2d\], which shows a plot of $y$ versus $x$ (the positions of the simulated particles) where the logarithm of the bin counts is colour-mapped. Again, note the units that are included by default on the axes. Similarly as for the one dimensional case, we can also display other statistics in bins as shows in the top right panel of Figure \[fig:viz2d\]. Here instead of showing the counts in bins, we display the standard deviation of the $z$ component of the velocity vector ($v_z$). This already shows some structure: a stream that is not fully phase mixed can be readily seen.
In the middle left panel of Figure \[fig:viz2d\], we create a selection (a rectangle in [E]{}and [L$_z$]{}space), and visualize the full dataset and the selection on top of each other. The default behaviour is to fade out the full dataset a bit so that the last selection stands out. Since selections are stored in the Dataset object, subsequent calculations and plots can refer to the same selection. We do this in the right panel of the middle row of the same Figure, where we show in a difference space ($y$ vs $x$) what the full dataset and the selection look like. Here we can clearly see that the clump selected in [E]{}and [L$_z$]{}space corresponds to a not fully phase-mixed stream, the same structure we noticed in the top right panel in this Figure.
On the bottom left panel of Figure \[fig:viz2d\], we use the same selection but a different visualization. We first repeat the same visualization as in the middle right panel, but overlay vectors, or a quiver plot. The method calculates a mean vector quantity on a (coarser) grid, and displays them using vectors. This can give better insight into vectorial quantities than for instance two separate density plots. The grid on which the vectorial quantity is computed is much courser to not clutter the visualization. Optionally, one can also colour-map the vectors in order to display a third component, for instance the mean velocity in the $z$ direction. Using a combination of the density map and the vector fields, one can plot up to five different dimensions on the same Figure.
Similarly to a vectorial quantity, we can visualize a symmetric two dimensional tensorial quantity by plotting ellipses. We demonstrate this on the bottom right panel in Figure \[fig:viz2d\] using , where we visualize the velocity dispersion tensor of the $x$ and $y$ components for the same selection. In this visualization a diagonal orientation indicates a correlation between the two velocity dispersion components and the size of the ellipse corresponds to the magnitudes of the velocity dispersions.
vaex-jupyter
------------
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![**Top:** Screenshot of a Jupyter notebook using the widget back-end to visualize the dataset interactively using panning, zooming and selection with the mouse. The back-end used here is `bqplot`. **Bottom:** A 3d visualization using the ipyvolume backend, showing the stream discussed before, where the vector visualize the mean velocity. *An interactive version of this visualization can be found in the online version*.[]{data-label="fig:jupyter"}](fig/vaex-jupyter-a "fig:")
![**Top:** Screenshot of a Jupyter notebook using the widget back-end to visualize the dataset interactively using panning, zooming and selection with the mouse. The back-end used here is `bqplot`. **Bottom:** A 3d visualization using the ipyvolume backend, showing the stream discussed before, where the vector visualize the mean velocity. *An interactive version of this visualization can be found in the online version*.[]{data-label="fig:jupyter"}](fig/vaex-jupyter-3d "fig:")
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The [`vaex`]{}library, especially the visualization tools described in Section \[sec:vaex-viz\] can also be used in combination with the Jupyter (formerly IPython) notebook [@PER-GRA:2007][^11] or Jupyter lab, which is a notebook environment in the web browser. A web browser offers more options for interactive visualization and thus exploration of data compared to the static images which are the default in [`matplotlib`]{}. The Jupyter environment allows [`vaex`]{}to work in combination with a variety of interactive visualization libraries, mostly built on `ipywidgets`[^12]. For two dimensional visualization `bqplot`[^13] allows for interactive zooming, panning and on-plot selections as shown in the top panel of Figure \[fig:jupyter\]. `ipympl`[^14] is an interactive back-end for [`matplotlib`]{}, but unlike `bqplot`, the visualization is rendered in the kernel as opposed to in the browser, giving a small delay when rendering a plot. The `ipyleaflet`[^15] library can be used to overlay and interact with a geographical map.
For displaying the data in three dimensions we use `ipyvolume`[^16], which offers volume and isosurface rendering, and quiver plots using WebGL. A big advantage of using WebGL in the Jupyter notebook is that it allows one to connect to a remote server while running the visualization on the local computer, a feature that is difficult to set up using OpenGL. The bottom panel of Figure \[fig:jupyter\] shows an example of three dimensional volume rendering including a quiver plot, created using the synergy between [`vaex`]{}and `ipyvolume` in the Jupyter environment. Note that in the three dimensional version, especially the interactive version on-line, gives a much clearer view on the orientation and direction of rotation of the stream compared to the two dimensional version shown in Figure \[fig:viz2d\].
vaex-ui {#sec:program}
-------
\[sec:vaex-ui\]
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---------------------------------- ----------------------------------
 
---------------------------------- ----------------------------------
The `vaex-ui` package provides a graphical user interface that can be used as a standalone program or used in the Jupyter environment. We focus now on the standalone program. Upon starting, the program shows the open tables on the left side, as shown in the top left panel on Figure \[fig:program\]. On the right side it shows metadata information of the selected table, and it contains buttons for opening windows that can visualize the dataset, give statistics on the data, show the ranking of the subspaces, or display the whole table. The next tabs shows the available columns, allows editing of units and UCDs, and addition and editing of virtual columns. The third tab shows the variables that can be used in expressions.
Similarly as in Section \[sec:vaex-viz\], one can do one dimensional visualization using histograms or statistics in regular bins. The top right panel in Figure \[fig:program\] shows an example of this, where we plot the histogram of [L$_z$]{}for the example dataset as presented in Section \[sec:vaex-viz\]. The plot shown on this Figure is interactive, allowing zooming and panning with the mouse and keyboard, as well as interactive selections. In the “x-axis” text box of this window, one can enter any valid mathematical [`Python`]{}expression, such as for example, where , and are columns in the dataset we use. In addition to the columns present in the dataset, one can also use any pre-defined virtual columns or variables to compute user defined expressions. Apart from the standard histograms that are based on the count statistic in Table \[tab:algo\], users can visualize other statistics, such as the mean or standard deviation of an expression per bin.
For two dimensional visualizations, the program displays a two dimensional histogram using an image, also similar to Section \[sec:vaex-viz\]. An example of this is shown in the bottom left panel of Figure \[fig:program\], which is a plot of the [E]{}versus the [L$_z$]{}, where the logarithm of the bin counts is colour-mapped. On this panel, one can see the individual satellites, each having its own distinct energy and angular momentum. As in the case for the one dimensional histogram, the entries for the x- and y-axis fields can be any valid mathematical [`Python`]{}expressions. Also similar to the one dimensional visualization, the statistics listed in Table \[tab:algo\] can be visualized in the bins, now in two dimensions. Vectorial and symmetric tensor quantities can be displayed as well, in a similar manner to what is described in Section \[sec:vaex-viz\].
The program also supports volume rendering using the OpenGL shading language. We supports multi-volume rendering, meaning we can display both the full dataset and a selection. In addition, one can over-plot vector fields in three dimensions. The users have access to custom settings for the lighting and for the transfer function. Navigation and selections are done in two dimensional projection plots displayed alongside the panel that shows the three dimensional rendering. An example of this visualization is shown in the bottom right panel of Figure \[fig:program\].
### Linked views
The program also supports linked views [@Goodman2012]. This feature makes all the active plots of the program linked to a single selection, allowing for complex and interactive exploration of a dataset. To demonstrate this concept, the left panel of Figure \[fig:linked\], shows a zoomed-in view of the bottom left panel of Figure \[fig:program\], where we have selected a particular cluster of stars in the subspaces spanned by [E]{}and [L$_z$]{}. This is similar to what we have done in the middle left panel of Figure \[fig:viz2d\], except that we can now do the selection interactively with the mouse. On the right panel of Figure \[fig:program\] we see the how the selection looks like in configuration space, and we can readily see that the selected clump in [E]{}and [L$_z$]{}space is in fact a stream. This is confirmed the velocities of its constituents stars, which are displayed with the help of a vector field overlaid on the same panel.
### Common features
The windows that display the plots contain a number of options to aid users in the exploration and visualization of their data. These include the setting of an equal axis ratio, keeping a navigation history, undoing and redoing selections, and numerous options to adjust the visualization settings. Users also have the option to display multiple dataset on top of each other on the same figure using the concept of layers, where where different blending operations can be set by the user. There are also options for exporting figures as raster (e.g. jpeg) or vector (e.g. pdf) graphics, as supported by the [`matplotlib`]{}library. It is also possible to export the binned data and with it a script that reproduces the figure as seen on the screen. This enables users to further customize their plots and make them publication ready.
------------------------------ --------------------- -------------------
  
------------------------------ --------------------- -------------------
### Subspace exploration / ranking {#sec:mi}
Many tables nowadays contain a large number of columns. Thus, inspecting how different quantities present in a dataset depend on each other, and finding which subspaces, or combinations of those quantities in two or more dimensions contain the most information via manual inspection by the user can be quite tedious, and sometimes not feasible at all. To aid users in finding subspaces that are rich in information, the program offers the option to rank subspaces according to two metrics: the Pearson’s correlation coefficient and the mutual information. The calculation and interpretation of the Pearson’s correlation coefficient is well documented in classical statistics, and we will not discuss it further.
The mutual information is a measure of the mutual dependence between two or more random variables. It measures the amount of information one can obtain about one random variable through the measurement or knowledge of another random variable. In [`vaex`]{}, the mutual information is obtained using the Kullback-Leibler Divergence [@kullback1951 KLD], and for two random variables it is calculated via the expression:
$$I(X;Y) = \int_Y \int_X p(x,y) \log{\left(\frac{p(x,y)}{p(x)\,p(y)} \right) } \; dx \,dy
\label{eq:kld}$$
where $p(x,y)$ is the joint probability distribution of the random variables $x$ and $x$, while $p(x)$ and $p(y)$ are their marginalized probability distributions. This quantity can be calculated for all, or a user defined subset of subspaces having two or more dimensions. The mutual information effectively measures the amount of information contained within a space spanned by two or more quantities. On the left panel of Figure \[fig:ranking\] we show the window listing all two dimensional subspaces in our example dataset, sorted by mutual information. In the right panel of this Figure we show two subspaces that have the highest and two subspaces that have the lowest rank according to their mutual information, in the top and bottom rows respectively. One can readily see that the spaces spanned by the integrals of motion ([E]{}, [L]{}and [L$_z$]{}) are found to contains the most information, as expected.
vaex-astro {#sec:vaex-astro}
----------
This subpackage contains functionality mostly useful for astronomical purposes.
### FITS
Although FITS is not as flexible as [`hdf5`]{}, and is not designed to store data in column based format, it is possible to do so by storing one row where each column contains a large vector. [`TOPCAT`]{}is using this strategy, and calls it the [`col-fits`]{}format. However, the BINTABLE extension for [`FITS`]{}mandates that the byte order to be big endian. This mean that the bytes need to be swapped before use, which in [`vaex`]{}gives a performance penalty of $\sim$30%. For compatibility with existing software such as [`TOPCAT`]{}, we support the [`col-fits`]{}format both for reading and writing.
### Common transformation
The conversion between different coordinate systems is common in astronomy. The `vaex-astro` package contains many of the most common coordinate transformations, such as the conversion between positions and proper motions between different spherical coordinate systems (Equatorial, Galactic, Ecliptic), as well as the conversion of positions and velocities between spherical, Cartesian and cylindrical coordinate systems. The transformations also include the full error propagation of the quantities in question. If the dataset contains metadata describing the units of the quantities, these will be automatically transformed as needed, minimizing possible sources of error.
### SAMP
[`Vaex`]{}offers interoperability with other programs via the Simple Application Messaging Protocol [SAMP @SAMP2009ivoa.spec.0421B]. It can communicate with a SAMP hub, for instance by running [`TOPCAT`]{}’s build in SAMP hub, which can then broadcast a selection or objects that are ‘picked’ to other programs. [`Vaex`]{}understand the ‘table.load.votable‘ message, meaning other programs can send tabular data in the VOTable format to it. Although this transfer mechanism is slow, it means that any data that can be read into [`TOPCAT`]{}can be passed on to [`vaex`]{}. For example, one can download a VOTable from VizieR using the TAP protocol [@std:TAP], which is fully implemented in [`TOPCAT`]{}, and than use SAMP to transfer it to [`vaex`]{}. The same is possible with any other program, application or web service that supports SAMP.
vaex-server {#sec:vaex-server}
-----------
A dataset that consists of two columns with 1 billion rows filled with double precision floating point values amounts to 16 GB of data. On the other hand, if this dataset is binned on a $256\times256$ grid, which also uses double precision floating points for its bin valuesm has the size of only 0.5 MiB. Using a suitable compression this can be reduced further by a factor of $\approx 10$. This makes it possible and practical to do calculations on a server and transfer only the final results to a client.
A working client/server model is implemented in the `vaex-server` package, making it possible to work with both local and remote datasets at the same time. Also the program provided by [`vaex-ui`]{}allows a users to connect to a remote server. The server itself is completely stateless, meaning it does not keep into memory the state of a remote client. This means that when a user requests a statistic for a subsets of the data defined by a selection, the client needs to send that selection to the server, and the server will compute the selection at each request. The server can cache selections, speeding up subsequent calculations of statistics for which the same subset of the data is required. The benefit of having it stateless is that the server can be less complex, and can be scaled vertically, meaning that more servers can be added with a load balancer in front to scale up the workload.
Furthermore, clients have the option to let the server determine how much of the data it will use, assuming the data is shuffled, to give a approximate answer. The server will then estimate how much of the data should be processed to return a result in a predefined amount of time, which is 1 second by default. Clients that want to use a specific amount of data, up to 100%, may need to wait longer. Using this mechanism, the `vaex-server` can handle up to 100 requests per second on a single computer.
vaex-distributed
----------------
Many of the statistics that [`vaex`]{}can compute, such as the mean or variance can be written as a linear combination of smaller chunks of the data used. Thus it is possible to distribute the computations to different computers, each performing a part of the work on a smaller data chunk, and combine the results at the end. The `vaex-distributed` package makes this possible. With its use, we manage to perform computations 10 billion $(10^{10})$ rows per second on a cluster of 16 low-end computers. This demonstrates that [`vaex`]{}can be scaled up to even larger datasets of the order of $\approx 10-100$ billion rows with the help of a computer cluster, even if such a cluster is not composed of modern computers. Note that not all functionality is supported, as this is only a proof of concept.
vaex-ml
-------
Build on top of vaex, is another proof of concept package called `vaex-ml`, which combines machine learning with [`vaex`]{}’ efficient data handling. Comparing `vaex-ml` a k-means clustering algorithm to that of sklearn [@sklearn], we are about $5 \times$ faster and have low memory impact since [`vaex`]{}does not need to load the dataset in memory nor copies it. In both cases all cores are used. For PCA we are almost $7 \times$ faster, but [`vaex`]{}by default uses multi-threading for the calculation of the covariance matrix. Note that the sklearn implementations of both PCA and k-means are limited to 10 million rows in order to avoid using the swapdisk, while [`vaex`]{}happily works through the $100$ million rows. We furthermore integrated [`vaex`]{}with `xgboost` [@xgboost] to make boosted tree models easily available. Note that the `vaex-ml` source code is available and is free for personal and academic usage.
Any large tabular dataset
-------------------------
We would like to emphasize that, even though the main motivation for creating [`vaex`]{}was to visualize and explore the large *Gaia* catalogue, [`vaex`]{}is an ideal tool to use when working with any other large tabular dataset. To illustrate this point, in Figure \[fig:other\] we visualize three large datasets. The leftmost panel is a density plot of $\sim 1$ billion drop-off locations made by the Yellow cab taxies in New York City for the period between 2009 and 2015. The middle panel on the same Figure shows the positions for 0.6 billion particles from the pure dark matter Aquarius simulation [Aq-A, level2 @Springel2008]. The right panel in Figure \[fig:other\] displays the Open street map GPS data over Europe, made by $\sim 2$ billion GPS coordinates. These plots demonstrate that [`vaex`]{}is a suitable tool for exploration and visualization of any large tabular datasets regardless whether they are related to astronomy or not.
Conclusions {#sec:conclusions}
===========
In the future datasets will grow ever larger, making the use of statistics computed on N-dimensional grids for visualization, exploration and analysis more practical and therefore more common. In this paper we introduced [`vaex`]{}, a tool that handles large datasets, and processes $\sim 1$ billion rows per second on a single computer. The [`vaex`]{}[`Python`]{}library has a similar API to that of [`Pandas`]{}, making for a shallow learning curve for new users, and a familiar, easy to understand style for more experiences users. Built on top of many of the [`vaex`]{}packages is [`vaex-ui`]{}, which provides a standalone program allowing data visualization in one, two and three dimensions, interactive exploration of the data such as panning and zooming, and visual selections. By combining the [`vaex`]{}program, which can be used for a quick look at the data, with the [`vaex`]{}library for more advanced data mining using custom computations, users can be quite flexible in the manner in which they explore and make sense of their large datasets.
In the era of big data, downloading a large dataset to a local machine may not always be the most efficient solution. Using `vaex-server`, a dataset can be worked on remotely, only sending the final products, such as the (binned) statistics to the user. Combining multiple server in `vaex-distributed` allows [`vaex`]{}to scale effortless to $\sim 10$ billion rows per second on small cluster of a dozen computers.
We have demonstrated many of the features of [`vaex`]{}using the example dataset from [@Helmi2000MNRAS]. Visualization of statistics in one, two and three dimensions, using the full and subsets (selections in [`vaex`]{}) of the data as well as the visualization of vectorial and (symmetric) tensorial quantities. All of these calculations and visualizations will scale to datasets with billions of rows making [`vaex`]{}the perfect tool for the Visualization And EXploration (vaex) of the *Gaia* catalogue [@GaiaDR1cat], and even more so for the upcoming data releases or future missions such as LSST. The first data release of *Gaia* is available in [`hdf5`]{}format at <http://vaex.io>, and we plan to do so as well for the second data release.
[`Vaex`]{}is open source, and available under the MIT license. Contributions are welcome by means of pull requests or issue reports on <https://github.com/maartenbreddels/vaex>. The main website for [`vaex`]{}is <https://vaex.io>.
Acknowledgments {#acknowledgments .unnumbered}
===============
MB and JV thank Amina Helmi for making this work possible. MB thanks Yonathan Alexander for pushing me to create a more user friendly API. MB and JV AH are grateful to NOVA for financial support. This work has made use of data from the European Space Agency (ESA) mission *Gaia* (<http://www.cosmos.esa.int/gaia>), processed by the *Gaia* Data Processing and Analysis Consortium (DPAC, <http://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the *Gaia* Multilateral Agreement.
Binning algorihm {#app:algo}
================
``` {.python}
# This is equivalent code for the c code, but written in Python for readability
# It is for 2d only, 0, 1, and >= 3 dimensional are a generalization of this
# but more difficult to read
import numpy
def operation_count(input, aux):
if aux is not None and numpy.isinf(aux):
return input
else:
return input+1
def operation_minmax(input, aux):
if numpy.isinf(aux):
return input
else:
return min(input[0], aux), max(input[1], aux)
def operation_moment_012(input, aux):
if numpy.isinf(aux):
return input
else:
return [input[0] + 1, input[1] + aux, input[2] + aux**2]
def statistic2d(grid, x, y, aux, xmin, xmax, ymin, ymax, operation):
grid_width, grid_height = grid.shape[:2] # get dimensions of the 2d grid
for i in range(len(x)): # iterator over all rows
# normalize the x and y coordinate
norm_x = (x[i] - xmin) / (xmax-xmin)
norm_y = (y[i] - ymin) / (ymax-ymin)
# check if the point lies in the grid
if ( (norm_x >= 0) & (norm_x < 1) & (norm_y >= 0) & (norm_y < 1) ):
# calculate the indices in the 2d grid
index_x = numpy.int(norm_x * grid_width);
index_y = numpy.int(norm_y * grid_height);
# apply the operation
grid[index_x, index_y] = operation(grid[index_x, index_y],
aux[i] if aux is not None else None)
# To make a 2d histogram of 10 by 20 cells:
# data_x and data_y are 1d numpy arrays containing the data, and
# xmin, xmax, ymin, ymax define the border of the grid
shape = (10,20)
counts = np.zeros(shape)
statistic2d(counts, data_x, data_y, None, xmin, xmax, ymin, ymax, operation_count)
# To get a 2d grid with a min and max value of data_x at each cell
minmax = np.zeros(shape + (2,))
minmax[...,0] = np.inf
# Infinity and -infinity are good initial values since they will always be bigger
# (or smaller) than any finite value.
minmax[...,1] = -np.inf
statistic2d(minmax, data_x, data_y, data_x, xmin, xmax, ymin, ymax, operation_minmax)
# calculate the standard deviation on a 2d grid for x by calculating the count, the
# sum of x and the sum of x**2 at each cell
moments012 = np.zeros(shape + (3,))
statistic2d(moments012, data_x, data_y, data_x, xmin, xmax, ymin, ymax, operation_moment_012)
# then calculate the raw moments
moments2 = moments012[...,2] / moments012[...,0]
moments1 = moments012[...,1] / moments012[...,0]
# and finally the standard deviation (non stable algorihm)
std = numpy.sqrt((moments2 - moments1**2))
```
The binning algorithm in [`vaex`]{}is a generalization of a one dimensional binning algorithm to N dimensions. It also supports custom operations per bin, on top of simply counting the number of samples that a bin contains. The algorithm itself is written in [`C`]{}and [`C++`]{}. For presentation purposes, we rewrote it in pure [`Python`]{}and how it in Figure \[code:algo\] below. We consider [`Python`]{}code to be equivalent to pseudo code and thus self explanatory. The example includes the calculation of the counts, the minimum and maximum statistics, as well as the standard deviation on a regular two dimensional grid.
[^1]: <https://github.com/bokeh/datashader>
[^2]: <https://dask.pydata.org/>
[^3]: Releasing Python’s Global Interpreter Lock when entering the [`C`]{}part to actually make use of the multi-threading.
[^4]: Although the theoretical bandwidth is 25 GB/s, we measured it to be 20 GB/s using the bandwidth program from <http://zsmith.co/bandwidth.html>
[^5]: We call a combination of 1 or more columns (or expression using columns) a subspace
[^6]: A popular Python distribution: <https://www.continuum.io/downloads>.
[^7]: A scalar, or single value.
[^8]: Otherwise we would be limited to half of the total memory bandwidth.
[^9]: The 3d positions.
[^10]: Although [L]{}is not strictly an integral of motion in an axi-symmetric system, see @Helmi2000MNRAS.
[^11]: Jupyter is the new front end to the IPython kernel
[^12]: IPython widgets, <https://ipywidgets.readthedocs.io/>)
[^13]: <https://github.com/bloomberg/bqplot>
[^14]: <https://github.com/matplotlib/jupyter-matplotlib>
[^15]: <https://github.com/ellisonbg/ipyleaflet>
[^16]: <https://github.com/maartenbreddels/ipyvolume>
|
---
abstract: 'We present a new collapse condition to describe the formation of dark halos via nonspherical gravitational clustering. This new nonspherical collapse condition is obtained by the logical generalization of the spherical model to the nonspherical one. By solving a diffusion-like random matrix equation with the help of the Monte Carlo method, we show that this nonspherical collapse condition yields the mass function derived by Sheth & Tormen (1999) which has been shown to be in excellent agreement with the recent N-body results of high resolution. We expect that this nonspherical collapse condition might provide us a deeper insight into the structure formation, and suggest that it should be widely applied to various cosmological issues such as the galaxy merging history, the galaxy bias, and so forth.'
author:
- Tzihong Chiueh
- Jounghun Lee
title: On the Nonspherical Nature of Halo Formation
---
INTRODUCTION
============
The mass function $n(M,z)$ in cosmology is defined to give the comoving number density of dark halos with mass $M$ at redshift $z$. It provides a useful analytical tool to understand the formation and evolution of the large-scale structure in the universe. The excursion set approach to the mass function provides the most direct and solid way to count the number density of dark halos. Bond et al. (1991) applied for the first time the excursion set theory to the Gaussian random density field, and recovered the popular Press-Schechter mass function (Press & Schechter 1974, hereafter PS) with a correct normalization factor of $2$.
Sheth, Mo, & Tormen (1999, hereafter SMT) suggested an extension of the excursion set approach to a nonspherical dynamical model. Although the PS mass function works fairly well at the high-mass section, recent hight-resolution N-body simulations have yielded less intermediate-mass and more low-mass halos than the PS prediction [@lac-col94; @tor98; @gov-etal99; @jen-etal00]. It has been suspected that this discrepancy of the numerical mass functions with the PS prediction must be due to the departure of the true dynamics from the idealistic spherical one [@gov-etal99]. The work of SMT was in fact so motivated, attempting to justify the empirical mass function derived by Sheth & Tormen (1999, hereafter ST). Indeed, current numerical results from high-resolution N-body simulations agree with the ST formula much better than the standard PS mass function [@jen-etal00].
Yet it is hard to claim that the ST mass function is anything beyond a phenomenological fitting formula. Although SMT claimed that the excursion set approach associated with their nonspherical collapse condition does produce the ST mass function to excellent approximation, a more careful analysis of the nonspherical collapse condition from a different perspective ought to be encouraged. In particular, the perspective of symmetry and smoothness of the collapse condition in the $\Lambda$-space (see $\S 2$) should be seriously considered, and can serve as a guiding principle in constraining the correct collapse condition.
In this paper, we investigate the idea of SMT in a more sophisticated manner to find a physically motivated ellipsoidal collapse condition which yields almost the same ST mass function. The new collapse condition demonstrates in a clear-cut manner how the initial nonspherical properties of proto-halos affect the gravitational process, yielding the halo abundance substantially deviating from the PS predictions based on the spherical model.
NONSPHERICAL COLLAPSE CONDITION
===============================
We use the following five hypotheses to evaluate the mass function: 1) All cold dark matter elements eventually collapse into gravitationally bound halos by self similar clustering. 2) The collaspe process is so rapid that the violent relaxation may completely erase the internal structure of the bound region. 3) The collapse condition can be expressed by the linearly extrapolated parameters of proto-halos. 4) The rms fluctuations $\sigma(M)$ of the linear density field smoothed over a mass scale of $M$ at the moment of collapse determines the mass of the bound halo. 5) The collapse occurs in an ellipsoidal way, for which the necessary and sufficient collapse condition is a function of the three eigenvalues, $\lambda_1,\lambda_2,\lambda_3$ of the initial deformation tensor (defined as the second derivative of the linear gravitational potential).
Note that the first four hypotheses are borrowed from the standard PS mass function theory. The discrepancy from the PS theory arises in the fifth hypothesis. For the spherical collapse model on which the PS theory is based, the evolution of an initial spherical overdense region is governed by the self gravity alone, so the collapse condition for the given region to form dark halos depends solely on its local average overdensity $\delta$ ($\delta \equiv \Delta\rho/{\bar \rho}$. ${\bar \rho}$: the mean mass density). While for the nonspherical collapse model, not only the self gravity but also the tidal interaction with the surrounding matter acts on the given region. Therefore, the evolution of the initial density inhomogeneities can be no longer described by the local average density alone once the simple constraint of the spherical symmetry on the initial region is released. It must be described by other parameters quantifying both the intrinsic self gravity and the extrinsic tidal coupling with the neighbor mass distribution. It is worth mentioning here that it is this tidal interaction which causes the rotational motion of dark halos. That is, the generation of the angular momentum of dark halos is a unique consequence of the nonspherical collapse.
Given the third hypothesis, one can expect that the linear parameters to determine the [*sufficient*]{} nonspherical collapse condition may be the three eigenvalues of the tidal shear tensor (i.e., the deformation tensor), suggesting the fifth hypothesis. Below we also explain why all the three eigenvalues determine the [*necessary*]{} nonspherical collapse condition. Our fifth hypothesis is qualitatively consistent with the peak-patch theory for the ellipsoidal dynamics proposed by Bond & Myers (1996). SMT followed the peak-patch prescriptions to determine their nonspherical collapse condition. However, we have noted that the collapse condition (eq. \[3\] in SMT) obtained from the peak-patch picture has some [*unphysical*]{} drawbacks. In our model, instead of relying on the peak-patch prescriptions fully, we suggest an original idea of the [*logical generalization*]{} of the spherical collapse condition into the nonspherical one (see $\S 4$).
The above set of five basic hypotheses leads us to view the whole halo-formation process as a diffusion-like process of random fields in the three dimensional functional space spanned by the three eigenvalues $\lambda_1, \lambda_2, \lambda_3$ of the random deformation tensor. Note that here we use an [*unordered*]{} set of eigenvalues rather than the ordered one. Thus, all three eigenvalues are equivalent. Let us consider a large-scale smoothed initial region where the local eigenvalues of the deformation tensor are given by $\Lambda = (\lambda_1, \lambda_2, \lambda_3)$, and the rms linear density fluctuations has a small initial value of $\sigma (M_0)$ with the corresponding mass scale of $M_0$. As one zooms in to look into the details, the rms fluctuations increases and the probability distribution of $\Lambda$ [@dor70] becomes broader, making the low-mass bound objects easier to collapse. When $\Lambda$ of a given rms fluctuations $\sigma (M)$ just satisfies the collapse condition, this region will collapse into an [*isolated bound halo*]{} with mass $M$, according to the fourth hypothesis. The isolated bound halo refers to the halo which has just collapsed with no larger halo enclosing it.
Since the change of $\Lambda$ as the smoothing scale decreases is random, one can look upon this change of the smoothing scale as special kind of random walk of a particle in the three dimensional $\Lambda$-space. A particle corresponds to a bound region, and its position in the $\Lambda$-space is the local eigenvalues $\Lambda = (\lambda_1, \lambda_2, \lambda_3)$ of the deformation tensor defined at the region. Each random step is assumed to be independent, which amounts to choosing a sharp k-space filter to smooth out the density field [@pea-hea90; @bon-etal91; @jem95]. This random walk process is restricted within some absorbing boundary which corresponds to the collapse condition. The number of random-walk steps before the particle first hits the absorbing boundary is directly proportional to $\sigma (M)$, a decreasing function of M. Thus, those particles that first hit the absorbing boundary in a small number of steps correspond to the high-mass halos while the opposite cases correspond to the low-mass halos.
The shape of the absorbing boundary is determined by the collapse condition which is in turn governed by the underlying dynamics. For the case of the top-hat spherical model, the collapse condition is given by $\delta = \delta_c$ where $\delta_c$ is the density threshold. The original top-hat spherical dynamics gives $\delta_c \approx 1.69$ for a flat universe [@pee93]. However, the more realistic treatment of spherical collapse given the rapid virialization due to the growth of small-scale inhomogeneities gives $\delta_c \approx 1.5$ [@sha-etal99]. In our approach, we use this realistic lowered value of $\delta_c$.
Since $\lambda_1 + \lambda_2 + \lambda_3 = \delta$, the boundary for the spherical model is an infinite flat plane (the PS plane) described by the equation of $\lambda_1 + \lambda_2 + \lambda_3 = \delta_c$ in the $\Lambda$ space. Note that the PS plane is smooth (i.e., continuous and differentiable everywhere) and also symmetric about the line of $\lambda_1 = \lambda_2 = \lambda_3$. The latter property yields a rotationally invariant boundary, that is, the boundary remains unchanged under the exchange of the three variables ($\lambda_i \leftrightarrow \lambda_j$). We take these two important properties of the PS plane as the general properties that a physical collapse condition must satisfy. Given that the spherical collapse is a special case of the ellipsoidal one satisfying $\lambda_1 = \lambda_2 = \lambda_3$, we expect that a physically meaningful ellipsoidal collapse condition must possess these two properties just as the spherical one does. The rotational invariance implies the isotropic nature of the initial matter distribution, while the smoothness implies the absence of any inherent singularity in the gravitational collapse as a general physical process.
Note also here that these two required properties of the absorbing boundary, the [*rotational invariance*]{} and the [*smoothness*]{}, explains why the sufficient and necessary nonspherical collapse condition must be expressed in terms of all the three eigenvalues. If the absorbing boundary is expressed as a function of only one or two eigenvalues among the three, then either the rotational invariance or the smoothness must break in the $\Lambda$-space.
At any rate, it has long been pointed out that the spherical collapse condition is far from being realistic, and the true gravitational process must be ellipsoidal [@kuh-etal96]. Consequently the collapse condition cannot be expressed simply just by the density alone. Unfortunately, there has been no simple ellipsoidal dynamical model that can describe the nonlinear regime adequately well, whereas the simplest top-hat spherical model can trace all stages of halo formation even into the highly nonlinear regime after the moment of turn-around. The complicated nature of the tidal coupling with the surrounding matter in the nonlinear regime makes it extremely difficult to construct an universal ellipsoidal model from the first principles.
Nonetheless, some qualitative considerations on the nature of ellipsoidal dynamics can give us a hint for the collapse condition. The spherical collapse model is in fact a special case of the ellipsoidal one, satisfying $\lambda_1 = \lambda_2 = \lambda_3$. If the gravitational collapse of a bound region were spherical, then the differences between the three eigenvalues of a bound region, $|\lambda_i -\lambda_j|$ would remain zero during the collapse process. It implies that the the nonzero values of $|\lambda_i -\lambda_j|$ should quantify the [*nonspherical*]{} aspects of the true collapse. Using the given probability density distributions of each $\lambda$’s (see Appendix A in Lee & Shandarin 1998), one can easily show that $|\lambda_i -\lambda_j| \propto \sigma (M)$ on average. Thus, as $\sigma$ increases, the average $|\lambda_i -\lambda_j|$ of a bound region also increases in proportional to $\sigma$. Or, as the mass scale decreases, the degree to which the collapse deviates from the spherical model (the [*nonsphericality*]{}) increases. This explains why the PS mass function based on the top-hat spherical model works fairly well at the high-mass section where the nonsphericality is small [@tor98], while it fails at the low-mass section where the nonsphericality is high.
This idea can be quantitatively embodied by the $\Lambda$-space diffusion-like process. As mentioned above, the spherical collapse condition is represented by an absorbing boundary of an infinite flat plane in the $\Lambda$-space, with its distance to the origin being $\delta_c$. One may expect that the shape of the boundary for the nonspherical collapse should also be a smooth curved surface [@she-etal99] that coincides with the PS plane at its apex, where $\lambda_1 = \lambda_2 = \lambda_3$. The underlying logic is as follows: The distance from the PS plane to the nonspherical collapse boundary must provide a measure of the nonsphericality. The nonsphericality of the system is small at the high-mass section as argued above. In fact Bernardeau (1994) has shown semi-analytically that the evolution of rare events (very high-mass halos) is quasi-spherical. The formation of a high-mass halo therefore corresponds to the particles that reach the boundary near the $\lambda_1 = \lambda_2 = \lambda_3$ axis in only a small number of random-walk steps, and it suggests that the nonspherical collapse boundary should be smoothly tangential to the PS plane at the apex of that axis.
Distant from the apex, the nonspherical collapse boundary should increasingly deviate from the PS plane. This is because the distant part of the boundary can be reached only by those particles that have undergone a large number of random steps, and they correspond to the low-mass halos, for which the nonsphericality is dominant. As the nonspherical collapse boundary should also possess the smoothness and rotational invariance as the PS boundary does, the analytic equation for this rotationally symmetric smooth surface can be constructed as follows. On the curved boundary, the ratio of $\delta (\equiv \lambda_1 + \lambda_2 + \lambda_3$) to $\delta_c$ equals unity only at the apex, $\Lambda_T$. This ratio becomes slightly larger than unity near $\Lambda_T$, and increasingly exceeds unity at distant points from $\Lambda_T$. Thus, the most general equation of the nonspherical collapse boundary can be written as $\delta/\delta_c = S(r)$, where $$r \equiv \frac{1}{3}[(\lambda_1 - \lambda_2)^2 +
(\lambda_2 - \lambda_3)^2 + (\lambda_1 - \lambda_3)^2].$$ and $(1/r)dS(r)/dr >0$. Here the form of $r$ does guarantee the smoothness and rotational invariance of $S(r)$. It is worth noting that the variable $r$ is proportional to the angular momentum square of the bound region provided that the principal axes of the inertia and the deformation tensors of the region are not perfectly aligned with each other [@hea-pea88; @cat-the96]. Thus, the above general equation for the nonspherical collapse accounts for the generation of the rotational motion of dark halos.
Specifically, we propose the following boundary equation for the nonspherical collapse: $$\frac{\delta}{\delta_c}
= S(r) = \left( 1 + \frac{r^2}{\beta} \right)^{\beta},$$ where $\beta$ is a positive constant to be determined by fitting to the ST mass function. Here note that the height of our absorbing boundary $S(r)$ scales like $r^2$ rather than $r$. It guarantees the flatter bottom of the absorbing boundary, making it closer to the PS boundary around the apex, $\Lambda_T$ ($r=0$). Since the random walks representing the nonspherical collapse tend to avoid the axes of symmetry where any pair of the eigenvalues are the same [@dor70], the walks quicly diffuse away from $\Lambda_{T}$, never hitting the bottom of the absorbing boundary at $\Lambda_{T}$. Therefore in order to reproduce a mass function quite similar to the PS one at the high-mass section, one needs an absorbing boundary with a flat bottom. We realize this flat-bottom absorbing boundary by expressing $S(r)$ scaled as $r^2$ rather than $r$.
ALGORITHM
=========
In this section, we describe the numerical algorithm for the Monte-Carlo simulation of the random-walk process in $\Lambda$-space given the above absorbing boundary (eq. \[2\]). This algorithm basically represents our nonspherical collapse of bound regions into dark halos out of the initial Gaussian density field, which is in fact closely related to, but simpler than the ellipsoidal collapse model developed by Eisenstein & Loeb (1995).
To simulate the initial random deformation tensor where the rms density fluctuations is $\sigma (M_0) \equiv \sigma_0$, we first generate six independent Gaussian variables with the dispersion of $\sigma_0$, say $y_1, y_2, \cdots, y_6$. The symmetric deformation tensor $(d_{ij})$ can be constructed by the linear transformation of $(y_i)$ such that $$\begin{aligned}
d_{11} &=& -\frac{1}{3}\left( y_1 + \frac{3}{\sqrt{15}}y_2 +
\frac{1}{\sqrt{5}}y_3 \right), \nonumber \\
d_{22} &=& -\frac{1}{3}\left(y_1 - \frac{2}{\sqrt{5}}y_3\right),
\nonumber \\
d_{33} &=& -\frac{1}{3}\left( y_1 - \frac{3}{\sqrt{15}}y_2 +
\frac{1}{\sqrt{5}}y_3 \right), \nonumber \\
d_{12} &=& d_{21} = \frac{1}{\sqrt{15}}y_4, \hspace{0.5cm}
d_{23} = d_{32} = \frac{1}{\sqrt{15}}y_5, \hspace{0.5cm}
d_{31} = d_{13} = \frac{1}{\sqrt{15}}y_6.\end{aligned}$$ One can show easily that this linear transformation does satisfy the correlations of the deformation tensor [@bar-etal86].
Using the similarity transformation, we diagonalize the deformation tensor to find the three eigenvalues. We then check whether the set of the three eigenvalues crosses the boundary or not. If not, we generate a new six dimensional Gaussian random vector with the same dispersion of $\sigma_0$, and add it to the previous random vector. Here the new random vector is assumed to be uncorrelated with the previous random vector, which is consistent with the use of the sharp k-space filter.
Using this accumulated random vector, we repeat the above process: linear transformation into the deformation tensor, similarity transformation into a diagonal matrix to get the eigenvalues, and finally checking the boundary crossing. This process is repeated until the first crossing over the boundary occurs. The number of the repetition ($N_s$) is proportional to the square of $\sigma (M)$ at the moment of the halo formation such that $\sigma^{2} = N_s \sigma_0^{2}$. After a particle crosses the boundary, we re-starts the whole process with a new particle.
We have simulated a ensemble of $120,000$ particles, and calculated the distribution of the number of particles which first cross the boundary at a range of $[\sigma, \sigma + d\sigma]$. The results are plotted in Figure 1. This distribution is nothing but the differential volume fraction $dF/d\sigma$, occupied by the halos with the corresponding mass of $M$, directly proportional to the mass function by $dF/d\sigma = [n(M,z)/\bar{\rho}]d{\rm ln}M/d\sigma$.
To show the robustness of our approach, we also simulated the PS differential volume fraction by the above Monte Carlo method with the flat boundary. The triangle dots represent the resulting PS differential volume fraction while the dashed line is the analytic standard PS formula. The numerical and analytical PS mass function agree with each other perfectly, which guarantees the robustness of the diffusion approach to the mass function as well as the accuracy of our numerical scheme.
The solid line is the ST formula which has been proved to fit the currently available N-body results of high resolution very well [@jen-etal00], while the square dots represent our simulation results with the choice of the best-fit parameter of $\beta = 0.15$ (and we used the realistic lowered value of $\delta_c = 1.5$ as mentioned in $\S 2$). Here we used the ST formula as our fitting standard to find this value of $\beta$. Of course, fine tuning of $\beta$ would be necessary if one is to use another fitting standard. As one can see, our result agrees strikingly well with the ST formula, suggesting that our collapse condition can replace the PS spherical collapse condition in many interesting applications of the mass function.
DISCUSSIONS AND CONCLUSIONS
===========================
Motivated by the inspiring practical success of the ST mass function in recent N-body tests [@jen-etal00], we have attempted here to provide a more sophisticated and robust way to determine the nonspherical collapse condition which produces the ST mass function to good approximation.
In fact, various nonspherical approaches to the mass function were already attempted by several authors in the past decades [@mon95; @aud-etal97; @lee-sh98]. Strictly speaking, however, their approaches were not appropriate in the sense that they all used the original PS formalism which always yields ill-normalized mass functions. Although the PS mass function can be properly normalized by multiplying a constant normalization factor of $2$ [@bon-etal91], the mass function from a nonspherical model can no longer be corrected just by a constant normalization factor. The normalization factor is scale-dependent in any nonspherical dynamical model, due to the complicated pattern of the scale-dependent occurrence of the cloud-in-clouds. For a detailed description of the cloud-in-cloud problem, see Bond et al. (1991), Jedamzik (1995), and Lee & Shandarin (1998). Due to this scale-dependent manner of the cloud-in-cloud occurrence in the nonspherical models, the shape of the mass function could be very different from the one obtained without considering the cloud-in-cloud occurrence correctly. For example, we have tested by our diffusion algorithm the condition that the bound objects form at the local maxima of the smallest eigenfield of the deformation tensor. We found that the resulting mass function deviates considerably from the one originally given by Lee & Shandarin (1998) who had used a constant normalization factor of $12.5$.
In order to correctly solve the normalization problem, one should not rely on the original PS formalism. The only viable alternative for the evaluation of the mass function without extra efforts of concerning about the normalization is the excursion set approach, or equivalently the diffusion approach. SMT employed the excursion set approach to justify the ST mass function in terms of the nonspherical collapse condition. They related the nonspherical collapse condition with a moving barrier and approximated its shape with the help of the peak-patch prescriptions for the ellipsoidal dynamics [@bon-mye96]. Their moving barrier (eq. \[3\] in SMT) is similar to our curved boundary in concept, but differs significantly in practice since the SMT boundary is not a rotationally invariant smooth surface in the $\Lambda$-space and has kinks. Given equation (3) in SMT, one can see that the kinks arise at $p = 0$ ($p$: the prolateness of the given ellipsoidal region, see Bardeen et al. 1986). As mentioned in SMT, however, $p = 0$ on average in a Gaussian random field. Thus, the kinks of the SMT boundary occur in so high probability regions that one may not ignore the presence of those kinks.
Here we did not attempt to provide a better ellipsoidal dynamical model to describe the gravitational collapse process. Rather we retained the general framework of the peak-patch theory and try to improve the SMT boundary collapse condition into a more physically meaningful one by the logical generalization of the spherical collapse model. Two superior features of our collapse condition can be summarized as follows: First, it has a sound physical meaning, in that equation (2) is expressed in terms of only one single variable $r$ which is in fact directly proportional to the halo angular momentum square [@hea-pea88; @cat-the96]. Any physically meaningful nonspherical collapse condition should be expressed in terms of such quantities as obviously represent the nonshperical nature of halo formation. As mentioned in $\S 3$, the rotational motion of dark halos is a unique consequence of the nonspherical collapse. In this respect, our nonspherical collapse condition shows explicitly and quantitatively how the nonspherical gravitational collapse leads to the acquisition of the angular momentum of dark halos. Second it has the desirable analytical property, [*rotational invariance and smoothness*]{}. This feature of our collapse condition, in common with the PS collapse condition, will make it easy to extend various important cosmological issues, such as the halo merging, the light-to-mass bias, and so on, from the spherical dynamics to a nonspherical one.
We thank Z. Fan for helpful discussions and useful comments. We also thank our referee, R. Sheth, who helped us to improve the original manuscript. This work has been supported by the Taida-ASIAA CosPA Project. T. Chiueh acknowledges the partial support from the National Science Council of Taiwan under the grant: NSC89-2112-M-002-065.
Audit, E., Teyssier, R., & Alimi, J. M. 1997, , 325, 439 Bardeen, J. M., Bond, J. R., Kaiser, N., & Szalay, A. S. 1986, , 304, 15 Bernardeau, F. 1994, , 427, 51 Bond, J. R., Cole, S., Efstathiou, G., & Kaiser, N. 1991, , 379, 440 Bond, J. R., & Myers, S. T. 1996, , 103, 1 Catelan, P., & Theuns, T. 1996, , 282, 436 Doroshkevich, A. G. 1970, Astrofizika, 6, 581 Eisenstein, D. J., & Loeb, A. 1995, , 439, 520 Governato, F., Babul, A., Guinn, T., Tozzi, P, Baugh, C. M., Katz, N. & Lake, G. 1999, , 307, 949 Heavens, A. F., & Peacock, J. A. 1988, , 232, 339 Jedamzik, K. 1995, , 448, 1 Jenkins, A., Frenk, C. S., White, S. D. M., Colberg, J. M., Cole, S., Evrard, A. E., & Yoshida, N. 2000, preprint (astro-ph/0005260) Lacey, C. & Cole, S. 1994, , 271, 676 Lee, J. & Shandarin, S. F. 1998, , 500, 14 Monaco, P. 1995, , 447, 23 Peacock, J. A., & Heavens, A. F. 1990, , 243, 133 Peebles, P. J. E. 1993, Principles of Physical Cosmology (Princeton: Princeton Univ. Press) Press, W. H., & Schechter, P. 1974, , 187, 425 Kuhlman, B., Melott, A. L., & Shandarin, S. F. 1996, , 470, L41 Shaprio, P. R., Iliev, I., & Raga, A. C. 1999, , 307, 203 Sheth, R. K., & Tormen, G. 1999, , 308, 119 Sheth, R. K., Mo, H. J., & Tormen, G. 1999, preprint (astro-ph/9907024) Tormen, G. 1998, , 297, 648
|
---
abstract: |
Extensions of the Standard Model Higgs sector involving weak isotriplet scalars are not only benchmark candidates to reconcile observed anomalies of the recently discovered Higgs-like particle, but also exhibit a vast parameter space, for which the lightest Higgs’ phenomenology turns out to be very similar to the Standard Model one. A generic prediction of this model class is the appearance of exotic doubly charged scalar particles. In this paper we adapt existing dilepton+missing energy+jets measurements in the context of SUSY searches to the dominant decay mode $H^{\pm\pm}\to
W^\pm W^\pm$ and find that the LHC already starts probing the model’s parameter space. A simple modification towards signatures typical of weak boson fusion searches allows us to formulate even tighter constraints with the 7 TeV LHC data set. A corresponding analysis of this channel performed at 14 TeV center of mass energy will constrain the model over the entire parameter space and facilitate potential $H^{{\pm\pm}}\to W^\pm W^\pm$ discoveries.
author:
- Christoph Englert
- Emanuele Re
- Michael Spannowsky
title: Pinning down Higgs triplets at the LHC
---
Introduction {#sec:intro}
============
The recent discovery [@:2012gk; @:2012gu; @newboundsa; @newboundsb] of the Higgs boson [@orig] provides an opportunity to check the phenomenological consistency of various scenarios of electroweak symmetry breaking with measurements for the first time. Higgs triplet models have received considerable attention recently as they can reconcile the possibly observed anomaly in the $H\to \gamma\gamma$ channel [@lit; @andrew; @us; @Killick:2013mya]. Whether this excess persists or future measurements of the diphoton partial decay width will return to the Standard Model (SM) values as suggested by recent CMS results [@cmsaa] is unclear at the moment. However, as demonstrated in [@us], there are certain models with Higgs triplets [@Georgi:1985nv; @chano] which posses a large parameter space where the resulting phenomenology is SM-like [@carmi; @Belanger:2013xza] even for larger triplet vacuum expectation values. A generic prediction of electroweak precision measurements in this case is the appearance of doubly charged scalar particles $H^{{\pm\pm}}$ with a mass of several hundred GeV that result from the weak triplet structure in the Higgs sector extension.
Due to the quantum numbers of the ${\text{SU}}(2)_L$ triplet, Majorana mass-type operators can induce a prompt decay of $H^{{\pm\pm}}$ into two leptons with identical charge [@classic]. This interaction has already been constrained at the LHC in multilepton searches [@searchhpp]. However, as soon as the mass of the doubly charged scalar exceeds twice the $W$ mass, the decay to gauge bosons is preferred. This can be seen from the scaling of the partial decay widths: $\Gamma(H^{\pm\pm}\to W^\pm W^\pm)/\Gamma(H^{\pm\pm}\to
\ell^\pm \ell^\pm) \sim m_{H^{\pm\pm}}^2/m_W^2$. Over the bulk of the parameter space this leads to a dominant decay of the doubly charged Higgs to $W$ bosons [@Gunion:1989ci]. Formulating a meaningful constraint of this model class must therefore not neglect $H^{{\pm\pm}}\to
W^\pm W^\pm$ [@Kanemura:2013vxa].
The production of single intermediate $H^{{\pm\pm}}$ boson can only proceed via weak boson fusion (WBF) diagrams (Fig. \[fig:graph\]) and crossed processes ([*[i.e.]{}*]{} Drell-Yan type production). Hence, $H^{{\pm\pm}}$ production inherits all the phenomenological advantages of WBF Higgs and diboson production [@dieter]. Producing the relatively heavy final state requires energetic initial state partons. The $t$-channel color singlet exchange results in relatively small scattering angles of the two outgoing jets at moderate transverse momentum and a central detector region essentially free of QCD radiation. Eventually, the typical signature is two isolated central leptons and missing energy, and two forward jets at large rapidity differences with high invariant mass. Phenomenological investigations of this signatures are helped by small irreducible SM backgrounds [@barbara; @giulia]. These signatures have already been investigated partially in Refs. [@vbftriplet; @Chiang], however neither including a parameter scan involving the Higgs candidate’s signal strengths nor constraints from electroweak precision data (EWPD).
![\[fig:graph\] Sample weak boson fusion diagram involved in the production of $H^{{\pm\pm}}$. We do not show the $H^{{\pm\pm}}$ decay. By crossing one of the up-flavor quarks to the final state and the non-connected down-flavor to the initial we recover the Drell-Yan-type production modes.](diag.pdf){width="23.00000%"}
To our knowledge, neither ATLAS nor CMS have performed a dedicated analysis of this final state in the triplet Higgs model context. However, there are searches for Supersymmetry in same-sign dilepton events with jets and missing energy [@Aad:2011vj; @cmsnewer], where the same-sign leptons arise from the decay chains of the pair-produced gluino or squark particles’ cascade decays [@sstheo]. Such a process is mediated by a non-trivial color exchange in the $s$ or $t$ channels, which results in large scattering angles of the energetic final state jets. This signature, characterized by large $H_T=\sum_{i\in \text{jets}}
p_{T,i}$, is different from the typical WBF phenomenology. On the other hand, since ${\text{Br}}(H^{{\pm\pm}}\to W^\pm W^\pm)$ is large, we might overcome the limitations of searches for light Higgs particles in $H\to VV,~V=Z,W^\pm$, especially because the $H^{{\pm\pm}}W^\mp W^\mp$ coupling can be enhanced in comparison to $HW^+W^-$ due to the model’s triplet character. Furthermore, Ref. [@cmsnewer], which reports a SUSY search employing the 7 TeV 4.98 fb$^{-1}$ data set, comprises signal regions with relatively small $H_T\geq 80$ GeV (compensated with a larger missing energy requirement) which can be exploited to formulate constraints on the triplet model. This will be the focus of Sec. \[sec:cms\]. Subsequently, in Sec. \[sec:vbf7\], we demonstrate that a slight modification of the search strategy of Ref. [@cmsnewer] is sufficient to obtain superior constraints on the triplet model even for a pessimistic estimate of reducible backgrounds and other uncertainties. We also discuss in how far these estimates can be improved by including the 8 TeV data set. In Sec. \[sec:14tev\] we discuss an analysis on the basis of a WBF selection at $\sqrt{s} = 14$ TeV center-of-mass energy, which will yield strong constraints on the triplet models’ parameter space.
As we will argue, the results of these sections are not specific to a particular triplet model and largely generalize to [*any*]{} model with Higgs triplets. Since the tree-level custodial symmetry preserving implementation of Higgs triplets exhibits a richer phenomenology, we specifically analyze the impact of the described searches in the context of the Georgi-Machacek (GM) model [@Georgi:1985nv] (which we quickly review in Sec. \[sec:mod\] to make this work self-contained). In particular, we input the direct search constraints for doubly charged scalars into a global scan of the electroweak properties, also taking into account EWPD. We give our summary in Sec. \[sec:conc\].
A consistent model of Higgs triplets {#sec:mod}
====================================
The Georgi Machacek model [@Georgi:1985nv] is a tree-level custodial isospin-conserving implementation of Higgs triplets based on scalar content $$\label{eq:higgsfields}
\Phi=\left(\begin{matrix} \phi_2^{\ast} & \phi_1 \\ -\phi_1^{\ast} & \phi_2
\end{matrix}\right), \quad
\Xi=\left(\begin{matrix} \chi_3^{\ast} & \xi_1 & \chi_1 \\
-\chi_2^{\ast} & \xi_2 & \chi_2 \\
\chi_1^{\ast} & -\xi_1^{\ast} & \chi_3 \\
\end{matrix}\right)\,.$$ $\Phi$ is a SM-like Higgs doublet necessary for introducing fermion masses, and $\Xi$ combines the complex $(\chi_1,\chi_2,\chi_3)$ and real $(\xi_1,\xi_2,-\xi_1^*)$ triplets such that an additional ${\text{SU}}(2)_R$ can act in the usual fashion ($\Xi\to U_L\Xi U^\dagger_R$ and $\Phi\to \tilde U_L\Phi \tilde U^\dagger_R$) leaving custodial isospin unbroken after $\Phi$ and $\Xi$ obtain vacuum expectation values (vevs) $\left\langle \Xi \right \rangle = v_\Xi \mathbbm{1}$, $\left\langle \Phi \right \rangle = v_\Phi \mathbbm{1}$.
For the purpose of this paper we choose a Higgs sector Lagrangian
\[eq:lag\] $$\begin{gathered}
{\cal{L}}={1\over 2} {\text{Tr}}\left[ D_{2,\mu} \Phi^\dagger D_2^\mu
\Phi \right] + {1\over 2}{\text{Tr}}\left[ {D}_{3,\mu} \Xi^\dagger
{D}_3^\mu \Xi \right] - V(\Phi,\Xi) \\+ {\text{$\Phi$ Yukawa
interactions}} \,,
\end{gathered}$$ where we introduce the potential that triggers electroweak symmetry breaking $$\begin{gathered}
\label{eq:pot}
V(\Phi,\Xi)={\mu_2^2\over 2} {\text{Tr}}\left( \Phi^c \Phi \right) +
{\mu_3^2\over 2} {\text{Tr}}\left( \Xi^c \Xi \right) + \lambda_1
\left[{\text{Tr}}\left( \Phi^c \Phi \right)\right]^2 \\ + \lambda_2 {\text{Tr}}\left( \Phi^c \Phi \right) {\text{Tr}}\left( \Xi^c \Xi \right) +
\lambda_3 {\text{Tr}}\left( \Xi^c
\Xi\, \Xi^c \Xi \right)\\
+ \lambda_4 \left[{\text{Tr}}\left( \Xi^c \Xi \right)\right]^2 -
\lambda_5 {\text{Tr}}\left(\Phi^c t_2^a \Phi t_2^b \right) {\text{Tr}}\left(\Xi^c t_3^a
\Xi t_3^b \right) \,.
\end{gathered}$$
This choice reflects the properties of the Higgs triplet model in a simplified way [@Georgi:1985nv] and can be motivated from imposing a ${\mathbb{Z}}_2$ symmetry [@chano].
$D_2,D_3$ are the gauge-covariant derivatives in the ${\text{SU}}(2)_L$ doublet and triplet representations. Hypercharge ${\text{U}}(1)_Y$ is embedded into ${\text{SU}}(2)_R$ as in the SM, the ${\mathfrak{su}}(2)$ generators in the triplet representation are $$\begin{gathered}
t^1_3={1\over \sqrt{2}} \left( \begin{matrix}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0 \end{matrix}\right)\,, \quad
t^2_3={i\over \sqrt{2}} \left( \begin{matrix}
0 & -1 & 0 \\
1 & 0 & -1 \\
0 & 1 & 0 \end{matrix}\right) \,,\\
t^3_3=\left( \begin{matrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1 \end{matrix}\right)\,.\end{gathered}$$
The masses of the electroweak bosons $m_W,m_Z$ after symmetry breaking follow from the sum of the Higgs fields’ vevs, constraining $$\label{eq:vev}
(246~\hbox{GeV})^2= v_\Phi^2+8v_\Xi^2\,.$$ Defining the mixing angles $$\label{eq:vevrot}
\begin{split}
\cos\theta_H =:&\;\,c_H={v_\Phi\over v_{\text{SM}}}\,, \\
\sin\theta_H =:&\;\,s_H={2\sqrt{2}v_\Xi\over v_{\text{SM}}}
\end{split}$$ turns out to be useful. Since custodial isospin is preserved, in the unitary gauge the Higgs masses group into two singlets, one triplet and one quintet (the quintet includes our doubly charge scalar $H_5^{{\pm\pm}}$, which we will indicate also without the subscript). Their masses are $$\begin{aligned}
\begin{split}
\label{eq:masses}
m_{H_0}^2 &= 2 ( 2 \lambda_1 v_\Phi^2 + 2 ( \lambda_3 + 3 \lambda_4 ) v_\Xi^2 + m_{\Phi\Xi}^2 )\,, \\
m_{H'_0}^2 &= 2 ( 2 \lambda_1 v_\Phi^2 + 2 ( \lambda_3 + 3 \lambda_4 ) v_\Xi^2 - m_{\Phi\Xi}^2 )\,, \\
m_{H_3}^2&= {1\over 2} \lambda_5 (v_\Phi^2+8v_\Xi^2)\,,\\
m_{H_5}^2&= {3\over 2} \lambda_5 v_\Phi^2 + 8 \lambda_3 v_\Xi^2\,,
\end{split}\end{aligned}$$ with short hand notation $$\begin{gathered}
m_{\Phi\Xi}^2 = \Big{[} 4 \lambda_1^2 v_\Phi^4 - 8 \lambda_1
(\lambda_3 +3 \lambda_4) v_\Phi^2 v_\Xi^2 \\
+ v_\Xi^2 \Big{(} 3 (2\lambda_2 - \lambda_5)^2 v_\Phi^2 + 4
(\lambda_3 + 3 \lambda_4)^2 v_\Xi ^2 \Big{)} \Big{]}^{1/2}\,.\end{gathered}$$ To reach Eq. we have diagonalized the singlet mixing by an additional rotation $$\begin{split}
H_0 &= \phantom{-} c_q H_\Phi + s_q H_\Xi\,, \\
H'_0 &= -s_q H_\Phi + c_q H_\Xi \,,
\end{split}$$ with angle $$\begin{gathered}
\label{eq:sq}
\sin \angle(H_\Phi,H_0) =:s_q\\=\frac{\sqrt{3}}
{\sqrt{3+\Big{[}\frac{2\lambda_1 v_\Phi^2 -2(\lambda_3
+3\lambda_4)v_\Xi^2 + m_{\Phi\Xi}^2}{{(2\lambda_2 -
\lambda_5)}v_\Phi v_\Xi}\Big{]}^2}}\,.\end{gathered}$$ Note that $m_{H'_0}< m_{H_0}$, and therefore $m_{H'_0}$ will be the observed Higgs boson.
We straightforwardly compute the couplings of the uncharged states to the SM fermions $f$ and gauge bosons $v$, normalized to the SM expectation, as $$\begin{split}
\label{eq:ctcv}
c_{f,H_0} &= \frac{c_q}{c_H}\,, {\nonumber}\\
c_{v,H_0} &= c_q c_H + \sqrt{8/3} \,s_q s_H\,, {\nonumber}\\
c_{f,H'_0}&= -\frac{s_q}{c_H}\,, {\nonumber}\\
c_{v,H'_0}&= -s_q c_H + \sqrt{8/3} \,c_q s_H\,.
\end{split}$$ The custodial triplet $(H_3^+,H_3^0,H_3^-)$ is gaugephobic and the quintet fermiophobic with the additional assumption of a vanishing leptonic Majorana operator. For the purpose of our analysis this does not pose any phenomenological restriction. Since $\left\langle
\Xi\right\rangle$ is the order parameter that measures the degree of triplet symmetry breaking, a measurement of the $H^{{\pm\pm}}\to W^\pm
W^\pm$ directly reflects the phenomenology’s triplet character. Indeed, the vertex we are predominantly interested in is given by $$\label{eq:dpvertex}
H^{{\pm\pm}}W^\mp_\mu W^\mp_\nu: \quad \sqrt{2}i g m_W s_H g_{\mu\nu}\,,$$ and, as we mentioned in Sec. \[sec:intro\], the relevant final states to study this vertex are therefore “${E_T^{\text{miss}}}+\ell^\pm\ell^\pm$” in association with at least 2 jets. The $2$ jets signature will play the more important role.
Note that Eq. implies that $H^{{\pm\pm}}$ can be enhanced by up to a factor of two compared to the WBF production of a neutral SM-like Higgs boson of the same mass. The enhanced couplings Eqs. and are a direct consequence of the larger isospin of the triplet that feeds into the interactions via the gauge kinetic terms.
At this stage it is important to comment on the relation of the Georgi-Machacek model with “ordinary” triplet Higgs extension, [ *e.g.*]{} when we just add a complex scalar field to the SM Higgs sector with hypercharge $Y=2$ [@classic]. Such models introduce a tree-level custodial isospin violation and consistency with EWPD imposes a hierarchy of the vevs ($s_H\ll 1$). Since we are forced to tune the model already at tree level the additional singly and doubly charged states tend to decouple from the phenomenology apart from loop-induced effects on branching ratios (see [*e.g.*]{} [@andrew] for reconciling the possibly observed excess in $H\to \gamma\gamma$ in this fashion). The Georgi-Machacek model is [*fundamentally*]{} different in this respect: due to the $SU(2)_R$ invariant extension of the Higgs potential there are no tree-level constraints on $v_\Xi$. In fact, only the generation of fermion masses requires the presence of another doublet, and $2\sqrt{2} v_\Xi\gg v_\Phi$ does not lead to tree-level inconsistencies in the gauge sector. At one loop, however, this picture changes. The presence of a triplet requires the explicit breaking of $SU(2)_R$ invariance to tune the $\rho$ parameter to the values consistent with EWPD [@Gunion:1990dt; @us] but still larger values of $v_\Xi$ remain allowed in comparison to the simple complex triplet extension, where recent upper bounds for the triplet vev read as $v_{\text{triplet}} < 0.03 \times (246\hbox{ GeV})$ [@andrew].
An analysis which measures $H^{{\pm\pm}}_5 \to W^\pm W^\pm$ is not specific to the underlying model as Eq. simply follows from the presence of a triplet Higgs in the particle spectrum that contributes to electroweak symmetry breaking. Since the Georgi Machacek model accommodates larger values of $s_H$ with a rich phenomenology we take this particular model as a benchmark for our parameter fit in Sec. \[sec:ewpd\]. Our results generalize to any triplet Higgs model implementation – they provide constraints on this branching ratio, which are model-independent statements as long as the narrow width approximation can be justified.
Re-interpreting SUSY searches {#sec:cms}
=============================
We are now ready to compute an estimate of the performance of the CMS analysis of Ref. [@cmsnewer] when re-interpreted in the Higgs triplet context.
We focus on the light lepton flavor channel of Ref. [@cmsnewer]; the additional $\tau$ lepton channels are subject to large fake background uncertainties and do not provide statistical pull for our scenario in the first place. The CMS analysis of Ref. [@cmsnewer] clusters anti-$k_T$ jets [@antikt] with $R=0.5$ as implemented in [@fastjet] and selects jets with $p_T> 40$ GeV in $|\eta|<2.5$. Leptons are considered as isolated objects if the hadronic energy deposit in within $\Delta R=[(\Delta \phi)^2+(\Delta
\eta)^2]^{1/2}=0.3$ is less than 15% of the lepton candidate’s $p_T$. The thresholds are $p_{T,\mu}> 5~{\text{GeV}}$, and $p_{T,e}> 10~{\text{GeV}}$, and there is a “high $p_T$” selection with $p_{T,\ell}> 10~{\text{GeV}}$ ($\ell=e,\mu$) with the hardest lepton having $p_T> 20~{\text{GeV}}$. All leptons need to fall within $|\eta|<2.4$. CMS requires at least two jets and two leptons and vetos events with three leptons when one of the leptons combines with one of the others to the $Z$ boson mass within $\pm$15 GeV. CMS defines $H_T$ to be the scalar sum of all jets’ $p_T$ whose angular separation to the nearest lepton is $\Delta
R>0.4$.
We have generated CKKW-matched [@ckkw] $t\bar t+W^\pm/Z$, $W^\pm
W^\pm jj$ and $W^\pm Zjj$ which constitute the dominant backgrounds using [@sherpa]. The QCD corrections to theses processes are known to be small [@Campanario:2013qba; @barbara; @giulia; @giuboz]. The signal events are produced with / [@mg5] using a [@Christensen:2008py] interface to our model implementation described in Ref. [@us].[^1] The signal events are subsequently showered and hadronized with [@herwig]. In the analysis we include gaussian detector smearing of the jets and leptons on the basis of Ref. [@atlastdr]: $$\begin{split}
\label{eq:resol}
\text{jets}:\quad & {\Delta E\over E} = {5.2\over E} \oplus
{0.16\over
\sqrt{E}} \oplus 0.033\,,\\
\text{leptons}:\quad & {\Delta {{E}} \over
{E}} = {0.02}\,,\\
\end{split}$$ and we include the missing energy response from recent particle flow fits of CMS [@pflow] via the fitted function [@why] [^2] $$\text{missing energy}:\quad {\Delta {{E}}^{\text{miss}}_T \over
{E}^{\text{miss}}_T} = {2.92 \over E^{\text{miss}}_T} - 0.07\,.$$ The jet resolution parameters can be improved by particle flow too, we however choose the more conservative parametrization to capture the effect of an increased jet energy scale uncertainty in the forward detector region, which especially impacts the WBF-like selection.
We use the background samples to generate an efficiency profile over the 8 CMS search regions ([*[cf.]{}*]{} Fig. \[fig:cmschannels\]) we are focusing on $$\label{eq:search}
\begin{split}
\hbox{region 1:}\quad & {\text{high $p_T$}}, H_T>80~{\text{GeV}},~E_T^{\text{miss}}>120~{\text{GeV}},\\
\hbox{region 2:} \quad & {\text{low $p_T$}}, H_T>200~{\text{GeV}},~E_T^{\text{miss}}>120~{\text{GeV}},\\
\hbox{region 3:} \quad& {\text{high $p_T$}}, H_T>200~{\text{GeV}},~E_T^{\text{miss}}>120~{\text{GeV}},\\
\hbox{region 4:} \quad& {\text{low $p_T$}}, H_T>450~{\text{GeV}},~E_T^{\text{miss}}>50~{\text{GeV}},\\
\hbox{region 5:}\quad & {\text{high $p_T$}}, H_T>450~{\text{GeV}},~E_T^{\text{miss}}>50~{\text{GeV}},\\
\hbox{region 6:} \quad& {\text{low $p_T$}}, H_T>450~{\text{GeV}},~E_T^{\text{miss}}>120~{\text{GeV}},\\
\hbox{region 7:} \quad& {\text{high $p_T$}}, H_T>450~{\text{GeV}},~E_T^{\text{miss}}>120~{\text{GeV}},\\
\hbox{region 8:}\quad & {\text{high $p_T$}}, H_T>450~{\text{GeV}},~E_T^{\text{miss}}>0~{\text{GeV}}\,,
\end{split}$$ which we apply to our signal hypothesis. [^3] To obtain CLS exclusion limits [@Read:2002hq] we perform a log likelihood hypothesis test as described in [@junk], where we marginalize over the background uncertainty quoted in [@cmsnewer] (and indicated in Fig. \[fig:cmschannels\]).
The result is shown in Fig. \[fig:cmscls\], where we plot the observed and expected 95% confidence level constraints on the signal strength $$\xi=\frac{\sigma(H^{{\pm\pm}}jj)\times {\text{BR}}(H^{{\pm\pm}}\to W^\pm W^\pm
\to {\text{leptons}})}{[\sigma(H^{{\pm\pm}}jj)\times {\text{BR}}(H^{{\pm\pm}}\to W^\pm W^\pm
\to {\text{leptons}})]_{\text{ref}}}$$ as function of the mass of $H^{{\pm\pm}}$. Since the total width is dominated by $H_5^{{\pm\pm}}\to W^\pm W^\pm$, we have $\xi\simeq
s_H^2$. $\xi$ sets a limit in reference to a point that we choose with values $$s_H=1/\sqrt{2}\,,~m_3=500~{\text{GeV}}$$ for the Higgs mixing and triplet mass, [*i.e.*]{} a $hW^+W^-$-like value of the $H^{{\pm\pm}}W^\pm W^\pm$ coupling. These are also values allowed by constraints from non-oblique corrections, in particular due to $Z\to b\bar{b}$ measurements [@Haber:1999zh]. All other parameters are chosen such that the 125 GeV Higgs state has a coupling to weak gauge bosons that agrees with the SM within 5%. Given the large triplet Higgs vev, this is an optimistic scenario, but we stress that it only serves to establish a baseline for the measurement of $\xi$.
We see that the CMS analysis, which cuts on $H_T$, [*i.e.*]{} central jet activity instead of WBF-type topologies, only starts to probe the model for $H^{{\pm\pm}}_5$ masses close to the $W^\pm$ threshold. The discriminative power always predominantly comes from the search region 1, which is closest to a typical WBF selection among the eight search channels of Eq. . As we will see in Sec. \[sec:ewpd\], once other constraints such as electroweak precision measurements and direct Higgs search constraints are included, the SUSY search does not provide a strong constraint on the parameter space of the Georgi-Machacek model.
As can be guessed from Fig. \[fig:cmschannels\], excluding the triplet via the CMS SUSY search is hampered by the large systematic uncertainties. If we omit the systematic uncertainties and compute the excluded signal strength only on the basis of statistical uncertainties, the CMS analysis excludes $\xi=0.68$ for $m_{H_5^{{\pm\pm}}}=200~{\text{GeV}}$. This enables a qualitative projection of the situation when the 8 TeV sample is included. Due to the larger data sample we can expect that the background uncertainty is reduced by a larger available set of subsidiary background measurements at higher statistics. CMS has an 8 TeV data sample of ${\cal{L}}\simeq
23~{\text{fb}}^{-1}$. With this sample and a systematic uncertainty reduced by 50%, CMS starts probing the triplet parameter space for $H^{{\pm\pm}}_5$ masses up to $m_H^{{{\pm\pm}}}\simeq 250$ GeV.
![\[fig:cmsclsvbf\] Expected exclusion limits for a more WBF-like analysis based on Ref. [@cmsnewer]. For details see text.](plot_cls_vbf.pdf){width="43.00000%"}
Towards a more WBF-like selection {#sec:vbf7}
---------------------------------
We modify the above analysis towards a more signal-like selection. The base cuts are identical, but this time we extend the jet clustering over the full HCAL range $|\eta|<4.5$ and add standard WBF cuts via $$m_{j_1j_2}>500~{\text{GeV}}~\hbox{and}~|y_{j_1}-y_{j_2}|>4\,.$$ This means that instead of exclusively clustering central jets, we also allow more forward jets, so the systematic uncertainties might by different compared to the CMS analysis we discussed above in Sec. \[sec:cms\]. The fake background contribution, in particular, can quantitatively only be assessed by the experiments themselves. To get a qualitative estimate, we simulate $W+$heavy flavor events[^4] that we match onto the CMS analysis region 1 and use a flat extrapolation to the signal region described above. This yields approximately an estimate of the background composition of again $\sim 60:40$ of fake:irreducible. To calculate confidence levels we assume a systematic uncertainty on the background of 75% (which is a rather conservative estimate in the light of the CMS search of the previous section). As expected, the WBF selection reduces the background without degrading the signal too much, therefore enhancing the signal vs. background ratio. The expected exclusion limit on the basis of these parameters is shown in Fig. \[fig:cmsclsvbf\]. We see that already with the $4.98~{\text{fb}}^{-1}$ data set we can expect limits on the model up to masses $m_{H^{{\pm\pm}}}\simeq 300~{\text{GeV}}$. If the background uncertainty is reduced by 50% the full 8 TeV data set probes triplet models up to masses $m_{H^{{\pm\pm}}}\simeq 420~{\text{GeV}}$ for our reference value $s_H=1/\sqrt{2}$.
Prospective sensitivity and discovery thresholds at 14 TeV {#sec:14tev}
==========================================================
Switching to higher center-of-mass energy changes the sensitivity to the model dramatically. WBF-like cross sections increase by a factor $\sim 5$ when doubling the available center-of-mass energy from 7 TeV to 14 TeV [@vbfnlo]. We can therefore introduce additional WBF criteria like a central jet veto to further suppress the QCD backgrounds, as well as lepton vetos to remove the $WZjj$ backgrounds.
Our event generation for the 14 TeV analysis follows the 7 TeV tool chain. We use the anti-$k_T$ jets with $R=0.5$, and lower the $p_T$ thresholds to 20 GeV in $|\eta_j|<4.5$. We enlarge the requirement on the tagging jets invariant mass to $m_{jj}>600~{\text{GeV}}$ and furthermore require that the jets fall in opposite detector hemispheres $y_{j_1}\cdot y_{j_2}\leq 0$. The leptons are required to be isolated from the jets by a distance $\Delta R_{\ell j}=0.4$. This time we veto events with a third lepton and a central jet which meets the above requirement. No restrictions on $E_T^{\text{miss}}$ are imposed. The result is a signal-dominated selection, which not only allows us to highly constrain $s_H$ over a wide range of $H_5^{{\pm\pm}}$ masses but also enables the approximate reconstruction of the $H_5^{{{\pm\pm}}}$ mass from a Jacobian peak in the transverse cluster mass distribution,
$$m_{T,c}^2=\left( \sqrt{(p_{\ell_1}+p_{\ell_2})^2+|\vec{p}_{T,\ell_1}+\vec{p}_{T,\ell_2}|^2} +
E_T^{\text{miss}}\right)^2 -
\Big{|}\vec{p}_{T,\ell_1}+\vec{p}_{T,\ell_2}+\vec{E}_{T}^{\text{miss}}\Big{|}^2$$
in case such a model is realized in nature. Due to detector resolution effects, missing energy uncertainty and IS radiation, the mass resolution of the Jacobian peak degrades significantly when considering heavier $H^{{\pm\pm}}$ masses, as shown in Fig. \[fig:invmassdis\]. A statistically significant measurement will still be possible, the mass parameter determination, however, will be poor.
The above event selection serves two purposes. Firstly, all QCD-induced backgrounds (which are characterized by central jet activity at moderate $m_{jj}$) are highly suppressed. We suppress the backgrounds further by imposing lepton and central jet vetos. Note that this also remove signal contributions which arise from other processes other than WBF. As a result we directly constrain $s_H$. After all cuts have been applied the irreducible background is completely dominated by the electroweak SM $pp\to (W^\pm W^\pm\to
\ell^\pm\ell^\pm+E_T^{\text{miss}}) jj$ contribution at ${\cal{O}}({\alpha}^6\alpha_s^0)$. This background is comparably small and under good perturbative control [@barbara].
The fake background contribution can quantitatively only be assessed by the experiments themselves. To get a qualitative estimate, we again simulate $W+$heavy flavor events that we match onto the CMS analysis region 1 and use a flat extrapolation to 14 TeV WBF selection described above as already done for the 7 TeV WBF selection criteria. This yields approximately an estimate of background composition of 50:50 of fake:irreducible. We furthermore assume a systematic shape uncertainty of the background of 35% (flat), which follows from 10% and 25% uncertainties on the irreducible and fake background, respectively.
In Fig. \[fig:pval\] we show the associated $p$ values for a search based on the observable $m_{T,c}$ of Fig. \[fig:invmassdis\] for the reference point $s_H=1/\sqrt{2}$. The signal cross section scales with $ s_H^2 \sim s_H^{2\;\text{ref}}
\sqrt{{\cal{L}}/{\cal{L}}^\text{ref}}$. In principle this implies that the LHC provides us enough sensitivity for discoveries down to $s_H^2\sim 0.1$ for heavy masses for the considered $H_5^{{\pm\pm}}$ mass range.
![\[fig:pval\]Associated $p$ values for a search based on the single discriminant $m_{T,c}$ as a function of $m_{H^{{\pm\pm}}}$ and the integrated luminosity. The curves, moving from left to right, correspond to $H_5^{{\pm\pm}}$ masses between 250 GeV and 850 GeV in steps of 50 GeV. ](plot_p_val.pdf){width="43.00000%"}
In Fig. \[fig:vbfcls14\] we show the expected 95% confidence level constraints as a function of the $H^{{\pm\pm}}_5$ mass for luminosities 5 fb$^{-1}$ and 600 fb$^{-1}$. The expected constraint on the signal strength $\xi$ can directly be interpreted as a limit on the $H_5^{{\pm\pm}}W^\mp W^\mp$ coupling Eq. . As can be seen from this figure, an analysis based on the WBF channel is a very sensitive search, eventually yielding constraints $s_H^2\lesssim ~ 0.05$ over the entire parameter range. On the one hand, since $s_H\ll 1$ is required by the $W/Z$ mass ratio in a complex triplet extension the expected constraint is not good enough to constrain the entire parameter space. On the other hand, it is possible to constrain the bulk of the parameter space in the context of the Georgi-Machacek model, which typically allows larger values for $s_H$ [@us].
![\[fig:vbfcls14\] Associated signal strength limits at 95% confidence level computed from a binned log likelihood hypothesis test on the basis of the single discriminant $m_{T,c}$, Fig. \[fig:mt2c\], using the CL$_S$ method [@Read:2002hq]. We show results for two luminosity values for running at 14 TeV center-of-mass energy, 5 fb$^{-1}$ and 600 fb$^{-1}$.](plot_cls_vbf_14.pdf){width="43.00000%"}
Combining direct $H_5^{{\pm\pm}}$ searches with other constraints {#sec:ewpd}
=================================================================
In this section we want to compare the exclusion potentials due to searches for a doubly charged scalar obtained in the previous sections with representative points for the parameter space still allowed for the GM model. In particular, in a previous study [@us], we have shown that this space is large enough to accommodate both the case where the 125 GeV Higgs boson has an enhanced $\gamma\gamma$ decay rate with respect to the SM value and the case where the couplings for the Higgs boson candidate are SM-like.
It is therefore natural to study if the (future, possible) non-observation of excesses in searches for doubly charged states has the potential to completely rule out these two scenarios, and hence the GM extension of the Higgs sector.
Before showing the results, we summarize the information included in the two sets of points we will use in the following. We list here only the aspects which are relevant for the present work, and we refer the reader to Ref. [@us] for a detailed explanation of how these results were obtained:
The points that we consider correspond to scenarios where the $H'_0$ scalar is the observed Higgs boson. Therefore we restrict to the case where the other singlet $H_0$ is heavier, and we require that neither $H_0$ nor $H_3^0$ violate the LHC exclusion limits on scalar production. This case has been discussed in Ref. [@us] in detail.
We require that the tree-level couplings of $H'_0$ with fermions and gauge bosons, and the loop-induced coupling with gluons, are such that $H'_0$ reproduce the observed total signal strength as well as the individual signal strengths for $WW$ ($\xi_{H\to WW}$) and $\gamma\gamma$ ($\xi_{H\to\gamma\gamma}$) decays. In particular, at this level we distinguish among a scenario where we have room to reproduce an excess in the photonic branching ratio and another where signal strengths agree with the SM values within 20%. For further details on the scan we refer the reader to Ref. [@us].
In our previous study we have also taken into account constraints from electroweak precision measurements. In particular we studied both cases where the $T$ parameter is used or not, since at one-loop the radiative corrections are not unambiguously defined. In this work we have decided not to consider this subtle but important issue, which we instead discussed at length in Ref. [@us]: therefore we used the sets of points labelled in our previous paper as “S. param included”, [*i.e.*]{} the results obtained here are independent of any $T$ parameter constraint or fine tuning [@Gunion:1990dt].
In our previous work we have not explicitly included constraints due to the fermionic coupling of the custodial-triplet charged states $H_3^\pm$. The presence of these states might change significantly several observables involving $b$-quarks, because of possibly large values for the $H_3^+ t b$ coupling. One of the more important observables to look at is $R_b$, defined as $\Gamma(Z\to
b\bar{b})/\Gamma(Z\to\mbox{hadrons})$. Changes in the SM value prediction of $R_b$ induced by the GM model have been computed in Ref. [@Haber:1999zh]. We have reproduced these results, and checked that a large portion of the points we will use in the following, that were considered still allowed in our previous paper, survive also the bounds from $Z\to b\bar{b}$. [^5]
As the above discussion shows, in our previous study we have taken into account essentially all the available constraints from direct and indirect searches. In particular, for this paper we also checked that the conclusions we reached in Ref. [@us] remain essentially unchanged also when non-oblique corrections (in the $Z\to b\bar{b}$ case) are included.
In Figs. \[fig:modela\] and \[fig:modelb\] we show the exclusion potential of the search strategies discussed in Sec. \[sec:cms\] and \[sec:14tev\], together with the surviving points for the two scenarios we just described. The standard color coding is used for the exclusion plots, which here are shown as a function of $s_H$ and $m_{H^{{\pm\pm}}}$. From these plots we conclude that the searches at 7 TeV, if extended with WBF-like selection cuts, start to be able to probe, and hence exclude, some of the surviving scenarios. The more relevant result, however, is that WBF searches on the 14 TeV data will have the potential to completely rule out all the points that survive all other constraints. This search has therefore the potential to become a decisive obstacle that models with Higgs triplets and large triplet-doublet mixing have to pass in order not to be excluded. As such, it would be very important for LHC experimental collaborations to look into these final states. In particular an analysis based on the same-sign lepton WBF channel serves to also constrain the parameter region which is allowed in other recent analyses such as Ref. [@Belanger:2013xza].
Summary {#sec:conc}
=======
Higgs triplets as implemented in the Georgi-Machacek Model provide a viable extension of the SM Higgs sector which can be efficiently probed at the LHC. We have demonstrated that while current analyses of same-sign lepton final states do not provide a strong enough constraint on the presence of doubly charged scalar bosons decaying to same-sign $W$’s on the basis of SUSY searches, the enlarged statistical sample of the 8 TeV 2012 run should start constraining this model via the non-adapted SUSY search strategy. Furthermore we have shown that a simple modification of these SUSY searches allows us to constrain the model already with 7 TeV data even for a conservative background estimate. The model can be ultimately verified or ruled out at the LHC with 14 TeV in a clean WBF selection.
Our results are quite general and at the same time realistic as far as models with triplets are considered. In particular, studying $H^{{\pm\pm}}\to W^\pm W^\pm$ rather than the more commonly considered case $H^{{\pm\pm}}\to\ell^\pm\ell^\pm$ seems to be more natural, because of the dominance of the former decay over the latter for the bulk of the parameter space independent of the considered triplet scenario. Moreover, the analysis strategies we studied in this work are quite standard, but at the same time can lead to conclusive results for a complete exclusion of Higgs sectors with triplets. We therefore think that it would be very important for the LHC experimental collaborations to consider these searches in addition to the already considered and simpler case $H^{{\pm\pm}}\to\ell^\pm\ell^\pm$.
CE acknowledges funding by the Durham International Junior Research Fellowship scheme. We also acknowledge Chris Hays for interesting discussions, and for comments on the draft.
[99]{}
The ATLAS collaboration, Phys. Lett. B [**716**]{} (2012) 1. The CMS collaboration, Phys. Lett. B [**716**]{} (2012) 30. The ATLAS collaboration, ATLAS-CONF-2012-170.
The CMS collaboration, CMS-PAS-HIG-12-045.
F. Englert and R. Brout, Phys. Rev. Lett. [**13**]{} (1964) 321. P. W. Higgs, Phys. Lett. [**12**]{} (1964) 132 and Phys. Rev. Lett. [**13**]{} (1964) 508. G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, Phys. Rev. Lett. [**13**]{} (1964) 585. A. Arhrib, R. Benbrik, M. Chabab, G. Moultaka and L. Rahili, JHEP [**1204**]{}, 136 (2012). S. Chang, C. A. Newby, N. Raj and C. Wanotayaroj, Phys. Rev. D [**86**]{} (2012) 095015. L. Wang and X. -F. Han, arXiv:1303.4490 \[hep-ph\]. A. G. Akeroyd and S. Moretti, Phys. Rev. D [**86**]{} (2012) 035015. L. Wang and X. -F. Han, Phys. Rev. D [**87**]{} (2013) 015015. C. Englert, E. Re and M. Spannowsky, Phys. Rev. D [**87**]{}, 095014 (2013). R. Killick, K. Kumar and H. E. Logan, arXiv:1305.7236 \[hep-ph\]. The CMS collaboration, CMS-PAS-HIG-13-001.
H. Georgi and M. Machacek, Nucl. Phys. B [**262**]{} (1985) 463. M. S. Chanowitz and M. Golden, Phys. Lett. B [**165**]{} (1985) 105. D. Carmi, A. Falkowski, E. Kuflik, T. Volansky and J. Zupan, JHEP [**1210**]{}, 196 (2012). G. Belanger, B. Dumont, U. Ellwanger, J. F. Gunion and S. Kraml, arXiv:1306.2941 \[hep-ph\]. W. Konetschny and W. Kummer, Phys. Lett. B [**70**]{} (1977) 433. J. Schechter and J. W. F. Valle, Phys. Rev. D [**22**]{} (1980) 2227. T. P. Cheng and L. -F. Li, Phys. Rev. D [**22**]{} (1980) 2860. A. Hektor, M. Kadastik, M. Muntel, M. Raidal and L. Rebane, Nucl. Phys. B [**787**]{} (2007) 198. The CMS collaboration, arXiv:1207.2666 \[hep-ex\]. The ATLAS collaboration, arXiv:1210.5070 \[hep-ex\]. J. F. Gunion, R. Vega and J. Wudka, Phys. Rev. D [**42**]{}, 1673 (1990). S. Kanemura, K. Yagyu and H. Yokoya, arXiv:1305.2383 \[hep-ph\]. D. L. Rainwater and D. Zeppenfeld, Phys. Rev. D [**60**]{} (1999) 113004 \[Erratum-ibid. D [**61**]{} (2000) 099901\], N. Kauer, T. Plehn, D. L. Rainwater and D. Zeppenfeld, Phys. Lett. B [**503**]{} (2001) 113, T. Figy, C. Oleari and D. Zeppenfeld, Phys. Rev. D [**68**]{} (2003) 073005, C. Oleari and D. Zeppenfeld, Phys. Rev. D [**69**]{} (2004) 093004, B. Jager, C. Oleari and D. Zeppenfeld, Phys. Rev. D [**73**]{} (2006) 113006, B. Jager, C. Oleari and D. Zeppenfeld, JHEP [**0607**]{} (2006) 015, M. Ciccolini, A. Denner and S. Dittmaier, Phys. Rev. Lett. [**99**]{} (2007) 161803, M. Ciccolini, A. Denner and S. Dittmaier, Phys. Rev. D [**77**]{} (2008) 013002, B. Jager, C. Oleari and D. Zeppenfeld, Phys. Rev. D [**80**]{} (2009) 034022. B. Jager and G. Zanderighi, JHEP [**1111**]{} (2011) 055. T. Melia, K. Melnikov, R. Rontsch and G. Zanderighi, JHEP [**1012**]{} (2010) 053. C. -W. Chiang, T. Nomura and K. Tsumura, Phys. Rev. D [**85**]{} (2012) 095023. C. -W. Chiang and K. Yagyu, JHEP [**1301**]{} (2013) 026. S. Godfrey and K. Moats, Phys. Rev. D [**81**]{}, 075026 (2010). K. Cheung and D. K. Ghosh, JHEP [**0211**]{} (2002) 048.
The CMS collaboration, Phys. Rev. Lett. [**109**]{}, 071803 (2012) and CMS-PAS-SUS-12-017. The ATLAS collaboration, JHEP [**1110**]{} (2011) 107. S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Phys. Rev. Lett. [**109**]{} (2012) 071803 R. M. Barnett, J. F. Gunion and H. E. Haber, Phys. Lett. B [**315**]{} (1993) 349. J. F. Gunion, R. Vega and J. Wudka, Phys. Rev. D [**43**]{}, 2322 (1991). M. Cacciari, G. P. Salam and G. Soyez, JHEP [**0804**]{} (2008) 063. M. Cacciari, G. P. Salam and G. Soyez, Eur. Phys. J. C [**72**]{} (2012) 1896. S. Catani, F. Krauss, R. Kuhn and B. R. Webber, JHEP [**0111**]{} (2001) 063. T. Gleisberg, S. Hoeche, F. Krauss, M. Schonherr, S. Schumann, F. Siegert and J. Winter, JHEP [**0902**]{}, 007 (2009). G. Bozzi, B. Jager, C. Oleari and D. Zeppenfeld, Phys. Rev. D [**75**]{} (2007) 073004. F. Campanario, M. Kerner, L. D. Ninh and D. Zeppenfeld, arXiv:1305.1623 \[hep-ph\]. J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, JHEP [**1106**]{} (2011) 128. N. D. Christensen and C. Duhr, Comput. Phys. Commun. [**180**]{} (2009) 1614. M. Bahr, S. Gieseke, M. A. Gigg, D. Grellscheid, K. Hamilton, O. Latunde-Dada, S. Platzer and P. Richardson [*et al.*]{}, Eur. Phys. J. C [**58**]{} (2008) 639. G. Aad [*et al.*]{} \[ATLAS Collaboration\], JINST [**3**]{} (2008) S08003; G. L. Bayatian [*et al.*]{} \[CMS Collaboration\], J. Phys. G [**34**]{} (2007) 995. The CMS collaboration, CMS PAS PFT-09/001.
C. Englert, M. Spannowsky and C. Wymant, Phys. Lett. B [**718**]{} (2012) 538. A. L. Read, CERN-OPEN-2000-205. A. L. Read, J. Phys. G [**G28** ]{} (2002) 2693-2704. G. Cowan, K. Cranmer, E. Gross and O. Vitells, Eur. Phys. J. C [**71**]{}, 1554 (2011). T. Junk, Nucl. Instrum. Meth. A [**434**]{} (1999) 435. T. Junk, CDF Note 8128 \[cdf/doc/statistics/public/8128\]. T. Junk, CDF Note 7904 \[cdf/doc/statistics/public/7904\]. H. Hu and J. Nielsen, in 1st Workshop on Confidence Limits’, CERN 2000-005 (2000). H. E. Haber and H. E. Logan, Phys. Rev. D [**62**]{} (2000) 015011. K. Arnold, M. Bahr, G. Bozzi, F. Campanario, C. Englert, T. Figy, N. Greiner and C. Hackstein [*et al.*]{}, Comput. Phys. Commun. [**180**]{} (2009) 1661. A. Freitas and Y. -C. Huang, JHEP [**1208**]{}, 050 (2012) \[Erratum-ibid. [**1305**]{}, 074 (2013)\]
[^1]: We note that the $2j+{E_T^{\text{miss}}}+\ell^+\ell^+$ signal also receives contributions from the vertices $H_0 W^\pm W^\pm$, $H'_0 W^\pm W^\pm
$ and $H^0_5 W^\pm W^\pm$ which are present in diagrams containing an internal $t$-channel neutral Higgs boson connecting the $W^+$’s emitted from the two quark lines. These diagrams are important to ensure unitarity in longitudinal weak boson scattering for high energy ($H_5^{{\pm\pm}}$ off-shell) scattering. We have checked that their numerical contribution is negligible in the $H_5^{{\pm\pm}}$ resonant region captured by Fig. \[fig:graph\], and therefore we have not included them explicitly in this work.
[^2]: The missing energy response might vary from Ref. [@pflow] when the analysis is performed by the experiments. This is clearly beyond the scope of this work, but we believe that our parametrization is well-justified for demonstration purposes.
[^3]: Since the CMS analysis does not tag on the number of jets, we have also considered production modes with same-sign dilepton and $E_T^{\text{miss}}$, but where more than 2 jets are produced. In a model with an extended Higgs sector, the cross section to produce such final states could potentially be very different from the SM rate. We have explicitly checked that production rates for $pp \to E_T^{\text{miss}}+
\ell^\pm \ell^\pm + (> 2j)$ when extra states are included are negligible with respect to the main contribution to the signal, *i.e.* $ pp\to W^\pm W^\pm jj$, with an $s$-channel exchanged $H_5^{{\pm\pm}}$, is the dominant process. To establish this, we have computed the impact of $pp\to Z\to H^{{\pm\pm}}H^{{\mp\mp}}\to W^\pm W^\pm
jjjj$, $pp\ ( \to W^\pm ) \to H^{{\pm\pm}}H_{3,5}^\mp \to W^\pm W^\pm
jjjj$, $pp\to H^{{\pm\pm}}\to W^\pm H_{3,5}^\pm \to W^\pm W^\pm jj$ and $gg\to H_0\to H^{{\pm\pm}}H^{{\mp\mp}}\to W^\pm W^\pm jjjj$ ($gg\to H_3^0$ would also be possible, but $H_3^0\to H^{{\pm\pm}}H^{{\mp\mp}}$ is forbidden) to the signal estimate, and found negligible contributions. More precisely, the only process that could have a marginal impact is $gg\to H_0\to H^{{\pm\pm}}H^{{\mp\mp}}\to W^\pm W^\pm jjjj$, when $m_{H_0}>2
m_{H_5^{{\pm\pm}}}$. While $H_0$ can be heavier than the quintet, the situation where it is heavy enough to have an open 2-body decay channel into a quintet pair is not very frequent. For example we have checked that this is the case by inspecting the points we considered in our previous study [@us]. In the present work, only for the template scenarios with light quintets ($m_{H_5}<250$ GeV) we have found that this is possible, and in such cases we have checked that the total contribution from this subprocess can enhance the signal by a factor $1.5$. This is not enough to change our estimates significantly. We are therefore confident that the approximations we are using for the simulation of signal and backgrounds are robust. We however note that the contributions discussed in this footnote are model-dependent because they explicitly probe the larger particle content and the Higgs interactions due to the potential.
[^4]: Following Ref. [@cmsnewer] this is expected to be the dominant contribution of the fake background.
[^5]: For the sake of completeness, we would like to point out that recent results in the computation of 2-loop corrections for the SM $Zb\bar{b}$ coupling lead to sizeable effects which have not been taken into account in previous literature [@Freitas:2012sy]. Including these effects goes however beyond the purpose of this study, although it could be potentially relevant for constraints only due to non-oblique corrections. We will however show that searches for WBF-produced doubly charged states are very powerful as exclusion tests for these models, and therefore our main results will hold, regardless of the relative size of these loop effects.
|
---
author:
- 'D. Tripathi, H. E. Mason, G. Del Zanna, P. R. Young'
bibliography:
- 'new\_ref.bib'
date: 'Received(16 December 2009); Accepted(10 May 2010)'
subtitle: Basic physical parameters and their temporal variation
title: Active Region Moss
---
Introduction\[intro\]
=====================
The high resolution images obtained by the Transition Region and Coronal Explorer [TRACE; @trace] revealed a new type of emission called “moss”. Moss regions are bright, finely textured, mottled, low lying emission above the active region plage area. Moss regions are seen best in the TRACE images obtained at /[x]{} $\lambda$171 [@schrijver; @berger_moss]. It has been shown that moss regions are always observed in plage regions in the vicinity of hot loops. These features are possibly the same phenomena observed by [@prg] using the Normal Incidence X-ray Telescope, where they found that many active regions were associated with low-lying areas of intense emission resembling plage regions seen in H$\alpha$ observations. Using observations from TRACE and the Soft X-ray Telescope [SXT; @sxt] it was suggested that the moss regions correspond to the footpoint locations of hot loops which are observed using X-ray images at 3-5 MK [@berger_moss; @martens; @antiochos].
Active regions on the Sun primarily comprise two types of loops; the loops seen in the hot and dense core of active regions in X-ray observations at 2-3 MK (and higher) and the larger loops seen on the periphery of active regions at 1MK . The loops seen on the periphery of active regions are termed “warm loops”. With high spatial resolution instruments such as TRACE, and the Extreme-ultraviolet Imaging Spectrometer [EIS; @eis] onboard Hinode [@hinode], the warm loops seem to be spatially well resolved. Using TRACE and EIS observations the plasma parameters (such as electron density, temperature and flows) in warm loops can be measured .
In contrast, the hot loops in the core of active regions appear quite small, diffuse and difficult to resolve with present day instrumentation. It has also been known for some time that the corona appears ’fuzzier’ at higher temperatures. [@tripathi_2009] showed that this was not simply an instrumental feature. This effect makes it very difficult to resolve a single isolated loop structure in the core of an active region. As a consequence, it is difficult to study the heating mechanism for individual hot loops in the core of active regions. A different approach is therefore required. Since it has been proposed that moss regions are the footpoints of hot loops, a detailed investigation of physical plasma parameters in moss regions and their variation with time should give some indication of the nature of the heating mechanism(s).
------------- --------------
Date Raster Start
Times (UT)
01-May-2007 11:53:13
02-May-2007 05:06:11
18:31:20
03-May-2007 14:01:52
14:21:12
14:40:31
14:59:51
04-May-2007 06:37:17
06:56:36
05-May-2007 05:24:09
07:25:56
07:45:16
08:04:35
------------- --------------
: Dates and raster start times of EIS data used in this study. \[data\]
In an earlier study [@tripathi_moss], using a single dataset from EIS, we measured the electron densities and magnetic field structures in moss regions. We found that the densities in moss regions were higher than the surrounding regions in the active region and varied within the range 10$^{10}$ - 10$^{10.5}$ cm$^{-3}$ from one moss region to the other. In addition, we found that the moss regions were primarily located in only one magnetic polarity region. In this paper, which is an extension of [@tripathi_moss], we use observations recorded by EIS to study physical plasma parameters (such as electron densities, temperatures, filling factors, and column depth) in different moss regions within the same active region. In particular, we study the variation of these parameters over short (one hour) and long (5 days) time periods. To the best of our knowledge this is the first time that a spectroscopic study has been carried out to study the variation of physical parameters in moss regions over a short and a long period of time.
The rest of the paper is organized as follows. In section \[obs\] we describe the observations used in this study. In section \[techs\] we briefly discuss the spectroscopic techniques used in this paper. We also revisit the question of moss regions being the footpoints of hot loops in section \[moss\_hot\_loops\] using data from TRACE, EIS and the X-Ray Telescope [XRT; @xrt]. We discuss the thermal structure of moss regions in section \[temp\] followed by a discussion of density, filling factors and column depth in section \[dens\]. We draw some conclusions in section \[con\].
--------- ------------ ------------------ ------------
Line ID Wavelength log(N$_e$) Range log(T$_e$)
([Å]{}) (cm$^{-3}$) (K)
186.60 $-$ 5.6
278.39 $-$ 5.8
280.75 8.0$-$11.0 5.8
275.35 $-$ 5.8
188.50 $-$ 6.0
188.23 $-$ 6.1
258.37 8.0$-$9.7 6.1
261.04 $-$ 6.1
186.88 7.0$-$12.0 6.1
195.12 $-$ 6.1
196.54 9.3$-$11.0 6.2
202.02 $-$ 6.2
203.83 8.0$-$10.5 6.2
264.78 8.0$-$11.0 6.3
274.20 $-$ 6.3
284.16 $-$ 6.4
--------- ------------ ------------------ ------------
: Spectral lines (first column) from the study sequence ’CAM\_ARTB\_CDS\_A’ chosen to derive the physical parameters in moss regions. \[lines\]
Observations\[obs\]
===================
For this study, we have primarily used observations recorded by EIS aboard Hinode. EIS has an off-axis paraboloid design with a focal length of 1.9 meter and mirror diameter of 15 cm. It consists of a multi-toroidal grating which disperses the spectrum onto two different detectors covering 40 [Å]{} each. The first detector covers the wavelength range 170-210 [Å]{} and the second covers 250-290 [Å]{} providing observations in a broad range of temperatures (log T $\approx$ 4.7-7.3). EIS has four slit/slot options available (1, 2, 40and 266). High spectral resolution images can be obtained by rastering with a slit.
![Top panel: A TRACE image (plotted in a negative intensity scale) taken at 171 [Å]{} showing the active region studied in this paper. The over-plotted rectangle shows the region which was rastered on May 01 using the 2$\arcsec$ slit of EIS. We note that a raster of this active region was obtained on 5 consecutive days with roughly the same coordinates of the boxed region. Bottom panel: an EIS image in $\lambda$195. The overplotted contours are from the TRACE intensity image. The vertical structure outlined with the contour in the middle of the image (also marked with arrows) is the moss region being discussed in the paper. \[context\]](13883fg1.eps){width="35.00000%"}
An active region *AR 10953*, which appeared on the east solar limb on April 27, 2007, was observed by Hinode/EIS as it crossed the visible solar disk. From May 1 till May 5th it was observed using the study sequence *CAM\_ARTB\_CDS\_A* designed by the authors. This study sequence takes about 20 minutes to raster a field of view of the Sun of 200 $\times$ 200 with an exposure time of 10 seconds using the 2slit. It has 22 windows and is rich in spectral lines, which allows us to derive the physical plasma parameters simultaneously at different temperatures. The top panel of Fig. \[context\] displays the active region imaged by TRACE in its 171 [Å]{} channel. The over-plotted box shows the portion of the active region which was scanned by EIS with its 2 slit. The lower panel shows an EIS image in $\lambda$195.12 line. The vertical structure in the middle of the image, outlined by the contour and also marked with arrows, locates the moss regions discussed throughout the paper.
The datasets comprise a couple of rasters each day, but these were not necessarily taken sequentially. On May 03, the study sequence was run four consecutive times with a cadence of 20 min each. This provides an excellent opportunity to study the physical characteristics of the moss over an hour. We have used these four datasets to study the variation of plasma parameters such as electron temperature, density and filling factor. In addition we have taken one raster each day from May 01 to May 05 to study the variation in moss over a period of 5 days. In total we have analyzed 13 EIS datasets. Table \[data\] contains dates and the start times of EIS rasters used in this study.
Table \[lines\] provides the list of spectral lines (formed at log T = 5.6 - 6.5) used in this study. Four lines are affected by blending, but for three of the lines the blending component can be accurately estimated. $\lambda$278.39 is blended with $\lambda$278.44 which has a fixed ratio relative to the unblended $\lambda$275.35 line and so can be easily evaluated [see e.g., @peter_eis]. $\lambda$274.20 is blended with $\lambda$274.18 which is generally much weaker. We estimate the $\lambda$274.18 contribution using $\lambda$275.35 which has its highest ratio of 0.25 in a density region of 10$^{10}$ cm$^{-3}$.
$\lambda$203.82 is partly blended with $\lambda$203.72 and the two components can be extracted by simultaneously fitting two Gaussians to the observed spectral feature. $\lambda$186.60 is blended with $\lambda$186.61 but it is not possible to estimate the blending contribution using the available data. Since is formed at around $\log\,T=6.6$, it will only be significant in the core of the active region, however this is where the moss regions are found and so can be expected to be a significant contributor to the line.
The $\lambda$186 and $\lambda$195 lines are self blends. For $\lambda$186, we have fitted both of the lines with one Gaussian and we have used both spectral lines in the CHIANTI v6.0 [@chianti_v1; @chianti_v6] model in the derivation of the density. The $\lambda$195.12 line is self-blended with the $\lambda$195.18 line [@fe12]. The ratio of these two lines is sensitive to density. This blend can safely be ignored for quiet solar active region conditions such as for quiescent active region loops. However, the blend cannot be ignored while studying the moss regions, where the electron density is well above 10$^{10}$ cm$^{-3}$ and the line at $\lambda$195.18 becomes $\sim$15% of the line at $\lambda$195.12. [@peter_dens] suggested that to deal with the $\lambda $195.18 blend a two Gaussian fit can be performed, where the stronger $\lambda$195.12 line has free parameters for the centroid, width and intensity, while $\lambda $195.18 is forced to be 0.06 [Å]{} towards the long wavelength side of $\lambda $195.12, and to have the same line width as $\lambda $195.12. However, the intensity of $\lambda$195.18 is free to vary. In this study, we have used the technique suggested by [@peter_dens] to de-blend $\lambda
$195.12 from $\lambda$195.18.
Spectroscopic techniques {#techs}
========================
In order to derive physical parameters such as temperature, electron density and filling factors, a number of different spectroscopic techniques can be applied to EIS observations. For a review of different spectroscopic techniques see e.g. [@dere_mason; @mason].
The intensity of an optically thin emission line can be written as
$$\label{main}
I = 0.83~Ab(z) \int_{h} G(T_{\rm{e}}, N_{\rm{e}})~N_{e}^{2}~dh$$
where Ab(z) is the elemental abundances, $T_{e}$ is the electron temperature, and $N_{e}$ is the electron density. The factor 0.83 is the ratio of protons to free electrons which is a constant for temperatures above $10^5$ K. G($T_{e}$, $N_{e}$) is the *contribution function* which contains all the relevant atomic parameters for each line, in particular the ionization fraction and excitation parameters and is defined as
$$G(T_{\rm{e}}, N_{\rm{e}}) = \frac{hc}{4{\pi} {\lambda_{i,j}}} \frac{A_{ji}}{N_{e}} \frac{N_{j}(X^{+m})}{N(X^{+m})} \frac{N(X^{+m})}{N(X)}$$
where i and j are the lower and upper levels, A$_{ji}$ is the spontaneous transition probability, $\frac{N_{j}(X^{+m})} {N(X^{+m})}$ is the population of level j relative to the total $N(X^{+m})$ number density of ion $X^{+m}$ and is a function of electron temperature and density, $\frac{N(X^{+m})}{N(X)}$ is the ionization fraction which is predominantly a function of temperature. The contribution functions for the emission lines considered here were computed with version 6 of the CHIANTI atomic database [@chianti_v6] using the CHIANTI ion balance calculations and the coronal abundances of [@coronal_abund].
Determination of electron temperature {#pott}
-------------------------------------
The solar plasma generally shows a continuous distribution of temperatures which is why such a broad range of ion species is seen in the solar spectrum. The distribution is usually expressed as an emission measure distribution that indicates the amount of plasma at each temperature. In some cases solar plasma is found to be very close to isothermal and an example is the quiet Sun plasma observed above the limb [@feldman_1999]. A method that is very effective for establishing if a plasma is isothermal is the so-called EM-loci method [see e.g. @jordan; @feldman_1999; @em_loci]. In this method, the ratios of observed intensities of different spectral lines with their corresponding contribution functions and abundances (i.e., I$_{obs}$/\[Ab(z) G(T$_{e}$, N$_{e}$)\]) are plotted as a function of temperature. If the plasma is isothermal along the line-of-sight (LOS) then all of the curves would cross at a single location indicating a single temperature.
An indication of temperature can be obtained using emission lines from ions with different ionization stages. As contribution functions are generally sharply peaked functions in log temperature then ratios of two contribution functions will be monotonic functions in temperature, allowing observed intensity ratios to be converted to a temperature estimate. The temperature is not physically meaningful if the plasma is multithermal. However if the two ions are formed close to the dominant emission temperature of the plasma then the ratio will accurately reveal those locations with more high temperature plasma and those with more low temperature plasma. For the present work we have used emission lines from and (Sect. \[temp\]).
As the moss studied here is found to be multithermal it is necessary to perform an emission measure analysis to determine the temperature distribution. Here we follow the approach of [@pottasch] whereby individual emission lines yield estimates of the emission measure at the temperature of formation for each spectral line. By considering lines formed over a wide range of temperatures, an emission measure distribution can be determined. The method requires the contribution function to be approximated by a simplified function such that $G$ is defined to be a constant, $G_0$, over the temperature range $\log\,T_{\rm
max}-0.15$ to $\log\,T_{\rm
max}+0.15$ where $T_{\rm max}$ is the temperature where the contribution function has its maximum. $G_0$ is evaluated as $$\label{em2}
G_0 = \frac{\int G(T_{e}, N_{e})~dT_{e}} {T_{\rm max} \times (10^{0.15} - 10^{-0.15})}.$$ The expression for the line intensity, Eq. \[main\], then becomes $$I=0.83Ab(z) G_0 \int N_{\rm e}^2 dh.$$ The emission measure for the emission line is then defined as $$\label{em-def}
EM= \int N_{\rm e}^2 dh$$ and so $$\label{em3}
EM = {I_{\rm obs} \over 0.83 Ab(z) G_0 }$$ thus the emission measure is defined entirely by the observed line intensity, the element abundances and the atomic parameters contained in $G_0$.
An IDL routine called *integral\_calc.pro* available in the CHIANTI software distribution is used here to compute the quantity $G_0$.
Determination of electron density, filling factor and column depth
------------------------------------------------------------------
The electron density of an astrophysical plasma can be derived by measuring two emission lines of the same ion that have different sensitivities to the plasma density, the ratio yielding a direct estimate of the density [e.g. @mason]. This method is independent of the emitting volume, element abundances or ionization state of the plasma, and depends solely on the atomic population processes within the ion.
![CHIANTI, v6.0, theoretical intensity ratios with respect to electron density for the spectral line ratios used in this paper. The spectral lines are labelled on the plot.\[chianti\]](13883fg2.eps){width="40.00000%"}
EIS provides access to a number of line ratio density diagnostics formed at different temperatures and Table \[lines\] lists the diagnostics observed with the observation study *CAM\_ARTB\_CDS\_A*. The theoretical variations of the line ratios with density are derived using version 6 of the CHIANTI database (Dere et al. 2009) and the curves are shown in Fig. \[chianti\].
The density can be used to derive the filling factor of the plasma. If we assume that the density is constant within the emitting volume for the ion then the emission measure (Eq. \[em-def\]) can be written as $N_{\rm e}^2 h$ where $h$ is the column depth of the emitting plasma. Rearranging Eq. \[em3\] then gives $$\label{coldepth}
h = {I_{\rm obs} \over 0.83 Ab(z) G_0 N_{\rm e}^2 }.$$ By inspecting images of the emitting plasma, it is possible to determine the apparent column depth of the plasma, $h_{\rm
app}$. In the present case this is done by studying images of the moss as the active region approaches the limb (Sect. \[dens\]). That is when the radial dimension of the moss is almost perpendicular to the line of sight and so its depth can be measured visually. The ratio of the spectroscopically derived column depth, $h$, to $h_{\rm app}$ then yields a value for the filling factor, $\phi$, of the plasma. i.e., $$\label{fill}
\phi = { EM \over N_{\rm e}^2 h_{\rm app} }.$$ $\phi$ essentially measures the fraction of the observed plasma volume that is actually emitting the emission line under study. Values less than one imply that the volume is not completely filled with emitting plasma.
Active region moss and hot loops {#moss_hot_loops}
================================
![Co-aligned TRACE $\lambda$171 (top left), EIS $\lambda$195 (top right), EIS $\lambda$284 (bottom left) and XRT Al\_poly filter images (bottom right) taken on May 01, 2007. The images are shown in negative intensity. The two stars indicated by two arrows in the TRACE image show moss regions. The two stars in the XRT image are located at the same position as those in the TRACE image showing the footpoints of hot loops. The arrow ’A’ locates a couple of 1MK loops rooted in moss regions. Arrow ’B’ in the XRT image shows the high temperature fuzzy emission in the moss regions.[]{data-label="cross_cor_image"}](13883fg3.eps){width="45.00000%"}
Based on the observations recorded from TRACE and SXT and using analytical calculations it has been proposed that the moss regions are the footpoints of the hot loops seen in the SXT images taken at 3-5 MK. However, we note that the spatial resolution of TRACE is a factor of 2.5 better than the high resolution SXT images. The X-ray images recorded by XRT aboard Hinode are of comparable resolution to that of TRACE images (1 arcsec per pixel). In addition, spectral images obtained using EIS provide further information at intermediate temperatures. Hence, we have revisited this relationship question in this paper using TRACE, XRT and EIS observations.
In order to compare the observations taken from XRT, TRACE and EIS, a coalignment of the images was performed. It is known that images taken using the two CCDs of EIS are shifted with respect to each other [@peter_artb]. To coalign the EIS spectral images obtained from the two detectors, we cross-correlated images obtained in $\lambda$195 and $\lambda$261. Since the peak formation temperature of these two lines are the same, they reveal the same structures. The TRACE $\lambda$171 and XRT Al\_poly images were then cross-correlated with the images obtained in $\lambda$195 and $\lambda$284 respectively.
Figure \[cross\_cor\_image\] displays co-aligned images recorded from TRACE $\lambda$171 (top left panel), EIS (top right), EIS (bottom left) and XRT (using the Al\_poly filter) (bottom right panel). The images are displayed in a negative intensity scale. The bright moss regions can be seen as dark regions located in the left half of the top left image, as also shown in the bottom panel of Fig. \[context\]. We have plotted two asterisks on the TRACE image, shown by two arrows in the top left image, locating moss regions. The two asterisks in the bottom right image (also shown by two arrows) correspond to the same locations as those in the top right image. This clearly demonstrates that those moss regions are essentially located at the footpoints of the hot loops as deduced previously [see e.g., @antiochos]. The arrow labelled as ’A’ in the top left panel locates an 1MK loop located in the moss regions and coexistent with high temperature loops seen in EIS and XRT images. Therefore, it appears reasonable to deduce that the moss regions are not just the footpoints of hot loops, rather there are warm loops at 1MK which are also rooted in the moss regions. However, we cannot rule out the possibility that these warm loops are those which are cooling down to 1MK from a temperature of 2-3MK i.e., from XRT temperatures to TRACE.
The arrow labelled as ’B’ in the XRT image shows hot fuzzy emission, which is located over moss regions when compared to the top left image. The loop structures are not clear and it is difficult to deduce if these moss regions are the footpoints of loops.
Thermal structure of moss {#temp}
=========================
![A temperature map derived using intensity ratios of $\lambda$188 and $\lambda$202. We used the ionization fraction from CHIANTI v6.0. \[temp\_map\]](13883fg4.eps){width="50.00000%"}
![Top panel: an image from an EIS raster. The overplotted boxes show the regions which were used for deriving plasma parameters. BG is the region which was used to subtract the background emission in section \[fill\_factor\]. Bottom panel: an EM-loci plot for region 1 using the ionization balance from CHIANTI v6.0 and the coronal abundances of [@coronal_abund]. The meaning of the different symbols are shown in the figure.\[em\_loci\]](13883fg5.eps){width="40.00000%"}
![An average emission measure plot for the five regions shown in the top panel of Fig \[em\_loci\]. CHIANTI v6.0 ionization equilibrium and the coronal abundances of [@coronal_abund] wewre used. The EM derived for is an upper limit and is marked with an arrow.\[emission\_measure\]](13883fg6.eps){width="45.00000%"}
The moss regions were originally noted in the images recorded using the 171 [Å]{} channel of TRACE, which primarily observes solar transition region plasma at a temperature of $\sim$ 1 MK. [@fletcher_moss], using a DEM study of an observation taken from SoHO/CDS, showed that the plasma in moss regions was multi-thermal. Recent studies using EIS data [see e.g., @warren_moss; @tripathi_moss] confirmed that moss regions are seen not only at 1 MK but in a range of temperatures. Therefore, in order to understand the physics of moss regions, it is important to understand the thermal structure of moss and its temporal variation.
Fig. \[temp\_map\] displays a temperature map of the active region rastered on May 01, 2007 which was derived using intensity ratios of $\lambda$188.2 and $\lambda$202.0 using the ionization fraction from CHIANTI v.6.0 and the coronal abundances of [@coronal_abund]. The temperature map shows that most of the moss regions (corresponding to the contoured regions in the bottom panel of Fig. \[context\]) is within the temperature range of log T = 6.2 - 6.3. This basically reflects the fact that by taking ratios, we are measuring a temperature common to the *contribution functions* of the two spectral lines. From the figure, however, it is evident that the moss regions are at a temperature of log T = 6.2. In addition, we find that moss regions are cooler than some of the surrounding regions. Indeed, for those regions, we found the existence of hot emission by investigating the spectral images obtained in and lines.
The bottom panel in Fig. \[em\_loci\] shows the ’EM-loci’ plots for one region (region 1) of the moss, labelled in the top panel. In order to compute the EM-loci plots, we have only used the spectral lines of iron, so that we can rule out any effects of abundance variations on the relative magnitudes of the emission measures obtained for different spectral lines. As can be deduced from the figure, the plasma along the LOS is multi-thermal. Most of the emission, however, is within the temperature range 1.2 MK to 1.8 MK. The peak of emission measure is at around log T = 6.2, suggesting a similar temperature for the moss to that derived from the line ratios. For all the five regions marked in Fig. \[em\_loci\] (top panel), the crossing point of the curves are very similar. However, the magnitude of the emission measure crossing point is different for different regions. In order to check the variation in the thermal structure of the moss regions, we generated EM-loci plots of all 13 datasets listed in Table \[data\]. The crossing points of the EM curves were similar to that shown in the bottom panel of Fig. \[em\_loci\], however the magnitude of the emission measure did vary from region to region. We also considered each raster for five consecutive days and traced a specific region in all of the rasters. The EM-loci plots obtained for each region for all five days showed remarkable similarities in terms of the crossing points of the curves. Therefore, we conclude that the thermal structure of the moss region remains fairly constant, at least for the active region studied in this paper and that most of the plasma in the moss region is in the temperature range 1.2 MK to 1.8 MK. Hence, the EM-loci plot presented in Fig. \[em\_loci\] can be taken as typical for all regions of moss in this study.
The EM-loci analysis indicates that the plasma along the LOS in the moss regions is multi-thermal. Therefore, in order to get a proper thermal structure, we need to perform an EM analysis. For this purpose we have used the Pottasch method as described in subsection \[pott\]. Figure \[emission\_measure\] shows a plot of the average EM for all of the five regions shown in the top panel of Fig. \[em\_loci\]. The EM was calculated using ionization fraction from CHIANTI v6.0 and the coronal abundances of [@coronal_abund]. In addition, for and we have used densities derived using (formed at a similar temperature), for and we have used densities derived from and for and we have used densities dervided from diagnostic line ratios within those ions. As is evident from the plot, most of the emission in moss regions is observed in in all cases. The emission starts to decrease in and . From the plot it appears that is the turning point of the emission measure curve. It is likely that the emission seen in is not just from the moss emission, but is possibly contaminated with emission from hot loops which are seen in and . The plot shows very little difference in the emission seen in and . This could be due to the fact that the $\lambda$186.6 line used in this study is blended with another line, $\lambda$186.61 formed at log T = 6.4, and could therefore be contaminated with some emission from hot loops overlying the moss regions. The plot shown in Fig. \[emission\_measure\] suggests that a temperature somewhat close to the formation temperature of and (log T = 6.1-6.3) is the characteristic temperature of the moss for this active region.
{width="80.00000%"}
To study the variation of thermal structure in the moss regions over a period of five days, we have considered one raster every day and performed an emission measure analysis in a specific region. For the coalignment we cross-correlated the rasters obtained on consecutive days. Fig. \[int\_five\_days\] shows the co-aligned intensity images for five consecutive days obtained in $\lambda$195. The data above the white lines in the last three images show the artifacts introduced due to cross-correlation and interpolation. We believe that we have achieved the co-alignment within a few arcsec. The overall structure of the active region stays almost the same. Fig. \[five\_days\_em\] shows emission measure as a function of temperature for the boxed region shown in the left image in Fig. \[int\_five\_days\]. As discussed earlier and shown in Fig. \[emission\_measure\] most of the emission in the moss region comes from and the emission starts to decrease in . It can be easily seen from the plot that the average emission measure for the boxed region remains fairly constant over the five day period for all five spectral lines. This suggests that the thermal structure in the moss region does not change significantly with its temporal evolution.
Densities and filling factors in moss {#dens}
=====================================
Densities in moss regions
-------------------------
Figure \[density\_may1\] gives the densities measured in five different moss regions (shown in the top panel of Fig. \[em\_loci\]) simultaneously at different temperatures using the spectral lines (log T = 5.8), (log T = 6.2), (log T = 6.25), (log T = 6.3). The uncertainties on the densities are calculated using 1-sigma errors in the intensities derived from a Gaussian fitting of the spectral lines and the photon statistics. In addition the errors for the derived electron densities from the theoretical CHIANTI curves are estimated. These are larger when the curves approach their high and low density limits. The plot demonstrates that the electron density in each moss region falls off with temperature except for that derived from . However, we note the large error bars on the densities. These large errors are due to the fact that the two lines in EIS active region spectra are very weak [see e.g., @peter_artb]. It is also worth pointing out that the densities obtained using lines are much higher that those obtained using and . The decrease in the densities with temperature seen in the figure is anticipated if we assume a constant pressure in a given moss region. However, considering the peak formation temperature for each line and the corresponding derived densities we find that the pressure for is substantially higher than those for and .
![An emission measure curve (obtained using the Pottasch method) for five days for the region shown in Fig. \[int\_five\_days\]. CHIANTI v6.0 ionization equilibrium and the coronal abundances of [@coronal_abund] were used. The EM for is an upper limit and is marked with an arrow. \[five\_days\_em\]](13883fg8.eps){width="45.00000%"}
![Electron density measured using , , and for the five different regions marked in the top panel in Fig \[em\_loci\]. \[density\_may1\]](13883fg9.eps){width="45.00000%"}
![The electron density variation in the moss regions over an hour derived from four consecutive EIS rasters. \[dens\_anhour\]](13883fg10.eps){width="45.00000%"}
![Electron density maps obtained using coaligned intensity maps shown in Fig. \[int\_five\_days\]. The bottom right panel displays the electron density variation over a period of five days measured using , and for the boxed region shown in the top left panel. \[dens\_five\_days\]](13883fg11.eps){width="45.00000%"}
We have studied the short term and long term temporal variations of electron densities in the moss regions. For this purpose we have considered four consecutive rasters taken 20 minutes apart on May 03, 2007. Fig. \[dens\_anhour\] displays the variation of electron densities derived using , , and for the five regions shown in the top left panel of the figure. Electron density values for different rasters are plotted with different symbols in each plot. The error bars are estimated as in Fig. \[density\_may1\]. The uncertainties in the densities obtained using are very large, so we have omitted from the plot. The density falls off with temperature in a similar way to the plot shown in Fig. \[density\_may1\]. It is also clear from the plot that the densities obtained using are consistently higher than those derived using and . The densities derived using are also always larger than those by . However, the small difference between the densities obtained from and could be real indicating constant pressure. The plots clearly demonstrate that there is almost no change in electron density over an hour at all three temperatures.
To study the evolution of electron density in the moss regions over a period of five days, we have considered the boxed region corresponding to the one shown in the left panel of Fig. \[int\_five\_days\]. The density maps corresponding to the intensity maps shown in Fig. \[int\_five\_days\] are shown in Fig. \[dens\_five\_days\]. The box in the top left image in Fig. \[dens\_five\_days\] corresponds to the region boxed in the intensity image shown in Fig. \[int\_five\_days\]. As with the intensity images, the data above the white lines in the last three density maps shows an artifact introduced due to cross-correlation. As can be seen from the figure, the overall density structure stays fairly similar as does the intensity structure (see Fig. \[int\_five\_days\]) with just a small enhancement in the center of the moss region. To show this quantitatively, we have studied the density variation in the boxed region shown in the top left image. The bottom right panel in Fig. \[dens\_five\_days\] displays the density variation showing that the electron density increases from May 1 to May 3 and then decreases on May 4, which is quite pronounced in and very slightly in and . Except for this enhancement, we find that the electron density remains fairly similar in the moss regions and does not show much variation in time. Although the reason for this enhancement in densities is not clear to us, we anticipate that this could be due to small scale dynamic activity taking place in the core of active regions seen in an XRT movie for this region. Using Coronal Diagnostic Spectrometer [CDS; @cds] and Michelson Doppler Imager [MDI; @mdi] data, [@soho17; @helen_book] showed that localized enhancements in electron densities were correlated with emerging and canceling flux regions. We also note that canceling flux regions are frequently observed near the polarity inversion line [see e.g., @thesis]. However, further investigation of this is needed.
In all of the measurements so far presented in this paper and those results presented in other papers for high density regions (that is densities greater than 10$^9$ cm$^{-3}$) cf [@tripathi_moss; @warren_moss; @peter_dens; @brendan], the densities measured using are reported to be larger than those obtained from and . It is worth mentioning here that if the $\lambda$186.8 line were blended and we lower the intensity by 20%, the electron densities obtained using would become consistent with those obtained using and . However, we cannot at present explain these discrepancies and so we leave this as an open question.
Filling factors in moss regions {#fill_factor}
-------------------------------
Equation \[fill\] gives the expression for deriving the filling factor from the emission measure, density and apparent column depth. The emission measure and density are derived directly from the spectroscopic data as described in the previous sections. To estimate the apparent column depth we follow the method of [@martens] and study images of the moss at the solar limb. The active region was observed close to the limb with EIS on 2007 May 7 and radial intensity profiles cutting through a particular moss region were studied in lines of , and . A sample intensity profile from $\lambda$195.12 is shown in Fig. \[thickness\] where a distinctive spike in emission corresponding to the moss region can be seen. We interpret the width of this spike to be the column depth of the moss, which is found to be 6 ($\sim$4000 km) in this case, in good agreement with the results of [@martens].
![Top panel: Negative intensity image for $\lambda$195 on 07-May-2007. Bottom Panel: the intensity profile for $\lambda$195 between the two lines marked in the top panel. The arrow in the top panel marks the moss region and in the bottom panel marks the intensity enhancement due to the moss region.\[thickness\]](13883fg12.eps){width="40.00000%"}
--------- ------------------------- --------------------- -------- ------------------------- ------------------------- -------- ------------------------- ------------------------- --------
log N$_e$ h $\phi$ log N$_e$ h $\phi$ log N$_e$ h $\phi$
(cm$^{-3}$) (km) (cm$^{-3}$) (km) (cm$^{-3}$) (km)
Region1 10.32$^{10.39}_{10.25}$ 140$^{100}_{200}$ 0.04 9.73$^{9.76}_{9.70}$ 3000$^{2700}_{3300}$ 0.8 9.54$^{9.60}_{9.48}$ 5600$^{4400}_{7100}$ 1.4
Region2 10.32$^{10.39}_{10.26}$ 160$^{120}_{210}$ 0.04 9.68$^{9.71}_{9.66}$ 4200$^{3800}_{4300}$ 1.0 9.57$^{9.62}_{9.51}$ 6600$^{5400}_{8400}$ 1.6
Region3 10.22$^{10.28}_{10.15}$ 280$^{220}_{390}$ 0.07 9.63$^{9.65}_{9.60}$ 6400$^{6000}_{6900}$ 1.6 9.45$^{9.50}_{9.40}$ 14000$^{11000}_{17000}$ 3.5
Region4 9.79$^{9.86}_{9.72}$ 1200$^{800}_{1500}$ 0.3 9.33$^{9.36}_{9.30}$ 14000$^{13000}_{15000}$ 3.5 9.21$^{9.29}_{9.12}$ 22000$^{16000}_{33000}$ 5.6
Region5 9.83$^{9.91}_{9.74}$ 680$^{500}_{1000}$ 0.2 9.37$^{9.41}_{9.34}$ 8800$^{7700}_{9500}$ 2.2 9.25$^{9.33}_{9.15}$ 16000$^{11000}_{24000}$ 3.9
Region1 11.00$^{11.08}_{10.93}$ 8$^{6}_{10}$ 0.002 10.58$^{10.63}_{10.54}$ 130$^{100}_{150}$ 0.03 10.04$^{10.08}_{10.00}$ 720$^{620}_{830} $ 0.2
Region2 10.88$^{10.95}_{10.82}$ 15$^{10}_{20}$ 0.004 10.17$^{10.19}_{10.14}$ 970$^{930}_{1100}$ 0.2 9.87$^{9.90}_{9.83}$ 1900$^{1700}_{2200}$ 0.5
Region3 10.60$^{10.65}_{10.55}$ 60$^{50}_{70}$ 0.01 9.90$^{9.92}_{9.88}$ 3200$^{3000}_{3300}$ 0.8 9.62$^{9.65}_{9.59}$ 6900$^{6200}_{7700}$ 1.7
Region4 10.17$^{10.22}_{10.12}$ 230$^{190}_{290}$ 0.1 9.55$^{9.57}_{9.53}$ 7500$^{7200}_{7900}$ 1.9 9.48$^{9.52}_{9.43}$ 7100$^{6100}_{8700}$ 1.8
Region5 10.91$^{11.00}_{10.82}$ 7$^{5}_{11}$ 0.002 10.39$^{10.45}_{10.34}$ 250$^{200}_{300}$ 0.1 9.71$^{9.75}_{9.67}$ 2300$^{2000}_{2600}$ 0.6
--------- ------------------------- --------------------- -------- ------------------------- ------------------------- -------- ------------------------- ------------------------- --------
Table. \[table:fill\] shows electron densities, filling factors and the column depths for five different regions shown in the top panel in Fig. \[em\_loci\] using the ions , and before and after background subtraction. We have used the region labelled ’BG’ in the top panel of Fig. \[em\_loci\] for background/foreground subtraction. The filling factor is estimated using equation \[fill\]. The column depth is estimated using equation \[coldepth\] assuming a filling factor equal to 1. The table clearly demonstrates the importance of background/foreground in the measurements of electron densities and filling factors. The electron densities for each ion have increased substantially after subtracting the background, and the filling factors and column depths have substantially decreased. This is the first time the importance of background/foreground emission has been demonstrated while estimating physical parameters such as density, filling factors and column depth in moss regions. After the background subtraction, we find substantial increases in the electron densities and meaningful results for filling factors. The filling factors derived for are very low i.e., much less than 1, whereas those for and are closer to 1, sometimes even more than 1. A filling factor greater than 1 does not give any meaningful information. However, in the present case it suggests that we have very likely underestimated the column depth by using the thickness of the moss measured using TRACE observations.
The column depth measurements presented in Table \[table:fill\], which are based on the assumption that the filling factor is 1, show that the moss seen in is a very thin region i.e., about the order of a fews tens of kilometers in the dense moss regions. At higher temperatures e.g., in and , the estimated column depth is larger than that estimated by , by a large factor of $\approx$10-20. One possible reason for this difference could be that the background/foreground is not completely removed. This could explain the higher filling factor and larger path length obtained for and . However, the question remains as to why we have such a small column depth for in comparison to that which is measured from the limb observations. We note that the densities observed using are too high in comparison to those derived using and and this is the most likely the reason for very low column depth. However, this complex issue involving atomic physics calculations needs further investigation in order to understand the discrepancies in densities, filling factors and column depth between and other ions.
It is worthwhile emphasizing here that column depths and filling factors are derived using the coronal abundances of [@coronal_abund]. These values are significantly different (a factor of $\sim$3-4 larger) when photospheric abundences are used.
Summary and Conclusions {#con}
=======================
Using Hinode/EIS observations, we have studied basic physical plasma parameters such as temperature, electron density, filling factors, and column depth in moss regions and the variation of these parameters over an hour and over a time period of five days. In addition, we have revisited the question of whether the moss regions are the footpoints of hot loops using observations from TRACE, EIS and XRT. The results are summarized below.
- Based on the TRACE, EIS and XRT observations we find that most of the moss regions are essentially located at the footpoints of hot loops. In some places we observed TRACE 171 (1 MK) loops rooted in the moss regions.
- Based on the line intensity ratios of $\lambda$188 and $\lambda$202, and an emission measure analysis, we find that the characteristic temperature of moss regions is about log T = 6.2. Emission measure analyses over a time period of one hour (Fig. \[emission\_measure\]) and over five days (Fig. \[five\_days\_em\]) reveal that the thermal structure of the moss regions does not change significantly with time.
- The electron densities measured using ratios are about 1-3 $\times$ 10$^{10}$ cm$^{-3}$ and about 2-4 $\times$ 10$^{9}$ cm$^{-3}$ using and . Work is in progress to try to resolve this discrepancy. It is worth emphasizing here that if the $\lambda$186.8 were blended and we lower its intensity by 20%, then the electron densities obtained using would become consistent with those obtained using and . The densities derived using and are similar to those derived by [@fletcher_moss] using line ratios observed by CDS. However, when we subtract the foreground/background emission we find a substantial increase (a factor of 3-4 or even more in some cases) in the densities.
- The electron densities only show small changes ($\sim$25%) over a period of an hour. There are large variations (an order of magnitude increase) in densities when measured over a period of five days. However, the variation in the densities obtained using and is only about 50-70%.
- The filling factor of the moss plasma is in the range 0.1-1 and the path length along which the emission originates is from a few 100 to a few 1000 kms long.
These new measurements of the thermal and density structure in moss regions should provide important constraints for the modelling of loops in the hot and dense core of active regions.
Acknowledgements
================
We thank the referee for the constructive and thoughtful comments. DT, HEM and GDZ acknowledge the support from STFC. We thank Brendan O’Dwyer for various discussions. Hinode is a Japanese mission developed and launched by ISAS/JAXA, collaborating with NAOJ as a domestic partner, NASA and STFC (UK) as international partners. Scientific operation of the Hinode mission is conducted by the Hinode science team organized at ISAS/JAXA. This team mainly consists of scientists from institutes in the partner countries. Support for the post-launch operation is provided by JAXA and NAOJ (Japan), STFC (U.K.), NASA, ESA, and NSC (Norway). The help and support of the Hinode/EIS team in particular is acknowledged.
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DESY 11-078\
[**May 2011**]{}\
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We use integrability to construct the *general* classical splitting string solution on $\mathbb{R} \times S^3$. Namely, given *any* incoming string solution satisfying a necessary self-intersection property at some given instant in time, we use the integrability of the worldsheet $\sigma$-model to construct the pair of outgoing strings resulting from a split. The solution for each outgoing string is expressed recursively through a sequence of dressing transformations, the parameters of which are determined by the solutions to Birkhoff factorization problems in an appropriate real form of the loop group of $SL_2(\mathbb{C})$.
Introduction
============
Solving a conformal field theory amounts to finding its spectrum of anomalous dimensions and the structure constants of its 3-point functions. At weak coupling, the former problem is equivalent to perturbatively diagonalizing the dilatation operator. When the CFT admits an AdS dual, the same problem at strong coupling requires finding the semiclassical energy spectrum of strings in AdS-space. In the case of $\mathcal{N} = 4$ SYM, dual to type IIB superstrings on $AdS_5 \times S^5$, the emergence of integrability in the planar limit of both theories has gradually led to a very elegant analytical solution to both of these problems (see [@Beisert:2010jr] for a recent review). Assuming integrability at all loops, this eventually culminated in a series of proposals for computing the full spectrum of anomalous dimensions at all values of the coupling [@GKV; @Bombardelli:2009ns; @AF].
In sharp contrast with these developments for the spectrum of $\mathcal{N} = 4$ SYM, comparatively little is known about the structure constants of its 3-point functions. A possible explanation for this shortcoming might be the apparent lack of integrability methods beyond the planar sector. And yet it was recently shown in [@Tayloring1], building on [@WeakIntegr], that the integrability of planar $\mathcal{N} = 4$ SYM, which was so crucial in the exact study of the spectrum, can also be used to systematically tackle the problem of 3-point functions at weak coupling. At strong coupling, on the other hand, a similar use of the classical integrability of superstrings on $AdS_5 \times S^5$, which is after all a local property on the worldsheet, has not yet been exploited. Indeed, the semiclassical study of 3-point functions [@HeavyHeavyLight] has thus far been restricted to cases where the path integral is dominated by some finite-gap solution with cylindrical worldsheet. In particular, a comparison [@Tayloring2] of the results of [@Tayloring1] with string theory in the Frolov-Tseytlin limit could only be considered in the case where two of the three string states are “heavy”, *i.e.* semiclassical, while the third is “light”. Nevertheless, an attractive proposal for computing 3-point functions at strong coupling using classical methods was put forward in [@Janik:2010gc], which relies on finding classical Minkowskian solutions splitting/joining in $S^5$.
The aim of this paper is to exploit the classical integrability of the superstring $\sigma$-model on $AdS_5 \times S^5$ so as to construct classical string solutions with Lorentzian worldsheets of more general topology than the cylinder. The simplest such worldsheet is the ‘pair of pants’ diagram, or three punctured sphere, describing the splitting or joining of strings. By focusing on the bosonic subspace $\mathbb{R} \times S^3$, we shall construct the most *general* splitting string solution on this background. Specifically, we assume that we are given a solution on the cylinder with the property that: at some given instant in time $\tau = 0$, two of its points $\sigma_1$ and $\sigma_2$ coincide in target space and their velocities agree; a simple example is the folded spinning string [@Frolov:2003xy]. The splitting of such a string comes from treating it as consisting of two individual strings and letting each of them evolve separately, see Figure \[fig: topology change\]. The problem therefore consists in solving a pair of Cauchy problems on the outgoing cylinders (the legs of the pair of pants) with Cauchy data specified by a portion of the original solution at time $\tau = 0$.
![The splitting is the result of a manual change in the topology of the closed string imposed at the instant $\tau = 0$ when the profile in space-time self-intersects. The full time evolution of the string is described by a worldsheet with the topology of a pair of pants.[]{data-label="fig: topology change"}](topology_change2.eps){height="35mm"}
The classical splitting of strings has been extensively studied in the literature [@Splitting], mostly in flat Minkowski space, but also on certain backgrounds with respect to which the equations of motion are linear in the fields. In each case, therefore, the general solution of the equations is given by a Fourier series which can be subjected to the relevant boundary conditions and Cauchy data in order to solve the Cauchy problem. More recently, there has been renewed interest in splitting strings in the context of the AdS/CFT correspondence [@SplittingAdS]. However, in this case the non-linearity of the equations of motion renders Fourier analysis unapplicable. But fortunately these non-linear equations are well known to be integrable, in the sense that they define a Lax connection: a locally defined 1-form on the worldsheet, meromorphic in some auxiliary complex parameter, and with the property of being flat. This will allow us to solve the Cauchy problems on the outgoing cylinders.
The outline of the paper is as follows. In section \[sec: setup\] we set up the general formalism for discussing classical interacting strings, treating in parallel the case of flat space and $\mathbb{R} \times S^3$. In section \[sec: flat\] we set up and solve the Cauchy problem for splitting strings in flat space. This serves as a warmup exercise since it will turn out that many features of the solution in flat space carry over to those of the Cauchy problem in $\mathbb{R} \times S^3$. Finally, section \[sec: S3\] deals with splitting strings in $\mathbb{R} \times S^3$. After setting up the Cauchy problem on the outgoing cylinders, we show that the smoothness property of the solution is the same as in flat space, resulting in a similar tessellation of the worldsheet into tile-shaped regions bounded by null rays. We then show how integrability can be used to recursively construct the solution to the Cauchy problem in each tile, given the solution in the previous tile.
Classical interacting strings {#sec: setup}
=============================
Our analysis of splitting strings on $\mathbb{R} \times S^3$ will be closely related to the corresponding analysis in flat space. In this section we therefore introduce both cases in parallel. Let $W$ denote the worldsheet of the string, equipped with a Lorentian metric $\gamma$. For the moment we impose no restriction on the topology of $W$.
#### In flat space.
Consider the embedding $X^{\mu} : W \to \mathbb{R}^{p,1}$ of $W$ into Minkowski space $\mathbb{R}^{p,1}$. In order to describe the classical motion of a string, this map should minimize the string action $\int_W dX^{\mu} \wedge \ast dX_{\mu}$, where $\ast$ denotes the Hodge dual relative to the worldsheet metric $\gamma$. The corresponding equations of motion for all the fields $(X^{\mu}, \gamma)$ read
\[string eom flat\] $$\begin{aligned}
\label{string eom flat a} X^{\mu}: &\qquad d \ast d X^{\mu} = 0,\\
\label{string eom flat b} \gamma: &\qquad G^{\alpha \beta} = \ha \gamma^{\alpha \beta} \gamma_{\rho \sigma} G^{\rho \sigma},\end{aligned}$$
where $G_{\alpha \beta} = \partial_{\alpha} X^{\mu} \partial_{\beta} X_{\mu}$ is the pull-back of the flat target space metric to $W$.
#### On $\mathbb{R} \times S^3$.
The embedding of a string with worldsheet $W$ into the target space $\mathbb{R} \times S^3$ with signature $(-1, +1, +1, +1)$ is described by a pair of fields $X_0 : W \rightarrow \mathbb{R}$ and $g : W \rightarrow SU(2)$. The action for all these fields can be written down succinctly as $$\label{string action}
S = \frac{\sqrt{\lambda}}{4 \pi} \int \left[ \frac{1}{2} \operatorname{tr}(j \wedge \ast j) + dX_0 \wedge \ast dX_0 \right],$$ where $j = - g^{-1} dg$ is the $\mathfrak{su}(2)$-valued current and $\ast$ denotes the Hodge dual relative to the worldsheet metric $\gamma$. The resulting equations of motion for the respective fields are
\[string eom\] $$\begin{aligned}
\label{string eom a} g: &\qquad d \ast j = 0, \quad dj - j \wedge j = 0,\\
\label{string eom b} X_0: &\qquad d \ast d X_0 = 0,\\
\label{string eom c} \gamma: &\qquad G^{\alpha \beta} = \ha \gamma^{\alpha \beta} \gamma_{\rho \sigma} G^{\rho \sigma},\end{aligned}$$
where the induced metric is $G_{\alpha \beta} = \ha \operatorname{tr}(j_{\alpha} j_{\beta}) + \partial_{\alpha} X_0 \partial_{\beta} X_0$.
Mandelstam diagrams
-------------------
In the context of Riemannian worldsheets $W$, the conformal class of the Riemannian metric $g$ endows $W$ with a complex structure, promoting it to a Riemann surface $(W, [g])$. Moreover, as shown in [@GiddingsWolpert], there is a 1 – 1 correspondence between string light-cone diagrams and Abelian differentials on $W$. Analogous statements to these can also be made in the Lorentzian setting [@Liu:1996daa]. In particular, the conformal class of the Lorentzian metric $\gamma$ endows $W$ with a causal structure and static gauge provides a Mandelstam diagram representation of $W$.
#### Conformal gauge.
The equations of motion or being invariant under conformal transformations of the worldsheet metric $\gamma \mapsto e^{\phi} \gamma$, only the conformal equivalence class $[\gamma]$ of $\gamma$ is physically relevant. Yet there is a 1 – 1 correspondence between conformal equivalence classes of Lorentzian metrics and causal structures on $W$, *i.e.* ordered pairs $(\F_+, \F_-)$ of transverse null foliations [@Liu:1996daa]. In other words, specifying the worldsheet metric $\gamma$ amounts to giving $W$ a causal structure, thereby promoting it to a Lorentz surface $(W, [\gamma])$. In terms of any local null coordinates $\tilde{\sigma}^{\pm} : U \subset W \to \mathbb{R}$, defined by $\partial_{\pm} \coloneqq \partial_{\tilde{\sigma}^{\pm}}$ being tangent to the foliation $\F_{\mp}$, the metric reads $\gamma = \gamma_{+-} d\tilde{\sigma}^+ d\tilde{\sigma}^-$.
#### Static gauge.
We can write $dX_0 = \mu_+ + \mu_-$ where the pair $(\mu_+, \mu_-)$ are transverse measures to the foliations $(\F_+, \F_-)$ respectively, which locally read $\mu_{\pm} = (\partial_{\pm} X_0) d\tilde{\sigma}^{\pm}$. In other words $\mu_{\pm}$ vanishes on tangent vectors to leaves of the foliation $\F_{\pm}$ (in particular $\mu_{\pm}$ must vanish at singular points of the foliation $\F_{\pm}$ where multiple leaves end). By working in local coordinates it is easy to see that $\ast \mu_{\pm} = \pm \mu_{\pm}$. But now $dX_0$ is harmonic (that is, closed and co-closed) by the equations of motion, which means that $\mu_{\pm}$ are both closed and hence locally read $\mu_{\pm} = f_{\pm}(\tilde{\sigma}^{\pm}) d\tilde{\sigma}^{\pm}$. Now integrating gives $$X_0(p) = \int^p (\mu_+ + \mu_-)$$ for $p \in W$, and since $X_0$ must be a well defined function we have $\int_C \mu_+ = - \int_C \mu_-$ for any $C \in \pi_1(W)$. This last condition is the requirement for rectifiability of $W$ into a Mandelstam diagram [@Liu:1996daa], *i.e.* for there to be a representative of $(W, [\gamma])$ which is a Mandelstam diagram. Indeed, the level curves of $X_0$ are transverse to the leaves of both null foliations $\F_{\pm}$ and those through the singularities of $\F_{\pm}$ provide an annuli decomposition of $W$. The representative of the class $[\gamma]$ is given by $\eta \coloneqq -4 \, \mu_+ \cdot \mu_- = -4\, d\sigma^+ d\sigma^- = - d\tau^2 + d\sigma^2$, where $$\label{global null coords}
\sigma^{\pm} \coloneqq \int^p \mu_{\pm} = \ha (\tau \pm \sigma)$$ define global null coordinates on (the universal cover of) $W$. In particular we have $X_0 = \tau$. Note that $\eta$ is degenerate at the singular points of the foliation $\F_{\pm}$.
#### Virasoro constraints.
After going to conformal static gauge, the dynamical equation for $X_0$ is solved and the metric is now fixed to the flat metric $\gamma = \eta$. The equations of motion then reduce to $$\begin{aligned}
\label{string eom conf flat}
\textbf{In flat space} \quad &\left\{ \begin{array}{l} \partial_+ \partial_- X^{\mu} = 0, \quad \mu \neq 0,\\
T_{\pm\pm} \coloneqq \partial_{\pm} X^{\mu} \partial_{\pm} X_{\mu} = 0 \end{array} \right.
\\
\label{string eom conf S3}
\textbf{On $\mathbb{R} \times S^3$} \quad &\left\{ \begin{array}{l} \partial_- j_+ = \ha [j_-, j_+], \quad \partial_+ j_- = \ha [j_+, j_-], \\
T_{\pm \pm} \coloneqq \ha \text{tr}\, j_{\pm}^2 + 1 = 0. \end{array} \right.\end{aligned}$$ In either case, we find using the first set of equations that $T_{\pm\pm} = T_{\pm\pm}(\sigma^{\pm})$ only depends on $\sigma^{\pm}$. Therefore, if it vanishes for all $\sigma$ at some $\tau$ then it must vanish for all $\tau$. It follows that when solving the Cauchy problem for the equations or , the Virasoro constraints $T_{\pm\pm} = 0$ will be automatically taken care of provided the Cauchy data satisfy them.
Pair of pants
-------------
Since our aim is to discuss splitting strings, from now on we shall focus on the case where $W$ has the topology of a pair of pants, see Figure \[fig: pair of pants\].
![The string worldsheet $W$ has the topology of a pair of pants, or three punctured sphere. The initial string O splits into two outgoing strings labelled I and II.[]{data-label="fig: pair of pants"}](pants_diagram.eps){height="30mm"}
There is a single singular point and the level curve of $X_0$ through it, which has the topology of a ‘figure of 8’, can be assumed to be at $\tau = 0$ without loss of generality. With this curve removed, the space $W \setminus \{\tau = 0\}$ consists of three cylinders, which can be parameterized as follows
\[3 cyl\] $$\begin{aligned}
\label{3 cyl O} W_{\rm O} &\coloneqq \{ (\sigma, \tau) \, | \, 0 < \sigma \leq 2 \pi, \, \tau < 0 \},\\
\label{3 cyl I} W_{\rm I} &\coloneqq \{ (\sigma, \tau) \, | \, 0 < \sigma \leq 2 a \pi, \, \tau > 0 \},\\
\label{3 cyl II} W_{\rm II} &\coloneqq \{ (\sigma, \tau) \, | \, 2 a \pi < \sigma \leq 2 \pi, \, \tau > 0 \}.\end{aligned}$$
where $a < 1$ and the $\sigma$-interval is periodically identified in each case.
It is clear that any map from such a worldsheet $W$ into spacetime describes a single string which splits off into two separate strings at $\tau = 0$. In particular, the map has the following important self-intersection property: $$\begin{aligned}
\label{self-intersecting flat}
\textbf{In flat space} \quad &\left\{ \begin{array}{c} \; X^{\mu}(0, 0) = X^{\mu}(2 a \pi, 0), \\
\partial_{\tau} X^{\mu}(0, 0) = \partial_{\tau} X^{\mu}(2 a \pi, 0) \end{array} \right.
\\
\label{self-intersecting S3}
\textbf{On $\mathbb{R} \times S^3$} \quad &\left\{ \begin{array}{c} g(0, 0) = g(2 a \pi, 0), \\
\partial_{\tau} g(0, 0) = \partial_{\tau} g(2 a \pi, 0). \end{array} \right.\end{aligned}$$
Splitting strings in flat space {#sec: flat}
===============================
Before studying splitting strings on $\mathbb{R} \times S^3$, we start by analyzing the splitting of strings in flat space in detail since many features of the solution will remain true in the nonlinear case and serve as a guideline there.
Cauchy problem
--------------
We would like to solve the Cauchy problem corresponding to the linear wave equation on the pair of pants $W$, for a given set of initial conditions on the incoming circle. However, it is clear that these conditions cannot be completely arbitrary since they need to be such that the self-intersection property holds. We shall therefore assume that we are given a solution to on the incoming cylinder $W_{\rm O}$ satisfying . To determine the complete solution on the rest of $W$, it remains to find solutions $X^{\mu}_{\rm I}$, $X^{\mu}_{\rm II}$ to
\[eom I and II flat\]
$$\begin{aligned}
\label{eom I flat} \partial_+ \partial_- X^{\mu}_{\rm I} &= 0, \qquad \text{on} \quad W_{\rm I} \\
\label{eom II flat} \partial_+ \partial_- X^{\mu}_{\rm II} &= 0, \qquad \text{on} \quad W_{\rm II} \end{aligned}$$
with initial conditions at $\tau = 0$ specified by the given solution $X^{\mu}$ on $W_{\rm O}$ as follows
\[ic I and II flat\]
$$\begin{aligned}
{2}
\label{ic I flat} \left. \begin{array}{rcl} X^{\mu}_{\rm I}(\sigma, 0) \!\!&=&\!\! X^{\mu}(\sigma, 0),\\
\partial_{\tau} X^{\mu}_{\rm I}(\sigma, 0) \!\!&=&\!\! \partial_{\tau} X^{\mu}(\sigma, 0) \end{array} \right\} \quad &\text{for} &\quad 0 < &\;\sigma \leq 2 a \pi,\\
\label{ic II flat} \left. \begin{array}{rcl} X^{\mu}_{\rm II}(\sigma, 0) \!\!&=&\!\! X^{\mu}(\sigma, 0),\\
\partial_{\tau} X^{\mu}_{\rm II}(\sigma, 0) \!\!&=&\!\! \partial_{\tau} X^{\mu}(\sigma, 0) \end{array} \right\} \quad &\text{for} &\quad 2 a \pi < &\;\sigma \leq 2 \pi.\end{aligned}$$
Note that for each outgoing string, the singularity lies at the end points of the initial interval. Since they live on $W_{\rm I}$ and $W_{\rm II}$ respectively, the solutions $X^{\mu}_{\rm I}$ and $X^{\mu}_{\rm II}$ defined for $\tau \geq 0$ are required to have new periodicity conditions, different from those of $X^{\mu}$, namely
\[periodicity I and II flat\]
$$\begin{aligned}
\label{periodicity I flat} X^{\mu}_{\rm I}(\sigma + 2 a \pi, \tau) &= X^{\mu}_{\rm I}(\sigma, \tau),\\
\label{periodicity II flat} X^{\mu}_{\rm II}(\sigma + 2 (1-a) \pi, \tau) &= X^{\mu}_{\rm II}(\sigma, \tau).\end{aligned}$$
Notice that is consistent with by virtue of the self-intersection property .
Absolute elsewhere of the singularity
-------------------------------------
D’Alembert’s solution to the linear wave equation is expressed as a sum of functions in $\sigma^+$ and $\sigma^-$. In particular, the solution given on the initial cylinder assumes this general form,
\[sol flat\] $$\label{sol initial}
X^{\mu}(\sigma, \tau) = X^{\mu}_+(\sigma^+) + X^{\mu}_-(\sigma^-).$$ Since $X^{\mu}_{\text{I}}$, $X^{\mu}_{\text{II}}$ are solutions of the same equation , they are also given by d’Alembert’s general form on their respective cylinders $$\label{sol I and II flat}
X^{\mu}_{\rm I}(\sigma, \tau) = X^{\mu}_{{\rm I}+}(\sigma^+) + X^{\mu}_{{\rm I}-}(\sigma^-), \qquad
X^{\mu}_{\rm II}(\sigma, \tau) = X^{\mu}_{{\rm II}+}(\sigma^+) + X^{\mu}_{{\rm II}-}(\sigma^-).$$
In terms of the initial conditions read, in their respective domains in $\sigma$, $$\begin{aligned}
{2}
X^{\mu}_{{\rm I}+}(\sigma) + X^{\mu}_{{\rm I}-}(\sigma) &= X^{\mu}_+(\sigma) + X^{\mu}_-(\sigma), &\quad \partial_{\sigma} X^{\mu}_{{\rm I}+}(\sigma) - \partial_{\sigma} X^{\mu}_{{\rm I}-}(\sigma) &= \partial_{\sigma} X^{\mu}_+(\sigma) - \partial_{\sigma} X^{\mu}_-(\sigma),\\
X^{\mu}_{{\rm II}+}(\sigma) + X^{\mu}_{{\rm II}-}(\sigma) &= X^{\mu}_+(\sigma) + X^{\mu}_-(\sigma), &\quad \partial_{\sigma} X^{\mu}_{{\rm II}+}(\sigma) - \partial_{\sigma} X^{\mu}_{{\rm II}-}(\sigma) &= \partial_{\sigma} X^{\mu}_+(\sigma) - \partial_{\sigma} X^{\mu}_-(\sigma).\end{aligned}$$ But then, differentiating the left set of equations with respect to $\sigma$ and combining them with the right set we obtain $$\begin{aligned}
{3}
\partial_{\sigma} X^{\mu}_{{\rm I} \pm}(\sigma) &= \partial_{\sigma} X^{\mu}_{\pm}(\sigma), &\qquad &\text{for} &\quad 0 < &\;\sigma \leq 2 a \pi \\
\partial_{\sigma} X^{\mu}_{{\rm II} \pm}(\sigma) &= \partial_{\sigma} X^{\mu}_{\pm}(\sigma), &\qquad &\text{for} &\quad 2 a \pi < &\;\sigma \leq 2 \pi.\end{aligned}$$ Integrating in $\sigma$ and using part of the initial conditions again, we find
\[sigma der flat\]
$$\begin{aligned}
{3}
X^{\mu}_{{\rm I} \pm}(\sigma) &= X^{\mu}_{\pm}(\sigma) \pm v^{\mu}_{\rm I}, &\qquad &\text{for} &\quad 0 < &\;\sigma \leq 2 a \pi\\
X^{\mu}_{{\rm II} \pm}(\sigma) &= X^{\mu}_{\pm}(\sigma) \pm v^{\mu}_{\rm II}, &\qquad &\text{for} &\quad 2 a \pi < &\;\sigma \leq 2 \pi,\end{aligned}$$
where $v^{\mu}_{\rm I}$ and $v^{\mu}_{\rm II}$ are constants. Plugging this into we obtain
\[sol diamond flat\]
$$\begin{aligned}
{3}
X^{\mu}_{\rm I}(\sigma, \tau) &= X^{\mu}(\sigma, \tau), &\qquad &\text{for} &\quad \tau < &\;\sigma \leq 2 a \pi - \tau,\\
X^{\mu}_{\rm II}(\sigma, \tau) &= X^{\mu}(\sigma, \tau), &\qquad &\text{for} &\quad 2 a \pi + \tau < &\;\sigma \leq 2 \pi - \tau.\end{aligned}$$
Note that the domains of validity here are determined by the requirement that $\sigma^{\pm}$ satisfies the same bounds as $\sigma$ does in . Hence, these domains are bounded by null rays emanating from the singularity, as shown in Figure \[fig: regions I and II, R3\]. That is, the original solution remains valid at points on the outgoing cylinders I and II which are space-like separated from the singularity.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
![When $\tau < 0$ the worldsheet has the topology of a cylinder, the picture being periodically identified in the $\sigma$-direction. For $\tau > 0$ the change in topology is indicated by the blue cuts emanating from the singularity at $0 \equiv 2 a \pi$: the right and left sides of the cut through $0$ are to be identified with the left and right sides of the cut through $2 a \pi$ respectively. In the shaded area, bounded by null rays (in red), the initial solution remains valid.[]{data-label="fig: regions I and II, R3"}](regions_I_II_R3.eps "fig:"){height="40.5mm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
Absolute future of the singularity
----------------------------------
To determine the solutions I and II beyond the limited region of Figure \[fig: regions I and II, R3\], we impose their respective periodicity conditions . Taking the derivatives of these conditions with respect to $\sigma^{\pm}$ and using the general form of the solutions leads to $$\partial_{\sigma} X^{\mu}_{{\rm I}\pm}(\sigma + a \pi) = \partial_{\sigma} X^{\mu}_{{\rm I}\pm}(\sigma), \qquad
\partial_{\sigma} X^{\mu}_{{\rm II}\pm}(\sigma + (1 - a) \pi) = \partial_{\sigma} X^{\mu}_{{\rm II}\pm}(\sigma).$$ After integrating we find $$X^{\mu}_{{\rm I}\pm}(\sigma + a \pi) = X^{\mu}_{\text{I}\pm}(\sigma) + x^{\mu}_{{\rm I}\pm}, \qquad
X^{\mu}_{{\rm II}\pm}(\sigma + (1 - a) \pi) = X^{\mu}_{\text{II}\pm}(\sigma) + x^{\mu}_{{\rm II}\pm}.$$ In other words, the functions $X^{\mu}_{{\rm I}\pm}$ and $X^{\mu}_{{\rm II}\pm}$ are not periodic but shift by constants under the translations $\sigma \mapsto \sigma + a \pi$ and $\sigma \mapsto \sigma + (1-a) \pi$ respectively. This leads to
\[sol extension flat\]
$$\begin{gathered}
\left\{
\begin{split}
X^{\mu}_{\rm I}(\sigma^+ + a \pi, \sigma^-) &= X^{\mu}_{\rm I}(\sigma^+, \sigma^-) + x^{\mu}_{\rm I},\\
X^{\mu}_{\rm I}(\sigma^+, \sigma^- + a \pi) &= X^{\mu}_{\rm I}(\sigma^+, \sigma^-) + x^{\mu}_{\rm I},
\end{split} \right. \\
\left\{
\begin{split}
X^{\mu}_{\rm II}(\sigma^+ + (1-a) \pi, \sigma^-) &= X^{\mu}_{\rm II}(\sigma^+, \sigma^-) + x^{\mu}_{\rm II},\\
X^{\mu}_{\rm II}(\sigma^+, \sigma^- + (1-a) \pi) &= X^{\mu}_{\rm II}(\sigma^+, \sigma^-) + x^{\mu}_{\rm II}.
\end{split} \right.\end{gathered}$$
That the *same* $x^{\mu}_{\rm I}$ (resp. $x^{\mu}_{\rm II}$) appears for shifts in both $\sigma^+$ and $\sigma^-$ of $X^{\mu}_{\rm I}$ (resp. $X^{\mu}_{\rm II}$) follows from taking the difference of both equations and using the periodicity conditions which can be written as $X^{\mu}_{\rm I}(\sigma^+ + a \pi, \sigma^-) = X^{\mu}_{\rm I}(\sigma^+, \sigma^- + a \pi)$ and similarly for $X^{\mu}_{\rm II}$.
The formulae now allow us to extend each solution $X^{\mu}_{\rm I}$, $X^{\mu}_{\rm II}$ beyond their restricted domains depicted in Figure \[fig: regions I and II, R3 extend\]. Together, equations and therefore define the functions $X^{\mu}_{\rm I}$ and $X^{\mu}_{\rm II}$ completely on the whole outgoing cylinders $W_{\rm I}$ and $W_{\rm II}$. Combined with the original solution $X^{\mu}$ on the incoming cylinder, this gives a complete description of the corresponding splitting string. Let us emphasize that since the construction assumed d’Alembert’s form for both functions $X^{\mu}_{\rm I}$ and $X^{\mu}_{\rm II}$, they automatically satisfy the equations of motion , in a distributional sense, despite not being differentiable on the forward null rays through the singularity.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
![The solution in the ‘null tiles’ labelled I and II, delimited by null rays through the singularity, is obtained by extending the original solution $X^{\mu}$. The solutions $X^{\mu}_{\rm I}$ and $X^{\mu}_{\rm II}$ on subsequent tiles are given by constant translates of the solution in the regions I and II.[]{data-label="fig: regions I and II, R3 extend"}](regions_I_II_extended_R3.eps "fig:"){height="51mm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
One immediate advantage of this construction is that the qualitative description of the motion of the outgoing strings in space time is very transparent. In particular, the form of the solution clearly shows that the singular point, where the splitting occurs, propagates at the speed of light along the worldsheet of each outgoing string I and II. Along these null rays the space time profile of strings I and II exhibits cusps, but moreover, the profile away from these cusps is given by some rigid translate of a portion of the initial string.
Splitting strings on $\mathbb{R} \times S^3$ {#sec: S3}
============================================
Cauchy problem
--------------
As in the flat space case, to solve the Cauchy problem on the pair of pants $W$ we shall assume that a solution $g(\sigma, \tau)$ of the equations on the incoming cylinder $W_{\rm O}$ is given, which satisfies the self-intersection property at $\tau = 0$. This solution will specify Cauchy data for separate Cauchy problems on each outgoing cylinder $W_{\rm I}$ and $W_{\rm II}$. However, since both problems are essentially equivalent we shall focus on one of the outgoing strings, say I.
The equations of motion for the embedding field $g_{\rm I} : W_{\rm I} \to SU(2)$ can be written as
\[eom g system\] $$\label{eom g}
\partial_{\tau}^2 g_{\rm I} - \partial_{\sigma}^2 g_{\rm I} = \partial_+ \partial_- g_{\rm I} = \ha \big( \partial_+ g_{\rm I} (g_{\rm I}^{-1} \partial_- g_{\rm I}) + \partial_- g_{\rm I} (g_{\rm I}^{-1} \partial_+ g_{\rm I}) \big) \eqqcolon f(g_{\rm I}, \partial_{\sigma} g_{\rm I}, \partial_{\tau} g_{\rm I}).$$ Since the string is closed, we impose periodic boundary conditions, *i.e.* $g_{\rm I}(0, \tau) = g_{\rm I}(2 a \pi, \tau)$. Equivalently we require $g_{\rm I}(\sigma, \tau)$ to be $2 a \pi$-periodic in $\sigma$. The initial conditions read $$\label{ic g S3}
g_{\rm I}(\sigma, 0) = g(\sigma, 0),\qquad
\partial_{\tau} g_{\rm I}(\sigma, 0) = \partial_{\tau} g(\sigma, 0),$$
where $g(\sigma, \tau)$ is the given initial string solution. By assumption it satisfies the self-intersection property so that are consistent with the $2 a \pi$-periodicity of $g_{\rm I}$. Moreover, we assume the incoming string solution to be smooth so that are both smooth except at the self-intersection point $\sigma = 0 \equiv 2 a \pi$ of the initial string, where they are only continuous. Such points of reduced regularity are referred to as ‘singularities’ and the relation between singularities of a solution and singularities of the corresponding initial data goes under the name of ‘propagation of singularities’.
To solve the Cauchy problem we will proceed in three steps. First, we make use of the theory of propagation of singularities to identify the global smoothness properties of the outgoing string. It turns out that despite the nonlinearity of the equations, the singularity of the initial data propagates along null trajectories, exactly as in the flat space case where the equations were linear. We then argue that the initial solution can be trivially extended to all points which are spacelike separated from the singularity. Finally, using this information we construct the remainder of the solution in the forward light-cone of the singularity by exploiting the integrability of the equations and using the dressing method.
Propagation of the singularity {#sec: prop sing S3}
------------------------------
The propagation of singularities in nonlinear Klein-Gordon type equations of the general form $\partial_{\tau}^2 u - \partial_{\sigma}^2 u = f(x, u, \partial_{\sigma} u, \partial_{\tau} u)$ was first studied in [@Reed] and further developed in [@ReedRauch1]. It turns out that despite the presence of nonlinear terms on the right hand side, the result is exactly the same as in the linear case where $f \equiv 0$. In other words, if the initial data is $C^n$ at $(\sigma, 0)$ then the solution will be $C^n$ with respect to $\partial_-$ along the right null ray $(\sigma + \tau, \tau)$ and $C^n$ with respect to $\partial_+$ along the left null ray $(\sigma - \tau, \tau)$. In particular, the solution will be smooth at a point $(\sigma, \tau)$ if its backward null rays intersect $\tau = 0$ only at non-singular points of the initial data. Note that the pair of null rays through any point are nothing but the characteristic lines of the second order hyperbolic differential operator $\partial_{\tau}^2 - \partial_{\sigma}^2$. This conclusion remains true more generally for coupled Klein Gordon equations such as in which the nonlinear coupling terms depend only on lower order derivatives $g_{\rm I}$, $\partial_{\sigma} g_{\rm I}$, $\partial_{\tau} g_{\rm I}$.
The propagation of singularities in the light-cone components of the current $j_{\rm I} = - g_{\rm I}^{-1} dg_{\rm I}$ itself will be more relevant later so we discuss it directly. The equations of motion read
\[eom j system\] $$\label{eom jpm}
\partial_- j_{{\rm I}+} = \ha [j_{{\rm I}-}, j_{{\rm I}+}], \qquad \partial_+ j_{{\rm I}-} = \ha [j_{{\rm I}+}, j_{{\rm I}-}],$$ and the corresponding initial conditions derived from are $$\label{ic j S3}
j_{{\rm I} \pm}(\sigma, 0) = - g(\sigma, 0)^{-1} ( \partial_{\tau} g(\sigma, 0) \pm \partial_{\sigma} g(\sigma, 0) ).$$
These functions are smooth away from the self-intersection point $\sigma = 0 \equiv 2 a \pi$, but exhibit a jump discontinuity there since $g(\sigma, 0)$ is only continuous at that point.
The general semilinear hyperbolic first order system with piecewise-smooth initial data having jump discontinuities only at a discrete set of points was studied in [@ReedRauch2]. The system is of this type but has only two characteristic directions at any point, namely the left and right null rays. It follows (see [@ReedRauch2] for details) that there can be no ‘anomalous’ singularities – these are singularities which are not present in the linearized system – appearing at the intersection of two singularity bearing characteristics. Let us denote $\S_-$ and $\S_+$ the forward left and right null rays emanating from the singularity $(a \pi, 0)$, see Figure \[fig: null characteristics\] $(a)$. Then a solution of in the distributional sense must in fact be smooth in $W_{\rm I} \setminus (\S_+ \cup \S_-)$.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![$(a)$ The self-intersection point of the initial string corresponds to a singularity in the initial conditions of each outgoing string, which propagates along both left and right null rays $\S_-$ and $\S_+$. The solution is therefore smooth everywhere except on these forward null rays. The component $j_{{\rm I} \mp}$ has a jump discontinuity across $\S_{\pm}$ whereas $j_{{\rm I} \pm}$ is continuous.\ $\qquad\qquad\qquad$ ![$(a)$ The self-intersection point of the initial string corresponds to a singularity in the initial conditions of each outgoing string, which propagates along both left and right null rays $\S_-$ and $\S_+$. The solution is therefore smooth everywhere except on these forward null rays. The component $j_{{\rm I} \mp}$ has a jump discontinuity across $\S_{\pm}$ whereas $j_{{\rm I} \pm}$ is continuous.\
$(b)$ The shaded region, representing points which are causally disconnected from the singularity, is unaffected by the splitting. The solution there is simply given by the original solution extended beyond $\tau = 0$ as if the splitting never occurred.[]{data-label="fig: null characteristics"}](Null_characteristics.eps "fig:"){width="25mm"} $(b)$ The shaded region, representing points which are causally disconnected from the singularity, is unaffected by the splitting. The solution there is simply given by the original solution extended beyond $\tau = 0$ as if the splitting never occurred.[]{data-label="fig: null characteristics"}](Space-like_separation.eps "fig:"){width="25mm"}
$(a)$ $(b)$
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
We can be more precise about the value of the jump discontinuities across $\S_{\pm}$. By taking the integral of the first equation in over a vanishingly small interval in the $\sigma^-$ direction which intersects $\S_+$, we find that the discontinuity of $j_{{\rm I}+}$ across this characteristic vanishes. That is, $j_{{\rm I}+}$ is continuous in the $\sigma^-$ direction. On the other hand, its discontinuity $\Delta^- j_{{\rm I}+}$ across the left null ray $\S_-$ satisfies $\partial_- (\Delta^- j_{{\rm I}+}) = \ha [j_{{\rm I}-}, \Delta^- j_{{\rm I}+}]$. A similar reasoning applied to the second equation in shows that $j_{{\rm I}-}$ is continuous in the $\sigma^+$ direction and its jump discontinuity $\Delta^+ j_{{\rm I}-}$ across the right null ray $\S_+$ satisfies $\partial_+ (\Delta^+ j_{{\rm I}-}) = \ha [j_{{\rm I}+}, \Delta^+ j_{{\rm I}-}]$. The upshot is that the jump discontinuities of $j_{{\rm I}+}$ and $j_{{\rm I}-}$ in the initial condition at $\tau = 0$ propagate along the *left* null ray $\S_-$ and *right* null ray $\S_+$, respectively.
Absolute elsewhere of the singularity {#sec: abs else S3}
-------------------------------------
Having identified the smoothness properties of the current $j_{\rm I}$ on each outgoing cylinder, we now proceed to actually construct these solutions. In the spirit of section \[sec: flat\], we will solve the equations successively in each ‘null-tile’, delimited by the null lines $\S_{\pm}$, where the solution is known to be smooth. The first tile in contact with the Cauchy surface requires little effort.
Indeed, in any hyperbolic system, the solution of the Cauchy problem at any point $p$ only depends on that part of the Cauchy data which lies within the domain of dependence of $p$, defined as the interior of the backward characteristic cone with apex $p$. Now consider the region on the outgoing cylinder $W_{\rm I}$ consisting of all points, the domain of dependence of which does not contain the singularity. This region is delimited by the forward null rays $\S_{\pm}$ through the singularity and the Cauchy surface $\tau = 0$, see Figure \[fig: null characteristics\] $(b)$. It is then clear that the original solution remains valid within this region, since the relevant Cauchy data is the same as if the splitting had never occurred.
Integrability {#sec: integr}
-------------
Extending the solution into the region of influence of the singularity is considerably harder since it requires explicitly solving the Cauchy problem. Fortunately, the equations of the prinicpal chiral model are well known to be integrable in the sense that they can locally be rewritten in the form of a zero curvature equation. This will enable us to make use of powerful factorization methods to construct their solutions.
#### Lax connection.
Introduce the following one complex-parameter family of $\mathfrak{sl}_2(\mathbb{C})$-valued 1-forms on the worldsheet $W_{\rm I}$, depending on the single parameter $x \in \mathbb{C}P^1$, $$\label{Lax connection}
J_{\rm I}(x) \coloneqq \frac{j_{{\rm I}+}}{1 - x} \, d\sigma^+ + \frac{j_{{\rm I}-}}{1 + x} \, d\sigma^-,$$ where recall that $j_{\rm I} = j_{{\rm I}+} d\sigma^+ + j_{{\rm I}-} d\sigma^-$. It has the remarkable property of being flat, *i.e.* $$\label{flatness}
dJ_{\rm I}(x) - J_{\rm I}(x) \wedge J_{\rm I}(x) = 0,$$ if and only if the equations of motion hold. In other words, flat $\mathfrak{sl}_2(\mathbb{C})$-connections $J_{\rm I}(x)$ on $W_{\rm I}$ with simple poles at $x = \pm 1$ and a zero at $x = \infty$ are in 1 – 1 correspondence with solutions $j_{\rm I} : W_{\rm I} \rightarrow \mathfrak{sl}_2(\mathbb{C})$ of the principal chiral model equations. Specifically, the Lax connection is constructed from $j_{\rm I}$ as in and the current is recovered from $J_{\rm I}(x)$ by $$\label{j from J}
j_{\rm I} = J_{\rm I}(0).$$
#### Extended solution.
Since the $\mathfrak{sl}_2(\mathbb{C})$-connection $J_{\rm I}(x)$ is flat, it can be trivialized over any simply connected domain of $U \subset W_{\rm I}$, namely we can write $$\label{aux lin sys}
J_{\rm I}(x) = \big( d \Psi_{\rm I}(x) \big) \Psi_{\rm I}(x)^{-1},$$ where $\Psi_{\rm I}$ is uniquely determined if we require $\Psi_{\rm I}(x, \sigma_0, \tau_0) = {\bf 1}$ at some point $(\sigma_0, \tau_0) \in U$. It follows from that $(d - \operatorname{tr}J_{\rm I}(x)) \det \Psi_{\rm I}(x) = 0$ and therefore $\det\Psi_{\rm I}(x)$ is constant since $\operatorname{tr}J_{\rm I}(x) = 0$. The initial condition then implies that $\Psi_{\rm I}(x)$ takes values in $SL_2(\mathbb{C})$.
Ultimately we are interested in the group element $g_{\rm I} \in SU_2$ rather than the current $j_{\rm I}$. Comparing the definition of $j_{\rm I} = -g_{\rm I}^{-1} dg_{\rm I}$ with that of $\Psi_{\rm I}(x)$ in and using , we see that the group element can be recovered succinctly from $\Psi_{\rm I}(x)$ as $$\label{g from Psi}
g_{\rm I} = \Psi_{\rm I}(0)^{-1}.$$ For this reason $\Psi_{\rm I}(x)$ is sometimes called the extended solution.
#### Gauge transformations.
The zero-curvature equation has a large gauge redundancy since given any $\tilde{g}(x, \sigma, \tau)$, the gauge transformed Lax connection $$\label{gauge transf}
J_{\rm I}(x) \mapsto \tilde{g} J_{\rm I}(x) \tilde{g}^{-1} + (d \tilde{g}) \tilde{g}^{-1}$$ also satisfies the zero-curvature equation. For generic choices of $\tilde{g}$, however, it will no longer admit the same pole structure as and therefore can no longer be interpreted as a Lax connection of the principal chiral model. Yet, when $\tilde{g}$ is carefully chosen to preserve the pole structure of the Lax connection, provides a powerful map between solutions.
#### Reality conditions.
To obtain $\mathfrak{su}_2$-valued currents $j_{\rm I}$ and $SU_2$-valued solutions $g_{\rm I}$, one must impose reality conditions on the extended solution $\Psi_{\rm I}(x)$. Sufficient conditions are $$\label{Psi reality}
\Psi_{\rm I}(x)^{\dag} = \Psi_{\rm I}(\bar{x})^{-1},$$ which imply $g_{\rm I}^{\dag} = g_{\rm I}^{-1}$. Furthermore, the ensuing reality condition $J_{\rm I}(x)^{\dag} = - J_{\rm I}(\bar{x})$ on the Lax connection which follows from then implies $j_{\rm I}^{\dag} = - j_{\rm I}$. It will be convenient to think of real extended solutions as fixed points of the complex antilinear involution $$\label{tau inv}
\hat{\tau}(\Psi_{\rm I})(x) \coloneqq \tau \big( \Psi_{\rm I}(\bar{x}) \big),$$ where $\tau(A) \coloneqq (A^{\dag})^{-1}$ for any $A \in SL_2(\mathbb{C})$, so that $SU_2 \subset SL_2(\mathbb{C})$ is the fixed point set of $\tau$.
Absolute future of the singularity
----------------------------------
When deriving exact solutions for the outgoing strings in the flat space case we made full use of the fact that d’Alembert’s general solution to the linear wave equation is expressed as a linear superposition of two independent functions of $\sigma^+$ and $\sigma^-$. A nonlinear analogue of this statement in integrable models can be obtained using the so called dressing method (see [@Manas] for a review in the context of the principal chiral model). The rough idea is that there exists a pair of gauge transformations with parameters $\tilde{g}_{{\rm I}\pm}(x, \sigma^+, \sigma^-)$ which bring the Lax connection into canonical forms depending solely on $\sigma^{\pm}$, respectively: $$\tilde{g}_{{\rm I}\pm} J_{\rm I}(x) \tilde{g}_{{\rm I}\pm}^{-1} + (d \tilde{g}_{{\rm I}\pm}) \tilde{g}_{{\rm I}\pm}^{-1} = \frac{j^0_{{\rm I}\pm}(\sigma^{\pm})}{1 \mp x}\, d\sigma^{\pm} \eqqcolon J_{{\rm I}\pm}(x).$$ But moreover, given two such ‘right and left moving’ Lax connections $J_{{\rm I}\pm}(x)$ we can recover the original Lax connection . Specifically, we have a pair of maps $$\xymatrix{
J_{\rm I}(x, \sigma^+, \sigma^-) \ar @<2pt> @^{->} [rr] ^----{\rm undress} & &
\ar @<2pt> @^{->} [ll] ^----{\rm dress} \big( J_{{\rm I}+}(x, \sigma^+), J_{{\rm I}-}(x, \sigma^-) \big)}$$ referred to as the undressing and dressing transformations. This is effectively the nonlinear counterpart of the (linear) correspondence $X^{\mu}_{\rm I}(\sigma^+, \sigma^-) \rightleftharpoons (X^{\mu}_{{\rm I}+}(\sigma^+), X^{\mu}_{{\rm I}-}(\sigma^-))$ in flat space.
In fact, the analogy with the flat space case goes even further. Suppose we ‘normalize’ the solution $X^{\mu}_{\rm I}(\sigma^+, \sigma^-)$ by requesting that $X^{\mu}_{\rm I}(0, 0) = 0$. This amounts to performing a constant translation on the solution, which is a symmetry of the equations, and the ‘unnormalized’ solution is recovered by adding back the original value of $x^{\mu}_0 \coloneqq X^{\mu}_{\rm I}(0, 0)$. Then the functions $X^{\mu}_{{\rm I}\pm}$ may be defined simply as $X^{\mu}_{{\rm I}+}(\sigma^+) \coloneqq X^{\mu}_{\rm I}(\sigma^+, 0)$ and $X^{\mu}_{{\rm I}-}(\sigma^-) \coloneqq X^{\mu}_{\rm I}(0, \sigma^-)$. Note that they inherit the ‘normalization’ of $X^{\mu}_{\rm I}$ since $X^{\mu}_{{\rm I}\pm}(0) = 0$. The full solution may be obtained from its values on the left and right null rays through the special point $(0, 0) \in W_{\rm I}$, $$X^{\mu}_{\rm I}(\sigma^+, \sigma^-) = X^{\mu}_{{\rm I}+}(\sigma^+) + X^{\mu}_{{\rm I}-}(\sigma^-).$$ In particular, the ‘unnormalized’ solution is obtained by adding the constant $x^{\mu}_0$.
The analogous construction in the $\mathbb{R} \times S^3$ case goes as follows. Consider the flat Lax connections $J_{{\rm I}\pm}(x)$ define above but with $j^0_{{\rm I}+}(\sigma^+) \coloneqq j_{{\rm I}+}(\sigma^+, 0)$ and $j^0_{{\rm I}-}(\sigma^-) \coloneqq j_{{\rm I}-}(0, \sigma^-)$. We introduce their local trivializations $\Psi_{{\rm I}\pm}(x, \sigma^+, \sigma^-)$ as in but *normalized* such that $\Psi_{{\rm I}\pm}(x, 0, 0) = {\bf 1}$. Then it turns out that the trivialization $\Psi_{\rm I}(x)$ of the original solution $J_{\rm I}(x)$ normalized by $\Psi_{\rm I}(x, 0, 0) = {\bf 1}$ can be obtained by applying a dressing transformation $$\xymatrix{
\big( \Psi_{{\rm I}+}(x), \Psi_{{\rm I}-}(x) \big) \ar [rr] ^----{\rm dress}
& & \Psi_{\rm I}(x).}$$ As in the flat space case, the ‘unnormalized’ extended solution with $\Psi_{\rm I}(x, 0, 0) = \Psi_0$ is obtained by multiplying $\Psi_{\rm I}(x, \sigma^+, \sigma^-)$ on the right by the constant matrix $\Psi_0$, which is a symmetry of the equations .
The purpose of the next subsection is to make these statements precise. We shall use them in the following subsection to obtain a complete description of the outgoing string I.
### Dressing and undressing {#sec: (un)dress}
Given initial conditions $j^0_{{\rm I}+}(\sigma^+)$ and $j^0_{{\rm I}-}(\sigma^-)$ on the pair of characteristics $\S_{\pm}$ through $(0, 0)$ we shall reconstruct the full solution $j_{{\rm I}\pm}(\sigma^+, \sigma^-)$.
We start by defining the following pair of flat connections
\[Psi pm def\] $$\label{Lax pm}
J_{{\rm I}+}(x) = \frac{j^0_{{\rm I}+}(\sigma^+)}{1 - x} d\sigma^+, \qquad
J_{{\rm I}-}(x) = \frac{j^0_{{\rm I}-}(\sigma^-)}{1 + x} d\sigma^-.$$ The corresponding extended solutions normalized at $(0, 0)$ are denoted respectively as $\Psi_{{\rm I}\pm}(x)$, namely $$\label{Psi pm eq}
\big( d\Psi_{{\rm I}\pm}(x) \big) \Psi_{{\rm I}\pm}(x)^{-1} = J_{{\rm I}\pm}(x), \qquad \Psi_{{\rm I}\pm}(x, 0, 0) = {\bf 1}.$$
#### Birkhoff factorization.
Consider two small circles $\C^{\pm}$ around $x = \pm 1$ on the Riemann sphere $\mathbb{C}P^1$ and let $I^{\pm}$ be their interiors and $E^{\pm}$ their respective exteriors. We also introduce $\C \coloneqq \C^+ \cup \C^-$, $I \coloneqq I^+ \cup I^-$ and $E \coloneqq E^+ \cap E^-$. The pair of functions $\Psi_{{\rm I}\pm}(x)$ can be viewed as defining a single function $\C \to SL_2(\mathbb{C})$ and the set of all such smooth maps forms a group $\Loop_{\C} SL_2(\mathbb{C})$ under pointwise matrix multiplication. Consider the Birkhoff factorization problem which consists in writing $\Psi_{{\rm I}\pm}(x)$ as a product of maps in $\Loop_{\C} SL_2(\mathbb{C})$ which extend holomorphically to maps $I \to SL_2(\mathbb{C})$ and $E \to SL_2(\mathbb{C})$, respectively. Specifically, $$\label{Birkhoff dress}
\Psi_{{\rm I}\pm}(x) = \tilde{g}_{\rm I}(x)^{-1} \Psi_{\rm I}(x), \qquad \text{for} \quad x \in \C^{\pm}$$ where $\tilde{g}_{\rm I}(x)$ is holomorphic in $I$ and $\Psi_{\rm I}(x)$ is holomorphic in $E$ with $\Psi_{\rm I}(\infty) = {\bf 1}$.
The Birkhoff factorization theorem [@PressleySegal] states that there exists an open dense subset of the identity component of $\Loop_{S^1} SL_2(\mathbb{C})$, called the “big cell”, in which the factorization into loops holomorphic inside and outside the unit circle $S^1 = \{ x \in \mathbb{C} \,|\, |x| = 1 \}$ is possible. In particular, the Birkhoff factorization always exists locally, and this statement remains true also for $\Loop_{\C} SL_2(\mathbb{C})$. Since $\Psi_{{\rm I}\pm}(x, 0, 0) = {\bf 1}$ trivially factorizes into a pair of identity matrices, the existence of a solution to is therefore guaranteed for small enough $\sigma^{\pm}$. We shall come back to the question of existence after discussing reality conditions.
If it exists, however, it is easy to see that the factorization is unique. For suppose $\Psi_{{\rm I}\pm}(x) = \tilde{g}'_{\rm I}(x)^{-1} \Phi_{\rm I}(x)$ gives another factorization then $\Phi_{\rm I}(x) \Psi_{\rm I}(x)^{-1} = \tilde{g}'_{\rm I}(x) \tilde{g}_{\rm I}(x)^{-1}$, where the left and right hand sides are holomorphic in $E$ and $I$, respectively. However, since they are equal on $\C$, together they define a matrix of holomorphic functions over $\mathbb{C}P^1$ which is therefore constant. But the normalization condition at $\infty \in E$ implies $\Phi_{\rm I}(\infty) \Psi_{\rm I}(\infty)^{-1} = {\bf 1}$ so that this constant is the identity matrix and hence $\tilde{g}'_{\rm I}(x) = \tilde{g}_{\rm I}(x)$ and $\Phi_{\rm I}(x) = \Psi_{\rm I}(x)$.
The Birkhoff factorization therefore provides a (local) map $\Psi_{{\rm I}\pm}(x) \mapsto \Psi_{\rm I}(x)$. Before exploiting this map, let us show that it is invertible. Since the coefficients of the system are holomorphic in $E^{\pm}$, so are its solutions $\Psi_{{\rm I}\pm}(x)$. Furthermore, $\Psi_{{\rm I}\pm}(\infty)$ is a constant matrix which must be the identity by the initial conditions. Therefore given $\Psi_{\rm I}(x)$ one can recover $\Psi_{{\rm I}\pm}(x)$ using the ‘reverse’ Birkhoff factorization problem $$\label{Birkhoff undress}
\Psi_{\rm I}(x) = \tilde{g}_{{\rm I}\pm}(x) \Psi_{{\rm I}\pm}(x), \qquad \text{for} \quad x \in \C^{\pm}$$ where $\tilde{g}_{{\rm I}\pm}(x)$ and $\Psi_{{\rm I}\pm}(x)$ are holomorphic in $I^{\pm}$ and $E^{\pm}$, respectively, and with $\Psi_{{\rm I}\pm}(\infty) = {\bf 1}$. This is just a rewriting of , where we now consider the matrix $\Psi_{\rm I}(x)$ as given and $\Psi_{{\rm I}\pm}(x)$ as unknowns. In particular, $\tilde{g}_{{\rm I}\pm}(x)$ is the restriction of $\tilde{g}_{\rm I}(x)$ to $I^{\pm}$.
#### Gauge transformation.
Making use of the second factor in we define the following flat connection 1-form, $$\label{Lax reconstruct}
J_{\rm I}(x) \coloneqq \big( d \Psi_{\rm I}(x) \big) \Psi_{\rm I}(x)^{-1}.$$ Comparing this with using the factorization we find $$\label{Lax gauge transf pm}
J_{\rm I}(x) = \tilde{g}_{{\rm I}\pm}(x) J_{{\rm I}\pm}(x) \tilde{g}_{{\rm I}\pm}(x)^{-1} + \big( d\tilde{g}_{{\rm I}\pm}(x) \big) \tilde{g}_{{\rm I}\pm}(x)^{-1}.$$ This shows that is related by a gauge transformation to each of the Lax connections , in the sense of with parameter $\tilde{g} = \tilde{g}_{{\rm I}\pm}(x)$. As discussed in section \[sec: integr\], in order for this gauge transformation to be of any use we must show that it preserves the analytic structure of the Lax connection.
It follows from its definition that $J_{\rm I}(x)$ is holomorphic in $E$ and vanishes at $x = \infty$. Its behaviour in $I$ can be deduced from the alternative expressions . Indeed, the second term in this equation is holomorphic in $I^{\pm}$ whereas the first has a simple pole at $x = \pm 1$ with residue proportional to $d\sigma^{\pm}$. By Mittag-Leffler’s theorem this information uniquely specifies $J_{\rm I}(x)$ so we can write $$J_{\rm I}(x) = \frac{j_{{\rm I}+}}{1 - x} d \sigma^+ + \frac{j_{{\rm I}-}}{1 + x} d \sigma^-,$$ for some functions $j_{{\rm I}\pm}(\sigma^+, \sigma^-)$. Since $J_{\rm I}(x)$ is flat by definition , it follows that $j_{{\rm I}\pm}$ satisfy the equations of the principal chiral model.
#### Cauchy data.
It remains to show that the initial data of $j_{{\rm I}\pm}$ along the characteristics $\S_{\pm}$ through $(0, 0)$ coincides with $j^0_{{\rm I}+}(\sigma^+)$ and $j^0_{{\rm I}-}(\sigma^-)$. To show this, consider in light-cone coordinates, $$\big( \partial_+ \Psi_{\rm I}(x) \big) \Psi_{\rm I}(x)^{-1} = \frac{j_{{\rm I}+}}{1 - x}, \qquad \big( \partial_- \Psi_{\rm I}(x) \big) \Psi_{\rm I}(x)^{-1} = \frac{j_{{\rm I}-}}{1 + x}.$$ Setting $\sigma^- = 0$ in the first equation, we see that $\Psi_{\rm I}(x, \sigma^+, 0)$ is holomorphic in $E^+$ since the coefficient of the equation are. But then the solution of the factorization problem when $\sigma^- = 0$ is simply given by $\Psi_{{\rm I}+}(x, \sigma^+, 0) = \Psi_{\rm I}(x, \sigma^+, 0)$ and $\tilde{g}_{{\rm I}+}(x, \sigma^+, 0) = {\bf 1}$. Likewise, setting $\sigma^+ = 0$ in the second equation, we find that $\Psi_{\rm I}(x, 0, \sigma^-)$ is holomorphic in $E^-$ which in turn implies $\Psi_{{\rm I}-}(x, 0, \sigma^-) = \Psi_{\rm I}(x, 0, \sigma^-)$. In particular, this yields the desired result $$j^0_{{\rm I}+}(\sigma^+) = j_{{\rm I}+}(\sigma^+, 0), \qquad
j^0_{{\rm I}-}(\sigma^-) = j_{{\rm I}-}(\sigma^-, 0).$$
#### Reality conditions.
Since the circles $\C^{\pm}$ are centered around $x = \pm 1$ they are invariant under conjugation $x \mapsto \bar{x}$. The involution $\hat{\tau}$ therefore sends $\Loop_{\C} SL_2(\mathbb{C})$ to itself and its fixed point subset defines the twisted loop group $$\Loop_{\C}^{\hat{\tau}} SL_2(\mathbb{C}) \coloneqq \{ \Psi \in \Loop_{\C} SL_2(\mathbb{C}) \, | \, \hat{\tau}(\Psi) = \Psi \}.$$ It is straightforward to show that the Birkhoff factorization restricts to this subgroup. Indeed, suppose $(\Psi_{{\rm I}+}, \Psi_{{\rm I}-}) \in \Loop_{\C}^{\hat{\tau}} SL_2(\mathbb{C})$, then applying $\hat{\tau}$ to yields the factorization $$\Psi_{{\rm I}\pm}(x) = \tau\big( \tilde{g}_{\rm I}(\bar{x}) \big)^{-1} \tau \big( \Psi_{\rm I}(\bar{x}) \big), \qquad \text{for} \quad x \in \C^{\pm},$$ where $\tau\big( \tilde{g}_{\rm I}(\bar{x}) \big)$ and $\tau \big( \Psi_{\rm I}(\bar{x}) \big)$ are holomorphic in $I$ and $E$, respectively, with $\tau \big( \Psi_{\rm I}(\infty) \big) = {\bf 1}$. Therefore, by the uniqueness of the Birkhoff factorization it follows that $\tilde{g}_{\rm I}$ and $\Psi_{\rm I}$ are also in $\Loop_{\C}^{\hat{\tau}} SL_2(\mathbb{C})$, as claimed.
#### Existence.
We are finally in a position to address the question of existence of the factorization . The reason for postponing this issue until now is that although the Birkhoff factorization in $\Loop_{S^1} SL_2(\mathbb{C})$ is only possible on a dense open subset, it turns out [@TUhl; @Brander] that for the fixed point subgroup $\Loop^{\hat{\tau}}_{S^1} SL_2(\mathbb{C})$ with respect to a complex anti-linear involution $\hat{\tau}$ of the type , the Birkhoff factorization *always* exists. In other words, $\Loop_{S^1}^{\hat{\tau}} SL_2(\mathbb{C})$ is connected and the “big cell” in this case is the whole of $\Loop_{S^1}^{\hat{\tau}} SL_2(\mathbb{C})$ so that the Birkhoff decomposition is *global*.
This can be used to prove the desired factorization as follows. First of all, consider linear fractional transformations $f^{\pm}$, with real coefficients, mapping $S^1$ to $\C^{\pm}$ and the unit disk $\{ x \in \mathbb{C} \, | \, |x| < 1 \}$ to $I^{\pm}$. This allows us to reduce the Birkhoff facotrization of $\Loop_{\C^{\pm}}^{\hat{\tau}} SL_2(\mathbb{C})$ to that of $\Loop_{S^1}^{\hat{\tau}} SL_2(\mathbb{C})$. In other words, we can decompose any $\Psi_{\pm} \in \Loop_{\C^{\pm}}^{\hat{\tau}} SL_2(\mathbb{C})$ as a product $\Phi^I_{\pm} \Phi^E_{\pm}$ where $\Phi^I_{\pm} \in \Loop_{\C^{\pm}}^{\hat{\tau}} SL_2(\mathbb{C})$ and $\Phi^E_{\pm} \in \Loop_{\C^{\pm}}^{\hat{\tau}} SL_2(\mathbb{C})$ extend holomorphically to $I^{\pm}$ and $E^{\pm}$ respectively, with $\Phi^E_{\pm}(\infty) = {\bf 1}$.
Let $(\Phi_+, \Phi_-) \in \Loop_{\C} SL_2(\mathbb{C})$ denote the loop over $\C = \C^+ \cup \C^-$ defined by the pair of loops $\Phi_{\pm} \in \Loop_{\C^{\pm}} SL_2(\mathbb{C})$ over $\C^{\pm}$. Then the element $(\Psi_{{\rm I}+}, \Psi_{{\rm I}-}) \in \Loop_{\C} SL_2(\mathbb{C})$ can be factorized as $$\begin{aligned}
(\Psi_{{\rm I}+}, \Psi_{{\rm I}-}) &= (\Psi_{{\rm I}+}, {\bf 1}) ({\bf 1}, \Psi_{{\rm I}-}) = (\Phi^I_+, {\bf 1}) (\Phi^E_+, {\bf 1}) ({\bf 1}, \Psi_{{\rm I}-})\\
&= (\Phi^I_+, {\bf 1}) \big( {\bf 1}, \Psi_{{\rm I}-} (\Phi^E_+)^{-1} \big) (\Phi^E_+, \Phi^E_+) = (\Phi^I_+, {\bf 1}) \big( {\bf 1}, \Phi^I_- \Phi^E_- \big) (\Phi^E_+, \Phi^E_+),\\
&= \big( \Phi^I_+ (\Phi^E_-)^{-1}, \Phi^I_- \big) (\Phi^E_- \Phi^E_+, \Phi^E_- \Phi^E_+),\end{aligned}$$ where in the first line we have introduced the factorization $\Psi_{{\rm I}+} = \Phi^I_+ \Phi^E_+$ in $\Loop^{\hat{\tau}}_{\C^+} SL_2(\mathbb{C})$, and in the second line the factorization $\Psi_{{\rm I}-} (\Phi^E_+)^{-1} = \Phi^I_- \Phi^E_-$ in $\Loop^{\hat{\tau}}_{\C^-} SL_2(\mathbb{C})$. The last line then gives the desired factorization since $\tilde{g}_{{\rm I}+} \coloneqq \Phi^I_+ (\Phi^E_-)^{-1}$, $\tilde{g}_{{\rm I}-} \coloneqq \Phi^I_-$ and $\Psi_{\rm I} \coloneqq \Phi^E_- \Phi^E_+$ are holomorphic in $I^+$, $I^-$ and $E$, respectively, with $\Psi_{\rm I}(\infty) = {\bf 1}$.
### Dressing the outgoing strings
Putting together the results of this section we obtain a recursive algorithm for constructing the outgoing string solution I, one null-tile at a time. Since the tiles are naturally ordered we label them by integers, the $0^{\rm th}$ tile being the (half) tile introduced in section \[sec: abs else S3\] and the $i^{\rm th}$ tile ($i \geq 1$) is defined by its lowest point being at (see Figure \[fig: inductive step\]$(a)$) $$p_i \coloneqq (\sigma^+_i, \sigma^-_i) = \left\{ \begin{array}{ll} ( k \, a \pi, (k-1) a \pi ) & \text{for} \;\; i = 2 k,\\
( k \, a \pi, k \, a \pi ) & \text{for} \;\; i = 2 k + 1. \end{array}
\right.$$ The outgoing string can now be constructed recursively as follows.
------- ---------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\qquad\qquad\qquad$ ![$(a)$ We enumerate the different tiles in the tessellation created by the null rays $\S_{\pm}$ emanating from the singularity. In particular, the $0^{\rm th}$ tile contains the Cauchy surface.\ $\qquad\qquad\qquad$ ![$(a)$ We enumerate the different tiles in the tessellation created by the null rays $\S_{\pm}$ emanating from the singularity. In particular, the $0^{\rm th}$ tile contains the Cauchy surface.\
$(b)$ The shaded region represents the part of the solution already determined. For illustration purposes, we have cut the cylinder in such a way that the $i^{\rm th}$ tile appears whole.\ $(b)$ The shaded region represents the part of the solution already determined. For illustration purposes, we have cut the cylinder in such a way that the $i^{\rm th}$ tile appears whole.\
$(c)$ The inductive step consists in solving the Cauchy problem on the $i^{\rm th}$ tile, taking for Cauchy data along $\S_{\pm}$ the value of $j^{(i-1)}_{{\rm I}\pm}$ on the boundary of the $(i-1)^{\rm st}$ tile.[]{data-label="fig: inductive step"}](inductive_step1.eps "fig:"){width="25mm"} $(c)$ The inductive step consists in solving the Cauchy problem on the $i^{\rm th}$ tile, taking for Cauchy data along $\S_{\pm}$ the value of $j^{(i-1)}_{{\rm I}\pm}$ on the boundary of the $(i-1)^{\rm st}$ tile.[]{data-label="fig: inductive step"}](inductive_step2.eps "fig:"){width="25mm"}
$(a)$ $(b)$ $(c)$
------- ---------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
#### Initial step.
From section \[sec: abs else S3\] we know that the solution on the $0^{\rm th}$ tile is simply given by extending the original solution of the incoming string, see Figure \[fig: null characteristics\]$(b)$.
#### Inductive step.
Now given the solution on the $(i-1)^{\rm st}$ tile, the solution on the $i^{\rm th}$ tile can be obtained as follows, see Figure \[fig: inductive step\]$(b)$-$(c)$. Its Cauchy data consists of the values of $j_{{\rm I}\pm}$ along the two null rays $\S_{\pm}$ connecting it to the $(i-1)^{\rm st}$ tile. Yet by section \[sec: prop sing S3\] we know that the components $j_{{\rm I}\pm}$ are continuous across the singular null rays $\S_{\pm}$, respectively. Therefore the Cauchy data for the $i^{\rm th}$ tile is completely specified by the solution on the $(i-1)^{\rm st}$ tile as $$\label{ith tile ic}
j^0_{{\rm I}+}(\sigma^+) = j^{(i-1)}_{{\rm I}+}(\sigma^+, \sigma^-_i), \qquad
j^0_{{\rm I}-}(\sigma^-) = j^{(i-1)}_{{\rm I}-}(\sigma^+_i, \sigma^-).$$ The solution of the corresponding Cauchy problem is now obtained by applying the dressing transformation of section \[sec: (un)dress\] with the point $(0, 0)$ there replaced by $(\sigma^+_i, \sigma^-_i)$.
The first step requires solving the system with $j^0_{{\rm I}\pm}(\sigma^{\pm})$ given by . However, it is easy to see that the solution can be expressed in terms of the extended solution on the previous $(i-1)^{\rm st}$ tile, since this satisfies the same equations but with a different normalization. Specifically, we have
\[recursive formula\] $$\label{recursive formula 1}
\begin{split}
\Psi^{(i)}_{{\rm I}+}(x, \sigma^+) &= \Psi^{(i-1)}_{\rm I}(x, \sigma^+, \sigma^-_i) \Psi^{(i-1)}_{\rm I}(x, \sigma^+_i, \sigma^-_i)^{-1},\\
\Psi^{(i)}_{{\rm I}-}(x, \sigma^-) &= \Psi^{(i-1)}_{\rm I}(x, \sigma^+_i, \sigma^-) \Psi^{(i-1)}_{\rm I}(x, \sigma^+_i, \sigma^-_i)^{-1}.
\end{split}$$ Next we perform the Birkhoff factorization of $\Psi^{(i)}_{{\rm I}\pm}(x, \sigma^{\pm})$, namely $$\Psi^{(i)}_{{\rm I}\pm}(x, \sigma^{\pm}) = \tilde{g}^{(i)}_{\rm I}(x, \sigma^+, \sigma^-)^{-1} \Psi^{(i)}_{{\rm I} \, n}(x, \sigma^+, \sigma^-), \qquad \text{for} \quad x \in \C^{\pm}.$$ The second factor on the right defines the ‘normalized’ extended solution on the $i^{\rm th}$ tile, see Figure \[fig: ith tile\]. Finally, the ‘unnormalized’ extended solution is now given by $$\label{recursive formula2}
\Psi^{(i)}_{\rm I}(x, \sigma^+, \sigma^-) = \Psi^{(i)}_{{\rm I} \, n}(x, \sigma^+, \sigma^-) \Psi^{(i-1)}_{\rm I}(x, \sigma^+_i, \sigma^-_i).$$
Equations provide the desired recursive formula expressing the extended solution on the $i^{\rm th}$ tile in terms of that on the $(i-1)^{\rm st}$ tile through the use of a Birkhoff factorization.
![The normalized extended solution $\Psi^{(i)}_{{\rm I} \, n}(x, \sigma^+, \sigma^-)$ at the point $(\sigma^+, \sigma^-)$ in the $i^{\rm th}$ tile is obtained from the Birkhoff factorization of the pair $\Psi^{(i)}_{{\rm I}\pm}(x, \sigma^\pm)$ defined at the boundary points $(\sigma^+, \sigma^-_i)$ and $(\sigma^+_i, \sigma^-)$, respectively.[]{data-label="fig: ith tile"}](ith_tile.eps){width="35mm"}
Conclusions and Outlook
=======================
The classical integrability of the superstring $\sigma$-model on $AdS_5 \times S^5$ has so far played a vital role in the classification of its finite-gap solutions [@Classification] as well as their reconstruction [@Reconstruction] in the subsector $\mathbb{R} \times S^3$. In this article we made a first step beyond solutions with cylindrical worldsheet by constructing the general splitting solution in $\mathbb{R} \times S^3$. Although the worldsheets of these new solutions have the topology of a pair of pants, the integrability of the $\sigma$-model also played an essential role in their construction. This is no surprise since after all the Lax connection is a local object on the worldsheet.
Specifically, given any string solution with cylindrical worldsheet on $\mathbb{R} \times S^3$, which satisfies the self-intersection property at some instant in time, we constructed the pair of outgoing strings resulting from the split. This was achieved by reducing the problem to factorization in a loop group, as is usual in classical integrable systems. It would be important to investigate further the possibility of solving these Birkhoff factorization problems more explicitly, for instance in terms of Riemann $\theta$-functions.
An example of initial string could be a finite-gap string, the moduli of which are encoded in a finite-genus algebraic curve. In fact, since the outgoing strings are uniquely determined by their Cauchy data which in turn is given by the incoming string, the entire splitting solution is uniquely characterized by the same algebraic curve as the initial string. The difference between these two solutions will show up in the behaviour of the angle variables, encoded in the algebro-geometric language as a divisor on the curve [@Dorey:2006mx], at the moment of the splitting. It would be interesting, though, to have a more algebraic characterization of the self-intersection property at the level of the curve and the divisor.
This brings up the curious observation that for a given initial string there can more than one possible evolution, depending on whether or not we choose the string to split at $\tau = 0$. This existence of multiple different solutions for the same set of Cauchy data at $\tau < 0$ is merely a consequence of the fact that the topology of the worldsheet is not determined by the dynamics but rather fixed by hand from the outset. Another way to phrase this is to note that since the metric is not dynamical, it must be fixed prior to solving the equations. Its conformal class then reflects the underlying topology of the worldsheet. For instance, on the cylinder the metric can be made globally flat, whereas on the pair of pants it must be degenerate at the singular point.
Throughout our construction we have assumed the initial string to be smooth at $\tau = 0$. Since the pair of outgoing strings are not smooth along the null rays through the splitting point, it is therefore not immediate how to describe their potential further splitting. This would first require a slight generalization of the construction to include initial solutions with a discrete set of singularities propagating along null lines. It would also be interesting to study the joining of two classical strings in a similar fashion, as well as classical solutions exhibiting more general worldsheet topology. This is a novel possibility on curved backgrounds such as $\mathbb{R} \times S^3$ since the products of a split can eventually meet again and join.
We emphasize that splitting strings are solutions of an initial value problem for a system of hyperbolic (Lorentzian) differential equations. Such solutions, which describe a complicated splitting process in $S^3 \subset S^5$, should therefore be relevant for the semiclassical computation of 3-point functions in the *Minkowskian* approach of [@Janik:2010gc]. By comparison, the problem of constructing minimal surfaces in *Euclidean* AdS ending at certain points on the boundary is a very different one. It can *a priori* be phrased as a boundary value problem for a system of elliptic (Euclidean) differential equations. However, the classical minimal surface dominating a 3-point correlation function at strong coupling should also contain extra information, on each leg, about the type of operator inserted at the boundary.
Another promising approach for computing strong coupling 3-point functions directly within the Euclidean formalism is the vertex operator approach [@Tseytlin:2003ac]. The main obstacle in this direction is the construction of vertex operators corresponding to finite-gap solutions. The insertion of three such operators in the path integral should produce the correct boundary conditions for the minimal surface mentioned above. Because finite-gap solutions have Lorentzian worldsheets while minimal surfaces have Euclidean signature, one would naively expect the vertex operator to create a Euclidean continuation of the finite-gap solution. Such a relation is currently only understood for 2-point functions [@Buchbinder:2010vw].
Finally, our construction should have a natural generalization to $AdS_5 \times S^5$ superstrings or more generally to $\mathbb{Z}_4$-graded supercoset $\sigma$-models [@Zarembo:2010sg]. Indeed, the loop group factorization discussed here is the global counterpart of the loop algebra decomposition discussed in [@Vicedo:2010qd].
Acknowledgements {#acknowledgements .unnumbered}
================
This work was motivated by a talk of R. Janik presented at the workshop “From Sigma Models to Four-dimensional QFT” at DESY, Hamburg. I would like to thank Y. Aisaka, T. W. Brown, N. Dorey, R. Janik, M. C. Reed, J. Teschner, K. P. Tod, A. A. Tseytlin and K. Zarembo for interesting discussions.
[99]{}
N. Beisert [*et al.*]{}, “Review of AdS/CFT Integrability: An Overview,” arXiv:1012.3982 \[hep-th\].
N. Gromov, V. Kazakov and P. Vieira, “Exact Spectrum of Anomalous Dimensions of Planar N=4 Supersymmetric Yang-Mills Theory,” Phys. Rev. Lett. [**103**]{} (2009) 131601 \[arXiv:0901.3753 \[hep-th\]\] $\bullet$ N. Gromov, V. Kazakov, A. Kozak and P. Vieira, “Exact Spectrum of Anomalous Dimensions of Planar N = 4 Supersymmetric Yang-Mills Theory: TBA and excited states,” Lett. Math. Phys. [**91**]{} (2010) 265 \[arXiv:0902.4458 \[hep-th\]\].
D. Bombardelli, D. Fioravanti and R. Tateo, “Thermodynamic Bethe Ansatz for planar AdS/CFT: A Proposal,” J. Phys. A [**42**]{} (2009) 375401 \[arXiv:0902.3930 \[hep-th\]\].
G. Arutyunov and S. Frolov, “String hypothesis for the AdS(5) x S\*\*5 mirror,” JHEP [**0903**]{} (2009) 152 \[arXiv:0901.1417 \[hep-th\]\] $\bullet$ G. Arutyunov and S. Frolov, “Thermodynamic Bethe Ansatz for the AdS(5) x S(5) Mirror Model,” JHEP [**0905**]{} (2009) 068 \[arXiv:0903.0141 \[hep-th\]\].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, “Tailoring Three-Point Functions and Integrability,” arXiv:1012.2475 \[hep-th\].
K. Okuyama and L. S. Tseng, “Three-point functions in N = 4 SYM theory at one-loop,” JHEP [**0408**]{} (2004) 055 \[arXiv:hep-th/0404190\] $\bullet$ R. Roiban and A. Volovich, “Yang-Mills correlation functions from integrable spin chains,” JHEP [**0409**]{} (2004) 032 \[arXiv:hep-th/0407140\] $\bullet$ L. F. Alday, J. R. David, E. Gava and K. S. Narain, “Structure constants of planar N = 4 Yang Mills at one loop,” JHEP [**0509**]{} (2005) 070 \[arXiv:hep-th/0502186\].
K. Zarembo, “Holographic three-point functions of semiclassical states,” JHEP [**1009**]{} (2010) 030 \[arXiv:1008.1059 \[hep-th\]\] $\bullet$ M. S. Costa, R. Monteiro, J. E. Santos and D. Zoakos, “On three-point correlation functions in the gauge/gravity duality,” JHEP [**1011**]{} (2010) 141 \[arXiv:1008.1070 \[hep-th\]\] $\bullet$ R. Roiban and A. A. Tseytlin, “On semiclassical computation of 3-point functions of closed string vertex operators in $AdS_5 \times S^5$,” Phys. Rev. D [**82**]{} (2010) 106011 \[arXiv:1008.4921 \[hep-th\]\] $\bullet$ R. Hernandez, “Three-point correlation functions from semiclassical circular strings,” J. Phys. A [**44**]{} (2011) 085403 \[arXiv:1011.0408 \[hep-th\]\] $\bullet$ J. G. Russo and A. A. Tseytlin, “Large spin expansion of semiclassical 3-point correlators in $AdS_5 \times S^5$,” JHEP [**1102**]{} (2011) 029 \[arXiv:1012.2760 \[hep-th\]\] $\bullet$ L. F. Alday and A. A. Tseytlin, “On strong-coupling correlation functions of circular Wilson loops and local operators,” arXiv:1105.1537 \[hep-th\] $\bullet$ C. Ahn and P. Bozhilov, “Three-point Correlation functions of Giant magnons with finite size,” arXiv:1105.3084 \[hep-th\].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, “Tailoring Three-Point Functions and Integrability II. Weak/strong coupling match,” arXiv:1104.5501 \[hep-th\].
R. A. Janik, P. Surowka and A. Wereszczynski, “On correlation functions of operators dual to classical spinning string states,” JHEP [**1005**]{} (2010) 030 \[arXiv:1002.4613 \[hep-th\]\].
T. Deck, “Classical string dynamics with nontrivial topology,” J. Geom. Phys. [**16**]{} (1995) 1 $\bullet$ H. J. de Vega, J. Ramirez Mittelbrunn, M. Ramon Medrano and N. G. Sanchez, “Classical splitting of fundamental strings,” Phys. Rev. D [**52**]{} (1995) 4609 \[arXiv:hep-th/9502049\] $\bullet$ R. Iengo and J. G. Russo, “Semiclassical decay of strings with maximum angular momentum,” JHEP [**0303**]{} (2003) 030 \[arXiv:hep-th/0301109\] $\bullet$ R. Iengo and J. Russo, “Black hole formation from collisions of cosmic fundamental strings,” JHEP [**0608**]{} (2006) 079 \[arXiv:hep-th/0606110\].
K. Peeters, J. Plefka and M. Zamaklar, “Splitting spinning strings in AdS/CFT,” JHEP [**0411**]{} (2004) 054 \[arXiv:hep-th/0410275\] $\bullet$ P. Y. Casteill, R. A. Janik, A. Jarosz and C. Kristjansen, “Quasilocality of joining/splitting strings from coherent states,” JHEP [**0712**]{} (2007) 069 \[arXiv:0710.4166 \[hep-th\]\] $\bullet$ E. M. Murchikova, “Splitting of folded strings in AdS$\null_3$,” arXiv:1104.4804 \[hep-th\].
S. Frolov and A. A. Tseytlin, “Rotating string solutions: AdS / CFT duality in nonsupersymmetric sectors,” Phys. Lett. B [**570**]{} (2003) 96 \[arXiv:hep-th/0306143\].
S. B. Giddings and S. A. Wolpert, “A Triangulation of Moduli Space from Light-Cone String Theory,” Commun. Math. Phys. [**109**]{} (1987) 177.
C. H. Liu, “Lorentz surfaces and Lorentzian CFT - with an appendix on quantization of string phase space,” arXiv:hep-th/9603198.
M. C. Reed, “Singularities in non-linear waves of Klein-Gordon type,” Springer Lecture Notes in Mathematics, vol. 648, 145-161 (1978) $\bullet$ M. C. Reed, “Propagation of singularities for non-linear wave equations in one dimension,” Comm. in Partial Differential Equations, 3(2), 153-199 (1978).
J. Rauch and M. C. Reed, “Propagation of singularities for semilinear hyperbolic equations in one space variable,” Annals of Mathematics, 111, 531-522 (1980).
J. Rauch and M. C. Reed, “Jump discontinuities of semilinear, strictly hyperbolic systems in two variables: creation and propagation,” Commun. Math. Phys. 81, 203-227 (1981).
M. Mañas, [*The principal chiral model as an integrable system*]{}, Friedrich Vieweg & Sohn Verlag.
A. Pressley and G. Segal, [*Loop groups*]{}, Oxford Math. Monographs, Clarendon Press, Oxford, 1986.
C-L. Terng, K. Uhlenbeck, “Bäcklund Transformations and Loop Group Actions”, Comm. Pure Appl. Math. [**53**]{}, 1 (2000) 1-75, \[arXiv:math/9805074\].
D. Brander, “Loop group decompositions in almost split real forms and applications to soliton theory and geometry,” J. Geom. Phys. [**58**]{} (2008) 1792-1800, \[arXiv:0805.1979\].
V. A. Kazakov, A. Marshakov, J. A. Minahan and K. Zarembo, “Classical/quantum integrability in AdS/CFT,” JHEP [**0405**]{} (2004) 024 \[arXiv:hep-th/0402207\] $\bullet$ N. Beisert, V. A. Kazakov, K. Sakai and K. Zarembo, “The Algebraic curve of classical superstrings on AdS(5) x S\*\*5,” Commun. Math. Phys. [**263**]{} (2006) 659 \[arXiv:hep-th/0502226\].
N. Dorey and B. Vicedo, “On the dynamics of finite-gap solutions in classical string theory,” JHEP [**0607**]{} (2006) 014 \[arXiv:hep-th/0601194\]. $\bullet$ B. Vicedo, “The method of finite-gap integration in classical and semi-classical string theory,” J. Phys. A [**44**]{} (2011) 124002.
A. A. Tseytlin, “On semiclassical approximation and spinning string vertex operators in AdS(5) x S\*\*5,” Nucl. Phys. B [**664**]{} (2003) 247 \[arXiv:hep-th/0304139\].
E. I. Buchbinder and A. A. Tseytlin, “On semiclassical approximation for correlators of closed string vertex operators in AdS/CFT,” JHEP [**1008**]{} (2010) 057 \[arXiv:1005.4516 \[hep-th\]\].
N. Dorey and B. Vicedo, “A Symplectic Structure for String Theory on Integrable Backgrounds,” JHEP [**0703**]{} (2007) 045 \[arXiv:hep-th/0606287\].
K. Zarembo, “Strings on Semisymmetric Superspaces,” JHEP [**1005**]{} (2010) 002 \[arXiv:1003.0465 \[hep-th\]\].
B. Vicedo, “The classical R-matrix of AdS/CFT and its Lie dialgebra structure,” Lett. Math. Phys. [**95**]{} (2011) 249 \[arXiv:1003.1192 \[hep-th\]\].
|
---
abstract: 'Using basic topology and linear algebra, we define a plethora of invariants of boundary links whose values are power series with noncommuting variables. These turn out to be useful and elementary reformulations of an invariant originally defined by M. Farber [@Fa2].'
address:
- |
School of Mathematics\
Georgia Institute of Technology\
Atlanta, GA 30332-0160, USA.
- |
Department of Mathematics\
Brandeis University\
Waltham, MA 02454-9110, USA.
author:
- Stavros Garoufalidis
- Jerome Levine
date: ' This edition: October 19, 2000. First edition: October 17, 2000.'
title: Analytic invariants of boundary links
---
[^1]
Introduction
============
History and Purpose.
--------------------
In a series of papers, M. Farber used homological methods to introduce an invariant of boundary links with values in a ring of rational functions with noncommuting variables [@Fa2]. A similar invariant to that of Farber was recently introduced by V. Retakh, C. Reutenauer and A. Vaintrob [@RRV] based on the notion of quasideterminants.
The purpose of this paper is to give an interpretation of Farber’s invariant as a simple invariant of the Seifert matrix of a boundary link, which is more elementary and makes calculation more straightforward. From this point of view we will, in fact, define a whole spectrum of invariants which take values in non-commutative power series rings. Although these invariants all turn out to be determined by Farber’s—see Theorem \[th.chi\]—it is useful to have the different formulations. An example is given by $\chiD$—see page —which has direct application to the study of the Kontsevich integral of a boundary link and its rationality properties, as will be explained in subsequent publications [@Ga; @GK]. $\chiD$ also gives a natural way to see that Farber’s invariant determines the natural analog of the Alexander polynomial for a boundary link (the classical Alexander polynomial of a boundary link is $0$). See Proposition \[prop.2\].
We would like to thank Michael Farber for useful discussions.
Boundary links and their refinements
------------------------------------
All manifolds will be oriented and all maps will be smooth and orientation preserving. A [*boundary link*]{} ($\bd$-link) in a 3-manifold is an oriented link which is the boundary of a disjoint union of connected surfaces, each with one boundary component. A choice of such surfaces is called a [*Seifert surface*]{} of the boundary link. It is well-known that in the case of boundary links (unlike the case of knots) the cobordism class, relative boundary, of a Seifert surface for a given link is not unique. There are at least two ways to overcome this difficulty, as was explained by Cappell-Shaneson [@CS] and Ko [@K2]:
- A [*$\S$-boundary link*]{} $L$ (or simply, a [*$\S$-link*]{}) in a 3-manifold $M$ is a choice, up to isotopy, of Seifert surface $\Sigma$ in $S^3$ such that $\partial \Sigma=L$ .
- An [*$F$-boundary link*]{} $L$ of $n$ components (or, simply, an [*$F$-link*]{}) is a link, up to isotopy, equipped with a map $\phi:
\pi_1(M \sminus L)
\to F$ where $F$ is the free group on $n$ letters and $\phi$ maps a choice of meridians of $L$ to a basis of $F$. $\phi$ is called a [*splitting map*]{} for $L$.
It turns out that $F$-links can be identified with the set of cobordism classes, rel boundary, (or [*tube equivalence*]{} classes) of Seifert surfaces— see Gutierrez and Smythe [@Gu; @Sm]. Let $A_n$ denote the group of automorphisms $\a$ of the free group $F(t_1,\dots,t_n)$ that satisfy $\a(t_i)=w_i t_i w_i^{-1}$ for some $w_i \in
F(t_1,\dots,t_n)$, for all $i$, [@CS; @K2]. $A_n$ acts on the set of $F$-links by composition with the splitting map $\phi$. In [@K2] a simple set of generators for $A_n$ was given, and the action of these generators was described geometrically as what was there called [*cocooning*]{}. It turns out that the set of equivalences classes of $F$-links, modulo the $A_n$ action, can be identified with the set of $\partial$-links.
We denote by $X_L^{\w}$ the $F$-covering of $S^3 -L$ associated with ${\operatorname{Ker}}\phi =\pi_{\w}$, the intersection of the lower central series of $\pi
=\pi_1 (S^3 -L)$.
Seifert matrices of boundary links
----------------------------------
There is an algebraic notion of a [*Seifert matrix*]{} associated to a $\S$-link of $n$ components, [@K1; @K2]. These matrices are partitioned into $n\times n$ blocks of matrices, corresponding to the link components. Let $\Sei(n)$ denote the set of matrices $A=(A_{ij})$ of square matrix blocks $A_{ij}$ for $i,j=1\dots,n$, with integer entries, satisfying the conditions $$A_{ij}'=A_{ji} \,\, \text{for $i \neq j$ and} \,\, \det(A_{ii}-A_{ii}')=1
\,\, \text{for all $i$}.$$ Let $\Sei$ denote the set of all Seifert matrices. The Seifert matrix associated to a $\S$-link (resp. $F$-link, $\partial$-link) is an element of $\Sei(n)$, well-defined up to $S_1$-equivalence (resp. $S_{12}$-equivalence, $S_{123}$ equivalence), where $S_1$ stands for [*congruence*]{}, $S_2$ stands for [*stabilization*]{} and $S_3$ stands for equivalence under an algebraic action of $A_n$ on $\Sei(n)$ defined by Ko [@K1; @K2] (see Section \[sub.all\] below). Note that $S_{123}=S_{12}$ for $n\le 2$, since $A_n$ consists entirely of inner automorphisms which act trivially on Seifert matrices. We have a commutative diagram
$$\divide\dgARROWLENGTH by2
\begin{diagram}
\node{\S-\text{links}}\arrow{e}\arrow{s}\node{\Sei(n)/(S_1)}\arrow{s} \\
\node{F-\text{links}}\arrow{e}\arrow{s}\node{\Sei(n)/(S_{12})}\arrow{s} \\
\node{\partial-\text{links}}\arrow{e}\node{\Sei(n)/(S_{123})}
\end{diagram}$$
Set $\La=\BQ[F]$ the group-ring with rational coefficients and $\Lhat$ its completion with respect to powers of the augmentation ideal. Then $A_L =H_1
(X_L^{\w},\BQ )$ is a $\La$-module. Let $\Laab
=\BQ[H]$, where $H$ is the free abelian group on generators $(t_1,\dots,t_n)$. If $X_L^{\ab}$ denotes the universal abelian covering of $S^3 -L$, then $H_1
(X_L^{\ab},\BQ )$ is a $\Laab$-module. Note that $\Lhat$ can be identified with the power series ring in the $n$ noncommuting variables $x_i =t_i -1$ and $\Lhatab$ with the power series ring in $n$ commuting variables $x_i =t_i
-1$. $\La$ (and also, $\Lhat, \Laab, \Lhatab$) are rings with (anti)-involution given by $g\to \bar g=g^{-1}$ for $g \in F$. Note that $\bar x_i=-(x_i+1)^{-1}
x_i$. The action of $A_n$ on $F$ extends naturally to $\La$ and $\Lhat$ and induces the trivial action on $\Laab$. Now, we can introduce analytic invariants of the set $\Sei$: Let $f \in \BQ\la\la x,z \ra\ra$ be a noncommutative power series in two variables. We will say $f$ is [*admissible*]{} if, for any non-negative integer $n$, there are only a finite number of terms in $f$ of total $x$-degree $n$. The admissible power series form a subring $\qad$ of $\BQ\la\la x,z
\ra\ra$. Now let $X=\text{diag}(x_1,\dots,x_n)$ be a (block) diagonal matrix. Then we let $$\chi_f: \Sei(n) \to \Lhat
\text{ be defined by } \,\,\, \chi_f(A)={\operatorname{tr}}(f(X,Z_A)-f(X,I_{1/2}))$$ where $Z_A=A(A-A')^{-1}$ and $I_{1/2}$ is the block diagonal matrix in which half of the diagonal entries in each diagonal block are $0$ and half are $1$. Note that $f(X,I_{1/2}))$ is independent of how the $0$’s and $1$’s are distributed
For all admissible $f$, $\chi_f$ descends to a map $$\Sei(n)/(S_{12})\to \Lhat.$$
If $f\in\BZ\la\la x,z\ra\ra$ then $\chi_f (A)$ has integer coefficients.
If $\a\in A_n$, $f\in\qad$ and $A$ a Seifert matrix, is $\chi_f (\a\cdot A)$ determined by $\chi_f (A)$ and $\a$?
Let $\R(a_1,\dots,a_n)$ denote the subring of $\BQ\la\la a_1,\dots,a_n\ra\ra$ consisting of the [*rational functions*]{} in the noncommuting variables $\{a_1,\dots,a_n\}$ (see [@B]). This can be defined as the smallest subring of $\BQ\la\la a_1,\dots,a_n\ra\ra$ containing the polynomials $\BQ [a_1,\dots,a_n ]$ and closed under the operation of taking inverses of [*special*]{} series, i.e. those $f$ with constant term $f(0,\dots
,0)=1$[@B p.6]. Let $\R_* (x,z)$ denote the smallest subring of $\R(x,z)$ containing the polynomials and closed under the operation of taking inverses of [*extra-special*]{} series, i.e. admissible $f(x,z)$ which satisfy $f(0,z)=1$. Clearly $\R_* (x,z)\sub\R (x,z)$. We also note that $\R_* (x,z)\sub\qad$ since it is not hard to see that if $f$ is special then $f\i$ is admissible if and only $f$ is extra-special.
Is $\R_* (x,z)=\R (x,z)\cap\qad$?
If $f(x,z)\in\R_* (x,z)$, then $\chi_f\in\R (x_1 ,\dots ,x_n )$.
$\chi_f$ satisfies a general duality property. Define anti-involutions $f\to\ti
f$ and $f\to\bar f$ on $\BQ\la\la x,z \ra\ra$ and $\BQ\la\la
x_1 ,\dots ,x_n \ra\ra$ by the properties $\ti x=x, \bar x=-x(1+x)\i , \ti z=\bar z =z, \ti x_i =x_i ,\bar x_i =-x_i
(1+x_i )\i $ and $\widetilde{fg}=\ti g\ti f ,\overline{fg}=\bar g\bar f$ , and an involution $f\to\hat f$ on $\BQ\la\la
x,z \ra\ra$ and $\BQ\la\la x_1 ,\dots ,x_n \ra\ra$, by $\hat x=\bar
x, \hat z=z ,\hat x_i =\bar x_i$ and $\widehat{fg}=\hat f\hat g$. Note that the composition of any two of the maps $f \to \tilde{f}, \bar f$ or $\hat{f}$ on $\BQ\la\la x_1 ,\dots ,x_n \ra\ra$ is equal to the third.
For admissible $f$, we have that $\ti\chi_{f(x,z)}=\chi_{\ti f(x,1-z)}$ and $\chi_{\hat f}=\hat\chi_f$. Therefore we also have $\bar\chi_{f(x,z)}=\chi_{\bar f(x,1-z)}$.
Note that if $f$ is admissible then
- $\ti f, \hat f$ are admissible, and
- $f(x,1-z)$ is defined (which is not true for every $f\in\BQ\la\la x,z \ra\ra$) and admissible.
Let $f_{\D}=\log\left(xz+1\right)
\in \BQ\la\la x,z\ra\ra$. Note that $f_{\D}$ is admissible—in fact any $f\in \BQ\la\la x,z\ra\ra$ of the form $f(x,z)=G(xz)$, where $G(y)\in\BQ\la\la
y\ra\ra$, is admissible. Let us denote $\chi_{f_{\D}}$ by $\chiD$. Our interest in $\chiD$ comes from the fact that it can be identified with the “wheels part” of a (version of) the Kontsevich integral of $F$-links, as will be explained in a separate publication, [@GK]. For now, let us explain the relation between $\chiD$ and the algebraic topology of the complement of a boundary link $L$.
Let $\D^b(L) \in \Laab/(\mathrm{units})$, denote the order of the torsion $\Laab$-module $A^{\ab}_L=\text{torsion}_{\Laab}H_1(X_L^{\ab}, \BQ)$.
It is well-known that $\D^b =\D^b
(L)$ satisfies:
1. $\D^b (1,\dots,1)=\pm 1$ and
2. $\D^b (t\i _1 ,\dots,t\i _n )$ is a unit multiple of $\D^b (t_1,\dots,t_n )$ in $\Laab$.
It follows that we can choose a unique (normalized) representative in $\Laab$ such that
1. $\D^b (1,\dots,1)=1$ and
2. $\D^b (t\i _1 ,\dots,t\i _n )=\D^b (t_1,\dots,t_n )$.
We call this normalized representative the [*torsion polynomial*]{} of $L$.
- (Abelianization) If $\chiD^{\ab}$ denotes the abelianization of $\chiD$, then $$\chiD^{\ab}=\log \D^b \in \Lhatab.$$ where $\D^b$ is the torsion polynomial.
- (Realization) For every element $\l\in\La$ with integer coefficients satisfying $\l
(1,\dots,1)=1$ and $\l=\bar \l$, there exists an $F$-link $L$ with $H_1 (X^{\omega}_L ,\BZ)\iso\La /(\l )$, where $(\l)$ denotes the left ideal generated by $\l$. As a consequence every element $\D$ of $\Laab$ satisfying (1’) and (2’) can be realized as the torsion polynomial of some boundary link.
- (Duality) $ \chiD= \bar{\chi}_{\Delta}$ in $\Lhat/(\cyclic)$, the quotient of $\Lhat$ by its [*subgroup*]{} generated by $(ab-ba)$, for $a,b \in \Lhat$.
Thus, $\chiD^{\ab}$ determines the torsion polynomial. In contrast, the classical multivariable Alexander polynomial of a boundary link vanishes, and in general it is not known which Laurent polynomials can be realized as the multivariable Alexander polynomials of a link.
For an $F$-link $L$, we can think of $\chiD$ as an analogue of the order of the $\La$-module $A_L$ (even though the notion of order does not make sense for $\La$-modules).
If $\Phi (x,z)=(xz+1)^{-1}x \in \BQ \la \la x,z \ra \ra$, then $\chi_{\Phi}$ is related to Farber’s $\chi$-function [@Fa2 Section 2.4] by the formula $$\chi -\chi_{\Phi}=\sum_{i=1}^n g_i \left(x_i - \bar x_i \right)$$ for some non-negative integers $g_i$.
It follows from Farber’s approach that $\chi$ only depends on the $\La$-module $A_L$, and, therefore, this is also true for $\chi_{\Phi}$, when $\Phi(x,z)=(xz+1)^{-1}x$, since the integers $g_i$ are half the ranks of the $x_i$-components of the [*minimal lattice*]{} in $A_L$, as is demonstrated in the proof of the above proposition.
Is there some way to see directly that $\chi_{\Phi}$ depends only on $A_L$?
In [@Fa2] it is shown that $\chi$ (and thus $\chi_{\Phi}$) satisfies the duality property $\chi +\bar\chi =0$. We reprove this using Proposition \[prop.compare\] and \[prop.dual\].
For any $F$-link, we have $$\chi_{\Phi}=-\bar\chi_{\Phi}$$
(Realization) Can every [*rational*]{} power series $\rho$, with integer coefficients, satisfying the duality property $\rho =-\bar\rho$ be realized as $\chi_\Phi(L)$ for some $F$-link $L$?
For the cyclic module in Proposition \[prop.2\](c), what is $\chi$?
It is interesting, if perhaps disappointing, that this array of invariants are actually all determined by the original $\chi$ of Farber.
For any $f\in\BQ\la\la x,z\ra\ra, \chi_f (L)$ is completely determined by $\chi
(L)$, and therefore depends only on $A_L$.
It is pointed out in [@Fa2 Prop. 5.2] that $\chi (L)$ determines $A_L$ when it is semi-simple but not otherwise. For example, it follows from [@Fa2 Prop. 2.5(c)] that $\chi$ is not sensitive to different extensions of the same modules. In particular, for a knot $K$, $\chi (K)$ is determined by the Alexander polynomial [@Fa2 Section 10.4] and it is well-known that there exist knots with the same Alexander polynomial but different Alexander modules.
Finally we consider some examples. If $L$ is an $F$-link, let $L'$ denote the reflection (sometimes called [*mirror image*]{}) of $L$ with the natural $F$-structure induced from that of $L$ by the automorphism of $F$ defined by $t_i\to t_i\i$. If $A$ is a Seifert matrix of $L$ then $A'$ is a Seifert matrix for $L'$. Note that $Z_{A'}=I-Z_A =SZ'_A S\i$.
For any $f\in\BQ\la\la x,z\ra\ra$, $\chi_f
(L')=\ti\chi_{\ti f}(L)$. In particular $\chi (L')=\ti\chi (L)$.
For $2$-component links we have not been able to find any examples such that $\chi (L')\not=\chi (L)$.
Is $\ti\chi (L)=\chi (L)$ for any $2$-component $F$-link?
On the other hand for $3$-component $F$-links it is not hard to find such examples.
There exist $3$-component $F$-links such that $\chi(L')=\ti\chi(L)\not=
\chi (L)$.
Proofs
======
Proof of Theorem \[thm.all\]
----------------------------
Let us introduce three moves on the set $\Sei$ of Seifert matrices:
- Replace $A$ by $PAP'$ for a block diagonal matrix $P=\text{diag}
(P_1,\dots,P_n)$ of unimodular matrices $P_i$ with integer entries.
- Replace $A$ by $$\left(
\begin{matrix}
A & \rho & 0 \\
\rho' & 0 & 1 \\
0 & 0 & 0
\end{matrix}
\right)
\,\,\, \text{ or }
\left(
\begin{matrix}
A & \rho & 0 \\
\rho' & 0 & 0 \\
0 & 1 & 0
\end{matrix}
\right)$$ for a column vector $\rho$, where, for some $i$, the two new rows are added to $A_{ij}, 1\le j\le n$ and the two new columns are added to $A_{ji},
1\le j\le n$.
- The move that generates $A_n$-equivalence, where the algebraic action of $A_n$ on $\Sei(n)$ is described in [@K1; @K2].
Note that $S_1,S_2$ generate the so-called $S$-equivalence of Seifert matrices.
Given a Seifert matrix $A$, we define $Z_A=A(A-A')^{-1}$ and $S_A=A-A'$ (or simply, $Z$ and $S$ in case $A$ is clear), following Seifert. Note that $S$ is block-diagonal. The behavior of $Z$ under $S$-equivalence of $A$ is described by the following elementary matrix calculation
If $A \Sone B=PAP'$, then $Z_B= PZ_AP^{-1}$. If $A \Stwo B$, then $$Z_B=
\left(
\begin{matrix}
Z_A & 0 & \star \\
\star & 1 & \star \\
0 & 0 & 0
\end{matrix}
\right)
\,\,\, \mathrm{ or } \,\,\,
\left(
\begin{matrix}
Z_A & \star & 0 \\
0 & 0 & 0 \\
\star & \star & 1
\end{matrix}
\right).$$
We need to show that $\chi_f$ is invariant under the moves $S_1$ and $S_2$. If $A \Sone B$, then $f(X,Z_B)=
P f(X,Z_A) P^{-1}$ thus $\chi_f(B)=\chi_f(A)$. If $A \Stwo B$, then the following identity $$\left(
\begin{matrix}
C & 0 & \star \\
\star & c & \star \\
0 & 0 & 0
\end{matrix}
\right)
\left(
\begin{matrix}
C' & 0 & \star \\
\star & c' & \star \\
0 & 0 & 0
\end{matrix}
\right)
=
\left(
\begin{matrix}
C C' & 0 & \star \\
\star & c c' & \star \\
0 & 0 & 0
\end{matrix}
\right)$$ implies that $\chi_f(B)=\chi_f(A)$.
Given an $F$-link $L$ with a Seifert matrix $A$, then $XZ_A+I$ is a presentation matrix for $A_L$ over $\La$, and for $A_L^{\ab}$ over $\La^{ab}$.
It is well-known (see [@K2]) that a presentation matrix for $A_L$ is $TA-A'$, and similarly for $A_L^{\ab}$. Since $TA-A'=(T-I)A+ (A-A')=(XZ+I)(A-A')$, the lemma follows.
Proof of Proposition \[prop.rat\]
---------------------------------
Let $\R '(x,z)$ denote the subring of $\qad$ consisting of all admissible $f$ such that, for any scalar matrix $Z$ of the appropriate size, $f(X,Z)$ is a matrix all of whose entries are rational in $\BQ\la\la x_1,\dots ,x_n\ra\ra$. It will suffice to show that $\R_* (x,z)\sub\R '(x,z)$. To prove this we need to show that if $f\in\R '(x,z)$ is extra-special, then $g=f\i\in\R'(x,z)$.
Consider the matrix equation $f(X,Z)g(X,Z)=I$. This defines a system of equations for the entries of $g(X,Z)$ of the form $$\sum_j a_{rj}y_j =b_r$$ where $a_{rj}, b_r\in\R (x_1 ,\dots ,x_n )$. Since $f$ is extra-special, $f(0,Z)=I$, which implies that, with the correct choice of numbering of the equations, $a_{rj}(0,\dots
,0)=\delta_{rj}$. Now we can apply [@FV Proposition 2.1] to conclude that the solutions $y_r$, which are the entries of $g(X,Z)$, are unique and rational.
Proof of Proposition \[prop.dual\]
----------------------------------
It follows from the definition of $Z$ that $Z+SZ'S^{-1}=I$, where $S=A-A'$. This implies that $$\lbl{eq.dual}
Z=S(I-Z')S^{-1}$$ Thus $${\operatorname{tr}}f(X,Z)={\operatorname{tr}}f(X,I-Z')=\ti{\operatorname{tr}}\ti f(X,1-Z)$$ using the facts that $S$ commutes with $X$, that ${\operatorname{tr}}Y={\operatorname{tr}}Y'$, for any square matrix $Y$ and that ${\operatorname{tr}}(WY)={\operatorname{tr}}(YW)$, if the entries of $W$ commute with the entries of $Y$. From this we deduce the first equality.
The second equality is clear.
Let $[\qad,\qad]$ denote the abelian subgroup of the ring $\qad$ generated by $fg-gf$ for $f,g \in \qad$. It is easy to see that for all $f \in
[\qad,\qad]$ we have $\chi_f=0 \in \Lhat/(\cyclic)$.
Proof of Proposition \[prop.2\]
-------------------------------
To prove (a) first note that the normalized $\Delta^b$ can be defined by the equation $$\Delta^b =\det (T^{1/2}A-T^{-1/2}A')=\det ((I+X)^{-1/2}(I+XZ))$$ Thus we have $$\begin{aligned}
\log\Delta^b &=&{\operatorname{tr}}\log ((I+X)^{-1/2}(I+XZ))\\
&=&{\operatorname{tr}}\log (I+X)^{-1/2} +{\operatorname{tr}}\log (I+XZ)\\
&=&{\operatorname{tr}}\log (I+XZ) -\tfrac{1}{2}{\operatorname{tr}}\log (I+X)=\chi_{\Delta}\end{aligned}$$ This uses the following lemma, which is probably well-known.
Suppose that $M, M_1 ,M_2$ are matrices of the form $I+N$ over a completed commutative power series ring, where $N$ has all entries of degree $>0$. Then we have the following identities. $$\begin{aligned}
{\operatorname{tr}}\log (M_1 M_2 )&=&{\operatorname{tr}}\log (M_1 )+{\operatorname{tr}}\log (M_2)\lbl{eq.mult}\\
\log \det(M)&=&{\operatorname{tr}}\log (M) \lbl{eq.detr}\end{aligned}$$
follows from the Campbell-Baker-Haussdorf formula and the fact that ${\operatorname{tr}}(AB)={\operatorname{tr}}(BA)$ if $A,B$ are matrices over a commutative ring.
To prove , first note that it is obvious if $M$ is triangular. Secondly, it follows from that if it is true for $M_1$ and $M_2$, then it is true for $M_1 M_2$. Thus if will follow from the fact that any such $M$ can be written $M=LU$, where $L$ is lower triangular (i.e. $l_{ij}=0$ if $i<j$) and $U$ is upper triangular. We prove this by induction on the size of $M$.
Write $M=\left(\begin{array}{rr} u & \b \\\a & \ti M\end{array}\right)$, where $\a$ is a column vector and $\b$ is a row vector. By induction we can write $\ti
M-u\i\a\b =\ti L\ti U$, for triangular matrices $\ti L,\ti U$. Now we define $$L=\left(\begin{array}{rr} u & 0\\\a & \ti L \end{array}\right)
\quad\text{and}\quad
U=\left(\begin{matrix}1 & u\i\b\\0 & \ti U \end{matrix}\right)$$ One checks immediately that $M=LU$.
\(b) follows from a general construction in [@Le]. Consider the trivial link $L_0\sub
S^3$ with $n$ components. Then the splitting map $\phi$ is an isomorphism. Consider the universal cover $X_{L_0}^{\w}$ of $S^3 -L_0$. Given $\l =\sum_{g\in F}a_g
g$ satisfying $\l =\bar\l$, we can construct a simple closed curve $\g$ in $S^3
-L_0$ which is null-homotopic and unknotted in $S^3$ such that, if $\ti\g$ is any lift of $\g$ in $X_{L_0}^{\w}$ then the linking numbers of $\ti\g$ and its translates is given by $$\text{lk}(\ti\g , g\ti\g )=a_g \text{\ if }g\not= 1$$ This construction is described in [@Le]. Now do a $+1$-surgery on $\g$ to produce $\S^3$, which, since $\g$ was unknotted, is diffeomorphic to $S^3$. Let $L\sub S^3$ be the link corresponding to $L_0\sub\S^3$ under such a diffeomorphism. Note that surgery on all the lifts of $\g$ produces an $F$-covering of $S^3 -L$ and so $L$ is canonically a $F$-link. The argument in [@Le] shows that $H_1 (X_{L}^{\w})\iso\La /(\l)$.
For (c), we will use Proposition \[prop.dual\] and Remark \[rem.dual\]. Since $f_\D(x,z)=\log(1+xz)$, it is easy to see that $f_\D(x,z)=f_\D(z,x) \bmod [\qad,\qad]$. On the other hand, we have $$\begin{aligned}
\ti f_\D(x,1-z) &=& \log(1+(1-z)x)=\log(1+x-zx)=\log((1+z\bar x)(1+x)) \\
&=&
\log(1+z\bar x)+\log(1+x) \bmod [\qad,\qad] \\ &=&
\bar f_\D(x,z)+\log(1+x) \bmod [\qad,\qad] \\
&=&
\log(1+\bar x z)+\log(1+x) \bmod [\qad,\qad] \\
&=&
\hat f_\D (x,z)+\log(1+x) \bmod [\qad,\qad].\end{aligned}$$ Proposition \[prop.dual\] and Remark \[rem.dual\] imply that $$\chi_{f_\D(x,z)}=\ti\chi_{\ti f_\D(x,1-z)}=\ti\chi_{\hat f_\D (x,z)}
+\chi_{\log(1+x)}=\bar\chi_{f_\D}+\chi_{\log(1+x)}
\in \Lhat/(\cyclic).$$ Since $\chiD(x_i)=\chi_{f_\D(x,z)}$ and $\chi_{\log(1+x)}=0$, it follows that $\chiD=
\bar\chi_\D
\in \Lhat/(\cyclic)$.
Proof of Proposition \[prop.compare\]
-------------------------------------
Let $A$ be any Seifert matrix for $L$. We can construct a higher-dimensional [*simple*]{} link $\ti L$ in $S^{4k+3}$, for some large $k$, which has a Seifert manifold $W$ yielding $A$ as its Seifert matrix (see, e.g. [@K2]). Since the Seifert matrix determines the link module, via the presentation matrix $TA-A'$ we have $A_L=H_1(X^\omega_L,\BQ)\iso A_{\ti L}= H_{2k+1} (X^\omega_{\ti L},\BQ)$. Therefore $\chi$ for $A_L$ is the same as $\chi$ for $A_{\ti L}$. Now we can do surgery on $W$ to obtain a [*minimal Seifert manifold*]{} $V$ for $\ti L$, whose components are $2k$-connected—see [@Fa1 Section 6.12] and [@Gu]. This determines a minimal lattice $J$ for $A_{\ti L}$, according to [@Fa2 p.563-4]. The Seifert matrix $B$ determined by $V$ is S-equivalent to $A$ and so $\chi_{\Phi}(L)= {\operatorname{tr}}((I+XZ)\i X)- {\operatorname{tr}}((I+XI_{1/2})\i X)$, where $Z=B(B-B')\i$. Note that $J=\bigoplus_i x_i J$ and each $x_i J$ is isomorphic to $H_{2q+1}(V_i )$, where $V_i$ is the $i$-th component of $V$. Then, if $2g_i =\text{rank }H_{2q+1}(V_i )$, it is straightforward to check that $ {\operatorname{tr}}((I+XI_{1/2})\i X)=\sum_i g_i
(x_i -\bar x_i )$. The proof will be completed if we show that $\chi = {\operatorname{tr}}((I+XZ)\i X)$.
Now $A_L$ is the $\La$-module with presentation matrix $XZ+I$, as in Lemma \[lem.presentation\]. The generators $\a_r$ of $A_L$, corresponding to the columns of $XZ+I$, span the minimal lattice $J$ as described in [@Fa2 p.564], since $B$ comes from a minimal Seifert manifold. If we let $M_i=x_i A_L$, then the generators corresponding to the $i$th column block of $XZ+I$ generate $M_i$ since, if $\a_r$ denotes a generator corresponding to a column in the $i$th column block, the $r$-th row of $XZ+I$ gives the relation $\a_r=-x_i \sum Z_{rs}\a_s$. Thus, $\pi_i$ is given by the matrix $P_i=\matt 0 {} {} {} I {} {} {} 0$ where $I$ is in the $(i,i)$ block, $z$ is given by the matrix $Z'$ and $\partial_i$ is given by the matrix whose $i$th column block is the $i$th column block of $-Z'$ and the other columns are zero—call this matrix $Z_i$. Now, $\chi$ is given by $$\chi =\sum_k \sum_n {\operatorname{tr}}(\pi_k \partial_{a_1} \cdots \partial_{a_n})
x_{a_n} \cdots x_{a_1} x_k.$$ But $\pi_k \partial_{a_1} \cdots \partial_{a_n}$ is given by the matrix $P_k Z_{a_1}' \cdots Z_{a_n}'$. Note that $Z_{a_1}'\cdots Z_{a_n}'$ is the matrix with only the $a_n$-th column block nonzero and the $(r,a_n)$ block is $(-1)^n Z_{r,a_1}' Z_{a_1,a_2}' \cdots
Z_{a_{n-1},a_n}'$. Multiply this by $P_k$, giving a matrix whose only nonzero entries are in the $(k,a_n)$ block, and equal to $ (-1)^n Z_{k,a_1}' Z_{a_1,a_2}' \cdots Z_{a_{n-1},a_n}'$. Thus, we have a nonzero trace only if $k=a_n$, giving $$\begin{aligned}
\chi &=&
\sum_n \sum_{a_1,\dots,a_n}
(-1)^n {\operatorname{tr}}( Z_{a_n,a_1}' Z_{a_1,a_2}' \cdots Z_{a_{n-1},a_n}')
x_{a_n} \cdots x_{a_1} x_{a_n} \\
&=&
\sum_n \sum_{a_1,\dots,a_n}
(-1)^n {\operatorname{tr}}( Z_{a_n,a_{n-1}} Z_{a_{n-1},a_{n-2}} \cdots Z_{a_1,a_n})
x_{a_n} \cdots x_{a_1} x_{a_n} \\
&=&
\sum_n \sum_{a_1,\dots,a_n,a'_n}
(-1)^n {\operatorname{tr}}( (XZ)_{a_n,a_{n-1}} (XZ)_{a_{n-1},a_{n-2}} \cdots (XZ)_{a_1,a'_n}
X_{a'_n,a_n}) \\
&=&
\sum_n (-1)^n {\operatorname{tr}}((XZ)^nX)={\operatorname{tr}}\left((XZ+I)^{-1}X\right). \end{aligned}$$
Proof of Proposition \[prop.far\]
---------------------------------
It is easy to see that $\ti\Phi(x,z)=\Phi (x,z)$. Furthermore, since ${\operatorname{tr}}\Phi(X,I_{1/2})$ satisfies the asserted duality statements, we can omit this part of the definition of $\chi_{\Phi}$ in the following. We have: $$\begin{aligned}
\ti\Phi(x,1-z) &=& (1+x(1-z))^{-1}x=(1+x-xz)^{-1}x=(1+\bar
xz)^{-1}(1+x)^{-1}x \\ &=&
-(1+\bar xz)^{-1}\bar x
= -\Phi (\bar x,z)=-\hat\Phi (x,z).\end{aligned}$$ Proposition \[prop.dual\] implies that $\chi_{\Phi}=\ti\chi_{\ti\Phi (x,1-z)}=
-\ti\chi_{\hat\Phi}=-\ti{\hat\chi}_{\Phi}=-\bar\chi_{\Phi}$.
Proof of Theorem \[th.chi\]
---------------------------
It suffices to consider the case where $f$ is a monomial, say $$f=x^{f_0}z^{e_1}x^{f_1}\cdots z^{e_k}x^{f_k}$$ where $e_i >0$ for $1\le i\le k$ and $f_i >0$ if $0<i<k$. Note that we have a general formula $$\lbl{eq.trace}
{\operatorname{tr}}f(X,Z)=\sum_{i_1 ,\dots ,i_k}{\operatorname{tr}}(Z^{e_1})_{i_1 i_2}(Z^{e_2})_{i_2
i_3}\cdots (Z^{e_k})_{i_k
i_1}x^{f_0}_{i_1}x^{f_1}_{i_2}\cdots x^{f_{k-1}}_{i_k}x^{f_k}_{i_1}$$ where $(Z^e )_{ij}$ denote the $(i,j)$-block of $Z^e$. Now we associate with $f$ another monomial $f'\in\BQ\la\la x,y,z\ra\ra$ by replacing each $z^{e_i}$ in $f$ by $(zy)^{e_i -1}z$, for every $1\le i\le k$ and replacing each $x^{f_i}$ by $x$, for every $0\le i\le k$ (even when $f_0$ or $f_k$ is zero). Now consider ${\operatorname{tr}}f'(X,Y,Z)$, where $Y=\text{diag}(y_1 ,\dots ,y_n )$ is a matrix identical to $X$ in which each $x_i$ is replaced by a new variable $y_i$. It is not hard to see, using equation , that ${\operatorname{tr}}f'(X,Y,Z)$ and $f$ determine ${\operatorname{tr}}f(X,Z)$ by replacing each $x_j$ in ${\operatorname{tr}}f'(X,Y,Z)$ with the appropriate power of $x_j$ and each $y_j$ by $1$. Furthermore, again using equation , ${\operatorname{tr}}f'(X,X,Z)$ and $f$ determine ${\operatorname{tr}}f'(X,Y,Z)$ since $f$ tells us which $x_i$ in $f'(X,X,Z)$ to replace by $y_i$ to obtain ${\operatorname{tr}}f'(X,Y,Z)$.
Finally we note that $f'(x,x,z)$ is a monomial of the form $(xz)^k x$, and so coincides, up to sign, with the degree $k+1$ part of $\Phi$. Thus ${\operatorname{tr}}f'(X,X,Z)$ is determined by ${\operatorname{tr}}\Phi (X,Z)$. This completes the proof.
Proof of Propositions \[prop.reflect\] and \[prop.refl\]
--------------------------------------------------------
Since ${\operatorname{tr}}f(X,SZ'S\i )={\operatorname{tr}}f(X,Z')=\ti{\operatorname{tr}}{\ti f }(X,Z)$, we conclude that $\chi_f (L')=\ti\chi_{\ti f}(L)$. Since $\ti\Phi =\Phi$, it follows from Proposition \[prop.compare\] that $\chi(L')=\ti\chi (L)$. This proves Proposition \[prop.reflect\].
For Proposition \[prop.refl\] let us consider the matrix $$A=\left(\begin{array}{rrr} M &-S &-S\\ S & M & -S \\ S & S &
M\end{array}\right)$$ where $S=\left(\begin{array}{rr} 0 & 1\\-1 & 0\end{array}\right)$ and $M$ is any $2\times 2$-matrix satisfying $M-M'=S$. This is a Seifert matrix for some $F$-link $L$.
Then $$Z_A =\left(\begin{array}{rrr} N&-I&-I\\I&N&-I\\I&I&N\end{array}\right)$$ where $N=MS$. From the general formula $${\operatorname{tr}}_{\Phi}(Z)=\sum{\operatorname{tr}}x_{i_1}Z_{i_1 i_2}x_{i_2}\cdots
x_{i_k}Z_{i_{k}i_1}x_{i_1}=\sum ({\operatorname{tr}}Z_{i_1 i_2}\cdots
Z_{i_{k}i_1})x_{i_1}x_{i_2}\cdots x_{i_k}x_{i_1}$$ which follows from equation , we can see that ${\operatorname{tr}}_{\Phi}(Z_A )=\sum_{\bf m}a_{\bf m}\bf m$, summing over non-commutative monomials ${\bf m}=x_{i_1}x_{i_2}\cdots x_{i_k}x_{i_1}$, where $a_{\bf m}=(-1)^r{\operatorname{tr}}M^s$ with $r=\#\{ j:i_{j+1}>i_j\}$ and $s=\#\{ j:i_j
=i_{j+1}\}$. Thus, for example if ${\bf m}=x_1 x_2 x_3 x_1$ then $a_{\bf m}=2$ whereas if ${\bf m}=x_1 x_3 x_2 x_1$ then $a_{\bf m}=-2$. Thus $\chi_{\Phi}(A)\not=\ti\chi_{\Phi}(A)$ and we have the desired example.
[\[EMSS\]]{}
J. Berstel and C. Reutenauer, [Rational series and their languages]{}, EATCS Monographs on theoretical computer science [**12**]{}, Springer-Verlag (1984).
S. Cappell and J. Shaneson, [*Link cobordism*]{}, Commentarii Math. Helv. [**55**]{} (1980) 20–49.
M. Duval, [*Forme de Blanchfield et cobordism d’enterlacs bords*]{}, Comment. Math. Helv. [**61**]{} (1986) 617–635.
M. Farber, [*Hermitian forms on link modules*]{}, Comment. Math. Helv. [**66**]{} (1991) 189–236.
[to3em]{}, [*Noncommutative rational functions and boundary links*]{}, Math. Annalen [**293**]{} (1992) 543–568.
[to3em]{}and P. Vogel, [*The Cohn localization of the free group ring*]{}, Math. Proc. Cambridge Phil. Soc. [**111**]{} (1992) 432–443.
S. Garoufalidis and A. Kricker, [*A rational noncommutative invariant of boundary links*]{}, preprint 2001.
[to3em]{}, [*Boundary links, localization and the loop move*]{}, in preparation.
M. Gutierrez, [*Boundary links and an unlinking theorem*]{}, Trans. Amer. Math. Soc. [**171**]{} (1972) 491–499.
K.H. Ko, [*Seifert matrices and boundary links*]{}, thesis, Brandeis University, 1984.
[to3em]{}, [*Seifert matrices and boundary link cobordisms*]{}, Trans. Amer. Math. Soc. [**299**]{} (1987) 657–681.
J. Levine, [*A method of generating link polynomials,*]{} Amer. J. Math., [**89**]{} (1967), 69–84.
V. Retakh, C. Reutenauer and A. Vaintrob, [*Noncommutative rational functions and Farber’s invariants of boundary links*]{}, AMS Transl. [**194**]{} (1999) 237–245.
N. Smythe, [*Boundary links*]{}, Topology Seminar (Wisconsin 1965) Princeton Univ. Press, Princeton, NJ 1966, 69–72.
[^1]: The authors were partially supported by NSF grants DMS-98-00703 and DMS-99-71802 respectively, and by an Israel-US BSF grant. This and related preprints can also be obtained at [http://www.math.gatech.edu/$\sim$stavros ]{} and [http://www.math.brandeis.edu/Faculty/jlevine/ ]{} 1991 [*Mathematics Classification.*]{} Primary 57N10. Secondary 57M25.
|
---
abstract: 'We consider an edge computing scenario where users want to perform a linear computation on local, private data and a network-wide, public matrix. The users offload computations to edge servers located at the edge of the network, but do not want the servers, or any other party with access to the wireless links, to gain any information about their data. We provide a scheme that guarantees information-theoretic user data privacy against an eavesdropper with access to a number of edge servers or their corresponding communication links. The proposed scheme utilizes secret sharing and partial replication to provide privacy, mitigate the effect of straggling servers, and to allow for joint beamforming opportunities in the download phase, in order to minimize the overall latency, consisting of upload, computation, and download latencies.'
author:
-
title: |
Private Edge Computing for Linear Inference\
Based on Secret Sharing
---
Introduction
============
Edge computing has established itself as a pillar of the 5G mobile network [@ETSI] to guarantee very low-latency and high-bandwidth computing services. The key idea is to move the computation power from the cloud closer to where data is generated, by pooling the available resources at the network edge.
Processing data in a distributed fashion over a number of edge servers poses significant challenges. In particular, edge servers may fail, be inaccessible, or straggle. The straggler problem has recently been addressed in the context of distributed computing in data centers (over the cloud), where coding has been shown to be a powerful tool to reduce the computational latency due to straggling servers [@Li; @Lee; @Albin1; @Albin2]. The idea is to generate redundant computations by means of an erasure correcting code such that the partial computations of a subset of the servers suffice to complete the whole computation, thus providing resiliency to straggling (and failing) servers. The same concept can be applied in edge computing. In this scenario, besides the computational latency due to straggling servers, the communication latency of uploading and downloading data to the servers is of utmost importance, due to severe bandwidth limitations.
To reduce the communication latency, in [@Tao; @TaoStudy] subtasks were replicated across edge servers to enable cooperation opportunities to send results back to the users via joint beamforming. More recently, [@Osvaldo; @Kuikui] combined both straggler coding using a maximum distance separable (MDS) code and joint beamforming to reduce the overall latency. Another important challenge when processing data over heterogeneous, untrusted edge servers is guaranteeing the privacy of the user data. Recently, this problem has been addressed in the context of distributed computing in data centers in the presence of straggling servers [@SalimStair; @SalimRateless]. These works use secret sharing ideas to provide both privacy and robustness against stragglers.
In this paper, we propose a privacy-preserving edge computing scheme that exploits straggler coding and partial replications across servers to reduce latency. To the best of our knowledge, this problem has not been considered before in the literature. In particular, we consider a similar scenario to the one in [@Osvaldo] where multiple users wish to perform a linear inference on some local data given a network-wide, public matrix. Practical examples where such a scenario arises include recommender systems via collaborative filtering.
For this scenario, we present a scheme that guarantees information-theoretic user data privacy against an eavesdropper with access to a number of edge servers or their corresponding communication links. The proposed scheme utilizes secret sharing to provide both privacy and mitigate the effect of straggling servers. Furthermore, by replicating computations across different servers the scheme allows for joint beamforming opportunities. The proposed scheme entails an inherent tradeoff between computational latency due to stragglers, communication latency, and user data privacy. For a given privacy level, we optimize the parameters of the scheme in order to minimize the overall latency incurred by the upload and download of data as well as the computation. For the lowest privacy level, i.e., privacy against a single untrusted server, the proposed scheme yields an increase in latency in the worst case by a moderate factor of about $2.4$ compared to the nonprivate scheme in [@Osvaldo] for the selected system parameters.
*Notation:* Vectors and matrices are written in lowercase and uppercase bold letters, respectively, e.g., $\boldsymbol{a}$ and $\boldsymbol{A}$. The transpose of vectors and matrices is denoted by $(\cdot)^\top$. $\text{GF}(q)$ denotes the finite field of order $q$ and $\mathbb{N}$ denotes the positive integers. We use the notation $[a]$ to represent the set of integers $\{0,1,\ldots,a-1\}$. Furthermore, $\left\lceil a/b\right\rceil$ is the smallest integer larger than or equal to $a/b$, $\left\lfloor a/b\right\rfloor$ is the largest integer smaller than or equal to $a/b$, and $(a)_b$ is the integer $a$ modulo $b$. We represent permutations in cycle notation, e.g., the permutation $\pi = (0\;2\;1\;3)$ maps $0\mapsto2$, $2\mapsto1$, $1\mapsto3$, and $3\mapsto0$. In addition, $\pi(i)$ is the image of $i$ under $\pi$, e.g., $\pi(0) = 2$. The expected value of a random variable $X$ is denoted by $\mathbb{E}[X]$.
System Model {#Sec: SystemModel}
============
We consider the system in \[fig:system\_model\] with $u$ users $\mathsf{u}_0, \ldots ,\mathsf{u}_{u-1}$, where the data of user $\mathsf{u}_i$ is represented by the vector ${\boldsymbol{x}}_i = (x_{i,0},\ldots,x_{i,r-1})^\top\in\mathrm{GF}{(q)}^r$. Each user $\mathsf{u}_i$ wants to perform a computation-intensive linear inference ${\boldsymbol{W}}{\boldsymbol{x}}_i$, where ${\boldsymbol{W}}\in\textrm{GF}{(q)}^{m\times r}$, in a distributed fashion over $e$ edge nodes (ENs) $\mathsf{e}_0,\ldots,\mathsf{e}_{e-1}$ located at the edge of the network. For ease of notation we will refer to the set $\{{\boldsymbol{W}}{\boldsymbol{x}}_i\mid i\in [u]\}$ as $\{{\boldsymbol{W}}{\boldsymbol{x}}_i\}$ and to $\{{\boldsymbol{x}}_i\mid i\in [u]\}$ as $\{{\boldsymbol{x}}_i\}$. The matrix ${\boldsymbol{W}}$ stays constant for a sufficiently long period of time, and each EN has a storage capacity corresponding to a fraction $\mu$, $0<\mu \leq 1$, of the matrix ${\boldsymbol{W}}$, which is assumed to be public. Moreover, we assume that each user is connected by $e$ unicast wireless links to the $e$ ENs.
Computation Runtime Model
-------------------------
The ENs may straggle, which is represented by a random setup time $\lambda_j$ for each EN $\mathsf{e}_j$. The setup time is the time it takes an EN to start computing after it has received the necessary data. As in [@Osvaldo; @Dean; @Mallick], we assume that the setup times are independent and identically distributed (i.i.d.) according to an exponential distribution with parameter $\eta$, such that $\mathbb{E}[\lambda_j] = 1/\eta$. The time it takes an EN to compute one inner product in $\text{GF}(q)^r$ for each of the users is deterministic and denoted by $\tau$. Thus, the latency incurred by EN $\mathsf{e}_j$ to compute $d$ inner products for each user ($u\cdot d$ inner products in total) is $$\mathsf{L}^\mathsf{comp}_{j} = \lambda_{j} + d\tau.$$ We define the *normalized computation* latency of EN $\mathsf{e}_j$ as $$\mathsf{\tilde{L}}^\mathsf{comp}_{j} = \frac{\mathsf{L}^\mathsf{comp}_{j}}{\tau} = \frac{\lambda_{j}}{\tau} + d.$$
Communication
-------------
Both the upload of data from the users to the ENs and the download of the results of the computations from the ENs to the users is considered. We denote by $\gamma$ the normalized communication latency of unicasting $u$ symbols from $\text{GF}(q)$ in the upload or download. In the uplink, each user unicasts its data vector for computation to the ENs. In the downlink, ENs having access to the same symbol can collaboratively transmit to multiple users at the same time and thereby reduce the communication latency by exploiting joint beamforming opportunities [@Tao; @Kuikui; @Osvaldo; @TaoStudy; @Zhang; @Naderializadeh]. In particular, a symbol available at $\rho$ ENs can be transmitted simultaneously to $\min\{\rho,u\}$ users with a normalized communication latency of $\gamma/\min\{\rho,u\}$ in the high signal-to-noise (SNR) region. The normalized communication latency, in the high SNR region, of transmitting $v$ symbols, where symbol $\alpha_i$, $i\in[v]$, is available at $\rho_i$ ENs, is $$\mathsf{\tilde{L}}^\mathsf{comm,down} = \gamma \sum_{i=0}^{v-1} \frac{1}{\min\{\rho_i,u\}}.$$
Privacy and Problem Formulation
-------------------------------
We consider a scenario where some of the ENs or their corresponding communication links are compromised. In particular, we assume the presence of an eavesdropper with access to any $z$ ENs or their corresponding communication links.
The goal is to offload computations to the (untrusted) ENs in such a way that they do not gain any information in an information-theoretic sense (zero mutual information) about neither the user data $\{{\boldsymbol{x}}_i\}$ nor the results of the computations $\{{\boldsymbol{W}}{\boldsymbol{x}}_i\}$, while minimizing the overall normalized latency, consisting of upload, computation, and download latencies.
Private Distributed Linear Inference
====================================
In this section, we present a distributed linear inference computation scheme that provides user data privacy against an eavesdropper with access to any $z$ ENs or their corresponding communication links. At the heart of the proposed scheme lies Shamir’s secret sharing scheme (SSS) [@Shamir]. An SSS with parameters $(n,k)$, $n\geq k$, ensures that some private data can be shared with $n$ parties in such a way that any $k-1$ colluding parties do not learn anything about the data. On the other hand, any set of $k$ or more parties can recover the data.
For each user $\mathsf{u}_i$, an $(n, k)$ SSS is used to compute $n$ shares of its private data ${\boldsymbol{x}}_i = (x_{i,0},\ldots,x_{i,r-1})^\top$. In particular, for user $\mathsf{u}_i$ we encode each data entry $x_{i,l}$ along with $k-1$ i.i.d. uniform random symbols $r_{i,l}^{(1)},\ldots,r_{i,l}^{(k-1)}$ from $\text{GF}(q)$ using an $(n,k)$ Reed-Solomon (RS) code to obtain $n$ coded symbols $s_{i,l}^{(0)},\ldots ,s_{i,l}^{(n-1)}$. For each $h\in[n]$, the $(h+1)$-th share of user $\mathsf{u}_i$ is $$\begin{aligned}
{\boldsymbol{s}}^{(h)}_i=\left(\begin{matrix}
s^{(h)}_{i,0}, \ldots, s^{(h)}_{i,r-1}\\
\end{matrix}\right)^\top.
\end{aligned}$$ Finally, define the matrix of shares $$\begin{aligned}
\label{eq:matrix_of_shares}
\bm S^{(h)}=\left(\begin{matrix}
{\boldsymbol{s}}^{(h)}_0, {\boldsymbol{s}}^{(h)}_1, \ldots, {\boldsymbol{s}}^{(h)}_{u-1}
\end{matrix}\right)\in\text{GF}{(q)}^{r\times u}
\end{aligned}$$ as the matrix collecting the $(h+1)$-th share of all users.
The following theorem proves that the original computations $\{{\boldsymbol{W}}{\boldsymbol{x}}_i\}$ of all users can be recovered from a given set of computations based on the matrices of shares $\boldsymbol{S}^{(0)},\ldots, \boldsymbol{S}^{(n-1)}$, while providing privacy against an eavesdropper with access to at most $k-1$ distinct matrices of shares.
\[Th: RecoverSecretComputation\] Consider $u$ users with their respective private data ${\boldsymbol{x}}_i\in\text{GF}(q)^r$, $i\in[u]$. Use Shamir’s $(n,k)$ SSS on each ${\boldsymbol{x}}_i$ to obtain the matrices of shares $\boldsymbol{S}^{(0)},\ldots, \boldsymbol{S}^{(n-1)}$ in . Let ${\boldsymbol{W}}\in\text{GF}{(q)}^{m\times r}$ be a public matrix and $\mathcal I\subseteq[n]$ a set of indices with cardinality $|\mathcal I|=k$. Then, the set of computations $\{{\boldsymbol{W}}\bm S^{(h)}\mid h\in\mathcal I\}$ allows to recover the computations $\{{\boldsymbol{W}}{\boldsymbol{x}}_i\}$ of all users. Moreover, for any set $\mathcal{J}\subseteq[n]$ with $|\mathcal{J}| < k$, $\{{\boldsymbol{W}}\bm S^{(h)}\mid h\in\mathcal J\}$ reveals no information about $\{{\boldsymbol{W}}{\boldsymbol{x}}_i\}$.
Let ${\mathcal{C}}$ be the $(n,k)$ RS code used in the SSS. For each $h\in[n]$, the entries of the rows of $\bm S^{(h)}$ are code symbols in position $h$ of codewords from ${\mathcal{C}}$ pertaining to different users. More precisely, for each user $\mathsf{u}_i$, each row of the matrix $\bigl({\boldsymbol{s}}^{(0)}_i, {\boldsymbol{s}}^{(1)}_i, \ldots,{\boldsymbol{s}}^{(n-1)}_{i}\bigr)$ of all $n$ shares of $\mathsf{u}_i$ is a codeword from ${\mathcal{C}}$. Since ${\mathcal{C}}$ is a linear code, each of the $m$ rows of the matrix $$\begin{aligned}
{\boldsymbol{W}}\left(\begin{matrix}
{\boldsymbol{s}}^{(0)}_i, {\boldsymbol{s}}^{(1)}_i,\ldots, {\boldsymbol{s}}^{(n-1)}_{i}
\end{matrix}\right)
\end{aligned}$$ is a codeword of ${\mathcal{C}}$. Furthermore, the messages obtained by decoding these codewords are the rows of $$\begin{aligned}
\left({\boldsymbol{W}}{\boldsymbol{x}}_i,{\boldsymbol{W}}\bm r_i^{(1)},\ldots,{\boldsymbol{W}}\bm r_i^{(k-1)}\right),
\end{aligned}$$ where $\{ \bm r_i^{(\kappa)}=(r^{(\kappa)}_{i,0},\ldots,r^{(\kappa)}_{i,r-1})^\top \mid\kappa \in [k]\backslash\{0\}\}$ is the set of vectors of uniform random symbols used by user $\mathsf{u}_i$ in the computation of the shares $\bm s_i^{(h)}$, $h \in [n]$. Then, decoding the vectors in the set $\{{\boldsymbol{W}}{\boldsymbol{s}}^{(h)}_i\mid h\in{\mathcal{I}}\}$ gives ${\boldsymbol{W}}{\boldsymbol{x}}_i$, and it follows that $\{{\boldsymbol{W}}\bm S^{(h)}\mid h\in\mathcal I\}$ gives $\{{\boldsymbol{W}}{\boldsymbol{x}}_i\}$.
From the properties of Shamir’s SSS it follows that the mutual information between $\{\bm S^{(h)}\mid h\in\mathcal J\}$ and $\{{\boldsymbol{x}}_i\}$ is zero. Subsequently, from the data processing inequality it follows that $\{{\boldsymbol{W}}\bm S^{(h)}\mid h\in\mathcal J\}$ reveals no information about $\{{\boldsymbol{x}}_i\}$.
The following corollary gives a sufficient condition to recover the private computations $\{{\boldsymbol{W}}{\boldsymbol{x}}_i\}$.
\[Cor: Recovery\] Consider an edge computing scenario, where the public matrix ${\boldsymbol{W}}$ is partitioned into $b$ disjoint submatrices ${\boldsymbol{W}}_l\in\text{GF}{(q)}^{\frac{m}{b}\times r}$, $l\in[b]$, and the private data is $\{{\boldsymbol{x}}_i \}$. Then, the private computations $\{{\boldsymbol{W}}{\boldsymbol{x}}_i \}$ can be recovered from the computations in the sets $$\begin{aligned}
\label{Eq: RecoveryCondition}
{\mathcal{S}}_l\triangleq\{{\boldsymbol{W}}_l\bm S^{(h)}\mid h\in\mathcal I\},\; l\in[b],
\end{aligned}$$ for any fixed set $\mathcal{I}\subseteq[n]$ with cardinality $|\mathcal{I}| = k$.
From \[Th: RecoverSecretComputation\], for a given $l\in[b]$, the computations in the set $\{{\boldsymbol{W}}_l{\boldsymbol{x}}_i\}$ can be recovered from the computations in the set ${\mathcal{S}}_l$. Then, we obtain $${\boldsymbol{W}}{\boldsymbol{x}}_i=\left(\begin{matrix}
({\boldsymbol{W}}_0{\boldsymbol{x}}_i)^\top, ({\boldsymbol{W}}_1{\boldsymbol{x}}_i)^\top, \ldots, ({\boldsymbol{W}}_{b-1}{\boldsymbol{x}}_i)^\top
\end{matrix}\right)^\top,~\forall i\in[u].$$
In the following, we present a scheme that fulfills the sufficient recovery condition in \[Cor: Recovery\]. Note that it may be beneficial to repeat shares over several ENs in order to exploit broadcasting opportunities during the download phase. This presents difficulties in the design of a private scheme, because repeating shares at different nodes results in a privacy level $z$ lower than that of the SSS ($k$). For example, if all ENs have access to two matrices of shares, the scheme only provides privacy against any $z=\lfloor (k-1)/2 \rfloor$ colluding ENs.
Given the underlying SSS, the proposed scheme can be broken down into two combinatorial problems. The first corresponds to the assignment of the submatrices $\{{\boldsymbol{W}}_l \mid l \in [b]\}$ to the $e$ ENs such that no EN stores more than a fraction $\mu$ of ${\boldsymbol{W}}$. The second corresponds to the assignment of the $n$ matrices of shares $\{\bm S^{(h)} \mid h \in [n]\}$ to the ENs such that the users are guaranteed to obtain the computations in \[Eq: RecoveryCondition\].
Assignment of ${\boldsymbol{W}}$ to the Edge Nodes {#Sec: Wassignment}
--------------------------------------------------
We start by explaining the assignment of the submatrices of ${\boldsymbol{W}}$ to the ENs such that no EN stores more than a fraction $\mu$ of ${\boldsymbol{W}}$, while the users are guaranteed to recover their computations $\{{\boldsymbol{W}}{\boldsymbol{x}}_i\}$. Additionally, we would like to allow for replications across different ENs to allow for joint beamforming in the download phase.
In order to satisfy the storage requirement, we select $p\in\mathbb{N}$ such that $p/e \leq \mu$ and partition ${\boldsymbol{W}}$ into $b=e$ submatrices as $$\begin{aligned}
{\boldsymbol{W}}=\left(\begin{matrix}
{\boldsymbol{W}}_0^\top, {\boldsymbol{W}}_1^\top, \ldots, {\boldsymbol{W}}_{e-1}^\top
\end{matrix}\right)^\top.
\end{aligned}$$ We then assign $p$ submatrices to each of the $e$ ENs. The assignment has the following combinatorial structure. Consider a cyclic permutation group of order $e$ with generator $\pi$. We construct an index matrix $$\begin{aligned}
\label{Eq: Iw}
\bm{I}_{\mathsf w} &\triangleq \left(\begin{matrix}
\pi^0(0) & \pi^0(1) & \cdots & \pi^0(e-1)\\
\pi^1(0) & \pi^1(1) & \cdots & \pi^1(e-1)\\
\vdots & \vdots & \ddots & \vdots\\
\pi^{p-1}(0) & \pi^{p-1}(1) & \cdots & \pi^{p-1}(e-1)\\
\end{matrix}\right)
\end{aligned}$$ and define the set of indices $$\label{Eq:setIw}
{\mathcal{I}}_j^{\mathsf w}=\{\pi^{0}(j),\ldots,\pi^{p-1}(j)\}$$ for $j\in[e]$ as the set containing the elements in column $j$ of $\bm I_{\mathsf w}$. Then, we assign the submatrices $\{{\boldsymbol{W}}_l\mid l\in{\mathcal{I}}_j^{\mathsf w}\}$ to EN $\mathsf{e}_j$. For example, if $\pi=(0\;e-1\; e-2\; \cdots\; 1)$, we have $$\begin{aligned}
\bm{I}_{\mathsf w}=\left(\begin{matrix}
0 & 1 & \cdots & e-1\\
e-1 & 0 & \cdots & e-2\\
\vdots & \vdots & \ddots & \vdots\\
e-p+1 & e-p+2 & \cdots & e-p
\end{matrix}\right),
\end{aligned}$$ and EN $\mathsf{e}_1$ stores ${\boldsymbol{W}}_1, {\boldsymbol{W}}_{0}, {\boldsymbol{W}}_{e-1},\ldots,{\boldsymbol{W}}_{e-p+2}$.
The ENs process the assigned submatrices of ${\boldsymbol{W}}$ in the same order as their indices appear in the rows of $\bm{I}_{\mathsf{w}}$, and we define $\phi_j^\mathsf{w}(l)$ for $l\in [p]$ to be the map to the index of the $(l+1)$-th assigned submatrix of EN $\mathsf{e}_j$.
Assignment of Shares to the Edge Nodes
--------------------------------------
Given the assignment of the submatrices of ${\boldsymbol{W}}$, we now have to assign the shares in such a way that we can guarantee that the users obtain the computations in \[Eq: RecoveryCondition\]. The users upload their shares to the $e$ ENs according to the following assignment. Given the generator $\pi$ used to assign the submatrices of ${\boldsymbol{W}}$ to the ENs, we construct a $(\beta+1)\times e$ index matrix $$\begin{aligned}
\label{Eq: Is}
\bm I_{\mathsf s}={\scalebox{0.972}{\ensuremath{\left(\begin{matrix}
\pi^{0}(0) & \pi^{0}(1) & \cdots & \pi^{0}(e-1)\\
\pi^{e-p}(0) & \pi^{e-p}(1) & \cdots & \pi^{e-p}(e-1)\\
\vdots & \vdots & \ddots & \vdots\\
\pi^{\beta(e-p)}(0) & \pi^{\beta(e-p)}(1) & \cdots & \pi^{\beta(e-p)}(e-1)
\end{matrix}\right)}}},
\end{aligned}$$ where $\beta=\left\lceil e/p\right\rceil-1$. Define the set of indices $$\label{Eq:setIs}
\mathcal{I}_j^{\mathsf{s}}=\{\pi^{0}(j),\ldots,\pi^{\beta(e-p)}(j)\}\backslash\{n, n+1,\ldots,e-1\}$$ as the subset of elements in column $j$ of $\bm I_{\mathsf s}$ that are in $[n]$. User $\mathsf{u}_i$ transmits the shares $\{\bm s_i^{(h)}\mid h\in\mathcal{I}_j^{\mathsf{s}}\}$ to EN $\mathsf{e}_j$. Thereby, every EN receives $a = \left\lceil \lceil e/p\rceil \cdot n/e\right\rceil$ shares from each user as we keep only a fraction $\left\lceil n/e \right\rceil$ of the shares corresponding to the $\beta + 1 = \left\lceil e/p\right\rceil$ used permutations in $\bm I_{\mathsf s}$, i.e., of the indices in $[e]$ we keep only those in $[n]$. As for the submatrices of ${\boldsymbol{W}}$, the shares are processed in the same order as their indices appear in the rows of $\bm{I}_{\mathsf{s}}$, and we define $\phi_j^\mathsf{s}(h)$ for $h\in [a]$ to be the map to the index of the $(h+1)$-th assigned matrix of shares of EN $\mathsf{e}_j$. For a given matrix of shares assigned to an EN, all assigned submatrices of ${\boldsymbol{W}}$ are processed before moving on to the next matrix of shares. What remains to be shown is that these combined assignments of submatrices and shares to the ENs allow all users to obtain enough partial computations from the $e$ ENs to retrieve their desired computations.
\[Th: recover\] Consider an edge computing network consisting of $u$ users and $e$ ENs, each with a storage capacity corresponding to a fraction $\mu$, $0 < \mu \leq 1$, of ${\boldsymbol{W}}$, and an $(n,k)$ SSS, with $n\leq e$. For $j\in[e]$, EN $\mathsf{e}_j$ stores the submatrices of ${\boldsymbol{W}}$ from the set $\{{\boldsymbol{W}}_l\mid l\in{\mathcal{I}}_j^{\mathsf w}\}$ with ${\mathcal{I}}_j^{\mathsf w}$ defined in \[Eq:setIw\]. Furthermore, it receives the matrices of shares from the set $\{\bm S^{(h)}\mid h\in\mathcal{I}_j^{\mathsf{s}}\}$ with $\mathcal{I}_j^{\mathsf{s}}$ defined in \[Eq:setIs\], and computes and returns the set $\{{\boldsymbol{W}}_l\bm S^{(h)} \mid l\in{\mathcal{I}}_j^{\mathsf w}, h\in\mathcal{I}_j^{\mathsf{s}} \}$ to the users. Then, all users can recover their desired computations $\{{\boldsymbol{W}}\bm x_i \}$.
Due to lack of space, we omit the proof of \[Th: recover\]. We motivate the theorem, however, with the following example.
Consider $e=n=5$, $p=3$, and $\pi=(0\;3\;1\;4\;2)$, the generator of a cyclic permutation group of order $5$. From \[Eq: Iw,Eq: Is\], we have $$\begin{aligned}
\bm I_{\mathsf w}&=\left(\begin{matrix}
0 & 1 & 2 & 3 & 4\\
3 & 4 & 0 & 1 & 2\\
1 & 2 & 3 & 4 & 0
\end{matrix}\right) \text{ and }
\bm I_{\mathsf s}&=\left(\begin{matrix}
0 & 1 & 2 & 3 & 4\\
1 & 2 & 3 & 4 & 0
\end{matrix}\right).
\end{aligned}$$ We focus on the matrix of shares $\bm S^{(0)}$. It is assigned to EN $\mathsf{e}_0$ and gets multiplied with the submatrices of $\bm W$ indexed by the elements of the set $$\begin{aligned}
{\mathcal{I}}_0^{\mathsf w} = \{\pi^0(0), \pi(0), \pi^2(0)\}=\{0,3,1\}.
\end{aligned}$$ Note that the set ${\mathcal{I}}_0^{\mathsf w}$ contains three recursively $\pi$-permuted integers of $0$ ($\pi^0(0)$, $\pi^1(0)$, and $\pi^2(0)$). Now, consider EN $\mathsf{e}_4$, which is also assigned the matrix of shares $\bm S^{(0)}$. We have $$\begin{aligned}
{\mathcal{I}}_4^{\mathsf w} = \{\pi^0(4), \pi(4), \pi^2(4)\}=\{4,2,0\}.
\end{aligned}$$ Notice that $\pi^0(4) = \pi^3(0)=4$ is the fourth recursively $\pi$-permuted integer of $0$. Hence, the set ${\mathcal{I}}_0^{\mathsf w}\cup{\mathcal{I}}_4^{\mathsf w}$ contains in total six recursively $\pi$-permuted integers of $0$, which is sufficient to give the set $[5]$, since the group generated by $\pi$ is transitive. In a similar way, it can be shown that the same property holds for all other matrices of shares. Therefore, each matrix of shares is multiplied with all submatrices of $\bm W$, and the sets in \[Eq: RecoveryCondition\] are obtained.
Communication and Computation Scheduling
========================================
In this section, we describe the scheduling of uploading the assigned shares to the ENs, performing the computations, and downloading a subset of $\{{\boldsymbol{W}}_l\boldsymbol{S}^{(h)} \mid l \in \mathcal{I}_j^{\mathsf{w}}, h\in\mathcal{I}_j^{\mathsf{s}},j \in [e]\}$. In the following, we refer to a single ${\boldsymbol{W}}_l\boldsymbol{S}^{(h)}$ as an intermediate result (IR).
Upload and Computation
----------------------
As ${\boldsymbol{W}}$ stays constant for a long time, the assignment of the submatrices $\{{\boldsymbol{W}}_l \mid l \in [e] \}$ can be done offline and does not affect the overall latency. The online phase starts with the upload of the shares. In contrast to the nonprivate scheme in [@Osvaldo], a user $\mathsf{u}_i$ can not broadcast one vector to all ENs. Instead, the user has to unicast a number of shares to each EN to assure that any $z$ ENs do not obtain any information about ${\boldsymbol{x}}_i$. In general, broadcasting a message to $e$ receivers is more expensive than transmitting a single unicast message to one receiver. As in [@Lee], we assume that broadcasting to $e$ receivers is a factor $\log(e)$ more expensive in terms of latency than a single unicast. Recall that the cost (or normalized latency) of unicasting $u$ symbols from $\text{GF}(q)$ is $\gamma$. Hence, in the nonprivate scheme the normalized latency of every user broadcasting one vector from $\text{GF}(q)^r$ to all $e$ ENs is $\mathsf{\tilde{L}}^\mathsf{up}_\mathsf{NP} = \gamma \cdot r\cdot \log(e)$.
In contrast, the normalized latency of unicasting $u$ shares, one from each user, which are elements in $\text{GF}(q)^r$, to one EN is $\gamma r$. Recall that each EN receives $a$ matrices of shares. We assume that each user can upload only one share to one EN at a time. The upload is illustrated in \[fig:schedule\], in which the blue segments correspond to the upload phase. We start by uploading the first matrix of shares to EN $\mathsf{e}_0$, continue with EN $\mathsf{e}_1$, and proceed until all ENs have received their first matrix of shares. This process is repeated with the remaining matrices of shares until EN $\mathsf{e}_j$ has received the $a$ matrices of shares $\{\boldsymbol{S}^{(h)} \mid h\in\mathcal{I}_j^{\mathsf{s}} \}$, $j \in [e]$. EN $\mathsf{e}_j$ receives its $(h+1)$-th matrix of shares $\bm S^{(\phi^{\mathsf{s}}_j(h))}$ at normalized time $$\mathsf{\tilde{L}}^\mathsf{up,h}_j = \gamma r(eh + j+1),$$ and the total normalized upload latency of the private scheme becomes $\mathsf{\tilde{L}}^\mathsf{up}_\mathsf{P} = \gamma \cdot r\cdot e\cdot a$.
After an EN has received its first matrix of shares, it enters the computation phase. As mentioned earlier, the ENs experience a random setup time before they can start their computations. This is illustrated by the red segments in \[fig:schedule\]. For EN $\mathsf{e}_j$ this phase incurs a normalized latency of $\lambda_j/\tau$. Once set up, the ENs start their computations on the first assigned matrix of shares. In total, $p$ IRs of the form ${\boldsymbol{W}}_l\bm S^{(h)}$ have to be computed for each assigned matrix of shares $\bm S^{(h)}$ by EN $\mathsf{e}_j$, where $l \in {\mathcal{I}}_j^{\mathsf w}$ and $h \in {\mathcal{I}}_j^{\mathsf s}$. This incurs a normalized latency of $p\cdot m/e$, because each ${\boldsymbol{W}}_l$ has $m/e$ rows and hence, the ENs compute $u\cdot m/e$ inner products for each of the $p$ IRs.
In the case an EN has not received another matrix of shares before finishing the currently assigned computations, it remains idle until it receives another matrix of shares to compute on. This can be seen in yellow in \[fig:schedule\]. For $h \in [a]$, the normalized time at which EN $\mathsf{e}_j$ starts to compute on the $(h+1)$-th assigned matrix of shares, i.e., on $\bm S^{(\phi^{\mathsf{s}}_j(h))}$, is $$\mathsf{\tilde{L}}^{\mathsf{start},h}_j = \max\left \{\mathsf{\tilde{L}}^{\mathsf{start},h-1}_j + p\frac{m}{e}~,~\mathsf{\tilde{L}}^\mathsf{up,h}_j\right\},\; \text{for $h > 0$},$$ with $$\mathsf{\tilde{L}}^{\mathsf{start},0}_j = \frac{\lambda_j}{\tau} + \mathsf{\tilde{L}}^\mathsf{up,0}_j.$$ The computational phase continues at least until the computations in \[Eq: RecoveryCondition\] are obtained, i.e., until there are at least $k$ distinct IRs of the form ${\boldsymbol{W}}_l \bm S^{(h)}$, $h \in [n]$, for each $l \in [e]$. This ensures that a given user $\mathsf{u}_i$ can recover ${\boldsymbol{W}}{\boldsymbol{x}}_i$. It can be beneficial to continue computing products to reduce the communication latency in the download phase, as we discuss next.
Download
--------
In the download phase we can make use of joint beamforming opportunities to reduce the latency by serving multiple users at the same time. An IR ${\boldsymbol{W}}_l\boldsymbol{S}^{(h)}$ that is computed at $\rho_{l,h}$ ENs incurs a normalized communication latency of $\gamma /\min\{\rho_{l,h},u\}$. Hence, a higher multiplicity of computed IRs across different ENs will reduce the communication latency in the download phase. At the same time, the repeated IRs have to be computed first, thereby increasing the computational latency. This tradeoff can be optimized to reduce the overall latency. Assume the optimum is reached after EN $\mathsf{e}_{j^*}$ has computed the IR ${\boldsymbol{W}}_{\phi^\mathsf{w}_j(l^*)}\boldsymbol{S}^{(\phi^\mathsf{s}_j(h^*))}$. This gives a normalized computation latency of $$\mathsf{\tilde{L}}^\mathsf{comp} = \mathsf{\tilde{L}}^{\mathsf{start},h^*}_{j^*} + (l^*+1) \frac{m}{e}.$$
After the computation phase has finished, the ENs cooperatively send the computed IRs ${\boldsymbol{W}}_l\boldsymbol{S}^{(h)}$ simultaneously to multiple users in descending order of their multiplicity $\rho_{l,h}$ until the computations in \[Eq: RecoveryCondition\] are available to the users. More precisely, for each ${\boldsymbol{W}}_l$ the ENs send the $k$ IRs with the highest multiplicities to the users. Then, a given user $\mathsf{u}_i$ can decode the SSS to obtain the desired computation ${\boldsymbol{W}}{\boldsymbol{x}}_i$. For a fixed $l$, let $\mathcal{H}_l^\mathsf{max} = \arg\max_{\mathcal{A}\subseteq[n],|\mathcal{A}| = k} \sum_{h\in\mathcal{A}} \rho_{l,h}$ be the set of indices $h$ of the $k$ largest $\rho_{l,h}$. This results in a normalized communication latency of $$\mathsf{\tilde{L}}^\mathsf{comm} = \gamma \sum_{l = 0}^{e-1}\sum_{h\in \mathcal{H}_l^\mathsf{max}} \frac{1}{\min\{\rho_{l,h},u\}},$$ and the overall normalized latency becomes $$\label{eq:overall_latency}
\mathsf{\tilde{L}} = \mathsf{\tilde{L}}^{\mathsf{start},h^*}_{j^*} + (l^*+1) \frac{m}{e} + \gamma \sum_{l = 0}^{e-1}\sum_{h\in \mathcal{H}_l^\mathsf{max}} \frac{1}{\min\{\rho_{l,h},u\}}.$$
![Upload and computing schedule. For each EN $\mathsf{e}_j$, the upload times $r\gamma$ are shown in blue, the random setup times in red, the times $pm/e$ to compute $p$ IRs in green, and possible idle times in yellow. All times are normalized.[]{data-label="fig:schedule"}](Figures/Latency-figure0.pdf){width="\columnwidth"}
Optimization and Numerical Results
==================================
We start by explaining how to choose the parameters of the proposed scheme so that the overall normalized latency $\mathsf{\tilde{L}}$ in , consisting of upload, computation, and download latencies, is minimized for a given privacy level $z$. To reduce the upload latency, it may be beneficial to contact fewer ENs than the maximum number of ENs available, denoted by ${e}_{\max}$, to which a user can connect. Additionally, storing fewer than $\mu e$ submatrices of ${\boldsymbol{W}}$ at the ENs can be advantageous, because the ENs will start computations sooner on the later shares. Thus, we can choose $p \leq \mu e$.
From the combinatorial designs, it follows that the number of shares $n$ per user can be at most equal to $e$, while the value of the SSS threshold $k$ is constrained by the choices of $z$, $e$, $n$, and $p$. First, recall that the total number of shares per user assigned to each EN is $a = \left\lceil \lceil e/p\rceil \cdot n/e\right\rceil$, which means that any $z$ ENs have access to $a\cdot z$ possibly distinct shares of each user. Given that this set of shares must not leak any information about the private data $\{{\boldsymbol{x}}_i\}$, we have to pick $k \geq a z + 1$. According to \[Cor: Recovery\], for a given ${\boldsymbol{W}}_l$, waiting for $k$ distinct products allows to recover the computation ${\boldsymbol{W}}_l{\boldsymbol{x}}_i$ for each user $\mathsf{u}_i$. Note that there is no reason to pick $k$ larger than $a z +1$, since then the users have to wait for more products, leading to reduced straggler mitigation and increased computational latency. Therefore, we set $k = a z + 1$. Finally, we need to verify that all constraints on $n$ are fulfilled for the chosen parameter values, i.e., $k\leq n\leq e$ (for the scheme to be feasible), $n\geq k$ (for the SSS to work), and $n\leq e$ (from the combinatorial designs).
We have chosen $\pi = (0\;e-1\;e-2\;\cdots\;1)$ and performed an exhaustive search for the minimum expected overall normalized latency $\mathsf{\tilde{L}}$ given in over all valid parameter tuples $(e, n, p)$ for a given privacy level $z$. For each tuple we varied the number of total (not necessarily distinct) IRs to wait for across all ENs for each ${\boldsymbol{W}}_l$, in order to minimize the latency. We generated $10^6$ instances of the random setup times $\{\lambda_j\}$ in the simulation of the scheme in order to obtain an accurate estimate of the expected overall normalized latency.
In \[fig:plot\_9\], we compare the expected overall normalized latency of the proposed private scheme with the nonprivate MDS-repetition scheme in [@Osvaldo]. We plot the overall normalized latency versus $\gamma$ for different privacy levels $z$. For the presented scenario, the users have access to a maximum of $e_{\max} = 9$ ENs, which can store up to a fraction of $\mu = 2/3$ of the matrix ${\boldsymbol{W}}$ with dimensions $m=600$ and $r=50$. The ENs need $\tau = 0.0005$ time units to compute one inner product over $\text{GF}(q)^{50}$ for each of the users, and the straggling parameter is set to $\eta = 0.8$. Providing privacy against a single EN ($z=1$) yields an increase in latency for $\gamma=8$ by a factor of about $2.4$ compared to the nonprivate MDS-repetition scheme in [@Osvaldo]. For $z=2$, the latency increases by a factor of about $3.5$, while it increases to $5.7$ and $10.0$ for $z=3$ and $4$, respectively.
table\[x=gamma, y=cost\][data/data\_MDS\_hybrid\_9\_0.67\_0.0005\_0.8\_600\_50.txt]{}; ;
table \[x=gamma,y=cost\][data/data\_private\_num\_prods\_corrected\_1000000\_1\_9\_0.6667\_0.0005\_0.8\_600\_50.txt]{}; ;
table \[x=gamma,y=cost\][data/data\_private\_num\_prods\_corrected\_1000000\_2\_9\_0.6667\_0.0005\_0.8\_600\_50.txt]{}; ;
table \[x=gamma,y=cost\][data/data\_private\_num\_prods\_corrected\_1000000\_3\_9\_0.6667\_0.0005\_0.8\_600\_50.txt]{}; ;
table \[x=gamma,y=cost\][data/data\_private\_num\_prods\_corrected\_1000000\_4\_9\_0.6667\_0.0005\_0.8\_600\_50.txt]{}; ;
One of the factors that leads to an increased latency is the upload. In the nonprivate scheme, the users can broadcast their private data vectors to all ENs simultaneously, whereas in the private scheme, the users have to unicast their shares to the ENs sequentially. In \[fig:upload\_cost\], we show the impact of the upload on the proposed private scheme and the nonprivate MDS-repetition scheme in [@Osvaldo]. For both schemes, for $\gamma = 8$, the upload takes about $13\%$ of the overall latency, which means that for the private scheme the latency increases by around $1500$ time units, whereas for the nonprivate scheme it increases by only $700$ time units.
table\[x=gamma, y=cost\][data/data\_MDS\_hybrid\_6\_0.67\_0.0005\_0.8\_600\_50.txt]{}; ;
table\[x=gamma, y=cost\][data/data\_MDS\_hybrid\_6\_0.67\_0.0005\_0.8\_600\_0.txt]{}; ;
table \[x=gamma,y=cost\][data/data\_private\_num\_prods\_corrected\_1000000\_1\_6\_0.6667\_0.0005\_0.8\_600\_50.txt]{}; ;
table \[x=gamma,y=cost\][data/data\_private\_num\_prods\_corrected\_1000000\_1\_6\_0.6667\_0.0005\_0.8\_600\_0.txt]{}; ;
Conclusion
==========
We presented a privacy-preserving scheme that allows multiple users in an edge computing network to offload computations to edge servers for distributed linear inference, while keeping their data private to a number of edge servers or their corresponding communication links. The proposed scheme uses secret sharing to provide user data privacy and mitigate the effect of straggling servers, and partial repetitions to enable joint beamforming opportunities in the download phase in order to reduce the communication latency. The parameters of the scheme were optimized in order to minimize the overall latency incurred by the upload of data to the servers, the computation, and the transmission of partial computations back to the users.
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Y. C. Hu, M. Patel, D. Sabella, N. Sprecher, and V. Young, “Mobile edge computing—a key technology towards 5G,” *ETSI white paper*, vol. 11, no. 11, pp. 1–16, Sep. 2015.
K. [Lee]{}, M. [Lam]{}, R. [Pedarsani]{}, D. [Papailiopoulos]{}, and K. [Ramchandran]{}, “Speeding up distributed machine learning using codes,” *IEEE Trans. Inf. Theory*, vol. 64, no. 3, pp. 1514–1529, Mar. 2018.
S. [Li]{}, M. A. [Maddah-Ali]{}, and A. S. [Avestimehr]{}, “A unified coding framework for distributed computing with straggling servers,” in *Proc. IEEE Globecom Workshops (GC Wkshps)*, Washington, DC, Dec. 2016.
A. [Severinson]{}, A. [Graell i Amat]{}, and E. [Rosnes]{}, “Block-diagonal and LT codes for distributed computing with straggling servers," *IEEE Trans. Commun.*, vol. 67, no. 3, pp. 1739–1753, Mar. 2019.
A. [Severinson]{}, A. [Graell i Amat]{}, E. [Rosnes]{}, F. [Lázaro]{}, and G. [Liva]{}, “A droplet approach based on Raptor codes for distributed computing with straggling servers," in *Proc. Int. Symp. Turbo Codes Iterative Inf. Processing (ISTC)*, Hong Kong, China, Dec. 2018.
K. [Li]{}, M. [Tao]{}, and Z. [Chen]{}, “Exploiting computation replication for mobile edge computing: A fundamental computation-communication tradeoff study," to app. *IEEE Trans. Wireless Commun.*.
K. [Li]{}, M. [Tao]{}, and Z. [Chen]{}, “A computation-communication tradeoff study for mobile edge computing networks,” in *Proc. IEEE Int. Symp. Inf. Theory (ISIT)*, Paris, France, Jul. 2019, pp. 2639–2643.
J. [Zhang]{} and O. [Simeone]{}, “On model coding for distributed inference and transmission in mobile edge computing systems,” *IEEE Commun. Lett.*, vol. 23, no. 6, pp. 1065–1068, Jun. 2019.
L. [Kuikui]{}, M. [Tao]{}, J. [Zhang]{}, and O. [Simeone]{}, “Multi-cell mobile edge coded computing: Trading communication and computing for distributed matrix multiplication," to app. *IEEE Int. Symp. Inf. Theory (ISIT)*, Los Angeles, CA, Jun. 2020.
R. [Bitar]{}, P. [Parag]{}, and S. [El Rouayheb]{}, “Minimizing latency for secure coded computing using secret sharing via staircase codes," to app. *IEEE Trans. Commun.*.
R. Bitar, Y. Xing, Y. Keshtkarjahromi, V. Dasari, S. El Rouayheb, and H. Seferoglu, “PRAC: Private and rateless adaptive coded computation at the edge", in *Proc. SPIE Defense + Commercial Sensing*, Baltimore, MD, May 2019.
J. Dean and L. A. Barroso, “The tail at scale,” *Commun. ACM*, vol. 56, no. 2, pp. 74–80, Feb. 2013.
A. Mallick, M. Chaudhari, U. Sheth, G. Palanikumar, and G. Joshi, “Rateless codes for near-perfect load balancing in distributed matrix-vector multiplication,” *Proc. ACM Meas. Anal. Comput. Syst.*, vol. 3, no. 3, pp. 58:1–58:40, Dec. 2019.
J. [Zhang]{} and O. [Simeone]{}, “Fundamental limits of cloud and cache-aided interference management with multi-antenna edge nodes,” *IEEE Trans. Inf. Theory*, vol. 65, no. 8, pp. 5197–5214, Aug. 2019.
N. [Naderializadeh]{}, M. A. [Maddah-Ali]{}, and A. S. [Avestimehr]{}, “Fundamental limits of cache-aided interference management," *IEEE Trans. Inf. Theory*, vol. 63, no. 5, pp. 3092–3107, May 2017.
A. Shamir, “How to share a secret,” *Commun. ACM*, vol. 22, no. 11, pp. 612–613, Nov. 1979.
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COLO-HEP-98/408\
hep-th/9806178\
June 1998
[On the Supergravity Gauge theory Correspondence and the Matrix Model]{}
[S. P. de Alwis[^1]]{}\
[*Department of Physics, Box 390, University of Colorado, Boulder, CO 80309.*]{}\
[**[Abstract]{}**]{}
We review the assumptions and the logic underlying the derivation of DLCQ Matrix models. In particular we try to clarify what remains valid at finite $N$, the role of the non-renormalization theorems and higher order terms in the supergravity expansion. The relation to Maldacena’s conjecture is also discussed. In particular the compactification of the Matrix model on $T_3$ is compared to the $AdS_5\times S_5$ ${\cal N}=4$ super Yang-Mills duality, and the different role of the branes in the two cases is pointed out.
Introduction
============
There appear to be two conjectures on the relation between gauge theory and gravity. One is the Matrix model [@bfss] which was originally proposed as a microscopic theory whose low-energy limit is 11 dimensional supergravity. The other is the more recent conjecture on the relation between gauge theory and supergravity [@ik],[@jm],[@gkp],[@ew] whose clearest manifestation is in the correspondence between ${\cal N} =4$ $SU(N)$ four dimensional Yang-Mills theory and supergravity (string theory?) on a $AdS_5\times S_5$ background. The Matrix model can also be compactified and in particular on a three torus, it is supposed to be represented by the same Yang-Mills theory . One of the purposes of this investigation is to elucidate the connection between the two conjectures [^2]. The other purpose is to understand why finite $N$ calculations work at least in certain cases.
In the next section we will review the arguments given in [@as], [@ns] for obtaining the Matrix model. In the course of the discussion we will try to be careful about the logic of these arguments by distinguishing between what is actually derived and that which is still conjecture. In particular by expanding on arguments given in [@sda] we will try to explain precisely what the connection to supergravity should be. We will also comment on exactly what is achieved by the recently proven non-renormalization theorems for the model in relation to the connection between gauge theory and gravity.
In the third section we will discuss the correspondence between the higher order terms in the supergravity expansion and the non-renormalization theorem. We will point out that the latter imposes certain regularities in the supergravity terms and we will also identify the supergravity terms from which certain non-diagonal terms (in the terminology of [@bbpt] ) in the Matrix model expansion arise. In the third section we will briefly review the recent work [@jm],[@gkp],[@ew] on the gauge theory/gravity connection. In particular we will compare and contrast this with the Matrix model conjecture. The natural place for this is clearly the $AdS_5\times S_5$ supergravity/string theory, ${\cal N}=4
$ four dimensional Yang-Mills correspondence. In particular we will argue that although in the interpretation of this connection given in [@ew] the gauge theory is located at the boundary of the space-time, in the Matrix model the whole space is supposed to be the moduli space of the gauge theory. In fact there is a singularity at the origin which is to be interpreted as a break down of the moduli space approximation and is to be replaced by the non-Abelian quantum dynamics. Alternatively from the supergravity point of view one may regard the singularity as being resolved by the branes which are sitting there.
On the Matrix model
===================
We begin by summarizing the arguments of Seiberg [@ns] which suggest a connection to D0 quantum mechanics of the Discrete Light Cone Quantization (DLCQ) (i.e. the quantization of the theory compactified on a null circle) of M-theory.
a\) A microscopic Lorentz invariant M-theory should include a framework for calculating scattering amplitudes of the fundamental degrees of freedom (the supergravitons ?). At low energies these amplitudes should yield 11 dimensional supergravity. (This is exactly what happens in string theory. There is a Lorentz covariant formulation, which yields by general arguments on the consistent coupling of spin two fields, the 10 D supergravity low energy effective action. The challenge in M theory is to find the analog of this.)
b\) Given a theory satisfying a) its compactification on a null circle will yield scattering amplitudes which at low energies become those of 11 D supergravity compactified on a null circle.
c\) The theory compactified on a null circle (of radius R) is related by an infinite boost to the theory compactified on a space-like circle. The study of states in DLCQ M theory (with Planck length $l_P$ and finite values of light cone energy $P_+$) is most conveniently done in terms of a $\tilde M$ theory compactified on a space-like circle with vanishing radius $R_s$ and a vanishing Planck length $\tilde l_P$ such that $$\label{seilimit}
{R_s\over\tilde l_P^2}={R\over l_P^2},~~{\tilde R_i\over \tilde l_P}=
{R_i\over
l_P}$$ where the right hand sides are fixed.
d\) This limit of $\tilde M$ theory is equivalent to string theory in a certain regime. Namely one where $$\label{limit}
l_s\rightarrow 0;~~ g^2_{YM}\equiv{1\over l_m^3}={g_s\over
l_s^3}={R^3\over
l_P^6}
{}~fixed,~~{ R_i\over l_s^2}={R_i\over l_ml_P}\equiv U_i~fixed.$$ In the above we have introduced the string scale $l_S$ and string coupling[^3] $g_S$ which are related to the $\tilde M$ quantites by $$\label{}
\tilde l_P= g_s^{1/3} l_s,~ R_s= l_s g_s$$ This limit is often referred to as the DKPS limit and we will use this name for it . Note that the radius of the null circle $R$ has no physical significance and we may conveniently set $R=l_P$ so that the length scale set by the gauge theory may be identified with the Planck length, $l_P=l_m$.
e\) String theory in the regime defined in d) is given by D0-brane quantum mechanics; i.e. $U(N)$ quantum mechanics with 16 supercharges where N is the number of D0-branes and this corresponds to the sector with $P_+=N/R$ in the original $M$ theory.
In the above list a) is clearly influenced by what happens in string theory and b) is certainly very plausible. c) on the other hand involves an infinite boost and thus may be problematic but for the purposes of this paper we will assume that it is meaningful. d) involves a hidden assumption that is normally not made explicit. The relation between M theory and string theory is established only at the level of the effective actions. What is assumed here is that this relation holds also at the microscopic level. However this is a standard and plausible assumption that we will not question here.
The real problem is e). The (perturbative) string action (i.e. the sigma model action) is not defined in this limit (\[limit\]). In fact all D-brane actions are also ill-defined in the limit (since the tensions become infinite) except for the D0-brane action. If one took the open string representation of the latter, it becomes the quantum mechanics action $$\label{action}
S_{QM} =-{1\over 4g^2_{YM}}\int_{W_{1}}\tr (D_{\a}X_iD^{\a}X_i
+{1\over 4}[X_i,X_j]^2)+fermion~terms.$$ since the higher order terms in $\a '$ disappear. Here the $X_i$ are the ten dimensional gauge fields which in this case are to be interpreted as operators governing the position fluctuations of the branes.
However it is the closed string representation of this action that is directly related to the Kaluza-Klein reduction of the 11 D graviton. (see for example [@pt]). This action for a D0-brane in a background field given by the metric $g$ and RR field $C$ is $$\label{daction}
S_{sugra}=-{1\over gl_s}\int dt e^{-\phi}\sqrt{\det g}+{1\over
l_s}\int C$$ One would then expect a relation of the form$$\label{gagrav}
\int dX'e^{iS_{QM}[U+X']} = \lim_{DKPS} e^{iS_{sugra}[U]}.$$ between these two when $g$ and $C$ are due to a cluster of D0 -branes and $S_{sugra}$ is the supergravity representation of the probe brane action when it is a distance $U=r/l_s^2$ (in units with mass dimension!) from the cluster and moving with velocity $\dot U=v/l_s^2$. It is precisely relations of this sort that must be established if the gauge theory gravity connection implied by the arguments of [@bfss], [@as], [@ns] is to be proven. The problem is that the supergravity form of the action is meaningful when a massless closed string representation is valid i.e. when $r/l_s>1$, whereas the DKPS limit takes us to $r/l_s = l_s U\rightarrow 0$.
The supergravity solution corresponding to N zero branes is given by [@hs] $$\begin{aligned}
\label{metten}
ds_{10}^2&=&-H_0^{-1/2}dt^2+H_0^{1/2}dx^idx^i \nn
e^{-\phi}&=&H_0^{-3/4},~~~C_t=H_0^{-1}-1.\end{aligned}$$ where $H=1+h,~h={Nc_0gl_s^7\over r^7}$ and $c_0$ is a known constant whose value is irrelevant for our purposes. If we lift this solution to 11 dimensions using the standard formulae (see for example [@pt]) then we get $$\begin{aligned}
\label{meteleven}
ds_{11}^2&=&-(1-h)dt^2-2hdx^{11}dt+(1+h)dx^{11~2}+dx^{i2} \nn
&=&2d\tau dx^{-}+hdx^{-2}+dx^{i2} \nn
&=&e^{-2\bar\phi /3}\bar{ds}_{10}^2+e^{4\bar\phi /3}(dx^-+\bar
C_{\tau}d\tau)^2\end{aligned}$$ where in the last equation, $$\label{}
\bar{ds}_{10}^2=-h^{-1/2}d\tau^2+h^{1/2}dx^{i~2},~e^{-2\bar\phi
/3}=h^{-1/2},
{}~\bar C_{\tau}
=h^{-1}.$$ In particular the ten dimensional metric above is just the (asymptotically) light like compactified Aichelburg-Sexl [@as] metric which can be rewritten as. $$\label{metll}
\bar{ds_{10}}^2=l_s^2(-\bar{h}^{-1/2}d\tau^2+\bar{h}^{1/2}(dU^{2}+U^2d
\O^2_8)),$$ where $$\label{barh}
\bar h =l_s^4 h={c_0Ng^2_{YM}\over U^7}.$$ The argument above was given in essence in [@bbpt] and elaborated on in [@kk].
On the other hand let us consider again the 10 dimensional metric (\[metten\]) and take the limit (\[limit\]). This limit also leads to the light-like compactified M-theory metric (\[metll\]) except that we now have $\tau\rightarrow t$. Thus we might expect that this fact on the supergravity side of the D0-brane metric is reproduced by the gauge theory on the D0-brane in the same limit. In other words what we should expect is (\[gagrav\]). However as mentioned earlier the problem is that this limit gives us a region of string theory which takes us to substring scales where supergravity is not expected to be valid. Thus it is far from obvious that all graviton scattering amplitudes should be reproduced by the Matrix model.
Let us now review the argument of [@sda] in the light of the above discussion. The idea is to explain the agreement of the calculation of [@bb], [@bbpt] by using string theory as the interpolating theory connecting supergravity and gauge theory. In the above mentioned references the gauge theory effective action was calculated in a background corresponding to a situation in which one brane is separated from the rest by a distance $r$ and moving with some velocity $v$. In terms of the variables in the gauge theory this means that a variable $U=r/l_s^2$ and $\dot U=v/l_s^2$ have acquired expectation values. In the limit $l_s\rightarrow 0$ with $U$ fixed, since $r\rightarrow 0$, the physical separation of the branes are below the string scale and are best described by the gauge theory. Using dimensional analysis the perturbative expansion is given by [@bbpt] $$\label{gauge}
C_{I,L}(N)g^{2L-2}_{YM}{\dot U^I\over U^{3L+2(I-2)}}=
C_{I,L}(N){\dot U^2\over g^2_{YM}}
\left ({g^2_{YM}\dot U^2\over U^{7}}\right )^L
\left ({\dot U\over U^2}\right )^{I-2L-2}.$$ Before we go onto discuss the argument further it is important to stress the meaning of the recently proven non-renormalization theorem[@pss] in this context. Firstly it is clear purely from the dimensional analysis that the numerical coefficient of a given $\dot U^I\over U^N$ term can get a contribution only from the $L=(N-2(I-2))/3$ loop level. In particular this means that $\dot U^4/U^7$ term only gets a contribution from one loop and that the $\dot U^6/U^{14}$ from two loops. There is no question of renormalization of these numerical coefficients and so the agreement of these with supergravity cannot possibly be affected by going to strong coupling. [*Thus the non-renormalization theorem is irrelevant for the purpose of explaining this numerical agreement with supergravity*]{}. What it does tell us is that the only power of $U$ which comes with the $\dot U^4$ term is $U^{-7}$ and that the only one which comes with $\dot U^{6}$ is $U^{-14}$. The relation of this fact to supergravity will be discussed in the next section. The numerical agreement with supergravity still needs to be explained and this is precisely what was done in [@sda].
The corresponding open string perturbation expansion is obtained by replacing the coefficients $C_{I.L}(N)$ by functions $C_{I.L}(N,l_sU)$ and it was argued in [@sda] that $C_{I.L}(N,0)=C_{I.L}(N)$ [^4]. On the other hand for $l_sU={r\over l_s}$ greater than some critical value (say 1) the physics can be described by closed string fields. In this region one typically writes the effective action in a power series in $l_s^2\cal R$ but one may expect it to be convergent giving some effective action functional $S[g,\phi, C,
l_s]$ ($C$ stands for the RR field). Now in this closed string formalism a D0-brane is represented by the action (\[daction\]).
In the configuration that we are considering the closed string fields have the solutions given in (\[metten\]) to lowest order in $l_s^2$. Suppose now the solution to the exact effective action $S$ is known. This solution when plugged into (\[daction\]) will have an expansion of the same form as (\[gauge\]) but with the coefficients $C_{I.L}(N)$ replaced by functions $C_{I.L}^{SG}(N,l_sU)$. These functions (since they are obtained from the exact action functional for closed string fields) would be analytic continuations of the corresponding power series obtained from the $\a '$ expansion. Thus they must be the same as $C_{I.L}(N,l_sU)$ in the region $l_sU<1$ and in particular at $l_sU=0$. However it turns out that the [*exact*]{} value of the so-called diagonal coefficients $C_{2L+2,L}(N)
=C_{2L+2,L}^{SG}(N,0)$ can be calculated simply from the leading term of the closed string expansion. To see this we first need to plug in the leading order supergravity solution into (\[daction\]) and then take the limit $l_s\rightarrow 0$. This gives the (finite!) result[^5] $$\label{sugra}
-{1\over g^2_{YM}}k^{-1}(\sqrt{1-k\dot{U}^2}-1) ,$$ where $k\equiv{cg_{YM}^2N\over U^7}$ with $c$ a known constant. Now the important point is that one expects the DKPS limit of the full $\a'$ expansion to go over into the light like compactification of the corresponding low energy M-theory expansion (this is now a quantum M-theory expansion). But purely on dimensional grounds none of the higher derivative terms in the expansion can contribute to correcting the numerical coefficients of the “diagonal terms" which occur in the expansion of (\[sugra\]) (see [@bbpt] and the discussion in the next section). Thus the analytically continued value of the diagonal functions $C_{2L+2.L}(N,l_sU)$ at the origin $l_s=0$ are given by the leading supergravity values obtained from (\[sugra\]). This argument then explains why the supergravity calculation agrees with the loop expansion calculation in gauge theory.
Now the above argument did not actually use large $N$. This is just as well since the calculations of [@bb], [@bbpt] were done for $N=2$ but they still agreed with supergravity. [*The reason is that regardless of the value of N only the leading term in the supergravity expansion contributes to the diagonal ($I=2L+2$) terms*]{}. Thus one does not need a suppression of the higher powers of $R$. However there are other comparisons between the gauge theory calculations and supergravity which involve at least two scales where finite $N$ calculations disagree with supergravity. The classic case is the calculation of Dine and Rajaraman [@dr]. In this case the argument used above does not apply directly (though there may be a generalization of it). The reason is that in the above discussion we have used the limit (\[limit\]) of the probe action in a background solution of supergravity corresponding to a cluster of coincident D0-branes which can be lifted to eleven dimensions and identified with the Aichelburg-Sexl metric (averaged over the light like circle). In the more complicated case of [@dr] (and also the cases considered in [@dos],[@kt]) there is no corresponding argument whence one can regard the scattering of three gravitons to three in terms of the action of one probe. However if recent work [@oy] which contradict [@dr] is correct, (see also [@ffi],[@tv]) there is possibly a more general argument than the one given above that shows agreement between the finite $N$ Matrix model and arbitrary supergravity processes in a background with one light like compactified circle. On the other hand there are processes [@kt] where the finite $N$ argument is definitely violated but agreement is obtained at large $N$. This does not necessarily mean that only the large $N$ result of the Matrix model is reliable. What it does mean is that both bound state effects and higher order supergravity terms must be taken into account when such comparisons are being made. The simple dimensional arguments that enabled us to conclude that only the leading order supergravity term contributes to the diagonal terms for instance may not be valid. In fact as we shall see in the next section agreement of even the one loop Matrix model calculation with supergravity for the two graviton to two graviton case requires taking into account the higher derivative terms in the supergravity side. Thus one should not in general expect agreement with just the contributions from the Einstein term.
On the non-renormalization theorem and supergravity
===================================================
In order to get some perspective on this issue[^6] it is necessary to recall some history. In the BFSS paper it was stated after their observation (based on the calculation of [@dkps]) that the $v^4/r^7$ term [^7] in the Matrix model agreed with the 11D supergravity calculation of two graviton scattering at zero momentum transfer, that a non-renormalization theorem was needed in order to protect this agreement. Since there was no discussion of $R^4$ and higher derivative terms on the supergravity side the point they were making presumably was that since on the supergravity side the calculation gave only the term $v^4/r^7$ at order $v^4$ (i.e. that there are no other powers of $1/r$) this should be the only contribution in the Matrix model as well. The situation is much more complicated however, since first of all the Matrix model (or string theory) one loop calculation has an infinite number of non-vanishing terms. Thus even for agreement with the one loop Matrix model calculation one needs on the supergravity side (an infinite number of) higher derivative terms. In fact we may reverse the logic that led to the above quoted statement from BFSS and ask what restrictions the non-renormalization theorems have on the supergravity expansion.
As pointed out in [@rt], comparison with type II strings implies that the M-theory low energy expansion has (very schematically) the following form, $$\label{R}
S\sim \sum^{\infty}_{r=0}l_p^{3r-9}\int ``R"^{3r+1}.$$ The inverted commas are a reminder of the fact that in general there may be covariant derivatives as well as Riemann tensors so that the counting is in powers of squared derivatives. The first term here is the Einstein term. The second term is the by now well-known $R^4$ derivative term [^8], $$\label{}
t^{\mu_1...\mu_8}t^{\nu_1...\nu_8}R_{\mu_1..\nu_2}\ldots
R_{\mu_7..\nu_8}.$$ Where $t$ is a rank eight tensor constructed out of the metric. It is important to note that at the eight derivative level there are no covariant derivative terms in the action.
First let us note that the structure of this series is exactly what is required for agreement with the Matrix model expansion[^9]. This is simply because the expansion is in integer powers of $l_p^3$ and therefore fits in with the expansion in $g_{YM}\equiv{1\over l_m^3}$ since $l_m$ is to be identified with the Planck length. The contribution of the Einstein term was discussed above and it gives exactly the diagonal $I=2L+2$ terms in the Matrix model expansion.
The comparison with the Matrix model, of contributions from this $R^4$ term, was made in [@kk2](see also [@bb2]) where the basic technique for going beyond the Einstein term was developed. Let us first briefly review their method. Write the metric as $$\label{}
ds^2=(\eta_{\mu\nu}+\D_{\mu\nu})dx^{\mu}dx^{\nu}$$ where $$\label{}
\D_{\mu\nu}=h_{--}\d_{\mu}^-\d_{\nu}^--\k f_{\mu\nu}$$ The first term on the right hand side is the Aichelberg-Sexl metric which is an exact solution to the string effective action (\[R\]) (see [@bbpt] and references therein). The second term is a small perturbation due to the probe. Thus we assume that $f<<1$ so that the metric does not change significantly. Substituting in (\[R\]) we keep only the quadratric terms. It is important to note that the linear terms vanish since $f=0$ gives the Aichelburg-Sexl metric whtich is an exact solution to the quantum corrected equations of motion. Now for small enough $f$ we can choose the transverse traceless gauge for $f$ so that in particular $(\mu =+,-,i, \tau =x^{+}/2 $ as in section two) only $f_{ij}\ne 0$. The contribution from the $R^4$ term is of the form (using the $SO(9)\times SO(1,1)$ symmetry of the configuration and the fact that $h$ depends only on $r=\sqrt{(x^i)^2}$) is schematically of the form $\pa_+^2f\pa_+^2f\pa_{\perp}^2h_{--}\pa_{\perp}^2h_{--}$ where the subscript $\perp$ denotes transverse components. Thus we have the equation of motion, $$\label{rfour}
(-\pa_+\pa_--\pa_{\perp}^2+h\pa_+^2)f_{ij}+
b\pa_+^4f_{ij}\pa^2_{\perp}h_{--}\pa^2_{\perp}h_{--}=0$$ Writing $f\sim e^{ixp}$ we have, solving iteratively for the Routhian, [^10] $$\label{}
L'=L-p_-\dot x^- =p_i\dot x^i+p_{\tau}={p_-\over
h}(1-\sqrt{1-h_{--}v_{\perp}^2})+\D L'$$ The first term here is the exact solution to the Einstein term alone and corresponds to the diagonal terms of the Matrix model expansion as discussed in the previous section. In the case considered here we have from (\[rfour\]) the result, $$\label{}
\D L'\sim {p_{\tau}^4\over
p_-}(\pa_{\perp}^2h_{--})^2={N_p^3N_s^2v_{\perp}^8\over R^7
r^{18}}+\ldots .$$ In the last step we’ve used the formulae $p_{\tau}\sim
{p_{\perp}^2\over
p_-}\sim p_-v^2$ which are valid to leading order in $h$ and $p_-={N\over R}$. This term is not ruled out by the non-renormalization theorem (which only restricts the $v^4$ and $v^6$ terms). However its $N$ depends disagrees with the naive perturbative N dependence which must go like $N_pN_s^2$. We will find more such disagreements later and we assume that such disagreements are to be expected since bound state effects will almost certainly affect the N dependence of the perturbation series[^11].
It is actually easy to see that these $R^4$ terms will not contribute to renormalizing the $v^4$ or $v^6$ terms. This is because, as can be seen from (\[rfour\]), in order to maintain the SO(1,1) invariance the term must have four powers of $\pa_+$ and this leads to at least eight powers of $v$. It should be stressed that the form (\[rfour\]) obtains, because of the absence of covariant derivative terms in the $R^4$ term.
At this point one might wonder from whence the infinite number of non-vanishing one-loop terms on the gauge theory side namely terms like $v^8/r^{15}$ etc.[^12] come. This term clearly does not arise from the $R^4$ term so it has to come from a $R^7$ term or higher order term. It is easy to see that this term cannot come from a pure (i.e. with no covariant derivatives) term. In fact it comes from a 14 derivative term of the form $R\na^2 R\na^6R$. This leads to a term of the form $$\label{}
\pa_{\perp}^2 f_{ij}\pa_{\perp}^2\pa_+^2f_{ij}\pa_{\perp}^8h_{--}\sim
{N_p^5N_sv^8\over R^7r^{15}}$$
Thus we establish that in order to agree even with the one loop Matrix model result the $R^7$ expression must have covariant derivative terms (unlike the $R^4$ term). The fact that such terms must exist starting at the 14 derivative level means that there is no simple argument on the supergravity side that would correspond to the Matrix model non-renormalization theorem. To put it another way the non-renormalization theorem on the Matrix model side implies that on the super gravity side certain types of terms involving covariant derivatives are not allowed. For instance a 14 derivative term of the form $R\na^{10}R$ gives a term $\pa_+f_{ik}\pa_+f_{jk}\pa^{10}_{\perp}\pa_i\pa_j h_{--}$ and this would give a contribution proportional to $v^4/r^{19}$ and hence if the Matrix model supergravity correspondence is valid, must vanish by the non-renormalization theorem [@pss]. Similarly a term of the form $R\na^8R^2$ gives a contribution $v^6/r^{17}$ and must also be absent. In general it appears that all terms of the form $R\na^{6r-2}R$ and $R\na^{6r-4}R^2$ must be absent in order to have agreement with the Matrix model non-renormalization theorem.
The Matrix model and supergravity on $AdS_5\times S_5$
======================================================
Now let us try to generalize the arguments of the first part of section 2 to the case of Matrix models on torii.
The supergravity solution for an (extremal) Dp-brane is given by $$\label{}
ds^2 =H_p^{-1/2}(-dt^2+\sum_{i =1}^{p}(dx^{i})^2)+H_p^{1/2}
(dr^2+r^2d\O^2_{8-p}).$$ for the metric with the dilaton and the RR field taking the values $$\label{}
e^{-2\phi}=g^{-2}H_p^{p-3\over 2},~~C_{0...p}=(H_p^{-1}-1).$$ In the above $$\label{}
H=1+{Ng\bar d_pl_s^{7-p}\over r^{7-p}},$$ with $\bar d_p$ a known $p$ dependent constant and $N$ the number of p-branes and $g$ is the string coupling. In the weak coupling limit the Dp-brane is described by some non-Abelian version of the Born-Infeld action whose exact form is currently unknown. However one can take the limit [@jm] $$\label{mald}
\a '\rightarrow 0, {\rm with}~ g^2_{YM}=(2\pi)^{p-2}g_s(\a
')^{p-3\over 2}
{\rm fixed}.$$ Note that in this limit the gauge field $A$ on the p-brane as well as the transverse position operator $U$(the 9-p dimensional scalar field on the brane which is really the transverse components of the 10 dimensional gauge field) are kept fixed. The effective dimensionless coupling constant of the gauge theory is $g_{eff}\simeq Ng^2_{YM}U^{p-3}$ and the theory is strongly coupled in the infra-red for $p< 3$ and is weakly coupled in the infrared for $p>3$ while at $p=3$ we have ${\cal N}=4$ super Yang-Mills which is a conformal field theory.
The same scaling may be done in the supergravity solution and gives $$\begin{aligned}
\label{metric}
{ds^2\over l_s^2}&=&{U^{7-p\over 2}\over g_{YM}\sqrt{d_pN}}
(-dt^2+\sum_{i=1}^p(dx^{i})^2)+{g_{YM}\sqrt{(2\pi)^{p-2}d_pN}\over
U^{7-p\over 2}}dU^2 \nn
&+&g_{YM}\sqrt{(d_pN}U^{p-3\over 2}d\O^2_{8-p}.\end{aligned}$$ where $d_p=(2\pi)^{p-2}\bar d_p$. These solutions are supposed to be valid if one can ignore both string loop effects and $\a '$ corrections. As discussed in the second paper of [@jm] this is possible if the following conditions are satisfied, $$\label{}
\a'{\cal R}\sim {1\over
g_{eff}}<<1,~~e^{\phi}\sim{g_{eff}^{7-p}\over N}<<1,~~
g^2_{eff}\equiv Ng^2_{YM}U^{p-3}$$
For the case $p=3$ this metric becomes that of $AdS_5\times S_5$. From such arguments (and the agreements that have been shown to exist between calculations in black hole physics and gauge theory such as those in [@ik]) Maldacena conjectured that gauge theory the large N limit is dual in some sense to supergravity in the above background. Also including the $O(1/g^2_{YM}N)$ corrections to the strong coupling expansion in the gauge theory should be equivalent to including the string corrections on the above supergravity background, while string loop corrections are governed by $g^2_{YM}$. Actually in this case it has been argued that there are no string correction to this background. [@kr] so one may even work with small $g^2_{YM}N$.
Let us now review the Matrix model argument for relating gauge theory and gravity after compactifying on a p-torus. One starts with the $p=0$ (D0-brane) case of the earlier discussion (see section 2). The limit one takes is the same as (\[mald\]) for $p=0$. As we reviewed in section 2 the theory thus obtained is then interpreted as a microscopic model of M-theory on a light like circle. Now while the limit for $p=0$ is the same as the one taken by Maldacena [@jm] eqn(\[mald\]) the interpretation in the other cases is somewhat different. On the one hand the higher dimensional branes in M-theory are supposed to be obtained as condensates of the D0-branes. Secondly the matrix theory description of M-theory compactified on a p-torus is obtained by T-dualizing the D0-brane theory [@wt],[@bfss]. Let us compare the latter procedure with the above discussion of duality.
Under compactification on a p-torus (with radii $r_i$) and T-dualization, $$\label{}
r_i\rightarrow \s_i={l_s^2\over r_i};~~~g\rightarrow g^{(p)}=
g\prod_{i=1}^p{l_s\over r_i}={l_s^{3-p}\over l_m^3}\prod\s_i.$$ where we have put $g_{YM}^2\equiv 1/l_m^3$. It is important to observe that the limit $\l_s\rightarrow 0$ in the compactified Matrix model means in addition to (\[limit\]) that we keep the radii of the dual torus $\s_i$ fixed. (This corresponds to holding $U=r/l_s^2 fixed))$. Doing this Matrix model rescaling in the supergravity solutions we get the following: $$\label{}
H_p=1+{Nd_p\prod^p\s_i\over l_s^4l_m^3}{1\over U^{7-p}}\rightarrow
{Nd_p\prod^p\s_i\over l_s^4l_m^3}{1\over X^{7-p}}.$$ where $X=l_m^2U$. Rescaling the metric $ds^2\rightarrow {l_m^2\over
l_s^2}ds^2$ we have $$\label{mmmetric}
{ds^2}\rightarrow {X^{7-p \over 2}\over R^{7-p\over 2}}
(-dt^2+\sum_{\a=1}^p(dx^{\a})^2)+{R^{7-p\over 2}\over X^{7-p\over 2}}
(dX^2+X^2d\O^2_{8-p}).$$ where $$\label{}
R^{p-7}={l_m^{-3}\over Nd_p\prod\s_i}.$$ It is instructive to compare this in the case $p=3$ to the $AdS_5\times S_5$ case considered in [@jm]. For this case the above becomes (rewriting $X\rightarrow U$ in order to conform to notation that seems to have become standard for AdS spaces), $$\label{mmmet}
{ds}\rightarrow {U^2\over R^2}
(-dt^2+\sum_{i=1}^p(dx^{i})^2)+{R^2\over U^2}
(dU^2+U^2d\O^2_{8-p}).$$ This metric is locally the same as the metric (for the case $p=3$) in (\[metric\]) but it is not the same globally. The reason is that in this Matrix model case one has actually divided out by a discrete symmetry which is a sub group of the (apparent) translation isometry (under $x^{\a}\rightarrow x^{\a}+a^{\a}$ $\a =0,i$) of the above metric. However the actual (freely acting) isometry group of $AdS_5$ is $SO(4,2)$. The translation isometry has a fixed point at $U=0$. To see this let us it is only necessary to observe that the above coordinates of the AdS metric are ill-defined at $U=0$. The $AdS_{p+1}$ space is defined as the hyperboloid $$\label{ads}
-UV+(X^{\a})^2=-R^2.$$ embedded in a $p+2$-dimensional space with metric $$\label{}
ds^2=-dUdV +(dX^{\a})^2.$$ The metric in the form (\[mmmet\]) is obtained by eliminating $V$ and defining the coordinates $x^{\a}={X_{\a}R\over U}$. The translation symmetry of the $x^{\a}$ clearly have a fixed point at $U=0$ and hence when dividing by a discrete subgroup of this symmetry in order to get a 3-torus one gets a singularity at $U=0$. Thus the space-time metric that is related to the Matrix model on $T_3$ is not $AdS_5\times S_5$ which is a smooth space but a space which locally looks like it away from $U=0$, but has a singularity at $U=0$.
However this singularity is just the point at which the moduli space approximation of the gauge theory breaks down. The singularity must in fact be replaced by full quantum non-abelian description. In contrast to the situation in the non-orbifolded case here it is unclear whether there is a holographic interpretation. The holographic interpretation in the case of $AdS_5\times S_5$ comes from the ansatz of [@ew] (see also [@gkp] for a slightly different interpretation) according to which the ${\cal N}=4$ superconformal field theory sits on the boundary of the AdS space and the correlation functions of the former are obtained from the bulk supergravity by using the relation (in a Euclidean signature) $$\label{wit}
\int [dA]e^{-S_{CFT}[\phi_0,A]}=e^{-S[\phi ]} .$$ The functional integral is over all gauge theory variables and $\phi$ is a classical fluctuation around the background AdS space which has boundary value $\phi_0$. The left hand side of this equation is the generating functional for connected correlation functions and $\phi_0$ is an external source which uniquely determines the bulk value $\phi$. Thus the theory in the bulk is uniquely determined by the theory on the boundary giving a holographic picture of bulk physics. It should also be noted that since the space has no singularity there is no need to have branes anywhere in the space.
By contrast in the Matrix model case the equation which replaces (\[wit\]) is the analog of (\[gagrav\]) $$\label{?}
\int dX'e^{-S_{MM}[U+X']} = \lim_{DKPS} e^{-S_{sugra}[U]}.$$ where (in the present case) $S_{MM}$ is the same gauge theory except it is now on a three torus and the right hand side is the supergravity representation of the probe D3 brane in the background space given by (\[mmmet\]) which is singular at the origin. The latter is effectively to be replaced by the branes (i.e. the Matrix model). Clearly it is not straightforward to give this a holographic interpretation.
Acknowledgements:
=================
I would like to thank Esko Keski-Vakkuri and Per Kraus for collaboration on the material in section 3 and helpful comments on the manuscript, and Nathan Seiberg, Savdeep Sethi and Edward Witten for discussions. I would also like to thank Edward Witten for hospitality at the Institute for Advanced Study where much of this work was done and the Institute for Theoretical Physics, Santa Barbara for hospitality during the workshop on Duality where this project originated. Finally I wish to thank the Council on Research and Creative Work of the University of Colorado for the award of a Faculty Fellowship. This work is partially supported by the Department of Energy contract No. DE-FG02-91-ER-40672.
[99]{} T. Banks, W. Fischler, S. Shenker, L. Susskind, Phys. Rev. D55 (1997) 5112; hep-th/9610043. I. Klebanov, Nucl. Phys. B496 (1997) 231: I. Klebanov and S. Gubser, Phys. Lett B413 (1997) 41, hep-th/9708005. J. Maldacena, “The Large N limit of Superconformal Field Theories And Supergravity", hep-th/9711200. N. Itzhaki, J.Maldacena, J. Sonnenshein, S. Yankielowicz, “Supergravity and The Large N Limit of Theories With Sixteen Supercharges" hep-th/9802042. S. Gubser, I. Klebanov and A. Polyakov,“Gauge Theory Correlators from Non-Critical String Theory", hep-th/9802109. E. Witten, “Anti-de Sitter Space And Holography". hep-th/9802150. S. Hyun, “The Background Geometry of DLCQ Supergravity", hep-th/9802026; “Background geometry of DLCQ M theory on a p-torus and holography", hep-th/9805136 A. Sen, “D0 Branes on $T^n$ and Matrix Theory" hep-th/9709220. N. Seiberg, Phys. Rev. Lett. 79 (1997) 3577, hep-th/9710009. S. de Alwis, ‘Phys.Lett. B423 (1998) 59, hep-th/9710219. K. Becker, M. Becker, J. Polchinski, A. Tseytlin, Phys.Rev. D56 (1997) 3174, hep-th/9706072. M. Douglas, D. Kabat, Pouliot and S. Shenker, Nucl. Phys. B485 (1997) 85, hep-th/9608024. G. Horowitz and A. Strominger, Nucl. Phys. B360, (1991) 197. P. Townsend,“Four Lectures on M-theory" hep-th/9612121. P. Aichelburg and R. Sexl, Gen. Rel. Grav. 2 (1971) 303. M. Becker and K. Becker, Nucl.Phys. B506 (1997) 48, hep-th/9705091. E. Keski-Vakkuri and P. Kraus, Nucl.Phys. B518 (1998) 212, hep-th/9709122. S. Paban, S. Sethi, M. Stern, “Constraints From Extended Supersymmetry in Quantum Mechanics", hep-th/9805018, “Supersymmetry and Higher Derivative Terms in the Effective Action of Yang-Mills Theories", hep-th/9806028. J. Maldacena, “Branes probing black holes", hep-th/9709099. M. Dine and A. Rajaraman, “Multigraviton Scattering in the Matrix Model", hep-th/9710174. M. Douglas, H. Ooguri and S. Shenker, Phys.Lett. B402 (1997) 36, hep-th/9702203. D. Kabat and W. Taylor, “Spherical membranes in Matrix theory" hep-th/9711078; “Linearized supergravity from Matrix theory", hep-th/9712185. Y. Okawa and T. Yoneya, “Multi-Body Interactions of D-Particles in Supergravity and Matrix Theory" hep-th/9806108. M. Fabbrichesi, G. Ferretti and R. Iengo, “Supergravity and matrix theory do not disagree on multi-graviton scattering" hep-th/9806018; R. Echols and J. Gray, “Comment on multigraviton scattering in the Matrix model" hep-th/9806109; J. McCarthy, L. Susskind, and A. Wilkins, “Large N and the Dine-Rajaraman problem" hep-th/9806136. W. Taylor IV and M. Van Raamsdonk, “Three-graviton scattering in Matrix theory revisited", hep-th/9806066. J. Russo and A. Tseytlin, Nucl.Phys. B508 (1997) 245, hep-th/9707134 V. Balasubramanium, R. Gopakumar and F. Larson, “Gauge Theory, Geometry and the Large N Limit", hep-th/9712077. E. Keski-Vakkuri and P. Kraus, “Short Distance Contributions to Graviton-Graviton Scattering: Matrix Theory versus Supergravity" hep-th/9712013. K. Becker and M. Becker, Phys.Rev. D57 (1998) 6464, hep-th/9712238 R. Kallosh and A. Rajaraman, “Vacua of M-theory and string theory" hep-th/9805041. W. Taylor, Phys.Lett. B394 (1997) 283, hep-th/9811042.
[^1]: e-mail: dealwis@gopika.colorado.edu
[^2]: Recently there have been two papers by S. Hyun [@sh] on this issue. While there is some overlap between the present work and those papers our conclusions are somewhat different especially with regard to the interpretation of the Matrix model on the three torus and the corresponding AdS picture.
[^3]: Strictly speaking we should consider these as quantities with tildes since they are related to $\tilde M$ theory rather than to M theory, but since we are not going to discuss the space like compactification or the M theory it is not essential to make the distinction.
[^4]: This fact is true only for configurations such as the one being considered with some unbroken supersymmetry, see [@sda].
[^5]: This was first observed in [@mal]
[^6]: I would like to acknowledge the collaboration of E. Keski-Vakkuri and P. Kraus in this section.
[^7]: For convenience in comparing with standard results in the literature we have reverted back to the standard notation where $\dot U\rightarrow v,~ U\rightarrow r$.
[^8]: See [@rt] for the original references to this.
[^9]: This seems to have been first observed in [@bgl].
[^10]: This is the correct object to compute in order to compare with the gauge theory calculation as argued in [@bbpt].
[^11]: We wish to thank S. Sethi for discussions on this.
[^12]: The coefficient of the $v^6/r^{11}$ vanishes in the one loop calculation and this can be explained by the non- renormalization theorem [@pss]
|
---
abstract: 'We theoretically study the low energy electromagnetic response of BCS type superconductors focusing on propagating collective modes that are accessible with THz near field optics. The interesting frequency and momentum range is $\omega < \Delta$ and $q < 1/\xi$ where $\Delta$ is the gap and $\xi$ is the coherence length. We show that it is possible to observe the superfluid plasmons, amplitude (Higgs) modes, Bardasis-Schrieffer modes and Carlson-Goldman modes using THz near field technique, although none of these modes couple linearly to far field radiation. Coupling of THz near field radiation to the amplitude mode requires particle-hole symmetry breaking while coupling to the Bardasis-Schrieffer mode does not and is typically stronger. The Carlson-Goldman mode appears in the near field reflection coefficient as a weak feature in the sub-THz frequency range. In a system of two superconducting layers with nanometer scale separation, an acoustic phase mode appears as the antisymmetric plasmon mode of the system. This mode leads to well defined resonance peaks in the near-field THz response and has strong anti crossings with the Bardasis-Schrieffer mode and amplitude mode, enhancing their response. In a slab of layered superconductor such as the high T$_c$ compounds, which can be viewed as a natural optical cavity, many branches of propagating Josephson plasmon modes couple to the THz near field radiation.'
author:
- Zhiyuan Sun
- 'M. M. Fogler'
- 'D. N. Basov'
- 'Andrew J. Millis'
bibliography:
- 'Superconductor\_collective\_modes.bib'
date:
-
-
title: Collective modes and THz near field response of superconductors
---
Introduction
============
Electromagnetic (EM) response is a fundamental property of superconductors. The response to static electric and magnetic fields (infinite conductivity and the Meissner effect) are the defining properties of the superconducting state. The response to the time dependent, very long wavelength transverse fields produced by far field radiation has been extensively studied [@Mattis1958; @Anderson1958b; @Zimmermann1991; @Tinkham.1974; @Basov2005a]. Superconductors are also characterized by a diversity of sub-gap collective modes [@Parks1969] including plasmons, acoustic phase modes, amplitude (Higgs) modes, the Carlson-Goldman modes, and the Bardasis-Schrieffer modes associated with fluctuations of subdominant order parameters, shown schematically in Fig. \[fig:parameter\_regime\]. For superconductors of current interest including cuprates [@Damascelli.2003; @Basov2005a], iron pnictides [@Stewart.2011], NbSe$_2$ [@Sooryakumar1980] and MgB$_2$ [@Buzea_2001] the gap values and the relevant collective modes are in the terahertz (THz) range. These modes couple weakly, if at all, to far field transverse photons.
Recent progress in cryogenic near field nano optics [@Ni2018] has enabled new generations of experiments probing the response of materials to short wavelength, primarily longitudinal, finite frequency electric fields [@Basov2014a; @Lundeberg2017; @Dias2018]. This information is encoded in the near field reflection coefficient $R_p(\omega,q)$ (see Appendix \[appendix:rp\]). The essential new feature of the nano optics experiments as compared to traditional far field optics is the excitation of charge fluctuations in the material under study. In this paper we calculate the nano optics linear response $R_p(\omega,q)$ of superconductors focusing on the contribution of collective modes. Each of the modes we consider couples to charge fluctuations and is therefore in principle observable in nano optics experiments. We calculate in detail the matrix elements coupling each mode to charge excitations and from this the signal of the nano optical response.
Charge fluctuations are constrained by the continuity equation, the proper treatment of which requires going beyong BCS mean field theory in a manner consistent with the relevant Ward Identities [@Anderson1958a; @Schrieffer1999; @Arseev2006; @Yang2018]. In this paper we employ a one loop effective action method based on a Hubbard-Stratonovich transformation of the fundamental interacting electron system. This methodology is an efficient way to take order parameter and charge fluctuations into account while respecting conservation laws. It is equivalent to a diagrammatic calculation including vertex corrections [@Schrieffer1999].
We now discuss the physics of the collective modes and their coupling to light. The phase (Anderson-Bogoliubov-Goldstone) mode (heavy blue dashed line in Fig. \[fig:parameter\_regime\]) is the order parameter phase fluctuation. It is accompanied by a superfluid density oscillation [@Bogoljubov1958] which in the presence of long range Coulomb interaction converts this mode into a plasmon [@Anderson1963]. In two dimensions (2D), the plasmon has a $\omega\sim \sqrt{q}$ dispersion, shown as the red line in Fig. \[fig:parameter\_regime\] and discussed in section \[sec:plasmons\]. The plasmons directly couple to near field radiation. In multi layer systems, mutual screening leads to branches of acoustic (linearly dispersing) plasmons (not shown in Fig. \[fig:parameter\_regime\]). These acoustic plasmons also couple to THz near field probes, see section \[sec:double\_layer\].
In the presence of abundant normal carriers as happens for example close to $T_c$, the Coulomb potential of the superfluid density fluctuation can be screened and as a result, the gapless phase mode known as the Carlson-Goldman (CG) mode [@Carlson1975] (purple line in Fig. \[fig:parameter\_regime\]), appears. This mode is discussed in section \[sec:CG\].
The amplitude (Higgs) mode (green line in Fig. \[fig:parameter\_regime\]) is the gap amplitude fluctuation which couples to EM linear response only when particle hole symmetry is broken [@Littlewood1982; @Cea2014; @Cea2015] and only at non-zero $q$ because electron density fluctuation is needed to locally perturb the density of states and then the gap. The coupling is suppressed by the small parameter $\Delta/E_F$, as discussed in section \[sec:higgs\]. Note that the Higgs mode does couple to far field radiation through third order nonlinear response which has been demonstrated experimentally [@Katsumi2018; @Nakamura2019; @Shimano2020].
The Bardasis-Schrieffer mode (BSM) [@Bardasis1961; @Maiti2015; @Maiti2016; @Allocca2019] is a fluctuation of subdominant pairing order parameters, e.g., $d$-wave fluctuations in a $s$-wave superconductor. Although it was proposed half a century ago [@Bardasis1961], the BSM has been very difficult to observe. Its signature was reported only recently in an iron based superconductor [@Bohm2014]. The BSM frequency is slightly below the gap for weak subdominant pairing and approaches zero as the subdominant pairing strength approaches the dominant one.
The rest of the paper is organized as follows. In section \[sec:Ginzburg\_Landau\] we perform the Hubbard-Stratonovich transformation of the BCS Hamiltonian to obtain the Ginzburg-Landau effective action which includes the collective modes. Section \[sec:EM\_response\] derives the linear EM response functions and their simple forms in the low energy limit. With the longitudinal optical conductivity, we analyse the properties of the collective modes in sections \[sec:plasmons\], \[sec:higgs\], \[sec:BS\] and \[sec:CG\] with a focus on 2D. Section \[sec:double\_layer\] discusses the acoustic plasmon mode in superconducting double layer which is promising to be observed in THz near field optics. We then discuss the cluster of hyperbolic Josephson plasmons in naturally layered superconductors in section \[sec:bulk\_layered\] and show that they are greatly affected by the nonlocal correction to the optical conductivity and the discrete nature across the layers. Section \[sec:discussion\] is a summary and conclusion, with pointers to the relevant equations and figures, for readers uninterested in the details of the derivations. Appendix \[app:correlation\_function\] contains the definition and explicit forms of the correlation functions. Appendix \[appendix:rp\] has the derivation of the reflection coefficients.
The effective Ginzburg-Landau action {#sec:Ginzburg_Landau}
====================================
The action of fermion, gap and EM fields
----------------------------------------
The starting point is the BCS Lagrangian of attracting electrons coupled to electromagnetic (EM) field: $$\begin{aligned}
L =& \int dr
\left\{
\psi^\dagger
\left[ \partial_\tau +
\xi(p-eA) + e\phi
\right]
\psi
\right\}
\notag\\
&- \int dr dr^\prime g(r,r^\prime) \psi^\dagger(r)\psi^\dagger(r^\prime) \psi(r^\prime)\psi(r)
\notag\\
& - \int dr \frac{1}{16\pi} F^{\mu\nu} F_{\mu\nu}
\label{eqn:hamiltonian}\end{aligned}$$ where $(\phi,\mathbf{A})=A$ is the EM field, $F_{\mu\nu} =\partial_\mu A_\nu-\partial_\nu A_\mu$ , $\psi$ is the electron annihilation operator, $\xi(p)=\varepsilon(p) -\mu$ and $g<0$ is the attractive interaction. We will be interested in relatively low frequency longitudinal EM fluctuations where the magnetic field can be neglected. Thus $\int dr \frac{1}{16\pi} F^{\mu\nu} F_{\mu\nu} \rightarrow \int dr \frac{1}{8\pi} E^2$ for 3D and $\rightarrow \sum_q \frac{1}{4\pi |q|} E_{-q} E_q$ for a 2D plane embedded in 3D space where $E_{q}$ is the Fourier component of the electric field on the 2D plane. Note that the EM field $A$ has an UV cutoff which is much smaller than the fermi momentum such that it mediates only the smooth part of the Coulomb interaction between the electrons. The high energy part of the photon is already integrated out which together with the phonons results in the effective interaction $g$ whose cutoff is the Debye frequency $\omega_D$. For simplicity, we don’t explicitly notate the photons in what follows.
Performing the Hubbard-Stratonovich transformation of the path integral $Z=\int D[\bar{\psi},\psi] e^{-S}$ in the pairing channel gives $$\begin{aligned}
Z = & \int D[A] D[\bar{\psi},\psi] D[\bar{\Delta},\Delta]
e^{-S[\psi, A, \Delta]}\end{aligned}$$ where the action $$\begin{aligned}
S = \int d\tau dr
\Bigg\{
&
\psi^\dagger
G_{A\Delta}^{-1}
\psi
+
\sum_l \frac{1}{g_l}|\Delta_l|^2
\Bigg\}
\label{eqn:HS_action}\end{aligned}$$ describes coupled dynamics of the fermion field $\psi$, the EM field $A$ and the gap $\Delta_l$ with $l$ denoting the pairing angular momentum. Note that we have neglected detailed structure in $g$ that is not important to our conclusions. The fermion kernel is $$\begin{aligned}
G_{A\Delta}^{-1}
=
\begin{pmatrix}
\partial_\tau + e\phi + \xi(p-eA) & \sum_l \Delta_l f_l(p) \\
\sum_l \bar{\Delta}_l \bar{f}_l(p) & \partial_\tau - e\phi - \xi(-p-eA)
\end{pmatrix}
\,\end{aligned}$$ where $f_l(p)$ describes the momentum dependence of the pairing function in each angular momentum channel. For simplicity, we consider only spin singlet pairing with the Nambu spinors being $\psi^\dagger = (\psi_{\uparrow}^\dagger,\psi_{\downarrow})$. In the BCS regime of two dimensional superconductors, we can choose $f_l=\cos (l\theta_k)$ or $\sin (l\theta_k)$ and the corresponding pairing interaction is $g_l= \frac{1}{2\pi} \int d\theta \cos(\theta) g\left(2 k_F \sin(\theta/2) \right)$. Note that for $l=0$, the $1/2\pi$ factor should be changed to $1/4\pi$. We assume that the $l=0$ component of the interaction is the strongest and thus the ground state has $s$-wave pairing. We successively integrate out the fermion field to obtain the Ginzburg Landau action for the gap and EM field, and then the gap to obtain the action for EM field which gives the information of the EM response functions.
Integrating out the fermions
----------------------------
Expanding in the EM field, the Lagrangian density is $$\begin{aligned}
\mathcal{L} =
\psi^\dagger G_\Delta^{-1} \psi + J^P_\mu A^\mu + \frac{1}{2} D_{ij} A_i A_j + O(A^3) + \sum_l \frac{1}{g_l}|\Delta_l|^2
\label{eqn:lagrangian}\end{aligned}$$ where the Gor’kov Green’s function for the Bogoliubov quasi particle is $$\begin{aligned}
G_\Delta^{-1}
&=
\begin{pmatrix}
\partial_\tau + \xi(p) & \sum_l \Delta_l f_l(p) \\
\sum_l \bar{\Delta}_l \bar{f}_l(p) & \partial_\tau - \xi(-p)
\end{pmatrix}
\,,\end{aligned}$$ the paramagnetic contribution to the current is $$\begin{aligned}
J^P_\mu
&=
e \psi^\dagger (\sigma_3,\, \mathbf{v} \sigma_0) \psi = (\rho, \mathbf{j}^P)
\,\end{aligned}$$ and the diamagnetic ‘Drude’ kernel is $$\begin{aligned}
D_{ij}
&=
e^2 \psi^\dagger \sigma_3 (\partial_{p_i p_j} \varepsilon) \psi
\,.\end{aligned}$$ Note that we have assumed inversion symmetry of $\varepsilon(p)$. After integrating out the fermions, the action becomes $$\begin{aligned}
S(\Delta,A) &= \mathrm{Tr} \ln G_{A\Delta} + \int d\tau dr \sum_l \frac{1}{g_l}|\Delta_l|^2
\notag\\
&= S_{\mathrm{mean} \,\, \mathrm{field}} + S_{\mathrm{fluctuation}}
\,
\label{eqn:GL_action}\end{aligned}$$ where the trace is over the frequency, momentum and spinor degrees of freedom of $\psi$.
Mean field as the saddle point
------------------------------
Assuming the ground state has $s$-wave pairing with $\Delta$ independent on momentum, the mean field free energy is $$\begin{aligned}
S_{\mathrm{mean} \,\, \mathrm{field}}/V
&= \frac{1}{g} \Delta \bar{\Delta} +
\sum_{\omega_n, k} \ln \left((i\omega_n)^2 - E_k^2 \right)
\notag\\
&= \frac{1}{g} \Delta \bar{\Delta} -
\sum_{k} \left[\frac{2}{\beta}\ln \left(1 + e^{-\beta E_k} \right) + E_k\right]
\,
\label{eqn:mean_field_energy}\end{aligned}$$ where $E_k = \sqrt{\xi_k^2 + |\Delta|^2} = \sqrt{(\varepsilon_k-\mu)^2 + |\Delta|^2}$ is the quasi particle energy with gap $\Delta$. Minimization of $S_{\mathrm{mean} \,\, \mathrm{field}}$ with respect to the gap renders the gap equation $$\begin{aligned}
\frac{1}{g}\Delta
= \sum_k \frac{\Delta}{E_k} (1-2f(E_k))
\,\end{aligned}$$ where $f$ is the fermi distribution function. At zero temperature, the integral in [Eq. ]{} can be done and we obtain the condensation energy relative to the normal state: $$\begin{aligned}
F &= S_{\mathrm{mean} \,\, \mathrm{field}}/V + \sum_{k} \xi_k
= \frac{1}{g} \Delta \bar{\Delta} + \sum_{k} (\xi_k - E_k)
\notag\\
&\approx \frac{1}{g} |\Delta|^2 - \nu |\Delta|^2 \ln \frac{2\omega_D}{|\Delta|}
\,\end{aligned}$$ where $\nu$ is the density of state at the fermi level of the normal state. The gap at zero temperature is thus $\Delta_0=2\omega_D e^{-\frac{1}{g\nu}}$ for $g\nu \ll 1$. Without loss of generality, we pick the mean field gap $\Delta$ to be real. The coherence length is defined as $\xi=v_F/\Delta$. Note that the free energy is non analytic around $\Delta=0$, i.e., the expansion coefficients in small $\Delta$ all diverge. The Ginzburg Landau expansion in powers of $\Delta$ is only possible at finite temperature and accurate close to $T_c$.
Fluctuations
------------
The fluctuations include those of the EM field, the $s$-wave gap and the subdominant pairing order parameters. The $s$-wave gap fluctuation can be separated into amplitude and phase: $\Delta = (\Delta_0 + \Delta(r,t))e^{i2\theta(r,t)}$. It is convenient to perform a local gauge transformation[@Altland.2010] $$\begin{aligned}
\psi \rightarrow
\begin{pmatrix}
e^{i\theta}& 0\\
0& e^{-i\theta}
\end{pmatrix}
\psi\end{aligned}$$ which changes the action [Eq. ]{} to its equivalent form with the EM field changed to the gauge invariant ones $$\begin{aligned}
e A^\mu
\rightarrow \partial_\mu \theta + A_\mu =
(e\phi+i\partial_\tau \theta, e\mathbf{A} - \nabla \theta)
\,.\end{aligned}$$ In the effection action, the EM field always comes together with the phase fluctuations in the above gauge invariant form and therefore couples directly to the phase mode. The appearance of the other modes in the EM response can be inferred simply from their coupling to the phase mode.
The action of fluctuations $\Delta_q$ around the mean field gap can be decomposed as $$\begin{aligned}
S_{\mathrm{fluctuation}} = S_{\theta} + S_{\Delta} + S_{BS}+S_{c}
+ \mathrm{nonlinear \,\, terms}
\label{eqn:s_fluctuation}\end{aligned}$$ where $$\begin{aligned}
S_{\theta} =& \sum_q K^{\mu \nu}(q) (\partial_\mu \theta + eA_\mu)_{-q} (\partial_\nu \theta + eA_\nu)_q \end{aligned}$$ is the phase action, $$\begin{aligned}
S_{\Delta} =& \sum_{q} G_a(q)^{-1}
\Delta_{-q} \Delta_q \end{aligned}$$ is the amplitude action and $$\begin{aligned}
S_{BS} =& \sum_q G_l(q)^{-1}
\Delta_l(-q) \Delta_l(q)\end{aligned}$$ is the action for the fluctuation of the sub dominant pairing channels, i.e., the Bardasis-Schrieffer modes. Finally, $$\begin{aligned}
S_{c} =& \sum_q \left( C^\mu(q) \Delta_{-q} + \sum_l B_l^\mu(q) \Delta_l(-q) \right)
\left(\partial_\mu \theta + e A_\mu \right)_{q}
\label{eqn:amplitude_phase_coupling}\end{aligned}$$ is the coupling between phase and amplitude/BSM fluctuations. Global $U(1)$ symmetry under $\theta \rightarrow \theta + \delta$ is manifest here since the action depends only on derivatives of the phase. This ensures charge conservation.
Phase action
------------
The quadratic kernel for the phase action is $$\begin{aligned}
K_{\mu\nu}(q) &= \langle \hat{T} J^P_\mu(x) J^P_\nu (0) \rangle \big|_q +
\begin{pmatrix}
0 & 0 \\
0 & \langle D_{ij} \rangle
\end{pmatrix}
\notag\\
&=
\begin{pmatrix}
\chi^{(0)}_{\rho \rho} & \chi^{(0)}_{\rho \mathbf{j}} \\
\chi^{(0)}_{\mathbf{j} \rho} & \chi^{(0)}_{\mathbf{j} \mathbf{j}} + \langle D_{ij} \rangle
\end{pmatrix}
\,\end{aligned}$$ where $\chi^{(0)}_{\rho \rho}$, $\chi^{(0)}_{\rho \mathbf{j}}$ and $\chi^{(0)}_{\mathbf{j} \mathbf{j}}$ are the density-density, density-current and current-current correlation functions evaluated at the mean field saddle point. In the case of quadratic band, $\varepsilon = p^2/(2m)$, the system has Galilean invariance and $D_{ij} = n/m \delta_{ij}$ where $n$ is the total carrier density.
*Low energy limit—* At zero temperature, for $\omega \ll \Delta$ and $q \ll 1/\xi$, we have $\chi^{(0)}_{\rho \rho} \rightarrow \nu$, $\chi^{(0)}_{\rho \mathbf{j}} \sim \omega \mathbf{q}$ and $\chi^{(0)}_{\mathbf{j} \mathbf{j}} \sim q^2$, thus to leading order we have $$\begin{aligned}
K_{\mu\nu}(q)
&=
\begin{pmatrix}
-\nu & 0 \\
0 & n/m
\end{pmatrix}
\,\end{aligned}$$ where $\nu$ is the normal state density of state at the fermi level. Therefore, the effective low energy Lagrangian of the phase fluctuation is [@Altland.2010] $$\begin{aligned}
\mathcal{L} = \frac{1}{2}\nu (\partial_t \theta + e\phi)^2 - \frac{n}{2m} (\nabla \theta - e\mathbf{A})^2 \,
\label{eqn:action_superfluid}\end{aligned}$$ which describes the Nambu-Goldstone mode with velocity $v_g=\sqrt{n/(m \nu)} = v_F/\sqrt{d}$ if EM field were not present, also known as the Anderson-Bogoliubov mode [@Anderson1958a; @Anderson1958b]. Here $d$ is the space dimension. Due to the long range Coulomb interaction (coupling to EM field), the Goldstone mode does not actually exist but is shifted to the high frequency plasmons through the Anderson-Higgs mechanism[@Anderson1963].
Amplitude action
----------------
The inverse propagator for amplitude fluctuation is $$\begin{aligned}
G^{-1}_a(q) &= \frac{1}{g} + \chi_{\sigma_1 \sigma_1}(q)
\,.\end{aligned}$$ At zero momentum $q=0$ and rotated to real frequency, it has the analytical form $$\begin{aligned}
G^{-1}_a(\omega) &= (4\Delta_{sc}^2 -\omega^2) F(\omega)
\,\end{aligned}$$ where $\Delta_{sc}=\Delta$ and $$\begin{aligned}
F(\omega) = \sum_{k} \frac{1}{E_k(4E_k^2 - \omega^2)}
=
\frac{\nu}{4\Delta^2} \frac{2\Delta}{\omega} \frac{\mathrm{sin}^{-1}\left(\frac{\omega}{2\Delta}\right)}{\sqrt{1-\left(\frac{\omega}{2\Delta}\right)^2}}\end{aligned}$$ describes the quasi-particle effects. Specifically, $F$ diverges as $\frac{1}{\sqrt{2\Delta - \omega}}$ as the frequency approaches $2\Delta$ and has an imaginary part above $2\Delta$ due to quasi-particle excitations. Thus $G^{-1}_a$ does not have a simple pole at $\omega=2\Delta$ and the Higgs amplitude mode is not well defined in the weak coupling BCS approximation [@Cea2015].
However, in systems with superconductivity coexisting with charge density wave (CDW), the quasi particle absorption gap $\Delta=\sqrt{\Delta_{sc}^2 + \Delta_{cdw}^2}$ is pushed beyond the Higgs frequency $2\Delta_{SC}$ and the latter becomes a well defined collective mode [@Cea2014]. In this case, $F$ is well behaved around the Higgs pole and we can approximate the propagator by $$\begin{aligned}
G^{-1}_a(\omega) &= \frac{\nu}{4\Delta^2}(4\Delta_{sc}^2 + \frac{1}{d}v_F^2 q^2 -\omega^2)
\,\end{aligned}$$ where $d$ is space dimension and the $O(q^2)$ expansion can be found in Appendix \[appendix:Higgs\]. Thus the Higgs mode frequency disperses roughly as $\omega_{hq}^2 = 4\Delta_{sc}^2 + \frac{1}{d}v_F^2 q^2$ as in Ref. [@Littlewood1982]. Moreover, coupling between the Higgs mode and a higher frequency CDW phonon may further lower the Higgs mode frequency[@Littlewood1982] as has been proposed for $2H$-NbSe$_2$ [@Sooryakumar1980].
BSM action
----------
The inverse propagators for the fluctuations of the higher angular momentum pairing channels are $$\begin{aligned}
G^{-1}_l(q) &= \frac{1}{g_l} + \chi_{f_l(k)\sigma_2, f_l(k)\sigma_2}(q)
\,\end{aligned}$$ where the correlator is defined in Appendix \[appendix:BS\]. Note that there are two directions for the fluctuations of the subdominant order parameters on the complex plane: perpendicular to the mean field gap (in the ‘imaginary’ direction) and parallel to it (in the ‘real’ direction). The BSMs [@Bardasis1961; @Maiti2016; @Allocca2019] are the ‘imaginary’ fluctuations, i.e., in the $\sigma_2$ channel. This channel of fluctuations have nonzero matrix element in exciting quasi particles on the gap edge, thus rendering $\chi_{\sigma_2,\sigma_2}(\omega)$ diverging as the frequency approaches the gap $2\Delta$ from below. As a result, the BSMs all have energies below $2\Delta$. The ‘real’ modes, the fluctuations in the $\sigma_1$ channel, don’t have poles and are not collective modes, as shown in Appendix \[appendix:BS\].
In this paper we focus on the $d$-wave BSM in an $s$-wave superconductor which couples to light already at quadratic order in $q$. In two dimension, there are two $d$-wave BSMs corresponding to $d_{x^2-y^2}$ and $d_{xy}$. Assume the momentum is along $x$, to leading order in momentum, the inverse propagator of the $d_{x^2-y^2}$ mode is $$\begin{aligned}
G^{-1}_{BS}(\omega,q) &= \frac{1}{g_d} + \chi_{cos(2\theta_k)\sigma_2, cos(2\theta_k)\sigma_2}(\omega,q)
\notag\\
&= \frac{1}{g_d}-\frac{1}{2g} -\frac{1}{2}\omega^2 F(\omega) + \frac{1}{16} \frac{\nu}{\Delta^2} v_F^2 q^2
\,.
\label{eqn:bs_propagator}\end{aligned}$$ For $g_d \in (0,\, 2g)$, at zero momentum, it has a pole below $2\Delta$ which gives the mode frequency $$\begin{aligned}
\omega_{BS}
=
2\Delta \left\{
\begin{array}{lc}
1- \frac{\pi^2}{32} (\nu g_d)^2
&
\,(g_d \ll 2g)
\\
\sqrt{2} \left( \frac{1}{ g_d \nu}- \frac{1}{2g \nu } \right)
& \,(g_d \rightarrow 2g)
\end{array}
\right.
\label{eqn:sigma_n}\end{aligned}$$ in the weak and strong BSM fluctuation limits. As $g_d$ changes from $0$ to $2g$, $\omega_{BS}$ goes from $2\Delta$ to $0$. For $g_d > 2g$, the ground state is no longer a $s$-wave one [@Maiti2015]. At finite momentum, the mode frequency shifts up as $q^2$ as shown in Appendix \[appendix:BS\].
Coupling terms
--------------
Under either time reversal or particle hole operation, the phase $\theta$ and BSM $\Delta_l$ are odd because they are fluctuations in the ‘imaginary’ direction on the complex plane while the amplitude fluctuation $\Delta$ is even. Therefore, the linear coupling between phase and amplitude fluctuations breaks particle hole symmetry while that between phase and BSM does not.
Due to inversion and time reversal symmetry, the coupling coefficient can be expanded as $$\begin{aligned}
C^\mu(q) = \left( C_0 + O(q^2),\,\, C_i \omega \mathbf{q} + O(q^3) \right)
\,\end{aligned}$$ where $C_0,C_i=0$ in a particle hole symmetric situation. Indeed these limits can be verified from their exact forms $$\begin{aligned}
C^\mu(q) = \left( \chi^{(0)}_{\rho \Delta},\,\, \chi^{(0)}_{\mathbf{j} \Delta} \right)
= \left( \chi^{(0)}_{\sigma_3 \sigma_1},\,\, \chi^{(0)}_{\mathbf{v} \sigma_0,\, \sigma_1} \right)
\,.
\label{eqn:phase_higgs_C}\end{aligned}$$ To study the linear EM response for $q \ll 1/\xi$, it is sufficient to keep the leading terms. The simplest way to break the particle hole symmetry is to use an energy dependent electronic density of state (DOS), e.g., as in the parabolic band electron gas in three dimension. Assuming the DOS is $g(\xi) = \nu (1+\lambda \xi/E_F)$ (note that $\xi=\varepsilon-\mu$ should not be confused with the coherence length), we obtain from $\chi^{(0)}_{\sigma_3 \sigma_1}$ that $$\begin{aligned}
C_0
\approx & - \lambda \nu \frac{\Delta}{2E_F}\sinh^{-1} \left( \frac{\omega_D}{\Delta} \right)
\,
\label{eqn:higgs_coupling_C0}\end{aligned}$$ and from $\chi^{(0)}_{\mathbf{j} \Delta}$ that $$\begin{aligned}
C_i =& \frac{1}{12d} \lambda \nu \frac{\Delta}{E_F} \left(\frac{v_F}{\Delta} \right)^2
\,,
\label{eqn:higgs_coupling_Ci}\end{aligned}$$ see Appendix \[appendix:coupling\_constants\] for detailed derivation. The small factor $\lambda \Delta/E_F$ characterizes the strength of particle hole symmetry broken.
In two dimension, from inversion, time reversal symmetry, that $\Delta_d$ being odd under time reversal and its $d$-wave character, the linear coupling coefficient between the phase and the $d_{x^2-y^2}$ BSM can be written as $$\begin{aligned}
B^\mu =(B_0 \omega q^2,\, B_i q_x,\, -B_i q_y )
= \left( \chi^{(0)}_{\sigma_3,\, \sigma_2 f_l(p)},\,\, \chi^{(0)}_{\mathbf{v} \sigma_0,\, \sigma_2 f_l(p)} \right)
\label{eqn:BS_phase_coupling}\end{aligned}$$ where $B_i = i \pi \Delta v_F^2 F(\omega)$ describes coupling between electric field and the BSM, see Appendix \[appendix:coupling\_constants\]. Since the angular momentum change is $\delta l=2$ in exciting an s bound state to a d state, an inhomogeneous electric field is required to overcome the selection rule and thus the $B_i$ terms exist only at finite momentum. It will be shown that in the optical conductivity, at leading ($O(q^2)$) order the $B_0$ term does not contribute and thus can be neglected. Note that coupling to BSM does not require breaking particle hole symmetry and is not suppressed by the small number $\Delta/E_F$. Thus in general, the BSM couples to EM more strongly than the Higgs mode.
Electrodynamics of superconductors {#sec:EM_response}
==================================
Linear EM response
------------------
To obtain the linear EM response functions, one just needs to obtain the fluctuation action quadratic in the EM fields. Integrating out the amplitude results in $$\begin{aligned}
S =& \sum_q K^{\mu \nu}(q) (\partial_\mu \theta + eA_\mu)_{-q} (\partial_\nu \theta + eA_\nu)_q
\label{eqn:phase_action}\end{aligned}$$ with the kernel modified to $$\begin{aligned}
K^{\mu \nu}(q) =
&
\begin{pmatrix}
\chi^{(0)}_{\rho \rho} & \chi^{(0)}_{\rho \mathbf{j}} \\
\chi^{(0)}_{\mathbf{j} \rho} & \chi^{(0)}_{\mathbf{j} \mathbf{j}} + \langle D_{ij} \rangle
\end{pmatrix}
-\frac{1}{4} G_a(q) C^\mu(q) C^\nu(q)
\notag\\
& -\frac{1}{4} G_{BS}(q) B^\mu(q) B^\nu(q)
\label{eqn:full_K}
\,.\end{aligned}$$ For longitudinal EM response, it is inappropriate to directly employ the ‘free’ optical conductivity $\chi^{(0)}_{\mathbf{j} \mathbf{j}} + \langle D_{ij}\rangle$ or the density response $\chi^{(0)}_{\rho \rho}$ obtained from the BCS mean field Hamiltonian which breaks global $U(1)$ gauge invariance. They actually do not satisfy charge conservation: $K^{\mu \nu} q_\nu \neq 0$. The reason is that longitudinal EM fields excite order parameter phase fluctuations that are not captured by Bogoliubov quasi-particles.
The solution is to take into account the phase fluctuations which ensures charge conservation since the Euler-Lagrangian equation from [Eq. ]{} is just the continuity equation. By integrating out the phase (or simply solving the Euler Lagrangian equation of the phase), one finally obtains the EM action $$\begin{aligned}
S =& \sum_q \Pi^{\mu \nu}(q) A_\mu(-q) A_\mu(q)\end{aligned}$$ where $$\begin{aligned}
\Pi^{\mu \nu}(q) = K^{\mu \nu}(q) - \frac{q_\alpha q_\beta K^{\alpha \nu} K^{\mu \beta} }{q_a q_b K^{a b}}
\label{eqn:EM_response_tensor}\end{aligned}$$ is the EM response tensor satisfying $J^\mu = \Pi^{\mu \nu} A_\nu$ and the continuity equation $\Pi^{\mu \nu} q_\nu = 0$. Specifically, $\Pi^{0 0} = \chi_{\rho\rho}$ is the irreducible density density response (polarization function) and $\frac{1}{i\omega}\Pi^{i j} = \sigma_{ij}$ is the optical conductivity. Note that ‘irreducible’ is with respect to Coulomb interaction, i.e., EM field. The diagrammatic representation of [Eq. ]{} is shown in Fig. \[fig:Feynman\_Diagram\]. This is equivalent to correcting the current vertex by electron electron interactions after which gauge invariance [@Anderson1958] and thus Ward Identity [@Schrieffer1999] are recovered. [Eq. ]{} should be viewed as the central result of this paper which contains all the information of linear coupling between EM field and the collective modes.
The major effect of disorder is to induce a nonzero $\sigma_1$ above twice the gap. The Mattis-Bardeen[@Mattis1958] theory for optical absorption completely relaxes momentum conservation in the quasi particle excitation process, and has proven accurate in various BCS type superconductors. In this paper, we implement the Mattis-Bardeen formula to describe the optical conductivity above the gap: $$\begin{aligned}
\sigma_1(\omega,q) = & \sigma_n(\omega) \Theta\left(\frac{\omega}{2\Delta}-1 \right)
\notag\\
& \left[
\left(1+ \frac{2\Delta}{\omega} \right)
E\left(\frac{\omega-2\Delta}{\omega+2\Delta} \right) -
\frac{4\Delta}{\omega}
K\left(\frac{\omega-2\Delta}{\omega+2\Delta} \right)
\right]
\,.
\label{eqn:Mattis-Bardeen}\end{aligned}$$ where $\sigma_n$ is the normal state conductivity and $E(x),\, K(x)$ are the complete elliptic integrals.
The low energy limit
--------------------
At low temperature compared to $T_c$, in the low energy limit of $\omega \ll \Delta$ and $q \ll 1/\xi$ as shown in the blue region of Fig. \[fig:parameter\_regime\], the electrodynamics can be described by the Lagrangian [Eq. ]{} which leads to the longitudinal optical conductivity: $$\begin{aligned}
\sigma_s = i \frac{D_s}{\omega-v_g^2 q^2/\omega}
\,,\quad D_s=n_s e^2 / m
\,.
\label{eqn:superfluid_conductivity}\end{aligned}$$ Here $n_s$ is the superfluid density and $n_s/m$ is the superfluid stiffness which in 3D is related to the magnetic penetration depth as $\lambda_B= \sqrt{\frac{c^2}{4\pi} \frac{m}{n_s e^2} }$. This form closely resembles that of a hydrodynamic electron fluid [@Sun.2018; @Torre2019] except that damping is completely suppressed here by the gap. [Eq. ]{} contains the whole crossover between the Drude limit $\omega\gg v_F q$ and the Thomas-Fermi limit $\omega\ll v_F q$. Note that the amplitude/BSM and quasiparticle excitation don’t enter here since they appear at higher energy. For a clean and isotropic BCS superconductor, $v_g=v_F/\sqrt{d}$ at zero temperature and gradually decreases to zero as temperature is raised to $T_c$.
At non-zero temperature, one should add the contribution of the normal carriers which makes the conductivity into the ‘two fluid’ form $$\begin{aligned}
\sigma(\omega,q) = \sigma_s + \sigma_n =
i \frac{D_s/\pi}{\omega-v_g^2 q^2/\omega} + \sigma_n
\,.
\label{eqn:two_fluid}\end{aligned}$$ For $\Delta \ll T$, an analytical formula for the normal fluid conductivity with non-zero scattering rate can be found from the Boltzmann equation[@Warren1960]. In the simple limits, $$\begin{aligned}
\sigma_n
=
\left\{
\begin{array}{lc}
\frac{ i D_n/\pi}{\omega + i\gamma} & \,(\omega \gg D_f q^2)
\\
-i \nu_n\frac{\omega}{q^2}
& \,(\omega \ll D_f q^2, v_F q)
\end{array}
\right.
\label{eqn:sigma_n}\end{aligned}$$ where $D_n = \pi n_n/m$, $\gamma$ is the scattering rate, $D_f=v_F^2/(d\gamma)$ is the normal state diffusion constant, $d$ is space dimension and $\nu_n$ is close to $\nu$ at temperate close to $T_c$.
In principle, this two fluid formula [Eq. ]{} can be obtained from the general derivation [Eq. ]{} with electron-impurity or electron-phonon scattering taken into account. We sketch the derivation of the two fluid model here by calculating the polarization function from [Eq. ]{}: $$\begin{aligned}
\chi_{\rho\rho} = \chi_{\rho\rho}^{(0)} - \frac{\left(\omega \chi_{\rho\rho}^{(0)} + \mathbf{q} \chi_{\rho\mathbf{j}}^{(0)} \right)^2}
{\omega^2 \chi_{\rho\rho}^{(0)} + q_i q_j \chi_{j_ij_j}^{(0)} + \frac{n}{m}q^2 + 2\omega \mathbf{q}\chi_{\rho\mathbf{j}}^{(0)} }
\,.
\label{eqn:chi}\end{aligned}$$ Close to $T_c$, we have $$\begin{aligned}
\chi_{\rho\rho}^{(0)} = \chi^{(0)} + \chi_s = \chi^{(0)} - \frac{\pi}{4} \frac{\Delta}{T_c} \nu
\,
\label{eqn:chi_rho}\end{aligned}$$ where $\chi^{(0)}$ come from the ‘intra band’ process among thermally excited quasiparticles while $\chi_s$ is the ‘interband’ contribution from exciting quasi particle pairs which has the interpretation of superfluid susceptibility (compressibility). Close to $T_c$, the $\chi^{(0)}$ should resemble the polarization function of a normal fermi liquid, i.e., the Lindhard function [@Mahan1990; @Warren1960] with non-zero scattering rate. Similarly [@Altland.2010; @Abrikosov:1975], $$\begin{aligned}
\chi_{j_xj_x}^{(0)} + \frac{n}{m} = -i\omega \sigma_n + \frac{n_s}{m}
\,
\label{eqn:chi_jj}\end{aligned}$$ where $\sigma_n $ is the ‘intraband’ part and $$\begin{aligned}
n_s &= n \int d\xi \left( \partial_E f(E) - \partial_\xi f(\xi) \right)
\notag\\
&= n\frac{\Delta^2}{2} \int d\xi \frac{1}{\xi} \partial^2_\xi f(\xi) + O\left(\frac{\Delta^4}{T^4}\right)
\notag\\
&\approx \frac{7\zeta(3)}{4\pi^2} \frac{\Delta^2}{T_c^2} n =2(1-T/T_c) n
\,
\label{eqn:chi_jj}\end{aligned}$$ is the superfluid density of a clean superconductor. Moreover, close to $T_c$, the intraband contributions to the correlation functions should approximately satisfy the continuity equations $\omega \chi^{(0)} + \mathbf{q} \chi_{\rho\mathbf{j}}^{(0)} =0$ and $q_j \sigma_{nij} + \omega \chi_{\rho j_i}^{(0)} =0$ since they are identical to those of the normal state at $\Delta=0$. With these simplifications [Eq. ]{} becomes $$\begin{aligned}
\chi_{\rho\rho} &= \chi^{(0)} + \chi_s - \frac{\left(\omega \chi_s \right)^2}
{\omega^2 \chi_s +\frac{n_s}{m}q^2 }
\notag\\
&=\chi^{(0)} + \frac{q^2 }
{\omega^2 - v_g^2 q^2 } D_s/\pi
\,.
\label{eqn:chi_two_fluid}\end{aligned}$$ The corresponding conductivity is just [Eq. ]{} with $\sigma_n = \chi^{(0)} i\omega/q^2$ and $v_g = \sqrt{\frac{n_s}{m}/\chi_s}$.
2D Plasmons {#sec:plasmons}
===========
For simplicity, we neglect the coupling to the amplitude mode in this section. The plasmons are the charge density fluctuations and can be found by the zeros of the dielectric function $$\begin{aligned}
\epsilon = 1- V_q \chi_{\rho\rho} = 1+ V_q \frac{i q^2}{\omega} \sigma =0 \,
\label{eqn:dielectric}\end{aligned}$$ where $V_q = 4\pi /q^2$ for three dimension and $V_q = 2\pi /|q|$ for 2D. Together with [Eq. ]{}, we obtain the plasmon dispersion $\omega_p=\sqrt{2D_s q + v_g^2 q^2}$ for two dimension. For three dimension, [Eq. ]{} predicts $\omega_p=\sqrt{4\pi n_se^2/m + v_g^2 q^2} \gg \Delta$ which lies in the high energy regime beyond the limit of validity of our theory although the correct plasma frequency $\omega_p^2 = 4\pi n_se^2/m$ is obtained for a clean superconductor.
In the low frequency limit $\omega \lesssim \omega_c$, the 2D plasmon dispersion $\omega_p=\sqrt{2D_sq}$ approaches the edge of the continuum of vacuum propagating photons $\omega=cq$ (recall that here $q$ is a two dimensional momentum and light modes disperse as $\omega=c\sqrt{q^2+k_z^2}$ for any $k_z$). For lower frequencies the analysis given here requires modification, because the electric fields associated with the plasmons begin to extend far from the 2D sheet, so that the plasmon couples much less strongly to near field radiation. The critical frequency can be estimated as $\omega_c=2D_s/c$ corresponding to the energy $\hbar\omega_c=\frac{e^2}{\hbar c}\frac{\hbar^2n_s}{m}\sim \frac{1}{137}E_F^\ast$ where $E_F^\ast$ is the fermi energy equivalent to a two dimensional superfluid stiffness $n_s/m$. For a clean, weakly correlated material $E_F^\ast$ is of eV-scale and the crossover frequency is of the order of $1$ THz. However, many superconductors of current interest [@Orenstein1990] have much lower $E_F^\ast$ so that the crossover frequency is well below the THz regime.
Assuming a doping level of $n=7\times 10^{13} {\,\mathrm{cm}}^{-2}$ (corresponding to the fermi momentum $k_F = 2\pi/(3 {\,\mathrm{nm}})$) and the fermi velocity of $v_F = 2.5\times 10^5 {\,\mathrm{m/s}}$, one obtains a wavelength of $\lambda \approx 180 {\,\mathrm{\mu m}}$ for the plasmon at $1 {\,\mathrm{THz}}$. This wavelength is close to that of the corresponding vacuum photon ($300 {\,\mathrm{\mu m}}$) although a substrate with large dielectric screening might make the plasmon wave length shorter.
Nevertheless, in a dirty superconductor with a large normal state scattering rate $\gamma \gg T_c$, the sub-gap plasmon frequency is mainly determined by the superfluid density which is only a part of the total density even at zero temperature: $n_s \sim n T_c/\gamma$. At the same THz frequency far below $\gamma$, the plasmon wavelength is smaller by the factor $T_c/\gamma$ and they become more confined to the 2D plane. Below the gap, weakly damped plasmons with dispersion $\omega = \sqrt{2D_s q}$ couple strongly to near field probe as shown by the near field reflection coefficient $$\begin{aligned}
R_p = -\frac{1}{\epsilon} +1 = -\frac{1}{1+\frac{i 2\pi q}{\omega} \sigma} +1 \,
\label{eqn:rp}\end{aligned}$$ in Fig. \[fig:Monolayer\_plasmon\_cg\] (a). For the ratio $T_c/\gamma =0.1$, the $1 {\,\mathrm{THz}}$ plasmon wave length is shrunk by the same factor to $18 {\,\mathrm{\mu m}}$.
Higgs mode {#sec:higgs}
==========
The Higgs mode couples to phase fluctuation in [Eq. ]{}, manifests itself in the second term of [Eq. ]{} and finally enters the EM response through [Eq. ]{}. The density response with the Higgs mode correction is thus $$\begin{aligned}
\chi_{\rho\rho}=\Pi^{00} = 2 \frac{- \frac{q^2}{\omega^2} \frac{n}{2m} (\nu/2 - G_\Delta C_0^2/4) -\frac{1}{8} \nu G_\Delta C_i^2 q^4 }
{\frac{\nu}{2} - \frac{1}{4} G_\Delta (C_0 +C_i q^2)^2 - \frac{q^2}{\omega^2} \frac{n}{2m}}
\,.
\label{eqn:dielectric_higgs}\end{aligned}$$ Since the $C_0$ and $C_i$ terms contribute terms at the same order, we take $C_i =0 $ to arrive at a simplified expression $$\begin{aligned}
\chi_{\rho\rho} = \frac{- \frac{q^2}{\omega^2} \frac{n}{m} (\nu/2 - G_\Delta C_0^2/4) }
{\frac{\nu}{2} - \frac{1}{4} G_\Delta C_0^2 - \frac{q^2}{\omega^2} \frac{n}{2m}}
\,.
\label{eqn:dielectric_higgs2}\end{aligned}$$ Thus the longitudinal optical conductivity is $$\begin{aligned}
\sigma(\omega,q) = i \frac{n_s e^2 / m}{\omega} \frac{1}{1- \frac{v_g^2 q^2}{\omega^2} \frac{1}{1- \kappa^2 \Delta^2/(\omega^2 - \omega_{hq}^2)}}
\,
\label{eqn:sigma_higgs}\end{aligned}$$ where $$\begin{aligned}
\kappa = \lambda \frac{\Delta}{E_F} \sqrt{\frac{1}{2}\sinh^{-1} \left( \frac{\omega_D}{\Delta} \right) }
\,
\label{eqn:kappa}\end{aligned}$$ is the dimensionless coupling constant of the Higgs mode to EM and $\omega_{hq}= \sqrt{4\Delta_{sc}^2+ v_F^2q^2/d}$ is the Higgs mode frequency. Since $\lambda$ is order one and $\sinh^{-1} \left( \frac{\omega_D}{\Delta} \right)$ is not a large number, this coupling is simply suppressed by the small number $\frac{\Delta}{E_F}$. Note that at $q=0$, the optical conductivity reduces to the Drude form and there is no signature of Higgs mode in it. Therefore it is impossible to observe it in far field THz linear response. In contrast, near field optical imaging technique has access to non-zero $q$ where the Higgs mode manifests itself through coupling to the plasmons.
Specifically, for a monolayer superconductor, the coupled collective modes can be found as the poles of [Eq. ]{}. The weight of the Higgs pole in $R_p$ scales as $W_{higgs} \sim \kappa^2 v_g^2 q^2/\Delta$ for $\omega_h \gg \omega_p$, i.e., well before the Higgs mode crosses the plasmon. Nevertheless, the most prominent signature of the Higgs mode is its anti crossing with the plasmon mode which happens roughly at $\omega_p(q)=\omega_{hq}$. A detailed solution of [Eq. ]{} gives the frequency splitting at the anti-crossing as $$\begin{aligned}
\delta \omega \approx \frac{\kappa \Delta}{2\omega_{hq}} \sqrt{\kappa^2 \Delta^2 + 4 v_g^2 q^2}
\label{eqn:higgs_splitting}\end{aligned}$$ where $q$ is the momentum at the anti-crossing. Therefore, the splitting will be bigger if the anti-crossing happens at larger momentum.
Bardasis-Schrieffer mode {#sec:BS}
========================
Optical excitation of the BSM can be viewed as transition from an s bound state of the cooper pair to a d bound state. This is forbidden in far field optics for two reasons: first, unlike the Hydrogen atom case, uniform electric field exerts the same force on the two electrons and does not change the internal structure; second, both s-state and d-state have even parity which forbids the transition due to optical selection rule. Thus it is necessary to go to nonzero momentum for its nonzero coupling to EM field. Indeed, the coupling constant is proportional to $\xi q$ which is appreciable when the electric field becomes substantially nonuniform on the scale of a cooper pair size.
Plugging [Eq. ]{} into [Eq. ]{} and [Eq. ]{} gives the appearance of the BSM in the longitudinal optical conductivity: $$\begin{aligned}
\sigma(\omega,q) = i \frac{n_s e^2 / m}{\omega} \frac{1}{\frac{1}{1 + \kappa_{BS}^2 v_g^2q^2/(\omega^2 - \omega_{BS}^2)}- \frac{v_g^2 q^2}{\omega^2} }
\,
\label{eqn:sigma_bs}\end{aligned}$$ where $\kappa_{BS}= \frac{\pi}{\sqrt{8}}v_F^2/v_g^2 \sim 1$ is the dimensionless coupling constant between BSM and EM. The $B_0$ terms are higher order in $q$ and are neglected. Note that if the momentum $q$ is along x, the BSM means the $d_{x^2-y^2}$ order parameter fluctuation.
The BSM couples to near field more strongly than the Higgs mode due to the absence of the $\Delta/E_F$ factor in the coupling constant $\kappa_{BS}$. Solving the pole equation, [Eq. ]{}, one obtains the frequency splitting at the anti-crossing between BS and plasmon $$\begin{aligned}
\delta \omega \approx \kappa_{BS} v_g q
\label{eqn:bs_splitting}\end{aligned}$$ which scales linearly with the momentum at the anti-crossing.
Carlson-Goldman mode {#sec:CG}
====================
The CG mode is a superfluid density fluctuation accompanied by the counter flow of normal carriers such that the Coulomb potential from the superfluid fluctuation is almost completely screened [@Carlson1975; @schmid.1975; @Pethick1979; @Artemenko1979; @Goldman2007]. This screening requires a large density of normal carriers which is typically found near T$_c$. The speed $v_g$ of the CG mode depends on the ratio between the superfluid density $n_s$ and superfluid susceptibility $\chi_s = \frac{\pi}{4} \frac{\Delta}{T_c} \nu$ and has different expressions in the clean and dirty limits: $$\begin{aligned}
v_g = \sqrt{\frac{n_s}{m} / \chi_s}
=
\frac{v_F}{\sqrt{d}}
\left\{
\begin{array}{lc}
\sqrt{2\Delta/\gamma} & \gamma \gg T_c \,(\text{Dirty})
\\
\sqrt{\frac{7\zeta(3)}{\pi^3} \frac{\Delta}{T}}
& \gamma \ll T_c \,(\text{Clean})
\end{array}
\right.
\label{eqn:vg}\end{aligned}$$ where $\zeta(x)$ is the Riemann Zeta function and we have used the fact that $n_s = n\frac{\pi \Delta^2}{2\gamma T_c}$ for dirty superconductors and $n_s = 2(1-T/T_c) n$ for clean superconductors close to $T_c$.
Its dispersion can be derived from the two fluid conductivity [Eq. ]{} by setting $\epsilon=0$ which yields $$\begin{aligned}
\omega^3 + i \frac{\omega_n^2}{\gamma}\omega^2 - (\omega_s^2 + v_g q^2) \omega - i\frac{\omega_n^2}{\gamma} v_g^2 q^2 =0
\,
\label{eqn:CG_equation}\end{aligned}$$ in the limit of $D_f q^2 \ll \omega \ll \gamma$. Note the plasma frequency $\omega_{s/n} = \sqrt{4D_{s/n}}$ in 2D and $\omega_{s/n} = \sqrt{2 D_{s/n}q}$ in 3D. Solving [Eq. ]{} in the case of $ \omega_s \gg \omega \gg \frac{\omega_s^2}{\omega_n^2} \gamma$ renders the CG mode $$\begin{aligned}
\omega = \sqrt{v_g^2 q^2 - \frac{1}{4}\frac{\omega_s^4}{\omega_n^4} \gamma^2} -i\frac{1}{2}\frac{\omega_s^2}{\omega_n^2} \gamma
\,.
\label{eqn:CG_solution}\end{aligned}$$ At even lower frequency $\omega \ll \frac{\omega_s^2}{\omega_n^2} \gamma$ in 2D, the solution to [Eq. ]{} gives the weakly damped plasmons $$\begin{aligned}
\omega = \sqrt{\omega_s^2-\frac{\omega_n^4}{4\gamma^2}} -i\frac{1}{2}\frac{\omega_n^2}{\gamma}
\,
\label{eqn:CG_plasmon}\end{aligned}$$ where the $v_g q$ contribution has been neglected. Note that the effective Drude weight is $D_s$ in the low frequency regime of the two fluid model [Eq. ]{}, similar to the collective mode called Demons in the hydrodynamic regime of the Dirac fluid[@Sun2016a; @Sun.2018]. The schematic dispersion of the CG mode and plasmons in 2D are depicted in Fig. \[fig:CG\]. The damping rate $\frac{\omega_s^2}{2\omega_n^2} \gamma$ of the CG mode is equal to $\frac{\pi}{4}\frac{\Delta^2}{T}$ in the dirty case and $(1-T/T_c)/\gamma$ in the clean case.
Note that the CG mode can be understood as a sound with the standard sound velocity $\sqrt{\frac{n_s}{m}/\chi_s}$ and $\chi_s$ being the superfluid compressibility. The latter is smaller than $\nu$, the compressibility of the whole fluid in the low frequency thermal dynamic limit, because the super and normal fluids move out of phase in this relative high frequency regime. The local accumulation of superfluid causes the local chemical potential to shift up leading to change of quasiparticle energy and charge. However, the quasiparticle occupation number relaxes too slowly and cannot adjust itself to this change[@Pethick1979], resulting in ‘branch imbalance’ [@Tinkham] as shown in Fig. \[fig:CG\_picture\]. Normal impurities cannot relax this branch imbalance because in $s$-wave superconductors, the elastic scattering matrix element $u_k u_{k^\prime}-v_k v_{k^\prime}$ vanishes between hole like and electron like states at the same energy. Inelastic scattering due to, e.g., phonons, does relax branch imbalance and cause extra damping to the CG mode but we assume it to be small. In $d$-wave superconductors, the same matrix element is non-zero due to anisotropy of the gap which allows normal impurities to relax the branch imbalance and bring extra damping to the CG mode [@Artemenko1997].
In clean superconductors the CG mode can cross the diffusion line before reaching the gap, entering the regime $\omega \ll D_f q^2$ where the normal fluid part of the [Eq. ]{} is in the Thomas-Fermi form [Eq. ]{}. The normal fluid still screens the CG mode but with a Thomas-Fermi screening character. The CG mode speed is slightly modified to $v_{CG} = \sqrt{v_g^2 + \frac{n_s}{\nu_n m}}$ in this regime but still remains close to $v_g$ since the second term is much smaller.
The original experiment using Josephson tunneling junctions by Carlson and Goldman[@Carlson1975] seems to be the only observation of this novel collective mode, the closest analogy to the Goldstone mode of $U(1)$ symmetry breaking in a superconductor. In the optical conductivity measured by far field optics, the CG mode might move part of the superfluid spectra weight to finite frequency due to smooth disorder[@Orenstein2003; @Barabash2003]. At non-zero momentum, being almost charge neutral, the CG mode appears as a very weak feature in the near field reflection coefficient: a one percent crossover of $\mathrm{Abs}[R_p]$ as shown by Fig. \[fig:Monolayer\_plasmon\_cg\](b) plotted for a typical dirty superconductor close to its $T_c$.
Double layer superconductor {#sec:double_layer}
===========================
Consider the system made of two superconducting layers separated by some small distance $a$, as shown in Fig. \[fig:double\_layer\_rp\]. We neglect the Josephson coupling between the layers which is weak for $a$ substantially larger than atomic scale. Each layer has an in plane conductivity described by [Eq. ]{} at low temperature. In the quasi static limit, the 2D plasmon dispersion can be obtained from the following eigenmode condition $$\begin{aligned}
\left( 1+\frac{2\pi i}{\omega} q \sigma \right)^2 + e^{-2aq} \left( \frac{2\pi q}{\omega} \sigma \right)^2 =0 \,
\label{eqn:double_layer_mode_equation}\end{aligned}$$ which leads to two plasmon branches $$\begin{aligned}
\omega_{\pm} = \sqrt{2Dq (1 \pm e^{-aq}) + v_g^2 q^2}
\,.
\label{eqn:double_layer_plasmon_dispersion}\end{aligned}$$ The upper branch is the symmetric mode whose dispersion follow the $\omega_{+} \sim \sqrt{q}$ law at small momentum. The lower (anti symmetric) branch is an acoustic mode which has the dispersion $$\begin{aligned}
\omega_{-} = \sqrt{2Da + v_g^2} \cdot q = v_{-} \cdot q \,
\label{eqn:acoustic_plasmon_dispersion}\end{aligned}$$ for $q \ll 1/a$. This acoustic mode is charge fluctuations of the two layers which are out of phase such that the net charge fluctuation is near zero if looked at far away. In other words, the Coulomb interaction is mutually screened and is modified to the effective short range form $V(q)=2\pi(1-e^{-aq})/q$ that makes the mode acoustic.
Both modes correspond to non-zero momentum oscillations of the phase of the superconducting order parameter. This acoustic plasmon can be viewed as the Goldstone mode which recovers its acoustic nature because Coulomb interaction is greatly weakened. Its speed still has a large contribution $\sqrt{2Da} \sim \sqrt{\alpha k_F a} v_F$ from the residual Coulomb interaction where $\alpha= e^2/(\hbar v_F)$ is the ‘fine structure constant’. In BSCCO 2212 at typical doping[@Chiao2000], $\alpha \approx 9$ since $v_F \approx 2.5 \times 10^5 {\,\mathrm{m/s}}$. For $k_F=2\pi/(10 {\,\mathrm{nm}})$ and $a=3 {\,\mathrm{nm}}$, the ratio between the speeds of this acoustic mode and the original Goldstone mode is $v_{-}/v_g \approx 6$ which means they are at the same order of magnitude. Therefore, an accurate measurement of the acoustic plasmon dispersion would contain the information of the ‘Goldstone mode’ speed.
In order for the acoustic mode to be observable to near field probe, it should have substantial spectral weight in the the near field reflection coefficient $$\begin{aligned}
R_p(\omega,q) = - \frac{2\pi i q \sigma}{\omega} \frac{\epsilon + e^{-2aq} \left( 1-\frac{2\pi i}{\omega} q \sigma \right) }
{\epsilon^2 + e^{-2aq} \left( \frac{2\pi q}{\omega} \sigma \right)^2 } \,
\label{eqn:r_p_double_layer}\end{aligned}$$ derived in Appendix \[appendix:rp\]. Given the same amplitude of charge density oscillation in each layer, the electric field generated by the two layers tend to cancel each other since they are opposite in sign. The remaining field is weaker than the symmetric plasmon mode by a factor of $qa/2$ and the near field spectra weight is weaker by $(qa/2)^{3/2}$. Nevertheless, the acoustic mode is still visible as shown by the $R_p$ plotted in Fig. \[fig:double\_layer\_rp\].
Moreover, since the acoustic plasmon has higher momentum given the same frequency in the THz range, it has stronger coupling to the Higgs/BSMs. Thus there is more prominent anti crossing feature between them, as shown in Fig. \[fig:double\_layer\_rp\]. Note that [Eq. ]{} and [Eq. ]{} apply to anti-crossings with both the symmetric and anti-symmetric modes. For example, the anti crossing of the BSM with the acoustic plasmon happens at a momentum roughly $20$ times that with the symmetric plasmon, rendering the energy splitting $20$ times larger than the latter.
Bulk layered superconductors {#sec:bulk_layered}
============================
For the class of layered superconductors, e.g., High T$_c$ cuprates, there is Josephson coupling between the layers and the low temperature and subgap collective modes are the Josephson plasmons [@Basov2005a]. Considering only the phase degree of freedom, the Lagrangian for an evenly spacing layered superconductor is $$\begin{aligned}
L = \int dr \sum_{n}
\bigg[& \frac{1}{2}\nu (\partial_t \theta_n + \phi_n)^2 - \frac{n_s}{2m} (\nabla \theta_n - \mathbf{A}_n)^2
\notag\\
&-E_c \cos\left(\theta_{n+1} - \theta_{n} - \int_{n+1}^{n} A dz \right)
\bigg]
\,
\label{eqn:action_layered}\end{aligned}$$ where $\theta_n(r)$, $\phi_n(r)$ and $A_n(r)$ are the phase, scalar and vector potentials on the nth layer and $E_c$ is the Josephson coupling energy per unit area and we have set $e=1$. For longitudinal fields we are interested in, we can choose tha gauge where $A_n(r)= 0$. Due to continuous translational symmetry in plane and discrete in z direction, it is convenient to Fourier transform the fields into the ‘Bloch’ form $$\begin{aligned}
\theta_{n}(r) = \sum_{k_z, q} \theta_{k_z,q} e^{i(qr + k_z n a)}
\,
\label{eqn:theta_bloch}\end{aligned}$$ where $a$ is the layer spacing, $q$ is the in plane momentum and $k_z \in (-\pi/a,\,\pi/a)$ is the lattice momentum in $z$ direction. The Lagrangian [Eq. ]{} is diagonalized as $$\begin{aligned}
L = \sum_{k_z, q}
\left[ \frac{1}{2}\nu (\partial_t \theta_n + \phi_n)_q^2 - \left(\frac{n_s}{2m} q^2
-E_c (1-\cos(ak_z)) \right) \theta_q^2
\right]
\,.
\label{eqn:action_layered_diag}\end{aligned}$$ Solving the Euler-Lagrange equation of the phase and making use of the expression of the charge density $\rho = \nu (\partial_t \theta -\phi)$, we obtain the ‘nonlocal’ polarization function $$\begin{aligned}
\chi_{\rho \rho}(k_z, q) = \frac{\frac{n_s}{m} q^2 + 2 E_c \left(1- \cos(ak_z) \right)}
{\omega^2 - v_g^2 q^2 - \frac{1}{\nu} 2 E_c\left(1- \cos(ak_z) \right) }
\,.
\label{eqn:polarization_layered_sc}\end{aligned}$$ Due to the discrete Fourier transform, the Coulomb potential kernel is modified to $$\begin{aligned}
V(k_z,q)= \frac{2\pi e^2}{q} \frac{\sinh(aq)}{\cosh(aq) -\cos(ak_z)}
\,.
\label{eqn:coulomb_kernel}\end{aligned}$$ The zeros of the dielectric function $\epsilon = 1- V(k_z, q)\chi_{\rho \rho}$ gives the dispersion of the collective modes $$\begin{aligned}
\omega^2 = \left(1/\nu + V(k_z,q) \right)
\left(\frac{n_s}{m}q^2 + 2 E_c\left(1- \cos(ak_z) \right) \right)
\,.
\label{eqn:layered_modes_discrete}\end{aligned}$$ In the long wave length limit $q,k_z \ll 1/a$, the Coulomb kernel reduces to that of the continuous limit and the mode dispersion simplifies to $$\begin{aligned}
\omega = \sqrt{\omega_p^2 \frac{q^2}{q^2 + k_z^2} + \omega_J^2 \frac{k_z^2}{q^2 + k_z^2} + v_g^2 q^2 + v_z^2 k_z^2
}
\,.
\label{eqn:layered_modes_continuous}\end{aligned}$$ where $\omega_J = \sqrt{4\pi E_c a^2} $ is the Josephson plasma frequency, $\omega_p = \sqrt{4\pi n_s/m}$ is the in plane plasma frequency, $v_g=v_F/\sqrt{2}$ is the in plane Goldstone mode speed and $v_z=\frac{\omega_J}{\omega_p} v_g$ is the z axis Goldstone mode speed. These are the hyperbolic Josephson plasmons (HJP) extensively studied in the literature[@Pokrovsky1996; @Stinson2014a; @Zhou2014; @Sun.2015; @Basov2016]. Indeed, [Eq. ]{} could be derived directly from the continuous limit of the nonlocal dielectric function $$\begin{aligned}
\epsilon(k_z,q) = 1- \frac{ \omega_p^2 \frac{q^2}{q^2 + k_z^2} + \omega_J^2 \frac{k_z^2}{q^2 + k_z^2}}
{\omega^2 - v_g^2 q^2 - v_z^2 k_z^2 }
\,
\label{eqn:sigma_layered_continuous}\end{aligned}$$ which is defined as the external electrical potential divided by the total potential.
Alternatively, the long wavelength response can be described by the anisotropic dielectric function $$\begin{aligned}
\epsilon_x(\omega,q,k_z) &= 1- \frac{ \omega_p^2 }
{\omega^2 - v_g^2 q^2 - v_z^2 k_z^2 }
\,, \notag\\
\epsilon_z(\omega,q,k_z) &= 1- \frac{ \omega_j^2 }
{\omega^2 - v_g^2 q^2 - v_z^2 k_z^2 }
\,
\label{eqn:dielectric_anisotropic}\end{aligned}$$ and the collective mode dispersion is determined by $q^2 \epsilon_x + k_z^2 \epsilon_z =0$. This formalism is more convenient for calculating the reflection coefficient of a slab. To include the effect of Higgs and BSMs, one just needs to modify the in plane response $\epsilon_x$ in similar fashions as Eqs. and .
For a superconducting slab in the continuous limit, the near field reflection coefficient is shown in Fig. \[fig:layered\_josephson\_plasmon\], taking into account the nonlocal corrections to the dielectric function due to the Goldstone mode. Note that due to high anisotropy of the EM response, the z direction wavelength $\lambda_z \sim \omega/\omega_p$ can easily get comparable to the layer spacing. In that case, the full form [Eq. ]{} should be used as the bulk mode dispersion and the number of hyperbolic plasmon branches is limited by the number of layers $N$. Due to Josephson coupling between the layers, the transfer matrix method does not apply and numerical diagonalization of a set of $N$ coupled linear equations will be needed to calculate the near field reflection coefficient.
Discussion {#sec:discussion}
==========
We studied the non local EM response properties of superconductors which are of great importance to the emerging field of THz near field experiments. With analytical formulas for the non local optical conductivity and plots of reflection coefficients, we have demonstrated that for monolayer or multilayer quasi two dimensional superconductors essentially all of the interesting collective modes (plasmons, hyperbolic interlayer or Josephson plasmons, the Carlson-Goldman mode, the amplitude(Higgs) mode and the Bardasis-Schrieffer mode) couple linearly to the THz EM fields produced by near field probes. As old arguments of Anderson show, the dispersion of the plasmon ($\sqrt{q}$) is essentially unaffected by superconductivity but the gap substantially suppresses the loss at low frequencies. Fig. \[fig:Monolayer\_plasmon\_cg\](a) shows the plasmon dispersion expected for a monolayer superconductor. In superconducting bilayers, an additional acoustic ($\omega \propto q$) plasmon (phase) mode exists and is also easily observable in near field experiments (Fig. \[fig:double\_layer\_rp\](b)). As the temperature becomes close to T$_c$, as shown by Fig. \[fig:Monolayer\_plasmon\_cg\](b), the Carlson-Goldman mode appears but as a very weak feature across the resonance since it has almost no net charge density fluctuation. Note that this mode is not enhanced in multilayer systems. The amplitude (Higgs) mode appears in the EM response because it couples to the phase fluctuation with a matrix element that is non-zero if there is no perfect particle hole symmetry ([Eq. ]{}). The ultimate coupling to THz near field is proportional to the square of the near field momentum $q$ ([Eq. ]{}), and is strongly enhanced by an anticrossing with the plasmon or phase modes. The Higgs mode is only weakly visible for monolayer materials because the $\sqrt{q}$ plasmon dispersion means that the anticrossing occurs at a very small momentum (Fig. \[fig:Monolayer\_plasmon\_cg\](a)). The feature is much more easily visible in bilayer systems as an anti crossing with the acoustic plasmon (phase) mode (Fig. \[fig:double\_layer\_rp\](b)). The coupling to Bardasis-Schrieffer (subdominant order parameter) mode is very similar to that of the Higgs mode, except that it does not require particle hole symmetry breaking. It is again most easily visible as a large $q$ anti-crossing with the phase (Fig. \[fig:double\_layer\_rp\](b)) or plasmon mode (Fig. \[fig:Monolayer\_plasmon\_cg\](a)).
In multilayer superconductors, a multiplicity of phase modes exist, coined the hyperbolic Josephson plasmons (Fig. \[fig:layered\_josephson\_plasmon\]). The plasmon dispersion is hyperbolic ($\epsilon <0$ for in plane and $\epsilon >0$ for out of plane), leading to total internal reflection (Fig. \[fig:layered\_josephson\_plasmon\](a)) and many plasmon branches with Higgs and BSMs visible as anti-crossings. The multiplayer nature means there are multiple branches of Higgs modes/BSMs, but they are weakly separated and may be difficult to resolve.
On the experimental side, detection of the collective modes offers useful information about both the ground state and the low lying excited states. On the theory side, knowledge of how to excite the collective modes are often the first step towards understanding non equilibrium dynamics. From the technological point of view, the low loss plasmonic modes are promising as information carriers in superconductor wave guides. The multiplayer systems described in Sections \[sec:double\_layer\] and \[sec:bulk\_layered\] can be viewed as a kind of naturally occuring photonic cavities which enhance light matter coupling.
The formalism presented here is for $s$-wave superconductors. For $d$-wave superconductors, the qualitative features of the EM response such as the two fluid model [Eq. ]{}, the plasmons and the Carlson-Goldman mode should be the same. Nevertheless, the CG mode might exist down to much lower temperature because of the large proportion of the normal fluid [@Ohashi2000] although it might be heavily damped by normal disorder[@Artemenko1997]. Due to the nodes in the $d$-wave gap, the THz plasmons might experience substantial damping even at zero temperature. The effect of disorder is not explicitly taken into account and would be a useful extension of the present research, e.g., disorder assisted Cherenkov radiation of plasmons by quasiparticles.
We acknowledge support from the Department of Energy under Grant DE-SC0018218. We thank W. Yang, R. Jing, Y. Shao, G. Ni and Y. He for helpful discussions.
Correlation functions {#app:correlation_function}
=====================
The correlation function $\chi_{\sigma_i \sigma_j}$ is defined as $$\begin{aligned}
\chi_{\sigma_i \sigma_j}(q) =
\left\langle \hat{T}
\left(\psi^\dagger \sigma_i \psi \right)_{(r,t)}
\left(\psi^\dagger \sigma_j \psi \right)_0
\right\rangle \bigg|_q
=
\sum_{\omega_n, k}
Tr\left[ G(k,i\omega_n) \sigma_i G(k+q,i(\omega_n+\Omega)) \sigma_j \right]
\,.
\label{eqn:chi_defi}\end{aligned}$$ where $\hat{T}$ is the time order symbol, $x=(\mathbf{r},t)$, $q=(\mathbf{q},i\Omega)$ and $$\begin{aligned}
G(k,i\omega_n) &= G_\Delta(k,i\omega_n) =
\left\langle \hat{T}
\psi (x)
\psi^\dagger(0)
\right\rangle \bigg|_{k,i\omega_n}
=\frac{1}{i\omega_n - \xi_k \sigma_3 -\Delta\sigma_1 }
\,\end{aligned}$$ is the electron Green’s function. Rotation from imaginary to real time makes the time ordered correlation functions into retarded ones. In the correlation functions involving the currents, one should change the $\sigma$ vertex to the current vertex. For example, $$\begin{aligned}
\chi_{j_l \sigma_m}(q) &=
\left\langle \hat{T}
\left(\psi^\dagger v_l\sigma_0 \psi \right)_x
\left(\psi^\dagger \sigma_m \psi \right)_0
\right\rangle \bigg|_q
=
\sum_{\omega_n, k}
\frac{1}{2} \left( v(k)+v(k+q) \right)
Tr\left[ G(k,i\omega_n) \sigma_0 G(k+q,i(\omega_n+\Omega)) \sigma_l \right]
\,.\end{aligned}$$ Evaluating the correlation function [Eq. ]{} renders $$\begin{aligned}
\chi_{\sigma_i \sigma_j}(q) &=
\sum_{\omega_n, k}
Tr\left[
\frac{
\left(i\omega_n + \xi \sigma_3 + \Delta \sigma_1 \right) \sigma_i
\left( i(\omega_n +\Omega) + \xi^\prime \sigma_3 + \Delta \sigma_1 \right) \sigma_j
}{\left( (i\omega_n)^2 - E^2) \right)
\left(
(i(\omega_n+\Omega))^2 - E^{\prime 2}) \right)
}
\right]
\notag\\
&= \frac{1}{4}
\sum_{k}
\Bigg\{
Tr\left[
\sigma_i \sigma_j
\right]
\left(
\frac{f(E^\prime)-f(E)}{i\Omega- (E-E^\prime)}+
\frac{1-f(E^\prime)-f(E)}{i\Omega- (E+E^\prime)} +
\frac{f(E^\prime)+f(E)-1}{i\Omega + (E+E^\prime)} +
\frac{f(E)-f(E^\prime)}{i\Omega- (E^\prime-E)}
\right)
\notag\\
&+
Tr\left[
\frac{\sigma_i (\xi^\prime \sigma_3 + \Delta \sigma_1) \sigma_j}{E^\prime}
\right]
\left(
-\frac{f(E^\prime)-f(E)}{i\Omega- (E-E^\prime)}+
\frac{1-f(E^\prime)-f(E)}{i\Omega- (E+E^\prime)} -
\frac{f(E^\prime)+f(E)-1}{i\Omega + (E+E^\prime)} +
\frac{f(E)-f(E^\prime)}{i\Omega- (E^\prime-E)}
\right)
\notag\\
&+
Tr\left[
\frac{(\xi \sigma_3 + \Delta \sigma_1) \sigma_i \sigma_j}{E}
\right]
\left(
-\frac{f(E^\prime)-f(E)}{i\Omega- (E-E^\prime)}
-\frac{1-f(E^\prime)-f(E)}{i\Omega- (E+E^\prime)} +
\frac{f(E^\prime)+f(E)-1}{i\Omega + (E+E^\prime)} +
\frac{f(E)-f(E^\prime)}{i\Omega- (E^\prime-E)}
\right)
\notag\\
&+
Tr\left[
\frac{(\xi \sigma_3 + \Delta \sigma_1) \sigma_i (\xi^\prime \sigma_3 + \Delta \sigma_1) \sigma_j}{E E^\prime}
\right]
\left(
\frac{f(E^\prime)-f(E)}{i\Omega- (E-E^\prime)}
-\frac{1-f(E^\prime)-f(E)}{i\Omega- (E+E^\prime)}
-\frac{f(E^\prime)+f(E)-1}{i\Omega + (E+E^\prime)} +
\frac{f(E)-f(E^\prime)}{i\Omega- (E^\prime-E)}
\right)
\Bigg\}
\label{eqn:chi_final}\end{aligned}$$ where $\xi$/$E$ means $\xi(k)$/$E(k)$ and $\xi^\prime$/$E^\prime$ means $\xi(k+q)$/$E(k+q)$. At zero temperature, rotating $i\Omega$ to $\omega$, [Eq. ]{} simplifies to $$\begin{aligned}
\chi_{\sigma_i \sigma_j}(\omega, q)
= \frac{1}{4}
\sum_{k}
\Bigg\{&
Tr\left[
\sigma_i \sigma_j
-
\frac{(\xi \sigma_3 + \Delta \sigma_1) \sigma_i (\xi^\prime \sigma_3 + \Delta \sigma_1) \sigma_j}{E E^\prime}
\right]
\frac{2(E+E^\prime)}{\omega^2 - (E+E^\prime)^2}
\notag\\
&+
Tr\left[
\frac{\sigma_i (\xi^\prime \sigma_3 + \Delta \sigma_1) \sigma_j}{E^\prime}
-
\frac{(\xi \sigma_3 + \Delta \sigma_1) \sigma_i \sigma_j}{E}
\right]
\frac{2\omega}{\omega^2 - (E+E^\prime)^2}
\Bigg\}
\label{eqn:chi_zero_t}\end{aligned}$$
The Higgs propagator {#appendix:Higgs}
--------------------
The Higgs propagator involves the correlation in $\sigma_1$ channel: $$\begin{aligned}
\chi_{\sigma_1 \sigma_1}(\omega, q)
&=
\sum_{k}
\Bigg\{
\left(
1- \frac{\Delta^2-\xi\xi^\prime}{EE^\prime}
\right)
\frac{E+E^\prime}{\omega^2 - (E+E^\prime)^2}
\Bigg\}
\label{eqn:chi_zero_t}\end{aligned}$$ At zero momentum, it becomes $$\begin{aligned}
\chi_{\sigma_1 \sigma_1}(\omega, 0)
&=
\sum_{k}
\frac{\xi^2}{E}
\frac{4}{\omega^2 - 4E^2}
\label{eqn:chi_sigma1}\end{aligned}$$ With the knowledge of the gap equation, the Higgs propagator is thus [@Cea2015] $$\begin{aligned}
G_a^{-1}(\omega)
&= \frac{1}{g} +
\chi_{\sigma_1 \sigma_1}(\omega, 0)
= (\omega^2 - 4\Delta^2)
\sum_{k}
\frac{1}{E(\omega^2 - 4E^2)}
=-(\omega^2 - 4\Delta^2) F(\omega)
\,.
\label{eqn:chi_11}\end{aligned}$$ where $$\begin{aligned}
F(\omega)
&=
\sum_{k}
\frac{1}{E(-\omega^2 + 4E^2)}
\approx \frac{1}{2}\nu \int d\xi \frac{1}{E(-\omega^2+4E^2)}
=\frac{\nu}{4\Delta^2} \frac{2\Delta}{\omega} \frac{\mathrm{sin}^{-1}\left(\frac{\omega}{2\Delta}\right)}{\sqrt{1-\left(\frac{\omega}{2\Delta}\right)^2}}
\notag\\
&=
\frac{\nu}{2\omega \Delta }
\left\{
\begin{array}{lc}
\frac{\sin^{-1}(\frac{\omega}{2\Delta})}{\sqrt{1-(\frac{\omega}{2\Delta})^2}} & \omega \le 2\Delta
\\
\frac{-\sinh^{-1}\left(\sqrt{-1+(\frac{\omega}{2\Delta})^2}\right)}
{\sqrt{-1+(\frac{\omega}{2\Delta})^2}}
+ i\frac{\pi}{2\sqrt{-1+(\frac{\omega}{2\Delta})^2}}
& \omega > 2\Delta
\end{array}
\right.
\,
\label{eqn:chi_11}\end{aligned}$$ is shown in Fig. \[fig:f\_omega\].
At non-zero momentum, the propagator is [@Littlewood1982] $$\begin{aligned}
G_a^{-1}(\omega, q)
&=
\frac{1}{2}\sum_{k}
\frac{E+E^\prime}{EE^\prime}
\frac{\omega^2 - (\xi-\xi^\prime)^2-4\Delta^2}{\omega^2 - (E+E^\prime)^2}
\label{eqn:chi_finite_q}\end{aligned}$$ whose $O(q^2)$ expansion gives $$\begin{aligned}
G_a^{-1}(\omega, q)
\approx
\left(-\omega^2 + 4\Delta^2 + \frac{1}{d}v_F^2q^2 \right)
F(\omega)
\,.
\label{eqn:chi_q_expand}\end{aligned}$$
The BS propagator {#appendix:BS}
-----------------
The total order parameter can be written as $\Delta_k=\Delta+ \sum_l\Delta_l(r,t) f_l(k)$ where we have chosen the mean field gap $\Delta$ to be real. The subdominant pairing order parameter fluctuations $\Delta_l$ can have two possible directions: 1, orthogonal to $\Delta$ on the complex plane or in the ‘imaginary’ direction; 2, parallel to $\Delta$ or in the ‘real’ direction. The ’imaginary’ fluctuations are the BSMs while the ‘real’ ones don’t have poles and are not collective modes.
We first consider the BSM correlator $$\begin{aligned}
\chi_{f_l(k)\sigma_2, f_l(k)\sigma_2}(i\Omega,q) =
\sum_{\omega_n, k} Tr\left[ G(k,i\omega_n) f_l(k)\sigma_2 G(k+q,i(\omega_n+\Omega)) f_l(k)\sigma_2 \right]
\,.\end{aligned}$$ In two dimension with rotational symmetry, the $d_{x^2-y^2}$ BSM correlator is in the $\cos(2\theta_k)\sigma_2$ channel: $$\begin{aligned}
\chi_{\cos(2\theta_k)\sigma_2,\cos(2\theta_k)\sigma_2}(\omega, 0)
&= \sum_{k}
\frac{4 \cos^2(2\theta_k) E_k }{\omega^2 - 4E^2}
=-\frac{1}{g} -\omega^2 F(\omega)
\,.
\label{eqn:bs_correlation}\end{aligned}$$ The BSM inverse propagator is $$\begin{aligned}
G_{BS}^{-1}(\omega) = \frac{1}{g_d} +
\chi_{\cos(2\theta_k)\sigma_2,\cos(2\theta_k)\sigma_2}(\omega, 0)
= \frac{1}{g_d} -\frac{1}{2g} - \frac{1}{2}\omega^2 F(\omega)
\,
\label{eqn:bs_propagator}\end{aligned}$$ which crosses zero at $\omega_{BS}$ below the gap, as shown by the left panel of Fig. \[fig:GBS\_omega\]. For momentum along $x$, extending the correlator to $O(q^2)$ gives $$\begin{aligned}
G_{BS}^{-1}(\omega) \approx \frac{1}{g_d} -\frac{1}{2g} - \frac{1}{2}\omega^2 F(\omega)
+\frac{1}{16} \frac{\nu}{\Delta^2} v_F^2 q^2
\,
\label{eqn:bs_propagator_q}\end{aligned}$$ in two dimension. In the case of $\omega_{BS} \ll 2\Delta$, the propagator is simplified to $$\begin{aligned}
G_{BS}^{-1}(\omega) \approx \frac{\nu}{8\Delta^2}
\left( \omega_{BS}^2 + \frac{1}{2} v_F^2 q^2 -\omega^2 \right)
\,
\label{eqn:bs_propagator_q_simple}\end{aligned}$$ where $\omega_{BS}^2=8\Delta^2 (\frac{1}{\nu g_d} - \frac{1}{2 \nu g})$. Thus the BSM frequency disperses as $\omega_{BS}(q)^2= \omega_{BS}^2 + \frac{1}{2} v_F^2 q^2$.
We now consider the correlator of the ‘real’ fluctuations: $$\begin{aligned}
\chi_{\cos(2\theta_k)\sigma_1,\cos(2\theta_k)\sigma_1}(\omega, 0)
&= \sum_{k}
\frac{\xi^2}{E}
\frac{4 \cos^2(2\theta_k)}{\omega^2 - 4E^2}
=\frac{1}{2} \left[ -\frac{1}{g}-(\omega^2 - 4\Delta^2) F(\omega) \right]
\,
\label{eqn:bs_real_correlation}\end{aligned}$$ which is different from the Higgs correlator [Eq. ]{} only by the $\cos^2(2\theta_k)$ factor. The resulting propagator is $$\begin{aligned}
G^{-1}_{\text{real}}(\omega, 0)
= \frac{1}{g_d} + \chi_{\cos(2\theta_k)\sigma_1,\cos(2\theta_k)\sigma_1}(\omega, 0)
= \frac{1}{g_d} -\frac{1}{2g} -\frac{1}{2}(\omega^2 - 4\Delta^2) F(\omega)
\,
\label{eqn:bs_real_correlation}\end{aligned}$$ which never crosses zero as shown by the right panel of Fig. \[fig:GBS\_omega\].
The linear coupling between phase and BSM/Higgs modes {#appendix:coupling_constants}
-----------------------------------------------------
The coupling between phase fluctuation and the Higgs mode requires particle hole symmetry breaking which we model using an energy dependent DOS $g(\xi) = \nu (1+\lambda \xi/E_F)$. The coupling constants are derived from the correlation functions in [Eq. ]{}. From the general formula [Eq. ]{} at zero temperature, the temporal part is $$\begin{aligned}
\chi_{\sigma_3 \sigma_1}(\omega, q)
&=
\sum_{k}
\Bigg\{ \Delta
\frac{\xi+\xi^\prime}{E E^\prime}
\frac{(E+E^\prime)}{(E+E^\prime)^2-\omega^2 }
\Bigg\}
\xrightarrow{q=0}
4\Delta\sum_{k}
\frac{\xi}{E \left(4E^2-\omega^2 \right) }
=4\Delta \int d\xi \frac{g(\xi) \xi}{E \left(4E^2-\omega^2 \right) }
\notag \\
&= \lambda \frac{\Delta}{2E_F} \nu
\left[
-\sqrt{\left(\frac{2\Delta}{\omega} \right)^2 -1}
\tan^{-1} \left( \frac{1}{\sqrt{\left(\frac{2\Delta}{\omega} \right)^2 -1} \sqrt{\left(\frac{\Delta}{\omega_D} \right)^2 +1} } \right)
+\sinh^{-1} \left( \frac{\omega_D}{\Delta} \right)
\right]
\notag\\
& \approx \lambda \nu \frac{\Delta}{2E_F}\sinh^{-1} \left( \frac{\omega_D}{\Delta} \right)
\,
\label{eqn:phase_Higgs_C0}\end{aligned}$$ which gives $C_0$ in [Eq. ]{}. The spatial part is $$\begin{aligned}
\chi_{v_i\sigma_0,\, \sigma_1}(\omega, q)
&=
\frac{\Delta}{2}
\sum_{k}
\Bigg\{
(v_i+v_i^\prime)
\frac{E-E^\prime}{E E^\prime}
\frac{\omega}{(E+E^\prime)^2-\omega^2 }
\Bigg\}
\approx
\Delta \omega q_j
\sum_{k}
\frac{ v_i v_j \xi}{\left(4 E^2-\omega^2\right) E^3 }
= \frac{1}{12d} \lambda \nu \frac{\Delta}{E_F} \left(\frac{v_F}{\Delta} \right)^2 \omega q_i
\,
\label{eqn:phase_Higgs_Ci}\end{aligned}$$ which gives $C_i$ in [Eq. ]{}.
The coupling of phase to the ‘real’ $d$-wave order parameter fluctuations is similar to [Eq. ]{} and except that another $f_d(k)$ term should be added to the momentum summation. The coupling constants are also suppressed by the small particle-hole breaking factor $\lambda \Delta/E_F$. We don’t calculate them here since the ‘real’ fluctuations are not collective modes.
We now calculate the coupling of phase to the $d$-wave BSM fluctuations which are in the $\sigma_2 f_d(k)$ channel. The temporal part is $$\begin{aligned}
\chi_{\sigma_3,\, \sigma_2 f_d(k)}(\omega, q)
&=
i\Delta \omega
\sum_{k}
\Bigg\{ f_d(k)
\frac{E+E^\prime}{E E^\prime}
\frac{-1}{(E+E^\prime)^2-\omega^2 }
\Bigg\}
\approx \frac{i}{4}\Delta \omega
\sum_{k}
f_d(k)
\frac{1}{E^6}
\left( -\frac{5}{4} \frac{\xi^2}{E}
+ \frac{3}{4} E
\right) (\mathbf{v} \mathbf{q})^2
\,.
\label{eqn:phase_BS_B0}\end{aligned}$$ The expansion to $O(q^2)$ is necessary because of the $d$-wave symmetry of $f_d(k)$. It proves the temporal term in [Eq. ]{} but we don’t calculate it since this term affects the EM response at higher orders in $q$. The spatial part is $$\begin{aligned}
\chi_{v_i \sigma_0,\, \sigma_2 f_d(k)}(\omega, q)
&=
i\Delta \sum_{k}
\Bigg\{ f_d(k) v_i
\frac{\xi-\xi^\prime}{E E^\prime}
\frac{E+E^\prime}{(E+E^\prime)^2-\omega^2 }
\Bigg\}
\approx
i2\Delta q_j \sum_{k}
f_d(k) v_i v_j
\frac{1}{\left(4E^2-\omega^2 \right) E }
\,.
\label{eqn:phase_BS_B1}\end{aligned}$$ There are two $d$-wave BSMs in two dimension: the $d_{x^2-y^2}$ and $d_{xy}$ modes which correspond to $f_{d1}=\cos 2\theta_k$ and $f_{d2}=\sin 2\theta_k$ respectively. Since they are different only by a $\pi/4$ rotation, we focus on the $d_{x^2-y^2}$ mode only. Replacing $f_d$ by $\cos 2\theta_k$ in [Eq. ]{} renders $$\begin{aligned}
\chi_{v_i \sigma_0,\, \sigma_2 f_d(k)}(\omega, q)
= i\pi \Delta v_F^2 F(\omega) M_{ij} q_j
\,
\label{eqn:phase_BS_B1_final}\end{aligned}$$ where $\hat{M}=\sigma_3$.
The density density correlation
-------------------------------
The density density correlation is in the $\sigma_3$ channel: $$\begin{aligned}
\chi^{(0)}_{\rho \rho} =
\chi_{\sigma_3 \sigma_3}(\omega, q)
&=
\frac{1}{2}\sum_{k}
\Bigg\{
\left(
1- \frac{\xi\xi^\prime-\Delta^2}{EE^\prime}
\right)
\frac{2(E+E^\prime)}{\omega^2 - (E+E^\prime)^2}
\Bigg\}
\label{eqn:chi_rho_rho}\end{aligned}$$ At zero momentum it becomes $$\begin{aligned}
\chi_{\sigma_3 \sigma_3}(\omega, 0)
&=
\sum_{k}
\frac{\Delta^2}{E}
\frac{4}{\omega^2 - 4E^2}
= -4\Delta^2 F(\omega)
\label{eqn:chi_pho_pho_zero_q}\end{aligned}$$ In the limit of $\omega \ll \Delta,\, q\ll \xi^{-1}$, we have $\chi_{\sigma_3 \sigma_3} =-\nu$.
Near field reflection coefficients {#appendix:rp}
==================================
Monolayer
---------
In the near field limit, there is only longitudinal electric field and no magnetic field. The incident and reflected fields can be described simply using electric potentials $\phi(r,t)$, as shown in Fig. \[fig:nano\_terahertz\_setup\](a). We write the electrical potential as $$\begin{aligned}
\phi_i(r,t)=e^{-i\omega t} \left(\phi_{i\uparrow}e^{iqx - q z}+\phi_{i\downarrow}e^{iqx + q z} \right)
\,,\end{aligned}$$ where $\phi_{i\uparrow}$/$\phi_{i\downarrow}$ are the amplitude of up going/down going fields in the ith vacuum medium. We have explicitly noted that the z direction momentum is $\pm iq$ due to the Laplace equation satisfied by $\phi$ in vacuum, i.e., the electric potentials are evanescent waves. The reflection problem is described by the boundary conditions of $E_\parallel$ being continuous across the 2D layer and $E_\perp$ satisfying Gauss’s law, or equivalently $$\begin{aligned}
\phi_{1\uparrow} + \phi_{1\downarrow} = \phi_{2\uparrow} + \phi_{2\downarrow}
\,,\quad
(q\phi_{1\uparrow} - q\phi_{1\downarrow}) - (q\phi_{2\uparrow} - q \phi_{2\downarrow}) = 4\pi \rho_{2D} = 4\pi \frac{q}{\omega} j_{2D}= 4\pi \frac{q}{\omega} \sigma(\omega,q) (-iq) (\phi_{1\uparrow} + \phi_{1\downarrow})
\,.
\label{eqn:bc}\end{aligned}$$ Written in matrix form, [Eq. ]{} becomes $$\begin{aligned}
\begin{pmatrix}
1 & 1 \\
q + \frac{i4\pi q^2}{\omega} \sigma & -q + \frac{i4\pi q^2}{\omega} \sigma
\end{pmatrix}
\begin{pmatrix}
\phi_{1\uparrow} \\
\phi_{1\downarrow}
\end{pmatrix}
=
\begin{pmatrix}
1 & 1 \\
q & -q
\end{pmatrix}
\begin{pmatrix}
\phi_{2\uparrow} \\
\phi_{2\downarrow}
\end{pmatrix}\end{aligned}$$ whose solution gives the linear relation between the fields each side of the 2D layer $$\begin{aligned}
\begin{pmatrix}
\phi_{1\uparrow} \\
\phi_{1\downarrow}
\end{pmatrix}
=
\frac{1}{-2q}
\begin{pmatrix}
-q + \frac{i4\pi q^2}{\omega} \sigma & -1 \\
-q - \frac{i4\pi q^2}{\omega} \sigma & 1
\end{pmatrix}
\begin{pmatrix}
1 & 1 \\
q & -q
\end{pmatrix}
\begin{pmatrix}
\phi_{2\uparrow} \\
\phi_{2\downarrow}
\end{pmatrix}
=
\begin{pmatrix}
1 - \frac{i2\pi q}{\omega} \sigma & -\frac{i2\pi q}{\omega} \sigma \\
\frac{i2\pi q}{\omega} \sigma & 1 + \frac{i2\pi q}{\omega} \sigma
\end{pmatrix}
\begin{pmatrix}
\phi_{2\uparrow} \\
\phi_{2\downarrow}
\end{pmatrix}
\equiv
\hat{M}
\begin{pmatrix}
\phi_{2\uparrow} \\
\phi_{2\downarrow}
\end{pmatrix}\end{aligned}$$ where $\hat{M}$ is the transfer matrix. Setting $\phi_{2\uparrow}=0$, one obtains the near field reflection coefficient for a 2D layer $$\begin{aligned}
R_p \equiv \frac{\phi_{1\uparrow}}{\phi_{1\downarrow}} = \frac{-\frac{i2\pi q}{\omega} \sigma}{ 1 + \frac{i2\pi q}{\omega} \sigma} = 1- \frac{1}{\epsilon_{2D}}\end{aligned}$$ where $\epsilon_{2D}= 1 + \frac{i2\pi q}{\omega} \sigma$ is the dielectric function in 2D.
Double layer
------------
As shown in Fig. \[fig:nano\_terahertz\_setup\](b), applying the reflection problem twice, one obtains $$\begin{aligned}
\begin{pmatrix}
\phi_{1\uparrow} \\
\phi_{1\downarrow}
\end{pmatrix}
=
\hat{M}
\begin{pmatrix}
\phi_{2\uparrow} \\
\phi_{2\downarrow}
\end{pmatrix}
=
\hat{M}
\begin{pmatrix}
e^{-qa} & 0 \\
0 & e^{qa}
\end{pmatrix}
\hat{M}
\begin{pmatrix}
\phi_{3\uparrow} \\
\phi_{3\downarrow}
\end{pmatrix}
\,.\end{aligned}$$ Setting $\phi_{3\uparrow}=0$ yields the reflection coefficient [Eq. ]{} for the double layer system. A characteristic plot of the reflection coefficient of the double layer system is Fig. \[fig:doublelayer\_rp\_q\] where resonances due to the symmetric and anti symmetric plasmons show up.
 at $\omega= 4 {\,\mathrm{THz}}$, i.e., $\mathrm{Im}[R_p(4 {\,\mathrm{THz}},q)]$ as a function of $q$.[]{data-label="fig:doublelayer_rp_q"}](doublelayer_rp_q.png){width="3.4"}
A slab with nonlocal optical response
-------------------------------------
If the polarization function is nonlocal, more unfortunately, if it depends also on the $z$ direction momentum $k_z$ such as that of the layered superconductor [Eq. ]{}, the near field reflection coefficient of the vacuum-infinite superconductor interface should be modified to $$\begin{aligned}
R_p(\omega,q) = \frac{iq-k_z \epsilon_z(\omega,q,k_z)}{iq+k_z \epsilon_z(\omega,q,k_z)}
\,
\label{eqn:r_p_bscco}\end{aligned}$$ where $q$ is the in-plane momentum which is a conserved quantity and $k_z$ is that in the nonlocal medium determined by the condition $\epsilon(\omega,q,k_z)=0$. For a slab with finite thickness $a$, as shown in Fig. \[fig:nano\_terahertz\_setup\](c), the transfer matrix method for solving the reflection problem renders $$\begin{aligned}
R_{slab}(\omega,q) = R_p\frac{1-e^{2ik_z d}}{1-e^{2ik_z d} R_p^2}
\,
\label{eqn:r_p_bscco_slab}\end{aligned}$$ where $R_p$ is from [Eq. ]{}.
Longitudinal optical conductivity of the normal fermi liquid
============================================================
In the low frequency hydrodynamic regime ($\omega \ll \Gamma_{ee}$, $q \ll l_{ee}^{-1}$) of a fermi liquid, the longitudinal optical conductivity reads [@Sun.2018] $$\begin{aligned}
\sigma(\omega,q) = i \frac{n e^2 / m}{\omega +i\Gamma_d -v_d^2 q^2/\omega} \,.
\label{eqn:hydrodynamic_conductivity}\end{aligned}$$ In the above formula, $n$ is the electron density, $m$ is the electron effective mass, $\Gamma_d$ is the momentum relaxation rate and $v_d=\sqrt{\frac{1}{m}\left(\frac{\partial P}{\partial n} \right)_{ise}}$ is the first sound velocity of a neutral fermi liquid. Neglecting the effect of the Landau parameter $F_{0s}$, $v_d=v_F/\sqrt{D}$ where $D$ is the space dimension. In the limit of $\omega \gg v_d q ,\, D_f q^2$ where $D_f = v_d^2/\Gamma_d$ is the diffusion constant, [Eq. ]{} becomes the Drude formula. In the opposite limit, $\omega \ll v_d q,\, D_f q^2$, it crossovers to the Thomas-Fermi case.
. The large/small period is due to the symmetric/antisymmetric mode. The dipole is polarized in z direction and is placed at $(x,y,z)=(0,0,30 {\,\mathrm{nm}})$ above the top layer. The parameters are $k_F = 2\pi/(3 {\,\mathrm{nm}})$, $v_F = 2.5\times 10^5 {\,\mathrm{m/s}} $, $\gamma= 30 {\,\mathrm{THz}}$, $a= 3 {\,\mathrm{nm}}$, $\Delta= 3.0 {\,\mathrm{THz}}$, $\kappa=0.2$ and $\kappa_{BS}=0.2$. Higgs/BSM frequencies are assumed to be $4.5 {\,\mathrm{THz}}$/$3.0 {\,\mathrm{THz}}$ at zero momentum. []{data-label="fig:nano_terahertz_setup"}](Near_field_launcher.png){width="5"}
|
---
abstract: |
Boosting combines weak (biased) learners to obtain effective learning algorithms for classification and prediction. In this paper, we show a connection between boosting and kernel-based methods, highlighting both theoretical and practical applications. In the context of $\ell_2$ boosting, we start with a weak linear learner defined by a kernel $K$. We show that boosting with this learner is equivalent to estimation with a special [*boosting kernel*]{} that depends on $K$, as well as on the regression matrix, noise variance, and hyperparameters. The number of boosting iterations is modeled as a continuous hyperparameter, and fit along with other parameters using standard techniques.\
We then generalize the boosting kernel to a broad new class of boosting approaches for more general weak learners, including those based on the $\ell_1$, hinge and Vapnik losses. The approach allows fast hyperparameter tuning for this general class, and has a wide range of applications, including robust regression and classification. We illustrate some of these applications with numerical examples on synthetic and real data.
author:
- |
Aleksandr Y. Aravkin saravkin@uw.edu\
Department of Applied Mathematics\
University of Washington\
Seattle, WA 98195-4322, USA Giulio Bottegal giulio.bottegal@gmail.com\
Department of Electrical Engineering\
TU Eindhoven\
Eindhoven, MB 5600, The Netherlands Gianluigi Pillonetto giapi@dei.unipd.it\
Department of Information Engineering\
University of Padova\
Padova, 35131, Italy
bibliography:
- 'boosting\_biblio.bib'
- 'references\_sasha.bib'
title: 'Boosting as a Kernel-Based Method'
---
Boosting; weak learners; Kernel-based methods; Reproducing kernel Hilbert spaces; robust estimation
Introduction
============
Boosting is a popular technique to construct learning algorithms [@schapire2003boosting]. The basic idea is that any *weak learner*, i.e. algorithm that is only slightly better than guessing, can be used to build an effective learning mechanism that achieves high accuracy. Since the introduction of boosting in Schapire’s seminal work [@schapire1990strength], numerous variants have been proposed for regression, classification, and specific applications including semantic learning and computer vision [@schapire2012boosting; @viola2001fast; @temlyakov2000weak; @tokarczyk2015features; @bissacco2007fast; @cao2014face]. In particular, in the context of classification, *LPBoost*, *LogitBoost* [@friedman2000additive], *Bagging and Boosting* [@lemmens2006bagging] and *AdaBoost* [@freund1997decision] have become standard tools, the latter having being recognized as the best off-the-shelf binary classification method [@breiman1998arcing; @zhu2009multi]. Applications of the boosting principle are also found in decision tree learning [@tu2005probabilistic] and distributed learning [@fan1999application]. For a survey on applications of boosting in classification tasks see the work of [@freund1999short]. For regression problems, *AdaBoost.RT* [@solomatine2004adaboost; @avnimelech1999boosting] and *$\ell_2$ Boost* [@buhlmann2003boosting; @tutz2007boosting; @champion2014sparse] are the most prominent boosting algorithms. In particular, in $\ell_2$ boosting the weak learner often corresponds to a kernel-based estimator with a heavily weighted regularization term. The fit on the training set is then measured using the quadratic loss and increases at each iteration. Hence, the procedure can lead to overfitting if it continues too long [@buhlmann2007boosting]. To avoid this, several stopping criteria based on model complexity arguments have been developed. [@hurvich1998smoothing] propose a modified version of Akaike’s information criterion (AIC); [@hansen2001model] use the principle of minimum description length (MDL), and [@buhlmann2003boosting] suggest a five-fold cross validation.\
In this paper, we focus on $\ell_2$ boosting and consider linear weak learners induced by the combination of a quadratic loss and a regularizer induced by a kernel $K$. We show that the resulting boosting estimator is equivalent to estimation with a special [*boosting kernel*]{} that depends on $K$, as well as on the regression matrix, noise variance, and hyperparameters. This viewpoint leads to both greater generality and better computational efficiency. In particular, the number of boosting iterations $\nu$ is a continuous hyperparameter of the boosting kernel, and can be tuned by standard fast hyper-parameter selection techniques including SURE, generalized cross validation, and marginal likelihood [@hastie2001elements]. In Section \[sec:experiments\], we show that tuning $\nu$ is far more efficient than applying boosting iterations, and non-integer values of $\nu$ can improve performance.\
We then generalize the boosting kernel to a wider class of problems, including robust regression, by combining the boosting kernel with piecewise linear quadratic (PLQ) loss functions (e.g. $\ell_1$, Vapnik, Huber). The computational burden of standard boosting is high for general loss functions, since the estimator at each iteration is no longer a linear function of the data. The boosting kernel makes the general approach tractable. We also use the boosting kernel in the context of regularization problems in reproducing kernel Hilbert spaces (RKHSs), e.g. to solve classification formulations that use the hinge loss.\
The organization of the paper is as follows. After a brief overview of boosting in regression and classification, we develop the main connection between boosting and kernel-based methods in the context of finite-dimensional inverse problems in Section \[sec:boosting\_kernel\]. Consequences of this connection are presented in Section \[sec:consequences\]. In Section \[sec:new\_class\_boosting\] we combine the boosting kernel with PLQ penalties to develop a new class of boosting algorithms. We also consider regression and classification in RKHSs. In Section \[sec:experiments\] we show numerical results for several experiments involving the boosting kernel. We end with discussion and conclusions in Section \[sec:conclusions\].
Boosting as a kernel-based method {#sec:boosting_kernel}
=================================
In this section, we give a basic overview of boosting, and present the boosting kernel.
Boosting: notation and overview {#SecBS}
-------------------------------
Assume we are given a model $g(\theta)$ for some observed data $y\in\mathbb{R}^n$, where $\theta\in\mathbb{R}^m$ is an unknown parameter vector. Suppose our estimator $\hat \theta$ for $\theta$ minimizes some objective that balances variance with bias. In the boosting context, the objective is designed to provide a [*weak estimator*]{}, i.e. one with low variance in comparison to the bias.
Given a loss function $\mathcal{V}$ and a kernel matrix $K\in\mathbb{R}^{m\times m}$, the weak estimator can be defined by minimizing the regularized formulation $$\label{Weak2}
\hat \theta := \arg\min_\theta \left\{ J(\theta; y) := \mathcal{V}(y-g(\theta)) + \gamma \theta^T K^{-1} \theta\right\},$$ where the regularization parameter $\gamma$ is large and leads to over-smoothing. Boosting uses this weak estimator iteratively, as detailed below. The predicted data for an estimator $\hat \theta$ are denoted by $\hat{y}=g(\hat{\theta})$.
[**[Boosting scheme:]{}**]{}
1. Set $\nu=1$ and obtain $\hat{\theta}(1)$ and $\hat{y}(1)=g(\hat{\theta}(1))$ using (\[Weak2\]);
2. Solve (\[Weak2\]) using the current residuals as data vector, i.e. compute $$\hat\theta(\nu) = \argmin_{\theta} \ J(\theta;y-\hat{y}(\nu)),$$ and set the new predicted output to $$\hat{y}(\nu+1) = \hat{y}(\nu) + g(\hat\theta(\nu)).$$
3. Increase $\nu$ by 1 and repeat step 2 for a prescribed number of iterations.
Using regularized least squares as weak learner
-----------------------------------------------
Suppose data $y$ are generated according to $$\label{MeasMod}
y = U\theta + v, \quad v \sim \mathcal N (0, \sigma^2 I),$$ where $U$ is a known regression matrix of full column rank. The components of $v$ are independent random variables, mean zero and variance $\sigma^2$.
We now use a quadratic loss to define the regularized weak learner. Let $\lambda$ to denote the kernel scale factor and set $\gamma=\sigma^2 / \lambda$ so that (\[Weak2\]) becomes $$\label{Eq1}
\hat \theta = \arg\min_{\theta} \|y - U\theta\|^2 + \frac{\sigma^2}{\lambda}\theta^T K^{-1}\theta.$$ We obtain the following expressions for the predicted data $\hat y = U \hat \theta$: $$\begin{aligned}
\nonumber
\hat{y} &=& \arg\min_{f} \left\{\|y - f\|^2 + \sigma^2 f^T P_{\lambda}^{-1} f \right\}\\ \label{Eq2}
&=& P_{\lambda} (P_{\lambda} + \sigma^2 I)^{-1}y,\end{aligned}$$ where $$\label{Eq3}
P_{\lambda} = \lambda U K U^T$$ is assumed invertible for the moment. This assumption will be relaxed later on.
The following well-known connection [@Wahba1990] between (\[Eq1\]) and Bayesian estimation is useful for theoretical development. Assume that $\theta$ and $v$ are independent Gaussian random vectors with priors $$\theta \sim \mathcal{N}(0, \lambda K), \quad v \sim \mathcal{N}(0, \sigma^2 I).$$ Then, (\[Eq1\]) and (\[Eq2\]) provide the minimum variance estimates of $\theta$ and $U \theta$ conditional on the data $y$. In view of this, we refer to diagonal values of $K$ as the [*prior variances*]{} of $\theta$.
The boosting kernel
-------------------
Define $$\label{eq:Slam}
S_{\lambda} = P_{\lambda} (P_\lambda + \sigma^2 I)^{-1}.$$ Fixing a small $\lambda$, the predicted data obtained by the weak kernel-based learner is $$\hat y(\nu = 1) = S_{\lambda}y,$$ where $\nu$ is the number of boosting iterations. According to the scheme specified in Section \[SecBS\], as $\nu$ increases, boosting refines the estimate as follows: $$\begin{aligned}
\nonumber
\hat y(2) &=& S_{\lambda}y+S_{\lambda}(I-S_{\lambda})y \\ \nonumber
\hat y(3) &=& S_{\lambda}y+S_{\lambda}(I-S_{\lambda})y + S_{\lambda}(I-S_{\lambda})^2y \\ \nonumber
&\vdots& \\ \label{BoostEst}
\hat y(\nu) &=& S_{\lambda}\sum_{i = 0}^{\nu-1} \left( I - S_{\lambda}\right)^{i} y. \end{aligned}$$
We now show that the boosting estimates $\hat y(\nu)$ are kernel-based estimators from the [*boosting kernel*]{}, which plays a key role for subsequent developments.
\[BoostKer\] The quantity $\hat y(\nu)$ is a kernel-based estimator $$\hat{y}(\nu) = S_{\lambda, \nu}y = P_{\lambda,\nu }(P_{\lambda,\nu} + \sigma^2 I)^{-1} y,$$ where $P_{\lambda,\nu }$ is the boosting kernel defined by $$\begin{aligned}
P_{\lambda, \nu} \nonumber
&=& \sigma^2 \left( I - P_{\lambda}\left(P_{\lambda} + \sigma^2 I\right)^{-1}\right)^{-\nu}-\sigma^2I \\ \label{BoostingKer}
& =& \sigma^2 \left( I - S_{\lambda}\right)^{-\nu}-\sigma^2I.\end{aligned}$$
First note that $S_\lambda$ satisfies $$\label{eq:SlamId}
S_\lambda = P_{\lambda}\left(P_{\lambda} + \sigma^2 I\right)^{-1}
=
I - \sigma^2\left(P_{\lambda} + \sigma^2 I \right)^{-1}.$$ This follows simply from adding the term $\sigma^2\left(P_{\lambda} + \sigma^2\right)^{-1}$ to and observing that expression reduces to $I$. Next, plugging in the expression for $P_{\lambda, \nu}$ into the right hand side of expression for $S_{\lambda, \nu}$, we have $$\begin{aligned}
S_{\lambda, \nu} &= I - \sigma^2\left(P_{\lambda,\nu} + \sigma^2 I \right)^{-1}\\
& = I - \sigma^2\left(\sigma^2 \left( I - S_{\lambda}\right)^{-\nu} \right)^{-1}\\
& = I - \left( I - S_{\lambda}\right)^{\nu} \\
& = S_{\lambda}\sum_{i = 0}^{\nu-1} \left( I - S_{\lambda}\right)^{i},
\end{aligned}$$ exactly as required by (\[BoostEst\]).
In Bayesian terms, for a given $\nu$, the above result also shows that boosting returns the minimum variance estimate of the noiseless output $f$ conditional on $y$ if $f$ and $v$ are independent Gaussian random vectors with priors $$\label{Eq5}
f \sim \mathcal{N}(0, P_{\lambda,\nu}), \quad v \sim \mathcal{N}(0, \sigma^2 I).$$
Consequences {#sec:consequences}
============
In this section, we use Proposition \[BoostKer\] to gain new insights on boosting and a new perspective on hyperparameter tuning.
Insights on the nature of boosting {#Sec3.1}
----------------------------------
We first derive a new representation of the boosting kernel $P_{\lambda, \nu}$ via a change of coordinates. Let $VDV^T$ be the SVD of $UKU^T$. Then, we obtain $$\begin{aligned}
\nonumber
P_{\lambda, v} &=& \frac{\sigma^2}{(\sigma^2)^\nu} \left(\lambda U K U^T + \sigma^2 I\right)^{\nu} - \sigma^2 I \\ \label{BoostingKer2}
& =& \sigma^2 V \left[ \left(\frac{\lambda D + \sigma^2 I}{\sigma^2}\right)^{\nu}-I\right]V^T\end{aligned}$$ and the predicted output can be rewritten as $$\hat{y}(\nu) = V\left(I- \sigma^{2\nu} \left(\lambda D + \sigma^2 I\right)^{-\nu}\right)V^Ty.$$ In coordinates $z = V^Ty$, the estimate of each component of $z$ is $$\label{BoostEstzi}
\hat{z}_i(\nu) = \left(1 - \frac{\sigma^{2\nu}}{\left(\lambda d_i^2 + \sigma^2\right)^\nu}\right)z_i,$$ and corresponds to the regularized least squares estimate induced by a diagonal kernel with $(i,i)$ entry $$\label{BoostPriorVar}
\sigma^2 \left(\frac{\lambda d_i^2}{\sigma^2}+1\right)^\nu - \sigma^2.$$ In Bayesian terms, (\[BoostPriorVar\]) is the prior variance assigned by boosting to the noiseless output $V^T U \theta$.
Eq. shows that boosting builds a kernel on the basis of the output signal-to-noise ratios $SNR_i = \frac{\lambda d_i^2}{\sigma^2}$, which then enter $\left(\frac{\lambda d_i^2}{\sigma^2}+1\right)^\nu$. All diagonal kernel elements with $d_i>0$ grow to $\infty$ as $\nu$ increases; therefore asymptotically, data will be perfectly interpolated but with growth rates controlled by the $SNR_i$. If $SNR_i$ is large, the prior variance increases quickly and after a few iterations the estimator is essentially unbiased along the $i$-th direction. If $SNR_i$ is close to zero, the $i$-th direction is treated as though affected by ill-conditioning, and a large $\nu$ is needed to remove the regularization on $\hat{z}_i(\nu)$.
This perspective makes it clear when boosting can be effective. In the context of inverse problems (deconvolution), $\theta$ in (\[MeasMod\]) represents the unknown input to a linear system whose impulse response defines the regression matrix $U$. For simplicity, assume that the kernel $K$ is set to the identity matrix, so that the weak learner (\[Eq1\]) becomes ridge regression and the $d_i^2$ in (\[BoostPriorVar\]) reflect the power content of the impulse response at different frequencies. Then, *boosting can outperfom standard ridge regression if the system impulse response and input share a similar power spectrum*. Under this condition, boosting can inflate the prior variances (\[BoostPriorVar\]) along the right directions. For instance, if the impulse response energy is located at low frequencies, as $\nu$ increases boosting will amplify the low pass nature of the regularizer. This can significantly improve the estimate if the input is also low pass.
Hyperparameter estimation {#Sec3.2}
-------------------------
In the classical scheme described in section \[SecBS\], $\nu$ is an iteration counter that only takes integer values, and the boosting scheme is sequential: to obtain the estimate $\hat{y}(\nu=m)$, one has to solve $m$ optimization problems. Using (\[BoostingKer\]) and (\[BoostingKer2\]), we can interpret $\nu$ as a kernel hyperparameter, and let it take real values. In the following we estimate both the scale factor $\lambda$ and $\nu$ from the data, and restrict the range of $\nu$ to $\nu \geq 1$.
The resulting boosting approach estimates $(\lambda,\nu)$ by minimizing fit measures such as cross validation or SURE [@hastie2001elements]. In particular, this accelerates the tuning procedure, as it requires solving a single problem instead of multiple boosting iterations. Consider estimating $(\lambda,\nu)$ using the SURE method. Given $\sigma^2$ (e.g. using an unbiased estimator), choose $$\label{SURE}
(\hat \lambda,\hat \nu) = \arg\min_{\lambda \geq 0, \nu \geq 1} \ \|y - \hat y(\nu)\|^2 + 2 \sigma^2 \mbox{trace}(S_{\lambda,\nu}).$$ Straightforward computations show that, for the cost of a single SVD, problem simplifies to $$\label{eq:SURE_2}
(\hat \lambda,\hat \nu) = \arg\min_{\lambda\geq 0,\nu\geq 1}
\sum_{i=1}^n \frac{z_i^2\sigma^{4\nu}}{\left(\lambda d_i^2 + \sigma^2\right)^{2\nu}} +
2 \sigma^2 n- \sum_{i=1}^n \frac{2 \sigma^{2\nu+2}}{(\lambda d_i^2 + \sigma^2)^\nu},$$ which is a smooth 2-variable problem over a box, and can be easily optimized.
We can also extract some useful information on the nature of the optimization problem (\[eq:SURE\_2\]). In fact, denoting $J$ the objective, we have $$\begin{aligned}
\label{eq:der_J}
\frac{\partial J}{\partial \nu} & = 2 \sum_{i=1}^n \log (\alpha_i) z_i^2 \alpha_i^{2\nu} - 2\sigma^2\sum_{i=1}^n \log (\alpha_i) \alpha_i^{\nu} \nonumber \\
& = 2 \sum_{i=1}^n \log (\alpha_i) \alpha_i^{\nu} ( z_i^2 \alpha_i^{\nu} - \sigma^2) \,,\end{aligned}$$ where we have defined $\alpha_i := \frac{\sigma^2}{\lambda d_i^2 + \sigma^2} \,.$ Simple considerations on the sign of the derivative then show that
- if $$\label{eq:lambda_min}
\lambda < \min_{i=1,\ldots,n}\frac{z_{ i}^2 - \sigma^2}{d_{ i}^2},$$ then $\hat \nu = +\infty$. This means that we have chosen a learner so weak that SURE suggests an infinite number of boosting iterations as optimal solution;
- if $$\label{eq:lambda_max}
\lambda > \max_{i=1,\ldots,n} \frac{z_{ i}^2 - \sigma^2}{d_{ i}^2},$$ then $\hat \nu = 1$. This means that the weak learner is instead so strong that SURE suggests not to perform any boosting iterations.
Numerical illustration
----------------------
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We illustrate our insights using a numerical experiment. Consider (\[MeasMod\]), where $\theta \in \mathbb{R}^{50}$ represents the input to a discrete-time linear system. In particular, the signal is taken from [@Wahba1990] and displayed in Fig. \[Fig1\] (thick red line). The system is represented by the regression matrix $U \in \mathbb{R}^{200 \times 50}$ whose components are realizations of either white noise or low pass filtered white Gaussian noise with normalized band $[0,0.95]$. The measurement noise is white and Gaussian, with variance assumed known and set to the variance of the noiseless output divided by 10.
We use a Monte Carlo of 100 runs to compare the following two estimators
- [**Boosting**]{}: boosting estimator with $K$ set to the identity matrix and with $(\lambda,\nu)$ estimated using the SURE strategy (\[SURE\]).
- [ **Ridge:**]{} ridge regression (which corresponds to boosting with $\nu$ fixed to 1).
Fig. \[Fig2\] displays the box plots of the 100 percentage fits of $\theta$, $100\left(1-\frac{\| \theta- \hat{\theta} \|}{\| \theta \|}\right)$, obtained by [**Boosting**]{} and [**Ridge**]{}. When the entries of $U$ are white noise (left panel) one can see that the two estimators have similar performance. When the entries of $U$ are filtered white noise (right panel) [**Boosting**]{} performs significantly better than [**Ridge**]{}. Furthermore, 36 out of the 100 fits achieved by [**Boosting**]{} under the white noise scenario are lower than those obtained adopting a low pass $U$, which is surprising since the conditioning of latter problem is much worse. The reasons are those previously described. The unknown $\theta$ represents a smooth signal. In Bayesian terms, setting $K$ to the identity matrix corresponds to modeling it as white noise, which is a poor prior. If the nature of $U$ is low pass, the energy of the $d_i^2$ are more concentrated at low frequencies. So, as $\nu$ increases, [**Boosting**]{} can inflate the prior variances associated to the low-frequency components of $\theta$. The prior variances associated to high-frequencies induce low $SNR_i$, so that they increase slowly with $\nu$. This does not happen in the white noise case, since the random variables $d_i^2$ have similar distributions. Hence, the original white noise prior for $\theta$ can be significantly refined only in the low pass context: it is reshaped so as to form a regularizer, inducing more smoothness. Fig. \[Fig1\] shows this effect by plotting estimates from [**Ridge**]{} and [**Boosting**]{} in a Monte Carlo run where $U$ is low pass.
Boosting algorithms for general loss functions and RKHSs {#sec:new_class_boosting}
========================================================
In this section, we combine the boosting kernel with piecewise linear-quadratic (PLQ) losses to obtain tractable algorithms for more general regression and classification problems. We also consider estimation in Reproducing Kernel Hilbert (RKHS) spaces.
Boosting kernel-based estimation with general loss functions
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(a) quadratic & (b) huber & (d) hinge\
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(e) quantile & (f) vapnik & (h) elastic net
In the previous sections, the boosting kernel was derived using regularized least squares (\[Eq1\]) as the weak learner. The sequence of resulting linear estimators then led to a closed form expression for $P_{\lambda,\nu}$. Now, we consider a kernel-based weak learner , based on a general (convex) penalty $\mathcal{V}$. Important examples include Vapnik’s epsilon insensitive loss (Fig. \[fig:SDRex\]f) used in support vector regression [@Vapnik98; @Hastie01; @Scholkopf00; @Scholkopf01b], hinge loss (Fig. \[fig:SDRex\]d) used for classification [@Evgeniou99; @Pontil98; @Scholkopf00], Huber and quantile huber (Fig \[fig:SDRex\]b,e), used for robust regression[@Hub; @Mar; @Bube2007; @Zou08; @KG01; @Koenker:2005; @aravkin2014qh], and elastic net (Fig. \[fig:SDRex\]f), a sparse regularizer that also finds correlated predictors [@ZouHuiHastie:2005; @EN_2005; @li2010bayesian; @de2009elastic]. The resulting boosting scheme is computationally expensive: $\hat{y}(\nu=m)$ requires solving a sequence of $m$ optimization problems, each of which must be solved iteratively. In addition, since the estimators $\hat{y}(\nu)$ are no longer linear, deriving a boosting kernel is no longer straightforward.
We combine general loss $\mathcal{V}$ with the regularizer induced by the boosting kernel from the linear case to define a new class of kernel-based boosting algorithms. More specifically, given a kernel $K$, let $VDV^T$ be the SVD of $UKU^T$. If $P_{\lambda,\nu}$ is invertible, the boosting output estimate is $\hat{y}(\nu) = U \hat{\theta}(\nu)$ where $$\begin{aligned}
\nonumber
\hat{\theta}(\nu) &=& \arg\min_{\theta} \ \mathcal{V}(y - U \theta) + \sigma^2 \theta^T U^T P_{\lambda,\nu}^{-1} U \theta \\ \label{Eq6}
&=& \arg\min_{\theta} \left\{ \mathcal{V}(y - U \theta) + \theta^T U^T V \left[ \left(\frac{\lambda D + \sigma^2 I}{\sigma^2}\right)^{\nu}-I\right]^{-1}V^T U \theta\right\},\end{aligned}$$ where the last line is obtained using (\[BoostingKer2\]). Note, here and also in the reformulations below, that the solution depends on $\lambda$ and $\sigma^2$ only through the ratio $\gamma=\sigma^2/\lambda$.\
If $P_{\lambda,\nu}$ is not invertible, the following two strategies can be adopted.
#### Approach I:
We use to obtain the factorization $$P_{\lambda,\nu} = \sigma^2 A_{\lambda,\nu} A_{\lambda,\nu}^T,$$ where $A_{\lambda,\nu}$ is full column rank and contains the columns of the matrix $$A_{\lambda,\nu} = V \left[ \left(\frac{\lambda D + \sigma^2 I}{\sigma^2}\right)^{\nu}-I\right]^{1/2}$$ associated to the $d_i>0$. Then, the output estimate is $\hat{y}(\nu)= A_{\lambda,\nu} \hat{a}(\nu)$ with $$\label{Eq7}
\hat{a}(\nu) = \arg\min_{a} \left\{ \mathcal{V}(y - A_{\lambda,\nu} a) + a^T a \right\}.$$ The estimate of $\theta$ is then given by $\hat{\theta} = U^{\dag}_{\lambda,\nu} \hat{y}(\nu)$, where $U^{\dag}_{\lambda,\nu}$ is the pseudo-inverse of $U_{\lambda,\nu}$. One advantage of the formulation (\[Eq7\]) is that the evaluation of $A_{\lambda,\nu}$ for different $\lambda$ and $\nu$ is efficient.\
#### Approach II:
Define the matrix $$B_{\lambda,\nu}=U P_{\lambda,\nu} U^T.$$ Then, it is easy to see that another representation for the output estimate is $\hat{y}(\nu)= B_{\lambda,\nu} \hat{b}(\nu)$ with $$\label{Eq8}
\hat{b}(\nu) = \arg\min_{b} \left\{ \mathcal{V}(y - B_{\lambda,\nu} b) + b^T B_{\lambda,\nu} b\right\}.$$
The new class of boosting kernel-based estimators defined by (\[Eq7\]) or (\[Eq8\]) keeps the advantages of boosting in the quadratic case. In particular, the kernel structure can decrease bias along directions less exposed to noise. The use of a general loss $\mathcal{V}$ allows a range of applications, with e.g. penalties such as Vapnik and Huber, guarding against outliers in the training set. Finally, the algorithm has clear computational advantages over the classic scheme described in Section \[SecBS\]. Whereas in the classic approach, $\hat{y}(\nu=m)$ require solving $m$ optimization problems, in the new approach, given any positive $\lambda$ and $\nu \geq 1$, the prediction $\hat{y}(\nu=m)$ is obtained by solving the single convex optimization problem (\[Eq6\]). This is illustrated in Section \[sec:experiments\].
New boosting algorithms in RKHSs {#sec:RKHS}
--------------------------------
We now show how the new class of boosting algorithms can be extended to the context of regularization in RKHSs. We start with $\ell_2$ Boost in RKHSs.
Assume that we want to reconstruct a function from $n$ sparse and noisy data $y_i$ collected on input locations $x_i$ taking values on the input space $\mathcal{X}$. Our aim now is to allow the function estimator to assume values in infinite-dimensional spaces, introducing suitable regularization to circumvent ill-posedness, e.g. in terms of function smoothness. For this purpose, we use $\mathcal{K}$ denote a kernel function $\mathcal{K}: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}$ which captures smoothness properties of the unknown function. We can then use $\ell_2$ Boost, with weak learner $$\label{WeakRKHS}
\argmin_{f \in \mathcal{H}} \ \sum_{i=1}^n \ \mathcal{V}_i(y_i-f(x_i)) + \gamma \| f \|^2_{\mathcal{H}},$$ where $\mathcal{V}_i$ is a generic convex loss and $\mathcal{H}$ is the RKHS induced by $\mathcal{K}$ with norm denoted by $\| \cdot \|_{\mathcal{H}}$. From the representer theorem of [@Scholkopf01], the solution of (\[WeakRKHS\]) is $\sum_{i=1}^n \ \hat{c}_i \mathcal{K}(x_i,\cdot)$ where the $\hat{c}_i$ are the components of the column vector $$\label{WeakRKHS2}
\argmin_{c \in \mathbb{R}^n} \ \sum_{i=1}^n \ \mathcal{V}_i(y_i- K_{i,\cdot} c ) + \gamma c^T K c,$$ and $K$ is the kernel (Gram) matrix, with $K_{i,j} = \mathcal{K}(x_i,x_j)$ and $K_{i,\cdot}$ is the $i$-th row of $K$. Using (\[WeakRKHS2\]), we extend the boosting scheme from section \[SecBS\] with (\[WeakRKHS\]) as the weak learner. In particular, repeated applications of the representer theorem ensure that, for any value of the iteration counter $\nu$, the corresponding function estimate belongs to the subspace spanned by the $n$ kernel sections $\mathcal{K}(x_i,\cdot)$. Hence, $\ell_2$ Boosting in RKHS can be summarized as follows.
[**[Boosting scheme in RKHS:]{}**]{}
1. Set $\nu=1$. Solve (\[WeakRKHS2\]) to obtain $\hat{c}$ and $\hat{f}$ for $\nu=1$, call them $\hat{c}(1) $ and $\hat{f}(\cdot,1)$.
2. Update $c$ by solving (\[WeakRKHS2\]) with the current residuals as the data vector: $$\hat{c}(\nu+1) = \hat{c}(\nu) + \argmin_{c \in \mathbb{R}^n} \ \sum_{i=1}^n \ \mathcal{V}_i(y_i- K_i \hat{c}(\nu) - K_i c ) + \gamma c^T K c,$$ and set the new estimated function to $$\hat{f}(\cdot, \nu+1) = \sum_{i=1}^n \ \hat{c}_i(\nu+1) \mathcal{K}(x_i,\cdot).$$
3. Increase $\nu$ by 1 and repeat step 2 for a prescribed number of iterations.
There is a fundamental computational drawback related to this scheme which we have already encountered in the previous sections. To obtain $\hat{f}(\cdot, \nu)$ we need to solve $\nu$ optimization problems, each of them requiring an iterative procedure. Now, we define a new computationally efficient class of regularized estimators in RKHS. The idea is to obtain the expansion coefficients of the function estimate through the new boosting kernel. Letting $\gamma=\sigma^2 / \lambda$ and $P_{\lambda}= \lambda K$, with $K$ the kernel matrix, define the boosting kernel $P_{\lambda,\nu}$ as in (\[BoostingKer\]). Then, we can first solve $$\label{Eq9}
\hat{b}(\nu) = \arg\min_{b} \left\{ \mathcal{V}(y - P_{\lambda,\nu} b) + b^T P_{\lambda,\nu} b\right\},$$ with $\mathcal{V}$ defined as the sum of the $\mathcal{V}_i$. Then, we compute $$\tilde{c}=K^{\dag}\tilde{y}(\nu) \quad \mbox{with} \quad \tilde{y}(\nu) = P_{\lambda,\nu} \hat{b}(\nu),$$ and the estimated function becomes $$\hat{f}(\cdot, \nu) = \sum_{i=1}^n \ \tilde{c}_i(\nu) \mathcal{K}(x_i,\cdot).$$ Note that the weights $\tilde{c}(\nu)$ coincide with $\hat{c}(\nu)$ only when the $\mathcal{V}_i$ are quadratic. Nevertheless, given any loss, (\[Eq9\]) preserves all advantages of boosting outlined in the linear case. Furthermore, as in the finite-dimensional case, given any $\nu$ and kernel hyperparameter, the estimator (\[Eq9\]) can compute $\tilde{c}(\nu)$ by solving a single problem, rather than iterating the boosting scheme.
#### Classification with the hinge loss.
Another advantage related to the use of the boosting kernel w.r.t. the classical boosting scheme arises in the classification context. Classification tries to predict one of two output values, e.g. 1 and -1, as a function of the input. $\ell_2$ Boost could be used using the residual $y_i-f(x_i)$ as misfit, e.g. equipping the weak learner (\[WeakRKHS\]) with the quadratic or the $\ell_1$ loss. However, in this context one often prefers to use the margin $m_i=y_if(x_i)$ on an example $(x_i,y_i)$ to measure how well the available data are classified. For this purpose, support vector classification is widely used [@scholkopf2002learning]. It relies on the hinge loss $$\mathcal{V}_i(y_i,f(x_i)) = | 1 - y_i f(x_i) |_+ =
\left\{
\begin{array}{lcl}
0, \quad & m > 1 \\
1-m, \quad & m \leq 1
\end{array}, \quad m=y_if(x_i),
\right.$$ which gives a linear penalty when $m<1$. Note that this loss assumes $y_i\in\{1,-1\}$. However, the classical boosting scheme applies the weak learner (\[WeakRKHS\]) repeatedly, and [**residuals will not be binary**]{} for $\nu>1$. This means that $\ell_2$ Boost cannot be used for the hinge loss.
This limitation does not affect the new class of boosting-kernel based estimators: support vector classification can be boosted by plugging in the hinge loss into (\[Eq9\]): $$\label{Eq10}
\hat{b}(\nu) = \arg\min_{b} \ \sum_{i=1}^n | 1 - y_i [P_{\lambda,\nu} b]_i |_+ + b^T P_{\lambda,\nu} b,$$ where we have used $[P_{\lambda,\nu} b]_i $ to denote the $i$-th component of $P_{\lambda,\nu} b$.
Numerical Experiments {#sec:experiments}
=====================
Boosting kernel regression: temperature prediction real data {#sec:RealData}
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To test boosting on real data, we use a case study in thermodynamic modeling of buildings. Eight temperature sensors produced by Moteiv Inc were placed in two rooms of a small two-floor residential building of about $80$ $\textrm{m}^2$ and $200$ $\textrm{m}^3$. The experiment lasted for 8 days starting from February 24th, 2011; samples were taken every 5 minutes. A thermostat controlled the heating systems and the reference temperature was manually set every day depending upon occupancy and other needs. The goal of the experiment is to assess the predictive capability of models built using kernel-based estimators.
We consider Multiple Input-Single Output (MISO) models. The temperature from the first node is the output ($y_i$) and the other 7 represent the inputs ($u^j_i$, $j=1,..,7$). The measurements are split into a training set of size $N_{id}=1000$ and a test set of size $N_{test}=1500$. The notation $y^{test}$ indicates the test data, which is used to test the ability of our estimator to predict future data. Data are normalized so that they have zero mean and unit variance before identification is performed.
The model predictive power is measured in terms of $k$-step-ahead prediction fit on $y^{test}$, i.e. $$100 \times \left(1-{\sqrt{\sum_{i=k}^{N_{test}}(y^{test}_i-\hat y_{i|i-k})^2}/ \sqrt{\sum_{i=k}^{N_{test}} (y^{test}_i)^2}}\right).$$
We consider ARX models of the form $$y_i = (g^1 \otimes y)_i + \sum_{j=1}^{7} (g^{j+1} \otimes u^j)_i + v_i,$$ where $\otimes$ denotes discrete-time convolution and the $\{g^j\}$ are 8 unknown one-step ahead predictor impulse responses, each of length 50. Note that when such impulse responses are known, one can use them in an iterative fashion to obtain any $k$-step ahead prediction. We can stack all the $\{g^j\}$ in the vector $\theta$ and form the regression matrix $U$ with the past outputs and the inputs so that the model becomes $y=U\theta + v$. Then, we consider the following two estimators:
- [**Boosting SS**]{}: this estimator regularizes each $g^j$ introducing information on its smoothness and exponential decay by the stable spline kernel [@SS2010]. In particular, let $P \in \mathbb{R}^{50 \times 50}$ with $(i,j)$ entry $\alpha^{\max(i,j)}, \ 0 \leq \alpha <1$. Then, we recover $\theta$ by the boosting scheme (\[Eq7\]) with $K=\mbox{blkdiag}(P,\ldots,P)$, and $\mathcal{V}$ set to the quadratic loss. Note that the estimator contains the three unknown hyperparameters $\nu,\alpha$ and $\gamma=\sigma^2/\lambda$. To estimate them, the training set is divided in half and hold-out cross validation is used.
- [**Classical Boosting SS**]{}: the same as above except that $\nu$ can assume only integer values.
- [**SS**]{}: this is the stable spline estimator described in [@SS2010] (and corresponds to [**Boosting SS**]{} with $\nu=1$) with hyperparameters obtained via marginal likelihood optimization.
For [**Boosting SS**]{}, we obtained $\gamma=0.02, \alpha=0.82$ and $\nu = 1.42$; note that it is not an integer. For [**Classical Boosting SS**]{}, we obtained $\gamma=0.03, \alpha=0.79$ and $\nu = 1$. In practice, this estimator gives the same results achieved by [**SS**]{} so that our discussion below just compares the performance of [**Boosting SS**]{} and [**SS**]{}.
The left panel of Fig. \[Fig3\] shows the prediction fits, as a function of the prediction horizon $k$, obtained by [**Boosting SS**]{} and [**SS**]{}. Note that the non-integer $\nu$ gives an improvement in performance. This means that in this experiment using a continuous $\nu$ improves also over the classical boosting. The right panel of Fig. \[Fig3\] shows sample trajectories of half-hour-ahead boosting prediction on a part of the test set.
Boosting kernel regression using the $\ell_1$ loss: Real data water tank system identification {#sec:real_experiment}
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We test our new class of boosting algorithms on another real data set obtained from a water tank system (see also [@bottegal2016robust]). In this example, a tank is fed with water by an electric pump. The water is drawn from a lower basin, and then flows back through a hole in the bottom of the tank. The system input is the voltage applied, while the output is the water level in the tank, measured by a pressure sensor at the bottom of the tank. [The setup represents a typical control engineering scenario, where the experimenter is interested in building a mathematical model of the system in order to predict its behavior and design a control algorithm [@Ljung]. To this end,]{} input/output samples are collected every second, comprising almost 1000 pairs that are divided into a training and test set. The signals are de-trended, removing their means. The training and test outputs are shown in the left and right panel of Fig. \[Fig4\]. One can see that the second part of the training data are corrupted by outliers caused by pressure perturbations in the tank; these are due to air occasionally being blown into the tank. [Our aim is to understand the predictive capability of the boosting kernel even in presence of outliers.]{}
We consider a FIR model of the form $$y_i = (g \otimes u)_i + v_i,$$ where the unknown vector $g \in \mathbb{R}^{50}$ contains the impulse response coefficients. It is estimated using a variation of the estimator [**Boosting SS**]{} described in the previous section: while the stable spline kernel is still employed to define the regularizer, the key difference is that $\mathcal{V}$ in (\[Eq7\]) is now set to the robust $\ell_1$ loss. The hyperparameter estimates obtained using hold-out cross validation are $\gamma=17.18,\alpha=0.92$ and $\nu=1.7$. The right panel of Fig. \[Fig4\] shows the boosting simulation of the test set. The estimate from [**Boosting SS**]{} predicts the test set with $76.2 \%$ fit. Using the approach $\mathcal{V}$ equal to the quadratic loss, the test set fit decreases to $57.8 \%$.
Boosting in RKHSs: Classification problem {#sec:class_rkhs}
-----------------------------------------
Consider the problem described in Section 2 of [@hastie2001elements]. Two classes are introduced, each defined by a mixture of Gaussian clusters; the first 10 means are generated from a Gaussian $\mathcal{N}([1 \ 0]^T, I)$ and remaining ten means from $\mathcal{N}([0 \ 1]^T, I)$ with $I$ the identity matrix. Class labels $1$ and $-1$ corresponding to the clusters are generated randomly with probability 1/2. Observations for a given label are generated by picking one of the ten means $m_k$ from the correct cluster with uniform probability 1/10, and drawing an input location from $\mathcal{N}(m_k, I/5)$. A Monte Carlo study of 100 runs is designed. At any run, a new data set of size 500 is generated, with the split given by $50\%$ for training and $25\%$ each for validation and testing. The validation set is used to estimate through hold-out cross-validation the unknown hyperparameters, in particular the boosting parameter $\nu$. Performance for a given run is quantified by computing percentage of data correctly classified.
We compare the performance of the following two estimators:
- [**Boosting+$\ell_1$ loss**]{}: this is the boosting scheme in RKHS illustrated in the previous section ($\nu$ may assume only integer values) with the weak learner (\[WeakRKHS\]) defined by the Gaussian kernel $$K(x,a) = \exp(-10 |x-a|^2), \quad | \cdot |=\mbox{Euclidean norm}$$ setting each $\mathcal{V}_i$ to the $\ell_1$ loss and using $\gamma=1000$.
- [**Boosting kernel+$\ell_1$ loss**]{}: this is the estimator using the new boosting kernel. The latter is defined by the kernel matrix built using the same Gaussian kernel reported above, with $\sigma^2=1,\lambda=0.001$ so that one still has $\gamma=1000$. The function estimate is achieved solving (\[Eq9\]) using the $\ell_1$ loss.
Note that the two estimators contain only one unknown parameter, i.e. $\nu$ which is estimated by the cross validation strategy described above. The top left panel of Fig. \[FigTib1\] compares their performance. Interestingly, results are very similar, see also Table \[Table1\]. This supports the fact that the boosting kernel can include classical boosting features in the estimation process. In this example, the difference between the two methods is mainly in their computational complexity. In particular, the top right panel of Fig. \[FigTib1\] reports some cross validation scores as a function of the boosting iterations counter $\nu$ for the classical boosting scheme. The score is linearly interpolated, since $\nu$ can assume only integer values. On average, during the 100 Monte Carlo runs the optimal value corresponds to $\nu=340$, so on average, problems (\[WeakRKHS\]) must be solved 340 times. After obtaining the estimate of $\nu$, to obtain the function estimate using the union of the training and validation data, another 340 problems must be solved.
In contrast, the boosting kernel used in (\[Eq9\]) does not require repeated optimization of the weak learner. Using a golden section search,estimating $\nu$ by cross validation on average requires solving 20 problems of the form (\[Eq9\]). Once $\nu$ is found, only one additional optimization problem must be solved to obtain the function estimate. Summarizing, in this example the boosting kernel obtains results similar to those achieved by classical boosting, but requires solving only 20 optimization problems rather than nearly 700. The computational times of the two approaches are reported in the bottom panel of Fig. \[FigTib1\].
Table \[Table1\] also shows the average fit obtained by other two estimators. The first estimator is denoted by [**Boosting SVC**]{}: it coincides with [**Boosting kernel+$\ell_1$ loss**]{}, except that the hinge loss replaces the $\ell_1$ loss in (\[Eq9\]). The other one is [**SVC**]{} and corresponds to the classical support vector classifier. It uses the same Gaussian kernel defined above with the regularization parameter $\gamma$ determined via cross validation on a grid containing 20 logarithmically spaced values on the interval $[0.01,100]$. One can see that the best results are obtained by boosting support vector classification. Recall also that the hinge loss cannot be adopted using the classical boosting scheme as discussed at the end of the previous section.
[cccc]{} & [**Boosting kernel+$\ell_1$**]{} & [**Boosting SVC**]{} & [**SVC**]{}\
78.91 % & 79.15 % & 79.73 % & 78.12 %\
[cc]{}\
Boosting in RKHSs: Regression problem {#sec:regression_rkhs}
-------------------------------------
Consider now a regression problem where only smoothness information is available to reconstruct the unknown function from sparse and noisy data. As in the previous example, our aim is to illustrate how the new class of proposed boosting algorithms can solve this problem using a RKHS with a great computational advantage w.r.t. the traditional scheme. For this purpose, we just consider a classical benchmark problem where the unknown map is the Franke’s bivariate test function $f$ given by the weighted sum of four exponentials [@Wahba1990]. Data set size is 1000 and is generated as follows. First, 1000 input locations $x_i$ are drawn from a uniform distribution on $[0,1] \times [0,1]$. The data are divided in the same way described in the classification problem. The outputs in the training and validation data are $$y_i = f(x_i) + v_i$$ where the errors $v_i$ are independent, with distribution given by the mixture of Gaussians $$0.9 \mathcal{N}(0,0.1^2) + 0.1 \mathcal{N}(0,1).$$ The test outputs $y^{test}_i$ are instead given by noiseless outputs $f(x_i^{test})$. A Monte Carlo study of 100 runs is considered, where a new data set is generated at any run. The test fit is computed as $$100 \left( 1-\frac{| y^{test} -\hat{y}^{test}|}{|y^{test}- \mbox{mean}(y^{test}) |}\right),$$ where $\hat{y}^{test}$ is the test set prediction.
Note that the mixture noise can model the effect of outliers which affect, on average, 1 out of 10 outputs. This motivates the use of the robust $\ell_1$ loss. Hence, the function is still reconstructed by [**Boosting+$\ell_1$ loss**]{} and [**Boosting kernel+$\ell_1$ loss**]{} which are implemented exactly in the same way as previously described. Fig. \[FigWahba1\] displays the results with the same rationale adopted in Fig. \[FigTib1\]. The fits are close each other but, at any run, the classical boosting scheme requires solving hundreds of optimization problems, while the boosting kernel-based approach needs to solve around 15 problems on average. The computational times of the two approaches are reported in the bottom panel of Fig. \[FigWahba1\].
Finally, Table \[Table2\] reports the average fits including those achieved by [**Gaussian kernel+$\ell_1$ loss**]{}, which is implemented as the estimator [**SVC**]{} described in the previous section except that the hinge loss is replaced by the $\ell_1$ loss. The best results are achieved by boosting kernel with $\ell_1$.
[cccc]{} & [**Boosting kernel+$\ell_1$**]{} & [**Gaussian kernel+$\ell_1$**]{}\
76.62 % & 76.75 % & 75.19 %\
[cc]{}\
Conclusion {#sec:conclusions}
==========
In this paper, we presented a connection between boosting and kernel-based methods. We showed that in the context of regularized least-squares, boosting with a weak learner can be interpreted using a boosting kernel. This connection was used for three main applications: (1) providing insight into boosting estimators and when they can be effective; (2) determining schemes for hyperparameter estimation using the kernel connection and (3) proposing a more general class of boosting schemes for general misfit measures, including $\ell_1$, Huber and Vapnik, which can use also RKHSs as hypothesis spaces.
The proposed approach combines generality with computational efficiency. In contract to the classic boosting scheme, treating boosting iterations $\nu$ as a continuous hyperparameter may improve prediction capability. Real data support the use of these generalized schemes in practice. Indeed, in some real experiments we obtained $\nu = 1.42$ as estimate improving on the classic scheme. In addition, this new viewpoint avoids sequential solutions. This turns out a particularly strong advantage for boosting using general losses $\mathcal{V}$, as each boosting run would itself require an iterative algorithm. This has been outlined also in the RKHS setting: the boosting kernel allows to obtain results similar (or also better) than the classical boosting scheme dramatically reducing the computational cost.
0.2in
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---
abstract: 'It is shown that a locally geometrical structure of arbitrarily curved Riemannian space is defined by a deformed group of its diffeomorphisms.'
author:
- 'Serhiy E. SAMOKHVALOV[^1]\'
date: 'April 22, 2007'
title: ' **Group-theoretic Description of Riemannian Spaces**'
---
[*Key words*]{}: deformed group of diffeomorphisms, parallel transports, curvature, covariant derivatives, Riemannian space
[*Mathematics Subject Classification (2000)*]{}: 53B05; 53B20; 58H05; 58H15\
Until recently it was thought impossible to realize Klein’s Erlangen Program [@1] for geometrical structures with arbitrary variable curvature; this Riemann-Klein antagonism, as it was figuratively called by E. Cartan [@2], could only be overcome at the cost of program’s modification and rejection of group structure of transformations which were used. Thus in [@3] categories are employed while in [@4] quasigroups are, and it is even stated that quasigroups are an algebraic equivalent of geometric notion of curvature.
In work [@5] it was shown that group-theoretic description of connections in fiber bundles with arbitrary variable curvature can be performed by means of deformed infinite Lie groups introduced out of physical considerations in work [@6], the structural equation follows from group axioms and is a necessary condition for existence of a group which defines given geometrical structure. This allowed realization of Klein’s Program for connections in fiber bundles.
The structure of (pseudo)Riemannian space $M$ is a special case of structure of affine connection in tangent bundle and therefore it can be specified similarly to arbitrary connection [@5]. At the same time it necessary to apply additional conditions of torsion absence and coordination of connection with metric. The group fulfilling this description acts in tangent bundle of space $M$ and is an infinite and specially deformed group which has the structure of semidirect product of diffeomorphisms group $\Gamma_T
= Diff\ M$ and gauge group $SO \left(m, n-m \right)^g$, where $n$ is space dimension [@6].
It was shown in [@6] that there exists a more natural way of group-theoretic description for (pseudo)Riemannian spaces, the one with the help of a narrower group, i.e. the deformed group $\Gamma^H_T$ of diffeomorphisms of space $M$. The generators of such group define on M (locally, within the bounds of coordinate chart) field of affine vielbein, multiplication law define the rule of parallel transport of vectors, in consideration of which torsion is automatically zeroed in view of group axioms, components of vielbeins field in coordinate basis as well as anholonomity and connection coefficients are expressed through auxiliary deformation functions by means of which group $\Gamma^H_T$ is built. Locally any space of torsion-free affine connection can be described in such fashion. With additional assumption that vielbeins field, defined by action of group $\Gamma^H_T$ is (pseudo)orthonormal, and in case of parallel transport of vectors they merely rotate, coefficients of affine connection in coordinate basis automatically become Christoffel symbols, i.e. they are defined through metric in a certain way, therefore there is no need to postulate this statement.
Publication [@6] had physical value and group-theoretical and geometrical aspects were only slightly touched upon there, while some important geometrical relations were neglected at all. This work makes up for this. Specifically, we show that definition of curvature tensor (which in our approach becomes a characteristic of group $\Gamma^H_T$) through connection coefficients follows from equation which comes from group axioms and is an essential condition for existence of group $\Gamma^H_T$.
With the help of groups $\Gamma^H_T$ Klein’s Erlangen Program is realized for (pseudo)Riemannian spaces of arbitrary variable curvature, in the most rational fashion at that. Groups $\Gamma^H_T$ act on $M$ and their transformations are interpreted as gauge translation in curved (pseudo)Riemannian spaces. It is due to this that group-theoretic description of (pseudo)Riemannian spaces through groups $\Gamma^H_T$ is important for gravitation theory, gravitation being interpreted as gauge theory of translations group [@7].
The work doesn’t deal with global topological problems and all relations are obtained within the bounds of a single coordinate chart. Besides, we perceive groups to be respective local groups.
**1.** Let’s specify the general procedure of building deformed infinite Lie groups [@5] for the case of deformed group of diffeomorphisms $\Gamma^H_T$. This time, opposite to [@5] we will use coordinate approach.
Let $O$ be a coordinate chart on manifold $M$ with coordinates $x^{\mu}$ (we use Greek alphabet for indices). We assume coordinates to be fixed and won’t change them further.
In $O$ there acts Abelian group of translations $T=\{\tilde{t} \}$ according to the formula: $$x'^{\mu}=x^{\mu}+\tilde{t}^{\mu}$$ In set $C_{\infty}(O, T)$ of smooth mappings of $O$ in $T$ let’s single out subset $\Gamma_T = \{\tilde{t}(x)\}$ with condition: $$det\{{\delta}^{\mu}_{\nu} + \partial_{\nu}
\tilde{t}^{\mu} (x)\} \ne 0,\ \forall x \in O,$$ where $\partial_\nu :=\partial /
\partial x^\nu$, and assign to it the multiplication law $\tilde{t''} = \tilde{t} \times
\tilde{t'}$: $$\label{eq1}
\tilde{t''}^\mu (x) = \tilde{t}^\mu (x) + \tilde{t'}^\mu (x'),$$ where $$\label{eq2}
{x'}^\mu = {x}^\mu + \tilde{t}^\mu (x).$$ With it the set $\Gamma_T$ becomes a local group. Group $\Gamma_T$ acts smoothly in chart $O$ according to the formula (\[eq2\]) and is a local group of diffeomorphisms of chart $O$ in additive parameterization. According to definition 1 from \[5\] group $\Gamma_T$ is *a group of undeformed chart $O$, or undeformed group*.
Let’s deform group $\Gamma_T$ by means of *deformation* $H$ which is defined by mapping $H : O \times T \rightarrow T$ with properties which in our case are described as follows:
$1H)\ H \in C_{\infty}(O, T)$;
$2H)\ H(x,0)=0,\ \forall x \in O$;
$3H)\ \exists\ {\rm mapping}\ K : O \times T \to T: K(x, H(x,
\tilde{t})) = \tilde{t},\ \forall x \in O,\ \tilde{t} \in T.$
Group $\Gamma^H_T=\{t(x)\}$ is obtained from group $\Gamma_T$ by isomorphism, which is specified by deformation $H$ according to the formula: $$\label{eq3}
t^m(x)=H^m(x,\tilde t (x)).$$ Functions $t^m(x)$ which parameterize group $\Gamma^H_T$ (we use indices from Latin alphabet for them), satisfy the condition: $$det\{\delta^{\mu}_{\nu}+d_{\nu}K^{\mu}(x, t(x))\}\ne 0,\ \forall
x \in O,$$ where $d_{\nu} :=d/dx^{\nu}$, and multiplication law $t'' = t*t'$ is defined by isomorphism (\[eq3\]): $$\label{eq4}
t''^m(x)= \varphi^m (x,t(x),t'(x')):= H^m(x,K(x,t(x))+K(x',t'
(x'))),$$ where $$\label{eq5}
{x'}^{\mu} = f^{\mu}(x,t(x)):= x^{\mu} + K^{\mu}(x,t(x)).$$ Group $\Gamma^H_T$ acts smoothly in the chart $O$ according to the formula (\[eq5\]). According to definition 3 from [@5] group $\Gamma^H_T$ is *a group of deformed chart* $O$, or *deformed group*.
Multiplication law (\[eq4\]) for deformed group $\Gamma^H_T$ explicitly depends on $x$, and, therefore, structural constants analogue for groups $\Gamma^H_T$ is structure functions $F(x)^n_{kl}$, which are defined by the formula: $$\label{eq6}
F(x)^n_{kl}:=
\left(\partial^2_{k,l'}-\partial^2_{l,k'}\right)\varphi^n(x,t,t')\Bigr|_{t=t'=0}.$$ (here and henceforth $\partial_k:=\partial / \partial
t^k$, primed index stands for differentiation with respect to $t'$).
**2.** Let’s introduce auxiliary functions: $$h(x)^m_{\mu} =
\frac{\displaystyle \partial}{\displaystyle
\partial \tilde t^{\mu}}H^m(x,\tilde t)\Bigr |_{\tilde t=0}.$$ Property $3H$ allows fulfillment of condition: $$\label{eq7}
det \left \{h(x)^m_{\mu}\right \}\ne 0,\ \ \forall x \in O,$$ wherefrom there follows existence of functions $h(x)^{\mu}_m$ of the type that $h(x)^{\mu}_nh(x)_{\mu}^m=\delta^m_n$, $\forall x \in O$. It is obvious that $h(x)^{\mu}_m=\partial_m K^{\mu}(x, t)\Bigr |_{t=0}$. With the help of these functions we will substitute Greek indices for Latin and vice versa.
Assuming parameters in multiplication law for group $\Gamma^H_T$ to be constant, let’s define functions: $$\label{eq8}
\mu(x, t)^m{}_n := \partial_{n'} \varphi^m(x,t',t) \Bigr |_{t'=0},$$ $$\label{eq9}
\lambda(x, t)^m{}_n := \partial_{n'} \varphi^m(x,t,t') \Bigr
|_{t'=0}.$$
The condition of multiplication law associativity in group $\Gamma^H_T$: $(t*t')*t''=t*(t'*t'')$ is fulfilled automatically for any deformation H in view of multiplication law (\[eq1\]) associativity in diffeomorphisms group. Let’s derive this condition for constant parameters of group $\Gamma^H_T$: $$\label{eq10}
\varphi^m(x,\varphi (x,t,t'),t'')=\varphi^m(x,t,\varphi
(x',t',t'')).$$ Differentiating it with respect to t in zero we obtain the equation: $$\label{eq11}
h(x)^{\mu}_k \partial_{\mu} \varphi^m(x,t,t')-\mu(x,t)^nk
\partial_n\varphi^m(x,t,t')=-\mu (x,\varphi(x,t,t'))^m{}_k,$$ and with respect to $t''$ in zero the equation: $$\label{eq12}
\lambda(x', t')^n{}_k \partial_{n'} \varphi^m(x,t,t')=\lambda
(x,\varphi (x,t,t'))^m{}_k.$$
The condition for their integrabilily is equation: $$h(x)^{\nu}_k \partial_{\nu}\mu (x,t)^m{}_l-\mu(x,t)^n{}_k\partial_n
\mu(x,t)^m{}_l - h(x)^{\nu}_l \partial_{\nu} \mu(x,t)^m{}_k+\mu
(x,t)^n{}_l \partial_n \mu (x,t)^m{}_k=$$ $$\label{eq13}
=F(x)^n_{kl}\mu(x,t)^m{}_n$$ and $$\label{eq14}
\lambda(x, t)^n{}_k \partial_{n}\lambda(x,t)^m{}_l-
\lambda(x,t)^n{}_l\partial_n
\lambda(x,t)^m{}_k=F(x')^n_{kl}\lambda(x,t)^m{}_n.$$ respectively.
Let’s call equations (\[eq11\]) and (\[eq12\]) *the left and the right Lie equation for groups* $\Gamma^H_T$, while equations (\[eq13\]) and (\[eq14\]) *the left and the right Maurer-Cartan equations for groups* $\Gamma^H_T$.
If the condition of associativity (\[eq10\]) is immediately differentiated with respect to $t'$ and $t''$ in zero with differing sequence we obtain the equation: $$\label{eq15}
h(x)^{\nu}_k \partial_{\nu}\lambda
(x,t)^m{}_l-\mu(x,t)^n{}_k\partial_n \lambda(x,t)^m{}_l +
\lambda(x,t)^{n}{}_l
\partial_{n} \mu(x,t)^m{}_k=0.$$
Let’s perform consequently two $\Gamma^H_T$-transformations with constant parameters $t$ and $t'$. Composition law of transformations results in equation: $$f^\mu (f(x,t),t')=f^\mu(x,\varphi(x,t,t')),$$ which is fulfilled automatically for any deformation $H$ in view of performance of composition law in the group of diffeomorphisms $\Gamma_T$. Differentiating it with respect to $t$ in zero we obtain the equation: $$\label{eq16}
h(x)^{\nu}_k \partial_{\nu}f^\mu (x,t)-\mu(x,t)^n{}_k\partial_n
f^\mu(x,t)=0,$$ and differentiating it with respect to $t'$ in zero the equation: $$\label{eq17}
h(x')^{\mu}_k -\lambda (x,t)^n{}_k\partial_n f^\mu(x,t)=0.$$ The condition of integrabilily of these equations in case of fulfillment of equations (\[eq13\]) and (\[eq14\]) is equation: $$\label{eq18}
h(x)^{\nu}_k \partial_\nu h(x)^\mu_l-h(x)^{\nu}_l \partial_\nu
h(x)^\mu_k =F(x)^{n}_{kl}h(x)^\mu_n.$$
We will call equations (\[eq16\]) and (\[eq17\]) *the left and the right Lie equations for groups $\Gamma^H_T$ transformations*, while equation (\[eq18\]) *the Maurer-Cartan equation for groups $\Gamma^H_T$ transformations*.
**3.** Let’s introduce differentiating operators: $$X^\tau_k=h(x)^{\nu}_k \partial_{\nu}-\mu(x,t)^n{}_k\partial_n,$$ $$X^\upsilon_k=\lambda(x,t)^{n}{}_k \partial_{n},$$ which we will call *generators of leftward and rightward shifts, or horizontal and vertical generators* of group $\Gamma^H_T$ respectively, as well as $$X_k=h(x)^{\nu}_k \partial_{\nu}$$ - *generators of action of group $\Gamma^H_T$ on* $O$. In terms of generators, equations (\[eq13\]) - (\[eq15\]) as well as equation (\[eq18\]) have quite an elegant form: $$\label{eq19}
\left[X^\tau_k,X^\tau_l\right]=F(x)^n_{kl}X^\tau_n,$$ $$\label{eq20}
\left[X^\upsilon_k,X^\upsilon_l\right]=F(x')^n_{kl}X^\upsilon_n,$$ $$\label{eq21}
\left[X^\tau_k,X^\upsilon_l\right]=0,$$ $$\label{eq22}
\left[X_k,X_l\right]=F(x)^n_{kl}X_n,$$ where square brackets stand for operators commutator. These equations follow from multiplication law associativity for group $\Gamma^H_T$, however, due to its infinity, generators commutators are expanded into generators not by means of structure constants as in finite parametric Lie groups, but by means of structure functions dependent on $x$.
The condition for integrability of equations (\[eq19\]) - (\[eq22\]) is the equation for structure functions of group $\Gamma^H_T$: $$\label{eq23}
h(x)^{\nu}_k
\partial_\nu F(x)^n_{lm}+F(x)^n_{kp}F(x)^p_{lm}+\mbox{cycle}(klm)=0,$$ which is derived from Jacobi’s identity for dual generators commutator.
**4.** Let’s study the expansion of functions defined by formulae (\[eq8\]), (\[eq9\]) according to group parameters with accuracy to the second order inclusive: $$\label{eq24}
\mu(x,t)^m{}_n=\delta^m_n+\gamma^m{}_{nk}t^k+\frac{1}{2}
\rho^m{}_{lkn}t^lt^k,$$ $$\label{eq25}
\lambda(x,t)^m{}_n=\delta^m_n+\gamma^m{}_{kn}t^k+\frac{1}{2}
\sigma^m{}_{lkn}t^lt^k.$$ The coefficients of these expansions $\gamma^m{}_{nk}$, $\rho^m{}_{nkl}$ and $\sigma^m{}_{nkl}$ depend on $x$ in general case; however, this dependence, defined by deformation functions, will be specified a little later and to make it shorter we will not show explicitly neither to the coefficients themselves nor to the functions they define. On inserting expansions (\[eq24\]) and (\[eq25\]) into formulae (\[eq13\]) and (\[eq14\]) we arrive at the result that in zeroth order with respect to $t$ the structure functions of group $\Gamma^H_T$ are defined by skew-symmetric part of coefficients $\gamma^m{}_{kn}$: $$\label{eq26}
F^m_{kn}=\gamma^m{}_{kn}-\gamma^m{}_{nk}.$$ This formula follows directly from definition (\[eq6\]) for structure functions of group $\Gamma^H_T$ and actually can be considered their definition. Let’s introduce functions: $$\label{eq27}
R^m{}_{lkn}:=\rho^m{}_{lkn}-\rho^m{}_{lnk},$$ $$\label{eq28}
S^m{}_{lkn}:=\sigma^m{}_{lkn}-\sigma^m{}_{lnk},$$ which we will call *tensors of left and right curvature of group* $\Gamma^H_T$ respectively. In the first order with respect to $t$ from formula (\[eq13\]) we derive: $$\label{eq29}
R^m{}_{lkn}=-\gamma^m{}_{sl}F^s_{kn}+h^\sigma_k\partial_\sigma
\gamma^m{}_{nl}-h^\sigma_n\partial_\sigma\gamma^m{}_{kl}+
\gamma^m{}_{ks}\gamma^s{}_{nl}-\gamma^m{}_{ns}\gamma^s{}_{kl},$$ and from formula (\[eq14\]) $$\label{eq30}
S^m{}_{lkn}=\gamma^m{}_{ls}F^s_{kn}+h^\sigma_l\partial_\sigma
F^m_{kn}+\gamma^m{}_{sk}\gamma^s{}_{ln}-\gamma^m{}_{sn}\gamma^s{}_{lk}.$$ Relation (\[eq15\]) yields: $$\sigma^m{}_{lkn}-\rho^m{}_{lnk}=h^\sigma_k\partial_\sigma
\gamma^m{}_{ln}+\gamma^m{}_{ks}\gamma^s{}_{ln}-\gamma^m{}_{sn}\gamma^s{}_{kl},$$ wherefrom follows: $$\label{eq31}
R^m{}_{lkn}+S^m{}_{lkn}=h^\sigma_k\partial_\sigma
\gamma^m{}_{ln}-h^\sigma_n\partial_\sigma
\gamma^m{}_{lk}+\gamma^m{}_{ks}\gamma^s{}_{ln}-\gamma^m{}_{sn}
\gamma^s{}_{kl}+\gamma^m{}_{sk}\gamma^s{}_{nl}-\gamma^m{}_{ns}\gamma^s{}_{lk}.$$ Taking into account formulae (\[eq26\]), (\[eq29\]) and (\[eq30\]), expression (\[eq31\]) is the result of condition (\[eq23\]).
**5.** The relations derived so far, follow solely from group axioms without consideration of deformation mode of building the group $\Gamma^H_T$ which allows their fulfillment. However, both multiplication law (\[eq4\]) and action (\[eq5\]) of deformed group $\Gamma^H_T$ in chart $O$ are defined by deformation $H$ with the help of which it is built. Let’s express auxiliary functions of group $\Gamma^H_T$ through deformation functions. To this end, let’s introduce matrices $H(x,t)^m_{\mu} =
\frac{\displaystyle \partial}{\displaystyle \partial \tilde
t^{\mu}}H^m(x,\tilde t)\Bigr |_{\tilde t=K(x,t)}$. Matrices $H(x,t)^{\mu}_m =\partial_m K^{\mu}(x,t)$ will be inverse to them. Direct use of the second equality in (4) in definitions (8) and (9) yields: $$\label{eq32}
\mu(x,t)^m{}_n=H(x,t)^m_\mu(\delta^\mu_\nu+\partial_\nu
K^\mu(x,t))h(x)^\nu_n,$$ $$\label{eq33}
\lambda(x,t)^m{}_n=H(x,t)^m_\mu h(x+K(x,t))^\mu_n,$$ or depending upon $\tilde t$: $$\mu(x,\tilde t)^m{}_n=\frac{\partial}{\partial\tilde
t^\mu}H^m(x,\tilde t)(\delta^\mu_\nu + \partial_\nu \tilde
t^\mu)h(x)^\nu_n,$$ $$\label{eq34}
\lambda(x,\tilde t)^m{}_n=\frac{\partial}{\partial\tilde
t^\mu}H^m(x,\tilde t)h(x+\tilde t)^\mu_n.$$
Let’s consider the expansion of functions of deformation $H$ up to the third order with respect to $\tilde t$ inclusive: $$\label{eq35}
H^m(x,\tilde t)=h^m_\mu(\tilde t^\mu+\frac{1}{2}\Gamma^\mu_{\nu
\rho}\tilde t^\nu\tilde t^\rho +\frac{1}{6}\Delta^\mu_{\nu \rho
\sigma}\tilde t^\nu \tilde t^\rho \tilde t^\sigma).$$ Coefficients $h^m_\mu$ satisfy condition (\[eq7\]) and $\Gamma^\mu_{\nu\rho}$, $\Delta^\nu_{\nu\rho\sigma}$ are symmetric in lower indices. On fulfilling these conditions, the coefficients of expansion (\[eq35\]) are arbitrary smooth functions of $x$. Applying them, with accuracy to the second order with respect to $t$ we derive: $$K^\mu(x,t)=h^\mu_k t^k-\frac{1}{2}\Gamma^\mu_{kl}t^k t^l,$$ $$H(x,t)^m_\mu=h^m_\mu +\Gamma^m_{\mu k}t^k
+\frac{1}{2}(\Delta^m_{\mu kl}-\Gamma^m_{\mu s}\Gamma^s_{kl})t^k
t^l.$$ In consideration of these expansions formulae (\[eq32\]) and (\[eq33\]) give the following expressions for coefficients of expansions (\[eq24\]) and (\[eq25\]) through coefficients of expansion (\[eq35\]): $$\label{eq36}
\gamma^m{}_{kn}=h^m_\mu (\Gamma^\mu_{kn}+h^\nu_k \partial_\nu
h^\mu_n),$$ $$\label{eq37}
\rho^m{}_{lkn}=h^m_\mu
(\Delta^\mu_{lkn}-\Gamma^\mu_{ns}\Gamma^s_{kl}-h^\nu_n
\partial_\nu \Gamma^\mu_{\kappa\lambda}h^\kappa_k h^\lambda_l),$$ $$\sigma^m{}_{lkn}=h^m_\mu
(\Delta^\mu_{lkn}-\Gamma^\mu_{ns}\Gamma^s_{kl}+h^\kappa_k
h^\lambda_l\partial^2_{\kappa\lambda}h^\mu_n-\Gamma^\nu_{kl}
\partial_\nu h^\mu_n + (\Gamma^\mu_{k\sigma}h^\nu_l +\Gamma^\mu_{l\sigma}
h^\nu_k)\partial_\nu h^\sigma_n).$$ Inserting these expressions into definitions (\[eq26\]) - (\[eq28\]) and taking into account the symmetry in lower indices of coefficients $\Gamma^\mu_{kn}$ and $\Delta^\mu_{lkn}$ we derive formula (18) for structure functions of group $\Gamma^H_T$, and for its curvature tensors the formulae as follows: $$\label{eq38}
R^m{}_{lkn}=h^m_\mu
(\partial_\kappa\Gamma^\mu_{\nu\lambda}-\partial_\nu\Gamma^\mu_{\kappa\lambda}+
\Gamma^\mu_{\kappa\sigma}\Gamma^\sigma_{\nu\lambda}-
\Gamma^\mu_{\nu\sigma}\Gamma^\sigma_{\kappa\lambda})h^\lambda_l
h^\kappa_k h^\nu_n,$$ $$S^m{}_{lkn}=h^m_\mu
(\Gamma^\mu_{\l\sigma}F^{\sigma}_{kn}+h^{\sigma}_l
\partial_{\sigma}F^\mu_{kn}+
\Gamma^\mu_{k \sigma}\Gamma^\sigma_{nl}-\Gamma^\mu_{n
\sigma}\Gamma^\sigma_{kl}+ \Gamma^\sigma_{nl}\partial_\sigma
h^{\mu}_k- \Gamma^\sigma_{kl}\partial_\sigma h^{\mu}_n+$$ $$h^{\sigma}_l (\Gamma^\mu_{k \sigma} \partial_{\lambda}
h^{\sigma}_n- \Gamma^\mu_{n \sigma}\partial_\lambda
h^{\sigma}_k+\partial_\lambda h^{\sigma}_n \partial_\sigma
h^{\mu}_k-\partial_\lambda h^{\sigma}_k \partial_\sigma
h^{\mu}_n)),$$ These formulae could be derived directly from formulae (\[eq29\]), (\[eq30\]) on inserting expressions (\[eq36\]) and (18) into them. The reason for this is that the condition of multiplication law associativity in groups $\Gamma^H_T$, which yields equations (\[eq13\]) and (\[eq14\]) wherefrom formulae (\[eq29\]) and (30) were derived, is fulfilled automatically through deformation mode of building groups $\Gamma^H_T$ which we apply.
**6.** The very form of tensor of left curvature of group $\Gamma^H_T$ (formulae (\[eq29\]) or (\[eq38\])) as well as that of its other characteristics indicates that groups $\Gamma^H_T$ possess ample geometric data which we proceed to study below.
Generators $X_m=h^\mu_m\partial_\mu$ of action of group $\Gamma^H_T$ specify on $O$ a field of affine vielbeins, auxiliary functions of deformation $h^\mu_m$ transfer from coordinate to affine bases. Elements $t$ of group $\Gamma^H_T$ specify on $O$ *vector fields* $t=t^mX_m$, parameters $t^m$ of group $\Gamma^H_T$ are components of these fields in basis $X_m$.
Structure functions of group $\Gamma^H_T$ with lower coordinate indices in view of formula (\[eq18\]) can be represented as: $$F^k_{\mu\nu}=\partial_\nu h^k_\mu -\partial_\mu h^k_\nu,$$ thus they have geometric meaning (with accuracy to factor -2) of *anholonomity object*.
Let’s study multiplication law $t*\tau$ in group $\Gamma^H_T$ for the case of infinitesimal second multiplier: $$(t*\tau)^m(x)=t^m(x)+\lambda(x,t(x))^m{}_n \tau^n (x'),$$ where $x'^\mu=f^\mu(x,t(x))$. Thus, this law gives the rule for composition of vectors fitted in different points, or the *rule of parallel transport* of vector field $\tau$ from point $x'$ to point $x$: $$\label{eq39}
\tau^m_\| (x)=\lambda(x,t(x))^m{}_n \tau^n(x').$$ Taking $t$ to be infinitesimal as well, and taking into account expansion (\[eq25\]) we have: $$\label{eq40}
\tau^m_\| (x)=\tau^m(x)+t^n (x) \nabla_n \tau^m (x),$$ where $$\nabla_n \tau^m (x)=h^\sigma_n
\partial_\sigma \tau^m (x) +\gamma^m{}_{nk} \tau^k (x)$$ by definition is a *covariant derivative* of vector field $\tau$ to the direction $X_n$. Thus, functions $\gamma^m{}_{nk}$ which define the second of parameters order of multiplication law in group $\Gamma^H_T$ get geometric meaning of *coefficients of affine connection in basis* $X_n$.
In coordinate basis, relation (\[eq39\]) in view of (\[eq34\]) becomes: $$\tau^\mu_\| (x)=\frac{\partial}{\partial\tilde t^\nu}H^\mu
(x,\tilde t)\tau^\nu (x+\tilde t),$$ or in case of infinitesimal $\tilde t$: $$\tau^\mu_\| (x)=\tau^\mu (x) +\tilde t^\nu (x) \nabla_\nu \tau^\mu (x),$$ in relation to which $$\nabla_\nu \tau^\mu (x)=\partial_\nu \tau^\mu (x)+\Gamma^\mu_{\sigma\nu}
\tau^\sigma (x).$$ Thus, coefficients $\Gamma^\mu_{\sigma\nu}$ which define the second order of expansion (\[eq35\]) of deformation functions get geometric meaning of *coefficients of affine connection in coordinate basis*. They are arbitrary smooth functions symmetric in lower indices, corresponding to arbitrary torsion-free affine connection. For undeformed group $\Gamma^H_T$ covariant derivatives is obviously congruent with partial derivatives.
It is in this meaning that, specifying the rule of parallel transport of vectors by its multiplication law (which is defined by deformation $H$), deformed groups of diffeomorphisms $\Gamma^H_T$ specify by their action a structure of torsion-free affine connection in tangent bundle of chart $O$; arbitrary torsion-free affine connection can be specified over $O$ in such fashion.
The same connection is specified by all groups $\Gamma^{H'}_T$ in which coefficients $\Gamma^\mu_{\sigma\nu}$ in the second order of expansion (\[eq35\]) of their function of deformation $H'$ are congruent, particularly if $$\label{eq41}
H'^m(x,\tilde t)=L(x)^m{}_n H^n (x,\tilde t).$$ where matrices $L(x)^m{}_n$, dependent upon $x$, belong to gauge group $GL(n)^g$. In transformation (\[eq41\]) the affine vielbeins field on $O$ changes: $X'_m =L^{-1} (x)^n{}_m
X_n$. The third and higher orders of parameter expansion of deformation functions do not influence the connection and can be arbitrary. This is related to the fact that definition (\[eq39\]) allows to make parallel transport of vector field from point $x'$ to point $x$ for *finite* distance $\tilde t
(x) = x'-x=K(x,t(x))$, though infinitesimal shifts are enough to specify a connection. There are, however, quite natural additional requirements to deformation functions which follow from geometric point of view and allow to completely fix deformation functions with respect to the first two orders of expansion (\[eq35\]), i.e. with respect to the affine vielbeins field and affine connection coefficients. They are related to the generation of finite parallel transports (\[eq39\]) with the help of integral sequence of infinitesimal transports (\[eq40\]), and we make plans to study this problem for the structure of affine connection in our next work [@8].
Let’s choose points $x_1=x+\tilde t_1$, $x_2=x+\tilde t_2$, $x_3=x+\tilde t_1+\tilde t_2$ ($\tilde t_1$ and $\tilde t_2$ we assume to be constant) and perform, according to formula (\[eq39\]), parallel transport of vector field $\tau (x)$ from point $x_3$ to point $x_1$, and then to point $x$ (first choice), as well as from point $x_3$ to point $x_2$ and then to point $x$ (second choice). The difference of the results obtained gives: $$\tau^m_\| (x)_1-\tau^m_\|
(x)_2=(\lambda(x,\tilde t_1)^m{}_k \lambda(x_1,\tilde
t_2)^k{}_n-\lambda(x,\tilde t_2)^m{}_k \lambda(x_2,\tilde
t_1)^k{}_n)\tau^n(x_3).$$ For infinitesimal $\tilde t_1$ and $\tilde t_2$ using formulae (\[eq34\]), (\[eq35\]) and (\[eq38\]) we derive: $$\tau^m_\| (x)_1-\tau^m_\|
(x)_2=R^m{}_{n\rho\sigma}\tau^n(x)\tilde t_1^\rho \tilde
t_2^\sigma.$$ Thus, the tensor $R^m{}_{n\rho\sigma}$ of the left curvature of group $\Gamma^H_T$, which according to the formula (\[eq27\]) is a skew-symmetric part of coefficients $\rho^m{}_{n\rho\sigma}$, which (partially) define the third of parameter order of multiplication law in group $\Gamma^H_T$, acquires the geometric meaning of *curvature tensor* of affine connection structure, which is specified in $O$ by the action of group $\Gamma^H_T$.
Let’s summarize the obtained results.
**Theorem 1.** *Deformed group $\Gamma^H_T$ of diffeomorphisms in chart $O$ specifies by its action on $O$ an affine vielbeins field and structure of torsion-free affine connection in tangent bundle over $O$. Geometric characteristics of space $O$, such as anholonomity object, affine connection coefficients, curvature tensor are defined by multiplication law in group $\Gamma^H_T$, which, in its turn, is defined by deformation $H$, with the help of which group $\Gamma^H_T$ is built.*
*Arbitrary torsion-free affine connection can be specified over $O$ in such fashion.*
Thus, geometric structure of torsion-free affine connection with arbitrary variable curvature can be referred to only in terms of deformed groups $\Gamma^H_T$ of diffeomorphisms, due to which Klein’s Erlangen Program is realized for such structure, the condition of torsion absence (\[eq26\]) is fulfilled in view of group axioms and there is no need for its additional application.
**7.** Let’s now assume that matrices $\lambda(x,t)^m{}_n$ belong to the gauge group $SO(m,n-m)^g$, so they satisfy the equation: $$\label{eq42}
\lambda(x,t)^k{}_m\lambda(x,t)^l{}_n \eta_{kl}=\eta_{mn},$$ where $\eta_{mn}$ is a flat metric (with the help of which we will lowering indices). This means that vielbein field $X_m$, specified by the action of group $\Gamma^H_T$ is (pseudo)orthonormal and in case of parallel transport of vectors (\[eq39\]) they merely (pseudo)rotate. Thus, the action of group $\Gamma^H_T$ in $O$ specifies the structure of (pseudo)Riemann space with metric $g_{\mu\nu}=h^m_\mu h^n_\nu \eta_{mn}$.
In the first order with respect to $t$ the equation (\[eq42\]) produces: $$\gamma^\centerdot_{ksl}+\gamma^\centerdot_{lsk}=0,$$ which allows, with the use of definition (\[eq26\]), to express affine connection coefficients in vielbein basis in terms of structure functions of group $\Gamma^H_T$: $$\label{eq43}
\gamma^\centerdot_{slk}=\frac{1}{2}\left(F^\centerdot_{slk}+
F^\centerdot_{ksl}+F^\centerdot_{lsk}\right).$$ Recalling geometric interpretation of structure functions we can see that coefficients $\gamma^s{}_{lk}$ in this case become *Ricci rotation coefficients*.
With the use of formula (\[eq34\]) equation (\[eq42\]) becomes equation directly for deformation functions: $$\label{eq44}
\frac{\partial}{\partial\tilde t^\mu} H^m (x,\tilde t)
\frac{\partial}{\partial\tilde t^\nu}H^n(x,\tilde
t)\eta_{mn}=g(x+\tilde t)_{\mu\nu}.$$ Besides, we have relation $2H$ for the function $H$: $$\label{eq45}
H^m(x,0)=0,$$ which we will consider a boundary condition for differential equation (\[eq44\]). The solution to the problem (\[eq44\]), (\[eq45\]) allows to find deformation functions $H^m(x,\tilde t)$ with whose help group $\Gamma^H_T$ is produced; the group specifies in $O$ a structure of (pseudo)Riemannian space, arbitrary (pseudo)Riemannian structure can be specified in $O$ in such fashion.
Equation (\[eq44\]) is invariant under given transformations: $$\label{eq46}
H'^m(x,\tilde t)=\Lambda(x)^m{}_nH^n(x,\tilde t),$$ where matrices $\Lambda(x)^m{}_n$, dependent upon $x$, belong to gauge group $SO(m,n-m)^g$, i.e. satisfy relation $\Lambda(x)^k{}_m\Lambda(x)^l{}_n\eta_{kl}=\eta_{mn}$. Thus, if the equation (\[eq44\]) is satisfied by deformation functions $H^m(x,\tilde t)$, it is also satisfied by functions $H'^m(x,\tilde t)$, which are defined by formula (\[eq46\]) with arbitrary $\Lambda(x)^m{}_n$ from the group $SO(m,n-m)^g$. All such groups $\Gamma^{H'}_T$ specify the same (pseudo)Riemann structure on $O$.
From geometric point of view, the field of (pseudo)orthonormal vielbeins: $X'_m=\Lambda^{-1}(x)^n{}_mX_n$ changes during transformations (\[eq46\]). By the field of (pseudo)orthonormal vielbeins $X_m$ from equation (\[eq44\]) deformation functions are *uniquely* defined (let’s recall that we assume coordinates in $O$ to be fixed).
Let’s point out that in our approach in (pseudo)Riemannian space with respect to the field of (pseudo)orthonormal vielbeins $X_m$ the rule of parallel transport of vectors to finite distance $\tilde t(x)=x'-x=K(x,t(x))$ is uniquely defined. On the other hand, in general case of curved space the result of parallel transport depends upon the curve along which it is performed. So there is a question to be asked: along which curve connecting points $x'$ and $x$ in the general case of curved (pseudo)Riemannian space during performance of integral sequence of infinitesimal transports (\[eq40\]) do we get the result which is given by formula (\[eq39\])? We plan to study this problem in our next publication [@8].
In the first order with respect to $\tilde t$ equation (\[eq44\]) produces: $$\label{eq47}
\Gamma^\centerdot_{\mu\nu\sigma}+\Gamma^\centerdot_{\nu\mu\sigma}=\partial_\sigma
g_{\mu\nu},$$ which, in view of symmetry of coefficients $\Gamma^\sigma_{\mu\nu}$ in lower indices, gives formula: $$\label{eq48}
\Gamma^\centerdot_{\sigma\mu\nu}=\frac{1}{2}(\partial_\mu
g_{\nu\sigma}+\partial_\nu g_{\mu\sigma}-\partial_\sigma
g_{\mu\nu}),$$ which, naturally, could be derived as the result of formula (\[eq43\]) in consideration of relation (\[eq36\]). Formula (\[eq48\]) indicates that $\Gamma^\centerdot_{\sigma\mu\nu}$ and $\Gamma^\sigma_{\mu\nu}$ in our case become *Christoffel symbols of the $I^{st}$ and $II^{nd}$ type* respectively.
In the second order with respect to $\tilde t$ it follows from equation (\[eq44\]) that: $$\Delta^\centerdot_{\mu\nu\sigma\rho}+\Delta^\centerdot_{\nu\mu\sigma\rho}=
\partial^2_{\sigma\rho}g_{\mu\nu}-\Gamma^\centerdot_{\tau\mu\sigma}\Gamma^\tau_{\nu\rho}-
\Gamma^\centerdot_{\tau\nu\sigma}\Gamma^\tau_{\mu\rho},$$ which, in view of symmetry of coefficients $\Delta^\sigma_{\mu\nu\rho}$ in lower indices and relation (\[eq47\]), gives: $$\Delta^\sigma_{\mu\nu\rho}=\frac{1}{3}(\partial_\rho
\Gamma^\sigma_{\mu\nu}+\partial_\nu
\Gamma^\sigma_{\mu\rho}+\partial_\mu
\Gamma^\sigma_{\nu\rho}+\Gamma^\sigma_{\tau\rho}\Gamma^\tau_{\mu\nu}+
\Gamma^\sigma_{\tau\nu}\Gamma^\tau_{\mu\rho}+\Gamma^\sigma_{\tau\mu}\Gamma^\tau_{\nu\rho}).$$ Inserting this expression into formula (\[eq37\]) and considering formula (\[eq38\]) we derive the expression $$\rho^\sigma{}_{\mu\nu\rho}=\frac{1}{3}(R^\sigma{}_{\mu\nu\rho}+R^\sigma{}_{\nu\mu\rho}),$$ the insertion of which into definition (\[eq27\]) produces a well-known identity for curvature tensor: $$R^\sigma{}_{\mu\nu\rho}+R^\sigma{}_{\nu\rho\mu}+R^\sigma{}_{\rho\mu\nu}=0.$$
Thus we have proved
**Theorem 2.** *Deformed group $\Gamma^H_T$ of diffeomorphisms of chart $O$, produced with the help of deformation, functions of which satisfy the equation (\[eq44\]), specifies by its action on $O$ a field of (pseudo)orthonormal vielbeins and structure of (pseudo)Riemannian space. In particular, coefficients of affine connection in coordinate basis become equal to Christoffel symbols. The same structure of (pseudo)Riemannian space is specified on $O$ by all the groups $\Gamma^{H'}_T$, the deformation functions of which are connected by transformations (\[eq46\]) from gauge group $SO(m,n-m)^g$.*
*Arbitrary (pseudo)Riemannian structure on O can be specified in such fashion.*
Through this theorem Klein’s Erlangen Program is realized for geometric structure of (pseudo) Riemannian space.
This work makes a group-theoretic description of geometric structures of torsion-free affine connection and (pseudo)Riemannian space locally within the bounds of a single coordinate chart. This limitation can be lifted provided Lie pseudogroups are studied.
The fact that relations derived from the conditions of existence of certain groups have profound geometric meaning is another confirmation of fundamentality of the ideas of Klein’s Erlangen Program in respect that geometry is defined completely by a group of congruencies.
[99]{} , [*Vergleichende Betrachtungen Uber Neuere Geometrische Forschungen (Erlanger Programm)*]{}, Erlangen, 1872. , 1925. , [*Differentialgeometrie und Faserbundel*]{}, Berlin, Veb Deutscher Verlag der Wissenschaften, 1972. , [*Methods of Nonassociativ Algebras in Differential Geometry,*]{} 1981. , [*About the Setting of Connections in Fiber Bundles by the Acting of Infinite Lie Groups,*]{} Ukrainian Math. J. **43** (1991) 1599-1603. , [*Group-Theoretic Description of Gauge Fields,*]{} Theor. Math. Phys. **76** (1988) 709-720. Class. Quantum Grav. **8** (1991) 2277 - 2282. , [*Canonical Deformed Groups of Diffeomorphisms and Finite Parallel Transports in Riemannian Spaces,*]{} arXive:0704.2980 \[math.DG\].
[^1]: [**e**]{}-[*mail*]{}: samokhval@dstu.dp.ua
|
---
abstract: 'I present a previously unpublished method for calculating and modeling multiple lens microlensing events that is based on the image centered ray shooting approach of @em_planet. It has been used to model all a wide variety of binary and triple lens systems, but it is designed to efficiently model high-magnification planetary microlensing events, because these high-magnification events are, by far, the most challenging events to model. It is designed to be efficient enough to handle complicated microlensing events, which include more than two lens masses and lens orbital motion. This method uses a polar coordinate integration grid with a smaller grid spacing in the radial direction than in the angular direction, and it employs an integration scheme specifically designed to handle limb darkened sources. I present tests that show that these features achieve second order accuracy for the light curves of a number of high-magnification planetary events. They improve the precision of the calculations by a factor of $>100$ compared to first order integration schemes with the same grid spacing in both directions (for a fixed number of grid points). This method also includes a $\chi^2$ minimization method, based on the Metropolis algorithm, that allows the jump function to vary in a way that allows quick convergence to $\chi^2$ minima. Finally, I introduce a global parameter space search strategy that allows a blind search of parameter space for light curve models without requiring $\chi^2$ minimization over a large grid of fixed parameters. Instead, the parameter space is explored on a grid of initial conditions for a set of $\chi^2$ minimizations using the full parameter space. While this method may be somewhat faster than methods that find the $\chi^2$ minima over a large grid of parameters, I argue that the main strength of this method is for events with the signals of multiple planets, where a much higher dimensional parameter space must be explored to find the correct light curve model.'
author:
- 'David P. Bennett'
title: |
An Efficient Method for Modeling High-Magnification\
Planetary Microlensing Events
---
Introduction {#sec-intro}
============
Gravitational microlensing has opened a new window on the study of extrasolar planets as it is the only method that is currently able to detect low-mass planets in orbits beyond $1\,$AU. In fact, six of the ten published microlensing planet discoveries have been of planets of less than Saturn’s mass [@ogle169; @gaudi-ogle109; @bennett-ogle109; @sumi-ogle368; @moa310] and two of these have masses below $10{{M_\oplus}}$ [@ogle390; @bennett-moa192]. The range of planetary separations probed by microlensing is particularly relevant to tests of the core accretion model for planet formation, as microlensing is particularly sensitive to planets just beyond the “snow-line" [@ida_lin; @kennedy-searth] where core accretion predicts that the most massive planets should form.
Seven of these ten published planetary microlensing events [@ogle71; @ogle169; @gaudi-ogle109; @bennett-moa192; @dong-moa400; @moa310] have been found in high-magnification events, which although rare, have a much higher planet detection probability [@griest_saf]. A corollary of this is that high-magnification events have significant sensitivity to events with signals from multiple planets [@gaudi_nab_sack] and to planets in stellar binary systems. This point was demonstrated with the discovery of the Jupiter-Saturn analog system OGLE-2006-BLG-109Lb,c [@gaudi-ogle109; @bennett-ogle109]. Furthermore, this event also demonstrated that microlensing can detect the orbital motion of a planet when the caustic structures are sufficiently large as can occur for a massive planet [@dong_ogle71] or a “resonant" caustic. (When a planet is close to the Einstein ring, the planetary and central caustics merge to form a “resonant" caustic.)
This sensitivity to orbital motion and to systems with more than two lens masses makes the modeling of these microlensing events significantly more challenging than other events. In fact, there is currently a backlog of planetary microlensing events that appear to have three or more lens masses. There are three such events discovered through the end of the 2008 Galactic bulge observing season, but the analysis is complete for only one of these [@gaudi-ogle109; @bennett-ogle109]. In contrast, the analysis is complete for 8 of the 9 planetary microlensing events discovered through the end of 2008 that can be modeled with a single planet and host star.
In this paper, we present a general method for light curve modeling that is designed to be able to model the most complicated and difficult high-magnification microlensing events. This method has gradually evolved from the first general method for calculating finite source light curves for binary lens events [@em_planet]. Versions of this modeling method have been used to analyze possible planetary events observed in the 1990’s [@rhie_ben96; @bennett-macho_planets; @mps-98blg35], to make the first successful real-time predictions of caustic crossings toward the Galactic bulge and Magellanic Clouds [@iauc96blg3; @macho-98smc1], and to analyze binary lensing events seen by the MACHO collaboration [@macho-lmc9; @macho-binaries]. More recent versions of this code have been used to model all of the known planetary microlensing events, and this code has played a major role in modeling 7 of the 10 published planetary microlensing signals [@bond-moa53; @ogle390; @ogle169; @gaudi-ogle109; @bennett-moa192; @bennett-ogle109; @sumi-ogle368].
This method has several unique features. In Section \[sec-int\], I present a numerical integration scheme with two new features that improve its precision by more than a factor of 100 (for a fixed number of integration grid points). The first feature is an integration scheme that is specifically designed to handle the singular derivative at the limb of a limb darkened source profile. This method also features a polar coordinate integration grid with a much larger grid spacing in the angular than in the radial direction to take advantage of the lensing distortion of the images. In Section \[sec-lc\_calc\], I present tests of this method using a number of previously analyzed planetary microlensing events. These tests confirm the dramatic improvement in precision for high-magnification events.
Next, in Section \[sec-markov\], I present an adaptive $\chi^2$ minimization recipe based on the @metrop algorithm that is designed to rapidly descend to a minimum of a complicated $\chi^2$ surface. This method, along with the integration scheme described in Section \[sec-int\], was critical for the analysis of the one double-planet microlensing event that has been successfully modeled [@gaudi-ogle109; @bennett-ogle109]. This event was unusually time consuming to model, due to the important effect of the orbital motion of one of the planets, but the optimizations described in Section \[sec-int\] and \[sec-markov\] made the modeling of this event tractable.
A global fit strategy, designed to find all the competitive $\chi^2$ minima for a microlensing event is presented in Section \[sec-global\], and in Section \[sec-global-ex\], I present examples of this method for a number of single-planet events. Finally, in Section \[sec-conclude\], I discuss ways in which this method can be improved and reach some conclusions.
Calculation of Planetary Microlensing Light Curves {#sec-int}
==================================================
Gravitational lensing by stars and planets can be approximated to extremely high accuracy by the lens equation for point-masses, $$w = z - \sum_i {\epsilon_i \over \bar{z} - \bar{x}_i} \ ,
\label{eq-mult_lens}$$ where $w$ and $z$ are the complex positions of the source and image, respectively, and $x_i$ are the complex positions of the lens masses. This equation uses dimensionless coordinates, normalized to the Einstein ring radius of the total lens system mass. The individual lens masses are represented by $\epsilon_i$, which is the mass fraction of the $i$th lens mass, so that $\sum_i \epsilon_i = 1$.
If we assume a point source, then we can derive a formula for the lensing magnification from the Jacobian determinant of the lens equation (and its complex conjugate): $$J = {\partial w\over \partial z} {\partial \bar{w}\over \partial \bar{z}}
- {\partial w\over \partial \bar{z}} {\partial \bar{w}\over \partial z}
= 1 - \left| {\partial w\over \partial \bar{z}} \right|^2 \ ,
\label{eq-J}$$ where $${\partial w\over \partial \bar{z}}
= \sum_i {\epsilon_i \over (\bar{z} - \bar{x}_i)^2} \ .
\label{eq-partw}$$ Because eq. \[eq-J\] gives the Jacobian determinant of the inverse mapping from the image plane to the source plane, the magnification of each image is given by $$A = {1\over |J|} \ ,
\label{eq-AJ}$$ evaluated at the position of each image. In order to use eq. \[eq-AJ\] to determine the magnification, we must solve the lens equation, \[eq-mult\_lens\]. For a lens with two point masses, eq. \[eq-mult\_lens\] can be inverted to yield a fifth order complex polynomial equation [@double-lens; @witt90]. The image locations for a given source position are roots of this polynomial, but there are either three or five solutions that correspond to physical image positions. These polynomial roots can be found with efficient numerical methods (e.g. @num-rec), and this provides a very quick calculation of binary microlensing light curves for point sources [@mao-pac]. The triple lens version of eq. \[eq-mult\_lens\] can be inverted to yield a tenth order polynomial equation, which corresponds to 4, 6, 8, or 10 physical image solutions [@rhie-3lens]. In general, the lens equation, eq. \[eq-mult\_lens\], for $n>1$ point masses has a minimum of $n+1$ images and a maximum of $5(n-1)$ images [@rhie_5n-1; @fund_alg].
The earliest investigations of the two-point mass lens system light curves have used this point source approximation [@double-lens; @mao-pac]. And these point-source solutions are an important aspect of the @em_planet method for calculating multiple-lens light curves, which I further develop in this paper. The triple lens solution to eq. \[eq-mult\_lens\] was critical for the modeling of the first double-planet microlensing event [@gaudi-ogle109; @bennett-ogle109]. However, for systems with extreme mass ratios, such as lens systems with masses similar to the Sun, Earth, and Moon, [@gest-sim] standard double precision (64-bit) arithmetic is insufficient to solve the tenth order polynomial. So, it may be necessary to resort to quadruple precision (128-bit) calculations, which can be up to 100 times slower than double precision in some compiler implementations.
Extensions of the point-source approximation have been provided by @pej_hey and @gould-hex, who have developed a power series corrections to the point-source approximation. The @gould-hex analysis goes to out to the hexadecapole term. This approximation is valid much closer to the caustics of the multiple lens light curve than the point-source approximation. This can result in a dramatic improvement in computation time for microlensing events with cusp-approaches, but no caustic crossings, such as OGLE-2005-BLG-71 [@dong_ogle71]. Events with a source trajectory that runs parallel to a caustic for a long period of time can also see a significant improvement. But for most events with caustic crossings, the expected improvement in computational time is much more modest - probably not exceeding a factor of two [@gould-hex].
The majority of the computational effort required to compute light curves for planetary microlensing events is devoted to the numerical integrations necessary for finite source calculations. The most obvious method would be to simply integrate the point-source magnification pattern over the disk of the source star. However, this method presents severe numerical difficulties due to the singularities in the point-source magnification profile. The point-source magnification pattern for a source crossing a fold caustic is $$\begin{aligned}
A &\approx C_1 + {C_2\over \sqrt{x - x_c}} \ \ \ \ {\rm for} \ \ x > x_c
\nonumber \\
& \approx C_1 \ \ \ \ {\rm for} \ \ x < x_c \ ,
\label{eq-fcaustic}\end{aligned}$$ where $C_1$ and $C_2$ are constants, and $x_c$ is the location of the fold caustic (which is assumed to be parallel with the $y$-axis). So, the point-source magnification is (formally) infinite and discontinuous on the line-like caustic curve. In the caustic exterior, there are also pole singularities in the magnification, $A\approx 1/r$, in the vicinity of cusps.
These singularities can be avoided with the ray-shooting method, originally developed by @ray-shoot1. For complicated lens systems that consist of more than just a few point masses, solving for the image positions for a given source position can be difficult, or even intractable. But, if we start with the image positions, we can always use eq. \[eq-mult\_lens\] to determine the source position from the image positions. By covering the image plane with light rays that are shot back towards the source it is possible to find all the images for a given source position. This is often referred to as the inverse ray-shooting method because without a solution to eq. \[eq-mult\_lens\], it is only possible to do the lens mapping in the inverse direction: from image to source.
For binary and triple lens systems, it is straight forward to solve the lens equation numerically, so it is not necessary to shoot the rays in the inverse direction. Instead, the advantage of ray shooting is that the integrands involved in the lens magnification calculations are less singular. The lens images have a surface brightness equal to the surface brightness of the source, so we can integrate in the image plane and avoid the strong singularities associated with caustic and cusp crossings in the source plane.
This leads to the basic strategy developed by @em_planet for the calculation of multiple lens light curves. The lens equation is solved to locate the images for a point-source located at the source center. If these images are sufficiently far from the critical curves and the source sufficiently far from the caustics, then the point-source approximation is used. (In most cases, the point source approximation is used if these separations are greater than the 7 source radii times the point source magnification.) When the point-source approximation cannot be used, we build integration grids in the image plane to cover each of the images to be integrated over. The image grids are increased until their boundaries are completely comprised of points that do not map onto the source. Some care in bookkeeping is required to ensure that images are not double counted as some grids can grow to include more than one image. There will be times when the source limb has crossed a caustic, while the source center remains on the exterior. To include these partial lensed images we must also build integration grids at the critical curve locations corresponding to the caustic points that overlap the source.
These finite source calculations become quite time consuming for high-magnification events due to the nature of the high-magnification images. As the lens and source approach perfect alignment, we approach the Einstein ring situation, and the images become large circular arcs with a length:thickness ratio that is approximately equal to the total magnification, $A$, which is typically in the range $100 {\lower.5ex\hbox{{$\; \buildrel <\over\sim \;$}}}A {\lower.5ex\hbox{{$\; \buildrel <\over\sim \;$}}}1000$. As I shall discuss below, it is these long, thin images that are very time consuming to integrate over, and so many of the features of my numerical integration scheme are designed to make these integrals more efficient.
For static lens systems (where the orbital motion of the lens system is not important) the image positions at different times in the light curve will often overlap, so we will often have to invoke the lens equation, eq. \[eq-mult\_lens\], many times at the same location. So, to minimize the number of lens equation calculations for high-magnification events, we store the lens equation solutions on a grid centered on the Einstein ring. (This feature is common to a number of other methods that use a version of ray-shooting [@wamb; @ratt; @dong-ogle343].)
This image centered ray shooting light curve calculation strategy [@em_planet] was the first method that was able to give precise calculations of planetary microlensing light curves including realistic finite source effects. Several potentially promising alternative approaches have been suggested [@wamb; @gould-stokes; @ratt; @dong-ogle343; @dominik-cont]. Several of these focus on finding solutions with fixed lens mass ratios and relative positions [@wamb; @ratt; @dong-ogle343] in order to map out $\chi^2$ as a function of these parameters. This is often used for an initial search for solutions, but it is not very efficient when these parameters are not fixed. Furthermore, the number of parameters that must be fixed increases from two to five when a triple lens system is considered, and the brute force method of mapping out the $\chi^2$ surface seems much less attractive if it must be done in five dimensions. The methods of @gould-stokes, @dominik-cont, and the “loop-linking" method of @dong-ogle343 are somewhat more flexible. Both @gould-stokes and @dominik-cont invoke a Stokes theorem approach that is much more efficient for sources that are not limb darkened. Of course, no limb-darkening is an unphysical approximation, but in many cases, the effect of limb darkening on microlensing light curves is relatively small. Thus, it might be sensible to develop a fast code based on the @gould-stokes method to search for approximate solutions without limb darkening. However, the errors due to the lack of limb darkening could be more serious in events like OGLE-2005-BLG-390 [@ogle390] and MOA-2007-BLG-400 [@dong-moa400] where planetary signal comes from a caustic curve that is much smaller than the source. The efficiency advantage of the Stokes theorem approach is lost when limb darkening is included, but this method may still be competitive. The basic method of @em_planet is not tied to specific fixed parameters and does not provoke these limb-darkening concerns, and therefore it seems sensible to continue with this basic approach.
Integration of Limb Darkened Profiles {#sec-int1}
-------------------------------------
In order to understand how to write a numerical integration scheme for gravitational lensing light curves, let us first consider the simpler question of one-dimensional numerical integration. There has been a lot of work in this field, and there are a number of numerical integration schemes that can give quite precise results for a small number of integration grid points if the integrand is smooth. A good discussion of these methods is given in @num-rec, and here I reproduce the relevant points.
For most numerical integration problems, the key to an efficient evaluation of the integrals, is to obtain high accuracy with as few evaluations of the integrand function as possible. This is often accomplished by invoking a higher order integration scheme, which means that the error can be expected to scale as a high power of the integration grid spacing, $h$. Of course, high order is no guarantee of high accuracy. A high order scheme can have a large coefficient in front of the error term that can render it less accurate than a lower order scheme at a given grid spacing. Furthermore, in our case, we are considering two dimensional integrals, so correlations between the numerical errors in different rows of one dimensional integration can have a significant effect on the overall accuracy of the integral. That is, there might be correlations that tend to make the numerical errors in different rows add coherently, instead of incoherently so that the relative error would fall as the square root of the number of integration rows. As a result, it is quite difficult to predict the accuracy of a numerical integration scheme with analytic arguments like the ones presented in this section. So, as is usually the case with numerical calculations, it will be the numerical tests of the method that will show which methods are most precise.
The integrands that we are concerned with are not very smooth, so I restrict the discussion to 2nd order integration schemes. For most numerical integration problems, there are two basic building blocks for the 2nd order integration schemes, the trapezoidal rule, $$\int_{x_1}^{x_2} f(x) dx = h\left({{1\over 2}}f_1 + {{1\over 2}}f_2 \right) + O(h^3 f^{\prime\prime}) \ ,
\label{eq-trap}$$ and the mid-point rule, $$\int_{x_{1/2}}^{x_{3/2}} f(x) dx = hf_1 + O(h^3 f^{\prime\prime}) \ .
\label{eq-midp}$$ These are both formulae for evaluating integrals over a single grid spacing, $h$, using values of the function calculated at integer multiples of the grid spacing. The function values are $f_i\equiv f(x_i)$. The error term $O(\ )$ indicates that the true answer differs from the estimate by an amount that is the product of some numerical coefficient times $h^3$ times the value of the second derivative of the function somewhere in the range of integration.
Now, these building block formulae can be strung together to build extended formulae that can be used over finite intervals. This yields the extended trapezoidal rule, $$\int_{x_1}^{x_N} f(x) dx = h\left({{1\over 2}}f_1 + f_2 + f_3 + ... + f_{N-1}
+ {{1\over 2}}f_N \right) + O\left( {(x_N-x_1)^3 f^{\prime\prime}\over N^2}\right) \ ,
\label{eq-trap_ext}$$ and the extended mid-point rule, $$\int_{x_{1/2}}^{x_{N+1/2}} f(x) dx = h\left( f_{1} + f_{2} + ... + f_{N-1}
+ f_{N} \right) + O\left( {(x_{N+1/2}-x_{1/2})^3 f^{\prime\prime} \over N^2}\right) \ .
\label{eq-midp_ext}$$ Usually, the extended mid-point rule (eq. \[eq-midp\_ext\]) is presented using integrand values evaluated at half integer grid points, but we have offset this grid by half a grid spacing so that most of the grid points coincide with those of the extended trapezoidal rule (eq. \[eq-trap\_ext\]). When written this way, the extended mid-point rule , the extended trapezoidal rule and the new integration formula presented below will require that the integrand only be evaluated at integer grid points in the interior. This makes it clear that the only difference between these integration formulae the treatment of the boundary.
One might imagine that we could implement something like eq. \[eq-trap\_ext\] or \[eq-midp\_ext\] in two dimensions give an integration scheme with a precision proportional to the inverse square of the total number of grid points. However, there are two difficulties with this procedure. First is the fact that our problem is slightly different from the normal numerical integration problem, because we calculate the $f_i$ values before we know where the boundary is. Thus, it is impossible for us to arrange that the boundaries are located at integer or half-integer values of the grid spacing. If we are interested in an integration scheme that is only first order accurate, then we only need to determine which points on the grid are inside the image boundary without attempting to locate the boundary. However, for a second order method, we do need to locate the boundary to a precision much greater than the grid spacing. I solve for the position of the boundary using the Brent’s method [@num-rec] to find the boundary to a precision of $0.1h^2$, which $h$ is the grid spacing. This increases the number of lens equation (eq. \[eq-mult\_lens\]) calculations that must be done per row by a factor of less than 1.5.
The second complication is that our integrands are not very smooth. Moving the finite source integration from the source plane to the image plane removes the singularities from the integrand, but for limb darkened sources, there are still singularities in the derivatives of the surface brightness in the image plane, and these will limit our attempt to do these lens magnification integrals efficiently.
The linear limb darkening law of @milne21 gives a reasonable approximation to the limb darkening for most stars. This linear law is given by $$I = I_0 \left[ 1 - c\left(1-\sqrt{1-\rho^2}\right)\right] \ ,
\label{eq-limbd}$$ where $I_0$ is the central intensity, $c$ is the linear limb darkening coefficient, and $\rho$ is the distance from the center toward the limb of the star at $\rho=1$. This limb darkening law has a first derivative that diverges at $\rho=1$. Also, when the source crosses a caustic, the fraction of the stellar profile that is inside the caustic has two additional images that meet on a critical curve in the image plane. These images have opposite parity, and the first derivative of the surface brightness profile will have a discontinuity on the critical curve, so the second derivative will diverge.
We should note that the linear limb darkening model is not a perfect match to model atmospheres, with an average difference of $> 1\,$% [@hey07] from Kurucz’s model atmospheres [@kurucz93a; @kurucz93b; @kurucz93c; @kurucz94]. This discrepancy can be reduced by going to more complicated limb darkening models [@claret2000]. However, the microlensing light curves involve integrals over the limb darkened profiles, and these are generally much more accurate than the limb darkened profiles themselves. Furthermore, there is no guarantee that these models are actually correct, so it is perhaps more sensible to compare with previous well sampled microlensing events that yielded high precision limb darkening measurements. The first such example is event MACHO 95-BLG-30 [@macho-95blg30] which is the first example of an event detected in progress which exhibited finite source effects. They employ a “quadratic" limb darkening model in place of eq. \[eq-limbd\], but due to sparse sampling, the $\chi^2$ improvement with respect to a model without limb darkening was only $\Delta\chi^2 = 9$, so it is likely that the improvement over the linear model is quite small. The high cadence follow-up observations of the PLANET collaboration yielded a number of binary lens caustic crossing events with a much stronger limb darkening signal. For example, PLANET found that limb darkening improved the fit for the binary microlensing event MACHO-97-BLG-28 [@planet-97blg28] by $\Delta\chi^2 = 393$. However, the additional improvement with the “square-root" limb darkening model was only $\Delta\chi^2 = 5$, which they argue is not statistically significant. One of the most spectacular binary microlensing events ever observed was event EROS-2000-BLG-5 [@planet-er2000b5], which had a very extended caustic crossing with a duration of $\sim 4\,$days followed by a cusp approach to within $0.1$ source radii from the stellar limb four days later. These features were measured with hundreds of photometric measurements while the source was magnified to a brightness ranging from $I = 13$ to $I = 15$ using several $1\,$m class telescopes. It is difficult to imagine circumstances that would allow a higher S/N measurement of limb darkening effects. However, the non-linear limb darkening parameters are found to be $< 0.1\sigma$ away from 0. So, this event does not yield a significant measurement of limb darkening parameters beyond the linear term. Similar results were also obtained for the well sampled single lens event OGLE-2004-BLG-254 [@cassan-ogle254].
It is perhaps somewhat more instructive to consider high-magnification events. since these are the events that are the focus of the method presented in this paper. Such events also show only weak evidence that non-linear limb darkening models improve the fits. @moa2002blg33 find marginal evidence for terms beyond the linear term of eq. \[eq-limbd\] for the $V$ band in the MOA-2002-BLG-33 light curve, an event with a very strong caustic crossing at the peak. But, there was no evidence for a term beyond the linear one in the $I$ band data, which dominate the light curve coverage for most events. Similarly, @dong-moa400 and @moa310 find only a modest improvement $\chi^2$ for non-linear limb darkening models planetary events with geometries that should maximize limb darkening effects. More importantly, the more complicated limb darkening models have no significant influence on the non-limb darkening parameters, so there is little reason to consider limb darkening models more complicated than eq. \[eq-limbd\] unless we are specifically interested in the limb darkening parameters.
For high-magnification events, it is the divergent first derivative at the limb that is, by far, the most serious problem. The images are highly extended parallel to the Einstein ring and compressed by a factor of about two in the radial direction, so they have a very large ratio of boundary to area. Thus, errors at the boundary make a large contribution to the total error in the integral. The caustic crossing features, on the other hand, are less singular than the limb darkening profile, and they also are also (usually) much shorter than the entire extent of the limb in the image plane.
While eq. \[eq-limbd\] describes the limb darkening in the source plane, the integrals are carried out in the image plane where the source brightness profile is distorted by the gravitational lens. However, the lowest order behavior near the limb is generically described by $I \sim C + D\sqrt{x}$ where $C$ and $D$ are constants and $x$ is the distance from the limb of the distorted image. The only case where the $\sqrt{x}$ behavior is removed by the lens distortion is when the stellar limb just touches the interior of a caustic. However, this will generally only occur at a single point of contact between the caustic and the limb, so the $\sqrt{x}$ behavior is generic.
The two building block formulae, eqs. \[eq-trap\] and \[eq-midp\], are derived by requiring that they be exact for low order power laws (as in a power series expansion of $f$), and in the second order case, the formulae are exact for $f = {\rm const.}$ and $f = x$. This fails for limb darkened profiles, because these cannot be expressed as a power series in $x-x_L$, where $x_L$ is the location of the limb. Instead, the distorted limb darkening profile can be expressed as a power series in $\sqrt{x-x_L}$. We can still demand that the our integration formula is exact for the two leading orders in the power series expansion of the integrand. In eqs. \[eq-trap\] and \[eq-midp\], the first term in the power series that does not vanish scales as $h^3$ (under the assumption that $f$ can be expanded in a power series in $x$). However, with half-integer powers of $h$ in addition to integer powers, there are more terms in the power series expansion. As a result, the first non-vanishing term in eqs. \[eq-trap\] and \[eq-midp\] scales as $h^{3/2}$ instead of $h^3$ when $f(x)$ has a limb darkened form like eq. \[eq-limbd\]. In order to cancel this $h^{3/2}$ error term, we will demand that our integration formula be exact for $f = {\rm const.}$ and $f = \sqrt{x-x_L}$, where $x_L$ is the location of the limb. This requirements lead us to replace eq. \[eq-midp\] by $$\int_{x_L}^{x_{3/2}} f(x) dx = h\left({{1\over 2}}+\delta\right)\left[(1-b)f_L + bf_1\right] \ ,
\label{eq-intlimb}$$ where $$b = {2\over 3} \sqrt{\delta + {{1\over 2}}\over \delta} \ ,
\label{eq-intlimbb}$$ and $\delta = (x_1-x_L)/h$. The $\delta$ in the numerator of eq. \[eq-intlimbb\] is somewhat worrisome because $\delta$ can become very small if the limb happens to come very close to a grid point. Conceivably, this could lead to a situation, where the error grows very large, even if it is formally of high order. Therefore, we introduce another parameter, $\delta_c$, such that eq. \[eq-intlimb\] is only invoked for $\delta \geq \delta_c$. When $\delta < \delta_c$, we invoke a standard “second order" method that will be converted to 1.5 order by the singular derivative at the limb. Any combination of $cf_L + df_1$ will satisfy this criteria as long as $c + d = 1/2 + \delta$. Experimentation with different $c$ and $d$ values indicates that $c = \delta_i/3$ and $d = 2\delta_i/3 + 1/2$ is a good choice, so it is used below. We will investigate the effect of this $\delta_c$ parameter in Section \[sec-lc\_calc\]. We can now write an extended numerical integration rule to take the place of eq. \[eq-midp\_ext\], $$\int_{x_{L1}}^{x_{L2}} f(x) dx = h\left( A_1 f_{L1} + B_1 f_{1} + f_{2} + ... + f_{N-1}
+ B_2 f_{N} + A_2 f_{L2} \right) \ ,
\label{eq-int_rule}$$ where $L1$ and $L2$ refer to the the stellar limbs at each limit of the $x$ coordinate integral, and the $A_i$ and $B_i$ coefficients are given by $$\begin{aligned}
A_i = \left({{1\over 2}}+ \delta_i \right)\left(1-b_i \right) \Theta (\delta_i-\delta_c)
+ {\delta_i \over 3} \Theta(\delta_c - \delta_i) \ , \nonumber \\
B_i = \left({{1\over 2}}+ \delta_i \right) b_i \Theta (\delta_i-\delta_c)
+ \left({2\over 3}\delta + {{1\over 2}}\right) \Theta(\delta_c - \delta_i) \ ,
\label{eq-AB}\end{aligned}$$ where $\Theta$ is the Heavyside step function and $\delta_i$ and $b_i$ refer to $\delta$ and $b$ for the each of the two stellar limbs (at $i = 1\,$, 2) on the image being integrated.
If we set $\delta_c = 0$, then eq. \[eq-int\_rule\] is accurate to second order, even though eq. \[eq-intlimb\] has a non-vanishing $h^2$ error term. The reason for this is because we only invoke eq. \[eq-int\_rule\] at the limbs, and we use eq. \[eq-midp\] for all the interior points. Since the limb darkened profile does have a power series expansion in $x$ away from the limb, the error for eq. \[eq-midp\] does scale as $h^3$ in the interior (except on a critical curve, where it has a $h^{5/2}$ contribution). Thus, it is only the $h^3$ error terms that get a $1/h$ contribution from the sum. So, formally, eq. \[eq-int\_rule\] is second order accurate (with $\delta_c = 0$), while eqs. \[eq-trap\_ext\] and \[eq-midp\_ext\] are only accurate to the three halves order for a limb darkened source. Of course, with $\delta_c > 0$, eq. \[eq-int\_rule\] also gains an error term that scales as $h^{3/2}$, but as we shall see, in some cases, even with $\delta_c > 0$, eq. \[eq-int\_rule\] can yield second order accurate results. In all cases, eq. \[eq-int\_rule\] with $\delta_c \sim 0.15$ is substantially more accurate than the first order or $\delta_c = 1.0$ calculations.
It is possible to derive integration formulae that are more complicated than eqs \[eq-int\_rule\] and \[eq-AB\] that are formally 2nd order accurate without the problem of any of the coefficients growing unreasonably large for any position of the boundary with respect to the grid spacing. However, experimentation with a number of such integration formulae has not found any such integration scheme that gives results as accurate as the scheme represented by eqs \[eq-int\_rule\] and \[eq-AB\].
Two-Dimensional Ray Shooting Integration {#sec-int2d}
----------------------------------------
In Section \[sec-int1\] we developed a one-dimensional numerical integration rule, eq. \[eq-int\_rule\], which is designed to improve the accuracy of the integration of limb darkened source profiles. But, of course, we will need to do two dimensional integrals to determine microlensing magnifications. The integral in the second dimension is not subject to the divergent integrand derivative at the boundary, because this is removed by the integral in the first direction. (Of course, the limb darkening has the same behavior in both directions. But the integral in the first direction is roughly proportion to the length of the row being integrated, and this generally goes to zero at the boundaries of the integral in the second direction.) But, we still must deal with the arbitrary location of the image boundary. I employ the following second order accurate formula for this integration $$\int_{y_L}^{y_{5/2}} F(y)dy = h\left[\left({3\over 8}+\eta + {\eta^2\over 2}\right)F_1
+ \left({9\over 8}- {\eta^2\over 2}\right)F_2\right] \ ,
\label{eq-inty}$$ where $\eta = (y_1-y_L)/h$ and $F(y)$ refers to the integral over the $x$ direction, which has a $y$ dependence that is not made explicit in eq. \[eq-int\_rule\]. With eqs. \[eq-int\_rule\] and \[eq-inty\] to handle the numerical integrations, we can now consider the coordinate system to use for the integrations. In this context, it is useful to consider the image geometry for high-magnification microlensing events. Consider a typical high-magnification event with a magnification of $A = 200$. If there are no companion planets or stars, then there will be two lensed images. The major image will have the shape of a circular arc with a magnification of $A_{\rm maj} = 100.5$, and it will be located just outside the Einstein ring. The minor image will be just inside the Einstein ring on the opposite side of the lens from the major image, and its magnification will be $A_{\rm min} = 99.5$. Each image will be compressed in the radial direction by about a factor of two, so the images will have the form of long, skinny arcs with a length-to-width ratio of about 200. Thus, the limb darkening profile will vary 200 times more rapidly in the radial direction than in the angular direction. This strong distortion of the images suggests that a polar coordinate grid is most appropriate for our problem, and it seems likely that we will require a much larger grid spacing in the angular direction than in the radial direction. In fact, the 200:1 distortion of the images for our example would seem to suggest that a 200:1 grid spacing ratio might be appropriate.
However, we must also consider the effect of the planetary lenses that are the primary motivation for observing high-magnification microlensing events. The planetary lenses will distort the single lens images, and if there are caustic crossings, new images will be produced that will not follow the Einstein ring as closely as the images that are not significantly influenced by the planet. So, during the planetary deviations, this image stretching in the angular direction may not be quite as severe as in our example. However, it is this image stretching in the angular direction that is responsible for the high magnification of these events, so we should expect that the optimal integration grid should include some extension in the angular direction.
Following the discussion above, I have arrived at the following two-dimentional integration strategy. An integration grid is set up in polar coordinates ($r$,$\theta$) with a larger grid spacing in the angular than in the radial direction. In Section \[sec-lc\_calc\], we study the effects of varying this axis ratio. The integration is done using using eq. \[eq-int\_rule\] in the radial direction with a fixed value of $\delta_c$. The integrand is given by the value of the limb darkening profile at the integration point, times $r$, to give the proper polar coordinate area element. However, the integration is done in the image plane, while the limb darkening is known in the source plane. Thus, we must apply the lens equation, eq. \[eq-mult\_lens\], to determine the appropriate source plane point and the limb darkened surface brightness that corresponds to the integration grid point in the image plane. The dependence of the light curve calculation precision on the grid size, the angular vs. radial grid spacing ratio, and on $\delta_c$ is investigated in Section \[sec-lc\_calc\].
Locating and Building the Integration Grids {#sec-grid}
-------------------------------------------
Since we don’t know the extend of the images when we start to calculate their magnification, we require a scheme to build an integration grid that covers each image, or at least each image that requires a finite source calculation. It most efficient to have integration grids that don’t extend far beyond the images because rays are shot from the image plane to the source plane at every grid point.
The method of @em_planet was to build a rectangular grid centered on each point source image, and to add rows and columns to each integration grid until the rows and columns at the boundaries of each do not contain any grid points inside an image. With Cartesian coordinates, this method is quite inefficient for high-magnification events, because the grid must cover almost the entire Einstein ring disk to integrate over thin images arcs that spread out over much of the Einstein ring. Polar coordinates are much more efficient in this case. Cartesian coordinates may be more efficient for low-magnification events. But the light curves of these events are less time consuming, so computational efficiency is less important. The complication of using different grid geometries for different events does not seem justified by the very minor improvement in efficiency for low magnification events with a Cartesian grid.
A somewhat more efficient method is to build the grid row-by-row, with each row extending just as far as the image does, an approach first implemented by @vermaak00. The grid is then extended row-by-row until we have a grid that is surrounded by a boundary of grid points that are outside of the image. Our tests have shown that this method is typically about a factor of two faster than building grids that are “rectangular" in polar coordinates, and it is this method that is used for our timing results presented in Section \[sec-global-ex\]. In both cases, one must check that no images are double-counted.
In addition to building integration grids around at the position of the point source images (when the point-source approximation cannot be used), we must also ensure that images associated with caustic crossing are included when the center of the source is outside the caustic. For static lens systems, this is most efficiently done by simply calculating the caustic curve location and building a grid at the location of any caustic point that is not included in another integration grid. For lens systems with orbital motion, this method can become inefficient because the caustics move and must be recalculated at every time step. Another method that can also be used is to calculate the number of images for source points on the boundary of the source. Then, a grid can be built at the location of any new image near the image boundary.
Finally, for static lens systems it is possible to speed up the calculations by storing the source positions corresponding to image positions in an annulus centered on the Einstein ring. This avoids the need to recalculate the lens equation, eq. \[eq-mult\_lens\], for the same image points for the magnification integration at different points on the light curve. This same optimization method is used more extensively in inverse ray-shooting [@wamb] and magnification map [@ratt; @dong-ogle343; @dong-moa400] methods.
What Light Curve Calculation Precision Is Necessary? {#sec-what_acc}
----------------------------------------------------
In Section \[sec-lc\_calc\], I will present the results of light curve calculation tests with different values of the grid spacing and different calculation parameters, but first, it will be helpful to consider how much light of light curve precision is needed for practical modeling calculations. In Section \[sec-int1\], we discussed the difference between the simple linear limb darkening models and more complicated models that do a better job of reproducing the limb darkening seen in stellar atmosphere calculations as well as a few microlensing events which have good light curve coverage at limb crossings. However, even these improved models have deviations from the stellar models that are a factor of a few smaller than the deviation from the linear model. Thus, we might still expect light curve errors at the level of ${\lower.5ex\hbox{{$\; \buildrel <\over\sim \;$}}}0.1$% at the limb crossings if these improved limb darkening models, are used, compared to errors of perhaps $\sim 0.3$% at the limb crossing with the linear model. These errors of ${\lower.5ex\hbox{{$\; \buildrel <\over\sim \;$}}}0.1$-0.3% are comparable to the level of systematic photometry errors that we expect in the microlensing light curves. These systematic errors are expected to affect the entire light curve, instead of just the limb crossings, although they are also likely to have large correlations that may allow some light curve features, such as weak caustic crossings, to be measured with a precision of $\sim 0.1$% [@ogle169].
These arguments might suggest that there is no need to calculate the light curves to a precision better than 0.1%, but in fact, it is usually the case that higher precision is needed. The reason for this is that the @metrop algorithm that is generally used for modeling multiple lens light curves is able to optimize the numerical errors, so that they tend to minimize $\chi^2$. If the light curves were calculated perfectly, the $\chi^2$ surface should usually be smooth over small distances in parameter space, and the main difficulty in finding the $\chi^2$ minima is to follow the steep and twisting valleys in $\chi^2$ to the local minimum. However, the numerical errors act to roughen the $\chi^2$ surface on extremely small scales. If the numerical calculation errors are similar to the photometric error bars at the light curve peak, then RMS variation in $\chi^2$ would be similar to the number of data points at the peak, which varies between events, but can often be 50 or more. But with parameters chosen by a $\chi^2$ minimization scheme, we might expect variations several times larger than this. Because of this, @dong-ogle343 advocate that the numerical precision of the light curve calculations be less than one third of the size of the error bars. However, numerical errors that are this large can still cause some difficulty. Since the modeling code tends to select parameters that allow the numerical error to minimize $\chi^2$, the variation in the $\chi^2$ seen during a modeling run tends to be much larger than the RMS value. Therefore, I recommend that random numerical errors be kept at ${\lower.5ex\hbox{{$\; \buildrel <\over\sim \;$}}}10^{-4}$ or at least ten times smaller than the smallest photometric error bars to avoid difficulties in locating local $\chi^2$ minima due to the roughness of the $\chi^2$ surface.
Light Curve Calculation Tests {#sec-lc_calc}
=============================
In order to determine the optimum light curve calculation parameters, I compare light curves seven different sets of model parameters, which are based on models of observed events. Three of these are relatively low magnification events, which are shown in Figure \[fig-lc\_lo\]. These are OGLE-2003-BLG-235, the first definitive planetary microlensing event [@bond-moa53], OGLE-2005-BLG-390 with a planet of $\sim 5{{M_\oplus}}$ [@ogle390], and MOA-2007-BLG-197, which has a brown dwarf secondary (Cassan, et al. in preparation).
The other four comparison events are high-magnification events, shown in Figure \[fig-lc\_hi\]. These include both cusp approach and caustic crossing models for MOA-2007-BLG-192 [@bennett-moa192], which includes a planet of $\sim 3\,{{M_\oplus}}$, OGLE-2005-BLG-169 [@ogle169], and MOA-2008-BLG-310 [@moa310]. In both of these figures, the red boxes indicate the regions of the light curves used in tests of light curve calculation precision. It is important only to use regions of the light curve where finite source calculations are done. Otherwise, the comparison of different integration parameters will be diluted by regions where the point source approximation is used (and the light curves are identical).
Note that these light curve calculation tests do not always use the published version of the data set for each event, so the resulting parameters will sometimes differ slightly from the published ones. In every case, the values published in the discovery or follow-up analysis papers should be considered definitive.
Figure \[fig-compare\] shows a comparison of the RMS fractional precision, $\sigma$, as a function of the geometric mean grid spacing, in units of the source star radius, for the 3 low-magnification events in the top two panels and the 4 high-magnification events in the bottom four panels. (The geometric mean is the square root of the product of the angular and radial grid spacings.) The short-dashed curves have $\delta_c = 1.00$, so that treatment of the limb-darkening profile is avoided, and the long-dashed black curve is is a first order integration with no attempt to locate the image boundaries on sub-grid-spacing scales. The blue, green, and red curves have $\delta_c = 0.017$, 0.05, and 0.15, respectively. All the curves use a angular grid spacing 4 times larger than the radial spacing, except for the black-dashed curves, which uses equal grid spacings. (In discussions of the grid spacing, we refer to the ratio for the grids at the Einstein ring radius. For the high-magnification events, this is very nearly the exact ratio, since the images are quite close to the Einstein ring, but for some low magnification events, like OGLE-05-390, the planetary deviation images are well outside the Einstein ring, so the actual grid spacing ratio is somewhat larger.) Note that for the low magnification events, the blue and green curves are often hidden under the red curve.
These comparisons are done with respect to calculations using $\delta_c = 0.15$ and an extremely fine grid, with 800 grid points per source radius in both the radial and angular directions. I use the RMS fractional deviation, $\sigma$, between these very high resolution calculations and the test calculations as the measure of the calculation precision. The maximum deviation is generally between 2 and $4\times\sigma$, so there appears to be no significant non-Gaussian tail in the error distribution.
The curves in Figure \[fig-compare\] mostly have a similar slope, which is close to the $h^{3/2}$ slope that was predicted in Section \[sec-int1\]. However, for the high-magnification events with $\delta_c = 0.15$, the RMS precision scales as $\sigma\sim h^2$. This may seem slightly surprising because it is only in the $\delta_c \rightarrow 0$ limit, where eqs. \[eq-int\_rule\] and \[eq-AB\] achieve second order accuracy. However, this analysis only applies to a single one-dimensional integral, and a full treatment of the accuracy of the two-dimensional integral must include a number of complications, such as correlations in the error terms for integrations over different rows of the two-dimensional domain of integration. Also, it is always possible for the integration accuracy to scale as a higher power of $h$ than expected, because the coefficient of the leading order term could be so small that a sub-leading term will dominate over the interesting range of $h$ values. Of course, this is more likely in situations, such as these limb-darkened source integrals, where the error terms are power laws in $h^{1/2}$ instead of in $h$. I expect that this is what has happened for the high-magnification events with $\delta_c = 0.15$, which have a $\sigma \sim h^2$ scaling despite the fact that the arguments presented in Section \[sec-int1\] suggest that the scaling should be $\sigma \sim h^{3/2}$. Thus, the arguments presented in Section \[sec-int1\] and \[sec-int2d\] should be considered to be only qualitative, and the comparison with much higher resolution calculations should be considered to be the definitive measure of the numerical errors.
The long-dashed black curves indicate that the first order calculations seem to do as well as, and often better than the short-dashed black curves, which represent an attempt at a second order correction without the limb darkening terms given in eqs. \[eq-int\_rule\] and \[eq-AB\]. This might seem somewhat surprising, since the analysis in Section \[sec-int1\] indicates that the one-dimensional integrals without the limb darkening correction should have errors that scale as $h^{3/2}$, whereas one-dimensional first order integration methods have errors that scale as $h$. @dong-ogle343 have also noted a $h^{3/2}$ error scaling in calculations with their method, which is also first order. In Appendix A.3 of this paper, they note the improvement from the $\sim h$ error term of the first order one dimensional integral to the $\sim h^{3/2}$ error term observed in the two dimensional integrals, and they attribute this factor of $h^{1/2}$ improvement in the fractional error to the $1/N^{1/2}$ Poisson decrease expected if the errors in each row are uncorrelated (where $N$ is the number of rows). This assumption of uncorrelated errors is plausible for a first order integration scheme, but it seems unlikely that a higher order scheme would also achieve this $h^{1/2}$ improvement when going from one to two dimensions. So, this might explain why the first order integration scheme has the same $h^{3/2}$ behavior as many of the attempted second order integration schemes. The first order schemes also have the minor advantage that they don’t require additional lens equation (eq. \[eq-mult\_lens\]) calculations to locate the boundaries on a scale smaller than the grid spacing, which implies a savings of a factor of up to 1.5 in computation time.
The numerical errors for the OGLE-05-390 calculation are significantly larger than for other events, but this is easily explained by the details of this event. It is the only event we consider that has a giant source star. In fact, the planetary caustic responsible for this planet detection [@ogle390] has a diameter that is 4-5 times smaller than the radius of the source, so the caustic crossing regions of the images are sampled more coarsely than the other events for the same grid size to source radius ratio.
For the high-magnification events, there is also a clear improvement from increasing the ratio of the angular to radial grid size from 1 to 4. This effect is demonstrated even more clearly by Figure \[fig-grid\_ratio\], which shows the effect of changing the grid size ratio from 1 to 4 to 16. The high-magnification event calculations show a clear improvement from the larger ratio of grid sizes, whereas the lower magnification events show the opposite effect (with the exception of OGLE-03-235).
In Figure \[fig-improve\], we show the factor by which $\sigma$ is improved compared to the first order calculations with an angular:radial grid size ratio of 1. In every case, the second order scheme with $\delta_c = 0.15$ is the most accurate, although the improvement is modest for the low magnification events. (Note that the blue and green curves for the low magnification events are often hidden under the red curve.) The low magnification events also benefit from having an angular:radial grid size ratio $= 1$. However, there is a dramatic improvement for the high-magnification events. The improvement ranges from a factor of 10 to a factor of nearly 300 at some grid sizes. The calculations with $\delta_c = 0.15$ and an angular:radial grid size ratio of 16 prove to be the most accurate for the high-magnification events. These calculations, represented by the cyan curves in Figure \[fig-improve\], provide a factor of 100-300 improvement in precision over the first order calculation case. If we were to try to reproduce the factor of $\sim 100$ improvement seen at the mean grid size of 0.1 by simply decreasing the grid size of the $\delta_c = 1$, grid ratio $= 1$ calculations, we would have to drop the grid size by a factor of 22. But since this is a 2-dimensional calculation, this means an increase in the number of calculations and hence the computing time by nearly a factor of 500.
So, in summary, it would seem that the new features that we have outlined have the potential to increase the computational efficiency of high-magnification event light curve calculations by a factor of several hundred.
$\chi^2$ Minimization Recipe {#sec-markov}
============================
In Section \[sec-int\], I developed an efficient method for the calculation of high-magnification planetary microlensing light curves, but we also require an efficient method to move through parameter space to find the $\chi^2$ minimum. Due to sharp light curve features like caustic crossings and cusp approaches, the $\chi^2$ surface for microlensing event models is not smooth enough to use a method, like Levenberg-Marquardt, that make use of the assumed smoothness of the $\chi^2$ surface. Instead, we must use a more robust method, similar to the Markov Chain Monte Carlo (MCMC) [@wmap_mcmc] or the simulated annealing method [@sim_anneal], which are both based on the @metrop algorithm. The Metropolis algorithm employs the Boltzmann factor from statistical physics to decide whether or not to accept the next proposed step through parameter space. If the next proposed step reduces $\chi^2$, ([[i.e.]{}]{} $\Delta\chi^2 \leq 0$), then it is always accepted, but if $\Delta\chi^2 > 0$, then it is accepted with probability, $e^{-\Delta\chi^2/(2T)}$, where the parameter $T$ is referred to as the temperature, in analogy with statistical physics.
Ideally, we would like to have a scheme that can find the global $\chi^2$ minimum automatically without having the specify any particular initial condition. In fact, the simulated annealing method was developed to find the global minimum in situations where there are many local minima. The basic idea is to start the method with a high temperature, $T$, in order to explore all of parameter space, and then to gradually decrease $T$ and allow the system to relax to the global $\chi^2$ minimum. While this method has been used to solve a number of difficult problems, it is not clear that it will work for planetary microlensing events. For some events, it might end up in a broad local minimum that is favored at high $T$ that is separated by a $\chi^2$ barrier from a narrow, but deeper global minimum. Also, it is unclear how one would design a schedule for modifying the temperature that would ensure the efficient location of the global $\chi^2$ for the observed wide variety of planetary microlensing events. Therefore, I do not attempt to use the Metropolis algorithm to find the global $\chi^2$ minimum.
A very important aspect of the Metropolis algorithm is the choice of the jump function. The jump function starts from the current set of model parameters and selects a new set of model parameters to be used to calculate $\chi^2$, which will yield a value of $e^{-\Delta\chi^2/(2T)}$ that will allow us to determine whether to move to this next set of model parameters.
As mentioned above, microlensing light curves are characterized by very sharp features due to caustic crossings and cusp approaches, so we expect that the $\chi^2$ surfaces we encounter will have steep valleys. For example, for a caustic crossing event, most directions in parameter space will cause the timing of a caustic crossing to change, which will induce a large change in $\chi^2$ if the photometric measurements sample the caustic very well. But there will also be some directions in parameter space that will leave the timing of the caustic crossing fixed. These will induce much smaller changes in $\chi^2$, so these will be the directions of the valleys in $\chi^2$ space. An efficient method of locating and exploiting these $\chi^2$ valleys has been presented by @cmbeasy. This involves calculating the correlation matrix, $$C_{ij} = \VEV{p_i p_j} = {1\over N} \sum_{k=1}^N p_i p_j \ ,
\label{eq-Cij}$$ of the last $N$ (accepted) sets of parameters, $\{p_i\}$. $C_{ij}$ is then diagonalized, and the diagonalized basis vectors can be considered to be a new set of parameters that are uncorrelated over these $N$ steps through parameter space. We then select new parameters at random from the most recently accepted parameter set with a Gaussian variance normalized by the elements of the diagonalized covariance matrix.
Of course, this scheme cannot be used until after $N$ steps have been accepted, so jump function scheme is needed for the first $N$ steps. For these initial steps, I specify initial uncertainty ranges for each parameter, and select new parameters with uniform probability within these ranges.
The use of the Metropolis algorithm in the way I have described is often referred to as a Markov Chain Monte Carlo (MCMC), with each accepted step considered a link in the chain. These MCMC runs can be used to estimate the parameter uncertainties, but this requires that the jump function be fixed during the MCMC run. But, such a strategy is not efficient when searching for a $\chi^2$ minimum, because the $\chi^2$ valleys often have many twists and turns, so the minimum is reached much more quickly when the jump function can be modified frequently.
I have found that the following recipe allows the Metropolis algorithm to quickly converge to a local $\chi^2$ minimum for a wide variety of planetary and binary microlensing events. The initial jump function is used until $N = \max(20,2N_{\rm par})$ steps have been taken, where $N_{\rm par}$ is the number of non-linear fit parameters. (Note that if we were to calculate $C_{ij}$ with $N < N_{\rm par}$, we would have a singular matrix, since we would not have enough points to span the $N_{\rm par}$-dimensional parameter space.) Then, the parameter correlation matrix, $C_{ij}$, is calculated and diagonalized. The new parameters are generated from the most recent step with a Gaussian probability distribution following the diagonalized parameter correlation matrix. Of course, these new parameters must be converted back to the original non-diagonal parameters for the light curve calculation and $\chi^2$ evaluation. The parameter correlation matrix, $C_{ij}$, is recalculated and diagonalized whenever the number of saved steps, $N$, increases by 4, until $N$ reaches 100. Once $N = 100$, the oldest saved parameter set is dropped each time a new one is added, so that the number of saved parameter sets to be used for $C_{ij}$ calculations remains fixed at $N = 100$, but the $C_{ij}$, is still recalculated and diagonalized every 4th time that a new parameter set is accepted.
This procedure allows the parameter correlation matrix to gradually adjust to twists and turns in the $\chi^2$ surface, as the modeling code travels toward the local $\chi^2$ minimum. However, sometimes this gradual modification of the parameter correlation matrix is not sufficient to keep up with the changing $\chi^2$ surface shape, and so I also have a procedure for modifying $C_{ij}$ more drastically. If the code attempts 40 consecutive parameter sets without a single one being accepted due to an improvement in $\chi^2$ or passing the Boltzmann probability test, then the oldest $3/8$ of the parameter sets are dropped, and if the $N \geq \max(20,2N_{\rm par})$ condition still holds, then $C_{ij}$ is recalculated, diagonalized and used to select the next set of parameters. If $N < \max(20,2N_{\rm par})$, then we revert to the initial procedure of selecting new parameters with uniform probability within the initially specified uncertainty ranges. Sometimes the reduction in the number of saved parameter sets, $N$, will not be sufficient to allow a new parameter set to be accepted in the next 40 steps, and in these cases, the number of saved parameter sets is again reduced by a factor of $5/8$. This procedure can even be invoked four times in a row to drop $N$ from 100 down to 14.
This $\chi^2$ minimization recipe has been extensively tested and has been shown to be robust and efficient for finding the local $\chi^2$ minima for a wide variety of microlensing events including all published planetary microlensing events and the four clear planet detections from the 2009 bulge observing season, as well as the orbiting two planet system, OGLE-2006-BLG-109 [@gaudi-ogle109; @bennett-ogle109]. Some examples are discussed in Section \[sec-global-ex\]. Note that this procedure of modifying the jump function should not be used for a Markov chain calculation that might be used to estimate parameter uncertainties.
Global Fit Strategy {#sec-global}
===================
In addition to a method for calculating planetary microlensing event light curves, we also need methods for efficiently moving through parameter space to find the best fit or fits (as there are sometimes degeneracies). For events with only two detectable lens masses and no orbital motion, this is fairly straight forward, and a number of methods have been demonstrated to work. For low magnification planetary events like OGLE-2005-BLG-390 (Beaulieu et al. 2006), it is possible to determine most of the parameters approximately by inspection of the light curve. For some high-magnification events, such as OGLE-2005-BLG-71 (Udalski et al. 2005), it is also possible to determine the parameters approximately by inspection, but it is more prudent to do a systematic search for solutions. Perhaps the best documented method is the grid search method of @dong-ogle343 [@dong-moa400]. In this method, the best fit is found for each point on a three dimensional grid over the mass ratio, $q$, lens separation, $d$, and angle, $\theta$, between the source trajectory and the lens axis. For each point on this grid, the remaining parameters, are adjusted using a standard fitting algorithm to find the $\chi^2$ minimum for the fixed values of $q$, $d$, and $\theta$. This method generally works quite well, although it can fail in certain instances, such as the case of MOA-2007-BLG-192, where there is a degeneracy involving the source star radius parameter, $t_\ast$, which is usually not one of the grid parameters. However, this problem is a result of the sparse light curve sampling for this particular event, and it seems likely that the magnification map method can be modified to model this event.
A more serious issue with this grid search method is that it is impractical to scale it up to systems with more parameters. If we add another lens mass, this adds three new parameters (the mass ratio and 2-d position of the additional mass), to the two parameters (the separation, $d$, and mass ratio, $q$) that are normally held fixed on the grid. If all five of these parameters are not held fixed, then the computational advantage of the inverse ray shooting method is lost because the same rays cannot be used throughout the calculation. But it is probably too computationally expensive to have more than three grid parameters. In some cases, it is possible to find an approximate solution using a simplified model with fewer parameters, and then to consider perturbations to this simplified model to find the full solution. This allows the grid search method to be used sequentially in stages, first to find the simplified model, and then to search for the perturbation solutions. This general strategy has proven to be quite useful for features such as microlensing parallax and lens orbital motion, as these usually produce only small light curve perturbations. It has also proven to be effective for some triple lens models. However, this method will not work for all triple lens events. Similarly, orbital motion also threatens to derail the computational advantage of this method, although some strategies to deal with such problems have been suggested [@gould-hex]. Thus, if we want a general method to model complicated lens systems, the approach I present here seems more promising.
I have developed the following method, which has been successfully tested on virtually all of the planetary microlensing events observed to date. First we identify the parameters which are obviously well constrained by the light curve. Typically, this would be the Einstein ring crossing time, $t_E$, the time, $t_0$, and distance, $u_0$, of lens-source closest approach, the impact parameter, and the source radius crossing time, $t_\ast$. For events with strong caustic crossings, it may be best to use one or two of the caustic-limb crossing times in place of $t_0$ and/or $t_E$. This is similar to the approach advocated by @cassan-meth, but it is simpler in that only the time variables $t_0$ and $t_E$ are modified. We then set up a coarse grid over the remaining parameters, and evaluate $\chi^2$ for all the grid points. The parameters that yield the best few $\chi^2$ values are then selected as initial conditions for fitting using a modified Markov Chain Monte Carlo (MCMC) routine. This procedure is repeated with the next best $\chi^2$ values from the initial grid search until we find that most of the fits are converging to the same final models. If the fits converge instead to different models with increasingly worse $\chi^2$ values, then we repeat this procedure with a denser initial conditions grid. This procedure still uses a grid for the initial conditions, but the modeling runs allow all the parameters to vary. Most of the computations for this method are done during these full modeling runs instead of the initial condition calculations. As a result, the computation time does not increase so dramatically with the number of model parameters.
Global Fit Strategy Examples {#sec-global-ex}
============================
In this section, I present several examples that demonstrate how the initial condition grid search method works. These examples are intended to show how to find an approximate solution for each event. These approximate solutions usually do not include all the solutions related by the well known light curve degeneracies, such as the $d \leftrightarrow 1/d$ degeneracy for high-magnification events [@dominik99]. I include one example that has not been used for our previous light curve calculation tests, OGLE-2005-BLG-71. The light curve for this event can be modeled reasonably well without the inclusion of a finite source [@ogle71]. Of course, the source must have a finite size, and the finite source effect is important for the complete analysis, which is able to determine the star and planet masses [@dong_ogle71].
This section does include the three high-magnification events that have been used in the light curve calculation tests, but of the low-magnification events used for the light curve calculation tests, only OGLE-2003-BLG-235 is included. No systematic effort to search for the correct light curve model is needed for OGLE-2005-BLG-390, because it is possible to get a very good estimate of the parameters by inspection. The single lens parameters are well determined by a single lens fit, and the single lens magnification at the time of the planetary deviation determines the separation. The shape of the deviation indicates a major image perturbation. This information is sufficient to specify initial parameters that will lead to the correct solution.
The other event of modest magnification that is not discussed in this section is MOA-2007-BLG-197. This event is not included because the primary analysis [@cassan-moa197] is not yet complete.
For all calculations in this section, we use the second order integration scheme given in eqs. \[eq-int\_rule\] and \[eq-AB\], with $\delta_c = 0.15$. The angular to radial grid-spacing ratio is adjusted to optimize the calculations based upon the characteristics of each individual event.
OGLE-2005-BLG-71 {#sec-ogle-71}
----------------
This event [@ogle71] is an $A_{\rm max} \approx 42$ event with a strong central caustic, cusp approach deviation due to a massive planet with mass ratio of $q = 7\times 10^{-3}$. The source passes on the side of the primary opposite the location of the planet, and so it approaches the two strong central caustic cusps. The interval between these cusp approaches is three days, and the real-time detection of the planetary signal plus good weather allowed OGLE to obtain good light curve coverage over the entire planetary deviation. The OGLE coverage is good enough to pin down the basic planetary parameters, so we include only the OGLE data in our search for the correct planetary model.
Based on a single lens fit to the OGLE data with the planetary deviation removed, we set $t_E = 80\,$days, $u_0 = 0.025$, and $t_0 = 3481.0\,$days (${\rm JD}-2450000$). This event has no obvious finite source features, so I fix the source radius crossing time, $t_\ast = 0$, and search for point-source models. (Finite source effects are [@dong_ogle71] detected in the full analysis of the data, but their inclusion does not significantly modify the other parameters.)
These cusp approach events are among the easiest planetary events to model as the fitting code will converge to the correct solution from a large range of initial condition parameters. So, I set the initial star-planet separation to $d = 0.7$, and the planetary mass fraction to either $\epsilon_1 = q/(1+q) = 5\times 10^{-3}$ or $10^{-2}$, and then scan over $\theta = 225^\circ ... 315^\circ$ at a $1^\circ$ interval. A range of only $90^\circ$ is needed for $\theta$ because we know by inspection the approximate source trajectory. The fitting code is then started at the parameters that yield the best initial $\chi^2$ value for each initial $\epsilon_1$ (or $q$) value. The runs for both initial $\epsilon_1$ values converge to essentially the same model with $\chi^2 = 280.18$ for 305 data points with $t_E = 70.96\,$days $t_0 = 3480.6683\,$days, $u_0 = 0.02352$, $d = 0.7626$, $\theta = 266.3^\circ$, $\epsilon_1 = 6.79\times 10^{-3}$ (or $q = 6.84\times 10^{-3}$). There is, of course, also a solution with $d \approx 1/0.7626 = 1.311$, that can easily be found using this solution plus the substitution $d \rightarrow 1/d$ as an initial condition.
Because these runs start far from the final solution, we find that it is most efficient to start at a high Metropolis algorithm temperature, $T$. I then reduce $T$ several times during the fit run. For this event, I started at $T = 50$, and then dropped it to 5, 0.5, and 0.05 to reach the final solution, which was reached after 110,559 $\chi^2$ calculations from the $\epsilon_1 = 5\times 10^{-3}$ starting point. The run starting from $\epsilon_1 = 10^{-2}$ required 393,328 $\chi^2$ calculations to approach this same solution. Despite the large number of $\chi^2$ calculations required, the entire solution search is fast because point-source calculations were used. The total calculation took less than 22 cpu minutes on a single cpu of a 3 GHz Quad-Core Intel Xeon processor (in a MacPro computer purchased in 2007 running Mac OS 10.5). The search over the initial condition grid took less than a cpu second.
OGLE-2003-BLG-235 {#sec-moa53}
-----------------
OGLE-2003-BLG-235Lb was the first definitive exoplanet discovery by microlensing [@bond-moa53]. This event reveals a giant planet with a mass ratio of $q = 3.9 \times 10^{-3}$ via a caustic crossing binary lens feature in an event with a modest stellar magnification of $A_{\rm max} \simeq 7.6$. While it is often the case that the basic parameters for these lower magnification events can be found by inspection, in this case, we have a so-called “resonant" caustic with $d \sim 1$, so that the planetary caustic is connected to the central caustic. In such cases, the caustics are weak, and it can be difficult to locate the caustic crossings if they are not directly observed. In the case of OGLE-2003-BLG-235, only the second caustic crossing was observed, so there is some uncertainty in the timing of the first caustic crossing.
In order to find candidate solutions for caustic crossing events, it is most efficient to change variables from $t_0$ and $t_E$ to the times of the first and second caustic crossings, $t_{cc1}$ and $t_{cc2} $. This is somewhat similar to, but simpler than, the scheme of @cassan-meth. The calculations for this event were done with a angular to radial grid spacing ratio of 4 although Figure \[fig-grid\_ratio\] indicates that a ratio of 1 would be more efficient. The mean grid spacing was 0.08 stellar radii.
For OGLE-2003-BLG-235, we fix the following parameters for the initial condition grid calculation: the second caustic crossing time, $t_{cc2} = 2842.04$ and $u_0 = -0.2216$. The remaining 5 parameters are allowed to vary over the following ranges: The planetary mass fraction takes the values $\epsilon_1 = q/(1+q) = 3.162\times 10^{-4}, 10^{-3}, 3.162\times 10^{-3}, 10^{-2}$. The separation $d$ takes the values 0.80, 0.84, 0.88, 0.92, 0.96, 1.04, 1.08, 1.12, 1.16, 1.20, 1.24, and the source trajectory angle ranges over $\theta = 0^\circ$, ..., $90^\circ$ at $3^\circ$ intervals. The initial condition grid also includes three source radius crossing times, $t_\ast = 0.04, 0.07, 0.1\,$days, and three first caustic crossing times, $t_{cc1} = 2833.7, 2835.25, 2835.9$. This gives a total of 12276 initial condition grid points for which I calculate $\chi^2$.
Next, the best initial condition for each $t_{cc1}$ value is selected, and used as an initial condition for $\chi^2$ minimization. This $\chi^2$ minimization is done with the usual time variables of $t_E$ and $t_0$ instead of $t_{cc1}$ and $t_{cc2}$. These minimizations are run with an initial value of $T = 0.5$, which is dropped to $T = 0.05$. Two of the final solutions match solutions given in @bond-moa53. The $t_{cc1} = 2835.25\,$days initial condition leads to the best fit model with $\chi^2 = 1641.63$ (for 1535 data points and 1524 degrees of freedom), $d = 1.119$, $\epsilon_1 = 3.9\times 10^{-3}$, $t_E = 61.78\,$days, and $t_\ast = 0.058\,$days. The $t_{cc1} = 2833.7\,$days initial condition yields the “early caustic" model of @bond-moa53, with $\chi^2 = 1649.35$, $d = 1.120$, $\epsilon_1 = 6.6\times 10^{-3}$, $t_E = 59.53\,$days, and $t_\ast = 0.060\,$days.
The modeling run with the $t_{cc1} = 2835.9\,$days initial condition yields a solution that was not reported in the @bond-moa53 discovery paper. This “late caustic" crossing model has $\chi^2 = 1646.25$, $d = 1.119$, $\epsilon_1 = 3.4\times 10^{-3}$, $t_E = 61.3\,$days, and $t_\ast = 0.055\,$days, so it is a somewhat better model than the “early caustic" crossing model reported in the paper. However, these parameters are within 1-$\sigma$ of the best solution values reported in the discovery paper, so it appears likely this model was included in the error bar calculations.
These $\chi^2$ minimization runs each included an average of 19,000 $\chi^2$ calculations and used about 1.4 cpu hours each. The total number of $\chi^2$ calculations needed to find these three models was slightly over 70,000, and these calculations were accomplished in 5.24 cpu hours.
MOA-2008-BLG-310 {#sec-moa310}
----------------
MOA-2008-BLG-310 [@moa310] is a high-magnification event in which the angular radius of the source star is larger than the width of the central caustic that it crosses. As can be seen from Figure \[fig-lc\_hi\], the planetary deviation has a maximum amplitude of only $\sim 5$% compared with the corresponding single lens model. Since this deviation occurs in a region of the light curve when the magnification due to the stellar lens is changing rapidly, it is difficult to see the planetary deviation in the raw light curve, before it is divided by the single lens model.
The light curve deviation due to the planet in Figure \[fig-lc\_hi\] does not resemble the light curve deviations for point sources or sources that are much smaller than the width of the central caustic. One might worry that this unfamiliar light curve deviation shape could be a sign that modeling such an event would be difficult. But, in fact, this is not the case. Events such as this or MOA-2007-BLG-400 [@dong-moa400] turn out to be relatively easy to model because there are few, if any, local $\chi^2$ minima besides the global minima for $d < 1$ and $d > 1$. As a result, a relatively sparse initial grid is all that is required.
In this case, I have selected an initial grid that is somewhat larger than is necessary. The single lens parameters, plus the source radius crossing time, are fixed to the values from the best fit single lens model: $t_E = 11.022\,$days, $u_0 = 0.002966$, $t_0 = 4656.399\,$days, and $t_\ast = 0.05485\,$days. The remaining three binary lens parameters are scanned over the following ranges: $\epsilon_1 = 10^{-5}, 10^{-4}, 10^{-3}$; $d = 0.5, 0.6, 0.7, 0.8, 0.9, 1.0$; and $\theta = 0^\circ$, ..., $356^\circ$ at $4^\circ$ intervals. Thus, the initial condition grid calculations require 1620 $\chi^2$ evaluations. The best $\chi^2$ values from this grid come from the parameter sets ($d = 0.9$, $\epsilon_1 = 10^{-4}$, $\theta = 120^\circ$) and ($d = 0.5$, $\epsilon_1 = 10^{-3}$, $\theta = 116^\circ$). $\chi^2$ minimization runs starting from these initial conditions converge to the same solution, which corresponds to the $d < 1$ solution of @moa310. The parameters of this solution are $t_E = 10.40\,$days, $t_0 = 4656.3997\,$days, $u_0 = 0.00322$, $d = 0.921$, $\theta = 112.7^\circ$, $\epsilon_1 = 3.38\times 10^{-4}$, and $t_\ast = 0.0546\,$days. The wide ($d > 1$) solution is easily found from this one. We note that these parameters differ slightly from those of @moa310 due to slight differences in the data sets used and a different treatment of the error bars.
These $\chi^2$ minimization runs each required approximately 18,000 $\chi^2$ evaluations, and the combination of the initial condition grid search and $\chi^2$ minimizations required about 7.5 cpu hours using a angular to radial grid spacing ratio of 16 and a mean grid size of 0.16 source radii.
OGLE-2005-BLG-169 {#sec-ogle169}
-----------------
OGLE-2005-BLG-169 [@ogle169] is a high-magnification caustic crossing event that is similar, in some ways, to MOA-2008-BLG-310. However, in this case the source is much smaller than the width of the central caustic. Like OGLE-2003-BLG-235, this event does have one well observed caustic crossing with a caustic crossing time that can be accurately estimated by inspection of the light curve. But, unlike the case of OGLE-2003-BLG-235, the stellar magnification peak at $A_{\rm max} =800$ is a much stronger feature of the light curve than the resolved caustic crossing. As a result, it is more sensible to use the standard time parameters, $t_E$ and $t_0$ instead of the caustic crossing times, $t_{cc1}$ and $t_{cc2}$.
One difficulty with modeling OGLE-2005-BLG-169 is the incomplete coverage of the light curve. While the light curve extending from the stellar magnification peak to $\sim 1.2$ magnitudes below the peak is very densely covered with observations from the 2.4m MDM telescope every 10 seconds, the rising portion of the light curve is only observed about once every two hours. As a result, there are several points in the light curve where the first caustic crossing could have occurred, and this complicates the search for a solution.
The caustic crossing observed for this event is quite weak, and this implies that the planet-star separation must be very close to the Einstein ring. In such a situation, the shape of the central caustic is a very sensitive function of the planet separation and mass fraction, so I use a denser-than-usual initial condition grid. The separation, $d$, spans the range from 0.97 to 1.03 at an interval of 0.015, and the planetary mass fraction ranges from $\epsilon_1 = 2\times 10^{-5}$ to $\epsilon_1 = 2\times 10^{-4}$ in logarithmic intervals of $\sqrt{2}$. For some source trajectories, these high-magnification, resonant caustic events do not have the usual $d \leftrightarrow 1/d$ symmetry, so it is prudent to search over both $d \leq 1$ and $d > 1$. The source trajectory angle spans the range $\theta = 0^\circ$, ..., $356^\circ$ with a $4^\circ$ interval. The observed caustic crossing could, in principle, be used to determine the source radius crossing time, $t_\ast$, but since the first caustic crossing is unobserved, the angle of the crossing is unknown, we can only be sure that $t_\ast \leq 0.03\,$days, although a value very much smaller than this would require an unreasonably shallow crossing angle. I allow $t_\ast$ to range from $0.01\,$days to $0.03\,$days in the initial condition grid. This parameter set yields an initial condition grid of 28,350 grid points. The remaining parameters are fixed at their single-lens fit values: $t_E = 41.63\,$days, $t_0 = 3491.8756\,$days, and $u_0 = 0.001256$.
The initial grid search generates four parameter sets that are used for the subsequent $\chi^2$ minimizations. These include two parameter sets with $\theta \sim 60^\circ$. These are $\epsilon_1 = 7\times 10^{-5}$, $d = 1.015$, $\theta = 64^\circ$ and $t_\ast = 0.0172\,$days, as well as $\epsilon_1 = 7\times 10^{-5}$, $d = 0.985$, $\theta = 56^\circ$, and $t_\ast = 0.015\,$days. $\chi^2$ minimization from these initial conditions leads to two solutions quite similar to the best fit solution of @ogle169. The best model has $\epsilon_1 = 8.35\times 10^{-5}$, $d = 1.0120$, $\theta = 62.8^\circ$, $t_E = 43.48\,$days, $t_0 = 3491.8756\,$days, and $t_\ast = 0.0185\,$days with $\chi^2 = 536.29$ for 605 data points and 588 degrees of freedom. The other model has a slightly worse $\chi^2 = 537.67$ with similar parameters except that $d = 0.9820$. These correspond to the best fit model of @ogle169. The remaining two models have $\theta \approx 95^\circ$, and they correspond to the secondary local minimal shown in Figure 2 of @ogle169. Note that @ogle169 use a different source trajectory angle, $\alpha$ that is related to our source trajectory angle by $\alpha = 180^\circ - \theta$.
The $\chi^2$ minimization runs for OGLE-2005-BLG-169 averaged about 14,000 $\chi^2$ evaluations and each took about 5.4 cpu hours to complete using an angular to radial grid spacing ratio of 16 and a mean grid spacing of 0.16 source star radii. The total number of $\chi^2$ evaluations needed for OGLE-2005-BLG-169 modeling is 85,400, and this required 32.3 cpu hours of computing time. This event is the most time consuming of our example events because of the high-magnification ($A_{\rm max} = 800$) and the large number of observations at the peak (although the MDM data are binned to give a sampling interval of 86.4 seconds.)
MOA-2007-BLG-192 {#sec-moa192}
----------------
This event is probably the most challenging of these example events to model because the sampling of the planetary deviation is quite sparse. It is a high-magnification event, like OGLE-2005-BLG-169, but slightly less than half of the peak region is covered with MOA survey observations with a sampling interval of about 50 minutes. This leads to considerable uncertainty in the planetary models [@bennett-moa192]. In fact, we cannot be sure if the data indicate a cusp approach or caustic approach solution. Additionally, the source radius crossing time, $t_\ast$, is not well constrained. As a result, a relatively large initial condition grid must be used. The initial parameters are: $\epsilon_1 = 10^{-5}, 3.16\times 10^{-5}, 10^{-4}, 3.16\times 10^{-4}$; $d = 0.5$, 0.6, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 0.96, 0.97, 0.98, 0.99, 1.00, 1.01, 1.02, 1.03, 1.04; $t_\ast = 0.013$, 0.03, 0.047, 0.064, 0.081, 0.099, 0.116, 0.133, $0.15\,$days; and $\theta = 0^\circ$, ..., $356^\circ$ at a $4^\circ$ interval. The remaining parameters are fixed to the values from a single lens fit with the planetary signal removed: $t_E = 61.08\,$days, $t_0 = 4245.401\,$days, and $u_0 = 0.00526$. The total number of $\chi^2$ evaluations in this initial condition grid is 55,080.
The 12 best results from the grid search were selected for $\chi^2$ minimization. 7 of these $\chi^2$ minimization runs converged to a variant of the cusp approach solution listed in Table \[tab-modpar\]; 4 converged to a variant of the caustic crossing solution, and the remaining run converged to another local minimum with a source trajectory angle, $\theta$, that differs from the cusp approach and caustic crossing values by almost $180^\circ$. This minimum has a $\chi^2$ value larger than the best fit value by $\Delta\chi^2 > 120$, so it is not considered to be a viable solution.
To get the final set of solutions for this event, we must consider both the cusp approach and caustic crossing solutions and their $d \leftrightarrow 1/d$ counterparts. Also, this event has a significant microlensing parallax signal that we do not investigate here, and this introduces other degeneracies [@bennett-moa192].
Discussion and Conclusions {#sec-conclude}
==========================
I have presented a previously unpublished general method for modeling multiple mass microlensing events that has been optimized for high-magnification events. This method has been developed over a number of years from the first general method for calculating binary lens light curves, the image centered ray shooting method of @em_planet. This method is specifically designed to be computationally efficient for the most demanding high magnification events, [[i.e.]{}]{}, those with more than two lens masses and/or orbital motion. There are three aspects to this method: an efficient method for numerical calculation of the integrals that are needed to calculation microlensing magnification, an adaptive version of the Metropolis algorithm to quickly find a $\chi^2$ minimum in parameter space, and a global search strategy that can find all the important local $\chi^2$ minima even in parameter spaces with many dimensions. The computational efficiency of the first two elements of this method was critical for finding the solution for the solution of the only triple-lens microlensing system yet to be published [@gaudi-ogle109], as well as the study of the lens masses and orbits that are consistent with the light curve data for this event [@bennett-ogle109].
This method has also been successfully tested on all eight published single planet microlensing events [@bond-moa53; @ogle71; @ogle390; @ogle169; @bennett-moa192; @dong-moa400; @moa310; @sumi-ogle368], as well as four planetary events from the 2009 observing season. For two of these 2009 events, this method found the correct solution before the planetary signal was completed, using data that spanned less than 25% of the planetary deviation (although these events did have characteristics that were relatively easy to model). The details of several of these test calculations were presented in Section \[sec-global-ex\].
In Section \[sec-lc\_calc\], I demonstrated the dramatic improvement in high-magnification light curve calculation precision given by my limb-darkening optimized integration method and polar coordinate grid with a much larger angular than radial grid spacing. For the high-magnification events tested, these features improve the light curve calculation precision by a factor of ${\lower.5ex\hbox{{$\; \buildrel >\over\sim \;$}}}100$. Nevertheless, it should still be possible to make significant additional improvements. One improvement would be to implement the hexadecapole approximation [@pej_hey; @gould-hex] to do finite source calculations, where the point source approximation does not quite work. This will reduce the number of lens magnification calculations that require the full finite source integrals. For events without caustic crossings, this can dramatically improve calculation efficiency [@dong_ogle71], but for most caustic crossing events, the improvement is likely to be only a factor of two or so.
Further modifications to the integration grid scheme I present here could provide a more dramatic improvement in computational efficiency. I have used a 16:1 grid spacing ratio for the most efficient high-magnification light curve calculations presented here. However, this is a compromise value. The integration of the bright images that are generated by the primary lens star could probably benefit from a larger ratio, but the smaller images (or parts of images) that are directly perturbed by the planet are done more efficiently with a more modest grid-spacing ratio. This compromise could be avoided by going to an adaptive grid scheme in which the dimensions of the grid are adjusted to match the distortion of the images over the entire image plane. Of course, it would be quite time consuming directly calculate the lens distortions over the entire lens plane, so this would require a simpler prescription to estimate the lens distortions. It might be that a direct calculation over a coarse grid would work.
Of course, the main goal of this method is to be able to model complicated microlensing events that have yet to be successfully modeled. So, the best demonstration of the strength of this method would be the successful modeling of a number of these events.
D.P.B. was supported by grants AST-0708890 from the NSF and NNX07AL71G from NASA.
Abe, F., et al. 2003, , 411, L493 Albrow, M. D., et al. 1999, , 522, 1011 Alcock, C., et al. 1997, , 491, 436 Alcock, C., et al. 1999, , 518, 44 Alcock, C., et al. 2000, , 541, 270 An, J., et al. 2002, , 572, 521 Beaulieu, J.-P., et al. 2006, , 439, 437 Bennett, D.P. & Rhie, S.H. 1996, , 472, 660 Bennett, D. P. & Rhie, S. H., 2002, , 574, 985 Bennett, D. P., et al. 1996a, , 6361, 1 Bennett, D. P., et al. 1996b, Nuc. Phys. B Proc. Sup., 51, 152 Bennett, D. P., et al. 1997, Planets Beyond the Solar System and the Next Generation of Space Missions, 119, 95 (arXiv:astro-ph/9612208) Bennett, D. P., et al. 2008, , 684, 663 Bennett, D. P., et al. 2010, , 713, 837 Bond, I.A., et al. 2004, , 606, L155 Cassan, A. 2008, , 491, 587 Cassan, A. 2006, , 460, 277 Cassan, A. 2010, in preparation Claret, A. 2000, A&A, 363, 1081 Dominik, M. 1999, , 349, 108 Dominik, M. 2007, , 377, 1679 Dong, S., et al. 2006, , , 642, 842 Dong, S., et al. 2009a, , 695, 970 Dong, S., et al. 2009b, , 698, 1826 Doran, M., & M[ü]{}ller, C. M. 2004, Journal of Cosmology and Astro-Particle Physics, 9, 3 Gaudi, B. S., Naber, R. M., & Sackett, P. D. 1998, , 502, L33 Gaudi, B. S., et al. 2008, Science, 319, 927 Gould, A. 2008, , 681, 1593 Gould, A., & Gaucherel, C. 1997, , 477, 580 Gould, A., et al. 2006, , 644, L37 Griest, K., & Safizadeh, N. 1998, , 500, 37 Heyrovsk[ý]{}, D. 2007, , 656, 483 Ida, S., & Lin, D.N.C. 2004, , 616, 567 Janczak, J., et al. 2010, , 711, 731 Kayser, R., Refsdal, S., & Stabell, R. 1986, , 166, 36 Kennedy, G. M., Kenyon, S. J., & Bromley, B. C. 2006, , 650, L139 Khavinson, D., & Neumann, G. 2006, Proc. Amer. Math. Soc., 134, 1077 Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. 1983, Science, 220, 671 Kurucz, R. 1993a, ATLAS9 Stellar Atmosphere Programs and 2 km/s grid. Kurucz CD-ROM No. 13. Cambridge, Mass.: Smithsonian Astrophysical Observatory, 1993., 13, Kurucz, R. 1993b, Limbdarkening for 2 km/s grid (No. 13): \[+1.0\] to \[-1.0\]. Kurucz CD-ROM No. 16. Cambridge, Mass.: Smithsonian Astrophysical Observatory, 1993., 16, Kurucz, R. 1993c, Limbdarkening for 2 km/s grid (No. 13): \[+0.0\] to \[-5.0\]. Kurucz CD-ROM No. 17. Cambridge, Mass.: Smithsonian Astrophysical Observatory, 1993., 17, Kurucz, R. 1994, Solar abundance model atmospheres for 0,1,2,4,8 km/s. Kurucz CD-ROM No. 19. Cambridge, Mass.: Smithsonian Astrophysical Observatory, 1994., 19, Mao, S. & [Paczy[ń]{}ski ]{}, B. 1991, , 374, L37 Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. 1953, , 21, 1087 Milne, E. A. 1921, , 81, 361 Pejcha, O., & Heyrovsk[ý]{}, D. 2009, , 690, 1772 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Cambridge: University Press, |c1992, 2nd ed. Rattenbury, N.J., Bond, I.A., Skuljan, J. & d Yock, P.C.M. 2002, , 335, 159 Rhie, S. H. 2002, arXiv:astro-ph/0202294 Rhie, S. H. 2003, arXiv:astro-ph/0305166 Rhie, S. H., & Bennett, D. P. 1996, Nuclear Physics B Proceedings Supplements, Vol. 51, 51, 86 Rhie, S. H. et al. 2000, , 533, 378 Schneider, P., & Weiss, A. 1986, , 164, 237 Sumi, T., et al. 2010, , 710, 1641 Udalski, A. et al. 2005, , 628, L109 Verde, L., et al. 2003, , 148, 195 Vermaak, P. 2000, , 319, 1011 Wambsganss, T. R. 1997, , 284, 172 Witt, H. J. 1990, , 236, 311
[lcccccccc]{}
MOA-07-197 & 64.45 & -11.241 & 26.759 & 0.0500 & 1.156 & 2.247 & $7.64\times 10^{-2}$ & 0.0490\
OGLE-03-235 & 61.52 & -13.863 & -5.063 & 0.1327 & 1.120 & 0.764 & $3.94\times 10^{-3}$ & 0.0593\
OGLE-05-390 & 11.03 & 8.769 & 11.269 & 0.3589 & 1.610 & 2.756 & $ 7.57\times 10^{-5}$ & 0.282\
OGLE-05-169 & 41.72 & -0.301 & 0.299 & 0.00125 & 1.020 & 1.020 & $8.77\times 10^{-5}$ & 0.0184\
MOA-08-310 & 10.47 & -0.200 & 0.200 & 0.00314 & 1.094 & 1.961 & $3.51\times 10^{-4}$ & 0.0546\
MOA-07-192a & 74.46 & -1.553 & 1.547 & 0.00360 & 1.120 & 4.262 & $1.25\times 10^{-4}$ & 0.0643\
MOA-07-192c & 75.05 & -1.562 & 1.538 & 0.00433 & 0.985 & 4.518 & $2.07\times 10^{-4}$ & 0.117\
|
---
author:
- 'David Jewitt, Amaya Moro-Martín and Pedro Lacerda'
title: The Kuiper Belt and Other Debris Disks
---
Introduction
============
Planetary astronomy is unusual among the astronomical sciences in that the objects of its attention are inexorably transformed by intensive study into the targets of other sciences. For example, the Moon and Mars were studied telescopically by astronomers for nearly four centuries but, in the last few decades, these worlds have been transformed into the playgrounds of geologists, geophysicists, meteorologists, biologists and others. Telescopic studies continue to be of value, but we now learn most about the Moon and Mars from in-situ investigations. This transformation from science at-a-distance to science up-close is a forward step and a tremendous luxury not afforded to the rest of astronomy. Those who study other stars or the galaxies beyond our own will always be forced by distance to do so telescopically.
However, the impression that the Solar system is now *only* geology or meteorology or some other science beyond the realm of astronomy is completely incorrect when applied to the outer regions. The Outer Solar system (OSS) remains firmly entrenched within the domain of astronomy, its contents accessible only to telescopes. Indeed, major components of the OSS, notably the Kuiper belt, were discovered (telescopically) less than two decades ago and will continue to be best studied via. astronomical techniques for the foreseeable future. It is reasonable to expect that the next generation of telescopes in space and on the ground will play a major role in improving our understanding of the Solar system, its origin and its similarity to related systems around other stars.
In this chapter, we present an up-to-date overview of the layout of the Solar system and direct attention to the outer regions where our understanding is the least secure but the potential for scientific advance is the greatest. The architecture of the Kuiper belt is discussed as an example of a source-body system that probably lies behind many of the debris disks of other stars. Next, the debris disks are discussed based on the latest observations from the ground and from Spitzer, and on new models of dust transport. Throughout, we use text within grey boxes to highlight areas where the Next Generation telescopes are expected to have major impact.
The Architecture of the Solar System {#Intro}
====================================
It is useful to divide the Solar system into three distinct domains, those of the Terrestrial planets, the giant planets, and the comets. Objects within these domains are distinguished by their compositions, by their modes of formation and by the depth and quality of knowledge we possess on each.
Terrestrial Planets
-------------------
The Terrestrial planets (Mercury, Venus, Earth and Moon, Mars and most main-belt asteroids) are found inside 3 AU. They have refractory compositions dominated by iron ($\sim$ 35% by mass), oxygen ($\sim$ 30%), silicon and magnesium ($\sim$15% each) and were formed by binary accretion in the protoplanetary disk. About 95% of the Terrestrial planet mass is contained within Venus and Earth ($\sim$1 M$_{\oplus}$ = 6$\times$10$^{24}$ kg, each). The rest is found in the small planets Mercury and Mars, with trace amounts ($\sim$3$\times$10$^{-4}$ M$_{\oplus}$) in the main-belt asteroids located between Mars and Jupiter. The largest asteroid is Ceres ($\sim$ 900 km diameter). The Terrestrial planets grew by binary accretion between solid bodies in the protoplanetary disk of the Sun. While not all details of this process are understood, it is clear that sticking and coagulation of dust grains, perhaps aided at first by hydrodynamic forces exerted from the gaseous component of the disk, produced larger and larger bodies up to the ones we see now in the Solar system. The gaseous component, judged mainly by observations of other stars, dissipated on timescales from a few to $\sim$10 Myr. Measurements of inclusions within primitive meteorites show that macroscopic bodies existed within a few Myr of the origin. The overall timescale for growth was determined, ultimately, by the sweeping up of residual mass from the disk, a process thought to have taken perhaps 40 Myr in the case of the Earth. No substantial body is found in the asteroid belt although it is likely that sufficient mass existed there to form an object of planetary class. This is thought to be because growth in this region was interrupted by strong perturbations, caused by the emergence of nearby, massive Jupiter (at $\sim$5 AU).
Giant Planets
-------------
Orbits of the giant planets (Jupiter, Saturn, Uranus and Neptune) span the range 5 AU to 30 AU. The giants are in fact of two compositionally distinct kinds.
### Gas Giants
Jupiter (310 M$_{\oplus}$) and Saturn (95 M$_{\oplus}$) are so-called because, mass-wise, they are dominated by hydrogen and helium. Throughout the bulk of each planet these gases are compressed, however, into a degenerate (metallic) liquid that supports convection and sustains a magnetic field through dynamo action. The compositional similarity to the Sun suggests to some investigators that the gas giants might form by simple hydrodynamic collapse of the protoplanetary gas nebula (Boss 2001). In hydrodynamic collapse the essential timescale is given by the free-fall time, and this could be astonishingy short (e.g. 1000 yrs). Details of this instability, especially related to the necessarily rapid cooling of the collapsing planet, remain under discussion (Boley et al. 2007, Boss 2007). In fact, measurements of the moment of inertia, coupled with determinations of the equation of state of hydrogen-helium mixtures at relevant pressures and temperatures, show that Saturn (certainly) and Jupiter (probably) have distinct cores containing 5 M$_{\oplus}$ to 15 M$_{\oplus}$ of heavy elements (the case of Jupiter is less compelling than Saturn because of its greater mass and central hydrostatic pressure, leading to larger uncertainties concerning the self-compressibility of the gas). The presence of a dense core is the basis for the model of formation through “ nucleated instability”, in which the core grows by binary accretion in the manner of the Terrestrial planets, until the gravitational escape speed from the core becomes comparable to the thermal speed of molecules in the gas nebula. Then, the core traps gas directly from the nebula, leading to the large masses and gas-rich compositions observed in Jupiter and Saturn.
Historically, the main sticking point for nucleated instability models has been that the cores must grow to critical size *before* the surrounding gas nebula dissipates (i.e. 5 M$_{\oplus}$ to 15 M$_{\oplus}$ cores must grow in much less than 10 Myr). The increase in the disk surface density due to the freezing of water as ice outside the snow-line is one factor helping to decrease core growth times. Another may be the radial jumping motion of the growing cores, driven by angular momentum and energy exchange with planetesimals (Rice and Armitage 2003). In recent times, a consensus appears to have emerged that Jupiter’s core, at least, could have grown by binary accretion from a disk with $\Sigma \sim$ 50 to 100 kg m$^{-2}$ on timescales $\sim$1 Myr (Rice and Armitage 2003). The collapse of nebular gas onto the core after this would have been nearly instantaneous.
Recent data show that the heavier elements (at least in Jupiter, the better studied of the gas giants) are enriched relative to hydrogen in the Sun by factors of $\sim$2 to 4 (Owen et al. 1999), so that wholesale nebular collapse cannot be the whole story (and may not even be part of it). The enrichment applies not only to species that are condensible at the $\sim$100 K temperatures appropriate to Jupiter’s orbit, but to the noble gases Ar, Kr and Xe, which can only be trapped in ice at much lower temperatures, $<$30 K (e.g. Bar-Nun et al. 1988) . Therefore, Owen et al. suggest that the gas giants incorporate substantial mass from a hitherto unsuspected population of ultracold bodies, presumably originating in the outer Solar system. In a variant of this model, cold grains from the outer Solar system trap volatile gases but drift inwards under the action of gas drag, eventually reaching the inner nebula when they evaporate in the heat of the Sun and enrich the gas (Guillot and Huesco 2006).
### Ice Giants
Uranus (15 M$_{\oplus}$) and Neptune (17 M$_{\oplus}$), in addition to being an order of magnitude less massive than the gas giants are compositionally distinct. These planets contain a few M$_{\oplus}$ of H and He, and a much larger fraction of the “ices” H$_2$O, CH$_4$ and NH$_3$. For this reason they are known as “ice giants”, but the name is misleading because they are certainly not solid bodies but are merely composed of molecules which, if they were much colder, would be simple ices. In terms of their mode of formation, the difference between the ice giants and the gas giants may be largely one of timescale. It is widely thought that the ice giants correspond to the heavy cores of Jupiter and Saturn, but with only vestigial hydrogen/helium envelopes accreted from the rapidly dissipating gaseous component of the protoplanetary disk.
While qualitatively appealing, forming Uranus and (especially) Neptune on the 10 Myr timescale associated with the loss of the gas disk has been a major challenge to those who model planetary growth. The problem is evident from a simple consideration of the collision rate between particles in a disk or surface density $\Sigma(R)$ kg m$^{-2}$, where $R$ is the heliocentric distance. The probability of a collision in each orbit varies in proportion to $\Sigma(R)$, while the orbital period varies as $R^{3/2}$. Together, this gives a collision timescale varying as $R^{3/2}$/$\Sigma(R)$ which, with $\Sigma(R) \propto R^{-3/2}$ gives $t_c \propto R^{3}$. A giant planet core that takes 1 Myr to form at 5 AU would take 6$^3\sim$200 Myr to form at 30 AU, and this considerably exceeds the gas disk lifetime. Suggested solutions to the long growth times for the outer planets include augmentation of the disk density, $\Sigma(R)$, perhaps through the action of aerodynamic drag, and formation of the ice giants at smaller distances (and therefore higher $\Sigma$) than those at which they reside. The latter possibility ties into the general notion that the orbits of the outermost three planets have expanded in response to the action or torques between the planets and the disk. Still another idea is that Uranus and Neptune are gas giants whose hydrogen and helium envelopes were ablated by ionizing radiation from the Sun or a nearby, hot star (Boss et al. 2002).
Comets
------
Comets are icy bodies which sublimate in the heat of the Sun, producing observationally diagnostic unbound atmospheres or “comae”. For most known comets, the sublimation is sufficiently strong that mass loss cannot be sustained for much longer than $\sim$10$^4$ yr, a tiny fraction of the age of the Solar system. For this reason, the comets must be continually resupplied to the planetary region from one or more low temperature reservoirs, if their numbers are to remain in steady state. In the last half century, at least three distinct source regions have been identified.
### Oort Cloud Comets
The orbits of long period comets are highly elliptical, isotropically distributed and typically large (Figure \[lpcs\]), suggesting a gravitationally bound, spheroidal source region of order 100,000 AU in extent (Oort 1950). Comets in the cloud are scattered randomly into the planetary region by the action of passing stars and also perturbed by the asymmetric gravitational potential of the galactic disk (e.g. Higuchi et al. 2007). Unfortunately, the cloud is so large that its residents cannot be directly counted. The population must instead be inferred from the rate of arrival of comets from the cloud and estimates of the external torques. A recent work gives the number larger than $\sim$1 km in radius as 5$\times$10$^{11}$, with a combined mass in the range 2M$_{\oplus}$ to 40M$_{\oplus}$ (Francis 2005).
![Orbits of the nearly 200 long period comets (orbital period, $P>200$ yrs) listed in the JPL small-body database. The highly eccentric orbits of many LPCs appear as nearly radial lines on the top (wide angle) view, showing a cube 20,000 AU on a side. The bottom panel shows a narrow angle view of a 100 AU cube. Prograde orbits are shown in cyan while retrograde orbits are in magenta. Numbers along the axes are distances in AU from the Sun at (0,0,0). The orbits of the giant planets are shown in white.[]{data-label="lpcs"}](lpcs.jpg){width="80.00000%"}
Oort cloud comets are thought to have originated in the Sun’s protoplanetary disk in the vicinity of the giant planets and were scattered out by interactions with the growing, migrating planetary embryos. Although most were lost, a fraction of the ejected comets, perhaps from 1% to 10%, was subsequently deflected by external perturbations that lifted the perihelia out of the planetary region, effectively decoupling these comets from the rest of the system (Hahn and Malhotra 1999). Over $\sim$1 Gyr, the orbits of trapped comets were randomized, converting their distribution from a flattened one reflecting their disk source to a spherical one, compatible with the random directions of arrival of long period comets. In this model, which is qualitatively unchanged from that proposed by Oort (1950), the scale and population of the Oort cloud are set by the external perturbations from nearby stars and the galactic disk. If the Sun formed in a cluster, then the average perturbations from cluster members would have been bigger than now and a substantial population of more tightly bound, so-called “inner Oort cloud” comets could have been trapped. It is possible that the Halley family comets, whose orbits are predominantly but not exclusively prograde (Figure \[hfcs\]) are delivered from the inner Oort cloud (Levison et al. 2001).
![Halley family comets. Their non-isotropic inclination distribution, with more prograde (cyan) orbits than retrograde (magenta), points to an origin in the inner Oort cloud, where stellar and galactic perturbations have been too small to randomize the orbits. Following the classical definition, we plot the 44 comets in the JPL small-body database having Tisserand parameters with respect to Jupiter $\le$2 and orbital periods $20<P\mathrm{(yrs)}<200$. The orbits of the giant planets are shown in white. []{data-label="hfcs"}](hfcs.jpg){width="\textwidth"}
Some 90% to 99% of the comets formed in our system eluded capture and now roam the interstellar medium. If this fraction applies to all stars, then $\sim$10$^{23}$ to $\sim$10$^{24}$ “interstellar comets” exist in our galaxy, containing several $\times$10$^5$ M$_{\odot}$ of metals. Interstellar comets ejected from other stars might be detected with all-sky surveys in the Pan STARRS or Large Synoptic Survey Telescope (LSST) class (Jewitt 2003). Such objects would be recognized by their strongly hyperbolic orbits relative to the Sun, quite different from any object yet observed.
### Kuiper Belt Comets
The orbits of the Jupiter family comets (JFCs) have modest inclinations, with no retrograde examples, and most have eccentricities much less than unity (Figure \[jfcs\]). They are dynamically distinct from the long period Oort comets, and they interact strongly with Jupiter. For many years it was thought that the JFCs were captured from the long period population by Jupiter but increasingly numerical detailed work in the 1980’s showed that this was not possible (Fernández 1980, Duncan et al. 1988). The source appears to be the Kuiper belt, although this is not an iron-clad conclusion and the particular region or regions in the Kuiper belt from which the JFCs originate has yet to be identified.
About 200 JFCs are numbered (meaning that their orbits are very well determined) and a further 200 are known. Their survival is limited by a combination of volatile depletion, ejection from the Solar system, or impact into a planet or the Sun. Dynamical interactions alone give a median lifetime near 0.5 Myr (Levison and Duncan 1994), which matches the volatile depletion lifetime for bodies smaller than $r\sim40$ km. The implication is that JFCs smaller than this size will become dormant before they are dynamically removed, ending up as bodies that are asteroidal in appearance but cometary in orbit. Some of these dead comets are suspected to exist among the near-Earth “asteroid” population; indirect evidence for a fraction $\sim10\%$ comes from albedo measurements (Fern[á]{}ndez et al. 2001) and dynamical models (Bottke et al. 2002).
![Perspective view of the Jupiter family comets (salmon) together with the orbits of the planets out to Saturn. The Sun is at $(x,y,z)=(0,0,0)$ and the distances are in AU. Plotted are 166 JFCs, selected from the JPL small-body database based on their Tisserand parameter with respect to Jupiter ($2<T_J<3$; see, e.g., Levison and Duncan 1994) and with orbital periods $P<20$ yr.[]{data-label="jfcs"}](jfcs1.jpg){width="\textwidth"}
### Main Belt Comets
At the time of writing (February 2008) three objects are known to have the dynamical characteristics of asteroids but the physical appearances of comets. They show comae and particle tails indicative of on-going mass loss (Hsieh and Jewitt, 2006; Figures \[mbcs\] and \[mbcae\]). These are the Main-Belt comets (MBCs), most directly interpreted as ice-rich asteroids.
In the modern Solar system, the MBCs are dynamically isolated from the Oort cloud and Kuiper belt reservoirs (i.e. they cannot be captured from these other regions given the present-day layout of the Solar system, c.f. Levison et al. 2006). The MBCs should thus be regarded as a third and independent comet reservoir. Two possibilities for their origin seem plausible. The MBCs could have accreted ice if they grew in place but outside the snow-line. Alternatively, the MBCs might have been captured from elsewhere if the layout of the Solar system were very different in the past, providing a dynamical paths from the Kuiper belt that do not now exist. Insufficient evidence exists at present to decide between these possibilities.
![Perspective view of the main-belt comets (orange) together with the orbits of 100 asteroids (thin, yellow lines) and planets out to Jupiter. []{data-label="mbcs"}](mbcs.jpg){width="\textwidth"}
These bodies escaped detection until now because their mass loss rates ($\ll$ 1 kg s$^{-1}$) are two to three orders of magnitude smaller than from typical comets. The mass loss is believed to be driven by the sublimation of water ice exposed in small surface regions, with effective sublimating areas of $\sim$1000 m$^{-2}$ and less. Although sublimation at 3 AU is weak, the MBCs are small and mass loss at the observed rates cannot be sustained for the age of the Solar system. Instead, a trigger for the activity (perhaps impact excavation of otherwise buried ices) is needed. This raises the possibility that the detected MBCs are a tiny fraction of the total number that will be found by dedicated surveys like *Pan STARRS* and *LSST*. Even more interesting is the possibility that many or even most outer belt asteroids are in fact ice-rich bodies which display only transient activity. Evidence from the petrology and mineralogy of meteorites (e.g. the CI and CM chondrites that are thought to originate in the outer belt) shows the past presence of liquid water in the meteorite parent bodies. The MBCs show that some of this water survived to the present day.
![Orbital semimajor axis vs. eccentricity for objects classified as asteroids (black) and comets (blue), together with the three known main-belt comets 133P/Elst-Pizarro, P/2005 U1 (Read), and 118401 (1999 RE$_{70}$) (plotted in red). Vertical dashed lines mark the semimajor axes of Mars and Jupiter and the 2:1 mean-motion resonance with Jupiter (commonly considered the outer boundary of the classical main belt), as labeled. Curved dashed lines show the loci of orbits with perihelia equal to Mars’s aphelion ($q=Q_{Mars}$) and orbits with aphelia equal to Jupiter’s perihelion ($Q=q_{Jup}$). Objects plotted above the $q>Q_{Mars}$ line are Mars-crossers. Objects plotted to the right of the $Q<q_{Jup}$ line are Jupiter-crossers. From Hsieh and Jewitt (2006). []{data-label="mbcae"}](mbcae.jpg){width="\textwidth"}
\
**Next generation** facilities will offer opportunities to study the physical properties of the satellite systems of the giant planets. The regular satellites occupy prograde orbits of small inclination and eccentricity, resulting from their formation in the accretion disks through which the planets grew. As such, the individual regular satellite systems offer information about the sub-nebulae that must have existed around the growing planets. These sub-nebulae had their own density and temperature structures very different from the local disk of the Sun. Studies of reflected and thermal radiation with *JWST* and *ALMA* will be able to determine compositional differences in even the fainter regular satellites of the ice giants, and may provide constraints on Io-like and Enceladus-like endogenous activity.
The study of comets will be greatly advanced by *JWST* spectroscopy of the nuclei of comets (in order to minimize the effects of scattering from the dust coma, these small objects must be observed when far from the Sun and therefore faint). Simultaneous measurements of reflected and thermal radiation will provide albedo measurements which, with high signal-to-noise ratio spectra, will help determine the nature and evolution of the refractory surface mantles of these bodies.
Kuiper Belt
===========
More than 1200 Kuiper belt objects (KBOs) have been discovered since the first example, 1992 QB1, was identified in 1992. The known objects have typical diameters of 100 km and larger. The total number of such objects, scaled from published surveys, is of order 70,000, showing that there is considerable remaining discovery space to be filled by future surveys. The number of objects larger than 1 km may exceed 10$^8$.
![Perspective view of the Kuiper belt. The Sun is at $(x,y,z)=(0,0,0)$ and the axes are marked in AU. White ellipses show the orbits of the four giant planets. Red orbits mark the classical KBOs. Their ring-like distribution is obvious. Blue orbits show resonant KBOs, while green shows the Scattered objects. For clarity, only one fifth of the total number of objects in each dynamical type is plotted. []{data-label="kurious3"}](kurious3.jpg){width="\textwidth"}
The KBO orbits occupy a thick disk or sheet outside Neptune’s orbit (Figure \[kurious3\]). A major observational result is the finding that the KBOs can be divided on the basis of their orbital elements into several, distinct groups (Jewitt et al. 1998). This is best seen in $a$ (semimajor axis) vs. $e$ (orbital eccentricity) space, shown here in Figure \[plotae\]. The major Kuiper belt dynamical groups in the figure are
![Orbital semimajor axis vs. eccentricity for KBOs known as of 2008 January 30. The objects are color-coded according to their dynamical type, as labeled in the Figure and discussed in the text. A vertical dashed line at $a$ = 30 AU marks the orbit of Neptune, the nominal inner-boundary of the Kuiper belt. Upper and lower solid arcs show the loci of orbits having perihelion distances $q$ = $a(1 - e)$ equal to 30 AU and 40 AU, respectively. []{data-label="plotae"}](plotae.jpg){width="\textwidth"}
- Classical Kuiper belt objects (red circles). These orbit between about $a$ = 42 AU and the 2:1 mean-motion resonance with Neptune at $a$ = 47 AU. Typical orbital eccentricities of the Classicals are $e \sim$ 0 to 0.2 while the inclination distribution appears to be bimodal with components near $i$ = 2$^{\circ}$ (so-called “Cold-Classicals) and $i$ = 20$^{\circ}$ (the “Hot Classicals”, Brown and Trujillo 2001, Elliot et al. 2005). Numerical integrations show that the orbits of the Classical objects are stable on timescales comparable to the age of the Solar system largely because their perihelia are always so far from Neptune’s orbit that no strong scattering occurs.\
- Resonant Kuiper belt objects (blue circles). A number of mean-motion resonances (MMRs) with Neptune are populated by KBOs, especially the 3:2 MMR at 39.3 AU and the 2:1 MMR at 47.6 AU (see Figure \[plotae\]). The 3:2 MMR objects are known as Plutinos, to recognize 134340 Pluto as the first known member of this population. The 2:1 MMR objects are sometimes called “twotinos” while those in 1:1 MMR are Neptune’s Trojans (Sheppard and Trujillo 2006). The resonant objects are dynamically stable by virtue of phase protections conferred by the resonances. For example, KBOs in 3:2 MMR can, like Pluto itself, have perihelia inside Neptune’s orbit, but their orbits librate under perturbations from that planet in such a way that the distance of closest approach to Neptune is always large. In fact, all objects above the upper solid arc in Figure \[plotae\] have perihelia interior to Neptune’s orbit.
The process by which KBOs became trapped in MMRs is thought to be planetary migration (Fern[á]{}ndez and Ip 1984). Migration occurs as a result of angular momentum transfer during gravitational interactions between the planets and material in the disk. At late stages in the evolution of the disk (i.e. later than $\sim$10 Myr, after the gaseous component of the disk has dissipated) the interactions are between the planets and individual KBOs or other planetesimals. In a one-planet system, the sling-shot ejection of KBOs to the interstellar medium would result in net shrinkage of the planetary orbit. In the real Solar system, however, KBOs can be scattered inwards from planet to planet, carrying energy and angular momentum with them as they go. Massive Jupiter then acts as the source of angular momentum and energy, ejecting KBOs from the system. In the process, its orbit shrinks, while those of the other giant planets expand. The timescale is the same as the timescale for planetary growth, and the distance through which a planet migrates depends on the mass ejected from the system. Outward migration of Neptune carries that planet’s MMRs outwards, leading to the sweep-up of KBOs (Malhotra 1995).\
![Same as Figure \[plotae\] but with a logarithmic x-axis extended to $a$ = 1000 AU to better show the extent of the scattered KBOs.[]{data-label="plotaexl"}](plotaexl.jpg){width="\textwidth"}
- Scattered Kuiper belt objects (green circles), also known as Scattered disk objects. These objects have typically eccentric and inclined orbits with perihelia in the 30 $\le q \le$ 40 AU range (Figures \[plotae\] and \[plotaexl\]). The prototype is 1996 TL66, whose orbit stretches from $\sim$35 AU to $\sim$130 AU (Luu et al 1997). The brightness of these objects varies strongly around the orbit, such that a majority are visible only when near perihelion. For example, the survey in which 1996 TL66 was discovered lacked the sensitivity to detect the object over 88% of the orbital period. Accordingly, the estimated population is large, probably rivaling the rest of the Kuiper belt (Trujillo et al. 2001).
Scattered KBOs owe their extreme orbital properties to continued, weak perihelic interactions with Neptune which excite the eccentricity to larger and larger values (Duncan and Levison 1997). The current aphelion record-holder is 87269 (2000 OO67) with $Q$ = 1123 AU. As the aphelion grows, so does the dynamical influence of external perturbations from passing stars and the asymmetric gravitational potential of the galactic disk.
- Detached Kuiper belt objects (black circles). The orbits of these bodies resemble those of the Scattered KBOs except that the perihelia are too far from Neptune, $q >$ 40 AU, for planetary perturbations to have excited the eccentricities. The prototype is 2000 CR105, with $q$ = 44 AU (Gladman et al. 2002) but a more extreme example is the famous Sedna with $q$ = 74 AU (Brown et al. 2004).
The mechanism by which the perihelia of the detached objects were lifted away from the influence of Neptune is unknown. The most interesting conjectures include the tidal action of external perturbers, whether they be unseen planets in our own system or unbound passing stars (Morbidelli and Levison 2004).\
The Kuiper belt is a thick disk (Figure \[plotai\]), with an apparent full width at half maximum, $FWHM \sim$ 10$^{\circ}$ (Jewitt et al. 1996). The apparent width is an underestimate of the true width, however, because most KBOs have been discovered in surveys aimed near the ecliptic, and the sensitivity of such surveys varies inversely with the KBO inclination. Estimates of the unbiased (i.e. true) inclination distribution give $FWHM \sim$ 25 to 30$^{\circ}$ (Jewitt et al. 1996, Brown and Trujillo 2001, Elliot et al. 2005 - see especially Figure 20b). Moreover, all four components of the Kuiper belt possess broad inclination distributions (Figure \[plotai\]). The inclination distribution of the Classical KBOs appears to be bimodal, with a narrow core superimposed on a broad halo (Brown and Trujillo 2001, Elliot et al. 2005).\
The Kuiper belt is not thin like the Sun’s original accretion disk and it is clear the the inclinations of the orbits of its members have been excited. The velocity dispersion amongst KBOs is $\Delta V$ = 1.7 km s$^{-1}$ (Jewitt et al. 1996, Trujillo et al. 2001). At these velocities, impacts between all but the largest KBOs lead to shattering and the production of dust, rather than to accretion and growth.
![Orbital semimajor axis vs. inclination for KBOs known as of 2008 January 30. The plot shows that the KBOs occupy orbits having a wide range of inclinations. Most of the plotted objects were discovered in surveys directed towards the ecliptic, meaning that an observation bias $\it{against}$ finding high inclination bodies is imprinted on the sample. []{data-label="plotai"}](plotai.jpg){width="\textwidth"}
**Next generation** facilities should provide unprecedented survey capabilities (through *LSST*) that will provide a deep, all-sky survey of the Solar system to 24th visual magnitude, or deeper. The number of objects for which reliable orbits exist will increase from $\sim$10$^3$, at present, by one to two orders of magnitude, depending on the detailed strategies employed by the surveys. Large samples are needed to assess the relative populations of the resonances and other dynamical niches that may place limits on the formation and evolution of the OSS. Objects much larger than Pluto, perhaps in the Mars or Earth class, may also be revealed by careful work.
Interrelation of the Populations
================================
Evidence that the small-body populations are interrelated is provided by dynamical simulations. The interrelations are of two basic types: a) those that occur through dynamical processes operating in the current Solar system and b) those that might have operated at an earlier epoch when the architecture of the Solar system may have been different from now.
As examples of the first kind, it is clear that objects in the Oort cloud and Kuiper belt reservoirs can be perturbed into planet crossing orbits, and that these perturbations drive a cascade from the outer Solar system through the Centaurs (bodies, asteroidal or cometary in nature) that are strongly interacting with the giant planets to the Jupiter family comets (orbits small enough for the Sun to initiate sublimation of near-surface ice and strongly interacting with Jupiter) to dead and dormant comets in the near-Earth “asteroid” population. In the current system, simulations show that it is *not* possible to capture the Trojans or the irregular satellites (Figure \[phoebe\]) of the giant planets from the passing armada of small bodies, and there is no known dynamical path linking comets from the Kuiper belt, for example, to comets in the main-belt (the MBCs).
![“Nice” model in which the architecture of the Solar system is set by the clearing of a massive (30 M$_{\oplus}$) Kuiper belt (stippled green region) when planets are thrown outwards by strong interactions between Jupiter (red) and Saturn (pink) at the 2:1 mean-motion resonance. (a) The initial configuration with the giant planets at 5.5, 8.2, 11.5 and 14.2 AU (b) Just before the 2:1 resonance crossing, timed to occur near 880 Myr from the start (c) 3 Myr after resonance crossing (note the large eccentricity of Uranus (purple) at this time and the placement of Neptune (blue) *in* the Kuiper belt) and (d) 200 Myr later, by which time the planetary orbits have assumed nearly their current properties. Adapted with permission from Gomes et al. 2005[]{data-label="gomes"}](GomesColor.jpg){width="\textwidth"}
The second kind of interrelation is possible because of planetary migration. The best evidence for the latter is deduced from the resonant Kuiper belt populations which, in one model, require that Neptune’s orbit expanded by roughly 10 AU, so pushing its mean-motion resonances outwards through the undisturbed Kuiper belt (Malhotra 1995). The torques driving the migration moved Jupiter inwards (by a few $\times$0.1 AU, since it is so massive) as the other giants migrated outwards. In one exciting model, the “Nice model”, this migration pushed Jupiter and Saturn across the 2:1 mean motion resonance (Gomes et al. 2005). The dynamical consequences of the periodic perturbations induced between the Solar system’s two most massive planets would have been severe (Figure \[gomes\]). In published models, these perturbations excite a 30 M$_{\oplus}$ Kuiper belt, placing large numbers of objects into planet crossing orbits and clearing the Kuiper belt down to its current, puny mass of $\sim$0.1 M$_{\oplus}$. During this clearing phase, numerous opportunities exist for trapping scattered KBOs in dynamically surprising locations. For example, the Trojans of the planets could have been trapped during this phase (Morbidelli et al. 2005). Some of the irregular satellites might likewise have been acquired at this time (Nesvorny et al. 2007; furthermore, any irregular satellites possessed by the planets *before* the mean-motion resonance crossing would have been lost). Other KBOs might have been trapped in the outer regions of the main-asteroid belt, perhaps providing a Kuiper belt source for ice in the MBCs.
![Saturn’s irregular satellite Phoebe, which might be a captured Kuiper belt object. The effective spherical radius is 107$\pm$1 km: the largest crater, Jason (at left), has a diameter comparable to the radius. The surface contains many ices (water, carbon dioxide) and yet is dark, with geometric albedo 0.08, as a consequence of dust mixed in the ice. Image from Cassini Imaging Team/NASA/JPL/Space Science Institute. []{data-label="phoebe"}](phoebe.jpg){width="\textwidth"}
Whether or not Jupiter and Saturn actually crossed the 2:1 resonance is unknown. In the Nice model, the 2:1 resonance crossing is contrived to occur at about 3.8 Gyr, so as to coincide with the epoch of the late-heavy bombardment (LHB). The latter is a period of heavy cratering on the Moon, thought by some to result from a sudden shower of impactors some 0.8 Gyr after the formation epoch. However, the interpretation of the crater age data is non-unique and reasonable arguments exist to interpret the LHB in other ways (Chapman et al. 2007). For example, the cratering rate could merely *appear* to peak at 3.8 Gyr because all earlier (older) surfaces were destroyed by an impact flux even stronger than that at the LHB. (The LHB would then be an analog to the epoch in the expanding early universe when the optical depth first fell below unity and the galaxies became visible).
Evidence concerning the birth environment
=========================================
Several lines of evidence suggest that the Sun was formed in a dense cluster.
The existence of widespread evidence for the decay products of short-lived isotopes in mineral inclusions in meteorites suggests that one or more supernovae exploded in the vicinity of the Sun, shortly before its formation. $^{26}$Al (half-life 0.7 Myr) was the first such unstable isotope to be identified (Lee et al. 1977) but others, like $^{60}$Fe (half-life 1.5 Myr) are known (and, unlike $^{26}$Al, cannot be produced by nuclear spallation reactions; Mostefaoui et al. 2004). It is possible, although not required, that the collapse formation of the protoplanetary nebula was triggered by shock compression from the supernova that supplied the unstable nuclei (Ouellette et al. 2007).
The sharp outer edge to the classical Kuiper belt could be produced by a stellar encounter having an impact parameter $\sim$150 AU to 200 AU (Ida et al. 2000), although this is only one of several possible causes. Such close encounters are highly improbable in the modern epoch but would be more likely in the dense environment of a young cluster. At the same time, the survival and regularity of the orbits of the planets suggests that no very close encounter occurred (Gaidos 1995).
As noted above, the existence of a dense inner Oort cloud, required in some models to explain the source of the Halley family comets (long-period comets which include retrograde examples but which are not isotropically distributed), can only be populated via the stronger mean perturbations exerted between stars in a dense cluster.
Adams and Laughlin (2001) conclude from these and related considerations that the Sun formed in a cluster of 2000$\pm$1000 stars.
Colors and Physical Properties
==============================
It has long been recognized that Kuiper belt objects exhibit a diversity in surface colors unparalleled among Solar system populations (Luu and Jewitt 1996). The distribution is relatively smooth from neutrally colored to extremely red objects. Indeed, a large fraction of KBOs (and Centaurs) is covered in “ultrared matter” (Jewitt 2002), the reddest material observed on small bodies. This material is absent in other populations, and is thought to be due to irradiated complex organics (Cruikshank et al. 2007). Further, the $UBVRIJ$ colors are mutually correlated (Jewitt and Luu 1998; Jewitt et al. 2007), which seems to indicate that the spread is caused by a single reddening agent.
The wide range of colors suggests a broad range of surface compositions. However, such extreme non-uniformity in composition is unlikely to be intrinsic given the uniform and low temperatures across the disk of the Kuiper belt; the compositional spread is probably the result of some evolutionary process. Early theories to explain the color scatter invoke a competition between (the reddening) long-term exposure to cosmic radiation and (the de-reddening) impact resurfacing (Luu and Jewitt 1996; Delsanti et al. 2004), but the implied rotational color variability and correlation between color and collision likelihood are not observed (Jewitt and Luu 2001; Th[é]{}bault and Doressoundiram 2003).
More recently it has been suggested that the diversity was emplaced when the small body populations were scattered by the outward migration of the ice giant planets (Gomes 2003), as a consequence of the mutual 2:1 resonance crossing by Jupiter and Saturn (see Nice model above). In other words, some of the objects now in the Kuiper belt may have formed much closer to the Sun, in the 10–20 AU region, where the chemistry would have been different and perhaps more diverse. Although appealing, the theory that the KB region was sprinkled with bodies from various heliocentric distances remains a non-unique explanation and the implicit assumption that KBOs formed closer to the Sun should be less red remains ad-hoc. The relative importances of this dynamical mixing and the reddening effect by cosmic irradiation are still poorly understood. Cosmic-ray reddening seems to be important as the Classical KBOs, supposedly formed locally at 40 AU and having passively evolving surfaces only subject to cosmic radiation (their circular orbits protect them from mutual collisions), are on average the reddest KBOs (Tegler and Romanishin 2000).
A powerful way to investigate the physical properties of KBOs is by the analysis of their rotational properties, usually inferred from their lightcurves (Sheppard and Jewitt 2002; Lacerda and Luu 2006). Lightcurves are periodic brightness variations due to rotation: as a non-spherical (and non-azimuthally symmetric) KBO rotates in space, its sky-projected cross-section will vary periodically, and thus modulate the amount of sunlight reflected back to the observer. The period $P$ and range $\Delta m$ of a KBO’s lightcurve provide information on its rotation period and shape, respectively. Multi-wavelength lightcurves may also reveal surface features such as albedo or color patchiness (Buie et al. 1992; Lacerda et al. 2008). These features are usually seen as second order effects superimposed on the principal, shape-regulated lightcurve. By combining the period and the range of a KBO lightcurve it is possible to constrain its density under the assumption that the object’s shape is mainly controlled by its self-gravity (Jewitt and Sheppard 2002; Lacerda and Jewitt 2007). Lightcurves can also uncover unresolved, close binary objects (Sheppard and Jewitt 2004; Lacerda and Jewitt 2007).
Interesting KBOs, whose lightcurves have been particularly informative include: 1) 134340 Pluto, whose albedo-controlled light variations have been used to map the distribution of ices of different albedos across the surface (Buie et al. 1992; Young et al. 1999), which is likely controlled by the surface deposition of frosts from Pluto’s thin atmosphere (Trafton 1989), 2) 20000 Varuna, whose rapid rotation ($P=6.34\pm0.01$ hr) and elongated shape ($\Delta m=0.42\pm0.03$ mag) indicate a bulk density $\rho\sim1000$ kg m$^{-3}$ and hence require an internally porous structure (Jewitt and Sheppard 2002), 3) 139775 2001 QG298, whose extreme lightcurve (large range $\Delta m=1.14\pm0.04$ mag and slow period $P\sim13.8$ hr) suggests an extreme interpretation as a contact or near-contact binary (Sheppard and Jewitt 2004; Takahashi and Ip 2004; Lacerda and Jewitt 2007), and finally 4) 136198 2003 EL61, exhibiting super-fast rotation ($P=3.9$ hr) that requires a density $\sim2500$ kg m$^{-3}$, and a recently identified surface feature both redder and darker than the average surface (Lacerda et al. 2008).
Statistically, the rotational properties of KBOs can be used to constrain the distributions of spin periods (Lacerda and Luu 2006) and shapes (Lacerda and Luu 2003), which in turn can be used to infer the importance of collisions in their evolution. For instance, most main-belt asteroids have been significantly affected by mutual collisions, as shown by their quasi-Maxwellian spin rate distribution (Harris 1979; Farinella et al. 1981) and a shape distribution consistent with fragmentation experiments carried out in the laboratory (Catullo et al. 1984). KBOs spin on average more slowly ($\langle
P_{KBO}\rangle\sim8.4$ hr vs. $\langle P_{ast}\rangle\sim6.0$ hr; Lacerda and Luu 2006) and are more spherical (as derived from the $\Delta m$ distribution; Luu and Lacerda 2003; Sheppard et al. 2008) than asteroids of the same size, both indicative of a milder collisional history. Typical impact speeds in the current Kuiper belt and main asteroid belt are respectively 2 and 5 km s$^{-1}$. However, three of the four KBOs listed in the previous paragraph show extreme rotations and shapes, likely the result of collision events. Because the current number density of KBOs is too low for these events to occur on relevant timescales, their rotations were probably aquired at an early epoch when the Kuiper belt was more massive and collisions were more frequent (Davis and Farinella 1997; Jewitt and Sheppard 2002).
**Next generation** telescopes will provide revolutionary new data on the physical properties of KBOs. Their colors and rotational properties can potentially reveal much about these objects’ surface and physical natures. *JWST*, in particular, will provide high quality near-infrared spectra that will place the best constraints on the (probably) organic mantles of the KBOs, thought to be some of the most primitive matter in the Solar system. Separate measurements of the albedos and diameters, obtained from optical/thermal measurements using *ALMA* and *JWST*, will give the albedos and accurate diameters, needed to fully understand the surface materials.
Survey data will permit the identification of $>$100 wide binaries, while the high angular resolution afforded by *JWST* will reveal a much larger number (thousands?) of close binaries. Orbital elements for each will lead to the computation of system masses through Kepler’s law. Diameters from optical/thermal measurements (using *ALMA* and *JWST*) will then permit the determination of system densities for KBOs over a wide range of diameters and orbital characteristics in the Kuiper belt. Density, as the “first geophysical parameter”, provides our best handle on accretion models of the Kuiper belt objects.
Solar System Dust
=================
Collisions between solids in the early Solar system generally resulted in agglomeration due to the prevailing low impact energies. However, at present times, high impact energies dominate and collisions result in fragmentation and the generation of dust from asteroids, comets and KBOs. Due to the effect of radiation forces (see $\S$ 7.2) these dust particles spread throughout the Solar system forming a dust disk.
Inner Solar System: Asteroidal and Cometary Dust
------------------------------------------------
The existence of dust in the inner Solar system (a.k.a. Zodiacal cloud) has long been known since the first scientific observations of the Zodiacal light by Cassini in 1683, correctly interpreted by de Duiliers in 1684 as produced by sunlight reflected from small particles orbiting the Sun. Other dust-related phenomena that can be observed naked eye are dust cometary tails and “shooting stars".
The sources of dust in the inner Solar system are the asteroids, as evident from the observation of dust bands associated with the recent formation of asteroidal families, and comets, as evident from the presence of dust trails and tails. Their relative contribution can be studied from the He content of collected interplanetary dust particles, which is strongly dependent on the velocity of atmospheric entry, expected to be low for asteroidal dust and high for cometary dust. Their present contributions are thought to differ by less than a factor of 10 (Brownlee et al. 1994), but they have likely changed with time. It is thought that due to the depletion of asteroids, the asteroidal dust surface area has slowly declined by a factor of 10 (Grogan et al. 2001) with excursions in the dust production rate by up to an order of magnitude associated with breakup events like those giving rise to the Hirayama asteroid families that resulted in the formation of the dust bands observed by [*IRAS*]{} (Sykes and Greenberg 1986). The formation of the Veritas family 8.3 Myr ago still accounts for $\sim$25% of the Zodiacal thermal emission today (Dermott et al. 2002). A major peak of dust production in the inner Solar system is expected to have occurred at the time of the LHB ($\S$4), as a consequence of an increased rate of asteroidal collisions and to the collisions of numerous impactors originating in the main asteroid belt (Strom et al. 2005) with the terrestrial planets.
The thermal emission of the Zodiacal cloud dominates the night sky between 5–500 $\mu$m and has a fractional luminosity of L$_{dust}$/L$_{Sun}$ $\sim$ 10$^{-8}$–10$^{-7}$ (Dermott et al. 2002). Studied by [*IRAS*]{}, [*COBE*]{} and ${\it ISO}$ space telescopes, it shows a featureless spectrum produced by a dominant population of low albedo ($<$0.08) rapidly-rotating amorphous forsterite/olivine grains that are 10–100 $\mu$m in size and are located near 1 AU. The presence a weak (6% over the continuum) 10 $\mu$m silicate emission feature indicates the presence of a small population of $\sim$1 $\mu$m grains of dirty crystalline olivine and hydrous silicate composition (Reach et al. 2003).
Interplanetary dust particles (IDPs) have been best characterized at around 1 AU by in situ satellite measurements, observations of micro-meteorite impact craters on lunar samples, ground radar observations of the ionized trails created as the particles pass through the atmosphere and laboratory analysis of dust particles collected from the Earth’s stratosphere, polar ice and deep sea sediments. Laboratory analysis of collected IDPs show that the particles are 1–1000$\mu$m in size, typically black, porous ($\sim$40%) and composed of mineral assemblages of a large number of sub-micron-size grains with chondritic composition and bulk densities of 1–3 g/cm$^3$. Their individual origin, whether asteroidal or cometary, is difficult to establish. The cumulative mass distribution of the particles at 1 AU follows a broken power-law such that the dominant contribution to the cross sectional area (and therefore to the zodiacal emission) comes from 10$^{-10}$ kg grains ($\sim$30 $\mu$m in radius), while the dominant contribution to the total dust mass comes from $\sim$ 10$^{-8}$ kg grains (Leinert and Grün 1990).
In situ spacecraft detections of Zodiacal dust out to 3 AU, carried out by Pioneer 8–11, Helios, Galileo and Ulysses, showed that the particles typically have $i < $30$^{\circ}$ and $e >$ 0.6 with a spatial density falling as $r^{-1.3}$ for $r <$ 1 AU and $r^{-1.5}$ for $r >$ 1 AU, and that there is a population of grains on hyperbolic orbits (a.k.a $\beta$-meteoroids), as well as stream of small grains origina jovian system (see review by Grün et al. 2001).
Outer Solar System: Kuiper Belt Dust
------------------------------------
As remarked above, collisions in the modern-day Kuiper belt are erosive, not agglomerative, and result in the production of dust. In fact, two components to Kuiper belt dust production are expected: (1) erosion of KBO surfaces by the flux of interstellar meteoroids (e.g. Grün et al. 1994), leading to the steady production of dust at about 10$^3$ to 10$^4$ kg s$^{-1}$ (Yamamoto and Mukai 1998); and (2) mutual collisions between KBOs, with estimated dust production of about (0.01–3)$\times$10$^{8}$ kg s$^{-1}$ (Stern 1996). For comparison, the dust production rate in the Zodiacal cloud, from comets and asteroids combined, is about 10$^3$ kg s$^{-1}$ (Leinert et al. 1983). The fractional luminosity of the KB dust is expected to be around L$_{dust}$/L$_{Sun}$ $\sim$ 10$^{-7}$–10$^{-6}$ (Stern 1996), compared to L$_{dust}$/L$_{Sun}$ $\sim$ 10$^{-8}$–10$^{-7}$ for the Zodiacal cloud (Dermott et al. 2002).
Observationally, detection of Kuiper belt dust at optical wavelengths is confounded by the foreground presence of cometary and asteroidal dust in the Zodiacal cloud. Thermally, these near and far dust populations might be distinguished on the basis of their different temperatures ($\sim$200 K in the Zodical cloud vs. $\sim$40 K in the Kuiper belt) but, although sought, Kuiper belt dust has not been detected this way (Backman et al. 1995). At infrared thermal wavelengths both foreground Zodiacal cloud dust *and* background galactic dust contaminate any possible emission from Kuiper belt dust. The cosmic microwave background radiation provides a very uniform source against which emission from the Kuiper belt might potentially be detected but, again, no detection has been reported (Babich et al. 2007).
Whereas remote detections have yet to be achieved, the circumstances for in-situ detection are much more favorable (Gurnett et al. 1997). The Voyager 1 and 2 plasma wave instruments detected dust particles via the pulses of plasma created by high velocity impacts with the spacecraft. Because the plasma wave detector was not built with impact detection as its primary purpose, the properties and flux of the impacting dust are known only approximately. Still, several important results are available from the Voyager spacecraft. Impacts were recorded continuously as the Voyagers crossed the (then unknown) Kuiper belt region of the Solar system. The smallest dust particles capable of generating measurable plasma are thought to be $a_0 \sim$2 $\mu$m in radius. Measured in the 30 AU to 60 AU region along the Voyager flight paths, the number density of such particles is $N_1 \sim$ 2$\times$10$^{-8}$ m$^{-3}$. Taking the thickness of the Kuiper belt (measured perpendicular to the midplane) as $H \sim$ 10 AU, this corresponds to an optical depth $\tau \sim \pi a_0^2 H N_1 \sim 4\times 10^{-7}$, roughly 10$^3$ times smaller than the optical depth of a $\beta$-Pictoris class dust circumstellar disk. An upper limit on the density of gravel ($cm$-sized) particles in the Kuiper belt is provided by the survival of a 20-cm propellant tank on the Pioneer 10 spacecraft (Anderson et al. 1998).
Dust Dynamics and Dust Disk Structure
-------------------------------------
After the dust particles are released from their parent bodies (asteroids, comets and KBOs) they experience the effects of radiation and stellar wind forces. Due to radiation pressure, their orbital elements and specific orbital energy change immediately upon release. If their orbital energy becomes positive ($\beta$ $>$ 0.5), the dust particles escape on hyperbolic orbits (known as $\beta$-meteoroids – Zook & Berg 1975). If their orbital energy remains negative ($\beta$ $<$ 0.5), their semi-major axis increases but they remain on bound orbits. Their new semimajor axis and eccentricity ($\it{a',e'}$) in terms of that of their parent bodies ($\it{a}$ and $\it{e}$) are $a'=a{1-\beta \over 1-2a\beta/r}$ and $e'={\mid 1 - { (1-2a\beta /r)(1-e^2)
\over (1-\beta^2)}\mid}^{1/2}$ (their inclination does not change), where $r$ is the particle location at release and $\beta$ is the ratio of the radiation pressure force to the gravitational force. For spherical grains orbiting the Sun, $\beta=5.7 Q_{pr}/\rho b$, where $\rho$ and $b$ are the density and radius of the grain in MKS units and $Q_{pr}$ is the radiation pressure coefficient, a measure of the fractional amount of energy scattered and/or absorbed by the grain and a function of the physical properties of the grain and the wavelength of the incoming radiation (Burns, Lamy & Soter 1979).
With time, Poynting-Rorbertson (P-R) and solar wind corpuscular drag (which result from the interaction of the dust grains with the stellar photons and solar wind particles, respectively) tend to circularize and decrease the semimajor axis of the orbits, forcing the particles to slowly drift in towards the central star until they are destroyed by sublimation in a time given by $
t_{PR} = 0.7 ({b \over \mu m}) ({\rho \over kg/m^3}) ({R \over AU})^2 ({L_\odot
\over L_*}) {1 \over 1+albedo} ~yr, $ where $R$ is the starting heliocentric distance of the dust particle and $\it{b}$ and $\rho$ are the particle radius and density, respectively (Burns, Lamy and Soter 1979 and Backman and Paresce 1993). If the dust is constantly being produced from a planetesimal belt, and because the dust particles inclinations are not affected by radiation forces, this inward drift creates a dust disk of wide radial extent and uniform density. Grains can also be destroyed by mutual grain collisions, with a collisional lifetime of $ t_{col} = 1.26 \times 10^4 ({R \over AU})^{3/2}
({M_\odot \over M_*})^{1/2} ({10^{-5} \over L_{dust}/L_*}) yr $ (Backman and Paresce 1993).
For dust disks with M$_{dust}$$>$10$^{-3}$ M$_{\oplus}$, $t_{col} < t_{PR}$, i.e. the grains are destroyed by multiple mutual collisions before they migrate far from their parent bodies (in this context, “destruction” means that the collisions break the grains into smaller and smaller pieces until they are sufficiently small to be blown away by radiation pressure). This regime is referred to as collision-dominated. The present Solar system, however, is radiation-dominated because it does not contain large quantities of dust and $t_{col} > t_{PR}$, i.e. the grains can migrate far from the location of their parent bodies. This is particularly interesting in systems with planets and outer dust-producing planetesimal belts because in their journey toward the central star the orbits of the dust particles are affected by gravitational perturbations with the planets via the trapping of particles in mean motion resonances (MMRs), the effect of secular resonances and the gravitational scattering of dust. This results in the formation of structure in the dust disk (Figure 12).
Dust particles drifting inward can become entrapped in exterior MMRs because at these locations the particle receives energy from the perturbing planet that can balance the energy loss due to P-R drag, halting the migration. This makes the lifetime of particles trapped in outer MMRs longer than in inner MMRs (Liou & Zook 1997), with the former dominating the disk structure. This results in the formation of resonant rings outside the planet’s orbit, as the vast majority of the particles spend most of their lifetimes trapped in exterior MMRs. In some cases, due to the geometry of the resonance, a clumpy structure is created. Figure \[KBdust\] (from Moro-Martín and Malhotra 2002) shows the effect of resonant trapping expected in the (yet to be observed) KB dust disk, where the ring-like structure, the asymmetric clumps along the orbit of Neptune, and the clearing of dust at Neptune’s location are all due to the trapping of particles in MMRs with Neptune (as seen in the histogram of semimajor axis). Neptune plays the leading role in the trapping of dust particles because of its mass and because it is the outermost planet and its exterior resonances are not affected by the interior resonances from the other planets. Trapping is more efficient for larger particles (i.e. smaller $\beta$ values) because the drag force is weaker and the particles cross the resonance at a slower rate increasing their probability of being captured. The effects of resonance trapping in hypothetical planetary systems each consisting of a single planet on a circular orbit and an outer planetesimal belt similar to the KB are shown in Figure \[FIX\]. More eccentric planets ($e < 0.6$) can also create clumpy eccentric rings and offset rings with a pair of clumps (Kuchner and Holman 2003). Even though, as mentioned in $\S$7.2, the KB dust disk has yet to be observed, this is not the case for the Zodical cloud, for which [*IRAS*]{} and [*COBE*]{} thermal observations show that there is a ring of asteroidal dust particles trapped in exterior resonances with the Earth at around 1 AU, with a 10% number density enhancement on the Earth’s wake that results from the resonance geometry (Dermott et al. 1994).
![Expected number density distribution of the KB dust disk for nine different particle sizes (or $\beta$ values). $\beta$ is a dimensionless constant equal to the ratio between the radiation pressure force and the gravitational force and depends on the density, radius and optical properties of the dust grains. If we assume that the grains are composed of spherical astronomical silicates ($\rho$=2.5, Weingartner & Draine 2001), $\beta$ values of 0.4, 0.2, 0.1, 0.05, 0.025, 0.0125, 0.00625, 0.00312, 0.00156 correspond to grain radii of 0.7, 1.3, 2.3, 4.5, 8.8, 17.0, 33.3, 65.9, 134.7 $\mu$m, respectively. The trapping of particles in MMRs with Neptune is responsible for the ring-like structure, the asymmetric clumps along the orbit of Neptune, and the clearing of dust at Neptune’s location (indicated with a black dot). The disk structure is more prominent for larger particles (smaller $\beta$ values) because the P-R drift rate is slower and the trapping is more efficient. The disk is more extended in the case of small grains (large $\beta$ values) because small particles are more strongly affected by radiation pressure. The histogram shows the relative occurrence of the different MMRs for different sized grains, where the large majority of the peaks correspond to MMRs with Neptune. The inner depleted region inside $\sim$ 10 AU is created by gravitational scattering of dust grains with Jupiter and Saturn. More details on these models can be found in Moro-Martín & Malhotra (2002, 2003).[]{data-label="KBdust"}](density_ahisto_SS_crop.pdf){width="100.00000%"}
![Same as Figure 12 but for nine hypothetical planetary systems around a solar type star consisting of a single planet with a mass of 1, 3 or 10M$_{Jup}$ in a circular orbit at 1, 5 or 30 AU, and a coplanar outer planetesimal belt similar to the KB. The models with 1 M$_{Jup}$ planet at 1 and 5 AU show that the dust particles are preferentially trapped in the 2:1 and 3:1 resonances, but when the mass of the planet is increased to 10 M$_{Jup}$, and consequently the hill radius of the planet increases, the 3:1 becomes dominant. The resonance structure becomes richer when the planet is further away from the star, and when the mass of the planet decreases (compare the model for a 1M$_{Jup}$ planet at 30 AU with the Solar system model in Figure 12, where the structure is dominated by Neptune). From Moro-Martín, Wolf & Malhotra (in preparation). []{data-label="singleplanet"}](density_ahisto_singleplanet_crop.png){width="100.00000%"}
Secular perturbations, the long-term average of the perturbing forces, act on timescales $>$0.1 Myr. If the planet and the planetesimal disk are not coplanar, the secular perturbations can create a warp in the dust disk as their tendency to align the orbits operates on shorter timescales closer to the star. A warp can also be created in systems with two non-coplanar planets. If the planet is in an eccentric orbit, the secular resonances can force an eccentricity on the dust particles and this creates an offset in the disk center with respect to the star that can result in a brightness asymmetry. Other effects of secular perturbations are spirals and inner gaps (Wyatt et al. 1999). The effect of secular perturbations can be seen in [*IRAS*]{} and [*COBE*]{} observations on the Zodiacal cloud and account for the presence of an inner edge around 2 AU due to a secular resonance with Saturn (that also explains the inner edge of the main asteroid belt), the offset of the cloud center with respect to the Sun, the inclination of the cloud with respect to the ecliptic, and the cloud warp (see review in Dermott et al. 2001).
The efficient ejection of dust grains by gravitational scattering with massive planets as the particles drift inward from an outer belt of planetesimals (Figures \[wind\] and \[efficiency\]) can result in the formation of a dust depleted region inside the orbit of the planet (Figures \[KBdust\] and \[singleplanet\]), such as the one expected inside 10 AU in the KB dust disk models (due to gravitational scattering by Jupiter and Saturn).
![Expected number density distribution of a KB dust disk composed of particles with $\beta$ = 0.2 with the trajectories of the particles ejected by Jupiter in white. The black dot indicates the position of Neptune and the circles correspond to the orbits of the Giant planets. In addition to the population of small grains with $\beta$ $>$ 0.5 blown-out by radiation pressure, the gravitational scattering by the giant planets (Jupiter and Saturn in the case of the Solar system) produces an outflow of large grains ($\beta$ $<$ 0.5) that is largely confined to the ecliptic (Moro-Martín & Malhotra 2005b). Interestingly, a stream of dust particles arriving from the direction of $\beta$ Pictoris has been reported by Baggaley (2000). []{data-label="wind"}](xyn_eject_beta02_crop.jpg){width="100.00000%"}
![Percentage of dust particles ejected from the system by gravitational scattering with the planet for the single planet models in Figure \[singleplanet\]. The particle size is fixed, corresponding to grains with $\beta$ = 0.044. Because gravitational scattering is independent of the particle size, the efficiency of ejection is fairly independent of $\beta$. [*Left*]{}: Dependency of the efficiency of ejection on the planet’s mass (x-axis) and the planet’s semimajor axis (indicated by the different colors). [*Right*]{}: Dependency of the efficiency of ejection on the planet semimajor axis (x-axis) and eccentricity (corresponding to the different colors). The models in the right panel correspond to a 1 M$_{Jup}$ mass planet on a circular orbit around a solar type star. Planets with masses of 3–10 M$_{Jup}$ at 1 AU–30 AU in a circular orbit eject $>$90% of the dust grains that go past their orbits under P-R drag; a 1 M$_{Jup}$ planet at 30 AU ejects $>$80% of the grains, and about 50%–90% if located at 1 AU, while a 0.3 M$_{Jup}$ planet is not able to open a gap, ejecting $<$ 10% of the grains. These results are valid for dust grains sizes in the range 0.7 $\mu$m–135 $\mu$m. From Moro-Martín and Malhotra 2005. []{data-label="efficiency"}](ejection_efficiency_crop_v2.pdf){width="95.00000%"}
**Next generation** facilities will probably be unable to detect diffuse emission from Kuiper dust because the optical depth is so low, and the effects of foreground and background confusion so large. However, small Kuiper belt objects are sufficiently numerous that there is a non-negligible chance that the collision clouds of recent impacts will be detected (Stern 1996). Such clouds can potentially be very bright, and evolve on timescales (days and weeks) that are amenable to direct observational investigation using *LSST* and *JWST*. Collision cloud measurements provide our best chance to understand the sub-kilometer population in the Kuiper belt. These objects are too small to be directly detected but are of special relevance as the precursors to the Centaurs and Jupiter family comet nuclei. Measurement of their number is of central importance in understanding the role of the Kuiper belt as the JFC source, and as the source of Kuiper dust.
Kuiper Belts of Other Stars
===========================
Radial velocity studies have revealed that $>$7% of solar-type stars harbor giant planets with masses $<$13 M$_{Jup}$ and semimajor axis $<$ 5 AU (Marcy et al. 2005). This is a lower limit because the duration of the surveys (6–8 years) limits the ability to detect planets long-period planets; the expected frequency extrapolated to 20 AU is $\sim$12% (Marcy et al. 2005). As of February 2008, 276 extra-solar planets have been detected with a mass distribution that follows d$\it{N}$/d$\it{M}$ $\propto$$\it{M}$$^{-1.05}$ from 0.3M$_{Jup}$ to 10 M$_{Jup}$ (the surveys are incomplete at smaller masses). A natural question arises whether these planetary systems, some of them harboring multiple planets, also contain planetesimals like the asteroids, comets and KBOs in the Solar system. Long before extra-solar planets were discovered we inferred that the answer to this question was yes: colliding planetesimals had to be responsible for the dust disks observed around mature stars. In $\S$8.1 we will discuss how these dust disks, known as [*debris disks*]{}, can help us study indirectly Kuiper belts around other stars; other methods by which extra-solar Kuiper belts might be found and characterized in the future will be discussed in $\S$8.2 and $\S$8.3.
Debris Disks
------------
### Evidence of Planetesimals
Theory and observations show that stars form in circumstellar disks composed of gas and dust that had previously collapsed from the densest regions of molecular clouds. For solar type stars, the masses of these disks are $\sim$ 0.01–0.10 M$_{\odot}$ and extend to 100s of AU, comparable to the minimum mass solar nebula ($\sim$0.015 M$_{\odot}$ , which is the mass required to account for the condensed material in the Solar system planets). Over time, these primordial or proto-planetary disks, with dust grain properties similar to those found in the interstellar medium, dissipate as the disk material accretes onto the star, is blown away by stellar wind ablation, photo-evaporation or high-energy stellar photons, or is stripped away by passing stars. The primordial gas and dust in these disks dissipate in less than 10$^{7}$ years (see e.g. Hartmann 2000 and references therein).
However, it is found that some main sequence stars older than $\sim$10$^{7}$ years show evidence of dust emission. In most cases, this evidence comes from the detection of an infrared flux in excess of that expected from the stellar photosphere, thought to arise from the thermal emission of circumstellar dust. In some nearby stars, like the ones shown in Figures \[debrisdisks\], \[aumic\] and \[vega\], direct imaging has confirmed that the emission comes from a dust disk. In $\S$7.3 we discussed the lifetimes of the dust grains due to radiation pressure, P-R drag and mutual grain collisions. It is found that in most cases these lifetimes are much shorter than the age of the star[^1], and therefore the observed dust cannot be primordial but is more likely produced by a reservoir of undetected dust-producing planetesimals, like the KBOs, asteroids and comets in the Solar system (see e.g. Backman and Paresce 1993). This is why these dust disks observed around mature main sequence stars are known as debris disks. Debris disks are evidence of the presence of planetesimals around other main sequence stars. In the core accretion model, these planetesimals formed in the earlier protoplanetary disk phase described above, as the ISM-like dust grains sedimented into the mid-plane of the disk and aggregated into larger and larger bodies (perhaps helped by turbulence) until they became planetesimals, the largest of which could potentially become the seeds out of which the giant planets form from the accretion of gas onto these planetary cores.
Even though these extra-solar planetesimals remain undetected, the dust they produce has a much larger cumulative surface area that makes the dust detectable in scattered light and in thermal emission. The study of these debris disks can help us learn indirectly about their parent planetesimals, roughly characterizing their frequencies, location and composition, and even the presence of massive planets.
### Spatially Resolved Observations
Most debris disk observations are spatially unresolved and the debris disks are identified from the excess thermal emission contributed by dust in their spectral energy distributions (SEDs). In a few cases (about two dozen so far), the disks are close enough and the images are spatially resolved. Figures \[debrisdisks\], \[aumic\] and \[vega\] show the most spectacular examples. These high resolution observations show a rich diversity of morphological features including warps (Au-Mic & $\beta$-Pic), offsets of the disk center with respect to the central star ($\epsilon$-Eri and Fomalhaut), brightness asymmetries (HD 32297 and Fomalhaut), clumpy rings (Au-Mic, $\beta$-Pic, $\epsilon$-Eri and Fomalhaut) and sharp inner edges (Fomalhaut), features that, as discussed in $\S$7.3 could be due to gravitational perturbations of massive planets, and that in some cases have been observed in the zodiacal cloud, while in other cases used to belong to the realm of KB disk models. Even though the origin of individual features is still under discussion and the models require further refinements (e.g. in the dust collisional processes and the effects of gas drag), the complexity of these features, in particular the azimuthal asymmetries, indicate that planets likely play a role in the creation of structure in the debris disks. This is of interest because the structure, in particular that created by the trapping of particles in MMRs, is sensitive to the presence of moderately massive planets at large distances (recall the KB dust disk models and the structure created by Neptune). This is a parameter space that cannot be explored with the present planet detection techniques, like the radial velocity and transient studies, and therefore the study of debris disk structure can help us learn about the diversity of planetary systems. In this context, high resolution debris disk observations over a wide wavelength range are of critical importance, like those to be obtained with [*Herschel*]{}, [*JWST*]{} and [*ALMA*]{}.
![Spatially resolved images of nearby debris disks showing dust emission from 10s to 100s of AU with a wide diversity of complex features including inner gaps, warps, brightness asymmetries, offsets and clumply rings, some of which may be due to the presence of massive planets ($\S$7.3). [*Left*]{} (from top to bottom): AU-Mic (Keck AO at 1.63 $\mu$m; Liu 2004), $\beta$-Pic (STIS CCD coronography at 0.2–1 $\mu$m; Heap et al. 2000), $\epsilon$-Eri (JCMT/SCUBA at 850 $\mu$m; Greaves et al. 2005) and HD 32297 (HST/NICMOS coronography at 1.1 $\mu$m; Schneider, Silverstone and Hines 2005). [*Right*]{}. All images correspond to Fomalhaut (7.7 pc away) and are on the same scale. From top to bottom: Spitzer/MIPS at 24 $\mu$m (Stapelfeldt et al. 2004); Spitzer/MIPS at 70 $\mu$m (Stapelfeldt et al. 2004); JCMT/SCUBA at 450 $\mu$m (Holland et al. 2003) and HST/ACS at 0.69–0.97 $\mu$m (Kalas et al. 2005). For this last panel, the annular disk in the scattered light image has an inner radius of $\sim$133 AU and a radial thickness of $\sim$25 AU and its center is offset from the star by about 15$\pm$1 AU in the plane, possibly induced by an unseen planet. Its sharp inner edge has also been interpreted as a signature of a planet (Kalas et al. 2005). The dust mass of the Fomalhaut debris disk in millimeter-sized particles is about 10$^{23}$ kg ($\sim$0.02 M$_{\oplus}$; Holland et al. 1998) but, if larger bodies are present, the mass could be 50 to 100 M$_{\oplus}$, or about 1000 times the mass of our Kuiper belt. []{data-label="debrisdisks"}](debrisdisksimages.jpg){width="90.00000%"}
![HST image of the AU Mic disk with the central regions obscured by a coronagraphic mask. AU Mic is a 12$_{-4}^{+8}$ Myr old M dwarf ($\sim$0.5 M$_{\odot}$) only 9.9$\pm$0.1 pc from Earth. Its excess thermal emission at submillimeter wavelengths suggests a dust mass near 0.01 M$_{\oplus}$ (Kalas et al. 2004) while, in scattered light, it shows a nearly edge-on disk about 100 AU in diameter with evidence for structure (Liu 2004). From Liu et al. (2004). []{data-label="aumic"}](AUMic.jpg){width="\textwidth"}
![Spatially resolved images of Vega from Spitzer/MIPS at 24, 70, 160 $\mu$m (Su et al. 2005) and from JCMT/SCUBA at 850 $\mu$m (Holland et al. 1998). All images are in the same scale. The instrument beam sizes (shown in white circles) indicate that the wide radial extent of the MIPS disk images compared to the SCUBA disk image is not a consequence of the instrumental PSF but due to a different spatial location of the particles traced by the two instruments. The sub-mm emission is thought to arise from large dust particles originating from a planetesimal belt analogous to the KB, while the MIPS emission is though to correspond to smaller particles with $\beta$$<$0.5 (may be due to porosity), produced by collisions in the planetesimal belt traced by the sub-mm observations, and that are blown away by radiation pressure to distances much larger than the location of the parent bodies (Su et al. 2005). This scenario would explain not only the wider extent of the MIPS disk but also its uniform distribution, in contrast with the clumply and more compact sub-mm disk. []{data-label="vega"}](vegaimages.jpg){width="80.00000%"}
### Spectral Energy Distributions
As we mentioned above, most of the debris disks observations are spatially unresolved and are limited to the study of the SED of the star+disk system. Even assuming that the dust is distributed in a disk (and not, for example, in a spherical shell) there are degeneracies in the SED analysis and the dust distribution cannot be unambiguously determined (e.g. Moro-Moro-Martín, Wolf & Malhotra 2005a). Nevertheless, a wealth of information can be extracted from the SED. [*IRAS*]{} and [*ISO*]{} made critical discoveries on this front, but the number of known debris disks remained too small for statistical studies. This changed recently with the unprecedented sensitivity of the [*Spitzer*]{} instruments, that allowed the detection of hundreds of debris disks in large stellar surveys that searched for dust around 328 single FGK stars (Hillenbrand et al. 2008, Meyer et al. 2008, Carpenter et al. in preparation), a different sample of 293 FGK stars (Trilling et al. 2008, Beichman et al. 2006a, 2006b, Bryden et al. 2006), 160 A single stars (Su et al. 2006, Rieke et al. 2005); 69 A3–F8 binary stars (Trilling et al. 2007), and in young stellar clusters (Gorlova et al. 2007, Siegler et al. 2007). As a result, we now possess information concerning their frequencies, their dependency with stellar type and stellar environment, their temporal evolution and the composition of the dust grains (see e.g. Moro-Martín et al. 2007 for a recent review).\
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[*Debris Disk Frequencies*]{}\
The [*Spitzer FEPS*]{}[^2] survey of 328 FGK stars found that the frequency of 24 $\mu$m excess is 14.7% for stars younger than 300 Myr and 2% for older stars, while at 70 $\mu$m, the excess rates are 6–10% (Hilllenbrand et al. 2008 and Carpenter et al. in preparation). These disks show characteristic temperatures of 60–180 K with evidence of a population of colder grains to account for the 70 $\mu$m excesses; the implied disk inner radii are $>$ 10 AU and extend over tens of AU (see Figure \[carpenter1\] – Carpenter et al. in preparation). Figure \[trilling11\] shows the debris disks incidence rates derived from a combined sample of 350 AFGKM stars from Trilling et al. (2008); for the 225 Sun-like (FG) stars in the sample older than 600 Myr, the frequency of the debris disks are 4.2$^{+2}_{-1.1}$% at 24 $\mu$m and 16.4$^{+2.8}_{-2..9}$% at 70 $\mu$m.
The above debris disks incidence rates compare to $\sim$20% of solar-type stars that harbor giant planets inside 20 AU (Marcy et al. 2005). Even though the frequencies seem similar, one should keep in mind that the sensitivity of the [*Spitzer*]{} observations is limited to fractional luminosities of L$_{dust}$/L$_*$$>$10$^{-5}$, i.e. $>$100 times the expected luminosity from the KB dust in our Solar system. Assuming a gaussian distribution of debris disk luminosities and extrapolating from [*Spitzer*]{} observations (showing that the frequency of dust detection increases steeply with decreasing fractional luminosity), Bryden et al. (2006) found that the luminosity of the Solar system dust is consistent with being 10 $\times$ brighter or fainter than an average solar-type star, i.e. debris disks at the Solar system level could be common. Observations therefore indicate that planetary systems harboring dust-producing KBOs are more common than those with giant planets, which would be in agreement with the core accretion models of planet formation where the planetesimals are the building blocks of planets and the conditions required for to form planetesimals are less restricted than those to form gas giants. Indeed, there is no apparent difference between the incidence rate of debris disks around stars with and without known planetary companions (Moro-Moro-Martín et al. 2007, Bryden et al. in preparation), although planet-bearing stars tend to harbor more dusty disks (Bryden et al. in preparation), which could result from the excitation of the planetesimals’ orbits by gravitational perturbations with the planet. Figure \[trilling11\] shows that there is no dependency on stellar type, neither in the frequency of debris disks, nor on the dust mass and location, indicating that planetesimal formation can take place under a wide range of conditions (Trilling et al. 2008).\
![Probability distribution for disk inner radii based on the analysis of the spectra (12–35 $\mu$m) of 44 debris disks around FGK stars from the [*FEPS*]{} survey. The dashed and grey histograms correspond to sources with and without 70 $\mu$m excess, respectively (with best fit parameters are shown as open and grey circles). Typical disk inner radius are $\sim$ 40 AU and $\sim$ 10 AU for disks with and without 70 $\mu$m excess, respectively, indicating that most of the debris disks observed are KB-like. Figure adapted with permission from Carpenter et al. (in preparation). []{data-label="carpenter1"}](carpenter1_v2.pdf){width="70.00000%"}
![The percentage of stars showing excess dust emission, i.e. with indirect evidence of the presence of dust-producing planetesimal belts, as a function of stellar type for ages $>$ 600 Myr (the mean ages within each type are shown at the top). The vertical bars correspond to binomial errors that include 68% of the probability (1 $\sigma$ for Gaussian errors). Black is for 24 $\mu$ excess emission (tracing warmer dust) and red is for 70 $\mu$ (tracing colder dust). The data are consistent with no dependence on spectral type. There seems to be a weak decrease with spectral type at 70 $\mu$ but so far this is statistically not significant and could be due to an effect of age. Further analysis indicates that percentage of stars showing excess is different in old A stars and in M stars than in FGK stars. The excess rate for old M stars is 0% with upper limits (binomial errors) of 2.9% at 24 $\mu$m and 12% at 70 $\mu$m (Gautier et al. 2007). The lack of distant disks around K stars may be an observational bias because their peak emission would be at $\lambda$ $>$ 70 $\mu$m and therefore remain undetected by [*Spitzer*]{}. The upcoming [*Herschel*]{} space telescope will provide the sensitivity to explore more distant and fainter debris disks. Figure adapted from Trilling et al. (2008) with data from Su et al. (2006), Trilling et al. (2007), Beichman et al. (2006b) and Gautier et al. (2007). []{data-label="trilling11"}](trilling_fg11.pdf){width="90.00000%"}
[*Debris Disk Evolution*]{}\
The [*FEPS*]{} survey of 328 FGK stars found that at 24 $\mu$m, the frequency of excess ($>$ 10.2% over the stellar photosphere) decreases from 14.7% at ages $<$ 300 Myr to 2% for older stars; at 70 $\mu$m, there is no apparent dependency of the excess frequency with stellar age, however, the amplitude of the 70 $\mu$m excess emission seems to decline from stars 30–200 Myr in age to older stars (Hillenbrand et al. 2008 and Carpenter et al. in preparation). Figures \[trilling13\] and \[trilling14\] from Trilling et al. (2008) show that for FGK type stars the debris disks incidence and fractional luminosity do not have a strong dependency with stellar age in the 1–10 Gyr time frame, in contrast with the 100–400 Myr evolution timescale of young (0.01–1 Gyr) stars seen in Figure \[siegler\]. Trilling et al. (2007) argues that this data suggests that the dominant physical processes driving the evolution of the dust disks in young stars might be different from those in more mature stars, and operate on different timescales: while the former might be dominated by the production of dust during transient events like the LHB in the Solar system or by individual collisions of large planetesimals (like the one giving rise to the formation of the Moon), the later might be the result of a more steady collisional evolution of a large population of planetesimals. The debris disks evolution observed by [*Spitzer*]{} for solar-type (Figure \[siegler\] – Siegler et al. 2007) and A-type stars (Rieke et al. 2005 and Su et al. 2006) indicate that both transient and more steady state dust production processes play a role; however, their relative importance and the question of how the dust production could be maintained in the oldest disks for billions of years is still under discussion.
![The percentage of stars showing excess dust emission, i.e. with indirect evidence of the presence of dust-producing planetesimal belts, as a function of age for the F0–K5 stars. The horizontal error bars are the age bins (not the age uncertainties). The vertical bars correspond to binomial errors that include 68% of the probability (1 $\sigma$ for Gaussian errors). The data are consistent with no dependency with age with a rate of $\sim$ 20%. The data seems to suggest an overall decrease but so far is statistically not significant, and if present may be due to an observational bias (because of the deficiency of excesses around the K stars in the oldest age bin). The number of stars in the bins are (from young to old): 24, 57, 60, 52, 33, and 7 (the high value of the oldest bin may be a small number statistical anomaly). For comparison, A-type stars evolve on timescales of 400 Myr (Su et al. 2006). Figure adapted from Trilling et al. (2008) with data from Beichman et al. (2006b). []{data-label="trilling13"}](trilling_fg13.pdf){width="90.00000%"}
![The fractional luminosity of the debris disks, L$_{dust}$/L$_{star}$, as a function of age for FGK stars. The open symbols show the means within each age bin; the horizontal and vertical error bars show the bin widths and 1 $\sigma$ errors, respectively. The data are consistent with no trend of L$_{dust}$/L$_{star}$ with age, but there seems to be a deficiency of disks with high L$_{dust}$/L$_{star}$ older than 6 Gyr. For comparison, the Solar system is 4.5 Gyr old and is expected to have a dust disk with L$_{dust}$/L$_{star}$ $\sim$ 10$^{-7}$–10$^{-6}$. Figure adapted from Trilling et al. (2008) with data from Beichman et al. (2006b). []{data-label="trilling14"}](trilling_fg14.pdf){width="90.00000%"}
![Ratio of the 24 $\mu$m Excess Emission over the expected stellar value for FGK stars as a function of stellar age ([*triangles*]{} for F0–F4 stars and [*circles*]{} for F5–K7 stars). The vertical alignments correspond to stars in clusters or associations. The data agree broadly with collisional cascade models of dust evolution (resulting in a 1/$t$ decay for the dust mass) punctuated by peaks of dust production due to individual collisional events. A-type stars show a similar behavior. Figure from Siegler et al. (2007) using data from Gorlova et al. (2004), Hines et al. (2006) and Song et al. (2005). []{data-label="siegler"}](siegler_24.pdf){width="80.00000%"}
An interesting example is HD 69830, one of the outliers in Figure \[hd69830\], a KOV star (0.8 M$_{\odot}$, 0.45 L$_{\odot}$) known to harbor three Neptune-like planets inside 0.63 AU. It shows a strong excess at 24 $\mu$m but no emission at 70 $\mu$m, indicating that the dust is warm and is located close to the star. The spectrum of the dust excess (Figure \[hd69830\]) shows strong silicate emission lines thought to arise from small grains of highly processed material similar to that of a disrupted P- or D-type asteroid plus small icy grains, likely located outside the outermost planet (Lisse et al. 2007). The observed levels of dust production are too high to be sustained for the entire age of the star, indicating that the dust production processes are transient (Wyatt et al. 2007).
![Spectrum of the dust excess emission from HD 69830 (Beichman et al. 2005 – [*top*]{}) compared to the spectrum of comet Hale-Bopp normalized to a blackbody temperature of 400 K (Crovisier et al. 1996 – [*bottom*]{}). HD 69830 is one of the outliers in Figure \[siegler\].[]{data-label="hd69830"}](HD69830.png){width="90.00000%"}
Whether the disks are transient or the result of the steady erosion of planetesimals is of critical importance for the interpretation of the statistics of the incidence rate of 24 $\mu$m excesses. For solar type stars, the 24 $\mu$m emission traces the 4–6 AU region. Terrestrial planet formation is expected to result in the production of large quantities of dust in this region, due to gravitational perturbations produced by large 1000 km-sized planetesimals that excite the orbits of a swarm of 1–10 km-size planetesimals, increasing their rate of mutual collisions and producing dust (Kenyon & Bromley 2005). This warm dust can therefore serve as a proxy of terrestrial planet formation. Figure \[meyer\] show the frequency of 24 $\mu$m emission for solar type (FGK) stars as a function of stellar age. This rate is $<$20% inside each age bin and decreases with age. If the dust-producing events are very long-lived, the stars that show dust excesses in one age bin will also show dust excesses at later times. In this case the frequency of warm dust (which indirectly traces the frequency of terrestrial planet formation) is $<$ 20%. However, if the dust-producing events are short-lived, shorter than the age bins, the stars showing excesses in one age bin are not the same as the stars showing excesses at other age bins, i.e. they can produce dust at different epochs, and in this case the overall frequency of warm dust is obtained from adding all the frequencies in all age bins, which results in $>$ 60% (assuming that each star only has one epoch of high dust production). If this is the case, the frequency of terrestrial planet formation would be high (Meyer et al. 2008). However, the interpretation of the data would change if the observed 24 $\mu$m excesses arise from the steady erosion of cold-KB-like disks (Carpenter et al., in preparation). Spatially resolved observations able to directly locate the dust would help resolve this issue.\
\
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**Next generation** facilities will offer high-sensitivity, high-resolution, multi-wavelength observations that should result in major breakthroughs in the study of debris disks. Debris disks are proxies for the presence of planetesimals around mature stars having a wide diversity of stellar types (A–K), suggesting that planetesimal formation is a robust process. The study of the warm dust can tell us about the frequency of terrestrial planet formation and the presence of asteroid-like bodies, while the study of the cold dust sheds light on the population of small bodies in KB-like regions. In addition, the study of debris disks around stars having a wide diversity of ages can help us learn about the evolution of planetary systems. However, the statistics so far are limited to dust disks 100–1000 $\times$ more luminous than that of our Solar system and the observations are generally spatially unresolved. High-sensitivity observations with future telescopes like [*Herschel*]{}, [*JWST*]{} and [*ALMA*]{} will be able to detect dust at the Solar system level, will help us improve our understanding of the frequency of planetesimals and, together with the result from planet searches, will show the diversity of planetary systems. Multi-wavelength observations are critical to help locate the dust in spatially unresolved disks and fundamental to interpret the debris disks statistics. High-resolution imaging observations are very important to directly locate the dust (circumventing the SED degeneracy), and to study the structure of the debris disks, perhaps serving as a planet-detection method sensitive to long-period Neptune-like planets that otherwise may be undetectable in the foreseeable future. Multi-wavelength observations also play a critical role in the interpretation of the structure because different wavelengths trace different particle sizes which have distinct dynamical characters that affect the disk morphology.
![Fraction of FGK type stars with 24 $\mu$m excess emission as a function of stellar age from a sample of 328 stars. The data points correspond to average values within a given age bin: (5/30) for stars 3–10 Myr, (9/48) for 10–30 Myr, (5/59) for 30–100Myr, (9/62) for 100–300Myr, (2/53) 300–1000 Myr. The widths of the age bins are shown by the horizontal bars, while the vertical bars show Poisson errors. Figure from Meyer et al. (2008).[]{data-label="meyer"}](meyer_08_fig2.png){width="90.00000%"}
Photospheric Pollution
----------------------
Dust produced collisionally in a Kuiper belt may spiral to the central star under the action of radiation and/or plasma drag, contaminating the photosphere with metal-enriched material. Separately, gravitational interactions and dynamical instabilities in a Kuiper belt may eject large objects (comets), causing some to impact the central star. Both processes operate in our Solar system but neither produces a spectrally distinctive signature, for example a metal enrichment, on the Sun. This is simply because the photosphere of the Sun already contains a large mass of metals and the addition of dust or macroscopic bodies makes only a tiny, fractional contribution.
However, the atmospheres of some white dwarf stars offer much more favorable opportunities for the detection of a photospheric pollution signal. First, many white dwarfs are naturally depleted in metals as a result of sedimentation of heavy elements driven by their strong gravitational fields. Since their atmospheres should be very clean, quite modest masses of heavy-element pollutants can be detected. About one fifth of white dwarfs expected to have pure hydrogen or pure helium atmospheres in fact show evidence for heavier elements, most likely due to pollution from external sources. Second, stellar evolution leading to the white dwarf stage includes the loss of stellar mass through an enhanced wind. As the central star mass decreases, the semimajor axes of orbiting bodies should increase, leading to dynamical instabilities caused by resonance sweeping and other effects (Debes and Sigurdsson 2002). Separately, white dwarf stars with Oort clouds should experience a steady flux of impacts from comets deflected inwards by stellar and galactic perturbations (Alcock et al. 1986).
An observed depletion of carbon relative to iron may suggest that the infalling material is cometary rather than of interstellar origin (Jura 2006).
Thermal Activation
------------------
The blackbody temperature, in Kelvin, of an object located at distance, $R_{AU}$, from a star of luminosity, $L_{\star}/L_{\odot}$, is $T_{BB}$ = 278 $R_{AU}^{-1/2} (L_{\star}/L_{\odot})^{1/4}$. In the Kuiper belt today, at $R_{AU}$ = 40, the blackbody temperature is $T_{BB}$ = 44 K. Water sublimation at this low temperature is negligible. However, the sublimation rate is an exponential function of temperature and, by $T_{BB}$ = 200 K, water sublimates rapidly (with a mass flux $\sim$10$^{-4}$ kg m$^{-2}$ s$^{-1}$, corresponding to ice recession at about 3 m yr$^{-1}$ for density 1000 kg m$^{-3}$). A 1 km scale body would sublimate away in just a few centuries. By the above relation, temperatures of 200 K are reached when $L_{\star}/L_{\odot}$ = 400, for the same 40 AU distance.
Stellar evolution into the red giant phase will drive the Sun’s luminosity to exceed this value after about 10 Gyr on the main-sequence, with an increase in the luminosity (by up to a factor of $\sim$10$^4$), and in the loss of mass through an enhanced stellar wind (from the current value, $\sim$10$^{-11}$ M$_{\odot}$ yr$^{-1}$, to $\sim$10$^{-7}$ M$_{\odot}$ yr$^{-1}$, or more). When this happens, the entire Kuiper belt will light up as surface ices sublimate and dust particles, previously embedded in the KBOs, are ejected into space. Not all KBOs will be destroyed by roasting in the heat of the giant Sun: observations of comets near the Sun show that these bodies can insulate themselves from the heat by the development of refractory mantles, consisting of silicate and organic-rich debris particles that are too large to be ejected by gas drag. Still, the impact of the red giant phase should be dramatic and suggests that the sublimated Kuiper belts of other stars might be detected around red-giants. The key observational signatures would be the thermal excess itself, at temperatures appropriate to Kuiper belt-like distances, and ring-like morphology. Sensitivity at thermal wavelengths combined with high angular resolution will lend *JWST* to this type of observation, although the overwhelming signal from the star itself will present a formidable observational limitation to any imaging studies.
Water vapor has been reported around carbon stars that are not expected to show water and interpreted as produced by sublimated comets (Ford and Neufeld 2001). In IRC +10216, the mass of water is estimated as 3$\times$10$^{-5}$ M$_{\odot}$ (Melnick et al. 2001). This is about 10 M$_{\oplus}$, or 100 times the mass of the modern Kuiper belt but perhaps comparable to the mass of the Kuiper belt when formed. A possible explanation is that the water derives from sublimated comets in an unseen Kuiper belt, but chemical explanations for this large water mass may also be possible (Willacy 2004). On the other hand, a search for (unresolved) thermal emission from dust around 66 first-ascent red giants proved negative, with limits on the Kuiper belt masses of these stars near 0.1 M$_{\oplus}$, the current mass of the Kuiper belt (Jura 2004).
DJ was supported by a grant from NASA’s Origins program, PL by an NSF Planetary Astronomy grant to DJ. A.M.M. is under contract with the Jet Propulsion Laboratory (JPL) funded by NASA through the Michelson Fellowship Program. A.M.M. is also supported by the Lyman Spitzer Fellowship at Princeton University.
[99.]{} Adams, F. C., & Laughlin, G. 2001: Constraints on the Birth Aggregate of the Solar System. Icarus, 150, 151
Alcock, C., Fristrom, C. C., & Siegelman, R.1986: On the number of comets around other single stars. Ap. J., 302, 462
Anderson, J. D., Lau, E. L., Scherer, K., Rosenbaum, D. C., & Teplitz, V. L. 1998: Kuiper Belt Constraint from Pioneer 10 Icarus, 131, 167
Babich, D., Blake, C. H., & Steinhardt, C. L.2007: What Can the Cosmic Microwave Background Tell Us about the Outer Solar System? Ap. J., 669, 1406
Backman, D. E., Dasgupta, A., & Stencel, R. E.1995: Model of a Kuiper Belt Small Grain Population and Resulting Far-Infrared Emission. Ap. J. Lett, 450, L35
Backman, D. E., & Paresce, F. 1993: Main-sequence stars with circumstellar solid material - The VEGA phenomenon.Protostars and Planets III, 1253
Baggaley, W. J. 2000: Advanced Meteor Orbit Radar observations of interstellar meteoroids. JGR, 105, 10353
Bar-Nun, A., Kleinfeld, I., & Kochavi, E. 1988: Trapping of gas mixtures by amorphous water ice. Phys. Rev. B., 38, 7749
Barrado y Navascu[é]{}s, D., Stauffer, J. R., Song, I., & Caillault, J.-P. 1999: The Age of beta Pictoris. Ap. J. Lett, 520, L123
Beichman, C. A., et al. 2006a: IRS Spectra of Solar-Type Stars: A Search for Asteroid Belt Analogs. Ap. J., 639, 1166
Beichman, C. A., et al. 2006b: New Debris Disks around Nearby Main-Sequence Stars: Impact on the Direct Detection of Planets.Ap. J., 652, 1674
Boley, A. C., Durisen, R. H., Nordlund, [Å]{}., & Lord, J. 2007: Three-Dimensional Radiative Hydrodynamics for Disk Stability Simulations: A Proposed Testing Standard and New Results. Ap. J., 665, 1254
Boss, A. P. 2001: Gas Giant Protoplanet Formation: Disk Instability Models with Thermodynamics and Radiative Transfer. Ap. J., 563, 367
Boss, A. P. 2007: Testing Disk Instability Models for Giant Planet Formation. Ap. J. Lett., 661, L73
Boss, A. P., Wetherill, G. W., & Haghighipour, N. 2002: Rapid Formation of Ice Giant Planets. Icarus, 156, 291
Bottke, W. F., Morbidelli, A., Jedicke, R., Petit, J.-M., Levison, H. F., Michel, P., & Metcalfe, T. S. 2002: Debiased Orbital and Absolute Magnitude Distribution of the Near-Earth Objects. Icarus, 156, 399
Brown, M. E. 2001: The Inclination Distribution of the Kuiper Belt. Astron. J., 121, 2804
Brown, M. E., Trujillo, C., & Rabinowitz, D. 2004: Discovery of a Candidate Inner Oort Cloud Planetoid. Ap. J., 617, 645
Brownlee, D. E., Joswiak, D. J., Love, S. G., Bradley, J. P., Nier, A. O., & Schlutter, D. J. 1994: Identification and Analysis of Cometary IDPs. Lunar and Planetary Institute Conference Abstracts, 25, 185
Bryden, G., et al. 2006: Frequency of Debris Disks around Solar-Type Stars: First Results from a Spitzer MIPS Survey. Ap. J., 636, 1098
Buie, M. W., Tholen, D. J., Horne, K. 1992:Albedo maps of Pluto and Charon - Initial mutual event results. Icarus 97, 21
Burns, J. A., Lamy, P. L., & Soter, S. 1979: Radiation forces on small particles in the solar system. Icarus, 40, 1
Catullo, V., Zappala, V., Farinella, P., Paolicchi, P. 1984: Analysis of the shape distribution of asteroids. Astron. Ap. 138, 464
Chapman, C. R., Cohen, B. A., & Grinspoon, D. H. 2007: What are the real constraints on the existence and magnitude of the late heavy bombardment? Icarus, 189, 233
Crovisier, J., et al. 1996: The infrared spectrum of comet C/1995 O1 (Hale-Bopp) at 4.6 AU from the Sun.. Astron. Ap., 315, L385
Cruikshank, D. P., Barucci, M. A., Emery, J. P., Fern[á]{}ndez, Y. R., Grundy, W. M., Noll, K. S., Stansberry, J. A. 2007:Physical Properties of Transneptunian Objects. Protostars and Planets V, 879
Davis, D. R., Farinella, P. 1997: Collisional Evolution of Edgeworth-Kuiper Belt Objects. Icarus 125, 50
Debes, J. H., & Sigurdsson, S. 2002: Are There Unstable Planetary Systems around White Dwarfs? Ap. J., 572, 556
Delsanti, A., Hainaut, O., Jourdeuil, E., Meech, K. J., Boehnhardt, H., Barrera, L. 2004: Simultaneous visible-near IR photometric study of Kuiper Belt Object surfaces with the ESO/Very Large Telescopes. Astron. Ap., 417, 1145
Dermott, S. F., Grogan, K., Durda, D. D., Jayaraman, S., Kehoe, T. J. J., Kortenkamp, S. J., Wyatt, M. C. 2001: Orbital evolution of interplanetary dust. Interplanetary Dust (Grum, Gustafson, Dermott, Fechtig eds.) Springer, A&A Library, pp. 569–639
Dermott, S. F., Jayaraman, S., Xu, Y. L., Gustafson, B. A. S., & Liou, J. C. 1994: A circumsolar ring of asteroidal dust in resonant lock with the Earth. Nature, 369, 719
Dermott, S. F., Kehoe, T. J. J., Durda, D. D., Grogan, K., & Nesvorn[ý]{}, D. 2002: Recent rubble-pile origin of asteroidal solar system dust bands and asteroidal interplanetary dust particles. Asteroids, Comets, and Meteors: ACM 2002, 500, 319
Duncan, M., Quinn, T., & Tremaine, S. 1988: The origin of short-period comets. Ap. J. Lett., 328, L69
Duncan, M. J., & Levison, H. F. 1997: A scattered comet disk and the origin of Jupiter family comets. Science, 276, 1670
Elliot, J. L., et al. 2005: The Deep Ecliptic Survey: Astron. J., 129, 1117
Farinella, P., Paolicchi, P., Zappala, V. 1981:Analysis of the spin rate distribution of asteroids. Astron. Ap. 104, 159
Fernández, J. A. 1980: On the existence of a comet belt beyond Neptune. MNRAS, 192, 481
Fernández, J. A., & Ip, W.-H. 1984: Some dynamical aspects of the accretion of Uranus and Neptune - The exchange of orbital angular momentum with planetesimals. Icarus, 58, 109
Fern[á]{}ndez, Y. R., Jewitt, D. C., Sheppard, S. S. 2001. Low Albedos Among Extinct Comet Candidates. Astrophysical Journal 553, L197-L200.
Ford, K. E. S., & Neufeld, D. A. 2001: Water Vapor in Carbon-rich Asymptotic Giant Branch Stars from the Vaporization of Icy Orbiting Bodies. Ap. J. Lett., 557, L113
Francis, P. J. 2005: The Demographics of Long-Period Comets. Ap. J., 635, 1348
Gaidos, E. J. 1995: Paleodynamics: Solar system formation and the early environment of the sun. Icarus, 114, 258
Gautier, T. N., III, et al. 2007: Far-Infrared Properties of M Dwarfs. Ap. J., 667, 527
Gladman, B., Holman, M., Grav, T., Kavelaars, J., Nicholson, P., Aksnes, K., & Petit, J.-M. 2002: Evidence for an Extended Scattered Disk. Icarus, 157, 269
Gomes, R. S. 2003: The origin of the Kuiper Belt high-inclination population. Icarus 161, 404
Gorlova, N., et al. 2004: New Debris-Disk Candidates: 24 Micron Stellar Excesses at 100 Million years. Ap. J. Supp., 154, 448 Gorlova, N., Balog, Z., Rieke, G. H., Muzerolle, J., Su, K. Y. L., Ivanov, V. D., & Young, E. T. 2007: Debris Disks in NGC 2547. Ap. J., 670, 516
Graham, J.R., Kalas, P.G., & Matthews, B. C. 2007: The Signature of Primordial Grain Growth in the Polarized Light of the AU Microscopii Debris Disk. Ap. J., 654, 595 Greaves, J. S., et al. 2005: Structure in the [$\epsilon$]{} Eridani Debris Disk. Ap. J. Lett., 619, L187
Grogan, K., Dermott, S. F., & Durda, D. D.2001: The Size-Frequency Distribution of the Zodiacal Cloud: Evidence from the Solar System Dust Bands. Icarus, 152, 251 Grün, E., Gustafson, B., Mann, I., Baguhl, M., Grun, E., Gustafson, B., Mann, I., Baguhl, M., dust in the heliosphere. Astron. Ap., 286, 915 3125 Grün, Baguhl, M., Svedhem, H. & Zook, H. A. 2001: In situ measurements of cosmic dust. Interplanetary Dust (E. Grün, B. A. S. Gustafson, S. F. Dermott, H. Fechtig, eds.), Springer A&A Library, p. 293. dust in the heliosphere. Astron. Ap., 286, 915
Guillot, T., & Hueso, R. 2006: The composition of Jupiter: sign of a (relatively) late formation in a chemically evolved protosolar disc. MNRAS, 367, L47
Gurnett, D. A., Ansher, J. A., Kurth, W. S., & system by the Voyager 1 and 2 plasma wave instruments. Geoph. R. Lett., 24, 3125 24, 3125
Hartmann L. 2000: Accretion Processes in Star Formation, Cambridge University Press.
Hahn, J. M., & Malhotra, R. 1999: Orbital Evolution of Planets Embedded in a Planetesimal Disk. A. J., 117, 3041
Harris, A. W. 1979: Asteroid rotation rates II. A theory for the collisional evolution of rotation rates. Icarus 40, 145
Heap, S. R., Lindler, D. J., Lanz, T. M., Cornett, R. H., Hubeny, I., Maran, S. P., & Woodgate, B. 2000: Space Telescope Imaging Spectrograph Coronagraphic Observations of [$\beta$]{} Pictoris. Ap. J., 539, 435
Higuchi, A., Kokubo, E., Kinoshita, H., & Mukai, T. 2007: Orbital Evolution of Planetesimals due to the Galactic Tide: Formation of the Comet Cloud. A. J., 134, 1693
Hillenbrand, L. A., et al. 2008: The Complete Census of 70-um-Bright Debris Disks within the FEPS (Formation and Evolution of Planetary Systems) Spitzer Legacy Survey of Sun-like Stars. ApJ, in press (arXiv:0801.0163).
Hines, D. C., et al. 2006: The Formation and Evolution of Planetary Systems (FEPS): Discovery of an Unusual Debris System Associated with HD 12039. Ap. J., 638, 1070
Holland, W. S., et al. 1998: Submillimetre images of dusty debris around nearby stars. Nature, 392, 788
Holland, W. S., et al. 2003: Submillimeter Observations of an Asymmetric Dust Disk around Fomalhaut. Ap. J., 582, 1141
Holland, W. S., et al. 1998: Submillimetre images of dusty debris around nearby stars. Nature, 392, 788
Holman, M. J., & Wisdom, J. 1993: Dynamical stability in the outer solar system and the delivery of short period comets. Astron. J., 105, 1987
Hsieh, H. H., & Jewitt, D. 2006: A Population of Comets in the Main Asteroid Belt. Science, 312, 561
Ida, S., Larwood, J., & Burkert, A. 2000: Evidence for Early Stellar Encounters in the Orbital Distribution of Edgeworth-Kuiper Belt Objects. Ap. J., 528, 351
Jewitt, D. C. 2002: From Kuiper Belt Object to Cometary Nucleus: The Missing Ultrared Matter. Astron. J. 123, 1039
Jewitt, D. 2003: Project Pan-STARRS and the Outer Solar System. Earth Moon and Planets, 92, 465
Jewitt, D., & Luu, J. 1993: Discovery of the candidate Kuiper belt object 1992 QB1. Nature, 362, 730
Jewitt, D., Luu, J. 1998: Optical-Infrared Spectral Diversity in the Kuiper Belt. Astron. J., 115, 1667
Jewitt, D. C., Luu, J. X. 2001: Colors and Spectra of Kuiper Belt Objects. Astron. J., 122, 2099
Jewitt, D., Luu, J., & Trujillo, C. 1998: Large Kuiper Belt Objects: The Mauna Kea 8K CCD Survey. Astron. J., 115, 2125
Jewitt, D., Peixinho, N., Hsieh, H. H. 2007:U-Band Photometry of Kuiper Belt Objects. Astron. J., 134, 2046
Jewitt, D. C., Sheppard, S. S. 2002: Physical Properties of Trans-Neptunian Object (20000) Varuna. Astron. J., 123, 2110
Jura, M. 2004: Other Kuiper Belts. Ap. J., 603, 729
Jura, M. 2006: Carbon Deficiency in Externally Polluted White Dwarfs: Evidence for Accretion of Asteroids. Ap. J., 653, 613
Kalas, P., Graham, J. R., & Clampin, M. 2005: A planetary system as the origin of structure in Fomalhaut’s dust belt. Nature, 435, 1067
Kalas, P., Graham, J. R., & Clampin, M. 2005: A planetary system as the origin of structure in Fomalhaut’s dust belt. Lee, T., Papanastassiou, D. A., & Wasserburg, G. J. 1977: Aluminum-26 in the early solar system - Fossil or fuel. Ap. J. Lett., 211, L107
Kalas, P., Liu, M. C., & Matthews, B. C. 2004: Discovery of a Large Dust Disk Around the Nearby Star AU Microscopii. Science, 303, 1990
Kenyon, S. J., & Bromley, B. C. 2005: Prospects for Detection of Catastrophic Collisions in Debris Disks. Astron. J., 130, 269
Kuchner, M. J., & Holman, M. J. 2003: The Geometry of Resonant Signatures in Debris Disks with Planets. Ap. J., 588, 1110
Lacerda, P., Jewitt, D. C. 2007: Densities of Solar System Objects from Their Rotational Light Curves. Astron. J., 133, 1393
Lacerda, P., Jewitt, D., Peixinho, N. 2008:High Precision Photometry of Extreme KBO 2003 EL61. In press in Astron. J.
Lacerda, P., Luu, J. 2003: On the detectability of lightcurves of Kuiper belt objects. Icarus 161, 174
Lacerda, P., Luu, J. 2006: Analysis of the Rotational Properties of Kuiper Belt Objects. Astron. J., 131, 2314
Leinert, C., Roser, S., & Buitrago, J. 1983: How to maintain the spatial distribution of interplanetary dust. Astron. Ap., 118, 345 Leinert, C., & Grün, E. 1990: Interplanetary Dust. Physics of the Inner Heliosphere I, 207
Levison, H. F., & Duncan, M. J. 1994:The long-term dynamical behavior of short-period comets. Icarus, 108, 18
Levison, H. F., Dones, L., & Duncan, M. J.2001: The Origin of Halley-Type Comets: Probing the Inner Oort Cloud. A. J., 121, 2253
Levison, H. F., Terrell, D., Wiegert, P. A., Dones, L., & Duncan, M. J. 2006:On the origin of the unusual orbit of Comet 2P/Encke. Icarus, 182, 161
Liou, J.-C., & Zook, H. A. 1997: Evolution of Interplanetary Dust Particles in Mean Motion Resonances with Planets. Icarus, 128, 354
Liu, M. C. 2004: Substructure in the Circumstellar Disk Around the Young Star AU Microscopii. Science, 305, 1442
Luu, J., Jewitt, D. 1996: Color Diversity Among the Centaurs and Kuiper Belt Objects. Astron. J., 112, 2310
Luu, J., Lacerda, P. 2003: The Shape Distribution Of Kuiper Belt Objects. Earth Moon and Planets 92, 221
Luu, J., Marsden, B. G., Jewitt, D., Trujillo, C. A., Hergenrother, C. W., Chen, J., & Offutt, W. B. 1997: A New Dynamical Class in the Trans-Neptunian Solar System. Nature, 387, 573
Marcy, G., Butler, R. P., Fischer, D., Vogt, S., Wright, J. T., Tinney, C. G., & Jones, H. R. A. 2005: Observed Properties of Exoplanets: Masses, Orbits, and Metallicities. Progress of Theoretical Physics Supplement, 158, 24
Malhotra, R. 1995: The Origin of Pluto’s Orbit: Implications for the Solar System Beyond Neptune. Astron. J., 110, 420
Melnick, G. J., Neufeld, D. A., Ford, K. E. S., Hollenbach, D. J., & Ashby, M. L. N. 2001: Discovery of water vapour around IRC+10216 as evidence for comets orbiting another star. Nature, 412, 160
Meyer, M. R., et al. 2008: Evolution of Mid-Infrared Excess around Sun-like Stars: Constraints on Models of Terrestrial Planet Formation. Ap. J. Lett., 673, L181
Moro-Mart[í]{}n, A., & Malhotra, R. 2002: A Study of the Dynamics of Dust from the Kuiper Belt: Spatial Distribution and Spectral Energy Distribution. Astron. J., 124, 2305
Moro-Mart[í]{}n, A., & Malhotra, R. 2003: Dynamical Models of Kuiper Belt Dust in the Inner and Outer Solar System.Astron. J., 125, 2255
Moro-Mart[í]{}n, A., Wolf, S., & Malhotra, R. 2005a: Signatures of Planets in Spatially Unresolved Debris Disks. Ap. J., 621, 1079
Moro-Mart[í]{}n, A., & Malhotra, R. 2005b: Dust Outflows and Inner Gaps Generated by Massive Planets in Debris Disks. Ap. J., 633, 1150
Moro-Mart[í]{}n, A., et al. 2007: Are Debris Disks and Massive Planets Correlated?. Ap. J., 658, 1312
Moro-Martin, A., Wyatt, M. C., Malhotra, R., & Trilling, D. E. 2008: Extra-Solar Kuiper Belt Dust Disks. The Solar System Beyond Neptune (A. Barucci, H. Boehnhardt, D. Cruikshank, A. Morbidelli, eds.) University of Arizona Press, Tucson. arXiv:astro-ph/0703383
Morbidelli, A., & Levison, H. F. 2004: Scenarios for the Origin of the Orbits of the Trans-Neptunian Objects 2000 CR105 and 2003 VB12 (Sedna). Astron. J, 128, 2564
Mostefaoui, S., Lugmair, G. W., Hoppe, P., & El Goresy, A. 2004: Evidence for live $^{60}$Fe in meteorites. New Astronomy Review, 48, 155
Oort, J. H. 1950: The structure of the cloud of comets surrounding the Solar System and a hypothesis concerning its origin. Bull. Astron. Inst. Neth., 11, 91
Ouellette, N., Desch, S. J., & Hester, J. J.2007: Interaction of Supernova Ejecta with Nearby Protoplanetary Disks. Ap. J., 662, 1268
Owen, T., Mahaffy, P., Niemann, H. B., Atreya, S., Donahue, T., Bar-Nun, A., & de Pater, I. 1999: A low-temperature origin for the planetesimals that formed Jupiter. Nature, 402, 269
Paresce, F. 1991: On the evolutionary status of Beta Pictoris. Astron. Ap., 247, L25
Peixinho, N., Doressoundiram, A., Delsanti, A., Boehnhardt, H., Barucci, M. A., Belskaya, I. 2003: Reopening the TNOs color controversy: Centaurs bimodality and TNOs unimodality. Astronomy and Astrophysics 410, L29.
Reach, W. T., Morris, P., Boulanger, F., & Okumura, K. 2003: The mid-infrared spectrum of the zodiacal and exozodiacal light. Icarus, 164, 384
Rice, W. K. M., & Armitage, P. J. 2003: On the Formation Timescale and Core Masses of Gas Giant Planets. Ap. J. Lett., 598, L55
Rieke, G. H., et al. 2005: Decay of Planetary Debris Disks. Ap. J., 620, 1010
Schneider, G., Silverstone, M. D., & Hines, D. C. 2005: Discovery of a Nearly Edge-on Disk around HD 32297. Ap. J. Lett., 629, L117 Siegler, N., Muzerolle, J., Young, E. T., Rieke, G. H., Mamajek, E. E., Trilling, D. E., Gorlova, N., & Su, K. Y. L. 2007: Spitzer 24 [$\mu$]{}m Observations of Open Cluster IC 2391 and Debris Disk Evolution of FGK Stars. Ap. J., 654, 580 Sheppard, S. S., Jewitt, D. C. 2002:Time-resolved Photometry of Kuiper Belt Objects: Rotations, Shapes, and Phase Functions. Astron. J., 124, 1757
Sheppard, S. S., Jewitt, D. 2004: Extreme Kuiper Belt Object 2001 QG298 and the Fraction of Contact Binaries. Astron. J., 127, 3023
Sheppard, S. S., Lacerda, P., & Ortiz, J.-L. 2008: Photometric Lightcurves of Transneptunian Objects and Centaurs: Rotations, Shapes, and Densities. In “The Solar System Beyond Neptune” (Eds. Barucci, A. et al.), Univ. Az. Press, Tucson, pp. 129
Sheppard, S. S., & Trujillo, C. A. 2006: A Thick Cloud of Neptune Trojans and Their Colors. Science, 313, 511
Song, I., Zuckerman, B., Weinberger, A. J., & Becklin, E. E. 2005: Extreme collisions between planetesimals as the origin of warm dust around a Sun-like star. Nature, 436, 363
Stapelfeldt, K. R., et al. 2004: First Look at the Fomalhaut Debris Disk with the Spitzer Space Telescope. Ap. J. Supp., 154, 458
Stern, S. A. 1996: Signatures of collisions in the Kuiper Disk.. Astron. Ap., 310, 999
Strom, R. G., Malhotra, R., Ito, T., Yoshida, F., & Kring, D. A. 2005: The Origin of Planetary Impactors in the Inner Solar System. Science, 309, 1847
Su, K. Y. L., et al. 2005: The Vega Debris Disk: A Surprise from Spitzer. Ap. J., 628, 487
Su, K. Y. L., et al. 2006: Debris Disk Evolution around A Stars. Ap. J., 653, 675
Sykes, M. V., & Greenberg, R. 1986: The formation and origin of the IRAS zodiacal dust bands as a consequence of single collisions between asteroids. Icarus, 65, 51
Tegler, S. C., Romanishin, W. 2000: Extremely red Kuiper-belt objects in near-circular orbits beyond 40 AU. Nature 407, 979
Th[é]{}bault, P., Doressoundiram, A. 2003:Colors and collision rates within the Kuiper belt: problems with the collisional resurfacing scenario. Icarus 162, 27
Trafton, L. M. 1989: Pluto’s atmosphere near perihelion. Geophysical Research Letters 16, 1213
Trilling, D. E., et al. 2007: Debris disks in main-sequence binary systems.. Ap. J., 658, 1289
Trilling, D. E., et al. 2008: Debris Disks around Sun-like Stars. Ap. J., 674, 1086
Trujillo, C. A., Jewitt, D. C., & Luu, J. X.2001: Properties of the Trans-Neptunian Belt. Astron. J. 122, 457
Willacy, K. 2004: A Chemical Route to the Formation of Water in the Circumstellar Envelopes around Carbon-rich Asymptotic Giant Branch Stars: Fischer-Tropsch Catalysis. Ap. J. Lett., 600, L87
Wyatt, M. C., Dermott, S. F., Telesco, C. M., Fisher, R. S., Grogan, K., Holmes, E. K., & Pi[ñ]{}a, R. K. 1999: How Observations of Circumstellar Disk Asymmetries Can Reveal Hidden Planets: Pericenter Glow and Its Application to the HR 4796 Disk. Ap. J., 527, 918
Wyatt, M. C., Smith, R., Greaves, J. S., Beichman, C. A., Bryden, G., & Lisse, C. M. 2007: Transience of Hot Dust around Sun-like Stars. Ap. J., 658, 569
Yamamoto, S., & Mukai, T. 1998: Dust production by impacts of interstellar dust on Edgeworth-Kuiper Belt objects. Astron. Ap., 329, 785
Young, E. F., Galdamez, K., Buie, M. W., Binzel, R. P., Tholen, D. J. 1999: Mapping the Variegated Surface of Pluto. Astron. J., 117, 1063
Zook, H. A., & Berg, O. E. 1975: A source for hyperbolic cosmic dust particles. Planetary and Space Sciences, 23, 183
[^1]: One needs to be cautious with this argument because the ages of main sequence stars are difficult to determine. For example, the prototype (and best studied) of debris disk $\beta$-Pictoris, was dated at $\sim$100 to 200 Myr (Paresce 1991) but later become only $\sim$20 Myr old (Barrado y Navascu[é]{}s et al. 1999).
[^2]: http://feps.as.arizona.edu/
|
---
abstract: 'A *transitive decomposition* of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.'
author:
- |
Alice Devillers\
Université Libre de Bruxelles\
Département de mathématiques\
Géométrie- CP 216\
Boulevard du Triomphe\
B-1050 Bruxelles Belgique\
\
Michael Giudici, Cai Heng Li and Cheryl E. Praeger\
School of Mathematics and Statistics\
The University of Western Australia\
35 Stirling Highway\
Crawley WA 6009\
Australia
title: 'Primitive decompositions of Johnson graphs[^1]'
---
Introduction
============
A *decomposition* of a graph is a partition of the edge set with at least two parts, which we interpret as subgraphs and call the *divisors* of the decomposition. If each divisor is a spanning subgraph we call the decomposition a *factorisation* and the divisors *factors*. Graph decompositions and factorisations have received much attention, see for example [@bosak; @heinrich]. Of particular interest [@iso1; @iso10] are decompositions where the divisors are pairwise isomorphic. These are known as *isomorphic decompositions*.
A *transitive decomposition* is a decomposition $\mathcal{P}$ of a graph $\Gamma$ together with a group of automorphisms $G$ which preserves the partition and acts transitively on the set of divisors. We refer to $(\Gamma,{\mathcal{P}})$ as a $G$-transitive decomposition. This is a special class of isomorphic decompositions and a general theory has been outlined in [@genpaper]. Sibley [@sibley] has described all $G$-transitive decompositions of the complete graph $K_n$ where $G$ is 2-transitive on vertices. This generalised the Cameron-Korchmaros classification in [@CamKorch] of the $G$-transitive 1-factorisations of $K_n$ (that is, the factors have valency 1) with $G$ acting 2-transitively on vertices. Note that a subgroup of $S_n$ is arc-transitive on $K_n$ if and only if it is 2-transitive. Also all $G$-transitive decompositions of graphs with $G$ inducing a rank three product action on vertices have been determined in [@BPP]. A special class of transitive decompositions called *homogeneous factorisations*, are the $G$-transitive decompositions $(\Gamma,{\mathcal{P}})$ such that the kernel $M$ of the action of $G$ on ${\mathcal{P}}$ is vertex-transitive. This implies that each divisor is a spanning subgraph and so ${\mathcal{P}}$ is indeed a factorisation. Homogeneous factorisations were first introduced in [@LP03] for complete graphs and extended to arbitrary graphs and digraphs in [@GLPP1].
The *Johnson graph* $J(n,k)$ is the graph with vertices the $k$-element subsets of an $n$-set $X$, two sets being adjacent if they have $k-1$ points in common. Note that $J(n,1)\cong K_n$ and $J(n,k)\cong J(n,n-k)$ so we always assume that $2\leq k\leq \frac{n}{2}$. Note that $J(4,2)\cong K_{2,2,2}$ while the complement of $J(5,2)$ is the Petersen graph. All homogeneous factorisations of $J(n,k)$ were determined in [@Cuaresma; @CGP]. Examples only exist for $J(q+1,2)$ for prime powers $q\equiv 1\pmod 4$, $J(q,2)$ and $J(q+1,3)$ for $q=2^{r^f}$ with $r$ an odd prime, and for $J(8,3)$. However, examples of transitive decompositions exist for all values of $n$ and $k$ (see Construction \[con:classic\]). Constructions \[con:classic\](1) and (2) were drawn to our attention by Michael Orrison. Both constructions were used in [@KK] to help determine maximal subgroups of symmetric groups while Construction \[con:classic\](1) was used in [@orrisonetal] for the statistical analysis of unranked data.
In this paper we determine all $G$-transitive decompositions of the Johnson graphs subject to two conditions on $G$. The first is that $G$ is arc-transitive while the second is that $G$ acts primitively on the set of divisors of the decomposition. We call $G$-transitive decompositions for which $G$ acts primitively on the set of divisors, *$G$-primitive decompositions.* We see in Lemma \[lem:primitive\] that any $G$-transitive decomposition is the refinement of some $G$-primitive decomposition. By Theorem \[Sn\], a subgroup $G\leqslant S_n$ acts transitively on the set of arcs of $J(n,k)$ if and only if $G$ is $(k+1)$-transitive, or $(n,k)=(9,3)$ and $G={\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$. Using this, we analyse the appropriate groups to determine all primitive decompositions. In particular we prove the following theorem.
\[thm:main\] Let $G$ be an arc-transitive group of automorphisms of $\Gamma=J(n,k)$ where $2\leq k\leq n/2$. If $(\Gamma,{\mathcal{P}})$ is a $G$-primitive decomposition then one of the following holds:
1. the divisors are matchings or unions of cycles,
2. the divisors are unions of $K_{n-k+1}, K_{k+1}$ or $K_3$, or
3. $(\Gamma,{\mathcal{P}})$ is given by one of the rows of Table \[tab:interesting\].
$\Gamma$ $G$ Divisor Comments
----------------------- ------------------------------------------------- --------------------------------------- ---------------------------------------------------
$J(6,3)$ $A_6$ or ${\langle}A_6, (1,2)\tau{\rangle}$ Petersen graph Example \[eg:An\](2)
$J(12,4)$ $M_{12}$ $2J(6,4)$ Construction \[con:design\] and \[con:primitive\]
$J(12,4)$ $M_{12}$ $\Sigma$ Construction \[con:M11graph\]
$J(24,4)$ $M_{24}$ $J(8,4)$ Construction \[con:design\]
$J(23,3)$ $M_{23}$ $J(7,3)$ Construction \[con:design\]
$J(11,3)$ $M_{11}$ $J(5,2)$ Construction \[con:design\]
$J(11,3)$ $M_{11}$ 2 Petersen graphs Construction \[con:petersen\]
$J(11,3)$ $M_{11}$ 11 Petersen graphs Construction \[con:PSL211\](2)
$J(11,3)$ $M_{11}$ $\Pi$ Construction \[con:PSL211\](1)
$J(9,3)$ ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$ ${\mathop{\mathrm{PSL}}}(2,8)$-orbits Construction \[con:pgammal\](1)
$J(9,3)$ ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$ Heawood graph Construction \[con:pgammal\](4)
$J(22,2)$ $M_{22}$ or ${\mathop{\mathrm{Aut}}}(M_{22})$ $J(6,2)$ Construction \[con:design\]
$J(2^d,2)$, $d\geq 3$ ${\mathop{\mathrm{AGL}}}(d,2)$ $2^{d-2}K_{2,2,2}$ Construction \[con:design\] and \[con:primitive\]
$J(16,2)$ $C_2^4\rtimes A_7$ $4K_{2,2,2}$ Construction \[con:design\] and \[con:primitive\]
$J(q+1,2)$ 3-transitive subgroup $J(q_0+1,2)$ Construction \[con:design\]
of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ $q=q_0^r$, $r$ prime
$J(q+1,2)$ 3-transitive subgroup ${\mathop{\mathrm{PSL}}}(2,q)$-orbits Construction \[con:PGL1\]
$q\equiv 1\pmod 4$ of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$
: $G$-primitive decompositions of $J(n,k)$ for Theorem \[thm:main\][]{data-label="tab:interesting"}
The divisor graphs $\Sigma$ and $\Pi$ of Table \[tab:interesting\] are investigated further in [@DGLP]. Construction \[con:design\] allows us to construct transitive decompositions of $J(n,k)$ with divisors isomorphic to $J(l,k)$ for any Steiner system $S(k+1,l,n)$ and this accounts for many of the examples in Table \[tab:interesting\]. Further constructions of transitive decompositions from Steiner systems are given in Section \[sec:cons\] and these have divisors isomorphic to unions of cliques or matchings.
General constructions {#sec:cons}
=====================
First we show that the study of transitive decompositions can be reduced to the study of primitive decompositions. We denote by $V\Gamma$, $E\Gamma$ and $A\Gamma$, the sets of vertices, edges and arcs respectively, of the graph $\Gamma$.
[ \[con:primitive\] Let $(\Gamma,{\mathcal{P}})$ be a $G$-transitive decomposition and let $\mathcal{B}$ be a system of imprimitivity for $G$ on ${\mathcal{P}}$. For each $B\in\mathcal{B}$, let $Q_B=\cup_{P\in B}P$ and let $\mathcal{Q}=\{Q_B\mid B\in\mathcal{B}\}$. Then $(\Gamma,\mathcal{Q})$ is a $G$-transitive decomposition. ]{}
\[lem:primitive\] Any $G$-transitive decomposition $(\Gamma,{\mathcal{P}})$ with $|{\mathcal{P}}|$ finite is the refinement of a $G$-primitive decomposition $(\Gamma,\mathcal{Q})$.
If $G^{{\mathcal{P}}}$ is primitive then we are done. If not, let $\mathcal{B}$ be a nontrivial system of imprimitivity for $G$ on ${\mathcal{P}}$ with maximal block size. Then $G^{\mathcal{B}}$ is primitive and ${\mathcal{P}}$ is a refinement of the partition $\mathcal{Q}$ yielded by Construction \[con:primitive\]. Thus $(\Gamma,\mathcal{Q})$ is a $G$-primitive decomposition.
------------------------------------------------------------------------
We have the following general construction of transitive decompositions.
[ \[con:general\] Let $\Gamma$ be a graph with an arc-transitive group $G$ of automorphisms. Let $e$ be an edge of $\Gamma$ and suppose that there exists a subgroup $H$ of $G$ such that $G_e<H<G$. Let $P=e^H$ and ${\mathcal{P}}=\{P^g\mid g\in G\}$. ]{}
\[lem:general\] Let $(\Gamma,{\mathcal{P}})$ be obtained as in Construction $\ref{con:general}$. Then $(\Gamma,{\mathcal{P}})$ is a $G$-transitive decomposition. Conversely, every $G$-transitive decomposition with $G$ arc-transitive arises in such a manner. Moreover, if the subgroup $H$ is maximal in $G$, then $(\Gamma,{\mathcal{P}})$ is a $G$-primitive decomposition.
Since $G$ is arc-transitive and $G_e<H<G$, then ${\mathcal{P}}$ is a partition of $E\Gamma$ which is preserved by $G$ and such that $G^{{\mathcal{P}}}$ is transitive. Thus $(\Gamma,{\mathcal{P}})$ is a $G$-transitive decomposition. Conversely, let $(\Gamma,{\mathcal{P}})$ be a $G$-transitive decomposition such that $G$ is arc-transitive. Let $e$ be an edge of $\Gamma$ and $P$ the divisor containing $e$. Since ${\mathcal{P}}$ is a system of imprimitivity for $G$ on $E\Gamma$ it follows that for $H=G_P$ we have $G_e<H<G$ and $P=e^H$. Moreover, ${\mathcal{P}}=\{P^g\mid g\in
G\}$ and so $(\Gamma,{\mathcal{P}})$ arises from Construction \[con:general\]. The last statement follows from the fact that $H$ is the stabiliser in $G$ of the divisor $P$.
------------------------------------------------------------------------
\[rem:2ways\] [Lemma \[lem:general\] implies that there are two possible ways to determine all $G$-transitive decompositions such that the divisor stabilisers are in a given conjugacy class $H^G$ of subgroups of $G$. One is to fix an edge $e$ and run over all subgroups conjugate to $H$ which contain the stabiliser of $e$. Note that different conjugates may give different partitions. The second is to run over all edges whose stabiliser is contained in $H$. Again, different edges may give different partitions. ]{}
We say that two decompositions $(\Gamma,{\mathcal{P}}_1)$ and $(\Gamma,{\mathcal{P}}_2)$ are *isomorphic* if there exists $g\in{\mathop{\mathrm{Aut}}}(\Gamma)$ such that ${\mathcal{P}}_1^g={\mathcal{P}}_2$. If both are $G$-transitive decompositions, then they are *isomorphic $G$-transitive decompositions* if there is such an element $g\in N_{{\mathop{\mathrm{Aut}}}(\Gamma)}(G)$. The following lemma gives us a condition for determining when different conjugates give the same decomposition.
\[lem:iso\] Let $(\Gamma,{\mathcal{P}}_1)$, $(\Gamma,{\mathcal{P}}_2)$ be two $G$-transitive decompositions with $G$ arc-transitive.
1. Let $e$ be an edge of $\Gamma$ and $P_1,P_2$ be the divisors of ${\mathcal{P}}_1$, ${\mathcal{P}}_2$ respectively that contain $e$. If there exists an automorphism $g\in N_{{\mathop{\mathrm{Aut}}}(\Gamma)}(G)$ fixing $e$ such that $G_{P_1}^g=G_{P_2}$ then $(\Gamma,{\mathcal{P}}_1)$ and $(\Gamma,{\mathcal{P}}_2)$ are isomorphic.
2. Let $e_1,e_2$ be two edges of $\Gamma$ with divisors $P_1=e_1^H$ and $P_2=e_2^H$ of ${\mathcal{P}}_1$, ${\mathcal{P}}_2$ respectively. If there exists an automorphism $g\in N_{{\mathop{\mathrm{Aut}}}(\Gamma)}(G)$ mapping $e_1$ onto $e_2$ such that $H^g=H$ then $(\Gamma,{\mathcal{P}}_1)$ and $(\Gamma,{\mathcal{P}}_2)$ are isomorphic.
<!-- -->
1. By Lemma \[lem:general\], $P_1=e^{G_{P_1}}$ and $P_2=e^{G_{P_2}}$. Thus $P_2=e^{g^{-1}G_{P_1}g}=e^{G_{P_1}g}=P_1^g$. Moreover, ${\mathcal{P}}_2=P_2^G=(P_1^g)^G=(P_1^G)^g={\mathcal{P}}_1^g$ and so $(\Gamma,{\mathcal{P}}_1)$ and $(\Gamma,{\mathcal{P}}_2)$ are isomorphic.
2. We have $P_2=e_2^H=(e_1^g)^H=(e_1^H)^g=P_1^g$. Hence we get the same conclusion.
------------------------------------------------------------------------
We also have the following useful lemma.
\[lem:restrict\] Let $(\Gamma,{\mathcal{P}})$ be a $G$-primitive decomposition, with $H$ the stabiliser of a divisor $P$. If $L\leqslant G$ is such that $L\not\leqslant H$, $L$ is arc-transitive on $\Gamma$ and $L\cap H$ is maximal in $L$, then $(\Gamma,{\mathcal{P}})$ is a $L$-primitive decomposition.
Since $L$ is arc-transitive and contained in $G$, it follows that $L$ acts transitively on ${\mathcal{P}}$. Moreover, since $H\cap L$ is the stabiliser in $L$ of a part, it follows that $L$ acts primitively on ${\mathcal{P}}$.
------------------------------------------------------------------------
We now describe some general methods for constructing transitive decompositions of Johnson graphs.
\[con:classic\]
Let $X$ be an $n$-set.
1. For each $(k-1)$-subset $Y$ of $X$, let $P_Y$ be the complete subgraph of $J(n,k)$ whose vertices are all the $k$-subsets containing $Y$. Then $${\mathcal P}_{\cap}=\{P_Y|Y \text{ is a $(k-1)$-subset of $X$}\}$$ is a decomposition of $J(n,k)$ with ${n\choose k-1}$ divisors, each isomorphic to $K_{n-k+1}$.
2. For each $(k+1)$-subset $W$ of $X$, let $Q_W$ be the complete subgraph whose vertices are all the $k$-subsets contained in $W$. Then $${\mathcal P}_{\cup}=\{Q_W|W \text{ is a $(k+1)$-subset of $X$}\}$$ is a decomposition of $J(n,k)$ with ${n\choose k+1}$ divisors, each isomorphic to $K_{k+1}$.
3. For each $\{a,b\}\subseteq X$, let $$M_{\{a,b\}}=\Big\{\big\{\{a\}\cup Y,\{b\}\cup Y\}\big\} \mid Y \text{ a
$(k-1)$-subset of } X\backslash\{a,b\}\Big\}.$$ Then $$\mathcal{P}_{\ominus}=\{M_{\{a,b\}}\mid \{a,b\}\subseteq X\}$$ is a decomposition of $J(n,k)$ with ${n\choose 2}$ divisors, each of which is a matching with ${n-2 \choose k-1}$ edges.
Given two sets $A$ and $B$ we denote the *symmetric difference* of $A$ and $B$ by $A\ominus B$.
\[lem:classic\] Let $G\leqslant S_n$ such that $\Gamma =J(n,k)$ is $G$-arc-transitive. Let $A$ and $B$ be two adjacent vertices of $\Gamma$. Then $(\Gamma, {\mathcal P}_{\cap})$, $(\Gamma, {\mathcal P}_{\cup})$, $(\Gamma, {\mathcal P}_{\ominus})$ are $G$-transitive decompositions. Moreover, if $G_{A\cap B}$, $G_{A\cup B}$, or $G_{A\ominus B}$ respectively is maximal in $G$, then the decomposition is $G$-primitive.
Since $P_Y^g=P_{Y^g}$, $Q_W^g=Q_{W^g}$ and $M_{\{a,b\}}^g=M_{\{a,b\}^g}$, it follows that $G$ preserves ${\mathcal P}_{\cap}$, ${\mathcal P}_{\cup}$ and ${\mathcal P}_{\ominus}$. Since $G$ is arc-transitive, all three decompositions are $G$-transitive. The divisor of ${\mathcal{P}}_\cap$, ${\mathcal{P}}_\cup$ or $P_\ominus$ containing $\{A,B\}$ is $P_{A\cap B}$, $Q_{A\cup B}$ or $M_{A\ominus B}$ respectively, and the stabiliser of this divisor is $G_{A\cap B}$, $G_{A\cup B}$, or $G_{A\ominus B}$ respectively. The last assertion follows.
------------------------------------------------------------------------
Another method for constructing transitive decompositions of $J(n,k)$ is to use Steiner systems with multiply transitive automorphism groups. A *Steiner system* $S(t,k,v)=(X,\mathcal{B})$ is a collection $\mathcal{B}$ of $k$-subsets (called *blocks*) of a $v$-set $X$ such that each $t$-subset of $X$ is contained in a unique block.
\[con:design\] [Let $\mathcal{D}=(X,\mathcal{B})$ be an $S(k+1,l,n)$ Steiner system with automorphism group $G$ such that $G$ is transitive on $\mathcal{B}$. For each $Y\in\mathcal{B}$, let $P_Y$ be the subgraph of $J(n,k)$ whose vertices are the $k$-subsets in $Y$ and let ${\mathcal{P}}=\{P_Y\mid Y\in \mathcal{B}\}$. ]{}
\[lem:design\] The pair $(J(n,k),{\mathcal{P}})$ yielded by Construction $\ref{con:design}$ is a $G$-transitive decomposition with divisors isomorphic to $J(l,k)$. Moreover, the decomposition is $G$-primitive if and only if the stabiliser of a block of $\mathcal{D}$ is maximal in $G$.
Let $\{A,B\}$ be an edge of $J(n,k)$. Then $A\cup B$ has size $k+1$ and so is contained in a unique block $Y$ of $\mathcal{D}$, and hence $\{A,B\}$ is contained in a unique part $P_Y$ of ${\mathcal{P}}$. Thus $(J(n,k),\mathcal{P})$ is a decomposition. Since $G$ is transitive on $\mathcal{B}$ the pair $(J(n,k),{\mathcal{P}})$ is $G$-transitive. Moreover, each $P_Y$ consists of all $k$-subsets of the $l$-set $Y$ and so is isomorphic to $J(l,k)$. Since the stabiliser in $G$ of $P_Y$ is $G_Y$, the last statement follows.
------------------------------------------------------------------------
\[con:design2\] [Let $\mathcal{D}=(X,\mathcal{B})$ be an $S(k+1,l,n)$ Steiner system with automorphism group $G$. Let $i=l-k-1$ and suppose that $G$ is $i$-transitive on $X$. For each $i$-subset $Y$ of $X$ let $$P_Y=\{\{A,B\}\mid |A|=|B|=k,|A\cap B|=k-1 \text{ and } A\cup B\cup Y \in
\mathcal{B}\}.$$ Define $${\mathcal{P}}=\{P_Y\mid Y \text{an $i$-subset of } X\}.$$ ]{}
\[lem:design2\] The pair $(J(n,k),{\mathcal{P}})$ yielded by Construction $\ref{con:design2}$ is a $G$-transitive decomposition with divisors isomorphic to $mK_{k+1}$, where $m$ is the number of blocks of $\mathcal{D}$ containing an $i$-set. Moreover, the decomposition is $G$-primitive if and only if the stabiliser of an $i$-set is maximal in $G$.
Let $\{A,B\}$ be an edge of $J(n,k)$. Then $A\cup B$ is contained in a unique block $W$ of $\mathcal{D}$ and the unique part of ${\mathcal{P}}$ containing $\{A,B\}$ is $P_Y$ where $Y=W\backslash (A\cup B)$. Each block containing $Y$ contributes a copy of $J(k+1,k)\cong K_{k+1}$ to $P_Y$, and since each $(k+1)$-subset is in a unique block, no two blocks containing $Y$ share a vertex of $P_Y$. Hence the $m$ copies of $K_{k+1}$ in $P_Y$, are pairwise vertex-disjoint, that is $P_Y\cong mK_{k+1}$. Since $G$ is $i$-transitive, it follows that $(J(n,k),{\mathcal{P}})$ is a $G$-transitive decomposition. Since the stabiliser in $G$ of $P_Y$ is $G_Y$, the last statement follows.
------------------------------------------------------------------------
\[con:design3\] [Let $\mathcal{D}=(X,\mathcal{B})$ be an $S(k+1,k+2,n)$ Steiner system with automorphism group $G$ such that $G$ acts $3$-transitively on $X$. For each $3$-subset $Y$ of $X$, let $$P_Y=\Big\{\big\{ Z\cup\{u\},Z\cup\{v\}\big\}\mid |Z|=k-1,Z\cup
Y\in\mathcal{B}, u,v\in Y\Big\}$$ and let ${\mathcal{P}}=\{P_Y\mid Y \text{ a $3$-subset of }$X$\}$. ]{}
The pair $(J(n,k),{\mathcal{P}})$ yielded by Construction $\ref{con:design3}$ is a $G$-transitive decomposition with divisors isomorphic to $mK_3$, where $m$ is the number of blocks of $\mathcal{D}$ containing a given $3$-set. Moreover, the decomposition is $G$-primitive if and only if the stabiliser of a $3$-subset is maximal in $G$.
Let $\{A,B\}$ be an edge of $J(n,k)$. Then $A\cup B$ is contained in a unique block $W$ of $\mathcal{D}$ and the unique part of ${\mathcal{P}}$ containing $\{A,B\}$ is $P_Y$ where $Y=W\backslash (A\cap B)$. Each block containing $Y$ contributes a copy of $K_3$ to $P_Y$, and since each $(k+1)$-subset is in a unique block, no two blocks containing $Y$ share a vertex of $P_Y$. Hence the $m$ copies of $K_3$ in $P_Y$ are pairwise vertex-disjoint, that is, $P_Y\cong mK_3$. Since $G$ is $3$-transitive, it follows that $(J(n,k),{\mathcal{P}})$ is a $G$-transitive decomposition. Since the stabiliser in $G$ of $P_Y$ is $G_Y$, the last statement follows.
------------------------------------------------------------------------
\[con:design4\] [Let $\mathcal{D}=(X,\mathcal{B})$ be an $S(k+1,k+2,n)$ Steiner system with $k$-transitive automorphism group $G$. For each $k$-subset $Y$ of $X$ let $$P_Y=\Big\{\big\{\{u\}\cup Z,\{v\}\cup Z\big\}\mid
Y\cup\{u,v\}\in\mathcal{B}, Z\subset Y, |Z|=k-1 \Big\}$$ and let ${\mathcal{P}}=\{P_Y\mid Y \text{ a k-subset of }$X$\}$. ]{}
The pair $(J(n,k),{\mathcal{P}})$ yielded by Construction $\ref{con:design4}$ is a $G$-transitive decomposition with divisors isomorphic to $mkK_2$, where $m$ is the number of blocks of $\mathcal{D}$ containing a given $k$-set. Moreover, the decomposition is $G$-primitive if and only if the stabiliser of a $k$-subset is maximal in $G$.
Let $\{A,B\}$ be an edge of $J(n,k)$. Then $A\cup B$ is contained in a unique block $W$ of $\mathcal{D}$ and the unique part of ${\mathcal{P}}$ containing $\{A,B\}$ is $P_Y$ where $Y=W\backslash (A\ominus B)$. Each block containing $Y$ contributes a copy of $kK_2$ to $P_Y$, and since each $(k+1)$-subset is in a unique block, no two blocks containing $Y$ share a vertex of $P_Y$. Hence the $m$ copies of $kK_2$ in $P_Y$, are pairwise vertex-disjoint, that is $P_Y\cong mkK_2$. Since $G$ is $k$-transitive, it follows that $(J(n,k),{\mathcal{P}})$ is a $G$-transitive decomposition. Since the stabiliser in $G$ of $P_Y$ is $G_Y$, the last statement follows.
------------------------------------------------------------------------
We end this section with a standard construction of arc-transitive graphs.
Let $G$ be a group with corefree subgroup $H$ and let $g\in G$ such that $g^2\in H$ and $g\notin N_G(H)$. Define the graph $\Gamma={\mathop{\mathrm{Cos}}}(G,H,HgH)$ with vertex set the set of right cosets of $H$ in $G$ and $Hx$ adjacent to $Hy$ if and only if $xy^{-1}\in HgH$. Then $G$ acts faithfully and arc-transitively on $\Gamma$ by right multiplication. We have the following lemma, see for example [@FP].
\[lem:cosetgraph\] Let $\Gamma$ be a $G$-arc-transitive graph with adjacent vertices $v$ and $w$. Let $H=G_v$, and let $g\in G$ interchange $v$ and $w$. Then $\Gamma\cong {\mathop{\mathrm{Cos}}}(G,H,HgH)$. The connected component of $\Gamma$ containing $v$ consists of all cosets of $H$ contained in ${\langle}H,g{\rangle}$. In particular, $\Gamma$ is connected if and only if ${\langle}H,g{\rangle}=G$.
Groups
======
In this section, we determine the groups $G$ such that $J(n,k)$ is $G$-vertex-transitive and $G$-arc-transitive.
[[@BCN Theorem 9.1.2]]{} \[thm:autgamma\] Let $n,k$ be positive integers and let $\Gamma=J(n,k)$. If $n>2k$ then ${\mathop{\mathrm{Aut}}}(\Gamma)=S_n$ with the action induced from the action of $S_n$ on $X$. For $n=2k\geq 4$, ${\mathop{\mathrm{Aut}}}(\Gamma)=S_n\times S_2={\langle}S_n,\tau {\rangle}$ where $\tau$ acts on $V\Gamma$ by complementation in $X$.
Given a subset $A$ of $X$ we denote the complement of $A$ in $X$ by $\overline{A}$. Also, if $|X|=n$ and $|A|=k$ then $\Gamma(A)$ denotes the set of neighbours of $A$ in the graph $J(n,k)$, that is, vertices $B$ such that $\{A,B\}$ is an edge.
[[@Cuaresma Proposition 3.2]]{} \[lemcond\] Let $\Gamma=J(n,k)$ and $G\leqslant S_n$. The graph $\Gamma$ is $G$-arc-transitive if and only if $G$ is $k$-homogeneous on $X$ and, for a $k$-subset $A$, $G_A$ is transitive on $A\times \overline{A}$.
Note that $G$ is arc-transitive if and only if $G$ is vertex-transitive and $G_A$ is transitive on $\Gamma(A)$. By definition, $\Gamma$ is $G$-vertex-transitive if and only if $G$ is $k$-homogeneous on $X$. Moreover, $G_A$ is transitive on $\Gamma(A)$ if and only if $G_A$ is independently transitive on the set of $(k-1)$-subsets of $A$ and on $\overline{A}$, that is, if and only if $G_A$ is transitive on $A\times \overline{A}$.
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\[Sncor\] If $G\leqslant S_n$ is $(k+1)$-transitive, then $\Gamma$ is $G$-arc-transitive. If $\Gamma$ is $G$-arc-transitive and $G\leqslant S_n$, then $G$ is $k$- and $(k+1)$-homogeneous.
\[Sn\] Let $n\geq 2k\geq 4$ and $G\leqslant S_n$. The graph $\Gamma=J(n,k)$ is $G$-arc-transitive if and only if $G$ is $(k+1)$-transitive on $X$ or $k=3$, $n=9$, and $G={\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$.
If $G$ is $(k+1)$-transitive, then by Corollary \[Sncor\], $\Gamma$ is $G$-arc-transitive. If $k=3$, $n=9$, and $G={\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$, then it is easy to check that $G$ is arc-transitive.
Suppose now that $\Gamma$ is $G$-arc-transitive. By Corollary \[Sncor\], $G$ is $k$- and $(k+1)$-homogeneous on $X$. If $G$ is not $(k+1)$-transitive, then, by [@Kantor; @LiWa] either $2k\leq n\leq 2k+1$, or $2\leq k\leq 3$ and $G$ is one of a small number of groups.
Suppose first that $k=2$. (This is an improvement on the proof of [@Cuaresma Proposition 3.3].) Since $G$ is $3$-homogeneous, it is transitive on $X$. For $A=\{a,b\}$, Lemma \[lemcond\] implies that $G_A$ is transitive on $A\times \overline{A}$. Therefore using elements of $G_A$ we can map $(a,c)$ onto $(a,d)$ for any $c,d\in \overline{A}$, and so $G_{a,b}$ is transitive on $\overline{A}$. Similarly, $G_{a,c}$ is transitive on $\overline{\{a,c\}}$ for any $c\in \overline{\{a,b\}}$. Hence $G_a$ is transitive on $\overline{\{a\}}$ and so $G$ is 3-transitive on $X$.
Next suppose that $k=3$. If $G$ is not 4-transitive then either $n=6,7$, or by [@Kantor], $G$ is one of ${\mathop{\mathrm{PGL}}}(2,8)$, ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$ (with $n=9$), or ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,32)$ (with $n=33$). Let $A=\{a,b,c\}$ and suppose that $G\neq {\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$.
Suppose first that $G={\mathop{\mathrm{PGL}}}(2,8)$. Then $G_A\cong S_3$ and $G_{A,a}=C_2$. Hence $G$ does not satisfy the arc-transitivity condition given in Lemma \[lemcond\]. Next suppose that $G={\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,32)$. Then $|G_{A,a}|=10$ and so again Lemma \[lemcond\] implies that $G$ is not arc-transitive.
If $n=6$, the only 3-homogeneous and 4-homogeneous group which is not 4-transitive is ${\mathop{\mathrm{PGL}}}(2,5)$. However, this does not satisfy the condition in Lemma \[lemcond\] for arc-transitivity. There are no 3-homogeneous and 4-homogeneous groups of degree 7 which are not 4-transitive.
Next suppose that $k=4$. If $G$ is not 5-transitive, then $n=8$ or $9$. Since $G$ is $4$-homogeneous and 5-homogeneous, either $G$ is $4$-transitive, or $G$ is one of ${\mathop{\mathrm{PGL}}}(2,8)$, ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$. However, these two groups are not arc-transitive as the stabiliser of a $4$-subset $A$ also stabilises a point in $\overline{A}$. The only 4-transitive groups of degree $n$ are $A_n$ and $S_n$ and they are also $5$-transitive.
If $k=5$ and $G$ is not 6-transitive, then $n=10$ or $11$. Since $G$ is $5$-homogeneous it is $5$-transitive and so $G$ contains $A_n$. Thus $G$ is also $6$-transitive. Finally, let $k\geq 6$. Since $G$ is $k$-homogeneous it is $k$-transitive. The only $k$-transitive groups for $k\geq 6$ are $A_n$ and $S_n$, which are also $(k+1)$-transitive.
------------------------------------------------------------------------
We need a couple of results for the case $n=2k$.
\[thm:2karctrans\] Let $\Gamma=J(2k,k)$ and suppose that $G\leqslant {\mathop{\mathrm{Aut}}}(\Gamma)=S_{2k}\times {\langle}\tau{\rangle}$ and $\Gamma$ is $G$-arc-transitive. Then either $G\cap S_{2k}$ is arc-transitive on $\Gamma$, or $k=2$, $G={\langle}A_4,(1,2)\tau{\rangle}$ and $G\cap S_4=A_4$ has two orbits on arcs.
Let $\hat{G}=G\cap S_{2k}$. If $\hat{G}=G$, we are done. Hence we can assume $\hat{G}$ is an index $2$ subgroup of $G$. The graph $\Gamma$ is connected and is not bipartite, as it contains 3-cycles. It follows that $\hat{G}$ cannot have two orbits on vertices and so $\hat{G}$ is vertex-transitive.
Suppose that $\hat{G}$ is not arc-transitive, and hence has two orbits of equal size on $A\Gamma$. Let $(A,B)\in A\Gamma$. Then $\hat{G}_{(A,B)}\leqslant G_{(A,B)}$ and $|G_A:G_{(A,B)}|=|\Gamma(A)|=k^2=2|\hat{G}_A:\hat{G}_{(A,B)}|=|G_A:\hat{G}_{(A,B)}|$. Hence $\hat{G}_{(A,B)}=G_{(A,B)}$ and $k$ is even.
Suppose first that $k\geq 6$. Since $\hat{G}$ is transitive on $V\Gamma$, $\hat{G}$ is $k$-homogeneous and therefore also $k$-transitive. Hence $A_{2k}\leqslant\hat{G}$, and so $\hat{G}$ is $(k+1)$-transitive. It follows from Theorem \[Sn\] that $\hat{G}$ is transitive on $A\Gamma$, which is a contradiciton. Thus $k=2$ or 4.
If $k=4$, then $\hat{G}$ is $k$-homogeneous. The only 4-homogeneous groups of degree 8 contain $A_8$, and so are also $5$-transitive. By Theorem \[Sn\], $\hat{G}$ is transitive on $A\Gamma$ in this case, and so $k=2$.
Since $\hat{G}$ is transitive on $V\Gamma$ and $(n,k)=(4,2)$ we have that $6$ divides $|\hat{G}|$. Since $\hat{G}$ is 2-homogeneous it follows that $A_4\leqslant \hat{G}$. Moreover, $S_4$ is arc-transitive and so $\hat{G}=A_4$. There are two groups $G\leqslant S_n\times S_2$ such that $\hat{G}=A_4$ and is of index 2 in $G$, namely ${\langle}A_4,\tau {\rangle}$ and ${\langle}A_4,(1,2)\tau {\rangle}$. It is easy to check that the second group is transitive on $A\Gamma$ but not the first one.
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We also have the following theorem about primitivity.
\[thm:2kreducetoSn\] Let $\Gamma=J(2k,k)$ and $G\leqslant {\mathop{\mathrm{Aut}}}(\Gamma)=S_{2k}\times {\langle}\tau{\rangle}$ such that both $G$ and $G\cap S_{2k}$ are arc-transitive. Suppose that $(\Gamma,{\mathcal{P}})$ is a $G$-primitive decomposition. Then $(\Gamma,{\mathcal{P}})$ is also $(G\cap S_{2k})$-primitive.
Let $\hat{G}=G\cap S_{2k}$, let $H$ be the stabiliser in $G$ of a divisor and $\hat{H}=H\cap \hat{G}=H\cap S_{2k}$. We may suppose that $G\neq\hat{G}$. Moreover, as $\hat{G}$ is arc-transitive it acts transitively on ${\mathcal{P}}$ and so $\hat{G}\not\leqslant H$. Since $H$ is maximal in $G$ it follows that $|H:\hat{H}|=2$.
Suppose first that $G=\hat{G}\times{\langle}\tau{\rangle}$. Now $H={\langle}\hat{H},\sigma\tau{\rangle}$ for some $\sigma\in \hat{G}$. Since $\hat{H}{\vartriangleleft}H$, the element $\sigma\tau$ (and hence also $\sigma$) normalises $\hat{H}$ and $\hat{H}$ contains $(\sigma\tau)^2=\sigma^2$. This implies that $H\leqslant{\langle}\hat{H},\sigma{\rangle}\times{\langle}\tau{\rangle}\leqslant G$. Since $H$ is maximal in $G$, either $H={\langle}\hat{H},\sigma{\rangle}\times{\langle}\tau{\rangle}$ or ${\langle}\hat{H},\sigma{\rangle}\times{\langle}\tau{\rangle}=G$. The first implies that $\sigma\in\hat{H}$ and hence $H=\hat{H}\times {\langle}\tau{\rangle}$. Thus $\hat{H}$ is maximal in $\hat{G}$ and so by Lemma \[lem:restrict\], $\mathcal{P}$ is $\hat{G}$-primitive. On the other hand, the second implies $\hat{G}={\langle}\hat{H},\sigma{\rangle}$. Since $\sigma^2\in\hat{H}$, we have $|\mathcal{P}|= |\hat{G}:\hat{H}|=2$ and so again $\hat{G}$ is primitive on ${\mathcal{P}}$.
Suppose now that $G={\langle}\hat{G},\sigma\tau{\rangle}$ for some $\sigma\in S_{2k}\backslash\{1\}$ and $\tau\notin G$. Then $\sigma$ normalises $\hat{G}$ and $\sigma^2\in\hat{G}$. Also, as $\tau\notin G$, we have $\sigma\notin\hat{G}$ and in particular $\hat{G}\neq S_{2k}$. By Theorem \[Sn\] and the fact that $n=2k$, the classification of $(k+1)$-transitive groups (see for example [@Cameron pp194–197]) implies that $\hat{G}=A_{2k}$ and $k\geq 3$. Let $\phi:S_{2k}\times{\langle}\tau{\rangle}\rightarrow S_{2k}$ be the projection of ${\mathop{\mathrm{Aut}}}(\Gamma)$ onto $S_{2k}$. Then $\phi_{\mid G}$ is an isomorphism. Moreover, for an edge $\{A,B\}$ contained in the divisor stabilised by $H$, $\phi(G_{A,B})= S_{k-1}\times S_{k-1}$. Since $k\geq 3$, there is a transposition in $\phi(G_{A,B})$ and so by [@wielandt Theorem 13.1] and since $\phi(G_{A,B})\subseteq \phi(H)$, $\phi(H)$ is not primitive. It follows that $\phi(H)$ is a maximal intransitive subgroup of $S_{2k}$ or a maximal imprimitive subgroup of $S_{2k}$ preserving a partition into at most 3 parts. Thus by [@LPS] and since $\hat{H}=\phi(H) \cap A_{2k}$, it follows that $\hat{H}$ is a maximal subgroup of $\hat{G}=A_{2k}$. Hence again $\hat{G}$ is primitive on ${\mathcal{P}}$.
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Alternating and symmetric groups {#sec:AnSn}
================================
We have already seen the $S_n$-transitive decompositions ${\mathcal{P}}_\cap,{\mathcal{P}}_\cup$ and ${\mathcal{P}}_\ominus$. Since $n\geq 2k$ it follows that $S_n$ always acts primitively on ${\mathcal{P}}_\cap$. Also, $S_n$ acts primitively on ${\mathcal{P}}_\cup$ if and only if $n\neq 2k+2$. When $n=2k+2$, applying Construction \[con:primitive\] to ${\mathcal{P}}_\cup$ we obtain an $S_n$-primitive decomposition with divisors isomorphic to $2K_{k+1}$. Finally $S_n$ acts primitively on ${\mathcal{P}}_\ominus$ if and only if $(n,k)\neq (4,2)$. This justifies the first four lines of Table \[tab:Sn\] below. We also have the following two examples.
\[eg:Sn\]
1. Let $G=S_4$, $H={\langle}(1,2,3,4),(1,3){\rangle}\cong D_8$, $A=\{1,2\}$ and $B=\{2,3\}$. Then $P=\{A,B\}^H$ is the 4-cycle $$\Big\{\big\{\{1,2\},\{2,3\}\big\},\big\{\{2,3\},\{3,4\}\big\},
\big\{\{3,4\},\{1,4\}\big\},\big\{\{1,4\}, \{1,2\}\big\}\Big\}.$$ Since $G_{\{A,B\}}={\langle}(1,3){\rangle}$ we have $G_{\{A,B\}}<H<G$ and so by Lemma \[lem:general\] $( (J(4,2),{\mathcal{P}})$ is a $G$-primitive decomposition with ${\mathcal{P}}=\{P^g\mid g\in G\}.$
2. Let $G=S_6$ and $H$ be the stabiliser in $G$ of the partition $$\big\{\{1,4\},\{2,3\},\{5,6\}\big\}$$ of $\{1,\ldots,6\}$. Let $A=\{1,2,3\}$ and $B=\{2,3,4\}$. Then $P=\{A,B\}^H$ is the matching $$\Big\{\big\{\{1,2,3\},\{2,3,4\}\big\},\big\{\{2,5,6\},\{3,5,6\}\big\},\big\{\{1,4,5\},\{1,4,6\}\big\},$$ $$\big\{\{1,5,6\},\{4,5,6\}\big\},\big\{\{2,3,5\},\{2,3,6\}\big\},\big\{\{1,4,2\},\{1,4,3\}\big\}
\Big\}.$$ Since $G_{\{A,B\}}<H<G$ it follows from Lemma \[lem:general\] that $((J(6,3),{\mathcal{P}})$ is a $G$-primitive decomposition with ${\mathcal{P}}=\{P^g\mid g\in G\}.$
We have now constructed all the $S_n$-primitive decompositions in Table \[tab:Sn\]. It remains to prove that these are the only ones.
${\mathcal{P}}$ $P$ $G_P$ $(n,k)$
--------------------------------------------------------- ----------------------- ------------------------------------ ------------------
${\mathcal{P}}_{\cap}$ $K_{n-k+1}$ $(k-1)$-set stabiliser
${\mathcal{P}}_\cup$ $K_{k+1}$ $(k+1)$-set stabiliser $n\neq 2k+2$
${\mathcal{P}}_\ominus$ ${n-2\choose k-1}K_2$ $2$-set stabiliser $(n,k)\neq(4,2)$
${\mathcal{P}}_\cup$ and Construction \[con:primitive\] $2K_{k+1}$ $S_{k+1}{\mathop{\mathrm{wr}}}S_2$ $n=2k+2$
Example \[eg:Sn\](1) $C_4$ $D_8$ $(n,k)=(4,2)$
Example \[eg:Sn\](2) $6K_2$ $S_2{\mathop{\mathrm{wr}}}S_3$ $(n,k)=(6,3)$
: $S_n$-primitive decompositions of $J(n,k)$[]{data-label="tab:Sn"}
\[G=Sn\] If $(J(n,k),{\mathcal{P}})$ is an $S_n$-primitive decomposition with $n\geq 2k$ then ${\mathcal{P}}$ is given by one of the rows of Table $\ref{tab:Sn}$.
Let $\Gamma=J(n,k)$, $X=\{1,\ldots, n\}$, and let $A=\{1,2,\ldots,k\}$ and $B=\{2,\ldots,k+1\}$ be adjacent vertices of $\Gamma$. Then $G_{\{A,B\}}=$\
${\mathop{\mathrm{Sym}}}(\{2,\ldots,k\})\times
{\mathop{\mathrm{Sym}}}(\{k+2,\ldots,n\})$. By Lemma \[lem:general\], to find all $G$-primitive decompositions of $\Gamma$, we need to determine all maximal subgroups $H$ of $G$ which contain $G_{\{A,B\}}$. Since $G_{\{A,B\}}$ contains a 2-cycle, [@wielandt Theorem 13.1] implies that there are no proper primitive subgroups of $G$ containing $G_{\{A,B\}}$. Hence $H$ is either imprimitive or intransitive.
Suppose first that $H$ is intransitive. Then $H$ is a maximal intransitive subgroup and hence it has two orbits $U,W$ on $X$ and $H={\mathop{\mathrm{Sym}}}(U)\times {\mathop{\mathrm{Sym}}}(W)$. Since $G_{\{A,B\}}\leqslant H$, the only possibilities for these two orbits are: $$\begin{array}{lll}
\{1,\ldots,k+1\} & \{k+2,\ldots,n\} & n\neq 2k+2\\
\{1,k+1\} & X\backslash\{1,k+1\} & (n,k)\neq (4,2)\\
\{2,\ldots,k\} & \{1,k+1,k+2,\ldots, n\}& \\
\end{array}$$
When $H={\mathop{\mathrm{Sym}}}(\{1,\ldots,k+1\})\times {\mathop{\mathrm{Sym}}}(\{k+2,\ldots,n\})=G_{A\cup B}$, we obtain the decomposition $(\Gamma,{\mathcal{P}}_{\cup})$, while $H={\mathop{\mathrm{Sym}}}(\{1,k+1\})\times{\mathop{\mathrm{Sym}}}(X\backslash\{1,k+1\})=G_{A\ominus B}$ yields the decomposition $(\Gamma,{\mathcal{P}}_\ominus)$. Finally, $H={\mathop{\mathrm{Sym}}}(\{2,\ldots,k\})\times{\mathop{\mathrm{Sym}}}(\{1,k+1,k+2,\ldots,n\})=G_{A\cap B}$ gives us the decomposition $(\Gamma,{\mathcal{P}}_{\cap})$.
If $H$ is transitive but imprimitive, then the possible systems of imprimitivity are: $$\begin{array}{ll}
\{1,\ldots,k+1\},\{k+2,\ldots,2k+2\} &\text{when } n=2k+2 \\
\{1,4\},\{2,3\},\{5,6\} & \text{when } (n,k)=(6,3) \\
\{1,3\},\{2,4\} & \text{when } (n,k)=(4,2)
\end{array}$$
In the first case, $P=\{A,B\}^H$ is the union of two cliques each of size $k+1$, and has as vertices all $k$-subsets of $\{1,\ldots,k+1\}$ and all $k$-subsets of , that is we get the decomposition obtained from applying Construction \[con:primitive\] to ${\mathcal{P}}_\cup$. The last two cases give us the two decompositions from Example \[eg:Sn\].
------------------------------------------------------------------------
By Theorem \[Sn\], $A_n$ is arc-transitive on $J(n,k)$ if and only if $n\geq 5$. Moreover, all the $S_n$-primitive decompositions in Table \[tab:Sn\] are $A_n$-primitive decompositions. We have the following extra examples for alternating groups.
\[eg:An\]
1. Let $(n,k)=(5,2)$, $G=A_5$, $A=\{1,2\}$ and $B=\{2,3\}$. Then $G_{\{A,B\}}={\langle}(1,3)(4,5){\rangle}$ and is contained in the maximal subgroup $H={\langle}(1,2,3,4,5),(1,3)(4,5){\rangle}\cong D_{10}$ of $G$. Letting $P=\{A,B\}^H$ and ${\mathcal{P}}=\{P^g\mid g\in G\}$, Lemma \[lem:general\] implies that $(J(5,2),{\mathcal{P}})$ is an $A_5$-primitive decomposition. Since $H_A\cong C_2$ it follows that the divisors are cycles of length $5$.
2. Let $(n,k)=(6,3)$, $G=A_6$, $A=\{1,2,3\}$ and $B=\{2,3,4\}$. Then $G_{\{A,B\}}={\langle}(2,3)(5,6),(1,4)(5,6){\rangle}$ and is contained in the maximal subgroup $H={\langle}(2,3)(5,6),(1,4,5)(2,3,6){\rangle}\cong {\mathop{\mathrm{PSL}}}(2,5)$ of $G$. Letting $P=\{A,B\}^H$ and ${\mathcal{P}}=\{P^g\mid g\in G\}$, Lemma \[lem:general\] implies that $(J(6,3),{\mathcal{P}})$ is an $A_6$-primitive decomposition. Now $P$ is a graph on $10$ vertices with valency $3$ admitting an arc-transitive action of $H\cong A_5$. Hence $P$ is the Petersen graph.
Let ${\mathcal{P}}$ be the decomposition of $J(6,3)$ given by Example $\ref{eg:An}(2)$. Then ${\mathcal{P}}$ is $G$-primitive if and only if $G=A_6$ or ${\langle}A_6, (1,2)\tau{\rangle}$ where $\tau$ is the complementation map as in Theorem \[thm:autgamma\].
As in the example, we take $A=\{1,2,3\}$, $B=\{2,3,4\}$ and $P=\{A,B\}^H$ for $H={\langle}(2,3)(5,6),(1,4,5)(2,3,6){\rangle}\cong A_5$.
If $G\leq S_6$, by Theorem \[Sn\], $G$ must be 4-transitive, so $A_6\leq G$. We have seen above that ${\mathcal{P}}$ is $A_6$-primitive. However, $S_6$ does not preserve the partition $\mathcal{P}$ of Example \[eg:An\](2), since $(1,4)$ preserves $\{A,B\}$ but not $P$. So assume $G\not\leq S_6$. By Theorems \[thm:2karctrans\] and \[thm:2kreducetoSn\], ${\mathcal{P}}$ is a $(G\cap S_6)$-primitive decomposition. Thus $G\cap S_6=A_6$ and so $G=G_1={\langle}A_6,\tau{\rangle}$ or $G=G_2={\langle}A_6, (1,2)\tau{\rangle}$. Thus $|G|=2|A_6|$ and so $|G_P:H|=2$. Then as $G_{\{A,B\}}\leqslant G_P$ it follows that $G_{\{A,B\}}$ normalises $H$. However, $(2,5)(3,6)\tau\in (G_1)_{\{A,B\}}$ and does not normalise $H$, so $G\neq G_1$. Now $(G_2)_{\{A,B\}}={\langle}(1,4)(2,5)(3,6)\tau, H_{\{A,B\}}{\rangle}$ does normalise $H$ and so fixes $P$. Thus ${\langle}H,(1,4)(2,5)(3,6)\tau{\rangle}=(G_2)_P\cong S_5$ which is a maximal subgroup of $G_2\cong S_6$. Hence ${\mathcal{P}}$ is a $G_2$-primitive decomposition.
------------------------------------------------------------------------
We now show that Example \[eg:An\] yields the only $A_n$-primitive decompositions which are not $S_n$-primitive.
Let $(J(n,k),{\mathcal{P}})$ be an $A_n$-primitive decomposition such that $A_n$ is arc-transitive and $n\geq 2k$. Then ${\mathcal{P}}$ is either an $S_n$-primitive decomposition, or $(n,k)=(5,2)$ or $(6,3)$ and ${\mathcal{P}}$ is isomorphic to a decomposition given in Example $\ref{eg:An}$.
Let $\Gamma=J(n,k)$. Since $G=A_n$ is arc-transitive it follows from Theorem \[Sn\] that $n\geq 5$. Let $X=\{1,\ldots,n\}$, $A=\{1,\ldots,k\}$ and $B=\{2,\ldots,k+1\}$. Then $$G_{\{A,B\}}=\big({\mathop{\mathrm{Sym}}}(\{1,k+1\})\times {\mathop{\mathrm{Sym}}}(\{2,\ldots,k\})\times {\mathop{\mathrm{Sym}}}(\{k+2,\ldots,n\})\big)\cap A_n.$$ We need to consider all maximal subgroups $H$ such that $G_{\{A,B\}}<H<G$. For each such $H$, $P=\{A,B\}^H$ is the edge-set of a divisor of the $G$-primitive decomposition.
Suppose first that $H$ is intransitive on $X$. Then $G_{\{A,B\}}$ has the same orbits on $X$ as $(S_n)_{\{A,B\}}$ and so $H$ is the intersection with $A_n$ of one of the maximal intransitive subgroups which we considered in the $S_n$ case in the proof of Theorem \[G=Sn\]. Moreover, we obtain the decompositions in rows 1–3 in Table \[tab:Sn\], and so $(\Gamma,{\mathcal{P}})$ is $S_n$-primitive.
Next suppose that $H$ is imprimitive on $X$. Since $G_{\{A,B\}}$ is primitive on both $A\cap B$ and $\overline{A \cup B}$, the only systems of imprimitivity preserved by $G_{\{A,B\}}$ are those discussed in the $S_n$ case. Thus $H$ is the intersection with $A_n$ of one of the maximal imprimitive subgroups considered in the $S_n$ case and we obtain the decompositions in rows 4 and 6 in Table \[tab:Sn\]. Thus $(\Gamma,{\mathcal{P}})$ is $S_n$-primitive.
Finally, suppose that $H$ is primitive on $X$. If $k-1\geq 3$ or $n-k-1\geq 3$, the edge stabiliser $G_{\{A,B\}}$, and hence $H$, contains a 3-cycle. Hence by [@wielandt Theorem 13.3], $H=A_n$, contradicting $H$ being a proper subgroup. Note that if $k\geq 4$ then $k-1\geq 3$, and so $(n,k)$ is one of $(5,2)$ or $(6,3)$.
If $(n,k)=(5,2)$ then $G_{\{A,B\}}={\langle}(1,3)(4,5){\rangle}$ and $H\cong D_{10}$. Since $A_5$ contains 15 involutions, $D_{10}$ contains 5 involutions and there are six subgroups $D_{10}$ in $A_5$, it follows that there are 2 choices for $H$ and these are $$H_1={\langle}(1, 2, 3, 4, 5),(1, 3)(4, 5){\rangle}$$ $$H_2={\langle}(1, 4, 5, 3, 2),(1, 3)(4, 5){\rangle}.$$ Note that $H_2=H_1^{(1,3)}$ and $(1,3)\in (S_n)_{\{A,B\}}$ and so by Lemma \[lem:iso\] the two decompositions obtained are isomorphic. Moreover, $H_1$ is the stabiliser of the divisor containing $\{A,B\}$ in the decomposition in Example \[eg:An\](1).
If $(n,k)=(6,3)$ then $G_{\{A,B\}}={\langle}(2,3)(5,6), (1,4)(5,6){\rangle}$ and $H\cong {\mathop{\mathrm{PSL}}}(2,5)$. A computation using [Magma]{} [@magma] showed that there are two choices for $H$ containing $G_{\{A,B\}}$ and these are: $$H_1={\langle}(2, 3)(5, 6), (1, 4, 5)(2, 3, 6){\rangle}$$ $$H_2={\langle}(2, 3)(5, 6), (1, 4, 5)(3, 2, 6){\rangle}.$$ Note that $H_2=H_1^{(2,3)}$ and $(2,3)\in (S_n)_{\{A,B\}}$ and so the two decompositions obtained are isomorphic. Moreover, $H_1$ is the stabiliser of the divisor containing $\{A,B\}$ in the decomposition in Example \[eg:An\](2).
------------------------------------------------------------------------
We now look at the case where $n=2k$ and $G$ is not a subgroup of $S_n$.
\[eg:A4tau\]
Let $(n,k)=(4,2)$ and $G={\langle}A_4,(1,2)\tau{\rangle}$. Let $A=\{1,2\}$ and $B=\{2,3\}$. Then $G_{\{A,B\}}={\langle}(2,4)\tau{\rangle}$.
1. Let $H_1={\langle}(1,2,4),(1,2)\tau{\rangle}$ and $$P=\{A,B\}^{H_1}=\bigg\{\big\{\{1,2\},\{2,3\}\big\},\big\{\{2,4\},\{3,4\}\big\},\big\{\{1,4\},\{1,3\}\big\}
\bigg\}.$$ Since $G_{\{A,B\}}\leqslant H_1$, it follows from Lemma \[lem:general\] that $(J(4,2),P^G)$ is a $G$-primitive decomposition, with divisors isomorphic to $3K_2$.
2. Let $H_2={\langle}(1,2)(3,4),(1,3)(2,4), (1,3)\tau{\rangle}$ and $P=\{A,B\}^{H_2}=$ $$\bigg\{\big\{\{1,2\},\{2,3\}\big\},
\big\{\{2,3\},\{3,4\}\big\}\big\{\{3,4\},\{1,4\}\big\},
\big\{\{1,4\},\{1,2\}\big\} \bigg\}.$$ Since $G_{\{A,B\}}\leqslant H_2$, it follows from Lemma \[lem:general\] that $(J(4,2),P^G)$ is a $G$-primitive decomposition, with divisors isomorphic to $C_4$. Notice that this decomposition is the one in Example \[eg:Sn\](1) and so is also $S_4$-primitive.
Let $\Gamma=J(n,k)$ with $n=2k$ and let such that $G$ is not contained in $S_n$. Further, suppose that $(\Gamma,{\mathcal{P}})$ is a $G$-primitive decomposition which is not $(G\cap S_n)$-primitive. Then $n=4$ and ${\mathcal{P}}$ is isomorphic to a decomposition given by Example $\ref{eg:A4tau}$.
By Theorems \[thm:2karctrans\] and \[thm:2kreducetoSn\], it follows that $k=2$ and $G={\langle}A_4,(1,2)\tau{\rangle}$, where $\tau$ is complementation in $X$. Let $A=\{1,2\}$ and $B=\{2,3\}$. Then $G_{\{A,B\}}={\langle}(2,4)\tau{\rangle}$. It is not hard to see that the only maximal subgroups of $G$ containing $G_{\{A,B\}}$ are the groups $H_1$ and $H_2$ from Example \[eg:A4tau\], and $H_3={\langle}(2,3,4),(2,3)\tau{\rangle}$. The first two give the two decompositions from Example \[eg:A4tau\]. Note that $(1,3)$ stabilizes $\{A,B\}$ and normalises $G$, and $H_3=H_1^{(1,3)}$. So by Lemma \[lem:iso\], this yields a decomposition isomorphic to the one in Example \[eg:A4tau\](1).
------------------------------------------------------------------------
The case $k\geq 4$
==================
By Theorem \[Sn\], if $k\geq 4$ then $G\leqslant S_n$ is arc-transitive on $J(n,k)$ if and only if $G$ is $(k+1)$-transitive on the $n$-set $X$. Hence by the Classificaton of finite 2-transitive permutation groups, other than $A_n$ or $S_n$, the only possibilities for $(n,G)$ when $k\geq 4$ are $(12, M_{12})$ and $(24,M_{24})$ with $k=4$.
First we state the following well known lemmas.
\[hexads-12\] Let $(X,\mathcal{B})$ be the Witt design $S(5,6,12)$. Then $\mathcal{B}$ contains $132$ elements, called *hexads*. Each point of $X$ is contained in $66$ hexads, each $2$-subset in $30$ hexads, each $3$-subset in $12$ hexads, each $4$-subset in $4$ hexads, and each $5$-subset in a unique hexad.
The number of hexads is given in [@atlas p 31] and then the number of hexads containing a given $i$-suset is calculated by counting $i$-subset–hexad pairs in two different ways.
------------------------------------------------------------------------
[[@sashasbook Lemma 2.11.7]]{} \[hexad12stab\] Suppose that $(X,\mathcal{B})$ is a Witt design $S(5,6,12)$ preserved by $G=M_{12}$ and let $h\in\mathcal{B}$ be a hexad. Then $G_h\cong
S_6$ and the actions of $G_h$ on $h$ and $X\backslash h$ are the two inequivalent actions of $S_6$ on six points.
Since the stabiliser of a 3-set or a 2-set is maximal in $G=M_{12}$, it follows from Lemma \[lem:classic\] that ${\mathcal{P}}_\cap$ and ${\mathcal{P}}_\ominus$ are $G$-primitive decompositions. Moreover, as $G$ acts primitively on the point set $X$ of the Witt design, Construction \[con:design2\] yields a $G$-primitive decomposition of $J(12,4)$. We also obtain a $G$-primitive decomposition from Construction \[con:design3\] as $G$ acts primitively on 3-subsets and one from Construction \[con:design4\] as $G$ acts primitively on 4-subsets. The $G$-transitive decomposition obtained from Construction \[con:design\] is not primitive as the stabiliser of a hexad is contained in the stabiliser of a pair of complementary hexads. However, applying Construction \[con:primitive\] we obtain a $G$-primitive decomposition with divisors isomorphic to $2J(6,4)$.
Before giving several more constructions arising from the Witt design, we need the following definition and lemma.
\[def:linked3\] [A *linked three* in $S(5,6,12)$ is a set of four triads (or 3-sets) such that the union of any two is a hexad. ]{}
\[lem:LTM12\] Let $A$, $B$ be two triads whose union is a hexad. Then there exists a unique linked three containing both $A$ and $B$.
By Lemma \[hexads-12\], there are exactly 12 hexads containing $A$. If such a hexad contains at least two points of $B$, then it is $A\cup B$. Let $b\in B$. Then there are 4 hexads containing $A$ and $b$, and so exactly 3 hexads meet $A\cup B$ in $A\cup \{b\}$. Therefore there are 9 hexads containing $A$ and meeting $A\cup B$ in a 4-set. Hence only two hexads contain $A$ and are disjoint from $B$. These yield two triads, $C$ and $D$, forming hexads with $A$. By Lemma \[hexad12stab\], the stabiliser of $A$ and $B$ is $S_3\times S_3$ which acts transitively on the remaining 6 points. Hence $C$ and $D$ must be disjoint. Since the complement of a hexad is a hexad, $C$ and $D$ must form hexads with $B$ too. It follows that $\{A,B,C,D\}$ is the unique linked three containing $A$ and $B$.
------------------------------------------------------------------------
\[con:124wittdesign\]
Let $(X,\mathcal{B})$ be the Witt design $S(5,6,12)$ and let $G=M_{12}$.
1. Let $T$ be a linked three as in Definition \[def:linked3\]. Let $$P_T=\Big\{\big\{\{u\}\cup Y,\{v\}\cup Y\big\}\mid Y\in T, \{u,v\} \text{ contained in some triad of }T\setminus Y\Big\}$$ and ${\mathcal{P}}=\{P_T\mid T \text{ is a linked three}\}$. Then $P_T\cong 12 K_3$, with each triad contributing $3K_3$. If $\{A,B\}$ is an edge of $J(12,4)$ then $A\cup B$ is contained in a unique hexad $A\cup B\cup\{x\}$ for some $x\in X$, and by Lemma \[lem:LTM12\], $\{A\cap B, \{x\}\cup (A\ominus B)\}$ is contained in a unique linked three $T$. For this $T$, $P_T$ is the unique part of ${\mathcal{P}}$ containing $\{A,B\}$. Since $G$ acts transitively on the set of linked threes and the stabiliser of a linked three is maximal, $(J(12,4),{\mathcal{P}})$ is a $G$-primitive decomposition.
2. Let $T$ be a linked three. A 4-set intersecting each triad of $T$ in a single point and such that its union with any triad is a hexad is called [*special*]{} for $T$. For fixed triads $T_1,T_2$ of $T$ and points $x_1\in T_1$, $x_2\in T_2$, these conditions imply that there is at most one special 4-set containing $\{x_1,x_2\}$ and existence of such a 4-set was confirmed by [Magma]{} [@magma]. Thus there are nine special 4-sets for $T$. Let $$P_T=\Big\{\big\{\{u,x,y,z\},\{v,x,y,z\}\big\}\mid \{x,y,z,t\}
\text{sp. 4-set for }T, \{u,v,t\}\in T \Big\}$$ and ${\mathcal{P}}=\{P_T\mid T \text{ is a linked three}\}$. Then $P_T\cong 36 K_2$, with each special 4-set contributing $4K_2$. If $\{A,B\}$ is an edge of $J(12,4)$ then $A\cup B$ is contained in a unique hexad $A\cup B\cup\{x\}$ for some $x\in X$, and there is a unique linked three $T$ such that $(A\cap B) \cup \{x\}$ is special for $T$ and $\{x\}\cup (A\ominus B)$ is a triad of $T$ (a [Magma]{} [@magma] calculation). Thus $P_T$ is the only part of ${\mathcal{P}}$ containing $\{A,B\}$. Since $G$ acts transitively on the set of linked threes and the stabiliser of a linked three is maximal, $(J(12,4),{\mathcal{P}})$ is a $G$-primitive decomposition.
\[con:M11graph\] [Let $G=M_{12}<S_{12}$ and let $H=M_{11}$ be a $3$-transitive subgroup of $G$. Then $H$ has an orbit of length $165$ on $4$-subsets and this orbit forms a $3-(12,4,3)$ design. Let $\Sigma$ be the subgraph of $J(12,4)$ induced on the orbit of length $165$. The graph $\Sigma$ was studied in [@DGLP], where it is seen that $\Sigma$ has valency $8$, is $H$-arc-transitive and given an edge $\{A,B\}$ we have $H_{\{A,B\}}\cong S_2\times S_3=G_{\{A,B\}}$. Thus Lemma \[lem:general\] and the fact that $H$ is maximal in $G$, imply that ${\mathcal{P}}=\Sigma^G$ is a $G$-primitive decomposition of $J(12,4)$. ]{}
We have now seen all the $M_{12}$-primitive decompositions listed in Table \[tab:M124\]. It remains to prove that these are the only ones.
${\mathcal{P}}$ $P$ $G_P$
---------------------------------------------------- -------------------- ------------------
${\mathcal{P}}_{\cap}$ $K_9$ $M_9\rtimes S_3$
${\mathcal{P}}_\ominus$ ${10\choose 3}K_2$ $M_{10}.2$
Constructions \[con:design\] and \[con:primitive\] $2J(6,4)$ $M_{10}.2$
Construction \[con:design2\] $66K_5$ $M_{11}$
Construction \[con:design3\] $12K_3$ $M_9\rtimes S_3$
Construction \[con:design4\] $16K_2$ $M_8\rtimes S_4$
Construction \[con:124wittdesign\](1) $12K_3$ $M_9\rtimes S_3$
Construction \[con:124wittdesign\](2) $36K_2$ $M_9\rtimes S_3$
Construction \[con:M11graph\] $\Sigma$ $M_{11}$
: $M_{12}$-primitive decompositions of $J(12,4)$[]{data-label="tab:M124"}
If $(J(12,4),{\mathcal{P}})$ is an $M_{12}$-primitive decomposition then ${\mathcal{P}}$ is given by one of the rows of Table $\ref{tab:M124}$.
Let $\Gamma=J(12,4)$ and $G=M_{12}$ acting on the point set $X$ of the Witt-design $S(5,6,12)$. Take adjacent vertices $A=\{1,2,3,4\}$ and $B=\{2,3,4,5\}$ and suppose that $h=\{1,2,3,4,5,6\}$ is the unique hexad containing $A\cup B$. Then $G_{\{A,B\}}=G_{\{1,5\},\{2,3,4\},\{6\}}\cong S_2\times S_3$, by Lemma \[hexad12stab\]. Since transpositions in the action of $G_h$ on $h$ act as a product of three transpositions on $X\backslash h$, and 3-cycles on $h$ act as a product of two 3-cycles on $X\backslash h$ it follows that $G_{1,5,6,\{2,3,4\}}\cong S_3$ acts regularly on $X\backslash h$, and so $G_{\{A,B\}}$ acts transitively on $X\backslash h$.
Let $H$ be a maximal subgroup of $G$ such that $G_{\{A,B\}}\leqslant H<G$. The maximal subgroups of $G$ are given in [@atlas p 33]. The orbit lengths of $G_{\{A,B\}}$ imply that $G_{\{A,B\}}$ does not preserve a system of imprimitivity on $X$ with blocks of size 2 or 4 and so $H\not\cong C_4^2\rtimes D_{12}, A_4\times S_3$, or $C_2\times S_5$. Moreover, $|H_6|$ is even and so $H\not\cong {\mathop{\mathrm{PSL}}}(2,11)$.
If $H$ is intransitive then $H$ is one of $G_{\{2,3,4,6\}}$, $G_{\{2,3,4\}}$, $G_{\{1,5,6\}}$, $G_{\{1,5\}}$ or $G_6$. (Note that $G_h$ is not maximal.) The first is the stabiliser of the divisor containing $\{A,B\}$ in the decomposition yielded by Construction \[con:design4\]. The second gives ${\mathcal{P}}_\cap$ while the third is the stabiliser of the divisor of the decomposition yielded by Construction \[con:design3\] containing $\{A,B\}$. If $H=G_{\{1,5\}}$ then we obtain the decomposition $\mathcal{P}_{\ominus}$ while if $H=G_6$ we obtain the decomposition yielded by Construction \[con:design2\].
The only hexad pair fixed by $G_{\{A,B\}}$ is $\{h,X\backslash
h\}$. Now $G_h$ is the stabiliser of the divisor of the decomposition yielded by Construction \[con:design\] containing $G_{\{A,B\}}$. Such a divisor is isomorphic to $J(6,4)$ and so $G_{\{h,X\backslash h\}}$ yields the decomposition with divisors isomorphic to $2J(6,4)$ obtained after applying Construction \[con:primitive\].
A calculation using [Magma]{} [@magma] shows that there is only one transitive subgroup of $G$ isomorphic to $M_{11}$ which contains $G_{\{A,B\}}$ and this yields Construction \[con:M11graph\].
By the list of maximal subgroups of $G$ given in [@atlas p 33], the only case left to consider is $H$ being the stabiliser of a linked three. If $T$ is a linked three preserved by $G_{\{A,B\}}$ then $\{1,5,6\}$ is a triad of $T$ and either $\{2,3,4\}$ is also a triad or $2$, $3$, and $4$ lie in distinct triads. Since a linked three is uniquely determined by any two of its triads (Lemma \[lem:LTM12\]), there is a unique linked three $T$ containing $\{1,5,6\}$ and $\{2,3,4\}$. Then $G_T$ is the stabiliser of the divisor of the decomposition yielded by Construction \[con:124wittdesign\](1) containing $\{A,B\}$. If $2$, $3$ and $4$ are in distinct blocks, a calculation using [Magma]{} [@magma] shows that there is a unique $H$ containing $G_{\{A,B\}}$ and we obtain the decomposition in Construction \[con:124wittdesign\](2).
------------------------------------------------------------------------
We need the following well known lemma to deal with the case where $G=M_{24}$.
\[octads\] [@sashasbook Lemma 2.10.1] Let $(X,\mathcal{B})$ be the Witt design $S(5,8,24)$. Then $\mathcal{B}$ contains $759$ elements, called *octads*. Each point of $X$ is contained in $253$ octads, each $2$-subset in $77$ octads, each $3$-subset in $21$ octads, each $4$-subset in $5$ octads, and each $5$-subset in a unique octad. Moreover, the stabiliser of an octad in $M_{24}$ is $C_2^4\rtimes A_8$ where $C_2^4$ acts trivially on the octad and transitively on its complement.
Then number of octads comes from [@sashasbook Lemma 2.10.1] and then the numbers of octads containing a given $i$-subset follows from basic counting. The statement about the stabiliser of an octad also comes from [@sashasbook Lemma 2.10.1].
------------------------------------------------------------------------
Since the stabilisers of a 3-set, of a 2-set, and of an octad are maximal in $G$, applying Constructions \[con:classic\], \[con:design\] and \[con:design2\], we get the list of $M_{24}$-primitive decompositions in Table \[tab:M244\].
${\mathcal{P}}$ $P$ $G_P$
------------------------------ -------------------- ----------------------------------------------
${\mathcal{P}}_{\cap}$ $K_{21}$ ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(3,4)$
${\mathcal{P}}_\ominus$ ${22\choose 3}K_2$ $M_{22}.2$
Construction \[con:design\] $J(8,4)$ $C_2^4\rtimes A_8$
Construction \[con:design2\] $21K_5$ ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(3,4)$
: $M_{24}$-primitive decompositions of $J(24,4)$[]{data-label="tab:M244"}
If $(J(24,4),{\mathcal{P}})$ is an $M_{24}$-primitive decomposition then ${\mathcal{P}}$ is given by one of the rows in Table $\ref{tab:M244}$.
Let $\Gamma=J(24,4)$ and $G=M_{24}$ acting on the point-set $X$ of the Witt-design $S(5,8,24)$. Take adjacent vertices $A=\{1,2,3,4\}$ and $B=\{2,3,4,5\}$ and suppose that $\Delta=\{1,2,3,4,5,6,7,8\}$ is the unique octad containing $A\cup B$. Then looking at the stabiliser of an octad given in Lemma \[octads\], we see that $G_{\{A,B\}}=G_{\{1,5\},\{2,3,4\},\{6,7,8\}}= C_2^4\rtimes ((S_2\times S_3^2)\cap A_8)$ with orbits in $\Delta$ of lengths 2, 3, 3. Since $G_{\{A,B\}}$ contains the pointwise stabiliser of the octad $\Delta$, which by Lemma \[octads\] acts regularly on $X\setminus\Delta$, it follows that $G_{\{A,B\}}$ is transitive on $X\setminus \Delta$.
Let $H$ be a maximal subgroup of $G$ such that $G_{\{A,B\}}\leqslant H<G$. The maximal subgroups of $G$ are given in [@atlas p 96], and comparing orders we see that $H\not\cong {\mathop{\mathrm{PSL}}}(2,7)$ or ${\mathop{\mathrm{PSL}}}(2,23)$. Since $G_{\{A,B\}}$ has an orbit of length 16 and an orbit of length 3 in $X$, it cannot fix a pair of dodecads. Similarly, if $H$ fixed a trio of disjoint octads, one of the three octads would be $\Delta$ and $G_{\{A,B\}}$ would interchange the other 2. However, all index 2 subgroups of $G_{\{A,B\}}$ are transitive on $X\setminus \Delta$ (a [Magma]{} calculation [@magma]) and so $H$ does not fix a trio of disjoint octads. Suppose next that $H$ fixes a sextet, that is, 6 sets of size 4 such that the union of any two is an octad. Then the $G_{\{A,B\}}$-orbit $X\setminus \Delta$ is the union of four of these sets. However, the remaining $G_{\{A,B\}}$-orbit lengths are incompatible with $H$ fixing a partition of $\{1,\ldots,8\}$ into two sets of size 4. Thus the list of maximal subgroups of $G$ in [@atlas p 96] implies that $H$ is intransitive on $X$, and so $H=G_{\{1,5\}}, G_{\{2,3,4\}}, G_{\{6,7,8\}}$, or $G_{\{1,2,3,4,5,6,7,8\}}$. By Lemma \[lem:classic\], the first gives the decomposition $\mathcal{P}_{\ominus}$ while the second gives $\mathcal{P}_{\cap}$. The third is the stabiliser of the divisor of the decomposition yielded by Construction \[con:design2\] containing $\{A,B\}$ while the fourth yields the decomposition obtained from Construction \[con:design\].
------------------------------------------------------------------------
The case $k=3$
==============
By Theorem \[Sn\], $G\leqslant S_n$ is arc-transitive on $J(n,3)$ if and only if $G$ is 4-transitive or $G={\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$ and $n=9$. Thus other than $A_n$ or $S_n$ the only possibilites for $(n,G)$ are $(11,M_{11}),(12,M_{12})$, $(23,M_{23})$, $(24,M_{24})$ and $(9,{\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8))$.
Since the stabiliser of a 2-subset is maximal in $M_{24}$, it follows that ${\mathcal{P}}_\cap$ and ${\mathcal{P}}_\ominus$ are $M_{24}$-primitive decompositions with divisors $K_{22}$ and ${22\choose 2}K_2$ respectively. We also have a construction involving sextets.
\[con:sextets\] [Let $S$ be a sextet, that is, a set of six $4$-subsets such that the union of any two is an octad, and define $P_S=\{\{A,B\}\mid A\cup B\in S\}$ and ${\mathcal{P}}=\{P_S\mid S\text{ a sextet}\}$. Then $P_S\cong 6J(4,3)\cong 6K_4$ with one copy of $K_4$ for each $4$-set in $S$. Let $\{A,B\}$ be an edge of $J(24,3)$. By [@sashasbook Lemma 2.3.3], $A\cup B$ is a member of a unique sextet $S$ and so $P_S$ is the only part of ${\mathcal{P}}$ containing $\{A,B\}$. Since $G$ acts primitively on the set of sextets, it follows that $(J(24,3),{\mathcal{P}})$ is an $M_{24}$-primitive decomposition. ]{}
If $(J(24,3),{\mathcal{P}})$ is an $M_{24}$-primitive decomposition then either ${\mathcal{P}}={\mathcal{P}}_\ominus$ or ${\mathcal{P}}_\cap$, or ${\mathcal{P}}$ arises from Construction $\ref{con:sextets}$.
Let $\Gamma=J(24,3)$ and $G=M_{24}$ acting on the point set $X$ of the Witt-design $S(5,8,24)$. Let $A=\{1,2,3\}$ and $B=\{2,3,4\}$ be adjacent vertices in $\Gamma$. Then $G_{\{A,B\}}=G_{\{1,4\},\{2,3\}}$ which is the stabiliser in ${\mathop{\mathrm{Aut}}}(M_{22})$ of a 2-subset and so by [@atlas p 39], $G_{\{A,B\}}\cong 2^5\rtimes S_5$. Since $G$ is 5-transitive on $X$, $G_{\{A,B\}}$ is transitive on $X\backslash \{1,2,3,4\}$.
Let $H$ be a maximal subgroup of $G$ such that $G_{\{A,B\}}\leqslant H<G$. The maximal subgroups of $G$ can be found in [@atlas]. Comparing orders we see that $H\not\cong {\mathop{\mathrm{PSL}}}(2,7)$, ${\mathop{\mathrm{PSL}}}(2,23)$, or the stabiliser of a trio of distinct octads. Now $G_{\{A,B\}}$ contains $G_{1,2,3,4}$ which is transitive on the remaining 20 points. Thus $G_{1,2,3,4}$ does not fix a pair of dodecads and so neither does $H$. Hence by the list of maximal subgroups of $G$ in [@atlas p 96], either $H$ is intransitive, or fixes a sextet. If $H$ is intransitive, then $H=G_{\{1,4\}}$ or $G_{\{2,3\}}$. By Lemma \[lem:classic\], the first gives $\mathcal{P}_{\ominus}$ while the second gives $\mathcal{P}_{\cap}$.
Suppose then that $H$ fixes a sextet. The orbit lengths of $G_{\{A,B\}}$ imply that $\{1,2,3,4\}$ is one of the blocks of the sextet. By [@sashasbook Lemma 2.3.3], $\{1,2,3,4\}$ is contained in a unique sextet $S$. Thus $H=G_S$ and is the stabiliser in $G$ of the divisor of the decomposition obtained from Construction \[con:sextets\] containing $\{A,B\}$.
------------------------------------------------------------------------
Before dealing with $G=M_{23}$ we need the following well known result which follows from Lemma \[octads\].
\[heptads\] Let $(X,\mathcal{B})$ be the Witt design $S(4,7,23)$. Then $\mathcal{B}$ contains $253$ elements, called *heptads*. Each point of $X$ is contained in $77$ heptads, each $2$-subset in $21$ heptads, each $3$-subset in $5$ heptads, and each $4$-subset in a unique heptad. Moreover, the stabiliser of a heptad is $C_2^4\rtimes A_7$ with the pointwise stabiliser of the heptad being $C_2^4$ which acts regularly on the $16$ points not in the heptad.
Since $(X,\mathcal{B})$ is derived from the set of all blocks of the Witt design $S(5,8,24)$ containing a given point, this follows from Lemma \[octads\].
------------------------------------------------------------------------
Using the Witt design $S(4,7,23)$ and the fact that the stabiliser of a 2-set is maximal in $M_{23}$ we get the $M_{23}$-primitive decompositions in Table \[tab:M233\]. These are in fact all such decompositions.
--------------------------------------------------------------------------------------------------
${\mathcal{P}}$ $P$ $G_P$
------------------------------ -------------------- ----------------------------------------------
${\mathcal{P}}_{\cap}$ $K_{21}$ ${\mathop{\mathrm{P}\Sigma\mathrm{L}}}(3,4)$
${\mathcal{P}}_\ominus$ ${21\choose 2}K_2$ ${\mathop{\mathrm{P}\Sigma\mathrm{L}}}(3,4)$
Construction \[con:design\] $J(7,3)$ $C_2^4\rtimes A_7$
Construction \[con:design2\] $5K_4$ $C_2^4\rtimes(C_3\times
A_5)\rtimes C_2$
--------------------------------------------------------------------------------------------------
: $M_{23}$-primitive decompositions of $J(23,3)$[]{data-label="tab:M233"}
If $(J(23,3),{\mathcal{P}})$ is an $M_{23}$-primitive decomposition then ${\mathcal{P}}$ is as in one of the lines of Table $\ref{tab:M233}$.
Let $\Gamma=J(23,3)$ and $G=M_{23}$ acting on the point-set $X$ of the Witt-design $S(4,7,23)$. Take adjacent vertices $A=\{1,2,3\}$ and $B=\{2,3,4\}$. By Lemma \[heptads\], $\{1,2,3,4\}$ is contained in a unique heptad, $h=\{1,2,3,4,5,6,7\}$ say, and so $G_{\{A,B\}}$ fixes $h$. Since the stabiliser of a heptad is isomorphic to $C_2^4\rtimes A_7$ (Lemma \[heptads\]), it follows that $G_{\{A,B\}}$ has order 192 and has orbits $\{1,4\}$, $\{2,3\}$, $\{5,6,7\}$ and $X\backslash h$.
Let $H$ be a maximal subgroup of $G$ such that $G_{\{A,B\}}\leqslant H<G$. The maximal subgroups of $G$ can be found in [@atlas]. By comparing orders, $H\not\cong C_{23}\rtimes
C_{11}$ and so $H$ is intransitive. Thus $H=G_{\{1,4\}}, G_{\{2,3\}},
G_{\{5,6,7\}}$ or $G_h$. By Lemma \[lem:classic\], the first two give the decompositions $\mathcal{P}_{\ominus}$ and $\mathcal{P}_{\cap}$ respectively. Also $G_{\{5,6,7\}}$ is the stabiliser of the divisor of the decomposition obtained from Construction \[con:design2\] containing $\{A,B\}$ while $G_h$ is the stabiliser of the divisor of the decomposition yielded by Construction \[con:design\].
------------------------------------------------------------------------
Since 4-set stabilisers and 2-set stabilisers are maximal in $M_{12}$, it follows from Lemma \[lem:classic\] that ${\mathcal{P}}_{\cup}$, ${\mathcal{P}}_{\cap}$ and ${\mathcal{P}}_{\ominus}$ are $M_{12}$-primitive decompositions with divisors isomorphic to $K_4$, $K_{10}$ and ${10 \choose 2}K_2$ respectively. We also have the following construction.
\[con:linkedfours\] [Let $(X,\mathcal{B})$ be the Witt design $S(5,6,12)$. Let $F$ be a *linked four*, that is a set of three mutually disjoint tetrads (sets of size 4) admitting a refinement into six duads (called duads of $F$) such that the union of any three duads coming from any two tetrads is a hexad. Let $$P_F=\Big\{\big\{\{x,u,v\},\{y,u,v\}\big\}\mid \{x,y,u,v\}\in F,
\{u,v\},\{x,y\} \text{ are duads of $F$} \Big\}$$ and let ${\mathcal{P}}=\{P_F\mid F \text{ a linked four}\}$. Then $P_F\cong 6K_2$ with one copy of $2K_2$ for each tetrad in $F$. Let $\{A,B\}$ be an edge of $J(12,3)$. It turns out ([Magma]{} calculation [@magma]) there is exactly one linked four $F$ having $A\cup B$ as a tetrad and $A\cap B$ as a duad of $F$, and so $P_F$ is the only part of ${\mathcal{P}}$ containing $\{A,B\}$. Since $G$ acts primitively on the set of linked fours, it follows that $(J(12,3),{\mathcal{P}})$ is an $M_{12}$-primitive decomposition. ]{}
If $(J(12,3),{\mathcal{P}})$ is an $M_{12}$-primitive decomposition then ${\mathcal{P}}={\mathcal{P}}_\cup,{\mathcal{P}}_\cap$ or ${\mathcal{P}}_\ominus$ or ${\mathcal{P}}$ is obtained from Construction $\ref{con:linkedfours}$.
Let $\Gamma=J(12,3)$ and $G=M_{12}$ acting on the point set $X$ of the Witt-design $S(5,6,12)$. Take adjacent vertices $A=\{1,2,3\}$ and $B=\{2,3,4\}$. The stabiliser in $G$ of a 4-set is $M_8\rtimes S_4$ such that the pointwise stabiliser $M_8$ of the 4-set acts regularly on the 8 remaining points. Hence $G_{\{A,B\}}=G_{\{1,4\},\{2,3\}}=M_8\rtimes (S_2\times S_2)$ which has order 32 and is transitive on the 8 points of $X\backslash\{1,2,3,4\}$.
Let $H$ be a maximal subgroup of $G$ such that $G_{\{A,B\}}\leqslant H<G$. The maximal subgroups of $G$ are given in [@atlas], and comparing orders we see that $H\not\cong M_{11}$, ${\mathop{\mathrm{PSL}}}(2,11)$, $M_9\rtimes S_3$, $C_2\times S_5$ and $A_4\times S_3$. Moreover, since $G_{\{A,B\}}$ has orbits of size 2,2 and 8 in $X$ it does not stabilise a hexad pair. If $H$ is intransitive then $H=G_{\{1,2,3,4\}}$, $G_{\{1,4\}}$ or $G_{\{2,3\}}$. These yield ${\mathcal{P}}_\cup$, ${\mathcal{P}}_{\ominus}$ and ${\mathcal{P}}_{\cap}$ respectively. Thus by [@atlas p 33] we are left to consider the case where $H\cong 4^2\rtimes D_{12}$. A [Magma]{} [@magma] calculation shows that there is a unique such $H$ containing $G_{\{A,B\}}$ and we obtain the decomposition from Construction \[con:linkedfours\].
------------------------------------------------------------------------
Before dealing with $G=M_{11}$ we need the following couple of lemmas, the first of which is well known.
\[pentads\] Let $(X,\mathcal{B})$ be the Witt design $S(4,5,11)$. Then $\mathcal{B}$ contains $66$ elements, called *pentads*. Each point of $X$ is contained in $30$ pentads, each $2$-subset in $12$ pentads, each $3$-subset in $4$ pentads, and each $4$-subset in a unique pentad. Moreover, the stabiliser of a pentad is isomorphic to $S_5$, which acts in its natural action on the pentad and as ${\mathop{\mathrm{PGL}}}(2,5)$ on the complementary hexad.
Since $(X,\mathcal{B})$ can be derived from the set of blocks of the Witt design $S(5,6,12)$ containing a given point, the first part follows from Lemma \[hexads-12\]. By [@atlas p 18], the stabiliser of a pentad is $S_5$ and has two orbits on $X$.
------------------------------------------------------------------------
\[lem:calcpentads\] Let $(X,\mathcal{B})$ be the Witt design $S(4,5,11)$ and $G=M_{11}$. Let $A=\{1,2,3\}$, $B=\{2,3,4\}$ and suppose that $p=\{1,2,3,4,5\}$ is the unique pentad containing $A\cup B$. Then $G_{\{A,B\}}\cong C_2^2$ and on $X\backslash p$ has an orbit $\{a,b\}$ of length $2$ and an orbit of length $4$. Moreover, $\{1,4,5,a,b\}$, $\{2,3,5,a,b\}$ and $X\backslash\{1,2,3,4,a,b\}$ are pentads.
By Lemma \[pentads\], $G_p$ induces $S_5$ on $p$, and since $G_{\{A,B\}}\leqslant G_p$ it follows that $G_{\{A,B\}}=G_{\{2,3\},\{1,4\}}\cong C_2^2$ and fixes the point $5$. By [@atlas], each involution of $G$ fixes precisely three points of $X$. Two of the involutions of $G_{\{A,B\}}$ fix three points of $p$ and so are fixed point free on $X\backslash p$. The third involution fixes the point 5 and fixes two points $a,b$ of $X\backslash p$. It follows that $G_{\{A,B\}}$ has an orbit of length two (namely, $\{a,b\}$) and an orbit of length 4 on $X\backslash p$.
Any four points lie in a unique pentad and by Lemma \[pentads\], any 3-subset is contained in 4 pentads. Hence $X\backslash p$ is divided into three sets of size two by the three pentads containing $\{1,4,5\}$ other than $\{1,2,3,4,5\}$. Similarly, $X\backslash p$ is partitioned by the three pentads containing $\{2,3,5\}$. Since $G_{\{A,B\}}$ fixes $\{1,4,5\}$ and $\{2,3,5\}$, it preserves both partitions and $\{a,b\}$ must be a block of both. Hence $\{1,4,5,a,b\}$ and $\{2,3,5,a,b\}$ are pentads. Moreover, since $X\backslash (\{a,b\}\cup p)$ is an orbit of length 4 of $G_{\{A,B\}}$ and is contained in a unique pentad, the fifth point of this pentad must also be fixed by $G_{\{A,B\}}$ and hence is 5. Thus $X\backslash\{1,2,3,4,a,b\}$ is a pentad.
------------------------------------------------------------------------
Since the stabiliser of a 2-set is maximal in $M_{11}$, it follows from Lemma \[lem:classic\] that ${\mathcal{P}}_\cap$ and ${\mathcal{P}}_\ominus$ are $M_{11}$-primitive decompositions. We also obtain $M_{11}$-primitive decompositions from Constructions \[con:design\], \[con:design2\], \[con:design3\] and \[con:design4\] by using the Witt design $S(4,5,11)$, since the stabilisers of a block, of a point and of a 3-subset are maximal subgroups of $M_{11}$.
\[con:M113\]
Let $(X,\mathcal{B})$ be the Witt design $S(4,5,11)$ and $G=M_{11}$. Let $A=\{1,2,3\}$ and $B=\{2,3,4\}$ be adjacent vertices of $J(11,3)$ and let $\{a,b\}$ be the orbit of length 2 of $G_{\{A,B\}}$ on $X\backslash \{1,2,3,4,5\}$ given by Lemma \[lem:calcpentads\].
1. For each $3$-subset $Y$ of $X$ let $$P_Y=\Big\{\big\{\{x,u,v\},\{y,u,v\}\big\}\mid \{x,y\}\cup Y, \{u,v\}\cup Y\in \mathcal{B} \Big\}$$ and let ${\mathcal{P}}=\{P_Y\mid Y \text{ a $3$-subset}\}$. By Lemma \[pentads\], $Y$ is contained in 4 pentads, and so $12K_2$. Let $Y=\{5,a,b\}$. By Lemma \[lem:calcpentads\], $\{A,B\}\in P_Y$ and $G_{\{A,B\}}\leqslant G_Y=G_{P_Y}$, which is a maximal subgroup of $G$. Hence by Lemma \[lem:general\], $(J(11,3),{\mathcal{P}})$ is an $M_{11}$-primitive decomposition.
2. Since $G$ is 4-transitive on $X$, Lemma \[lem:calcpentads\] implies that the stabiliser in $G$ of two $2$-subsets of $X$ fixes a third. For each $2$-subset $Y$ let $$P_Y=\Big\{\big\{\{x,u,v\},\{y,u,v\}\big\}\mid u,v,x,y\in X\backslash Y, G_{Y,\{x,y\}} =G_{Y,\{u,v\}}\Big\}$$ and let ${\mathcal{P}}=\{P_Y\mid Y \text{ a 2-subset}\}$. Then each $P_Y\cong {9\choose 2}K_2$. Moreover, by Lemma \[lem:calcpentads\] any edge of $J(11,3)$ is contained in a unique part of ${\mathcal{P}}$ ($\{A,B\}\in P_{\{a,b\}}$) and so $(J(11,3),{\mathcal{P}})$ is an $M_{11}$-primitive decomposition.
3. For each $Y\in\mathcal{B}$ let $$P_Y=\Big\{\big\{\{x,u,v\},\{y,u,v\}\big\}\mid x,y\in Y, \{u,v\}\cup(Y\backslash\{x,y\}) \in\mathcal{B}\Big\}$$ and let ${\mathcal{P}}=\{P_Y\mid Y\in\mathcal{B}\}$. By Lemma \[pentads\], each $3$-subset of $Y$ is contained in three more pentads and so each part of ${\mathcal{P}}$ is isomorphic to $3{5\choose 2}K_2=30K_2$. By Lemma \[lem:calcpentads\], $\{A,B\}\in P_Y$ for $Y=\{1,4,5,a,b\}$. Moreover, $G_{\{A,B\}}$ fixes $Y$ and so $G_{\{A,B\}}<G_Y=G_{P_Y}$. Thus Lemma \[lem:general\] and the fact that $G$ acts primitively on $\mathcal{B}$, imply that $(J(11,3),{\mathcal{P}})$ is a $G$-primitive decomposition.
4. For each $Y\in\mathcal{B}$ let $$P_Y=\Big\{\big\{\{x,u,v\},\{y,u,v\}\big\}\mid u,v\in Y, \{x,y\}\cup(Y\backslash\{u,v\}) \in\mathcal{B}\Big\}$$ and let ${\mathcal{P}}=\{P_Y\mid Y\in\mathcal{B}\}$. By Lemma \[pentads\], each $3$-subset of $Y$ is contained in three more pentads and so each part of ${\mathcal{P}}$ is isomorphic to $3{5\choose 2}K_2=30K_2$. By Lemma \[lem:calcpentads\], $\{A,B\}\in P_Y$ for $Y=\{2,3,5,a,b\}$ and $G_{\{A,B\}}<G_Y=G_{P_Y}$. Thus Lemma \[lem:general\] and the fact that $G$ acts primitively on $\mathcal{B}$, imply that $(J(11,3),{\mathcal{P}})$ is a $G$-primitive decomposition.
\[con:PSL211\]
Let $H={\mathop{\mathrm{PSL}}}(2,11)<M_{11}=G$. Then $H$ has an orbit of length $55$ on $3$-subsets and this orbit forms a $2-(11,3,3)$ design known as the Petersen design. The remaining $3$-subsets form an orbit of length 110 and a $2-(11,3,6)$ design [@PSL].
1. Let $\Pi$ be the subgraph of $J(11,3)$ induced on the orbit of length 55. The graph $\Pi$ was studied in [@DGLP] and is $H$-arc-transitive of valency 6. Given an edge $\{A,B\}$ of $\Pi$ we have $H_{\{A,B\}}=C_2^2=G_{\{A,B\}}$. Thus letting ${\mathcal{P}}=\{\Pi^g\mid g\in G\}$, it follows by Lemma \[lem:general\] that $(J(11,3),{\mathcal{P}})$ is a $G$-primitive decomposition.
2. Let $\Delta$ be the subgraph of $J(11,3)$ induced on the orbit of length 110. Then $\Delta$ has valency $15$ and given a vertex $A$, $H_A\cong S_3$ has orbits of length $3$, $6$ and $6$ on the neighbours of $A$. Let $B$ be a neighbour of $A$ in the orbit of length $3$ and let $P=\{A,B\}^H$. Let $g\in H$ which interchanges $A$ and $B$. Then by Lemma \[lem:cosetgraph\], $P\cong
{\mathop{\mathrm{Cos}}}(H,H_A,H_AgH_A)$. Moreover, ${\langle}H_A,g{\rangle}\cong A_5$ and so $P$ has 11 connected components, each with 10 vertices and isomorphic to the Petersen graph. Since $|H_{\{A,B\}}|=4=|G_{\{A,B\}}|$, it follows from Lemma \[lem:general\] that $(J(11,3),{\mathcal{P}})$ is a $G$-primitive decomposition with ${\mathcal{P}}=P^G$.
\[con:petersen\] [Let $A=\{1,2,3\}$ and $B=\{2,3,4\}$. By Lemma \[lem:calcpentads\], $Y=X\backslash\{1,2,3,4,a,b\}$ is a pentad fixed by $G_{\{A,B\}}$. Let $H=G_Y$ and $P=\{A,B\}^H$. Then by Lemma \[pentads\], $H$ induces $S_5$ on $Y$ and ${\mathop{\mathrm{PGL}}}(2,5)$ on $\{1,2,3,4,a,b\}$. Thus $H_A\cong S_3$ and is a maximal subgroup of $A_5\cong {\mathop{\mathrm{PSL}}}(2,5)$. Moreover, which interchanges $A$ and $B$ induces even permutations on $Y$ and so for such a $g$ we have ${\langle}H_A,g{\rangle}=A_5$. By Lemma \[lem:cosetgraph\], $P\cong {\mathop{\mathrm{Cos}}}(H,H_A,H_AhH_A)$. Since $|H:H_A|=20$ and ${\langle}H_A,g{\rangle}\cong A_5$, it follows that $P$ has two disconnected components with 10 vertices each. Since $|H_A:G_{A,B}|=3$ it follows that $P$ is a copy of two Petersen graphs. Let ${\mathcal{P}}=P^G$. Then as $G_{\{A,B\}}<H$, it follows from Lemma \[lem:general\] that $(J(11,3),{\mathcal{P}})$ is a $G$-primitive decomposition. ]{}
${\mathcal{P}}$ $P$ $G_P$
-------------------------------- ---------------------- ---------------------------------
${\mathcal{P}}_{\cap}$ $K_9$ $M_9\rtimes C_2$
${\mathcal{P}}_\ominus$ ${9\choose 2}K_2$ $M_9\rtimes C_2$
Construction \[con:design\] $J(5,3)\cong J(5,2)$ $S_5$
Construction \[con:design2\] $30K_4$ $M_{10}$
Construction \[con:design3\] $4K_3$ $M_8\rtimes S_3$
Construction \[con:design4\] $12K_2$ $M_8\rtimes S_3$
Construction \[con:M113\](1) $12K_2$ $M_8\rtimes S_3$
Construction \[con:M113\](2) ${9\choose 2}K_2$ $M_9\rtimes C_2$
Construction \[con:M113\](3) $30K_2$ $S_5$
Construction \[con:M113\](4) $30K_2$ $S_5$
Construction \[con:PSL211\](1) $\Pi$ ${\mathop{\mathrm{PSL}}}(2,11)$
Construction \[con:PSL211\](2) 11 Petersen graphs ${\mathop{\mathrm{PSL}}}(2,11)$
Construction \[con:petersen\] 2 Petersen graphs $S_5$
: $M_{11}$-primitive decompositions of $J(11,3)$[]{data-label="tab:M113"}
If $(J(11,3),{\mathcal{P}})$ is an $M_{11}$-primitive symmetric decomposition then ${\mathcal{P}}$ is given by Table $\ref{tab:M113}$.
Let $\Gamma=J(11,3)$ and $G=M_{11}<{\mathop{\mathrm{Sym}}}(X)$, and consider $X$ as the point set of the Witt-design $S(4,5,11)$ with automorphism group $G$. Let $A=\{1,2,3\}$ and $B=\{2,3,4\}$ be adjacent vertices. Suppose that $p=\{1,2,3,4,5\}$ is the unique pentad of the Witt design containing $\{1,2,3,4\}$ and let $H$ be a maximal subgroup of $G$ containing $G_{\{A,B\}}=G_{\{2,3\},\{1,4\}}$. The maximal subgroups of $G$ are given in [@atlas p 18].
If $H$ is the stabiliser of a point then $H=G_5$ and so we obtain the decomposition yielded by Construction \[con:design2\]. Next suppose that $H$ is the stabiliser of a duad. Then $H$ is one of $G_{\{2,3\}}, G_{\{1,4\}}$ or $G_{\{a,b\}}$ where $\{a,b\}$ is the orbit of length two of $G_{\{A,B\}}$ on $\{6,7,\ldots,11\}$. The first gives ${\mathcal{P}}_{\cap}$ the second gives ${\mathcal{P}}_{\ominus}$. Finally, if $H=G_{\{a,b\}}$ then $H$ is the stabiliser of the divisor of the decomposition obtained from Construction \[con:M113\](2) containing $\{A,B\}$.
Next suppose that $H$ is the stabiliser of a triad. Then $H$ stabilises $\{1,4,5\}$, $\{2,3,5\}$ or $\{5,a,b\}$. If $H=G_{\{1,4,5\}}$ then $H$ is the stabiliser of the divisor of the decomposition from Construction \[con:design3\] containing $\{A, B\}$. Also $H=G_{\{2,3,5\}}$ is the stabiliser of the divisor of the decomposition yielded by Construction \[con:design4\] containing $\{A,B\}$. Finally, $H=G_{\{5,a,b\}}$ is the stabiliser of the divisor of the decomposition obtained from Construction \[con:M113\](1) containing $\{A,B\}$.
Next suppose that $H$ is the stabiliser of a pentad. Since $G_{\{A,B\}}$ has only one orbit of odd length, it follows that $5$ is in the pentad. Combining $5$ with two orbits of $G_{\{A,B\}}$ of length two we get that $G_{\{A,B\}}$ fixes the pentads $\{1,2,3,4,5\}$, $\{1,4,5,a,b\}$, $\{2,3,5,a,b\}$ and $X\backslash\{1,2,3,4,a,b\}$ (by Lemma \[lem:calcpentads\], these 5-sets are actually pentads). Thus there are four choices for $H$. If $H=G_{\{1,2,3,4,5\}}$ then we obtain the decomposition from Construction \[con:design\]. If $H=G_{\{1,4,5,a,b\}}$, then $H$ is the stabiliser of the divisor of the decomposition from Construction \[con:M113\](3) containing $\{A,B\}$ while $H=G_{\{2,3,5,a,b\}}$ is the stabiliser of the divisor of the decomposition yielded by Construction \[con:M113\](4). Finally, if $H=G_{X\backslash\{1,2,3,4,a,b\}}$ then $H$ is the stabiliser of the divisor of the decomposition produced by Construction \[con:petersen\] containing $\{A,B\}$.
We are left to consider $H\cong {\mathop{\mathrm{PSL}}}(2,11)$. By a calculation using [Magma]{} [@magma], there are two such $H$ containing $G_{\{A,B\}}$. These give us the two decompositions in Construction \[con:PSL211\].
------------------------------------------------------------------------
We now give constructions for ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$-primitive decompositions of $J(9,3)$.
\[con:pgammal\]
Let $G={\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$ and $X={\mathop{\mathrm{GF}}}(8)\cup\{\infty\}$, where ${\mathop{\mathrm{GF}}}(8)$ is defined by the relation $i^3=i+1$.
1. By Theorem \[Sn\], $T={\mathop{\mathrm{PSL}}}(2,8)$ is not arc-transitive on $J(9,3)$ and so as $T{\vartriangleleft}G$ and has index three, $T$ has three equal sized orbits on edges. Thus the partition ${\mathcal{P}}=\{P_1,P_2,P_3\}$ given by these three orbits is a $G$-primitive decomposition. Since $T$ is vertex-transitive, this is in fact a homogeneous factorisation and appears in [@Cuaresma].
2. Let $x\in X$. Then $G_x={\mathop{\mathrm{A}\Gamma\mathrm{L}}}(1,8)$ and preserves the structure of an affine space ${\mathop{\mathrm{AG}}}(3,2)$ (with plane-set $\mathcal{B}$) on $X\backslash\{x\}$. Let $$P_x=\Big\{\{A,B\}\mid A\cup B\in\mathcal{B}\Big\}$$ and ${\mathcal{P}}=\{P_x\mid x\in X\}$. Then since each 3-subset lies in a unique plane, $P_x\cong 14K_4$. Moreover, $G_x$ acts transitively on the set $\mathcal{B}$ of affine planes and for $Y\in\mathcal{B}$ we have $G_{x,Y}$ induces $A_4$ on $Y$. Thus $G_x$ acts transitively on the set of edges in $P_x$ and so given $\{A,B\}\in P_x$ we have $|G_{x,\{A,B\}}|=2=|G_{\{A,B\}}|$. Thus $G_{\{A,B\}}\leqslant H$ and so by Lemma \[lem:general\], ${\mathcal{P}}=P_x^G$ is a $G$-primitive decomposition of $J(9,3)$.
3. Let $A=\{\infty,0,1\}$ and $B=\{\infty,0,i\}$. Then $G_{\{A,B\}}={\langle}g{\rangle}\cong C_2$ where $x^g=ix^{-1}$ and has orbits $\{0,\infty\}$, $\{1,i\}$, $\{i^2,i^6\}$, $\{i^3,i^5\}$ and $\{i^4\}$. Thus $G_{\{A,B\}}\leqslant G_{\{i^2,i^6\}}=H$ ($H$ has order 42) and so by Lemma \[lem:general\], letting $P=\{A,B\}^H$ and ${\mathcal{P}}=P^G$ we obtain a $G$-primitive decomposition of $J(9,3)$. Now $H_A={\langle}h{\rangle}$ where $x^h=x+1$, which has order two and so $P$ has $21$ vertices and valency $2$. Moreover, ${\langle}H_A,g{\rangle}=D_{14}$ and so by Lemma \[lem:cosetgraph\], $P$ has three connected components. Thus $P\cong 3C_7$.
4. Let $A=\{\infty,0,1\}$ and $B=\{\infty,0,i\}$. Then $G_{\{A,B\}}\leqslant G_{\{i^3,i^5\}}=H$ and so by Lemma \[lem:general\], letting $P=\{A,B\}^H$ and ${\mathcal{P}}=P^G$ we obtain a $G$-primitive decomposition of $J(9,3)$. Then $H_A={\langle}h{\rangle}$ where $x^h=(x^4+1)^{-1}$, which has order three. Thus $P$ has $14$ vertices and valency 3. Since $g$ and $h$ do not commute, ${\langle}H_A,g{\rangle}=H$ and so $P$ is a connected graph. Moreover, $P$ is $H$-arc-transitive and so by [@graphatlas p167], $P$ is the Heawood graph.
\[con:D18\]
Let $K={\mathop{\mathrm{GF}}}(64)$, with $\xi$ a primitive element of $K$, and let $F=\{0\}\cup\{(\xi^{9})^l|l=0,1,\ldots ,6\}\cong {\mathop{\mathrm{GF}}}(8)$. One can consider the projective line $X$ on which $G$ acts as the elements of $K$ modulo $F$. Then $H={\langle}\hat{\xi},\sigma,\tau{\rangle}\cong D_{18}\rtimes C_3$ where $\hat{\xi}:x\rightarrow \xi x \pmod F$, $\sigma:x\rightarrow x^8=x^{-1} \pmod F$, and $\tau:x\rightarrow x^4 \pmod F$.
1. Let $A=\{1,\xi,\xi^2\}$ and $B=\{\xi,\xi^2,\xi^3\}$. Then $\{A,B\}$ is an edge of $J(9,3)$ whose ends are interchanged by $\hat{\xi}^6\sigma\in H$. Thus letting $P=\{A,B\}^H$ and ${\mathcal{P}}=P^G$, Lemma \[lem:general\] implies that $(J(9,3),{\mathcal{P}})$ is a $G$-primitive decomposition. Now $H_A={\langle}\hat{\xi}^7\sigma{\rangle}$ and so $P$ has 27 vertices. Moreover, $H_{A,B}=1$ and so $P$ has valency 2. Since ${\langle}\hat{\xi}^6\sigma,\hat{xi}^7\sigma {\rangle}=D_{18}$ it follows from Lemma \[lem:cosetgraph\] that $P$ has 3 connected components and so $P\cong 3C_9$.
2. Let $A=\{1,\xi,\xi^3\}$ and $B=\{1,\xi,\xi^7\}$. Then $\{A,B\}$ is an edge of $J(9,3)$ whose ends are interchanged by $\hat{xi}^8\sigma\in H$. Thus letting $P=\{A,B\}^H$ and ${\mathcal{P}}=P^G$, Lemma \[lem:general\] implies that $(J(9,3),{\mathcal{P}})$ is a $G$-primitive decomposition. Now $\mid H_A\mid=1$ and so $P$ is a matching of 27 edges.
3. Let $A=\{1,\xi,\xi^3\}$ and $B=\{\xi,\xi^3,\xi^4\}$. Then $\{A,B\}$ is an edge of $J(9,3)$ whose ends are interchanged by $\hat{xi}^5\sigma\in H$. Thus letting $P=\{A,B\}^H$ and ${\mathcal{P}}=P^G$, Lemma \[lem:general\] implies that $(J(9,3),{\mathcal{P}})$ is a $G$-primitive decomposition. Now $\mid H_A\mid=1$ and so $P$ is a matching of 27 edges.
4. Let $A=\{1,\xi,\xi^3\}$ and $B=\{1,\xi^2,\xi^3\}$. Then $\{A,B\}$ is an edge of $J(9,3)$ whose ends are interchanged by $\hat{xi}^6\sigma\in H$. Thus letting $P=\{A,B\}^H$ and ${\mathcal{P}}=P^G$, Lemma \[lem:general\] implies that $(J(9,3),{\mathcal{P}})$ is a $G$-primitive decomposition. Now $\mid H_A\mid=1$ and so $P$ is a matching of 27 edges.
${\mathcal{P}}$ $P$ $G_P$
--------------------------------- --------------------------------------- ----------------------------------------------
${\mathcal{P}}_{\cap}$ $K_7$ $D_{14}\rtimes C_3$
${\mathcal{P}}_\ominus$ ${7\choose 2}K_2$ $D_{14}\rtimes C_3$
Construction \[con:pgammal\](1) ${\mathop{\mathrm{PSL}}}(2,8)$-orbits ${\mathop{\mathrm{PSL}}}(2,8)$
Construction \[con:pgammal\](2) $14K_4$ ${\mathop{\mathrm{A}\Gamma\mathrm{L}}}(1,8)$
Construction \[con:pgammal\](3) $3C_7$ $D_{14}\rtimes C_3$
Construction \[con:pgammal\](4) Heawood graph $D_{14}\rtimes C_3$
Construction \[con:D18\](1) $3C_9$ $D_{18}\rtimes C_3$
Construction \[con:D18\](2) $27K_2$ $D_{18}\rtimes C_3$
Construction \[con:D18\](3) $27K_2$ $D_{18}\rtimes C_3$
Construction \[con:D18\](4) $27K_2$ $D_{18}\rtimes C_3$
: ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$-primitive decompositions of $J(9,3)$[]{data-label="tab:pgammal"}
If $(J(9,3),{\mathcal{P}})$ is a ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$-primitive decomposition then ${\mathcal{P}}$ is as in Table $\ref{tab:pgammal}$.
Let $G={\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,8)$ act on $\{\infty\}\cup {\mathop{\mathrm{GF}}}(8)$ and suppose that ${\mathop{\mathrm{GF}}}(8)$ has primitive element $i$ such that $i^3=i+1$. Let $A=\{\infty,0,1\}$ and $B=\{\infty,0,i\}$ be adjacent vertices in $\Gamma=J(9,3)$. Then $G_{\{A,B\}}=G_{\{\infty,0\},\{1,i\}}={\langle}g{\rangle}\cong C_2$, where $x^g=ix^{-1}$, which fixes the point $i^4$ and has 4 orbits of size 2. Let $H$ be a maximal subgroup of $G$ containing $G_{\{A,B\}}$. The maximal subgroups of $G$ are given in [@atlas p 6].
If $H={\mathop{\mathrm{PGL}}}(2,8)$ then we obtain the decomposition in Construction \[con:pgammal\](1) while if $H$ is a point stabiliser then $H=G_{i^4}$ and we obtain the decomposition in Construction \[con:pgammal\](2).
Suppose now that $H\cong D_{14}\rtimes C_3$ is the stabiliser of a 2-subset. Then $H=G_{\{\infty,0\}}$, $H=G_{\{1,i\}}$, $H=G_{\{i^2,i^6\}}$, or $H=G_{\{i^3,i^5\}}$. In the first case we get the decomposition $\mathcal{P}_{\cap}$, while the second yields $\mathcal{P}_{\ominus}$. The third case gives Construction \[con:pgammal\](3) and the fourth gives the decomposition in Construction \[con:pgammal\](4).
Let $H={\langle}\hat{\xi},\sigma,\tau{\rangle}\cong D_{18}\rtimes C_3$ as given in Construction \[con:D18\]. Instead of finding all conjugates of $H$ containing $G_{\{A,B\}}$, we (equivalently) find all edge orbits $\{C,D\}^H$ such that $H$ contains $G_{\{C,D\}}$. Note that, for such an edge, $C$ and $D$ lie in the same $H$-orbit on vertices. One sees easily that $H$ has three orbits on vertices of $J(9,3)$, of sizes 3 ($\{1,\xi^3,\xi^6\}^{{\langle}\hat{xi}{\rangle}}$), 27 ($\{1,\xi,\xi^2\}^{{\langle}\hat{xi}{\rangle}}\cup\{1,\xi^2,\xi^4\}^{{\langle}\hat{xi}{\rangle}}\cup\{1,\xi^4,\xi^8\}^{{\langle}\hat{xi}{\rangle}}$), and 54 (all the other vertices). The orbit of size 3 contains no edges. In the orbit of size 27, if we fix the vertex $C=\{1,\xi,\xi^2\}$, we find two vertices $D$, namely $\{1,\xi,\xi^8\}$ and $\{\xi,\xi^2,\xi^3\}$, such that the unique involution switching $C$ and $D$ is in $H$. Moreover, these two vertices are interchanged by $H_C$. Hence this vertex orbit yields one orbit of edges whose stabilisers are contained in $H$ and we get the decomposition in Construction \[con:D18\](1).
In the orbit of size 54, if we fix the vertex $C=\{1,\xi,\xi^3\}$, we find three vertices $D$, namely $\{1,\xi,\xi^7\}$, $\{\xi,\xi^3,\xi^4\}$ and $\{1,\xi^2,\xi^3\}$, such that the unique involution switching $C$ and $D$ is in $H$. Since $H$ acts regularly on this orbit, each choice of $D$ gives a different $H$-orbit on edges and we get the three decompositions of Constructions \[con:D18\](2,3,4).
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The case $k=2$
==============
By Theorem \[Sn\], a subgroup $G$ of $S_n$ is arc-transitive on $J(n,2)$ if and only if $G$ is 3-transitive. Hence other than $A_n$ or $S_n$, the possibilities for $(n,G)$ are $(11,M_{11})$, $(12,M_{11})$, $(12,M_{12})$, $(22,M_{22})$, $(22,{\mathop{\mathrm{Aut}}}(M_{22}))$, $(23,M_{23})$, $(24,M_{24})$, $(2^d,{\mathop{\mathrm{AGL}}}(d,2))$ for $d>2$ , $(16,C_2^4\rtimes A_7)$, and $(q+1,G)$ where $G$ is a 3-transitive subgroup of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ with $q\geq 4$. We treat all but the last case in this section and deal with the subgroups of ${\mathop{\Gamma\mathrm{L}}}(2,q)$ in Section \[sec:PSL\].
If $(J(11,2),{\mathcal{P}})$ is an $M_{11}$-primitive decomposition then ${\mathcal{P}}$ is ${\mathcal{P}}_{\cap}$, ${\mathcal{P}}_{\cup}$, or ${\mathcal{P}}_{\ominus}$.
Let $G=M_{11}$ act on the point set $X$ of the Witt design $S(4,5,11)$, and let $A=\{1,2\}$, $B=\{2,3\}$ be adjacent vertices. Then $G_{\{A,B\}}=G_{2,\{1,3\}}$ and since $G$ is strictly 4-transitive it follows that $|G_{\{A,B\}}|=16$ and has one orbit on the 8 remaining points. Let $H$ be a maximal subgroup of $G$ containing $G_{\{A,B\}}$. Comparing orders and the maximal subgroups of $G$ given in [@atlas p 18] we see that $H\not\cong {\mathop{\mathrm{PSL}}}(2,11)$ or $S_5$. It follows that $H$ stabilises either a point, a pair or a 3-subset. In the first case $H=G_2$ and so ${\mathcal{P}}=\mathcal{P}_{\cap}$. In the second case, $H=G_{\{1,3\}}$ and we obtain the decomposition $\mathcal{P}_{\ominus}$, while in the last case $H=G_{\{1,2,3\}}$ and so we get the decomposition $\mathcal{P}_{\cup}$.
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Since the stabilisers of a point and a 2-subset are maximal in $M_{11}$ it follows from Lemma \[lem:classic\] that ${\mathcal{P}}_\cap$ and ${\mathcal{P}}_\ominus$ are $M_{11}$-primitive decompositions of $J(12,2)$. In order to give more constructions for $M_{11}$-primitive decompositions of $J(12,2)$, we will need the following lemma.
\[lem:linked3\] Let $G=M_{11}$ act $3$-transitively on the point set $X$ of the Witt design $S(5,6,12)$. As seen in Construction \[con:M11graph\], $G$ has an orbit of length $165$ on $4$-subsets, forming a $3-(12,4,3)$ design with block set $\mathcal{D}$. In this design, each $3$-set $S$ determines uniquely another $3$-set $S_
\mathcal{D}$, namely the set of fourth points of the $3$ blocks of $\mathcal{D}$ containing $S$. We have $(S_ \mathcal{D})_
\mathcal{D}=S$ and $S\cup S_\mathcal{D}$ is a hexad of $S(5,6,12)$. Moreover if $\{S, S_\mathcal{D}, U,V\}$ is the unique linked three containing $S$ and $S_\mathcal{D}$ as triads (see Lemma $\ref{lem:LTM12}$), then $U_\mathcal{D}=V$.
For any 3-set $S$, the set $S_\mathcal{D}$ is obviously well defined by the properties of the $3-(12,4,3)$ design. Now, an element of $G$ stabilising $S$ must also stabilise $S_\mathcal{D}$. Therefore $G_S\leqslant G_{S_\mathcal{D}}$. Since $S_\mathcal{D}$ is also a 3-set and $G$ is 3-transitive, we must have $|G_S|=|G_{S_\mathcal{D}}|$. Therefore $G_S=G_{S_\mathcal{D}}$. By a computation using [Magma]{} [@magma] we find that $G_S\cong S_3\times S_3$ has orbits of lengths 3, 3 and 6 on $X$. Hence $(S_\mathcal{D})_\mathcal{D}=S$.
Let $u,v$ be two points of $S_\mathcal{D}$. Then $S\cup\{u,v\}$ is contained in a unique hexad $h$. Since $G_S$ preserves the set of hexads containing $S$, and acts transitively on the 3 points of $S_\mathcal{D}$ and on the 6 points of $X\backslash (S\cup S_\mathcal{D})$, it follows that the sixth point of $h$ must also lie in $S_\mathcal{D}$. Hence $S\cup S_\mathcal{D}$ is a hexad. Let $T=\{S, S_\mathcal{D}, U,V\}$ be the unique linked three containing $S$ and $S_\mathcal{D}$ as triads (Lemma \[lem:LTM12\]). Since $G_S$ preserves $T$ and is transitive on $U\cup V$, it follows that $G_S$ has an index 2 subgroup $G_{S,U}$ with orbits $S, S_\mathcal{D}, U$ and $V$. Since the orbits of $G_{S,U}$ are a refinement of the orbits of $G_U$, $U_\mathcal{D}$ must be one of these orbits of size 3. Since $U_\mathcal{D}$ cannot be $S$ nor $S_\mathcal{D}$, it follows that $U_\mathcal{D}=V$.
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\[con:M11122\]
Let $G=M_{11}$ act 3-transitively on the point set $X$ of the Witt design $S(5,6,12)$. We use the notation of Lemma \[lem:linked3\].
1. Let $Y\in\mathcal{D}$. Let $$P_Y=\Big\{ \big\{\{u,x\},\{x,v\}\big\}\mid \{x,u,v\}_\mathcal{D}=Y\backslash\{x\} \Big\}$$ and ${\mathcal{P}}=\{P_Y\mid Y\in \mathcal{D}\}$. Then $P_Y\cong 4K_2$. Let $\{\{u,x\},\{x,v\}\}$ be an edge of $J(12,2)$. Then it is in a unique $P_Y$, with $Y=\{x\} \cup \{x,u,v\}_\mathcal{D}$. Since $G_Y$ is maximal in $G$, it follows that $(J(12,2),{\mathcal{P}})$ is a $G$-primitive decomposition.
2. Let $T$ be a $\mathcal{D}$-linked three, that is, a linked three for the $S(5,6,12)$ such that, for any $X\in T$, $X_\mathcal{D}$ is a triad of $T$. Let $$P_T=\Big\{\big\{\{u,x\},\{x,v\}\big\}\mid \{x,u,v\} \in T \Big\}$$ and ${\mathcal{P}}=\{P_T\mid T \text{ is a $\mathcal{D}$-linked three}\}$. Then $P_T\cong 4K_3$, with each triad contributing $K_3$. Let $\{\{u,x\},\{x,v\}\}$ be an edge of $J(12,2)$. Then $\{u,v,x\}$ and $\{u,v,x\}_\mathcal{D}$ must be triads of $T$. By Lemma \[lem:linked3\], the unique linked three containing these two triads is a $\mathcal{D}$-linked three. It follows that there is exactly one $\mathcal{D}$-linked three $T$ such that $P_T$ contains a given edge. Since the stabiliser in $G$ of a $\mathcal{D}$-linked three is maximal in $G$, it follows that $(J(12,2),{\mathcal{P}})$ is a $G$-primitive decomposition.
Thus we have the $M_{11}$-primitive decompositions listed in Table \[tab:M112\].
${\mathcal{P}}$ $P$ $G_P$
-------------------------------- ---------- ---------------------------------
${\mathcal{P}}_{\cap}$ $K_{11}$ ${\mathop{\mathrm{PSL}}}(2,11)$
${\mathcal{P}}_\ominus$ $10K_2$ $S_5$
Construction \[con:M11122\](1) $4K_2$ $M_8\rtimes S_3$
Construction \[con:M11122\](2) $4K_3$ $M_9\rtimes C_2$
: $M_{11}$-primitive decompositions of $J(12,2)$[]{data-label="tab:M112"}
If $(J(12,2),{\mathcal{P}})$ is an $M_{11}$-primitive decomposition then ${\mathcal{P}}$ is given by Table $\ref{tab:M112}$.
Let $G=M_{11}$ act transitively on the point set $X$ of the Witt design $S(5,6,12)$ and let $\mathcal{D}$ be the block set of the $3-(12,4,3)$ design described in Construction \[con:M11graph\] (see above). Take adjacent vertices $A=\{1,2\}$ and $B=\{2,3\}$. Then $G_{\{A,B\}}=G_{2,\{1,3\}}\cong
D_{12}$ which has an orbit of length 3 (namely, $\{1,2,3\}_\mathcal{D}$) and an orbit of length 6 on the remaining 9 points of $X$. Let $H$ be a maximal subgroup of $G$ containing $G_{\{A,B\}}$. Since $M_{10}$ contains no elements of order 6, it follows that $H\not\cong M_{10}$. If $H$ is a point stabiliser, then $H=G_2$ and we get the decomposition $\mathcal{P}_{\cap}$. If $H$ is a pair stabiliser then $H=G_{\{1,3\}}$, and we get the decomposition $\mathcal{P}_{\ominus}$. If $H\cong M_8\rtimes S_3$ then $H$ is the stabiliser of a block in $\mathcal{D}$. There is a unique such block, namely the union of $\{2\}$ with $\{1,2,3\}_\mathcal{D}$. Hence $H$ is the stabiliser of the divisor of the decomposition obtained from Construction \[con:M11122\](1) containing $\{A,B\}$.
Now let $H\cong M_9\rtimes S_3$. Then $H$ is a $\mathcal{D}$-linked three stabiliser, namely the only one containing $\{1,2,3\}$ as a triad (see the construction). Hence $H$ is the stabiliser of the divisor of the decomposition obtained from Construction \[con:M11122\](2) containing $\{A,B\}$.
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If $(J(12,2),{\mathcal{P}})$ is an $M_{12}$-primitive decomposition, then ${\mathcal{P}}$ is ${\mathcal{P}}_{\cup}$, ${\mathcal{P}}_{\cap}$ or ${\mathcal{P}}_{\ominus}$.
Let $G=M_{12}$ act on the point set $X$ of the Witt-design $S(5,6,12)$ and take adjacent vertices $A=\{1,2\}$ and $B=\{2,3\}$. Then $G_{\{A,B\}}=G_{2,\{1,3\}}$ which has order 144 and is 2-transitive on the 9 remaining points since $G$ is 5-transitive on $X$. Let $H$ be a maximal subgroup of $G$ containing $G_{\{A,B\}}$. The maximal subgroups of $G$ are given in [@atlas], and comparing orders we see that $H\not\cong {\mathop{\mathrm{PSL}}}(2,11)$, $2\times S_5$, $4^2: D_{12}$, $M_8.S_4$ or $A_4\times S_3$. Since $G_{\{A,B\}}$ fixes a point but not a hexad it follows that $H$ is not the stabiliser of a hexad pair, and since $G_{\{A,B\}}$ is 2-transitive on $X\backslash\{1,2,3\}$ we also have that $H$ is not the stabiliser of a linked three. In the action of $M_{11}$ on 12 points, ${\mathop{\mathrm{PSL}}}(2,11)$ is the stabiliser of a point. Since $144$ does not divide $|{\mathop{\mathrm{PSL}}}(2,11)|$ and $G_{\{A,B\}}$ fixes the point 2, it follows that $H$ is not a transitive copy of $M_{11}$. Thus $H=G_2, G_{\{1,3\}}$ or $G_{\{1,2,3\}}$. In the first case we get the decomposition ${\mathcal{P}}_\cap$, the second case yields ${\mathcal{P}}_\ominus$ while the third gives ${\mathcal{P}}_\cup$.
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Before dealing with $G=M_{22}$ we need the following well known result which follows from Lemma \[heptads\].
\[hexads-22\] Let $(X,\mathcal{B})$ be the Witt design $S(3,6,22)$. Then $\mathcal{B}$ contains $77$ elements, called *hexads*. Each point of $X$ is contained in $21$ hexads, each $2$-subset in $5$ hexads, and each $3$-subset in a unique hexad. Moreover, the stabiliser of a hexad is $C_2^4\rtimes A_6$ with the pointwise stabiliser of the hexad being $C_2^4$ which acts regularly on the $16$ points not in the hexad.
Since $(X,\mathcal{B})$ can be derived from the set of blocks of the Witt design $S(4,5,23)$ containing a given point, this follows from Lemma \[heptads\].
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If $(J(22,2),{\mathcal{P}})$ is an $M_{22}$-primitive decompositions then ${\mathcal{P}}={\mathcal{P}}_{\cap}$ or ${\mathcal{P}}_{\ominus}$, or ${\mathcal{P}}$ is obtained from Construction \[con:design\] and has divisors isomorphic to $J(6,2)$.
Let $G=M_{22}$ act on the point-set $X$ of the Witt design $S(3,6,22)$ and take adjacent vertices $A=\{1,2\}$ and $B=\{2,3\}$. Moreover, suppose that $h=\{1,2,3,4,5,6\}$ is the unique hexad of the Witt design containing $\{1,2,3\}$. By Lemma \[hexads-22\], $G_h=C_2^4\rtimes A_6$, where $C_2^4$ acts trivially on $h$ and transitively on $X\backslash h$. It follows that $G_{\{A,B\}}=G_{2,\{1,3\},\{4,5,6\}}$ had order 96 and acts transitively on $X\backslash h$.
Let $H$ be a maximal subgroup of $G$ containing $G_{\{A,B\}}$. Comparing orders and the maximal subgroups of $G$ given in [@atlas] we see that $H\not\cong {\mathop{\mathrm{PSL}}}(2,11)$, $A_7$ or $M_{10}$. Since $G_{\{A,B\}}$ does not stabilise an octad, it follows that $H$ is either $G_2$, $G_{\{1,3\}}$ or $G_h$. The first gives the decomposition $\mathcal{P}_{\cap}$, while the second yields $\mathcal{P}_{\ominus}$. Finally $G_h$ is the stabiliser of the part of the decomposition obtained from Construction \[con:design\] containing $\{A,B\}$ and has divisors isomorphic to $J(6,2)$.
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All ${\mathop{\mathrm{Aut}}}(M_{22})$-primitive decompositions of $J(22,2)$ are $M_{22}$-primitive decompositions.
By [@atlas], a maximal subgroup of ${\mathop{\mathrm{Aut}}}(M_{22})$ is either $M_{22}$ or arises from a maximal subgroup of $M_{22}$. Since $M_{22}$ is arc-transitive it does not give a decomposition. In all other cases, Lemma \[lem:restrict\] implies that we get $M_{22}$-primitive decompositions.
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If $(J(23,2),{\mathcal{P}})$ is an $M_{23}$-primitive decomposition then ${\mathcal{P}}$ is ${\mathcal{P}}_{\cap}$, ${\mathcal{P}}_{\ominus}$ or ${\mathcal{P}}_{\cup}$.
Let $G=M_{23}$ act on the point-set $X$ of the Witt design $S(4,7,23)$ and take adjacent vertices $A=\{1,2\}$ and $B=\{2,3\}$. Then $G_{\{A,B\}}=G_{2,\{1,3\}}\cong 2^4\rtimes S_5$ (see [@atlas p 71]) and since $G$ is 4-transitive, $G_{\{A,B\}}$ is transitive on $X\backslash\{1,2,3\}$. Let $H$ be a maximal subgroup of $G$ containing $G_{\{A,B\}}$. Since $|G_{\{A,B\}}|$ does not divide $23.11$, it follows from [@atlas p 71] that $H$ is intransitive. Hence $H$ is $G_2$, $G_{\{1,3\}}$ or $G_{\{1,2,3\}}$. These give us the decompositions $\mathcal{P}_{\cap}$, $\mathcal{P}_{\ominus}$ and $\mathcal{P}_{\cup}$ respectively.
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If $(J(24,2),{\mathcal{P}})$ is an $M_{24}$-primitive symmetric decompositions then ${\mathcal{P}}$ is ${\mathcal{P}}_\cap$, ${\mathcal{P}}_\ominus$ or ${\mathcal{P}}_\cup$.
Let $G=M_{24}$ acting on the point-set $X$ of the Witt design $S(5,8,24)$ and take adjacent vertices $A=\{1,2\}$ and $B=\{2,3\}$. Then $G_{\{A,B\}}=G_{2,\{1,3\}}\cong {\mathop{\mathrm{P}\Sigma\mathrm{L}}}(3,4)$ (see [@atlas p 96]). Note that $G_{\{A,B\}}$ is transitive on $X\backslash\{1,2,3\}$ since $G$ is 5-transitive on $X$. Let $H$ be a maximal subgroup of $G$ containing $G_{\{A,B\}}$. Looking at the maximal subgroups of $G$ in [@atlas], it follows that $H$ is either $G_2$, $G_{\{1,3\}}$ or $G_{\{1,2,3\}}$. Thus we obtain the decompositions $\mathcal{P}_{\cap}$, $\mathcal{P}_{\ominus}$ and $\mathcal{P}_{\cup}$ respectively.
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Let $G={\mathop{\mathrm{AGL}}}(d,2)$ acting on the set $X$ of vectors of a $d$-dimensional vector space over ${\mathop{\mathrm{GF}}}(2)$. Since the stabiliser of a vector is maximal in $G$, Lemma \[lem:classic\] implies that ${\mathcal{P}}_\cap$ is a $G$-primitive decomposition. The set of affine planes in the affine space ${\mathop{\mathrm{AG}}}(d,2)$ yields an $S(3,4,2^d)$ Steiner system with each point contained in $\frac{(2^d-1)(2^{d-1}-1)}{3}$ planes. In both cases, $G$ acts transitively on planes hence we can use Construction \[con:design\]. However, $G$ is not primitive on planes as it preserves parallelness. Applying now Construction \[con:primitive\] yields line 2 of Table \[tab:AGL\]. As $G$ is transitive on points and the stabiliser of a point is maximal in $G$, applying Construction \[con:design2\] yields line 3 of Table \[tab:AGL\]. As $G$ is 2-transitive, we can use Construction \[con:design4\]. However, $G$ acts imprimitively on $2$-subsets as $2$-subsets correspond to lines and again $G$ preserves parallelness. Thus we also apply Construction \[con:primitive\] and obtain line 4 of Table \[tab:AGL\]. Indeed the divisors are indexed by lines of the affine plane and are isomorphic to $2^{d-2}K_2$. Each pair $Y_1,Y_2$ of parallel lines yields a $C_4$ in the $J(4,2)$ induced on $Y_1\cup Y_2$. As a parallel class of lines contains $2^{d-1}$ lines, we have $\frac{2^{d-1}(2^{d-1}-1)}{2}$ pairs of parallel lines in the imprimitivity class.
When $d=4$ the group $\overline{G}=C_2^4\rtimes A_7<{\mathop{\mathrm{AGL}}}(4,2)$ is 3-transitive on $X$ and hence, by Corollary \[Sncor\], is arc-transitive on $J(2^4,2)$. Thus the four $G$-primitive decompositions in Table \[tab:AGL\] are also $\overline{G}$-transitive. The stabiliser in $\overline{G}$ of a point is $A_7$ which is maximal in $\overline{G}$. Hence the partitions in Rows 1 and 3 are $\overline{G}$-primitive. The stabilisers of 2-spaces and 1-spaces in $A_7$ are maximal in $A_7$ and so the remaining two partitions are also $\overline{G}$-primitive.
Before showing that these are the only $G$-primitive decompositions with $G\leqslant {\mathop{\mathrm{AGL}}}(d,2)$ we need a lemma.
\[lem:affinemax\] Let $G=N\rtimes G_0$ where $N\cong C_p^d$ for some prime $p$ and $G_0$ acts irreducibly on $N$. Suppose that $H$ is a maximal subgroup of $G$. Then either $H$ is a complement of $N$, or $M=N\rtimes H_0$ for some maximal subgroup $H_0$ of $H$.
Since $H$ normalises $N$ we have $H\leqslant NH\leqslant G$. Thus as $H$ is maximal, either $NH=H$ or $NH=G$. The first case implies that $N\leqslant H$ and so $H=N\rtimes H_0$ for some maximal subgroup $H_0$ of $G_0$. Suppose now that $NH=G$. Then $H/(H\cap N)\cong G_0$, and so for each $g\in G_0$, there exists $n\in N$ such that $ng\in H$. Since $N$ is abelian, it follows that $H$ induces $G_0$ in its action on $N$ by conjugation. Since $G_0$ acts irreducibly on $N$ and $H$ normalises $H\cap N$, it follows that $H\cap N=1$ or $N$. However, $H\cap N=N$ implies that $H=G$ which is not the case. Hence $H\cap N=1$ and $H\cong G_0$, that is $H$ is a complement of $N$.
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${\mathcal{P}}$ $P$ $G_P$
---------------------------------------------------- --------------------------------------- ----------------------------------------------
${\mathcal{P}}_{\cap}$ $K_{2^d-1}$ $G_0$
Constructions \[con:design\] and \[con:primitive\] $2^{d-2}J(4,2)\cong 2^{d-2}K_{2,2,2}$ $C_2^d\rtimes (G_0)_{{\langle}v,w{\rangle}}$
Construction \[con:design2\] $\frac{(2^d-1)(2^{d-1}-1)}{3}K_3$ $G_{v+w}$
Construction \[con:design4\] and \[con:primitive\] $2^{d-2}(2^{d-1}-1)C_4$ $C_2^d\rtimes (G_0)_{{\langle}v+w{\rangle}}$
: $G$-primitive decompositions of $J(2^d,2)$ for $G={\mathop{\mathrm{AGL}}}(d,2)$ with $d\geq 3$, or $G=C_2^4\rtimes A_7$ with $d=4$[]{data-label="tab:AGL"}
Let $d\geq 3$ and $G={\mathop{\mathrm{AGL}}}(d,2)$, or $d=4$ and $G=C_2^4\rtimes A_7$. If $(J(2^d,2),{\mathcal{P}})$ is a $G$-primitive decomposition then ${\mathcal{P}}$ is given by Table $\ref{tab:AGL}$.
We can identify $X$ with the vectors of a $d$-dimensional vector space over ${\mathop{\mathrm{GF}}}(2)$. Let $A=\{0,v\}$ and $B=\{0,w\}$ where $v,w$ are distinct non-zero elements of $X$. Thus $G_{\{A,B\}}=(G_0)_{\{v,w\}}$ which is an index 3 subgroup of $(G_0)_{{\langle}v,w{\rangle}}$. Moreover, $G_{\{A,B\}}$ fixes the vector $v+w$ and is transitive on $X\setminus{\langle}v,w{\rangle}$.
Let $H$ be a maximal subgroup of $G$ containing $G_{\{A,B\}}$. By Lemma \[lem:affinemax\], either $H$ is a complement of $N={\mathop{\mathrm{soc}}}(G)$ or $H=N\rtimes H_0$ for some maximal subgroup $H_0$ of $G_0$.
Suppose first that $H$ is a complement. By a [Magma]{} [@magma] calculation, $C_2^4\rtimes A_7$ has a unique conjugacy class of complements. If $d\geq 4$ then there is a unique class of complements of $N$ in ${\mathop{\mathrm{AGL}}}(d,2)$, while in ${\mathop{\mathrm{AGL}}}(3,2)$ there are two classes (see for example [@dempwolff]). Hence either $H$ is the stabiliser of a vector or $d=3$ and $H$ is transitive. In the second case $H={\mathop{\mathrm{PSL}}}(2,7)$ acting transitively on $V$. However, a Sylow 2-subgroup of $H$ is then regular on $V$, and hence $H$ cannot contain $G_{\{A,B\}}\cong D_8$ (fixing the point 0). Thus $H$ is the stabiliser of a vector and so $H=G_0$ or $G_{v+w}$. The first case yields the decomposition ${\mathcal{P}}_{\cap}$, while the second is the stabiliser of the divisor of the decomposition obtained from Construction \[con:design2\] containing $\{A,B\}$.
Suppose now that $H=N\rtimes H_0$ for some maximal subgroup $H_0$ of $G_0$. First let $G={\mathop{\mathrm{AGL}}}(d,2)$. Since $G_{\{A,B\}}$ is an index 3 subgroup of the stabiliser in ${\mathop{\mathrm{GL}}}(d,2)$ of the 2-space ${\langle}v,w{\rangle}$, it contains a Sylow 2-subgroup of ${\mathop{\mathrm{GL}}}(d,2)$. Thus $H_0$ contains a Sylow 2-subgroup of ${\mathop{\mathrm{GL}}}(d,2)$ and it follows from a Lemma of Tits (see for example [@seitz (1.6)]) that $H_0$ is a parabolic subgroup and hence is a subspace stabiliser. Now let $G=C_2^4\rtimes A_7$. Since $G_{\{A,B\}}\cong S_4$ fixes a nonzero vector it is contained in a subgroup ${\mathop{\mathrm{PSL}}}(2,7)$ of $A_7$ and hence by [@atlas p 10], the elements of order 3 in $G_{\{A,B\}}$ are from the conjugacy class $3B$, that is, in the representation of $A_7$ on 7 points they are products of two 3-cycles. By [@atlas p 10], $A_7$ has 5 conjugacy classes of maximal subgroups. The elements of order 3 in a maximal $S_5$ subgroup are from the conjugacy class $3A$ ([@atlas p 10]), instead of $3B$ and so $H_0\not\cong S_5$. If $H_0\cong A_6$ then $A_6\cong {\mathop{\mathrm{PSp}}}(4,2)'$ and contains two conjugacy classes of $S_4$ subgroups. One is the stabiliser of a vector and has orbit lengths 1, 6 and 8 on nonzero vectors and the other is the stabiliser of a totally isotropic 2-space with orbit sizes 3 and 12. Hence none of them stabilises the pair $\{v,w\}$ and so $H_0\not\cong A_6$. The remaining three conjugacy classes of maximal subgroups of $A_7$ are stabilisers of subspaces. Thus for both groups $G$, $H_0$ is a subspace stabiliser. The only proper, nontrivial subspaces fixed by $G_{\{A,B\}}$ are ${\langle}v+w{\rangle}$ and ${\langle}v,w{\rangle}$. If $H_0=(G_0)_{{\langle}v,w{\rangle}}$ then $H$ is the stabiliser of the class of planes parallel to ${\langle}v,w{\rangle}$ and so $H$ is the stabiliser of the divisor containing $\{A,B\}$ of the decomposition in Row 2 of Table \[tab:AGL\]. Similarly, if $H_0= (G_0)_{{\langle}v+w{\rangle}}$ then $H$ is the stabiliser of the class of lines parallel to ${\langle}v+w{\rangle}$ and so is the stabiliser of the divisor containing $\{A,B\}$ of the decomposition in Row 4 of Table \[tab:AGL\].
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Completing the case $k=2$: $G\leqslant{\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ {#sec:PSL}
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In this section we determine all $G$-primitive decompositions of $J(q+1,2)$ where $G$ is a 3-transitive subgroup of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ for $q=p^f\geq 4$ with $p$ a prime. The group ${\mathop{\mathrm{PGL}}}(2,q)$ is the group of all fractional linear transformations $$t_{a,b,c,d}: z\mapsto \frac{az+b}{cz+d}, \hspace{1cm} ad-bc\neq 0$$ of the projective line $X=\{\infty\}\cup{\mathop{\mathrm{GF}}}(q)$ with the conventions $1/0=\infty$ and $(a\infty+b)/(c\infty+d)=a/c$. Note that $t_{a,b,c,d}=t_{a',b',c',d'}$ if and only if $(a,b,c,d)=\lambda(a',b',c',d')$ for some $\lambda\neq 0$. The group ${\mathop{\mathrm{PSL}}}(2,q)$ is then the set of all $t_{a,b,c,d}$ such that $ad-bc$ is a square in ${\mathop{\mathrm{GF}}}(q)$. The Frobenius map $\phi:z\mapsto z^p$ also acts on $X$ and $\phi^{-1}t_{a,b,c,d}\phi=t_{a^p,b^p,c^p,d^p}$. Then ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)={\langle}{\mathop{\mathrm{PGL}}}(2,q),\phi{\rangle}$. Another interesting family of subgroups of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ occurs when $p$ is odd and $f$ is even. In this case we can define for each divisor $s$ of $f/2$, the group $M(s,q)={\langle}{\mathop{\mathrm{PSL}}}(2,q), \phi^st_{\xi,0,0,1}{\rangle}$, where $\xi$ is a primitive element of ${\mathop{\mathrm{GF}}}(q)$. Each $g \in{\mathop{\mathrm{PGL}}}(2,q)\setminus{\mathop{\mathrm{PSL}}}(2,q)$ can be written as $t_{\xi,0,0,1}h$ for some $h\in{\mathop{\mathrm{PSL}}}(2,q)$, and so $\phi^sg\in M(s,q)$. It was shown in [@GPZ Theorem 2.1] that $G$ is a 3-transitive subgroup of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ if and only if either $G$ contains ${\mathop{\mathrm{PGL}}}(2,q)$, or $G=M(s,q)$ for some $s$.
We begin with the following construction.
\[con:PGL1\] [[@Cuaresma] Let $X=\{\infty\}\cup {\mathop{\mathrm{GF}}}(q)$ be the projective line, $H={\mathop{\mathrm{PSL}}}(2,q)$ and $q\equiv 1\pmod 4$. Then $H$ is has two equal sized orbits on edges, namely $P_\square=\{\{\infty,0\},\{\infty,1\}\}^H$, and $P_{\not\square}=\{\{\infty,0\},\{\infty,t\}\}^H$, with $t$ not a square in $ {\mathop{\mathrm{GF}}}(q)$. Thus the partition ${\mathcal{P}}=\{P_\square,
P_{\not\square}\}$ is a $G$-primitive decomposition of $J(q+1,2)$ for any 3-transitive subgroup $G$ of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$. The divisors are complementary spanning graphs $\Theta$ of valency $q-1$. ]{}
\[prop:PSL\] Let $G$ be a $3$-transitive subgroup of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ and let ${\mathcal{P}}$ be a $G$-primitive decomposition of $J(q+1,2)$ such that ${\mathop{\mathrm{PSL}}}(2,q)$ fixes a part. Then $q\equiv 1\pmod 4$ and ${\mathcal{P}}$ is obtained from Construction $\ref{con:PGL1}$.
The graph $J(q+1,2)$ contains $\frac{q(q^2-1)}{2}$ edges. For $q$ even, $|{\mathop{\mathrm{PSL}}}(2,q)| =q(q^2-1)$ and an edge stabiliser has order 2, so ${\mathop{\mathrm{PSL}}}(2,q)$ is transitive on edges. Thus $q$ is odd and so $|{\mathop{\mathrm{PSL}}}(2,q)|=\frac{q(q^2-1)}{2}$. Whenever $(q-1)/2$ is odd, the stabiliser in ${\mathop{\mathrm{PSL}}}(2,q)$ of a point of $X$ has odd order. Since the stabiliser of the edge $\{\{x,y\},\{x,z\}\}$ fixes $x$ and interchanges $y$ and $z$, it follows that no nontrivial element of ${\mathop{\mathrm{PSL}}}(2,q)$ fixes an edge and so ${\mathop{\mathrm{PSL}}}(2,q)$ is edge-transitive. Hence $(q-1)/2$ is even and ${\mathop{\mathrm{PSL}}}(2,q)$ has two equal length orbits on edges, giving the $G$-primitive decomposition of Construction \[con:PGL1\] for any 3-transitive subgroup $G$ of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$.
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To classify all $G$-primitive decompositions with $G$ a 3-transitive subgroup of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ we require knowledge of the maximal subgroups of all such $G$. First we note the following theorem.
[[@PSLmax Corollary 1.2]]{} \[thm:dropdown\] Let ${\mathop{\mathrm{PGL}}}(2,q)\leqslant G\leqslant {\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ and suppose that $H$ is a maximal subgroup of $G$ not containing ${\mathop{\mathrm{PSL}}}(2,q)$. Then $H\cap {\mathop{\mathrm{PGL}}}(2,q)$ is maximal in ${\mathop{\mathrm{PGL}}}(2,q)$.
Theorem \[thm:dropdown\] and Lemma \[lem:restrict\] imply that we only need to find all ${\mathop{\mathrm{PGL}}}(2,q)$-primitive and all $M(s,q)$-primitive decompositions. We now state all maximal subgroups of these two groups. The first is well known and follows from Dickson’s classification [@dickson] of subgroups of ${\mathop{\mathrm{PSL}}}(2,q)$, see also [@PSLmax].
\[thm:PGLmax\] Let $G={\mathop{\mathrm{PGL}}}(2,q)$ with $q\geq 4$ a power of the prime $p$. Then the maximal subgroups of $G$ are:
1. $[q]\rtimes C_{q-1}$.
2. $D_{2(q-1)}$, $q\neq 5$.
3. $D_{2(q+1)}$.
4. $S_4$ if $q=p\equiv \pm 3\pmod 8$.
5. ${\mathop{\mathrm{PGL}}}(2,q_0)$ where $q=q_0^r$ with $q_0> 2$, $r$ is a prime and $r$ is odd if $q$ odd.
6. ${\mathop{\mathrm{PSL}}}(2,q)$, $q$ odd.
[[@PSLmax Theorem 1.5]]{} \[thm:Msqmax\] Let $G=M(s,q)$ with $q=p^f\geq 3$ for $p$ odd and $f$ even, and $s$ a divisor of $f/2$. Then the maximal subgroups of $G$ which do not contain ${\mathop{\mathrm{PSL}}}(2,q)$ are:
1. stabiliser of a point of the projective line,
2. $N_G(D_{q-1})$,
3. $N_G(D_{q+1})$,
4. $N_G({\mathop{\mathrm{PSL}}}(2,q_0))$ where $q=q_0^r$ with $r$ an odd prime.
We require the following knowledge about the stabiliser of an edge.
\[lem:Gedge\] Let $e=\{\{\infty,0\},\{\infty,1\}\}$. Then
1. ${\mathop{\mathrm{PGL}}}(2,q)_e={\langle}t_{-1,1,0,1}{\rangle}$,
2. ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)_e={\langle}t_{-1,1,0,1},\phi{\rangle}$ of order $2f$, and
3. $M(s,q)_e= {\langle}t_{-1,1,0,1},\phi^{2s}{\rangle}$ of order $f/s$.
Since ${\mathop{\mathrm{PGL}}}(2,q)$ is sharply 3-transitive, ${\mathop{\mathrm{PGL}}}(2,q)_e={\langle}g{\rangle}$ where $g$ fixes $\infty$ and interchanges $0$ and $1$. Thus ${\mathop{\mathrm{PGL}}}(2,q)_e$ is as in the lemma. Since $\phi$ fixes $\infty$, 0 and 1, the second claim follows. By [@GPZ Corollary 2.2], $M(s,q)_{\infty,0,1}={\langle}\phi^{2s}{\rangle}$ and since $q$ is an even power of a prime we have $q\equiv 1\pmod 4$. Thus $t_{-1,1,0,1}\in{\mathop{\mathrm{PSL}}}(2,q)$ and so $M(s,q)_e$ is as given by the lemma.
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Instead of finding all maximal subgroups $H$ containing the stabiliser of a fixed edge $\{A,B\}$ we solve the equivalent problem of choosing a representative $H$ from each conjugacy class of maximal subgroups and finding all edges whose edge stabiliser is contained in $H$. See Remark \[rem:2ways\].
\[con:PGL2\] [Let $X=\{\infty\}\cup {\mathop{\mathrm{GF}}}(q)$ be the projective line with $q$ odd and let $H={\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)_\infty={\mathop{\mathrm{A}\Gamma\mathrm{L}}}(1,q)$. Let $e=\{\{0,1\},\{0,-1\}\}$. The stabiliser in ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ of $e$ is ${\langle}\phi,t_{-1,0,0,1}{\rangle}$, which is contained in $H$. Moreover $H$ is a maximal subgroup of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$. Thus by Lemma \[lem:general\], letting $$P=e^H=\Big\{\big\{\{i,i+j\},\{i,i-j\}\big\}\mid
i,j\in{\mathop{\mathrm{GF}}}(q), i\neq j\Big\}$$ and ${\mathcal{P}}=P^{{\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)}$, we obtain a ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$-primitive decomposition of . The divisors have valency 2 and hence are a union of cycles. Since ${\mathop{\mathrm{GF}}}(q)$ has characteristic $p$ it follows that each cycle has length $p$ and so the divisors are isomorphic to $\frac{q(q-1)}{2p}C_p$. For any 3-transitive group $G$ with socle ${\mathop{\mathrm{PSL}}}(2,q)$, $H\cap G$ is maximal in $G$ and so ${\mathcal{P}}$ is $G$-primitive by Lemma \[lem:restrict\]. ]{}
\[lem:AGL\] Let $(J(q+1,2),\mathcal{P})$ be a $G$-primitive decomposition with $G$ a $3$-transitive subgroup of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ such that, for $P\in{\mathcal{P}}$, $G_P$ is the stabiliser of a point of the projective line. Then either ${\mathcal{P}}={\mathcal{P}}_{\cap}$ with divisors $K_q$ or $q$ is a power of an odd prime $p$ and ${\mathcal{P}}$ is obtained by Construction $\ref{con:PGL2}$.
Let $P\in{\mathcal{P}}$ and $\Gamma=J(q+1,2)$. Then without loss of generality we may suppose that $H=G_P$ is the stabiliser of the point $\infty$ of $X=\{\infty\}\cup {\mathop{\mathrm{GF}}}(q)$. We recall that $G$ either contains ${\mathop{\mathrm{PGL}}}(2,q)$ or is $M(s,q)$ for some $s$. Thus $H$ acts 2-transitively on ${\mathop{\mathrm{GF}}}(q)$ and so the orbits of $H$ on $V\Gamma$ are $O_1=\{\{\infty,x\}\mid x\in{\mathop{\mathrm{GF}}}(q)\}$ and $O_2=\{\{x,y\}\mid x,y\in {\mathop{\mathrm{GF}}}(q)\}$. If $\{A,B\}\in P$ then $H$ contains the stabiliser in $G$ of $\{A,B\}$ and so either $\{A,B\}\subseteq O_1$ or $\{A,B\}\subseteq O_2$. Note that $P=\{A,B\}^H$.
Since $H$ is 2-transitive on ${\mathop{\mathrm{GF}}}(q)$ it follows that $H$ acts transitively on the set of arcs between vertices of $O_1$ and so $H$ contains the stabiliser in $G$ of every edge between vertices of $O_1$. Thus if $\{A,B\}\subseteq O_1$ then $$\{A,B\}^H=\Big\{\big\{\{\infty,x\},\{\infty,y\}\big\}\mid
x,y\in{\mathop{\mathrm{GF}}}(q)\Big\} \cong K_q.$$ Hence ${\mathcal{P}}= \mathcal{P}_{\cap}$.
Suppose now that $\{A,B\}\subseteq O_2$. We may suppose that $A=\{0,1\}$ and $B=\{0,b\}$ for some $b\in{\mathop{\mathrm{GF}}}(q)\backslash\{0,1\}$. Let $g=t_{0,b,1-b,b}\in {\mathop{\mathrm{PGL}}}(2,q)$. Then $g$ maps $\infty\rightarrow 0\rightarrow1\rightarrow b$ and so $G_{\{A,B\}}=G_{\{\{\infty,0\},\{\infty,1\}\}}^g$ (this is obvious if $G$ contains ${\mathop{\mathrm{PGL}}}(2,q)$ and follows from the fact that $M(s,q){\vartriangleleft}{\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ for $G=M(s,q)$). By Lemma \[lem:Gedge\], $t_{-1,1,0,1}^g\in G_{\{A,B\}}\leqslant H=G_\infty$, and since $g$ does not fix $\infty$ and the only fixed points of $t_{-1,1,0,1}$ are $\infty$ and $2^{-1}$ (only if $q$ is odd), it follows that $q$ is odd and $g:2^{-1}\rightarrow\infty$. This implies that $b=-1$. Hence $\phi^g$ fixes $\infty$ and so by Lemma \[lem:Gedge\], $G_{\{\{0,1\},\{0,-1\}\}}\leqslant H$ in all cases. Hence ${\mathcal{P}}$ is the decomposition of Construction \[con:PGL2\].
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$D_{q-1}$ subgroups
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\[con:PGL3\]
Let $X=\{\infty\}\cup {\mathop{\mathrm{GF}}}(q)$ be the projective line where $q=p^f$ for some odd prime $p$ and let $\xi$ be a primitive element of ${\mathop{\mathrm{GF}}}(q)$. Then ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)_{\{0,\infty\}}={\langle}t_{\xi,0,0,1},t_{0,1,1,0},\phi{\rangle}\cong D_{2(q-1)}\rtimes C_f$.
1. Let $H={\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)_{\{0,\infty\}}$ and $e=\{\{0,1\},\{0,-1\}\}$. Then $t_{-1,0,0,1}\in H$ interchanges the two vertices of $e$ while $\phi$ fixes each of the vertices of $e$. Hence $H$ contains the stabiliser in ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ of $e$ and $H$ is a maximal subgroup of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ for $q\neq 5$. Thus by Lemma \[lem:general\], letting $$P=e^H= \Big\{\big\{\{x,y\},\{x,-y\}\big\}\mid
x\in\{0,\infty\},y\in{\mathop{\mathrm{GF}}}(q)\backslash\{0\}\Big\}$$ and ${\mathcal{P}}=P^{{\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)}$, we obtain a ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$-primitive decomposition of $J(q+1,2)$. The divisors are isomorphic to $(q-1)K_2$ since the stabiliser of the vertex $\{0,1\}$ in $H$ is ${\langle}\phi{\rangle}$, which fixes $\{0,-1\}$. For any $3$-transitive subgroup $G$ of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$, we have $H\cap G$ is maximal in $G$ and so ${\mathcal{P}}$ is a $G$-primitive decomposition by Lemma \[lem:restrict\].
2. Let $i<\frac{q-1}{2}$ and $l$ be an integer such that $\phi^l$ fixes the set $\{\xi^i,\xi^{-i}\}$. Let $G={\langle}{\mathop{\mathrm{PGL}}}(2,q),\phi^l{\rangle}$ and $H=G_{\{\infty,0\}}={\langle}t_{\xi,0,0,1},t_{0,1,1,0},\phi^l{\rangle}$. The automorphism of ${\mathop{\mathrm{PGL}}}(2,q)$ switching the vertices of the edge $e=\{\{1,\xi^i\},\{1,\xi^{-i}\}\}$ is $t_{0,1,1,0}$, while either $\phi^l$ or $t_{0,1,1,0}\phi^l$ fixes both vertices of $e$. Hence $G_e<H$ and $H$ is a maximal subgroup of $G$ for $q\neq 5$. Hence by Lemma \[lem:general\], letting $$P=e^H=\Big\{\big\{ \{x,\xi^ix\}, \{x,\xi^{-i}x\} \big\}\mid x\in{\mathop{\mathrm{GF}}}(q)\backslash\{0\}\Big\}$$ and ${\mathcal{P}}=P^G$, we obtain a $G$-primitive decomposition of $J(q+1,2)$. The divisors have valency 2 and hence are a union of cycles. These cycles have length the order of $\xi^i$, which is $\frac{q-1}{(q-1,i)}$. Thus each divisor is isomorphic to $(q-1,i)C_{\frac{q-1}{(q-1,i)}}$. In fact for any 3-transitive subgroup $\overline{G}$ of $G$, $H\cap \overline{G}$ is maximal in $\overline{G}$ and so ${\mathcal{P}}$ is a $\overline{G}$-primitive decomposition.
\[lem:D2(q-1)\] Let $(J(q+1,2),\mathcal{P})$ be a $G$-primitive decomposition such that ${\mathop{\mathrm{PGL}}}(2,q)\leqslant G \leqslant {\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ and for $P\in{\mathcal{P}}$ we have $G_P=N_G(D_{2(q-1)})$. Then either ${\mathcal{P}}={\mathcal{P}}_{\ominus}$, or $q$ is odd and ${\mathcal{P}}$ is obtained by Construction $\ref{con:PGL3}(1)$, or ${\mathcal{P}}$ is obtained by Construction $\ref{con:PGL3}(2)$.
Let $P\in{\mathcal{P}}$. Since $G_P\cap {\mathop{\mathrm{PGL}}}(2,q)$ is a maximal subgroup of ${\mathop{\mathrm{PGL}}}(2,q)$, by Lemma \[lem:restrict\], ${\mathcal{P}}$ is a ${\mathop{\mathrm{PGL}}}(2,q)$-primitive decomposition. Thus we may suppose that $G={\mathop{\mathrm{PGL}}}(2,q)$ and $H=G_P={\langle}t_{\xi,0,0,1},t_{0,1,1,0}{\rangle}\cong
D_{2(q-1)}$. The orbits of $H$ on vertices are $\{\{0,\infty\}\}$, $$O_0=\{\{x,y\}\mid x\in\{0,\infty\},y\in{\mathop{\mathrm{GF}}}(q)\backslash\{0\}\}$$ and $$O_i=\{\{x,\xi^ix\}\mid x\in{\mathop{\mathrm{GF}}}(q)\backslash\{0\}\}$$ for each $i\leq \frac{q-1}{2}$. Note that $|O_0|=2(q-1)$. When $q$ is even there are $q/2-1$ orbits $O_i$, each having length $q-1$. When $q$ is odd there are $\frac{q-3}{2}$ of length $q-1$ and one, $O_{\frac{q-1}{2}}$, of length $\frac{q-1}{2}$.
If $\{A,B\}\in P$ then $H$ contains the stabiliser in $G$ of $\{A,B\}$ and so $\{A,B\}$ is contained in one of the orbits of $H$ on vertices. Note that $P=\{A,B\}^H$.
Suppose first that $\{A,B\}\subseteq O_0$. Without loss, let $A=\{0,1\}$. Then the neighbours of $A$ in $O_0$ are $\{\infty,1\}$ and $\{0,y\}$ such that $y\in{\mathop{\mathrm{GF}}}(q)\backslash\{0\}$. The only ones which can be interchanged with $A$ by an element of $H$ are $\{\infty,1\}$, by $t_{0,1,1,0}$ and $\{0,-1\}$, by $t_{-1,0,0,1}$, when $q$ is odd. Thus the only edges between vertices of $O_0$ whose stabiliser in $G$ is contained in $H$ are those in the orbits $\{A,\{\infty,1\}\}^H$ and $\{A,\{0,-1\}\}^H$. The first gives the matching $\{\{\{0,y\},\{\infty,y\}\}\mid y\in{\mathop{\mathrm{GF}}}(q)\backslash\{0\}\}$ and hence the decomposition $\mathcal{P}_{\ominus}$ while the second gives the matching $\{\{\{x,y\},\{x,-y\}\}\mid x\in\{0,\infty\},y\in{\mathop{\mathrm{GF}}}(q)\backslash\{0\}\}$ and hence Construction \[con:PGL3\](1). Both matchings have $q-1$ edges and the second only occurs for $q$ odd. Note also that both orbits are preserved by ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)_{\{0,\infty\}}$ and so both decompositions are also ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$-decompositions.
Note that when $q$ is odd the orbit $O_{\frac{q-1}{2}}$ contains no edges. Thus suppose next that $\{A,B\}\subseteq O_i$ for $i<\frac{q-1}{2}$. Without loss of generality, let $A=\{1,\xi^i\}$. Then the neighbours of $A$ in $O_i$ are $\{1,\xi^{-i}\}$ and $\{\xi^i,\xi^{2i}\}$ and these are interchanged by $H_A={\langle}t_{0,\xi^i,1,0}{\rangle}\cong C_2$. Hence $H$ acts transitively on the set of edges between vertices of $O_i$. Moreover, ${\langle}t_{0,1,1,0}{\rangle}$ is the stabiliser $H$ of the edge $\{\{1,\xi^i\},\{1,\xi^{-i}\}\}$ and so $H$ contains the stabiliser in $G$ of an edge between two vertices of $O_i$. Thus ${\mathcal{P}}$ is obtained by Construction \[con:PGL3\](2). Moreover, an overgroup $\overline{G}={\langle}{\mathop{\mathrm{PGL}}}(2,q),\phi^l{\rangle}$ of ${\mathop{\mathrm{PGL}}}(2,q)$ in ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ preserves ${\mathcal{P}}$ if and only if $\overline{G}_{\{0,\infty\}}={\langle}H,\phi^l{\rangle}$ fixes $O_i$. Since $\phi^l$ fixes $1$, it follows that $\phi^l$ fixes $O_i$ if and only if $\phi^l$ fixes $\{\xi^i,\xi^{-1}\}$ and so $\overline{G}$ is as stated in Construction \[con:PGL3\](2).
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\[con:MsqDq-1\]
Let $G=M(s,q)$ and $\xi$ be a primitive element of ${\mathop{\mathrm{GF}}}(q)$ with $q=p^f$ for some odd prime $p$ and even integer $f$. Let $i$ be an integer and assume that either
- $s=f/2$ and $(\xi^i)^{{\langle}\phi^s{\rangle}}$ has length 2 and does not contain $\xi^{-i}$, or
- $s=f/4$ and $(\xi^i)^{{\langle}\phi^s{\rangle}}$ has length 4 and does contain $\xi^{-i}$.
Let $H=G_{\{0,\infty\}}={\langle}{\mathop{\mathrm{PSL}}}(2,q)_{\{0,\infty\}},\phi^st_{\xi,0,0,1}{\rangle}$ and note that ${\mathop{\mathrm{PSL}}}(2,q)_{\{0,\infty\}}={\langle}t_{\xi^2,0,0,1},t_{0,1,1,0}{\rangle}$.
1. Suppose that $i$ is even and let $e=\{\{1,\xi^i\},\{1,\xi^{-i}\}\}$ and $P=e^H$. Then $$\begin{aligned}
P =&\Big\{\big\{\{x^2,x^2\xi^i\},\{x^2,x^2\xi^{-i}\}\big\}\mid x\in{\mathop{\mathrm{GF}}}(q)\setminus\{0\}\Big\}\\
&\cup \Big\{\big\{\{y,y\xi^{ip^s}\},\{y,y\xi^{-ip^s}\}\big\}\mid
y= \not\square\Big\}\end{aligned}$$ Then $P$ has valency 2 (as the two neighbours of $\{1,\xi^i\}$ are $\{1,\xi^{-i}\}$ and $\{\xi^i,\xi^{2i}\})$ and so is a union of cycles. Each cycle has length the order of $\xi^i$ and so $P\cong
(q-1,i)C_{\frac{q-1}{(q-1,i)}}$.
Now $|\{1,\xi^i\}^H|=q-1$ and by Lemma \[lem:Gedge\], $|G_e|=f/s$. Since $|H|=(q-1)f/s$ it follows that $|H_e|=f/s$ and so $H_e=G_e$. Hence by Lemma \[lem:general\] and the fact that $H$ is maximal in $G$, letting ${\mathcal{P}}=P^G$ we get that ${\mathcal{P}}$ is a $G$-primitive decomposition.
2. Suppose now that $i$ is odd and let $e=\{\{1,\xi^i\},\{1,\xi^{-i}\}\}$ and $P=e^H$. Then $$\begin{aligned}
P =&\Big\{\big\{\{x^2,x^2\xi^i\},\{x^2,x^2\xi^{-i}\}\big\}\mid
x\in{\mathop{\mathrm{GF}}}(q)\backslash\{0\}\Big\} \\
&\cup \Big\{\big\{y,y\xi^{ip^s}\},\{y,y\xi^{-ip^s}\}\big\}
\mid y=\not\square\Big\}\end{aligned}$$ Then $|P|=q-1$ and so $|H_e|=f/s=|G_e|$, by Lemma \[lem:Gedge\]. The only neighbour of $\{1,\xi^i\}$ in $P$ is $\{1,\xi^{-i}\}$ and so $P=(q-1)K_2$. By Lemma \[lem:general\] and the fact that $H$ is maximal in $G$, letting ${\mathcal{P}}=P^G$ we get that ${\mathcal{P}}$ is a $G$-primitive decomposition.
\[lem:MsqD2(q-1)\] Let $(J(q+1,2),\mathcal{P})$ be a $G$-primitive decomposition with $G=M(s,q)$ for some $s$ such that for $P\in{\mathcal{P}}$, $G_P=N_G(D_{q-1})$. Then either ${\mathcal{P}}={\mathcal{P}}_{\ominus}$, or ${\mathcal{P}}$ arises from Construction $\ref{con:PGL3}(1)$, $\ref{con:PGL3}(2)$ or $\ref{con:MsqDq-1}$.
A subgroup $N_G(D_{q-1})$ of $G$ is a pair-stabiliser in $G$. Without loss of generality we may suppose that $H=G_{\{0,\infty\}}={\langle}{\mathop{\mathrm{PSL}}}(2,q)_{\{0,\infty\}},\phi^s t_{\xi,0,0,1}{\rangle}$. Note that $q\equiv 1\pmod 4$ and so ${\mathop{\mathrm{PSL}}}(2,q)_{\{0,\infty\}}={\langle}t_{\xi^{2},0,0,1},t_{0,1,1,0}{\rangle}$. Since $G$ is 3-transitive it follows that $$O_0=\{\{x,y\}\mid x\in\{0,\infty\},y\in{\mathop{\mathrm{GF}}}(q)\backslash\{0\}\}$$ is an $H$-orbit on vertices and as in the proof of Lemma \[lem:D2(q-1)\], if $\{A,B\}\subset O_0$ is an edge whose stabiliser in $G$ is contained in $H$ we obtain either ${\mathcal{P}}={\mathcal{P}}_{\ominus}$ or ${\mathcal{P}}$ is obtained by Construction \[con:PGL3\](1).
Now suppose $\{A,B\}\not\subset O_0$. Since $H$ is transitive on ${\mathop{\mathrm{GF}}}(q))\backslash\{0\}$, we can assume that $A=\{1,\xi^i\}$ where $1\leq i\leq q-2$ and that $A\cap B=\{1\}$, say $B=\{1,t\}$. We need to find the neighbours $B$ of $A$ such that $G_{\{A,B\}}\leqslant H$. Let $g\in {\mathop{\mathrm{PGL}}}(2,q)$ map $\{\{\infty,0\},\{\infty,1\}\}$ onto $\{A,B\}$. Then $G_{\{A,B\}}={\langle}t_{-1,1,0,1},\phi^{2s}{\rangle}^g$ by Lemma \[lem:Gedge\]. Hence $t_{-1,1,0,1}$ and $\phi^{2s}$ must stabilise $\{0,\infty\}^{g^{-1}}$. Note that $\infty^g\neq \infty$ (since $\infty\notin A$) and $\infty^g\neq 0$ (since $O\notin A$).
Since $B=\{1,t\}$, we can take $g=t_{a,\xi^i,a,1}$ where $a=\frac{\xi^i-t}{t-1}$, and then $\{0,\infty\}^{g^{-1}}=\{-\frac{\xi^i}{a},-\frac{1}{a}\}$. Recall that $t_{-1,1,0,1}$ stabilises this set. Now $t_{-1,1,0,1}$ fixes only the points $\infty, 2^{-1}$, and if $\{0,\infty\}^{g^{-1}}=\{\infty,2^{-1}\}$ we would have which is not the case. Hence $t_{-1,1,0,1}$ interchanges $-\frac{\xi^i}{a}$ and $-\frac{1}{a}$. Thus $-\frac{\xi^i}{a}=1+\frac{1}{a}$, that is $a=-1-\xi^i=\frac{\xi^i-t}{t-1}$, and so $t=\xi^{-i}$. For this value of $t$, $\{0,\infty\}^{g^{-1}}=\{\frac{\xi^i}{\xi^i+1},\frac{1}{\xi^i+1}\}$ and this set is stabilised by $t_{-1,1,0,1}$ and $\phi^{2s}$. The equality $\{\frac{\xi^i}{1+\xi^i},\frac{1}{1+\xi^i}\}^{\phi^{2s}}=\{\frac{\xi^i}{1+\xi^i},\frac{1}{1+\xi^i}\}$, is equivalent to either $\frac{\xi^{ip^{2s}}}{1+\xi^{ip^{2s}}}=\frac{\xi^i}{1+\xi^i}$ and $\frac{1}{1+\xi^{ip^{2s}}}=\frac{1}{1+\xi^i}$, or $\frac{\xi^{ip^{2s}}}{1+\xi^{ip^{2s}}}=\frac{1}{1+\xi^i}$ and $\frac{1}{1+\xi^{ip^{2s}}}=\frac{\xi^i}{1+\xi^i}$. In the first case $\xi^{ip^{2s}}=\xi^i$; in the second case $\xi^{ip^{2s}}=\xi^{-i}$. That means $O=(\xi^i)^{{\langle}\phi^s{\rangle}}$ has length 1,2 or 4.
Set $e=\{A,\{1,\xi^{-1}\}\}$. If $O$ has length 1, or $O$ has length 2 and $(\xi^i)^{\phi^s}=\xi^{-i}$, then $e^H$ yields a decomposition in Construction \[con:PGL3\](2). If $O$ has length 2 and $(\xi^i)^{\phi^s}\neq \xi^{-i}$, or $O$ has length 4 and $\xi^{ip^{2s}}=\xi^{-i}$, then $e^H$ yields a decomposition Construction \[con:MsqDq-1\](1) if $i$ is even and in Construction \[con:MsqDq-1\](2) if $i$ is odd.
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$D_{q+1}$ subgroups
-------------------
Before dealing with the case where $H\cap {\mathop{\mathrm{PSL}}}(2,q)=D_{q+1}$ we need a new model for the group action. Let $K={\mathop{\mathrm{GF}}}(q^2)$ for $q=p^f$ with primitive element $\xi$, and let $F=\{0\}\cup\{(\xi^{q+1})^l\mid l=0,1,\ldots,q-2\}\cong{\mathop{\mathrm{GF}}}(q)$. The element $\xi$ acts on $K$ by multiplication and induces an $F$-linear map. Moreover, under the induced action of$F$, $K$ is a 2-dimensional vector space over $F$. The field automorphism $\varphi$ of $K$ of order $2f$ mapping each element of $K$ to its $p^{\mathrm{th}}$ power is $F$-semilinear, that is, $\varphi$ preserves addition and for each $x\in K$, $\lambda\in F$, we have $(\lambda x)^{\varphi}=\lambda^p x^{\varphi}$. Then ${\mathop{\Gamma\mathrm{L}}}(2,q)={\langle}{\mathop{\mathrm{GL}}}(2,q),\varphi{\rangle}$. Note that $\varphi^f$ is an $F$-linear map so $\varphi^f\in{\mathop{\mathrm{GL}}}(2,q)$.
We can identify the projective line $X$ on which ${\mathop{\mathrm{PGL}}}(2,q)$ acts with the elements of $K$ modulo $F$, that is, $X=\{\xi^iF\mid i=0,1,\ldots,q\}$. Then ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)={\langle}{\mathop{\mathrm{PGL}}}(2,q),\varphi{\rangle}$. Multiplication by $\xi$ induces the map $\hat\xi$ of order $q+1$ and ${\langle}\hat{\xi}{\rangle}$ is normalised by $\varphi$. Moreover, for each $i$, $(\xi^i F)^{\varphi^f}=\xi^{iq}F=\xi^{-i}F$ and so $\varphi^f$ inverts $\hat\xi$. Hence ${\langle}\hat{\xi},\varphi^f{\rangle}\cong D_{2(q+1)}$.
\[con:PGL4\] [Let $X$ be the projective line modelled as above. Let $1\leq i<\frac{q+1}{2}$ and $e=\{\{1F,\xi^iF\},\{1F,\xi^{-i}F\}\}$ and let $s$ be a positive integer dividing $f$ such that ${\langle}\varphi^s{\rangle}$ has $\{\xi^iF,\xi^{-i}F\}$ as an orbit on $X$. Let $G={\langle}{\mathop{\mathrm{PGL}}}(2,q),\varphi^s{\rangle}$ and $H={\langle}\hat{\xi},\varphi^s{\rangle}\cong C_{q+1}\rtimes C_{2f/s}$. Now ${\langle}\varphi^s{\rangle}$ fixes $e$ and has order $2f/s$, which by Lemma \[lem:Gedge\] is the order of $G_e$. Hence $G_e<H$ and $H$ is a maximal subgroup of $G$. Thus by Lemma \[lem:general\], letting $$P=e^H=\Big\{\big\{\{xF,x\xi^iF\},\{xF,x\xi^{-i}F\}\big\}\mid
x\in{\mathop{\mathrm{GF}}}(q)\backslash\{0\}\Big\}$$ and ${\mathcal{P}}=P^G$, we obtain a $G$-primitive decomposition of $J(q+1,2)$. The divisors have valency 2 and hence are unions of cycles. These cycles have length the order of $\xi^iF$, which is $\frac{q+1}{(q+1,i)}$. Thus each divisor is isomorphic to $(q+1,i)C_{\frac{q+1}{(q+1,i)}}$. ]{}
\[lem:D2(q+1)\] Let $(J(q+1,2),\mathcal{P})$ be a $G$-primitive decomposition such that ${\mathop{\mathrm{PGL}}}(2,q)\leqslant G \leqslant{\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ and, for $P\in{\mathcal{P}}$, $G_P=N_G(D_{2(q+1)})$. Then ${\mathcal{P}}$ is obtained by Construction $\ref{con:PGL4}$.
Since ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)={\langle}{\mathop{\mathrm{PGL}}}(2,q),\varphi{\rangle}$ and $\varphi^f\in{\mathop{\mathrm{PGL}}}(2,q)$ we have $G={\langle}{\mathop{\mathrm{PGL}}}(2,q),\varphi^s{\rangle}$ for some $s$ dividing $f$. Let $L={\langle}\hat{\xi},\varphi^f{\rangle}\cong D_{2(q+1)}$. Then $N_G(L)={\langle}\hat{\xi},\varphi^s{\rangle}\cong C_{q+1}\rtimes C_{2f/s}$ and we may assume that $H=G_P=N_G(L)$. Let $e\in P$. Since $H$ is transitive on $X$ we may also assume that $e=\{\{1F,\xi^iF\},\{1F,\xi^jF\}\}$ for some integers $i$ and $j$. Since $H_{1F}={\langle}\varphi^s{\rangle}$ and by Lemma \[lem:Gedge\], $|G_e|=2f/s$, it follows that $G_e\leqslant H$ if and only if ${\langle}\varphi^s{\rangle}$ has $\{\xi^iF,\xi^jF\}$ as an orbit on $X$. Since $\varphi^f\in{\langle}\varphi^s{\rangle}$ and maps $\xi^iF$ to $\xi^{-i}F$ it follows that $j=-i$. Since $\xi^{-i}F=\xi^{q+1-i}F$ we may assume that $1\leq i\leq (q+1)/2$. Moreover, if $i=(q+1)/2$ then $q$ is odd and $\xi^{-(q+1)/2}F=\xi^{(q+1)/2}F$. Thus we may further assume that $1\leq i<(q+1)/2$. Hence ${\mathcal{P}}$ arises from Construction \[con:PGL4\].
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Next we need the following lemma about the normaliser in $M(s,q)$ of a subgroup $D_{q+1}$ in ${\mathop{\mathrm{PSL}}}(2,q)$.
\[lem:msqq+1dihedral\] Suppose $q=p^f$ where $f$ is even and $p$ is an odd prime. Let $L={\langle}\hat{\xi},\varphi^f{\rangle}\cap {\mathop{\mathrm{PSL}}}(2,q)$ and $G=M(s,q)$ for some divisor $s$ of $f/2$. Then
1. $L={\langle}\hat{\xi}^2,\varphi^f{\rangle}\cong D_{q+1}$.
2. If $p\equiv 1\pmod 4$ or $s$ is even then $N_G(L)= {\langle}\hat{\xi}^2,\varphi^s\hat{\xi}{\rangle}$, and is transitive on the projective line.
3. If $p\equiv 3\pmod 4$ and $s$ is odd then $N_G(L)={\langle}\hat{\xi}^2,\varphi^s {\rangle}$, and has two equal sized orbits on the projective line.
Now $\{1,\xi^{(q+1)/2}\}$ is a basis for $K$ over $F$. Define $\phi:K\rightarrow K$ such that, for all $\lambda_1,\lambda_2\in F$, $(\lambda_1+\lambda_2\xi^{(q+1)/2})^{\phi}=\lambda_1^p+\lambda_2^p\xi^{(q+1)/2}$. Then ${\mathop{\Gamma\mathrm{L}}}(2,q)={\langle}{\mathop{\mathrm{GL}}}(2,q),\phi{\rangle}$. Since also ${\mathop{\Gamma\mathrm{L}}}(2,q)={\langle}{\mathop{\mathrm{GL}}}(2,q),\varphi{\rangle}$, we must have $\varphi=\phi g$ for some $g\in {\mathop{\mathrm{GL}}}(2,q)$. Since $\varphi$ and $\phi$ fix 1, so does $g$. Moreover, $\phi$ fixes $\xi^{(q+1)/2}$ while $(\xi^{(q+1)/2})^{\varphi}=\xi^{p(q+1)/2}=\xi^{\frac{(p-1)(q+1)}{2}}\xi^{\frac{q+1}{2}}$. Note that $\xi^{\frac{(p-1)(q+1)}{2}}\in F$ and so $\xi^{(q+1)/2}$ is an eigenvector for $g$. Thus with respect to the basis $\{1,\xi^{(q+1)/2}\}$, the element $g$ is represented by the matrix $$\left(\begin{array}{cc} 1 & 0 \\
0 & \xi^{\frac{(p-1)(q+1)}{2}}\end{array}\right),$$ and $\det(g)=\xi^{\frac{(p-1)(q+1)}{2}}$ is a square in ${\mathop{\mathrm{GF}}}(q)$ if and only if $p\equiv 1\pmod 4$. Furthermore, $\varphi^f$ is represented by the matrix $$\left(\begin{array}{cc} 1 & 0 \\
0 & -1\end{array}\right).$$
Recall that an element of ${\mathop{\mathrm{GL}}}(2,q)$ induces an element of ${\mathop{\mathrm{PSL}}}(2,q)$ if and only if its determinant is a ${\mathop{\mathrm{GF}}}(q)$-square. Since $q\equiv 1\pmod 4$ it follows that $\varphi^f\in{\mathop{\mathrm{PSL}}}(2,q)$. Now ${\langle}\hat{\xi^2}{\rangle}\cong C_{(q+1)/2}$ and $\hat{\xi}^2\in{\mathop{\mathrm{PSL}}}(2,q)$, and since $\varphi^f$ inverts $\hat{\xi}$ it also inverts $\hat{\xi}^2$. Hence $L$ is as in part (1) of the lemma. Moreover, $L$ has two orbits on the projective line $X$, these being $\{1F,\xi^2F,\ldots,\xi^{q-1}F\}$ and $\{\xi F,\xi^3F,\ldots,\xi^qF\}$.
Now $\varphi=\phi g$ and $g\in{\mathop{\mathrm{PSL}}}(2,q)$ if and only if $p\equiv 1\pmod 4$. By definition it follows that $G=M(s,q)={\langle}{\mathop{\mathrm{PSL}}}(2,q),\phi^st{\rangle}$ for any $t\in{\mathop{\mathrm{PGL}}}(2,q)\setminus{\mathop{\mathrm{PSL}}}(2,q)$. Suppose first that $p\equiv 1\pmod 4$. Then $\varphi=\phi g$ with $g\in{\mathop{\mathrm{PSL}}}(2,q)$ and so $G={\langle}{\mathop{\mathrm{PSL}}}(2,q),\varphi^s\hat{\xi}{\rangle}$. When $p\equiv 3\pmod 4$ we have $\varphi=\phi g$ with $g\in{\mathop{\mathrm{PGL}}}(2,q)\setminus {\mathop{\mathrm{PSL}}}(2,q)$. Thus for odd $s$ we have $G={\langle}{\mathop{\mathrm{PSL}}}(2,q),\varphi^s{\rangle}$ while for even $s$ we have $G={\langle}{\mathop{\mathrm{PSL}}}(2,q),\varphi^s\hat{\xi}{\rangle}$. Now $(\varphi^f)^{\varphi^s\hat{\xi}}=(\varphi^f)^{\hat{\xi}}=\varphi^f\hat{\xi}^{-p^s+1}\in L$. Hence for $p\equiv 1\pmod 4$ or $s$ even we have $N_G(L)={\langle}\hat{\xi}^2,\varphi^s\hat{\xi}{\rangle}$. Since $\varphi^s\hat{\xi}$ interchanges the two $L$-orbits on $X$, $N_G(L)$ is transitive on $X$ and so we have proved part (2). For $p\equiv 3\pmod 4$ and $s$ odd we have $N_G(L)={\langle}\hat{\xi}^2,\varphi^s {\rangle}$. Since $\varphi^s$ fixes each $L$-orbit it follows that $N_G(L)$ has two orbits and the proof is complete.
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\[con:MsqDq+11\]
Let $q=p^f$ where $p$ is odd and $f$ even and let $G=M(s,q)$ for some divisor $s$ of $f/2$. Suppose that either $p\equiv 1\pmod 4$ or $s$ is even. Let $1\leq i<(q+1)/2$ such that ${\langle}\varphi^{2s}{\rangle}$ has $\{\xi^iF,\xi^{-i}F\}$ as an orbit on $X$. Let $H={\langle}\hat{\xi}^2,\varphi^s\hat{\xi}{\rangle}$ and $e=\{\{1F,\xi^iF\},\{1F,\xi^{-i}F\}\}$. Now ${\langle}\varphi^{2s}{\rangle}$ fixes $e$, lies in $G$, and has order $f/s$. Since this is the same order as $G_e$ (Lemma \[lem:Gedge\]) it follows that $G_e<H$. Hence by Lemma \[lem:general\], letting $P=e^H$ and ${\mathcal{P}}=P^G$ we obtain a $G$-primitive decomposition.
1. Suppose first that $i$ is even. Then $H_{\{1F,\xi^iF\}}={\langle}\varphi^f\hat{\xi}^i,\varphi^{4s}{\rangle}$ whose orbit containing $\{1F,\xi^{-i}F\}$ is $\{\{1F,\xi^{-i}F\},\{\xi^iF,\xi^{2i}F\}\}$. Thus $P$ has valency 2 and so is a union of cycles of length the order of $\hat{\xi}^i$, that is, $P\cong (q+1,i)C_{\frac{q+1}{(q+1,i)}}$.
2. Suppose now that $i$ is odd. An element of $H$ mapping $1F$ to $\xi^iF$ is of the form $h=\varphi^{st}\hat{\xi}^i$ with $t$ odd. Since ${\langle}\varphi^{2s}{\rangle}$ has $\{\xi^iF,\xi^{-i}F\}$ as an orbit on $X$, we have that $h$ maps $\xi^iF$ onto $\xi^{i(1+p^s)}F$ or onto $\xi^{i(1-p^s)}F$, according as $t\equiv 1$ or $3\pmod 4$ respectively. Hence, for $h$ to map $\xi^iF$ onto $1F$, we need $q+1$ to divide $i(1+p^s)$ or $i(1-p^s)$ respectively. Since $p^{2s}-1$ divides $p^f-1=q-1$, it follows that $\gcd(q+1,p^s+1)=2$ and $\gcd(q+1,p^s-1)=2$, and so $\frac{q+1}{2}$ must divide $i$, which is not possible since $1\leq i < \frac{q+1}{2}$. Thus $(\xi^iF)^h\neq 1F$. Hence $H_{\{1F,\xi^iF\}}=H_{1F,\xi^iF}={\langle}\varphi^{4s}{\rangle}$, which also fixes $\xi^{-i}F$ and hence fixes $e$. Thus $P$ is a matching with $q+1$ edges.
\[con:MsqDq+12\]
Let $p\equiv 3\pmod 4$ and let $G=M(s,q)$ for $q=p^f$ and $s$ an odd divisor of $f/2$. Let $1\leq i< (q+1)/2$ such that ${\langle}\varphi^{2s}{\rangle}$ has $\{\xi^iF,\xi^{-i}F\}$ as an orbit on $X$. Let $H= {\langle}\hat{\xi}^2,\varphi^s {\rangle}$ and $e=\{\{1F,\xi^iF\},\{1F,\xi^{-i}F\}\}$. Now ${\langle}\varphi^{2s}{\rangle}$ fixes $e$, lies in $G$ and has order $f/s$. Since this is the same order as $G_e$ (Lemma \[lem:Gedge\]) it follows that $G_e<H$ and so by Lemma \[lem:general\], letting $P=e^H$ and ${\mathcal{P}}=P^G$, we obtain a $G$-primitive decomposition.
1. Suppose first that $i$ is even. Then $H_{\{1F,\xi^iF\}}={\langle}\varphi^f\hat{\xi}^i, \varphi^{4s}{\rangle}$ and the $H$-orbit containing $\{1F,\xi^{-i}F\}$ has length 2. Thus $P$ is a union of cycles of length the order of $\hat{\xi}^i$, so $P\cong (q+1,i)C_{\frac{q+1}{(q+1,i)}}$.
2. If $i$ is odd then $1F$ and $\xi^iF$ lie in different $H$-orbits and so $H_{\{1F,\xi^iF\}}=H_{1F,\xi^iF}={\langle}\varphi^{4s}{\rangle}$ which also fixes $\xi^{-i}F$ and hence fixes $e$. Thus $P$ is a matching with $q+1$ edges.
\[con:MsqDq+13\]
Let $p\equiv 3\pmod 4$ and let $G=M(s,q)$ for $q=p^f$ and $s$ an odd divisor of $f/2$. Let $1\leq i<\frac{q+1}{2}$ such that ${\langle}\hat{\xi}^{-1}\varphi^{2s}\hat{\xi}{\rangle}$ has $\{\xi^{i+1}F,\xi^{-i+1}F\}$ as an orbit on $X$. Let $H= {\langle}\hat{\xi}^2,\varphi^s {\rangle}$ and $e=\{\{\xi F,\xi^{i+1}F\},\{\xi F,\xi^{-i+1}F\}\}$. Now ${\langle}\hat{\xi}^{-1}\varphi^{2s}\hat{\xi}{\rangle}\leqslant H$, fixes $e$, and has the same order as $G_e$. Thus $G_e<H$ and so by Lemma \[lem:general\], letting $P=e^H$ and ${\mathcal{P}}=P^G$, we obtain a $G$-primitive decomposition.
1. Suppose first that $i$ is odd. Then $\xi F$ and $\xi^{i+1}F$ lie in different $H$-orbits. Hence $H_{\{\xi F,\xi^{i+1}F\}}=H_{\xi F,\xi^{i+1}F}={\langle}\hat{\xi}^{-1}\varphi^{4s}\hat{\xi}{\rangle}$ which also fixes $\xi^{-i+1}F$ and so $P$ is a matching with $q+1$ edges.
2. If $i$ is even then $\varphi^f\hat{\xi}^{i+2}\in H$ interchanges $\xi F$ and $\xi^{i+1} F$, and so $H_{\{\xi F,\xi^{i+1}F\}}$ $={\langle}\hat{\xi}^{-1}\varphi^{4s}\hat{\xi},\varphi^f\hat{\xi}^{i+2}{\rangle}$, whose orbit containing $\{\xi F,\xi^{-i+1} F\}$ has size 2. Hence $P$ is a union of cycles of length the order of $\hat{\xi}^i$. Thus $P=(q+1,i)C_{\frac{q+1}{(q+1,i)}}$.
Let ${\mathcal{P}}$ be an $M(s,q)$-primitive decomposition of $J(q+1,2)$ with divisor stabiliser $N_{M(s,q)}(D_{q+1})$. Then ${\mathcal{P}}$ can be obtained from Construction $\ref{con:MsqDq+11}$, $\ref{con:MsqDq+12}$ or $\ref{con:MsqDq+13}$.
Let $G=M(s,q)$ and suppose first that $q=p^f$ where $p\equiv 1\pmod 4$ or $s$ is even. We may assume that $H={\langle}\hat{\xi}^2,\varphi^s\hat{\xi}{\rangle}$ by Lemma \[lem:msqq+1dihedral\]. Let $e\in P\in{\mathcal{P}}$. By Lemma \[lem:msqq+1dihedral\] again, $H$ is transitive on $X$ and so we can assume that $e=\{\{1F,\xi^iF\},\{1F,\xi^jF\}\}$ for some $i$ and $j$. Now $H_{1F}={\langle}\varphi^{2s}{\rangle}$, which has order $f/s$. By Lemma \[lem:Gedge\], this is the same order as $G_e$. Hence $G_e<H$ if and only if $H_{1F}=G_e$, which holds if and only if $\{\xi^iF,\xi^jF\}$ is an orbit of ${\langle}\varphi^{2s}{\rangle}$. Since $\varphi^f\in{\langle}\varphi^{2s}{\rangle}$ and maps $\xi^iF$ to $\xi^{-i}F$ it follows that $j=-i$ and we may assume as before that $1\leq
i<(q+1)/2$. Thus ${\mathcal{P}}$ comes from Construction \[con:MsqDq+11\].
Suppose now that $p\equiv 3\pmod 4$ and $s$ is odd. Then by Lemma \[lem:msqq+1dihedral\], we may assume that $H={\langle}\hat{\xi}^2,\varphi^s{\rangle}$. Let $e\in P\in{\mathcal{P}}$. By Lemma \[lem:msqq+1dihedral\], $H$ has 2 orbits on $X$ and so we may assume that $e=\{\{1F,\xi^iF\},\{1F,\xi^jF\}\}$ or $\{\{\xi F,\xi^{i+1}F\},\{\xi F,\xi^{j+1}F\}\}$. Suppose that $e$ is the first edge. Now $H_{1F}={\langle}\varphi^s{\rangle}$ which has order $2f/s$ while $G_e$ has order $f/s$ by Lemma \[lem:Gedge\]. Since $H_{1F}$ has a unique subgroup of order $f/s$ it follows that $G_e<H$ if and only if $G_e={\langle}\varphi^{2s}{\rangle}$, that is, if and only if ${\langle}\varphi^{2s}{\rangle}$ has $\{\xi^i F,\xi^j F\}$ as an orbit on $X$. Since $\varphi^f\in{\langle}\varphi^{2s}{\rangle}$ we have $j=-i$ and may assume $1\leq i<(q+1)/2$. It follows that ${\mathcal{P}}$ is as constructed in Construction \[con:MsqDq+12\]. If on the other hand $e=\{\{\xi F,\xi^{i+1}F\},\{\xi F,\xi^{j+1}F\}\}$, then $H_{\xi F}={\langle}\hat{xi}^{-1}\varphi^s\hat{\xi}{\rangle}$ which has order $2f/s$. Its only index two subgroup is ${\langle}\hat{\xi}^{-1}\varphi^{2s}\hat{\xi}{\rangle}$ and so by order arguments again this must have $\{\xi^{i+1} F,\xi^{j+1} F\}$ as an orbit. Since $\hat{\xi}^{-1}\varphi^f\hat{\xi}\in {\langle}\hat{\xi}^{-1}\varphi^{2s}\hat{\xi}{\rangle}$ and maps $\xi^{i+1}F$ to $\xi^{-i+1}F$ it follows that $j=-i$. Once again we have $1\leq i<\frac{q+1}{2}$. Hence ${\mathcal{P}}$ is as given by Construction \[con:MsqDq+13\].
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$S_4$-subgroups
---------------
First we have the following lemma on the orbit lengths of a subgroup $S_4$ of ${\mathop{\mathrm{PGL}}}(2,q)$ which we have adapted from [@PGLsubs].
\[lem:S4orbits\][[@PGLsubs Lemma 10]]{} Let $q=p\equiv\pm 3\pmod 8$, $q>3$, $G={\mathop{\mathrm{PGL}}}(2,q)$ acting on the projective line $X$, and $H$ a subgroup of $G$ isomorphic to $S_4$. Then $H$ has the following orbits of length less than $24$ on $X$.
1. If $q\equiv 5\pmod {24}$, then $H$ has one orbit of length $6$.
2. If $q\equiv 11\pmod {24}$, then $H$ has one orbit of length $12$.
3. If $q\equiv 13\pmod {24}$, then $H$ has one orbit of length $6$ and one of length $8$.
4. If $q\equiv 19\pmod {24}$, then $H$ has one orbit of length $8$ and one of length $12$.
\[con:PGL5\]
Let $X=\{\infty\}\cup {\mathop{\mathrm{GF}}}(q)$ be the projective line.
1. Let $q\equiv\pm 3\pmod 8$ be a prime ($q>3$) and $H=S_4$. Choose $x,y_1,y_2\in X$ such that $(|x^H|,|y_1|^H)=(6,8), (6,24),
(12,8)$ or $(12,24)$, and there exists in $H_x$ an element switching $y_1$ and $y_2$. Let $P=\{\{\{x,y_1\},\{x,y_2\}\}^H$ and ${\mathcal{P}}=P^{{\mathop{\mathrm{PGL}}}(2,q)}$. Then by Lemma \[lem:general\], $(J(q+1,2),{\mathcal{P}})$ is a ${\mathop{\mathrm{PGL}}}(2,q)$-primitive decomposition. Since , the stabiliser in $H$ of $\{x,y_1\}$ is trivial. Hence the divisors are isomorphic to $12K_2$.
2. Let $q\equiv 5\pmod 8$ be a prime and $H=S_4$. Let $P=\{\{x,y_1\},\{x,y_2\}\}^H$ where $x,y_1,y_2$ all lie in an $H$-orbit of length 6 and there exists in $H_x$ an element switching $y_1$ and $y_2$. By Lemma \[lem:S4orbits\], there is a unique orbit of $O_6$ of length 6. The group $H$ acts imprimitively on $O_6$ with blocks of size $2$, and $H_x\cong C_4$ contains an element interchanging $y_1$, $y_2$ if and only if $\{y_1,y_2\}$ is a block not containing $x$. Moreover, $P\cong 3C_4$. Let ${\mathcal{P}}=P^{{\mathop{\mathrm{PGL}}}(2,q)}$. Then by Lemma \[lem:general\] $(J(q+1,2),{\mathcal{P}})$, is a ${\mathop{\mathrm{PGL}}}(2,q)$-primitive decomposition.
3. Let $q\equiv 3\pmod 8$ be a prime and $H=S_4$. Let $P=\{\{x,y_1\},\{x,y_2\}\}^H$ where $x,y_1,y_2$ all lie in an $H$-orbit of length 12 and and there exists in $H_x$ an element switching $y_1$ and $y_2$. By Lemma \[lem:S4orbits\], there is a unique orbit $O_{12}$ of length 12. We can see this action as $S_4$ acting on ordered pairs, denoted by $[a,b]$. Then for $x=[1,2]\in O_{12}$, $H_x$ is the transposition $(3,4)$ in $S_4$. It fixes one remaining point of $O_{12}$, namely $[2,1]$ and interchanges the 5 pairs $\{[2,3],[2,4]\}$, $\{[3,1],[4,1]\}$, $\{[1,3],[1,4]\}$, $\{[3,2],[4,2]\}$, and $\{[3,4],[4,3]\}$. If we take $\{y_1,y_2\}$ as in the first two cases, then the stabiliser in $H$ of $\{x,y_1\}$ is trivial and so we get a matching $12K_2$ in each case. In the last three cases, the stabiliser in $H$ of $\{x,y_1\}$ has order 2, and we get unions of cycles. It is easy to see that in the third and fourth case, we get $4C_3$, while in the last case we get $3C_4$. Let ${\mathcal{P}}=P^{{\mathop{\mathrm{PGL}}}(2,q)}$. Then by Lemma \[lem:general\], $(J(q+1,2),{\mathcal{P}})$ is a ${\mathop{\mathrm{PGL}}}(2,q)$-primitive decomposition.
\[lem:S4\] Let $(J(q+1,2),\mathcal{P})$ be a $G$-primitive decomposition with $G={\mathop{\mathrm{PGL}}}(2,q)$ for $q=p\equiv\pm 3\pmod 8$ with $q\geq 5$ and given $P\in{\mathcal{P}}$ we have $G_P\cong S_4$. Then $P$ is obtained by Construction $\ref{con:PGL5}(1)$, $(2)$ or $(3)$.
Let $P\in {\mathcal{P}}$ and $H=G_P\cong S_4$. If $\{x,y\}\subseteq X$ with $x$ and $y$ in different $H$-orbits of length 24 then $|\{x,y\}^H|=24$ and that orbit contains no edges of $J(q+1,2)$. Thus if $x$ and $y$ come from different $H$-orbits $O_1$ and $O_2$ respectively, we may assume by Lemma \[lem:S4orbits\], that $|O_1|<|O_2|$ and so $\{x,y\}^H$ has length ${\mathop{\mathrm{lcm}}}(|O_1|,|O_2|)$ and contains edges. Moreover, $H$ contains the stabiliser in $G$ of such an edge $\{\{x,y_1\},\{x,y_2\}\}$ if and only if $H_x$ contains an element interchanging $y_1$ and $y_2$. If $x$ is in an orbit of size 8 then $|H_x|=3$ and so no such element exists, and if $x$ is in an orbit of size 24 then $|H_x|=1$ and so no such element exists. Thus the possibilities for $(|O_1|,|O_2|)$ are $(6,8)$, $(6,24)$, $(8,12)$ or $(12,24)$. In the first two cases $x$ must be in the orbit of length 6 and in the last two cases $x$ must be in the orbit of length 12. Thus we get the decomposition of Construction \[con:PGL5\](1).
Suppose now $e=\{\{x,y_1\},\{x,y_2\}\}$ is an edge such that $x,y_1,y_2\}$ lie in the same $H$-orbit $O_i$. Then $H$ contains $G_e$ if and only if $H_x$ interchanges $y_1$ and $Y_2$. Thus $|H_x|$ is even and so $|O_i|\neq 8,24$. If $q\equiv 5\pmod 8$ and $O_i$ is the unique orbit of size 6 then we obtain the decomposition in Construction \[con:PGL5\](2). If $q\equiv 3\pmod 8$ and $O_i$ is the unique orbit of size 12 then we obtain the decompositions in Construction \[con:PGL5\](3).
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Subfield subgroups
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Suppose now that $q=q_0^r$. Then $S=\{\infty\}\cup {\mathop{\mathrm{GF}}}(q_0)$ is a subset of the projective line $X=\{\infty\}\cup {\mathop{\mathrm{GF}}}(q)$ which is an orbit of the subgroup ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q_0)$ of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$. Notice that $\phi$ fixes the set $S$. Moreover, by [@handbook I, Example 3.23], if $\mathcal{B}=S^{{\mathop{\mathrm{PGL}}}(2,q)}$ then $(X,\mathcal{B})$ is a $S(3,q_0+1,q+1)$ Steiner system. Since $\phi$ fixes $S$ and ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)={\langle}{\mathop{\mathrm{PGL}}}(2,q),\phi{\rangle}$ it follows that $\mathcal{B}=S^{{\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)}$. Thus by Lemma \[lem:design\], we can construct a ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$-transitive decomposition of $J(q+1,2)$ with divisors isomorphic to $J(q_0+1,2)$. The stabiliser of a divisor is ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q_0)$. Moreover, this decomposition is $G$-transitive for any 3-transitive subgroup $G$ of ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$. For further constructions we need the orbits of ${\mathop{\mathrm{PGL}}}(2,q_0)$ on ${\mathop{\mathrm{GF}}}(q)\setminus {\mathop{\mathrm{GF}}}(q_0)$.
\[lem:subfieldorbits\] [[@PGLsubs Lemma 14]]{} Let $q=q_0^r$ for some prime $r$ and let $H=\{t_{a,b,c,d}\mid a,b,c,d\in{\mathop{\mathrm{GF}}}(q_0),ad-bc\neq 0\}$. If $r$ is odd then $H$ acts semiregularly on ${\mathop{\mathrm{GF}}}(q)\setminus{\mathop{\mathrm{GF}}}(q_0)$, while if $r=2$ then $H$ is transitive on ${\mathop{\mathrm{GF}}}(q)\setminus {\mathop{\mathrm{GF}}}(q_0)$.
\[con:PGL6\] [Let $X=\{\infty\}\cup {\mathop{\mathrm{GF}}}(q)$ be the projective line. Let $q=q_0^r$, where $q_0>2$, $r$ is a prime and $r$ is odd if $q$ is odd. Let $e=\{\{\infty,w_1\},\{\infty,w_2\}\}$ such that $w_1,w_2\in {\mathop{\mathrm{GF}}}(q)\setminus{\mathop{\mathrm{GF}}}(q_0)$ but $w_1+w_2\in{\mathop{\mathrm{GF}}}(q_0)$. Let $l$ be a positive integer such that $\phi^l$ fixes $\{w_1,w_2\}$. Then let $G={\langle}{\mathop{\mathrm{PGL}}}(2,q),\phi^l{\rangle}$ and $H={\langle}{\mathop{\mathrm{PGL}}}(2,q_0),\phi^l{\rangle}$. Let $P=e^H$ and ${\mathcal{P}}=P^G$. Then by Lemma \[lem:Gedge\], $G_e={\langle}t_{-1,w_1+w_2,0,1},\phi^l{\rangle}$ which is in $H$. Therefore by Lemma \[lem:general\], $(J(q+1,2),{\mathcal{P}})$ is a $G$-primitive decomposition. The stabiliser $H_{\{\infty,w_1\}}$ fixes $\infty$ and $w_1$ as they are in different $H$-orbits. We claim that ${\mathop{\mathrm{PGL}}}(2,q_0)_{\infty,w_1}=1$. Indeed, an element in that subgroup must be of the form $t_{a,b,0,1}$ with $a,b\in {\mathop{\mathrm{GF}}}(q_0)$, whose only fixed point is $\frac{b}{1-a}\in{\mathop{\mathrm{GF}}}(q_0)$ if it is not the identity. Hence there is a unique element of ${\mathop{\mathrm{PGL}}}(2,q_0)_{\infty}$ interchanging $w_1$ and $w_2$, this being $ t_{-1,w_1+w_2,0,1}$. Then as $\phi^l$ fixes $\{w_1,w_2\}$ and $\infty$, it follows that $H_{\infty,w_1}$ fixes $w_2$. Hence $P$ is isomorphic to $\frac{q_0(q_0^{2}-1)}{2}K_2$. ]{}
\[lem:PGLq0\] Let $(J(q+1,2),\mathcal{P})$ be a $G$-primitive decomposition such that ${\mathop{\mathrm{PGL}}}(2,q)\leqslant G\leqslant {\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$ and for $P\in{\mathcal{P}}$, $G_P\cong N_G({\mathop{\mathrm{PGL}}}(2,q_0))$ where $q=q_0^r$, $q_0>2$, $r$ is a prime and $r$ is odd if $q$ is odd. Then ${\mathcal{P}}$ is obtained by Construction $\ref{con:design}$ or $\ref{con:PGL6}$.
By Theorem \[thm:dropdown\], $\mathcal{P}$ is also a ${\mathop{\mathrm{PGL}}}(2,q)$-primitive decomposition so we may suppose that $G={\mathop{\mathrm{PGL}}}(2,q)$ and $H=G_P=\{t_{a,b,c,d}\mid a,b,c,d\in{\mathop{\mathrm{GF}}}(q_0), ad-bc\neq 0\}$. We have already seen that $H$ has an orbit $\{\infty\}\cup {\mathop{\mathrm{GF}}}(q_0)$ of length $q_0+1$ on $X$. Moreover, by Lemma \[lem:subfieldorbits\], when $r$ is odd, $H$ has $q_0^{r-3}+q_0^{r-5}+\cdots+q_0^{2}+1$ other orbits, all of length $q_0(q_0^{2}-1)$, while when $r=2$, $H$ is transitive on ${\mathop{\mathrm{GF}}}(q)\backslash{\mathop{\mathrm{GF}}}(q_0)$.
Suppose that $H$ contains the stabiliser in $G$ of the edge $e=\{\{v,w_1\},\{v,w_2\}\}$. Then $H_v$ contains the unique nontrivial element interchanging $w_1$ and $w_2$ (see Lemma \[lem:Gedge\]). Now $v$ must lie in the unique $H$-orbit of length $q_0+1$. For, if $r$ is odd and $v$ lies in an $H$-orbit of length $q_0(q_0^{2}-1)$ then $H_v=1$, while if $r=2$ and $v$ lies in ${\mathop{\mathrm{GF}}}(q)\setminus {\mathop{\mathrm{GF}}}(q_0)$, then $|H_v|=q_0+1$ which is odd. Without loss of generality we may suppose that $v=\infty$.
Then $G_e={\langle}t_{-1,w_1+w_2,0,1}{\rangle}$, so $G_e\leqslant H$ if and only if $w_1+w_2\in {\mathop{\mathrm{GF}}}(q_0)$. If $w_1$ and $w_2$ lie in the orbit of length $q_0+1$, that is, are in ${\mathop{\mathrm{GF}}}(q_0)$ then we obtain the decomposition from Construction \[con:design\], which is in fact preserved by ${\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)$. If $w_1\notin{\mathop{\mathrm{GF}}}(q_0)$ and $w_2=a-w_1$ with $a\in{\mathop{\mathrm{GF}}}(q_0)$, then we get a decomposition obtained from Construction \[con:PGL6\].
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For a primitive element $\mu$ of ${\mathop{\mathrm{GF}}}(q_0)$, $t_{\mu,0,0,1}\in {\mathop{\mathrm{PGL}}}(2,q)\setminus {\mathop{\mathrm{PSL}}}(2,q)$. Thus $\phi^st_{\mu,0,0,1}\in M(s,q)$ and normalises ${\mathop{\mathrm{PSL}}}(2,q_0)$. Hence $N_{M(s,q)}({\mathop{\mathrm{PSL}}}(2,q_0))={\langle}{\mathop{\mathrm{PSL}}}(2,q_0),\phi^st_{\mu,0,0,1}{\rangle}$.
We will need the following lemma.
\[lem:subfieldedge\] Let $G=M(s,q)$ with $q=q_0^r=p^f$ for some odd primes $r$ and $p$, and even integer $f$, and let $H={\langle}{\mathop{\mathrm{PSL}}}(2,q_0),\phi^st_{\mu,0,0,1}{\rangle}$ where $\mu$ is a primitive element of ${\mathop{\mathrm{GF}}}(q_0)$. Let $e=\{\{\infty,w_1\},\{\infty,w_2\}\}$.
1. Then $G_e\leqslant H$ if and only if both $w_1+w_2$ and $(w_2-w_1)^{p^{2s}-1}$ lie in ${\mathop{\mathrm{GF}}}(q_0)$.
2. There exist $w_1,w_2\notin{\mathop{\mathrm{GF}}}(q_0)$ such that $w_1+w_2$ and $(w_2-w_1)^{p^{2s}-1}$ lie in ${\mathop{\mathrm{GF}}}(q_0)$ if and only if $\gcd(\frac{q-1}{q_0-1},p^{2s}-1)\neq 1$.
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1. By Lemma \[lem:Gedge\], $G_e={\langle}t_{-1,w_1+w_2,0,1},(\phi^{2s})^g{\rangle}$ where $g=t_{w_2-w_1,w_1,0,1}$. Since $f$ is even and $q=q_0^r$ with $r$ odd, $q_0$ is an even power of $p$ and hence $-1$ is a square in ${\mathop{\mathrm{GF}}}(q_0)$. Thus $t_{-1,w_1+w_2,0,1}\in H$ if and only if $w_1+w_2\in{\mathop{\mathrm{GF}}}(q_0)$. Moreover, $$\begin{aligned}
g^{-1}\phi^{2s}g &=t_{1,-w_1,0,w_2-w_1}\phi^{2s}t_{w_2-w_1,w_1,0,1}\\
&=\phi^{2s}t_{1,-w_1^{p^{2s}},0,(w_2-w_1)^{p^{2s}}}t_{w_2-w_1,w_1,0,1}\\
&=\phi^{2s}t_{w_2-w_1,-(w_2-w_1)w_1^{p^{2s}}+w_1(w_2-w_1)^{p^{2s}},0,(w_2-w_1)^{p^{2s}}}\\
&=\phi^{2s}t_{1,w_1(w_2-w_1)^{p^{2s}-1}-w_1^{p^{2s}},0,(w_2-w_1)^{p^{2s}-1}}.\end{aligned}$$ Let $h=t_{1,w_1(w_2-w_1)^{p^{2s}-1}-w_1^{p^{2s}},0,(w_2-w_1)^{p^{2s}-1}}$. As $\phi^{2s}\in H$, it follows that $(\phi^{2s})^g \in H$ if and only if $h\in{\mathop{\mathrm{PSL}}}(2,q_0)$. Now if $h\in{\mathop{\mathrm{PSL}}}(2,q_0)$ then $(w_2-w_1)^{p^{2s}-1}\in{\mathop{\mathrm{GF}}}(q_0)$. Thus if $G_e\leqslant H$ then both $w_1+w_2$ and $(w_2-w_1)^{p^{2s}-1}$ lie in ${\mathop{\mathrm{GF}}}(q_0)$. Conversely, suppose that $w_1+w_2=a\in{\mathop{\mathrm{GF}}}(q_0)$ and $w_2-w_1=u$ with $u^{p^{2s}-1}=b\in{\mathop{\mathrm{GF}}}(q_0)$. Then writing $\frac{1}{2}$ for $2^{-1}\in{\mathop{\mathrm{GF}}}(p)$ and noting that $2^{p^{2s}}=2$, $w_1(w_2-w_1)^{p^{2s}-1}-w_1^{p^{2s}}
=\frac{a-u}{2}b-\frac{a^{p^{2s}}-u^{p^{2s}}}{2^{p^{2s}}}=\frac{ab-a^{p^{2s}}}{2}\in{\mathop{\mathrm{GF}}}(q_0)$. Thus $h\in{\mathop{\mathrm{PGL}}}(2,q_0)$, and since $p^{2s}-1$ is even $h\in{\mathop{\mathrm{PSL}}}(2,q_0)$.
2. Let $\xi$ be a primitive element of ${\mathop{\mathrm{GF}}}(q)$. Then ${\mathop{\mathrm{GF}}}(q_0)=\{0\}\cup\{\xi^{i\frac{q-1}{q_0-1}}|i=1,\ldots,q_0-1\}$ and we can choose $\mu=\xi^{\frac{q-1}{q_0-1}}$. For $w_2-w_1=\xi^j\in {\mathop{\mathrm{GF}}}(q)\setminus\{0\}$, if $(w_2-w_1)^{p^{2s}-1}$ lies in ${\mathop{\mathrm{GF}}}(q_0)$, that means that $\xi^{j(p^{2s}-1)}=\xi^{i\frac{q-1}{q_0-1}}$ for some integer $i$. If $\gcd(\frac{q-1}{q_0-1},p^{2s}-1)=1$, we must have $j$ a multiple of $(q-1)/(q_0-1)$, and so $w_2-w_1\in {\mathop{\mathrm{GF}}}(q_0)$. If we also have $w_1+w_2\in{\mathop{\mathrm{GF}}}(q_0)$, then this implies that $w_1,w_2\in{\mathop{\mathrm{GF}}}(q_0)$. Hence if $w_1,w_2\notin{\mathop{\mathrm{GF}}}(q_0)$ such that $w_1+w_2$ and $(w_2-w_1)^{p^{2s}-1}$ lie in ${\mathop{\mathrm{GF}}}(q_0)$ then $\gcd(\frac{q-1}{q_0-1},p^{2s}-1)\neq 1$. Conversely, suppose $\gcd(\frac{q-1}{q_0-1},p^{2s}-1)=d\neq 1$ and choose $j=(q-1)/d(q_0-1)$. Then take $w_2=\xi^j/2$ and $w_1=-\xi^j/2$. We obviously have $w_1+w_2\in{\mathop{\mathrm{GF}}}(q_0)$ and $w_1,w_2\notin{\mathop{\mathrm{GF}}}(q_0)$. Moreover $(w_2-w_1)^{p^{2s}-1}=\xi^{\frac{p^{2s}-1}{d}\frac{q-1}{q_0-1}}\in{\mathop{\mathrm{GF}}}(q_0)$.
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\[con:Msqsubfield\]
Let $G=M(s,q)$ and let $X=\{\infty\}\cup {\mathop{\mathrm{GF}}}(q)$ be the projective line. Let $q=q_0^r=p^f$ for some odd primes $r$ and $p$, and $f$ an even integer, and let $H={\langle}{\mathop{\mathrm{PSL}}}(2,q_0), \phi^st_{\mu,0,0,1}{\rangle}$ where $\mu$ is a primitive element of ${\mathop{\mathrm{GF}}}(q_0)$. Assume $\gcd(\frac{q-1}{q_0-1},p^{2s}-1)\neq 1$, so that by Lemma \[lem:subfieldedge\], there exist $w_1,w_2\notin {\mathop{\mathrm{GF}}}(q_0)$ such that $w_1+w_2,(w_2-w_1)^{p^{2s}-1} \in{\mathop{\mathrm{GF}}}(q_0)$. Let $e=\{\{\infty,w_1\},\{\infty,w_2\}\}$. By Lemma \[lem:Gedge\], $G_e={\langle}t_{-1,w_1+w_2,0,1},(\phi^{2s})^g{\rangle}$, where $g=t_{w_2-w_1,w_1,0,1}$, and by Lemma \[lem:subfieldedge\], $G_e\leqslant H$. Thus letting $P=e^H$ and ${\mathcal{P}}=P^G$, $(J(q+1,2),{\mathcal{P}})$ is a $G$-primitive decomposition by Lemma \[lem:general\].
We claim that the divisors of ${\mathcal{P}}$ are either matchings or unions of cycles. Since ${\mathop{\mathrm{PSL}}}(2,q_0)_\infty{\vartriangleleft}H_\infty={\langle}{\mathop{\mathrm{PSL}}}(2,q_0)_\infty,\phi^st_{\mu,0,0,1}{\rangle}$, $H_\infty$ (of order $\frac{q_0(q_0-1)}{2}\frac{f}{s}$) acts on the set of ${\mathop{\mathrm{PSL}}}(2,q_0)_\infty$-orbits. Now $t_{-1,w_1+w_2,0,1}\in {\mathop{\mathrm{PSL}}}(2,q_0)_{\infty}$ interchanges $w_1$ and $w_2$, and hence $w_1,w_2$ lie in the same ${\mathop{\mathrm{PSL}}}(2,q_0)_{\infty}$-orbit, $\theta$ say. By Lemma \[lem:subfieldorbits\], ${\mathop{\mathrm{PGL}}}(2,q_0)$ acts semiregularly on ${\mathop{\mathrm{GF}}}(q)\backslash{\mathop{\mathrm{GF}}}(q_0)$ and hence $|\theta|=|{\mathop{\mathrm{PSL}}}(2,q)_{\infty}|=\frac{q_0(q_0-1)}{2}$. Note that $H_{\{\infty,w_1\}}=H_{\infty,w_1}$ and $H_{\{\infty,w_2\}}=H_{\infty,w_2}$. Also $H_{\{\infty,w_1\},\{\infty,w_2\}}={\langle}(\phi^{2s})^g{\rangle}$ has order $\frac{f}{2s}$. Notice that $(\phi^st_{\mu,0,0,1})^2=\phi^{2s}t_{\mu^{p^s+1},0,0,1}$. Hence ${\langle}{\mathop{\mathrm{PSL}}}(2,q_0)_\infty,(\phi^{2s})^g{\rangle}$ has index 2 in $H_\infty$ and fixes $\theta$. Therefore $H_\infty$ either fixes $\theta$ or switches it with another ${\mathop{\mathrm{PSL}}}(2,q_0)_\infty$-orbit $\theta'$. In the first case, $H_{\{\infty,w_1\}}$ has order $\frac{f}{s}$, while $H_{\{\infty,w_1\},\{\infty,w_2\}}$ has order $\frac{f}{2s}$, hence the divisor has valency 2 and is a union of cycles. In the second case, $H_{\{\infty,w_1\}}$ and $H_{\{\infty,w_1\},\{\infty,w_2\}}$ both have order $\frac{f}{2s}$, and so the divisor is a matching $\frac{q_0(q_o^2-1)}{2}K_2$.
[We have not determined the length of the cycles occuring in the first case of Construction \[con:Msqsubfield\]. This case happens if and only if there exists $w\in {\mathop{\mathrm{GF}}}(q)\setminus{\mathop{\mathrm{GF}}}(q_0)$ such that $w^{\phi^st_{\mu,0,0,1}}=w^{p^s}\mu\in \{a^2 w+b|a,b\in{\mathop{\mathrm{GF}}}(q_0)\}$. We have not been able to find any instances where this condition holds. ]{}
Let $(J(q+1,2),\mathcal{P})$ be a $G$-primitive decomposition with $G=M(s,q)$ and for $P\in{\mathcal{P}}$ we have that $G_P=N_G({\mathop{\mathrm{PSL}}}(2,q_0))$ where $q=q_0^r$ for some odd prime $r$. Then ${\mathcal{P}}$ is obtained by Construction $\ref{con:design}$ or $\ref{con:Msqsubfield}$.
Let $q=p^f$ with $p$ a prime and $f$ an even integer. As seen in the discussion before Lemma \[lem:subfieldedge\], $H:=G_P={\langle}{\mathop{\mathrm{PSL}}}(2,q_0),\phi^st_{\mu,0,0,1}{\rangle}$ where $\mu$ is a primitive element of ${\mathop{\mathrm{GF}}}(q_0)$. Let $X=\{\infty\}\cup{\mathop{\mathrm{GF}}}(q)$. Then one orbit of $H$ on $X$ is $\{\infty\}\cup{\mathop{\mathrm{GF}}}(q_0)$. Since $H$ is maximal in $G$, $H$ is exactly the stabiliser in $G$ of $\{\infty\}\cup{\mathop{\mathrm{GF}}}(q_0)$.
Suppose that $H$ contains $G_e$ for some edge $e=\{\{v,w_1\},\{v,w_2\}\}$.
Then by Lemma \[lem:Gedge\], $H$ contains an element of ${\mathop{\mathrm{PSL}}}(2,q)$, and hence of ${\mathop{\mathrm{PSL}}}(2,q_0)$, which fixes $v$ and interchanges $w_1$ and $w_2$. Since, by Lemma \[lem:subfieldorbits\], ${\mathop{\mathrm{PSL}}}(2,q_0)$ acts semiregularly on ${\mathop{\mathrm{GF}}}(q)\setminus{\mathop{\mathrm{GF}}}(q_0)$, it follows that $v\in\{\infty\}\cup{\mathop{\mathrm{GF}}}(q_0)$. Without loss of generality we may suppose that $v=\infty$. By Lemma \[lem:subfieldedge\], this means that both $w_1+w_2$ and $(w_2-w_1)^{p^{2s}-1}$ lie in ${\mathop{\mathrm{GF}}}(q_0)$. This is of course satisfied if $w_1,w_2\in{\mathop{\mathrm{GF}}}(q_0)$, and then we get Construction \[con:design\] using $\mathcal{B}=(\{\infty\}\cup{\mathop{\mathrm{GF}}}(q_0))^{{\mathop{\mathrm{P}\Gamma\mathrm{L}}}(2,q)}$, as $G$ is transitive on $\mathcal{B}$. Now assume $w_1,w_2\notin {\mathop{\mathrm{GF}}}(q_0)$. Then by Lemma \[lem:subfieldedge\], $\gcd(\frac{q-1}{q_0-1},p^{2s}-1)\neq 1$. Moreover, $P=e^H$ and ${\mathcal{P}}=P^G$ are as obtained in Construction \[con:Msqsubfield\].
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[1]{}
J. Bamberg, G. Pearce and C. E. Praeger, Transitive decompositions of graph products: rank 3 product action type, J. Group Theory, to appear.
J. Bosák, Decompositions of Graphs, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp. 24 3/4 (1997), 235–265. Also see the [Magma]{} home page at http://www.maths.usyd.edu.au:8000/u/magma/.
A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-regular graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) \[Results in Mathematics and Related Areas (3)\], 18. Springer-Verlag, Berlin, 1989.
F. Buekenhout, P. Cara, K. Vanmeerbeek, Geometries of the group ${\mathop{\mathrm{PSL}}}(2,11)$. Geom. Dedicata 83 (2000), 169–206.
P. J. Cameron, Permutation Groups, London Math. Soc, Student Texts 45, 1999.
P. J. Cameron and G. Korchmáros, One-factorizations of complete graphs with a doubly transitive automorphism group. Bull. London Math. Soc. 25 (1993), 1–6.
P. J. Cameron, G. R. Omidi and B. Tayeh-Rezaie, $3$-Designs from ${\mathop{\mathrm{PGL}}}(2,q)$, Electron. J. Combin. 13 (2006), \#R50.
C. J. Colbourn and J. H. Dinitz (Editors), The CRC Handbook of Combinatorial Designs, CRC Press Series on Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 1996.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985.
M. C. Cuaresma, Homogeneous Factorisations of Johnson Graphs, PhD Thesis, University of the Philippines, 2004.
M. C. Cuaresma, M. Giudici and C. E. Praeger, Homogeneous factorisations of Johnson graphs, submitted.
A. Devillers, M. Giudici, C.H. Li and C. E. Praeger, A remarkable Mathieu graph tower, submitted.
U. Dempwolff, On the second chomology of ${\mathop{\mathrm{GL}}}(n,2)$, J. Austral. Math. Soc. 16 (1973), 207–209.
L. E. Dickson, Linear groups: [W]{}ith an exposition of the [G]{}alois field theory, Dover Publications Inc., New York, 1958.
J. D. Dixon and B. Mortimer, Permutation Groups, Graduate Texts in Mathematics 163, Springer, New York, 1996.
X. G. Fang and C. E. Praeger, Finite two-arc-transitive graphs admitting a Suzuki simple group, Comm. Alg. 27 (1999), 3727–3754.
A. Gardiner, C. E. Praeger and S. Zhou, Cross ratio graphs. J. London Math. Soc. (2) 64 (2001), 257–272.
Michael Giudici, Maximal subgroups of almost simple groups with socle ${\mathop{\mathrm{PSL}}}(2,q)$, preprint: math.GR/0703685.
M. Giudici, C.H. Li, P. Potočnik and C. E. Praeger, Homogeneous factorisations of graphs and digraphs. European J. Combin. 27 (2006), no. 1, 11–37.
M. Giudici, C. H. Li and Cheryl E. Praeger, Symmetrical covers, decompositions and factorisations of graphs, in preparation.
F. Harary, R. W. Robinson and N. C. Wormald, Isomorphic factorisations. I. Complete graphs, Trans. Amer. Math. Soc. 242 (1978), 243–260.
F. Harary and R. W. Robinson, Isomorphic factorisations X: unsolved problems, J. Graph Theory 9 (1985), 67–86.
K. Heinrich, Graph decompositions and designs, in: The CRC Handbook of Combinatorial Designs, Charles J. Colbourn and Jeffrey H. Dinitz, (Editors), CRC Press Series on Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 1996, 361–366.
B. Huppert, Endliche [G]{}ruppen. [I]{}, in *Die Grundlehren der Mathematischen Wissenschaften, Band 134* (Springer-Verlag, Berlin, 1967).
A. A. Ivanov, Geometry of sporadic groups I: Petersen and tilde geometries, Cambridge University Press, Cambridge, 1999.
L. A. Kalužnin and M. H. Klin, On some maximal subgroups of symmetric and alternating groups, Mat. Sbornik 87 (1972), 91–121 (in Russian).
W. M. Kantor, $k$-homogeneous groups, Math. Z. 124 (1972), 261–265.
C. H. Li and C. E. Praeger, On partitioning the orbitals of a transitive permutation group, Trans. Amer. Math. Soc. 355 (2003), 637–653.
M. W. Liebeck, C. E. Praeger and J. Saxl, A classification of the maximal subgroups of the finite alternating and symmetric groups. J. Algebra 111 (1987), no. 2, 365–383.
D. Livingstone and A. Wagner, Transitivity of finite permutation groups on unordered sets, Math. Z. 90 (1965), 393–403.
D. K. Maslen, M. E. Orrison and D. N. Rockmore, Computing isotypic projections with the Lanczos iteration, SIAM J. Matrix Anal. Appl. 25 (2004), 784–803.
R. C. Read and R. J. Wilson, An atlas of graphs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.
H. Wielandt, Finite permutation groups. Academic Press, New York-London, 1964.
T. Sibley, On classifying finite edge colored graphs with two transitive automorphism groups. J. Combin. Theory Ser. B 90 (2004), 121–138.
G. M. Seitz, Flag-transitive subgroups of Chevalley groups. Ann. of Math. (2) 97 (1973), 27–56.
[^1]: The first author is a Postdoctoral Researcher of the Fonds National de la Recherche Scientifique (Belgium). This research was supported under the Australian Research Council’s Discovery Projects funding scheme (project number DP0449429). The second author is a recipient of an ARC Postdoctoral Fellowship while the third author holds an ARC Queen Elizabeth II Fellowship.
|
---
abstract: 'This paper reports results of the third-year campaign of monitoring super-Eddington accreting massive black holes (SEAMBHs) in active galactic nuclei (AGNs) between $2014-2015$. Ten new targets were selected from quasar sample of Sloan Digital Sky Survey (SDSS), which are generally more luminous than the SEAMBH candidates in last two years. H$\beta$ lags ($\tauhb$) in five of the 10 quasars have been successfully measured in this monitoring season. We find that the lags are generally shorter, by large factors, than those of objects with same optical luminosity, in light of the well-known $\rblrl$ relation. The five quasars have dimensionless accretion rates of $\mathdotM=10-10^3$. Combining measurements of the previous SEAMBHs, we find that the reduction of H$\beta$ lags tightly depends on accretion rates, $\tauhb/\taurl\propto\mathdotM^{-0.42}$, where $\taurl$ is the H$\beta$ lag from the normal $\rblrl$ relation. Fitting 63 mapped AGNs, we present a new scaling relation for the broad-line region: $\rhb=\alpha_1\ell_{44}^{\beta_1}\,\min\left[1,\left(\mathdotM/\mathdotM_c\right)^{-\gamma_1}\right]$, where $\ell_{44}=L_{5100}/10^{44}\,\ergs$ is 5100 Å continuum luminosity, and coefficients of $\alpha_1=(29.6_{-2.8}^{+2.7})$lt-d, $\beta_1=0.56_{-0.03}^{+0.03}$, $\gamma_1=0.52_{-0.16}^{+0.33}$ and $\mathdotM_c=11.19_{-6.22}^{+2.29}$. This relation is applicable to AGNs over a wide range of accretion rates, from $10^{-3}$ to $10^3$. Implications of this new relation are briefly discussed.'
author:
- |
Pu Du, Kai-Xing Lu, Zhi-Xiang Zhang, Ying-Ke Huang, Kai Wang, Chen Hu, Jie Qiu, Yan-Rong Li,\
Xu-Liang Fan, Xiang-Er Fang, Jin-Ming Bai, Wei-Hao Bian, Ye-Fei Yuan,\
Luis C. Ho and Jian-Min Wang\
(SEAMBH collaboration)
title: |
Supermassive Black Holes with High Accretion Rates in Active Galactic Nuclei. V.\
A New Size-Luminosity Scaling Relation for the Broad-Line Region
---
Introduction
============
This is the fifth paper of the series reporting the ongoing large campaign of monitoring Super-Eddington Accreting Massive Black Holes (SEAMBHs) in active galaxies and quasars starting from October 2012. One of the major goals of the campaign is to search for massive black holes with extreme accretion rates through reverberation mapping (RM) of broad emission lines and continuum. Results from the campaigns in 2012–2013 and 2013–2014 have been reported by Du et al. (2014, 2015, hereafter Papers I and IV), Wang et al. (2014, Paper II) and Hu et al. (2015, Paper III). This paper carries out the results of SEAMBH2014 sample, which was monitored from September 2014 to June 2015. With the three monitoring years of observations, we build up a new scaling relation of the broad-line region (BLR) in this paper.
[lllllccrr]{}\[!ht\]\
Mrk 335 & 00 06 19.5 & $+$20 12 10 & 0.0258 & Oct., 2012 $-$ Feb., 2013 & 91 & & $ 80^{\pp}.7$ & $ 174.5^{\circ}$\
Mrk 1044 & 02 30 05.5 & $-$08 59 53 & 0.0165 & Oct., 2012 $-$ Feb., 2013 & 77 & & $207^{\pp}.0$ & $-143.0^{\circ}$\
IRAS 04416+1215 & 04 44 28.8 & $+$12 21 12 & 0.0889 & Oct., 2012 $-$ Mar., 2013 & 92 & & $137^{\pp}.9$ & $ -55.0^{\circ}$\
Mrk 382 & 07 55 25.3 & $+$39 11 10 & 0.0337 & Oct., 2012 $-$ May., 2013 &123 & & $198^{\pp}.4$ & $ -24.6^{\circ}$\
Mrk 142 & 10 25 31.3 & $+$51 40 35 & 0.0449 & Nov., 2012 $-$ Apr., 2013 &119 & & $113^{\pp}.1$ & $ 155.2^{\circ}$\
MCG $+06-26-012$ & 11 39 13.9 & $+$33 55 51 & 0.0328 & Jan., 2013 $-$ Jun., 2013 & 34 & & $204^{\pp}.3$ & $ 46.1^{\circ}$\
IRAS F12397+3333 & 12 42 10.6 & $+$33 17 03 & 0.0435 & Jan., 2013 $-$ May., 2013 & 51 & & $189^{\pp}.0$ & $ 130.0^{\circ}$\
Mrk 486 & 15 36 38.3 & $+$54 33 33 & 0.0389 & Mar., 2013 $-$ Jul., 2013 & 45 & & $193^{\pp}.8$ & $-167.0^{\circ}$\
Mrk 493 & 15 59 09.6 & $+$35 01 47 & 0.0313 & Apr., 2013 $-$ Jun., 2013 & 27 & & $155^{\pp}.3$ & $ 98.5^{\circ}$\
\
SDSS J075101.42+291419.1 & 07 51 01.4 & $+$29 14 19 & 0.1208 & Nov., 2013 $-$ May., 2014 & 38 & & $133^{\pp}.3$ & $ -41.3^{\circ}$\
SDSS J080101.41+184840.7 & 08 01 01.4 & $+$18 48 40 & 0.1396 & Nov., 2013 $-$ Apr., 2014 & 34 & & $118^{\pp}.8$ & $ -98.2^{\circ}$\
SDSS J080131.58+354436.4 & 08 01 31.6 & $+$35 44 36 & 0.1786 & Nov., 2013 $-$ Apr., 2014 & 31 & & $100^{\pp}.0$ & $ 145.2^{\circ}$\
SDSS J081441.91+212918.5 & 08 14 41.9 & $+$21 29 19 & 0.1628 & Nov., 2013 $-$ May., 2014 & 34 & & $ 79^{\pp}.0$ & $ 73.9^{\circ}$\
SDSS J081456.10+532533.5 & 08 14 56.1 & $+$53 25 34 & 0.1197 & Nov., 2013 $-$ Apr., 2014 & 27 & & $164^{\pp}.5$ & $-172.9^{\circ}$\
SDSS J093922.89+370943.9 & 09 39 22.9 & $+$37 09 44 & 0.1859 & Nov., 2013 $-$ Jun., 2014 & 26 & & $175^{\pp}.1$ & $-139.0^{\circ}$\
\
SDSS J075949.54+320023.8 & 07 59 49.5 & $+$32 00 24 & 0.1880 & Sep., 2014 $-$ May., 2015 & 27 & & $109^{\pp}.2$ & $ -48.3^{\circ}$\
SDSS J080131.58+354436.4 & 08 01 31.6 & $+$35 44 36 & 0.1786 & Oct., 2014 $-$ May., 2015 & 19 & & $139^{\pp}.2$ & $ -85.3^{\circ}$\
SDSS J084533.28+474934.5 & 08 45 33.3 & $+$47 49 35 & 0.3018 & Sep., 2014 $-$ Apr., 2015 & 18 & & $205^{\pp}.5$ & $-126.4^{\circ}$\
SDSS J085946.35+274534.8 & 08 59 46.4 & $+$27 45 35 & 0.2438 & Sep., 2014 $-$ Jun., 2015 & 26 & & $169^{\pp}.8$ & $ -89.1^{\circ}$\
SDSS J102339.64+523349.6 & 10 23 39.6 & $+$52 33 50 & 0.1364 & Oct., 2014 $-$ Jun., 2015 & 26 & & $123^{\pp}.2$ & $ 108.1^{\circ}$
Reverberation mapping (RM) technique, measuring the delayed echoes of broad lines to the varying ionizing continuum (Bahcall et al. 1972; Blandford & McKee 1982; Peterson 1993), is a powerful tool to probe the kinematics and geometry of the BLRs in the time domain. Countless clouds, which contribute to the smooth profiles of the broad emission lines (e.g., Arav et al. 1997), are distributed in the vicinity of supermassive black hole (SMBH), composing the BLR. As an observational consequence of photonionization powered by the accretion disk under the deep gravitational potential of the SMBH, the profiles of the lines are broadened, and line emission from the clouds reverberate in response to the varying ionizing continuum. The reverberation is delayed because of light travel difference between H$\beta$ and ionizing photons and is thus expected to deliver information on the kinematics and structure of the BLR. The unambiguous reverberation of the lines, detected by monitoring campaigns from ultraviolet to optical bands since the late 1980s, supports this picture of the central engine of AGNs (e.g., Clavel et al. 1991; Peterson et al. 1991, 1993; Maoz et al. 1991; Wanders et al. 1993; Dietrich et al. 1993, 1998, 2012; Kaspi et al. 2000; Denney et al. 2006, 2010; Bentz et al. 2009, 2014; Grier et al. 2012; Papers I-IV; Barth et al. 2013, 2015; Shen et al. 2015a,b). The $\rblrl$ relation was first discussed by Koratkar & Gaskell (1991) and Peterson (1993). Robust RM results for 41 AGNs in the last four decades lead to a simple, highly significant correlation of the form $$\rblr\approx \alpha_0\,\ell_{44}^{\beta_0},$$ where $\ell_{44}=L_{5100}/10^{44}\ergs$ is the 5100 Å luminosity in units of $10^{44}\ergs$ (corrected for host galaxy contamination) and $\rblr=c\tauhb$ is the emissivity-weighted radius of the BLR (Kaspi et al. 2000; Bentz et al. 2013). We refer to this type of correlation as the normal $\rblrl$ relationship. The constants $\alpha_0$ and $\beta_0$ differ slightly from one study to the next, depending on the number of sources and their exact luminosity range (e.g., Kilerci Eser et al. 2015). For sub-Eddington accreting AGNs, $\alpha_0=35.5$ ltd and $\beta_0=0.53$, but for SEAMBHs they are different (see Paper IV).
As reported in Paper IV, some objects from the SEAMBH2012 and SEAMBH2013 samples have much shorter H$\beta$ lags compared with objects with similar luminosity, and the $\rblrl$ relation has a large scatter if they are included. In particular, the reduction of the lags increases with the dimensionless accretion rate, defined as $\mathdotM=\dot{M}_{\bullet}/L_{\rm Edd}c^{-2}$, where $\dot{M}_{\bullet}$ is the accretion rate, $L_{\rm Edd}$ is the Eddington luminosity and $c$ is the speed of light. Furthermore, it has been found, so far in the present campaigns, that SEAMBHs have a range of accretion rates from a few to $\sim 10^3$. This kind of shortened H$\beta$ lags was discovered in the current SEAMBH project (a comparison with previous campaigns is given in Section 6.5). Such high accretion rates are characteristic of the regime of slim accretion disks (Abramowicz et al. 1988; Szuszkiewicz et al. 1996; Wang & Zhou 1999; Wang et al. 1999; Mineshige et al. 2000; Wang & Netzer 2003; Sadowski 2009). These interesting properties needed to be confirmed with observations. We aim to explore whether we can define a new scaling relation, $\rhb=\rhb(L_{5100},\mathdotM)$, which links the size of the BLR to both the AGN luminosity [*and*]{} accretion rate.
We report new results from SEAMBH2014. We describe target selection, observation details and data reduction in §2. H$\beta$ lags, BH mass and accretion rates are provided in §3. Properties of H$\beta$ lags are discussed in §4, and a new scaling relation of H$\beta$ lags is established in §5. Section 6 introduces the fundamental plane, which is used to estimate accretion rates from single-epoch spectra, for application of the new size-luminosity scaling relation of the BLR. Brief discussions of the shortened lags are presented in §7. We draw conclusions in §8. Throughout this work we assume a standard $\Lambda$CDM cosmology with $H_0=67~{\rm km~s^{-1}~Mpc^{-1}}$, $\Omega_{\Lambda}=0.68$ and $\Omega_{\rm M}=0.32$ (Ade et al. 2014).
[rlcrllcrlcrll]{} 29.374 & $17.373\pm 0.007$ & & 76.365 & $ 2.773\pm 0.029$ & $ 2.469\pm 0.038$ & & 60.324 & $17.757\pm 0.010$ & & 112.294 & $ 2.096\pm 0.016$ & $ 0.853\pm 0.027$\
30.360 & $17.375\pm 0.008$ & & 80.319 & $ 2.864\pm 0.017$ & $ 2.461\pm 0.031$ & & 62.314 & $17.720\pm 0.010$ & & 116.394 & $ 2.076\pm 0.016$ & $ 0.893\pm 0.029$\
32.352 & $17.406\pm 0.009$ & & 83.314 & $ 2.724\pm 0.038$ & $ 2.366\pm 0.041$ & & 63.299 & $17.734\pm 0.010$ & & 119.336 & $ 2.106\pm 0.021$ & $ 0.831\pm 0.031$\
33.339 & $17.416\pm 0.009$ & & 86.422 & $ 2.795\pm 0.011$ & $ 2.535\pm 0.023$ & & 68.397 & $17.721\pm 0.013$ & & 135.324 & $ 2.086\pm 0.026$ & $ 0.807\pm 0.033$\
34.331 & $17.408\pm 0.009$ & & 89.378 & $ 2.888\pm 0.012$ & $ 2.402\pm 0.032$ & & 77.318 & $17.746\pm 0.009$ & & 139.319 & $ 2.074\pm 0.028$ & $ 0.862\pm 0.035$\
Observations and data reduction
===============================
Target Selection
----------------
We followed the procedures for selecting SEAMBH candidates described in Paper IV. We used the fitting procedures to measure H$\beta$ profile and 5100 Å luminosity of SDSS quasar spectra described by Hu et al. (2008a,b). Following the standard assumption that the BLR gas is virialized, we estimate the BH mass as $$\bhm=\fblr \frac{\rblr V_{\rm FWHM}^2}{G}=1.95\times 10^6~\fblr V_{3}^2\tau_{10}~\sunm,$$ where $\rblr=c\tauhb$, $\tauhb$ is the H$\beta$ lag measured in the rest frame, $\tau_{10}=\tauhb/10$days, $G$ is the gravitational constant, and $V_{3}=V_{\rm FWHM}/10^3\kms$ is the full-width-half-maximum (FWHM) of the H$\beta$ line profile in units of $10^3\kms$. We take the virial factor $\fblr=1$ in our series of papers (see some discussions in Paper IV).
In order to select AGNs with high accretion rates, we employed the formulation of accretion rates derived from the standard disk model of Shakura & Sunyaev (1973). In the standard model it is assumed that the disk gas is rotating with Keplerian angular momentum, and thermal equilibrium is localized between viscous dissipation and blackbody cooling. Observationally, this model is supported from fits of the so-called big blue bump in quasars (Czerny & Elvis 1987; Wandel & Petrosian 1988; Sun & Malkan 1989; Laor & Netzer 1989; Collin et al. 2002; Brocksopp et al. 2006; Kishimoto et al. 2008; Davis & Laor 2011; Capellupo et al. 2015). The dimensionless accretion rate is given by $$\mathdotM=20.1\left(\frac{\ell_{44}}{\cos i}\right)^{3/2}m_7^{-2},
\label{eq:SS_2}$$ where $m_7=\bhm/10^7\sunm$ (see Papers II and IV) and $i$ is the inclination angle to the line of sight of the disk. We take $\cos i=0.75$, which represents a mean disk inclination for a type 1 AGNs with a torus covering factor of about 0.5 (it is assumed that the torus axis is co-aligned with the disk axis). Previous studies estimate $i\approx0-45^{\circ}$ \[e.g., Fischer et al. (2014) find a inclination range of $i\approx10^{\circ}-45^{\circ}$, whereas Pancoast et al. (2014) quote $i\approx 5^{\circ}-45^{\circ}$; see also supplementary materials in Shen & Ho (2014)\], which results in $\Delta \log \mathdotM=1.5\Delta \log \cos i \lesssim 0.15$ from Equation (3). This uncertainty is significantly smaller than the average uncertainty on $\mathdotM$ ($\sim 0.3-0.5$ dex) in the present paper, and is thus ignored. Equation (3) applies to AGNs that have accretion rates $10^{-2}\lesssim\mathdotM\lesssim 3\times 10^3$, namely excluding the regimes of advection-dominated accretion flows (ADAF; Narayan & Yi 1994) and of flows with hyperaccretion rates ($\mathdotM\ge 3\times 10^3$; see Appendix A for the validity of Equation 3 for SEAMBHs).
Using the normal $\rblrl$ relation (Bentz et al. 2013), we fitted all the quasar spectra in SDSS Data Release 7 by the procedures in Hu et al. (2008a, b) and applied Equations (2) and (3) to select high$-\mathdotM$ targets. We ranked quasars in terms of $\mathdotM$ and chose ones as candidates with the highest $\mathdotM$. We found that the high$-\mathdotM$ quasars are characterized by 1) strong optical lines; 2) relatively narrow H$\beta$ lines ($\lesssim 2000\kms$); 3) weak lines; and 4) steep 2–10 keV spectra (Wang et al. 2004). These properties are similar to those of NLS1s (Osterbrock & Pogge 1987; Boroson & Green 1992), but most of the candidates have more extreme accretion rates (a detailed comparison of SEAMBH properties with normal quasars will be carried out in a separate paper). Considering that the lags of all targets should be measured within one observing season, and taking into consideration the limitations of the weather of the Lijiang Station of Yunnan Observatory (periods between June and September are raining seasons there), we only chose objects with maximum estimated lags of about 100 days or so (the monitoring periods should be at least a few times the presumed lags). Also, to ensure adequate signal-to-noise ratio (S/N) for measurements of light curves, we restricted the targets to a redshift range of $z=0.1-0.3$ and magnitudes $r^{\prime}\le 18.0$. The fraction of radio-loud objects with $\mathdotM>3$ is not high. We discarded radio-loud objects[^1] based on available FIRST observations, in order to avoid H$\beta$ reverberations potentially affected by nonthermal emission from relativistic jets, or optical continuum emission strongly contaminated by jets. We chose about 20 targets for photometry monitoring, which served as a preselection to trigger follow-up spectroscopic monitoring. The photometric monitoring yielded 10 targets with significant variations ($\gtrsim 0.1$ magnitudes), and time lags were successfully measured for 5 objects (Table 1). For an overview of our entire ongoing campaign, Table 1 also lists samples from SEAMBH2012 and SEAMBH2013.
To summarize: we have selected about 30 targets for spectroscopic monitoring during the last three years (2012–2014). The successful rate of the monitoring project is about 2/3. Our failure to detect a lag for the remaining 1/3 of the sample are either due to low-amplitude variability or bad weather that leads to poor monitoring cadence. In particular, the SEAMBH2014 observations were seriously affected by the El Niño phenomenon.
[rlcrllcrlcrll]{}\[!h\] 29.417 & $17.803\pm 0.008$ & & 75.402 & $ 1.728\pm 0.026$ & $ 1.145\pm 0.047$ & & 30.428 & $17.401\pm 0.008$ & & 90.382 & $ 2.651\pm 0.015$ & $ 1.693\pm 0.024$\
30.398 & $17.797\pm 0.008$ & & 80.395 & $ 1.721\pm 0.019$ & $ 1.085\pm 0.033$ & & 33.403 & $17.399\pm 0.007$ & & 97.433 & $ 2.593\pm 0.033$ & $ 1.728\pm 0.038$\
33.376 & $17.777\pm 0.009$ & & 84.365 & $ 1.680\pm 0.020$ & $ 1.046\pm 0.035$ & & 36.410 & $17.406\pm 0.005$ & & 104.293 & $ 2.392\pm 0.024$ & $ 1.835\pm 0.034$\
36.368 & $17.780\pm 0.009$ & & 91.326 & $ 1.673\pm 0.016$ & $ 1.086\pm 0.028$ & & 48.357 & $17.422\pm 0.008$ & & 111.261 & $ 2.583\pm 0.022$ & $ 1.595\pm 0.037$\
38.411 & $17.816\pm 0.017$ & & 104.409 & $ 1.681\pm 0.015$ & $ 1.069\pm 0.029$ & & 51.353 & $17.440\pm 0.006$ & & 116.448 & $ 2.512\pm 0.025$ & $ 1.697\pm 0.027$\
Photometry and Spectroscopy
---------------------------
The SEAMBH project uses the Lijiang 2.4m telescope, which has an alt-azimuth Ritchey-Chrétien mount with a field de-rotator that enables two objects to be positioned along the same long slit. It is located in Lijiang and is operated by Yunnan Observatories. We adopted the same observational procedures described in detail in Paper I, which also introduces the telescope and spectrograph. We employed the Yunnan Faint Object Spectrograph and Camera (YFOSC), which has a back-illuminated 2048$\times$4608 pixel CCD covering a field of $10^{\prime}\times 10^{\prime}$. During the spectroscopic observation, we put the target and a nearby comparison star into the slit simultaneously, which can provide high-precision flux calibration. As in SEAMBH2013 (Paper IV), we adopted a $5^{\pp}$-wide slit to minimize the influence of atmospheric differential refraction, and used Grism 3 with a spectral resolution of 2.9 Å/pixel and wavelength coverage of 3800–9000 Å. To check the accuracy of spectroscopic calibration, we performed differential photometry of the targets using some other stars in the same field. We used an SDSS $r^{\prime}$-band filter for photometry to avoid the potential contamination by emission lines such as H$\beta$ and H$\alpha$. Photometric and spectroscopic exposure times are typically 10 and 60 min, respectively.
The reduction of the photometry data was done in a standard way using [IRAF]{} routines. Photometric light curves were produced by comparing the instrumental magnitudes to those of standard stars in the field (see, e.g., Netzer et al. 1996, for details). The radius for the aperture photometry is typically $\sim4^{\pp}$ (seeing $\sim1.5^{\pp}-2^{\pp}$), and background is determined from an annulus with radius $8^{\pp}.5$ to $17^{\pp}$. The uncertainties on the photometric measurements include the fluctuations due to photon statistics and the scatter in the measurement of the stars used.
The spectroscopic data were also reduced with IRAF. The extraction width is fixed to $8^{\pp}.5$, and the sky regions are set to $7^{\pp}.4-14^{\pp}.1$ on both sides of the extracted region. The average S/N of the 5100 Å continuum of individual spectra are from $\sim$16 to $\sim$22, except for J085946, which only has S/N $\approx $12. The flux of spectroscopic data was calibrated by simultaneously observing a nearby comparsion star along the slit (see Paper I). The fiducial spectra of the comparison stars are generated using observations from several nights with the best weather conditions. The absolute fluxes of the fiducial spectra are calibrated using additional spectrophotometric standard stars observed in those nights. Then, the in-slit comparison stars are used as standards to calibrate the spectra of targets observed in each night. The sensitivity as a function of wavelength is produced by comparing the observed spectrum of the comparison star to its fiducial spectrum. Finally, the sensitivity function is applied to calibrate the observed AGN spectrum[^2]. The procedures adopted here resemble the method used by Maoz et al. (1990) and Kaspi et al. (2000). In order to illustrate the invariance of the comparison stars, we show the light curves from the differential photometry of the comparison stars in Appendix B. It is clear that their fluxes are very stable and the variations are less than $\sim$1%.
The calibration method of van Groningen & Wanders (1992), based on the emission line and popularily used in many RM campaigns (e.g., Peterson et al. 1998; Bentz et al. 2009; Denney et al. 2010; Grier et al. 2012; Barth et al. 2015), is not suitable for SEAMBHs. $\lambda5007$ tends to be weak in SEAMBHs (especially for the objects in SEAMBH2013-2014), and, even worse, is blended with strong around 5016 Å. Applying the calibration method to SEAMBHs results in large statistical (caused by the weakness of ) and systematic (caused by the variability of ; see Paper III) uncertainties. The method based on in-slit comparison stars, used in our campaign, does not rely on and provides accurate flux calibration for the spectra of SEAMBHs. For comparison, in Paper I we measured the fluxes in the calibrated spectra of three objects in SEAMBH2012 with relatively strong ; the variation of their flux is on the order of $\sim$3%. This clearly demonstrates the robustness of our flux calibration method based on in-slit comparison stars.
The procedures to measure the 5100 Å and H$\beta$ flux are nearly the same as those given in Paper I. The continuum beneath H$\beta$ line is determined by interpolation of two nearby bands (4740–4790 Å and 5075 -5125 Å) in the rest frame. These two bands have minimal contamination from emission lines. The flux of H$\beta$ is measured by integrating the band between 4810 and 4910 Å after subtraction of the continuum; the H$\beta$ band is chosen to avoid the influence from lines. The 5100 Å flux is taken to be the median over the region 5075-5125 Å. Detailed information of the observations is provided in Table 1. All the photometry and continuum and H$\beta$ light curves for the five objects with successfully detected lags are listed in Tables $2-4$ and shown in Figure 1. We also calculated the mean and RMS (root mean square) spectra and present them in Appendix C.
{width="48.00000%"} {width="48.00000%"}\
{width="48.00000%"} {width="48.00000%"}
Host Galaxies
-------------
Like the SEAMBH2013 sample, we have no observations that can clearly separate the host galaxies of the AGNs in SEAMBH2014. Shen et al. (2011) propose the following empirical relation to estimate the fractional contribution of the host galaxy to the optical continuum emission: $L_{5100}^{\rm host}/L_{5100}^{\rm AGN}=0.8052-1.5502x+0.912x^2-0.1577x^3$, for $x<1.053$, where $x=\log \left(L_{5100}^{\rm tot}/10^{44}{\rm erg~s^{-1}}\right)$ and $L_{5100}^{\rm tot}$ is the total emission from the AGN and its host at 5100 Å. For $x>1.053$, $L_{5100}^{\rm host}\ll L_{5100}^{\rm AGN}$, and the host contamination can be neglected. The host fractions at 5100 Å for the objects (J075949, J080131, J084533, J085946 and J102339) are (27.5%, 37.1%, 14.2%, 19.0% and 31.8%). The values of $L_{5100}$ listed in Table 5 are the host-subtracted luminosities. We note that this empirical relation is based on SDSS spectroscopic observations with a $3^{\prime\prime}$ fiber, whereas we used a $5^{\prime\prime}$-wide slit. It should apply to our observations reasonably well (see Paper IV for additional discussions on this issue). We will revisit this issue in the future using high-resolution images that can more reliably separate the host.
Measurements of H$\beta$ Lags, Black Hole Masses and Accretion Rates
====================================================================
Lags
----
As in Papers I–IV, we used cross-correlation analysis to determine H$\beta$ lags relative to photometric or 5100 Å continuum light curves. We use the centroid lag for H$\beta$. The uncertainties on the lags are determined through the “flux randomization/random subset sampling" method (RS/RSS; Peterson et al. 1998, 2004). The cross-correlation centroid distribution (CCCD, described in Appendix E) and cross-correlation peak distribution (CCPD) generated by the FR/RSS method (Maoz & Netzer 1989; Peterson et al. 1998, 2004; Denney et al. 2006, 2010, and references therein) are shown in Figure 1. We used the following criteria to define a successful detection of H$\beta$ lag: 1) non-zero lag from the CCF peak and 2) a maximum correlation coefficient larger than 0.5. Data for the light curves of the targets are given in Tables 2–4. All the measurements of the SEAMBH2014 sample are provided in Table 5.
The $r^{\prime}$-band light curves are generally consistent with the 5100 Å continuum light curve, but the former usually have small scatter, as shown in Figure 1. We calculated CCFs for the H$\beta$ light curves with both $r^{\prime}$-band photometry and with 5100 Å spectral continuum for all objects. The quality of the H$\beta-r^{\prime}$ CCFs is usually better than the H$\beta-F_{5100}$ CCFs. We show the H$\beta-r^{\prime}$ CCFs for all objects in Figure 1, except for J080131. We use the H$\beta-r^{\prime}$ lags in the following analysis. For J080131, the $r^{\prime}$-band light curve between 200 and 220 days does not match the 5100 Å continuum light curve, even though H$\beta$ does follow 5100 Å continuum tightly. Notes to individual sources are given in Appendix D.
Black Hole Masses and Accretion Rates
-------------------------------------
There are two ways of calculating BH mass, base either on the RMS spectrum (e.g., Peterson et al. 2004; Bentz et al. 2009; Denney et al. 2010; Grier et al. 2012) or on the mean spectrum (e.g., Kaspi et al. 2005; Papers I–IV). Different studies also adopt different measures of the line width, typically either the line dispersion $\sigma_{\rm line}$ (second moment of the line profile) or the FWHM. In this study, we choose to parameterize the line width using FWHM, as measured in the mean spectra. The narrow H$\beta$ component may influence the measurement of FWHM. We adopt the same procedure as in Paper I to remove the narrow H$\beta$. We fix narrow H$\beta$/$\lambda5007$ to 0.1, and measure FWHM from the mean spectra with narrow H$\beta$ subtracted. Then we set H$\beta$/$\lambda5007$ to 0 and 0.2 and repeat the process to obtain lower and upper limits to FWHM. The relatively wide slit employed in our campaign ($5^{\prime\prime}$) significantly broadens the emission lines by $V_{\rm inst} \approx 1200\,\kms$, where $V_{\rm inst}$ is the instrumental broadening that can be estimated from the broadening of selected comparison stars. As in Paper IV, we obtain the intrinsic width of the mean spectra from ${\rm FWHM}=\left({\rm FWHM}_{\rm obs}^2-V_{\rm inst}^2\right)^{1/2}$. The FWHM simply obtained here is accurate enough for BH mass estimation. Our procedure for BH mass estimation is based on FWHM measured from the mean spectrum (see explanation in Papers I and II). As shown recently in Woo et al. (2015), the scatter in the scaling parameter ($f_{\rm BLR}$) derived in this method is very similar to the scatter in the method based on the RMS spectrum. We use Equations (2) and (3) to calculate accretion rates and BH masses for the five sources listed in Table 5. For convenience and completeness, Table 5 also lists H$\beta$ lags, BH masses and accretion rates for the sources from SEAMBH2012 and SEAMBH2013. Our campaign has successfully detected H$\beta$ lags for 18 SEAMBHs since October 2012.
{width="48.00000%"}
[Figure 1 [*continued.*]{}]{}
As described in Paper II, there are some theoretical uncertainties in identifying a critical value of $\mathdotM$ to define a SEAMBH (Laor & Netzer 1989; Beloborodov 1998; Sadowski et al. 2011). Following Paper II, we classified SEAMBHs as those objects with $\eta\mathdotM \geq 0.1$. This is based on the idea that beyond this value, the accretion disk becomes slim and the radiation efficiency is reduced mainly due to photon trapping (Sadowski et al. 2011). Since we currently cannot observe the entire spectral energy distribution, we have no direct way to measure $\er$, and this criterion is used as an approximate tool to identify SEAMBH candidates. To be on the conservative side, we chose the lowest possible efficiency, $\eta=0.038$ (retrograde disk with $a=-1$; see Bardeen et al. 1972). Thus, SEAMBHs are objects with $\mathdotM=2.63$. For simplicity, in this paper we use $\mathdotM_{\rm min}=3$ as the required minimum (Papers II and IV). We refer to AGNs with $\mathdotM\ge 3$ as SEAMBHs and those with $\mathdotM<3$ as sub-Eddington ones. Paper IV clearly shows that the properties of the $\rblrl$ relation for $\mathdotM\ge 3$ and $\mathdotM<3$ are significantly different.
[rlcrll]{}\[!b\] 54.400 & $16.759\pm 0.009$ & & 102.431 & $ 5.455\pm 0.056$ & $ 2.340\pm 0.049$\
56.431 & $16.776\pm 0.011$ & & 105.309 & $ 5.430\pm 0.022$ & $ 2.429\pm 0.050$\
62.333 & $16.762\pm 0.008$ & & 111.450 & $ 5.443\pm 0.034$ & $ 2.325\pm 0.046$\
72.412 & $16.820\pm 0.007$ & & 115.406 & $ 5.374\pm 0.041$ & $ 2.438\pm 0.064$\
76.438 & $16.800\pm 0.021$ & & 119.396 & $ 5.397\pm 0.027$ & $ 2.548\pm 0.044$\
Figure 2 plots distributions of $L_{5100}$, EW(H$\beta$), $\mathdotM$ and $\bhm$ of all the mapped AGNs (41 from Bentz et al. 2013 and the 18 SEAMBHs from our campaign; see Table 7 in Paper IV and Table 5 here). As shown clearly in the diagrams, SEAMBH targets are generally more luminous by a factor of 2–3 compared to previous RM AGNs (Figure 2[*a*]{}). The BH masses of SEAMBHs are generally less smaller by a factor of 10 compared to previous samples, whereas, as a consequence of our selection, the accretion rates of SEAMBHs are higher by 2–3 orders of magnitude (Figures 2[*c*]{} and 2[*d*]{}). However, EW(H$\beta$) of SEAMBHs are not significantly smaller (Figure 2[*d*]{}). On average, the high$-\mathdotM$ sources have lower mean EW(H$\beta$), consistent with the inverse correlation between EW(H$\beta$) and $L_{\rm bol}/L_{\rm Edd}$ (e.g., Netzer et al. 2004).
[lccccccrccc]{}\
Mrk 335 & $ 8.7_{- 1.9}^{+ 1.6} $ & $ 2096\pm170 $ & $1470\pm 50$ & $ 6.87_{-0.14}^{+0.10} $ & $ 1.28_{-0.30}^{+0.37} $ & $ 43.69\pm 0.06 $ & $ 42.03\pm 0.06 $ & $ 110.5\pm 22.3 $\
Mrk 1044 & $ 10.5_{- 2.7}^{+ 3.3} $ & $ 1178\pm 22 $ & $ 766\pm 8$ & $ 6.45_{-0.13}^{+0.12} $ & $ 1.22_{-0.41}^{+0.40} $ & $ 43.10\pm 0.10 $ & $ 41.39\pm 0.09 $ & $ 101.4\pm 31.9 $\
Mrk 382 & $ 7.5_{- 2.0}^{+ 2.9} $ & $ 1462\pm296 $ & $ 840\pm 37$ & $ 6.50_{-0.29}^{+0.19} $ & $ 1.18_{-0.53}^{+0.69} $ & $ 43.12\pm 0.08 $ & $ 41.01\pm 0.05 $ & $ 39.6\pm 9.0 $\
Mrk 142 & $ 7.9_{- 1.1}^{+ 1.2} $ & $ 1588\pm 58 $ & $ 948\pm 12$ & $ 6.59_{-0.07}^{+0.07} $ & $ 1.65_{-0.23}^{+0.23} $ & $ 43.56\pm 0.06 $ & $ 41.60\pm 0.04 $ & $ 55.2\pm 9.5 $\
IRAS F12397 & $ 9.7_{- 1.8}^{+ 5.5} $ & $ 1802\pm560 $ & $1150\pm122$ & $ 6.79_{-0.45}^{+0.27} $ & $ 2.26_{-0.62}^{+0.98} $ & $ 44.23\pm 0.05 $ & $ 42.26\pm 0.04 $ & $ 54.2\pm 8.4 $\
Mrk 486 & $ 23.7_{- 2.7}^{+ 7.5} $ & $ 1942\pm 67 $ & $1296\pm 23$ & $ 7.24_{-0.06}^{+0.12} $ & $ 0.55_{-0.32}^{+0.20} $ & $ 43.69\pm 0.05 $ & $ 42.12\pm 0.04 $ & $ 135.9\pm 20.3 $\
Mrk 493 & $ 11.6_{- 2.6}^{+ 1.2} $ & $ 778\pm 12 $ & $ 513\pm 5$ & $ 6.14_{-0.11}^{+0.04} $ & $ 1.88_{-0.21}^{+0.33} $ & $ 43.11\pm 0.08 $ & $ 41.35\pm 0.05 $ & $ 87.4\pm 18.1 $\
IRAS 04416 & $ 13.3_{- 1.4}^{+13.9} $ & $ 1522\pm 44 $ & $1056\pm 29$ & $ 6.78_{-0.06}^{+0.31} $ & $ 2.63_{-0.67}^{+0.16} $ & $ 44.47\pm 0.03 $ & $ 42.51\pm 0.02 $ & $ 55.8\pm 4.7 $\
\
SDSS J075101 & $ 33.4_{- 5.6}^{+15.6} $ & $ 1495\pm 67 $ & $1055\pm 32$ & $ 7.16_{-0.09}^{+0.17} $ & $ 1.34_{-0.41}^{+0.25} $ & $ 44.12\pm 0.05 $ & $ 42.25\pm 0.03 $ & $ 68.1\pm 8.6 $\
SDSS J080101 & $ 8.3_{- 2.7}^{+ 9.7} $ & $ 1930\pm 18 $ & $1119\pm 3$ & $ 6.78_{-0.17}^{+0.34} $ & $ 2.33_{-0.72}^{+0.39} $ & $ 44.27\pm 0.03 $ & $ 42.58\pm 0.02 $ & $ 105.5\pm 8.3 $\
SDSS J080131 & $ 11.5_{- 3.6}^{+ 8.4} $ & $ 1188\pm 3 $ & $ 850\pm 12$ & $ 6.50_{-0.16}^{+0.24} $ & $ 2.46_{-0.54}^{+0.38} $ & $ 43.98\pm 0.04 $ & $ 42.08\pm 0.03 $ & $ 64.0\pm 7.0 $\
SDSS J081441 & $ 18.4_{- 8.4}^{+12.7} $ & $ 1615\pm 22 $ & $1122\pm 11$ & $ 6.97_{-0.27}^{+0.23} $ & $ 1.56_{-0.57}^{+0.63} $ & $ 44.01\pm 0.07 $ & $ 42.42\pm 0.03 $ & $ 132.0\pm 23.7 $\
SDSS J081456 & $ 24.3_{-16.4}^{+ 7.7} $ & $ 2409\pm 61 $ & $1438\pm 32$ & $ 7.44_{-0.49}^{+0.12} $ & $ 0.59_{-0.30}^{+1.03} $ & $ 43.99\pm 0.04 $ & $ 42.15\pm 0.03 $ & $ 74.4\pm 7.6 $\
SDSS J093922 & $ 11.9_{- 6.3}^{+ 2.1} $ & $ 1209\pm 16 $ & $ 835\pm 11$ & $ 6.53_{-0.33}^{+0.07} $ & $ 2.54_{-0.20}^{+0.71} $ & $ 44.07\pm 0.04 $ & $ 42.09\pm 0.04 $ & $ 53.0\pm 6.7 $\
\
SDSS J075949 & $ 55.0_{-13.1}^{+17.0} $ & $ 1807\pm 11 $ & $1100\pm 3$ & $ 7.54_{-0.12}^{+0.12} $ & $ 0.70_{-0.29}^{+0.29} $ & $ 44.20\pm 0.03 $ & $ 42.48\pm 0.02 $ & $ 97.5\pm 9.1 $\
SDSS J080131 & $ 11.2_{- 9.8}^{+14.8} $ & $ 1290\pm 13 $ & $ 800\pm 5$ & $ 6.56_{-0.90}^{+0.37} $ & $ 2.29_{-0.80}^{+1.87} $ & $ 43.95\pm 0.04 $ & $ 41.96\pm 0.05 $ & $ 52.3\pm 7.7 $\
SDSS J084533 & $ 15.2_{- 6.3}^{+ 3.2} $ & $ 1243\pm 13 $ & $ 818\pm 10$ & $ 6.66_{-0.23}^{+0.08} $ & $ 2.98_{-0.22}^{+0.52} $ & $ 44.54\pm 0.04 $ & $ 42.58\pm 0.05 $ & $ 55.9\pm 7.5 $\
SDSS J085946 & $ 34.8_{-26.3}^{+19.2} $ & $ 1718\pm 16 $ & $1031\pm 14$ & $ 7.30_{-0.61}^{+0.19} $ & $ 1.51_{-0.43}^{+1.27} $ & $ 44.41\pm 0.03 $ & $ 42.51\pm 0.02 $ & $ 63.1\pm 5.2 $\
SDSS J102339 & $ 24.9_{- 3.9}^{+19.8} $ & $ 1733\pm 29 $ & $1139\pm 19$ & $ 7.16_{-0.08}^{+0.25} $ & $ 1.29_{-0.56}^{+0.20} $ & $ 44.09\pm 0.03 $ & $ 42.14\pm 0.03 $ & $ 57.0\pm 5.9 $
Properties of H$\beta$ Lags in SEAMBHs
======================================
The $\rhb-L_{5100}$ correlation was originally presented by Peterson (1993; his Figure 10, only nine objects). It was confirmed by Kaspi et al. (2000) using a sample of 17 low-redshift quasars. Bentz et al. (2013) refined the $\rblrl$ relation through subtraction of host contamination and found that its intrinsic scatter is only 0.13 dex. Paper IV (Table 7) provides a complete list of previously mapped AGNs, based on Bentz et al. (2013); we directly use these values[^3]. As in Paper IV, for objects with multiple measurements of H$\beta$ lags, we obtain the BH mass from each campaign and then calculate the average BH mass. Using the averaged BH mass, we apply it to get accretion rates of the BHs during each monitoring epoch, which are further averaged to obtain the mean accretion rates of those objects (Kaspi et al. 2005; Bentz et al. 2013). We call this the “average scheme." On the other hand, we may consider each individual measurement of a single object as different objects (e.g., Bentz et al. 2013). We called this the “direct scheme." Although the two approaches are in principle different, we obtain very similar results (see a comparison in Paper IV).
All correlations of two parameters shown in this paper are calculated with the FITEXY method, using the version adopted by Tremaine et al. (2002), which allows for intrinsic scatter by increasing the uncertainties in small steps until $\chi^2$ reaches unity (this is typical for many of our correlations). We also emplot the BCES method (Akristas & Bershady 1996) but prefer not to use its results because it is known to give unreliable results in samples containing outliers (there are a few objects with quite large uncertainties of $\mathdotM$).
The $\rblrl$ relation
---------------------
As shown in Paper IV, the H$\beta$ lags of the SEAMBH2013 sample were found to significantly deviate from the normal $\rblrl$ relation, by a factor of a few. We plot the $\rblrl$ relation of all samples in Figure 3. For sub-Eddington AGNs ($\mathdotM\le 3$) in the direct scheme, $\log \left(\rhb/{\rm ltd}\right)=(1.54\pm0.03)+(0.53\pm0.03)\log \ell_{44},$ with an intrinsic scatter of $0.15$ (see Paper IV). Using FITEXY, we have $$\begin{split}
&\hspace{-0.5cm}\log \left(\rhb/{\rm ltd}\right)=\\
&\left\{\begin{array}{ll}
(1.30\pm 0.05)+(0.53\pm 0.06)\log \ell_{44} & (\mathdotM\ge 3),\\ [0.8em]
(1.44\pm 0.03)+(0.49\pm 0.03)\log \ell_{44} & ({\rm for~ all~\mathdotM}),
\end{array}\right.
\end{split}$$ with intrinsic scatters of $\sigma_{\rm in}=(0.24,0.21)$. Clearly, the intrinsic scatter of SEAMBHs is much larger than the sample of sub-Eddington AGNs. In the averaged scheme, we have $\log \left(\rhb/{\rm ltd}\right)=(1.55\pm0.04)+(0.53\pm0.04)\log \ell_{44}$ for sub-Eddington AGNs, with an intrinsic scatter of $0.16$ (see Paper IV), and
$$\begin{split}
&\hspace{-0.5cm}\log \left(\rhb/{\rm ltd}\right)= \\
&\left\{\begin{array}{ll}
(1.32\pm 0.05)+(0.52\pm 0.06)\log \ell_{44} & (\mathdotM\ge 3),\\ [0.8em]
(1.44\pm 0.03)+(0.49\pm 0.03)\log \ell_{44} & ({\rm for~ all~\mathdotM}),
\end{array}\right.
\end{split}$$
with intrinsic scatters of $\sigma_{\rm in}=(0.22,0.21)$. The slope of the correlation for the SEAMBH sample is comparable to that of sub-Eddington AGNs, but the normalization is significantly different. It is clear that the SEAMBH sources increase the scatter considerably, especially over the limited luminosity range occupied by the new sources.
As in Paper IV, we also tested the correlation between H$\beta$ lag and H$\beta$ luminosity, namely, the $\rhb-\Lhb$ relation. The scatter of the $\rhb-\Lhb$ correlation is not smaller than that of the $\rhb-L_{5100}$ correlation, and we do not consider it further.
{width="100.00000%"}
$\mathdotM-$dependent BLR Size
------------------------------
To test the dependence of the BLR size on accretion rate, we define a new parameter, $\Delta \rhb=\log\left(\rhb/\prblr\right)$, that specifies the deviation of individual objects from the $\rblrl$ relation of the subsample of $\mathdotM<3.0$ sources (i.e., $\prblr$ as given by Equations 4b and 5b for $\mathdotM<3$ AGNs in Paper IV). The scatter of $\Delta \rhb$ is calculated by $\sigma_{_{\rhb}}=\left[\sum_i\left(\Delta R_{{\rm H\beta}, i}-\langle\Delta \rhb\rangle\right)^2/N\right]^{1/2}$, where $N$ is the number of objects and $\langle \Delta R_{{\rm H\beta}}\rangle$ is the averaged value. Figure 3 provides $\Delta\rhb$ plots for comparison.
Figure 4 shows $\Delta \rhb$ versus $\mathdotM$, as well as $\Delta \rhb$ distributions for the $\mathdotM\ge3$ and $\mathdotM<3$ subsamples in the direct (panels a and b) and averaged (panels c and d) schemes. A Kolmogorov-Smirnov (KS) test shows that the probability that the two subsamples are drawn from the same parent distributions is $p_{_{\rm KS}}=0.00029$ for the direct scheme and $p_{_{\rm KS}}=0.0094$ for the averaged scheme. This provides a strong indication that the main cause of deviation from the normal $\rblrl$ relation is the extreme accretion rate. Thus, a single $\rblrl$ relation for all AGNs is a poor approximation for a more complex situation in which both the luminosity and the accretion rate determine $R_{{\rm H\beta}}$. From the regression for $\mathdotM\ge 3$ AGNs, we obtain the dependence of the deviations of $\rhb$ from the $\rblrl$ relation in Figure 4: $$\begin{split}
&\Delta \rhb=\\
&\left\{\begin{array}{ll}
(0.39\pm0.09)-(0.47\pm0.06)\log \mathdotM & ({\rm direct~scheme}),\\ [0.8em]
(0.34\pm0.09)-(0.42\pm0.07)\log \mathdotM & ({\rm averaged ~scheme}),
\end{array}\right.
\end{split}$$
with $\sigma_{\rm in}=(0.01,0.05)$, respectively. We have tested the above correlations also for $\mathdotM<3$. The FITEXY regressions give slopes near 0, with very large uncertainties: $\Delta \rhb\propto \mathdotM^{-0.055\pm 0.032}$ and $\Delta \rhb\propto \mathdotM^{-0.095\pm0.050}$ for Figure 4a and 4c, respectively, implying that $\Delta\rhb$ does not correlate with $\mathdotM$ for the $\mathdotM<3$ group. All this confirms that $\mathdotM$ is an additional parameter that controls the $\rblrl$ relation in AGNs with high accretion rates.
{width="80.00000%"}
A New Scaling Relation for the BLR
==================================
We provide evidence H$\beta$ lags depend on luminosity [*and*]{} accretion rate. There are a total of 28 SEAMBHs (including those discovered in other studies). We now have an opportunity to define a new scaling relation for the BLR, one that properly captures the behavior of sub-Eddington and super-Eddington AGNs. Considering the dependence of $\Delta \rhb\propto \mathdotM^{-0.42}$ (Equation 6), a unified form of the new scaling law can take the form[^4] $$\rhb=\alpha_1\, \ell_{44}^{\beta_1}
\min\left[1,\left(\frac{\mathdotM}{\mathdotM_{\rm c}}\right)^{-\gamma_1}\right],$$ where $\mathdotM_c$ is to be determined by data. Equation (7) reduces to the normal $\rblrl$ relation for sub-Eddington AGNs and to $\rhb=\alpha_1\ell_{44}^{\beta_1}\left(\mathdotM/\mathdotM_c\right)^{-\gamma_1}$ for AGNs with $\mathdotM\ge \mathdotM_c$. There are four parameters to describe the new scaling relation, but only two ($\mathdotM_c$ and $\gamma_1$) are new due to the inclusion of accretion rates; the other two are mainly determined by sub-Eddington AGNs. The critical value of $\mathdotM_c$, which is different from the criterion of SEAMBHs, depends on the sample of SEAMBHs.
{width="90.00000%"}
In order to determine the four parameters simultaneously, we define $$\chi^2=\frac{1}{N}\sum_{i=1}^N\frac{\left(\rhb-\rhb^i\right)^2}{\left(\Delta_{\rhb}^i\right)^2},$$ where $\Delta_{\rhb}^i$ is the error bar of $\rhb^i$. Minimizing $\chi^2$ among all the mapped AGNs and employing a bootstrap method, we have $$\alpha_1=29.6_{-2.8}^{+2.7};~~~\beta_1=0.56_{-0.03}^{+0.03};~~~\gamma_1=0.52_{-0.16}^{+0.33};
~~~\mathdotM_c=11.19_{-6.22}^{+2.29}.$$ This new empirical relation has a scatter of $0.19$, smaller than the scatter (0.26) of the normal $\rblrl$ relation for all the mapped AGNs. The new scaling relation is plotted in Figure 5.
Equation (7) shows the dependence of the BLR size on accretion rates, but it cannot be directly applied to single-epoch spectra for BH mass without knowning $\mathdotM$. Iteration of Equation (7) does not converge. The reason is due to the fact that larger $\mathdotM$ leads to smaller $\rhb$ and higher $\mathdotM$, implying that the iteration from Equation (1) does not converge. Du et al. (2016b) devised a new method to determine $\mathdotM$ from single-epoch spectra. Beginning with the seminal work of Boroson & Green (1992), it has been well-known that $\RFe \equiv F_{\rm FeII}/F_{\rm H\beta}$, the flux ratio of broad optical to H$\beta$, correlates strongly with Eddington ratio (Sulentic et al. 2000; Shen & Ho 2014). At the same time, the shape of broad H$\beta$, as parameterized by $\Dhb={\rm FWHM}/\sighb$, where $\sighb$ is the line dispersion, also correlates with Eddington ratio (Collin et al. 2006). Combining the two produces produces a strong bivariate correlation, which we call the fundamental plane of the BLR, of the form $$\log\mathdotM=\alpha_2+\beta_2\Dhb+\gamma_2\RFe,$$ where $$\alpha_2=2.47\pm0.34;~ \beta_2=-1.59\pm0.14;~ \gamma_2=1.34\pm0.20.$$
Discussion
==========
Normalized BLR Sizes
--------------------
In order to explore the relation between BLR size and accretion rate, we define a dimensionless radius for the BLR, $\rrhb=\rhb/\Rg$, where $\Rg=1.5\times 10^{12}m_7$cm is the gravitational radius. As in Paper IV, we insert Equation (3) into $\rrhb$ to replace $\ell_{44}$, to obtain $\rrhb=1.9\times 10^4~\mathdotM^{0.35}m_7^{-0.29}$. This relation implies that $\rrhb$ increases with accretion rates as $\rrhb\propto \mathdotM^{0.35}$ for sub-Eddington AGNs, whereas in SEAMBHs $\rrhb\propto \mathdotM^{0.29\pm0.08}$ (as shown in Figure 6[*a*]{}) and $\rrhb$ and tends toward a maximum saturated value of $\rrhb^{\rm max}=\fblr^{-1}\left(c/V_{\rm min}\right)^2
=9\times 10^4\,\fblr^{-1}V_{\rm min,3}^{-2}$, where $V_{\rm min,3}=V_{\rm min}/
10^3\kms$ is the minimum velocity width of H$\beta$ (see Equation 15 in Paper IV). We note that the minimum observed FWHM values of H$\alpha$ (which is comparable to H$\beta$) is $\sim 10^3\, \kms$ among low-mass AGNs (Greene & Ho 2007; Ho & Kim 2016). Indeed, this limit is consistent with the saturation trend of $\rrhb$ (Figure 6[*a*]{}).
We note that the relatively large scatter in Figure 6[*a*]{} is mostly due to the uncertainties in BH mass. In order to better understand the relation between the BLR and the central engine, we define, as in Paper IV, the parameter $Y=m_7^{0.29}\rrhb$, which reduces to $$Y=1.9\times 10^4~\mathdotM^{0.35}.$$ We would like to point out that Equation (12) describes the coupled system of the BLR and the accretion disks. It is therefore expected that $Y$ is a synthetic parameter describing the photoionization process including ionizing sources.
It is easy to observationally test Equation (12) using RM results. Figure 6[*b*]{} plots $Y$ versus $\mathdotM$. It is very clear that the observed data for objects with $\mathdotM<3$ agree well with Equation (12). Furthermore, there is a clear saturation of $Y$ for objects with $\mathdotM\ge3$ objects. All these results strengthen the conclusions drawn in Paper IV. As in that work, we define an empirical relation $$Y=Y_{\rm sat}\min\left[1,\left(\frac{\mathdotM}{\mathdotM_b}\right)^{b}\right],$$ where $$Y_{\rm sat}=(3.5_{-0.5}^{+0.6})\times 10^4,~~~
\mathdotM_b=15.6_{-9.1}^{+22.0},~~~
b=0.27_{-0.04}^{+0.04}.$$ In fact, we can get $Y$ by inserting Equation (3) into (7), and find that it is in agreement with Equation (13). From the saturated$-Y$, we have the maximum value of $$r_{_{\rm H\beta,sat}}=(3.5_{-0.5}^{+0.6}) \times 10^4\,m_7^{-0.29}~~{\rm or}~~
R_{{\rm H\beta,sat}}=(19.9_{-2.8}^{+3.4}) m_7^{0.71}\,{\rm ltd}.$$ This result provides a strong constraint on theoretical models of super-Eddington accretion onto BHs.
The Shortened Lags
------------------
The shortened H$\beta$ lags is the strongest distinguishing characteristic so far identified between super- and sub-Eddington AGNs. Two factors may lead to shortened lags for SEAMBHs. First, Wang et al. (2014c) showed that, in the Shakura-Sunyaev regime, retrograde accretion onto a BH can lead to shorter H$\beta$ lags. The reason is due to the suppression of ionizing photons in retrograde accretion compared with prograde accretion. The second factor stems from self-shadowing effects of the inner part of slim disks (e.g., Li et al. 2010), which efficiently lower the ionizing flux received by the BLR (Wang et al. 2014c). When $\mathdotM$ increases, the ratio of the disk height to disk radius increases due to radiation pressure; the radiation field becomes anisotropic (much stronger than the factor of $\cos i$) due to the optically thick funnel of the inner part of the slim disk. In principle, the radiation from a slim disk saturates ($\propto \ln
\mathdotM$), and the total ionizing luminosity slightly increases with accretion rate, but the self-shadowing effects efficiently suppress the ionizing flux to the BLR clouds. For face-on disks of type 1 AGNs, observers receive the intrinsic luminosity. If the ionization parameter is constant, the ionization front will significantly shrink, and hence the H$\beta$ lag is shortened in SEAMBHs compared with sub-Eddington AGNs of the same luminosity.
The shortened H$\beta$ lag observed in SEAMBHs cannot be caused by retrograde accretion. However, the strong dependence on accretion rate of the deviation from the standard lag-luminosity relation implies that the properties of the ionizing sources are somehow different from those in sub-Eddington AGNs. According to the standard photoionization theory, the observed $\rhb\propto L_{5100}^{1/2}$ relation can be explained if $L_{5100}\propto L_{\rm ion}$ and $Q_c=Un_e\pepsilon$ is constant, where $U=L_{\rm ion}/4\pi \rhb^2cn_e\pepsilon$, $L_{\rm ion}$ is the ionizing luminosity, $n_e$ is gas density of BLR clouds, and $\pepsilon$ is the average energy of the ionizing photons (Bentz et al. 2013). The relation $L_{5100}\propto L_{\rm ion}$ holds for sub-Eddington AGNs, and the constancy of $Q_c$ is determined by the clouds themselves. $Q_c$ is not expected to vary greatly as a function of Eddington ratio. Therefore, $$\rhb=\frac{L_{\rm ion}^{1/2}}{Q_c}=\calS\rhb^0,$$ where the factor $\calS=\left(L_{\rm ion}/L_{\rm ion,0}\right)^{1/2}$ describes the anisotropy of the ionizing radiation field, $L_{\rm ion}$ is the shadowed ionizing luminosity received by the BLR clouds, and $\rhb^0$ is the BLR radius corresponding to $L_{\rm ion,0}$, the unshadowed luminosity. Based on the classical model of slim disks, Wang et al. (2014c) showed that, for a given accretion rate, $\calS$ strongly depends on the orientation of the clouds relative to the disk, and that it range from 1 to a few tens. Therefore, the reduction of the H$\beta$ lag can, in principle, reach up to a factor of a few, even 10, as observed.
Furthermore, the saturated-$Y$ implies that the ionizing luminosity received by the BLR clouds gets saturated. The theory of super-Eddington accretion onto BHs is still controversial. Although extensive comparison with models is beyond the scope of this paper, we briefly discuss the implications of the current observations to the theory. Two analytical models, which reach diametrically extreme opposite conclusions, have been proposed. Abramowicz et al. (1988) suggested a model characterized by fast radial motion with sub-Keplerian rotation and strong photon-trapping. Both the shortened lags and saturated$-Y$ may be caused by self-shadowing effects and saturated radiation from a slim disk. Both features are expected from the Abramowicz et al. model (Wang et al. 2014b). On the other hand, photon-bubble instabilities may govern the disk structure and lead to very high radiative efficiency (Gammie 1998). If super-Eddington accretion can radiate as much as $L/L_{\rm Edd}\gtrsim 470\,
m_7^{6/5}$ (Equation 14 in Begelman 2002), the disk remains geometrically thin. In such an extreme situation, self-shadowing effects are minimal, H$\beta$ lags should not be reduced, and $Y-$saturation disappears.
![ The best fit of the new scaling relation for all mapped AGNs. We find that $\alpha_1=(29.6_{-2.8}^{+2.7})$lt-d, $\beta_1=0.56_{-0.03}^{+0.03}$, $\gamma_1=0.52_{-0.16}^{+0.33}$ and $\mathdotM_c=11.19_{-6.22}^{+2.29}$. The scatter of the BLR size is greatly reduced to $\sigma=0.19$. []{data-label="r-l"}](fig5.eps){width="48.00000%"}
Recent numerical simulations that incorporate outflows (e.g., Jiang et al. 2014) and relativistic jets (Sadowski et al. 2015) also suggest that super-Eddington accretion flows can maintain a high radiative efficiency. However, most AGNs with high accretion rates are radio-quiet (Ho 2002; Greene & Ho 2006), in apparent contradiction with the numerical simulation predictions. Furthermore, evidence for $Y$-saturation also does not support the models with high radiative efficiency. Recent modifications of the classical slim disk model that include photo-trapping appear promising (e.g., Cao & Gu 2015; Sadowski et al. 2014), but the situation is far from settled. Whatever the outcome, the results from our observations provide crucial empirical constraints on the models.
Inclination Effects on $\mathdotM$
----------------------------------
If the BLR is flattened, its inclination angle to the observer will influence $\bhm$, and hence $\mathdotM$ (see Equation 3). To zero-order approximation, the observed width of the broad emission lines follow $$\Delta V_{\rm obs}\approx \left[\left(\frac{H_{\rm BLR}}{R}\right)^2+\sin^2i\right]^{1/2}V_{\rm K},$$ where $V_{\rm K}$ is the Keplerian velocity and $H_{\rm BLR}$ is the height of the flatten BLR (e.g., Collin et al. 2006). For a geometrically thin BLR, $H_{\rm BLR}/R\ll1$, $\Delta V_{\rm obs}\approx V_{\rm K}\sin i$, and hence $\mathdotM\propto (\sin i)^{-4}$, which is extremely sensitive to the inclination can be severely overestimated for low inclinations. On the other hand, many arguments (e.g., Goad & Korista 2014) support $H_{\rm BLR}/R\lesssim 1$, and the inclination angle significantly influences $\bhm$ only for $\sin i\gtrsim H_{\rm BLR}/R$. Currently, the values of $H_{\rm BLR}/R$ are difficult to estimate, but detailed modelling of RM data suggests $H_{\rm BLR}/R\sim 1$ (Li et al. 2013; Pancoast et al. 2014). If true, this implies that the BLR is not very flattened, and hence the inclination angle only has a minimal influence on $\bhm$ and $\mathdotM$.
Comparison with Previous Campaigns
----------------------------------
The objects in our SEAMBH sample are very similar to NLS1s. As previous RM AGN samples include NLS1s, why have previous studies not noticed that NLS1s deviate from the $\rblrl$ relation (e.g., Figure 2 in Bentz 2011)? We believe that the reason is two-fold. First, the number of NLS1s included in previous RM campaigns was quite limited (Denney et al. 2009, 2010; Bentz et al. 2008, 2009; see summary in Bentz 2011). The level of optical variability in NLS1s is generally very low (Klimek et al. 2004), and many previous attempts at RM have proved to be unsuccessful (e.g., Giannuzzo & Stirpe 1996; Giannuzzo et al. 1999). Second, not all NLS1s are necessarily highly accreting. Our SEAMBH sample was selected to have high accretion rates (see $\mathdotM$ listed in Table 7 of Paper IV), generally higher than that of typical NLS1s previously studied successfully through RM. As discussed in Wang et al. (2014b) and in Paper IV, high accretion rates lead to anisotropic ionizing radiation, which may explain the shortened BLR lags.
{width="95.00000%"}
SEAMBHs as Standard Candles
---------------------------
Once its discovery, quasars as the brightest celestial objects in the Universe had been suggested for cosmology (Sandage 1965; Hoyle & Burbidge 1966; Longair & Scheuer 1967; Schmidt 1968; Bahcall & Hills 1973; Burbidge & O’Dell 1973; Baldwin et al. 1978). Unfortunately, the diversity of observed quasars made these early attempts elusive. After five decades since its discovery, quasars are much well understood: accretion onto supermassive black holes is powering the giant radiation, in particular, the BH mass can be reliably measured. Quasars as the most powerful emitters renewed interests for cosmology in several independent ways: 1) the normal $\rblrl$ relation (Horn et al. 2003; Watson et al. 2011; Czerny et al. 2013); 2) the linear relation between BH mass and luminosity in super-Eddington quasars (Wang et al. 2013; Paper-II); 3) Eddington AGNs selected by eigenvector 1 (Marziani & Sulentic 2014); 4) X-ray variabilities (La Franca et al. 2014) and 5) $\alpha_{_{\rm OX}}-L_X$ relation (Risaliti & Lusso 2015). These parallel methods will be justified for cosmology by their feasibility of experiment periods and measurement accuracy.
The strength of SEAMBHs makes its application more convenient for cosmology. Selection of SEAMBHs only depends on single epoch spectra through the fundamental plane (Equation 10). BH mass can be estimated by the new scaling relation (Equation 7). We will apply the scheme outlined by Wang et al. (2014a) to the sample of selected SEAMBHs for cosmology in a statistical way (in preparation). On the other hand, the shortened H$\beta$ lags greatly reduce monitoring periods if SEAMBHs are applied as standard candles, in particular, the reduction of lags govern by super-Eddington accretion can cancel the cosmological dilltion factor of $(1+z)$. Otherwise, the monitoring periods of sub-Eddington AGNs should be extended by the same factor of $(1+z)$ for measurements of H$\beta$ lags. Such a campaign of using the normal $\rblrl$ relation for cosmology will last for a couple of years, even 10 years for bright high$-z$ quasars. Similarly to extension of the $\rblrl$ relation to - and -lines (Vestergaard & Peterson 2006), we can extend Equation (7) to and lines for the scaling relations with luminosity as $R_{\rm MgII}(L_{3000},\mathdotM)$ and $R_{\rm CIV}(L_{3000},\mathdotM)$, respectively, where $R_{\rm MgII}$ and $R_{\rm CIV}$ are sizes of the and regions, and $L_{3000}$ is the 3000 Å luminosity. Such extended relations conveniently allow us to investigate cosmology by making use of large samples of high$-z$ quasars without time-consuming RM campaigns. It is urgent for us to make use of kinds of standard candles to test the growing evidence for dynamical dark energy (e.g., Zhao et al. 2012; Ade et al. 2015).
Conclusions
===========
We present the results of the third year of reverberation mapping of super-Eddington accreting massive black holes (SEAMBHs). H$\beta$ lags of five new SEAMBHs have been detected. Similar to the SEAMBH2012 and SEAMBH2013 samples, we find that the SEAMBH2014 objects generally have shorter H$\beta$ lags than the normal $\rblrl$ relation, by a factor of a few. In total, we have detected H$\beta$ lags for 18 SEAMBHs from this project, which have accretion rates from $\mathdotM\sim 10$ to $\lesssim 10^3$. The entire SEAMBH sample allows us to establish a new scaling relation for the BLR size, which depends not only on luminosity but also on accretion rate. The new relation, applicable over a wide range of accretion rates from $\mathdotM \approx 10^{-3}$ to $10^3$, is given by $\rhb=\alpha_1\ell_{44}^{\beta_1}\,\min\left[1,\left(\mathdotM/\mathdotM_c\right)^{-\gamma_1}\right]$, where $\ell_{44}=L_{5100}/10^{44}\,\ergs$ is 5100 Å continuum luminosity, and coefficients of $\alpha_1=(29.6_{-2.8}^{+2.7})$lt-d, $\beta_1=0.56_{-0.03}^{+0.03}$, $\gamma_1=0.52_{-0.16}^{+0.33}$ and $\mathdotM_c=11.19_{-6.22}^{+2.29}$.
.
{width="90.00000%"}
Validity of Equation (3)
========================
The validity of Equation (3) can be justified for application to SEAMBHs. Solutions of slim disks are transonic and usually given only by numerical calculations (Abramowicz et al. 1988). When the accretion rate of the disk is high enough, the complicated structure of the disk reduces to a self-similar, analytical form (Wang & Zhou 1999). Using the self-similar solutions (Wang & Zhou 1999; Wang et al. 1999), we obtained the radius of disk region emitting optical (5100 Å) photons, $$\frac{R_{5100}}{R_{\rm Sch}}\approx 4.3\times 10^3\,m_7^{-1/2},$$ and the photon-trapping radius $$\frac{R_{\rm trap}}{R_{\rm Sch}}\approx 144\left(\frac{\mathdotM}{10^2}\right).$$ We used the blackbody relation $kT_{\rm eff}=hc/\lambda$, where $k$ is the Boltzmann constant, $T_{\rm eff}$ is the effective temperature of the disk surface, $h$ is the Planck constant, and $R_{\rm Sch}=3.0\times 10^{12}m_7$cm is the Schwartzschild radius. Equation (3) holds provided $R_{5100}\gtrsim R_{\rm trap}$, namely $$\mathdotM\lesssim 3\times 10^3\,m_7^{-1/2}.$$ In this regime, optical radiation is not influenced by photon-trapping effects. We would also like to point out that the BH spin only affects emission from the innermost regions of the accretion disk rather than the regions emitting 5100 Å photons. In the present campaign, no SEAMBH so far has been found to exceed this critical value. Beyond this critical value of accretion rate, optical photons are trapped by the accretion flow. We call this the hyper-accretion regime.
Here the cited $10^{-2}$ below Equation (3) is not a strict value of the ADAF threshold since it depends on several factors, such as viscosity and outer boundary conditions. There are a few mapped AGNs with $\mathdotM\lesssim10^{-2}$ (NGC 4151, NGC 5273, 3C 390.3 and NGC 5548; see Table 7 in Paper IV), but we do not discuss them in this paper because they do not influence our conclusions.
Recently, reprocessing of X-rays (e.g., Frank et al. 2002; Cackett et al. 2007) has been found to play an important role in explaining the variability properties of NGC 5548 (e.g., Fausnaugh et al. 2015). The fraction of X-ray emission to the bolometric luminosity strongly anti-correlates with the Eddington ratio (see Figure 1 in Wang et al. 2004). This result is usually interpreted to mean that the hot corona becomes weaker with increasing accretion rate, as a result of more efficiently cooling of the corona by UV and optical photons from the cold disk. This suggests that AGNs with high accretion rates will have less reprocessed emission, such that Equation (3) would be more robust in SEAMBHs.
Light curves of comparison stars
================================
In order to avoid selecting variable stars as comparison stars, we examined their variability. To test the invariance of the comparison stars used in our spectroscopic observation, we performed differential photometry by comparing them with other stars in the same field. We typically use six stars for the differential photometry. The light curves of the stars are shown in Figure 7. On average, the standard deviations in the light curves of the comparison stars are 1%. This guarantees that they can be used as standards for spectral calibration. Figure 7 shows the light curves of the comparison stars for each SEAMBH targets.
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Averaged and RMS spectra
========================
The averaged and RMS spectra of the SEAMBH2014 sample are provided in this Appendix. Following the standard way, we calculated the averaged spectrum as $$\bar{F}_{\lambda}=\frac{1}{N}\sum_{i=1}^NF_{\lambda}^i,$$ and the RMS spectrum as $$S_{\lambda}=\left[\frac{1}{N}\sum_{i=1}^N\left(F_{\lambda}^i-\bar{F}_{\lambda}\right)^2\right]^{1/2},$$ where $F_{\lambda}^i$ is the $i-$th observed spectrum and $N$ is the total number of observed spectra. They are shown in Figure 8. We note that both the averaged and RMS spectra are affected by the broadening effects of the $5^{\prime\prime}$-slit on the observed profiles. Using the Richards-Lucy iteration, we can correct the observed profiles (averaged and RMS) for velocity-resolved mapping, which will be carried out in a separate paper (Du et al. 2015c).
Notes on individual objects
===========================
We briefly remark on individual objects for which H$\beta$ lags have been successfully measured. We failed in getting lag measurements for the other five objects because either their flux variations are very small or the data sampling rate was inadequate.
[*J075949*]{}: The detected H$\beta$ lag arises from two major peaks in the light curves.
[*J080131*]{}: The monitoring observations during $2013-2014$ did not yield a well-determined H$\beta$ lag because of the lack of H$\beta$ response to the second continuum flare (see Paper IV). During $2013-2014$, the first reverberation of H$\beta$, which can be clearly seen during the first 70 days of the light curves, yields a very significant lag, as shown in the CCF with a rest-frame centroid lag of $11.5_{-3.6}^{+8.4}$ days (with a very high coefficient of $r_{\rm max}=0.81$). We monitored this object again in this observing season (Figure 1). We successfully measure $\tauhb=11.2_{-9.8}^{+14.8}$ days, consistent with last season’s result.
[*J084533*]{}: Its continuum slightly decreased before being monitored for $\sim 70$ days, and steadily increased until $\sim 200$ days and then decreased again. Although the CCF has a very flat peak close to $\sim 0.9$, Monte Carlo simulations show that H$\beta$ lag is around 20 days, which arises from the two peaks in the H$\beta$ and $r^{\prime}$-band light curves.
[*J085946*]{}: The CCF peaks near 0.6, which results from the two major dips in the H$\beta$ and $r^{\prime}$-band light curves. There are two peaks with roughly the same correlation coefficients around $\sim$20 days and $\sim$70 days in the observed frame, respectively. Considering the relatively poor data quality of this object, it is difficult to distinguish which is the true response. The centroid lag represents the average of these two peaks (responses), and its uncertainties cover the distribution (Figure 1) obtained in FR/RSS method. So, we use it in the analysis of main text.
[*J102339*]{}: The detected lag is from the dip feature around $\sim 150$ days in light curves.
Description of CCCD
===================
For multiple-peaked CCFs with similar correlation coefficients, it is ambiguous as to which peak should be used to calculate the final lag. In such cases, we use the CCCD to determine the lag. However, there are two approaches to calculate the centroid time lag in the CCF, as illustrated in Figure 9.
- Approach 1 calculates the centroid using all peaks above some criterion, such as $0.8r_{\rm max}$.
- Approach 2 only uses the highest peak.
In Approach 1, the CCCD tends to be smoother than the CCPD (see Figure 1), whereas in in Approach 2 the CCCD and CCPD always have a similar distribution. We adopt Approach 1 in our analysis. If the CCF has two or even three peaks, the two approaches give different centroid lags. However, when the quality of the data is high and the CCF is unimodal, the two approaches yield the same results. It should be pointed out that Approach 2 is often employed in the literature.
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[99]{}
Abramowicz, M. A., Czerny, B., Lasota, J.-P., & Szuszkiewicz, E. 1988, , 332, 646
Ade, P. A. R., Aghanim, N., Armitage-Caplan, C. et al. 2014, , 571, 16
Ade, P. A. R., Aghanim, N., Arnaud, M. et al. (Planck collaboration), 2015, arXiv:1502.01590
Akritas, M. G. & Bershady, M. A. 1996, , 470, 706
Arav, N., Barlow, T. A., Laor, A., & Blandford, R. D. 1997, , 288, 1015
Bahcall, J. N., Kozlovsky, B.-Z. & Salpeter, E. E. 1972, , 171, 467
Bahcall, J. N. & Hills, R. E. 1973, , 179, 699
Baldwin, J. A., Burke, W. L., Gaskell, C. M. & Wampler, E. J. 1978, , 273, 431
Bardeen, J. M., Press, W. H. & Teukolsky, S. A. 1972, , 178, 347
Barth, A. J., Bennert, V. N., Canalizo, G. et al. 2015, , 217, 26
Barth, A. J., Pancoast, A., Bennert, V. N. et al. 2013, , 769, 128
Begelman, M. C. 2002, , 568, L97
Beloborodov, A. M. 1998, , 297, 739
Bentz, M. C. 2011, in Narrow-Line Seyfert 1 Galaxies and Their Place in the Universe, ed. L. Foschini et al., p33
Bentz, M. C., Walsh, J. L., Barth, A. J. et al. 2008, , 689, L21
Bentz, M. C. et al. 2009, , 705, 199
Bentz, M. C., Denney, K. D., Grier, C. J. et al. 2013, , 767, 149
Bentz, M. C., Horenstein, D., Bazhaw, C. et al. 2014, , 796, 8
Bentz, M. C., Walsh, J. L., Barth, A. J. et al. 2009, , 705, 199
Blandford, R. D. & McKee, C. F. 1982, , 255, 419
Boroson, T. A. & Green, R. F. 1992, , 80, 109
Brocksopp, C., Starling, R. L. C., Schady, P. et al. 2006, , 366, 953
Burbidge, G. R. & O’dell, S. L. 1973, , 183, 759
Clavel, J., Reichert, G. A., Alloin, D. et al. 1991, , 366, 64
Capellupo, D. M., Netzer, H., Lira, P. et al. 2015, , 446, 3427
Cao, X. & Gu, W.-M. 2015, , 448, 3514
Collier, S. J., Horne, K., Kapsi, S. et al. 1998, , 500, 162
Collin, S., Boisson, C., Mouchet, M. et al. 2002, , 388, 771
Collin, S., Kawaguchi, T., Peterson, B. M. & Vestergaard, M. 2006, , 456, 75
Czerny, B., Hryniewicz, K., Maity, I. et al. 2013, , 556, 97
Czerny, B. & Elvis, M. 1987, , 321, 305
Davis, S. W. & Laor, A. 2011, , 728, 98
Denney, K. D., Bentz, M. C., Peterson, B. M. et al. 2006, , 653, 152
Denney, K. D. et al. 2009, , 702, 1353
Denney, K. D., Peterson, M. C., Pogge, R. W. et al. 2010, , 721, 715
Dietrich, M., Kollatschny, W., Peterson, B. M., et al. 1993, , 408, 416
Dietrich, M., Peterson, B. M., Albrecht, P. et al. 1998, , 115, 185
Dietrich, M., Peterson, B. M., Grier, C. J. et al. 2012, , 757, 53
Du, P., Hu, C., Lu, K.-X., et al. 2014, , 782, 45 (Paper I)
Du, P., Lu, K.-X., Hu, C., et al. 2015, , 820, 27 (Paper VI)
Du, P., Hu, C., Lu, K.-X., et al. 2016a, , 806, 22 (Paper IV)
Du, P., Wang, J.-M., Hu, C., Ho, L. C., Li, Y.-R. & Bai, J.-M., 2016b, ApJL, 818, L14
Fischer, T. C., Crenshaw, D. M., Kraemer, S. B., et al. 2014, , 785, 25
Gammie, C. F. 1998, , 297, 929
Giannuzzo, M. Z., Mignoli, M., Stirpe, G. M. & Comastri, A. 1998, , 330, 894
Giannuzzo, M. Z. & Stirpe, G. M. 1996, , 314, 419
Goad, M. R. & Korista, K. T. 2014, , 444, 43
Greene, J. & Ho, L. C. 2006, , 636, 56
Greene, J. & Ho, L. C. 2007, , 670, 92
Grier, C. J., Peterson, B. M., Pogge, R. W. et al. 2012, , 755, 60
Ho, L. C. 2002, , 564, 120
Ho, L. C. 2008, , 46, 475
Ho, L. C., & Kim, M. 2016, , in press (arXiv:1603.00057)
Horne, K., Korista, K. T. & Goad, M. G. 2003, , 339, 367
Hoyle, F. & Burbidge, R, 1966, , 210, 1346
Hu, C., Du, P., Lu, K.-X. et al. 2015, , 804, 138 (Paper III)
Hu, C., Wang, J.-M. & Ho, L. C. et al. 2008a, , 687, 78
Hu, C., Wang, J.-M. & Ho, L. C. et al. 2008b, , 683, L115
Jiang, Y.-F., Stone, J. M. & Davis, S. W. 2014, , 796, 106
Kaspi, S., Maoz, D., Netzer, H. et al. 2005, , 629, 61
Kaspi, S., Smith, P. S., Netzer, H. et al. 2000, , 533, 631
Kilerci Eser, E., Vestergaard, M., Peterson, B. M. et al. 2015, , 801, 8
Klimek, E. S., Gaskell, C. M. & Hedrick, C. H. 2004, , 609, 69
Kishimoto M., Antonucci R., Blaes O. et al. 2008, , 454, 492
Koratkar, A. P. & Gaskell, C. M. 1991, , 370, L61
La Franca, F., Bianchi, S. & Ponti, G. et al. 2014, , 787, L12
Laor, A. & Netzer, H. 1989, , 238, 897
Li, G.-X., Yuan, Y.-F. & Cao, X., 2010, , 715, 623
Li, Y.-R., Wang, J.-M., Ho, L. C., Du, P. & Bai, J.-M. 2013, , 779, 110
Longair, M. S. & Scheuer, P. A. G. 1967, , 215, 919
Maoz, D. & Netzer, H. 1989, , 236, 21
Maoz, D., Netzer, H., Leibowitz, E., et al. 1990, , 351, 75
Maoz, D., Netzer, H., Mazeh, T. et al. 1991, , 367, 493
Marziani, P. & Sulentic, J. 2014, , 442, 1211
Mineshige, S., Kawaguchi, T., Takeuchim M., & Hayashida, K. 2000, , 52, 499
Narayan, R. & Yi, I. 1994, , 428, L13
Netzer, H., Heller, A., Loinger, F. et al., 1996, , 279, 429
Netzer, H., Shemmer, O., Maiolino, R. et al. 2004, , 614, 558
Osterbrock, D. E. & Pogge, R. W. 1987, , 323, 108
Pancoast, A., Brewer, B. J., Treu, T. et al. 2014, , 445, 3073
Peterson, B. M., Balonek, T. J., Barker, E. S. et al. 1991, , 368, 119
Peterson, B. M. 1993, , 105, 247
Peterson, B. M., Ferrarese, L., Gilbert, K. M. et al. 2004, , 613, 682
Peterson, B. M., Grier, C. J., Horne, K. et al. 2014, , 795, 149
Peterson, B. M., Wanders, I., Bertram, R. et al. 1998, , 501, 82
Risaliti, R. & Lusso, E. 2015, arXiv:1505.07118
Sadowski, A. 2009, , 183, 171
Sadowski, A., Abramowicz, M. & Bursa, M. 2011, , 527, A17
Sadowski, A., Narayan, R., McKinney, J. C., & Tchekhovskoy, A. 2014, , 439, 503
Sadowski, A., Narayan, R., Tchekhovskoy, A. et al. 2015, , 447, 49
Sandage, A. 1965, , 141, 1560
Schmidt, M. 1968, , 151, 393
Shakura, N. I., & Sunyaev, R. A. 1973, , 24, 337
Shen, Y. & Ho, L. C. 2014, , 513, 210
Shen, Y, Brandt, W. N., Dawson, K. S. et al. 2015, , 216, 4
Shen, Y., Horne, K., Grier, C. J., et al. 2015b, arXiv:1510.02802
Shen, Y., Richards, G. T., Strauss, M. A., et al. 2011, , 194, 45
Sulentic, J., Marziani, P. & Dultzin-Hacyan, D. 2000, , 38, 521
Sun, W.-H. & Malkan, M. A. 1989, , 346, 68
Szuszkiewicz, E., Malkan, M. A., & Abramowicz, M. A. 1996, , 458, 474
Tremaine, S., Gebhardt, K., Bender, R., et al. 2002, , 574, 740
Vestergaard, M. & Peterson, B. M. 2006, , 641, 689
van Groningen, E., & Wanders, I. 1992, PASP, 104, 700
Wanders, I., van Groningen, E., Alloin, D. et al. 1993, , 269, 39
Wandel, A. & Petrosian, V. 1988, , 329, L11
Wang, J.-M., Watarai, K. & Mineshige, S. 2004, , 607, L107
Wang, J.-M., Du, P., Hu, C., et al. 2014a, , 793, 108 (Paper II)
Wang, J.-M., Du, P., Valls-Gabaud, D., Hu, C., & Netzer, H. 2013, , 110, 081301
Wang, J.-M., Qiu, J., Du, P. & Ho, L. C. 2014b, , 797, 65
Wang, J.-M., Du, P., Li, Y.-R., Ho, L. C., Hu, C. & Bai, J.-M. 2014c, , 792, L13
Wang, J.-M., & Netzer, H. 2003, , 398, 327
Wang, J.-M., Szuszkiewicz, E., Lu, F.-J., & Zhou, Y.-Y. 1999, , 522, 839
Wang, J.-M., & Zhou, Y.-Y. 1999, , 516, 420
Watson, D., Denney, K. D., Vestergaard, M., & Davis, T. M. 2011, , 740, L49
Woo, J.-H., Yoon, Y., Park, S., Park, D. & Kim, S. C. 2015, , 801, 38
Zhao, G.-B., Crittenden, R. G., Pogosian, L. & Zhang, X. 2012, , 109, 1301
[^1]: It has been realised that high-accretion rate AGNs are usually radio-quiet (Greene & Ho 2006), although there are a few NLS1s reported to be radio-loud. The fraction of radio-loud AGNs decreases with increasing accretion rate (Ho 2002, 2008).
[^2]: The uncertainty of our absolute flux calibration is $\lesssim$10%. We multiply the fiducial spectra of the in-slit stars with the bandpass of the SDSS r$^{\prime}$ filter and compare their synthesized magnitudes with the magnitudes found in the SDSS database. The maximum difference is $\lesssim$10%.
[^3]: NGC 7469 was mapped twice by Collier et al. (1998) and Peterson et al. (2014). While their H$\beta$ lags are consistent, the FWHM of H$\beta$ is very different. We only retain the later observation in the analysis.
[^4]: We have tried $\rhb=\alpha_1 \ell_{44}^{\beta_1}\left[1+\left(\mathdotM/\mathdotM_c\right)^{\gamma_1}\right]^{\delta_1}$, which is continuous for the transition from sub- to super-Eddington sources. The fitting also yields a very rapid transition at $\mathdotM_{\rm c}\sim10$, with $\gamma_1 = 0.025$ and $\delta_1 = 21.02$ (the present sample is still dominated by sub-Eddington AGNs, with a ratio of 35/63). We prefer the form given by Equation (7).
|
---
abstract: 'In this work, we propose a novel mobile rescue robot equipped with an immersive stereoscopic teleperception and a teleoperation control. This robot is designed with the capability to perform safely a casualty-extraction procedure. We have built a proof-of-concept mobile rescue robot called ResQbot for the experimental platform. An approach called “loco-manipulation” is used to perform the casualty-extraction procedure using the platform. The performance of this robot is evaluated in terms of task accomplishment and safety by conducting a mock rescue experiment. We use a custom-made human-sized dummy that has been sensorised to be used as the casualty. In terms of safety, we observe several parameters during the experiment including impact force, acceleration, speed and displacement of the dummy’s head. We also compare the performance of the proposed immersive stereoscopic teleperception to conventional monocular teleperception. The results of the experiments show that the observed safety parameters are below key safety thresholds which could possibly lead to head or neck injuries. Moreover, the teleperception comparison results demonstrate an improvement in task-accomplishment performance when the operator is using the immersive teleperception.'
author:
- Roni Permana Saputra
- Petar Kormushev
title: |
ResQbot: A Mobile Rescue Robot\
with Immersive Teleperception\
for Casualty Extraction
---
Introduction
============
Catastrophic events, disasters, or local incidents generate hazardous and unstable environments in which there is an urgent need for timely and reliable intervention, mainly to save lives. A multi-storey building fire disaster—such as the recent Grenfell Tower inferno in London, United Kingdom [@6:grenfell-tower]—is an example of such a scenario. Responding to such situations is a race against time—immediate action is required to reach all potential survivors in time. Such responses, however, are limited to the availability of trained first responders as well as prone to potential risks to their lives. The high risk to the lives of rescue workers means that, in reality, fast response on-site human intervention may not always be a possibility.
Using robots in search-and-rescue (SAR) missions offers a great alternative by potentially minimising the danger for the first responders. Moreover, it is more flexible—the number of these robots can be expanded to perform faster responses. Thus, various robotic designs have been proposed to suit several specific SAR applications [@11:yoo2011military; @2:Murphy:2008SAR]. These robots are designed to assist with one or sometimes multiple tasks as part of a SAR operation—including reconnaissance, exploration, search, monitoring, and excavation.
Our aim in this study is to develop a mobile rescue robot system (see Fig. \[fig:prototype\]a) that is capable of performing a casualty-extraction procedure. This procedure includes loading and transporting a human victim—a.k.a. casualty—smoothly, which is essential for ensuring the victim’s safety, via teleoperation mode. We also aim to develop more immersive teleperception for the robot’s operators to improve their performance and produce higher operation accuracy and safer operations.
\[fig:resqbot-platform\] \[fig:resqbot-tele\]
Related Work
============
Wide-ranging robotics research studies have been undertaken in the area of search, exploration, and monitoring, specifically with applications in SAR scenarios [@3:Shen:2012uavindoor; @4:Waharte:2010supportingUAVs; @5:Goodrich:2008supporting]. Despite the use of the term ’rescue’ in SAR, little attention has been given to the development of a rescue robot that is capable of performing a physical rescue mission, including loading and transporting a victim to a safe zone—a.k.a. casualty extraction.
Several research studies have been conducted to enable the use of robots in the rescue phase of SAR missions. The majority of these studies focused on developing mobile robots—mainly tracked mobile robots—with mounted articulated manipulators [@7:Telemax:online; @8:Schwarz:2017nimbro] or other novel arm mechanisms, such as elephant trunk-like arms [@9:Wolf:2003elephant] or snake-like arms [@2:Murphy:2008SAR]. Such designs enable the mobile robots to move debris by using their manipulators and also perform other physical interventions during the rescue mission, such as a casualty-extraction procedure.
Battlefield Extraction Assist Robot (BEAR) is one of the most sophisticated robot platforms designed and developed specifically for casualty-extraction procedures [@11:yoo2011military; @10:theobald2010mobile]. This robot was developed by Vecna Technologies and intended for the U.S. Army [@12:robotics2010bear]. It has a humanoid form with two independent, tracked locomotion systems. It is also equipped with a heavy-duty, dual-arm system that is capable of lifting and carrying up to 227-kg loads [@13:newatlas2010bear]. This robot is capable of performing casualty-extraction procedures by lifting up and carrying the casualty using its two arms. However, such a procedure could cause additional damage to the already injured victim—such as a spinal or neck injury from the lack of body support during the procedure.
Several investigations have been conducted on the development of rescue robots capable of performing safer and more robust casualty-extraction routines. These robots were designed to be more compact by using stretcher-type constructions or litters [@14:6106797; @15:5981556; @16:6343761; @17:sahashi2011study]. The robots presented in these studies are intended for performing safe casualty extractions with simpler mechanisms compared to arm mechanisms. The use of the stretcher-type design on this robot also ensures the victim’s safety during transportation.
The mobile rescue robot demonstrated by the Tokyo Fire Department is one of the robots that uses this design concept [@18:ota2011robocue; @19:tele-rescue]. This robot is equipped with a belt-conveyor mechanism and also a pair of articulated manipulators. It uses its manipulators to lift up the casualty and place it onto the conveyor during the casualty-extraction procedure [@20:popular-rescue]. Then the belt conveyor pulls up the casualty into the container inside the robot for safe transportation. Compared to BEAR, this robot offers a safer casualty-transportation process. However, lifting up the casualty during the loading process is still a procedure that is highly likely to cause additional damage to the victim.
Iwano et al. in [@14:6106797; @15:5981556; @16:6343761] proposed a mobile rescue robot platform capable of performing casualty extraction without a “lifting” process. This robot loads the casualty merely using a belt-conveyor mechanism. This belt conveyor pulls the casualty from the ground onto the mobile platform, while the mobile platform synchronises the movement toward the body \[15\]. Since there is no “lifting” process during the casualty-extraction procedure, this method is expected to be safer than the methods applied in BEAR and the Tokyo Fire Department’s robot. However, to the best of our knowledge, no safety evaluation of performing this extraction method has been published.
In terms of controlling method, these robots are still manually operated or teleoperated by human operators. The BEAR robot and the rescue robot demonstrated by the Tokyo Fire Department are teleoperated by human operators using a conventional teleoperation control setup [@19:tele-rescue]. On the other hand, based on the reports presented in [@14:6106797; @15:5981556; @16:6343761], the rescue robot developed by Iwano et al. still requires human operators to be present on the scene to perform casualty-extraction procedures.
Research Contributions
======================
In this work, we present a mobile rescue robot we developed that is capable of performing a casualty-extraction procedure using the method presented in [@15:5981556]. Moreover, we equip this robot with teleoperation control and an immersive stereoscopic teleperception. As a proof of concept, we have designed and built a novel mobile rescue robot platform called ResQbot [@21:resqbot] shown in Fig. 1. This robot is equipped with an onboard stereoscopic camera rig to provide immersive teleperception that is transmitted via a virtual-reality (VR) headset to the operator.
The contributions to this work are:
1. Proof-of-concept ResQbot, including teleoperation mode with immersive teleperception via HTC Vive[^1] headset;
2. Preliminary evaluation of loco-manipulation-based casualty extraction using the ResQbot platform, in terms of task accomplishment and safety;
3. Evaluation of the proposed immersive teleperception compared with the other teleperception methods, including the conventional teleperception setup and direct observation as a baseline.
Casualty Extraction via Loco-Manipulation Approach
==================================================
To perform a casualty-extraction procedure using the ResQbot platform, we propose using a loco-manipulation approach. By using this approach, the robot can implicitly achieve a manipulation objective—which is loading of a victim onto the robot—through a series of locomotive manoeuvres. We utilise the conveyor module mounted on the mobile robot base to create a simple mechanism for the loading of a victim onto the conveyor surface by solely following a locomotive routine.
Figure \[fig:method\] illustrates the proposed casualty-extraction operation using the loco-manipulation technique. By removing the need for high-complexity robotic manipulators or mechanisms, this technique greatly simplifies the underlying controls required for conducting complex casualty extraction in rescue missions. This simplicity is highly beneficial for intuitive teleoperation by human operators.
This casualty-extraction procedure involves four major phases:
1. **Relative pose adjustment**: the robot aligns its relative pose with respect to the victim in preparation for performing the loco-manipulation routine.
2. **Approaching**: the robot gently approaches the victim to safely make contact with the victim’s head for initiating the loading process.
3. **Loading**: by using a balance between the locomotion of the base and the motion of the belt conveyor, the robot smoothly loads the victim onboard. Smooth operation at this stage is crucial in order to minimise traumatic injury caused by the operation. The victim is fully onboard when the upper body is fully loaded onto the stretcher bed. We consider the upper body to be from the head to the hip, thus protecting the critical parts of the body, including the head and spinal cord.
4. **Fastening**: once the victim is fully onboard, the strapping mechanism fastens the victim using a stretcher-strap mechanism in preparation for safe transportation. The conveyor surface serves as a stretcher bed for transporting the victim to a safe zone where paramedics can provide further medical care.
![Illustration of casualty extraction via loco-manipulation technique. From the top to the bottom: 1) ResQbot gently approaching the victim, 2) the belt conveyor on the stretcher bed moving at the synchronized speed of the mobile base, 3) the belt conveyor pulling the victim onto stretcher bed while the mobile base move to the opposite direction, 4) the victim fully onboard the stretcher bed.[]{data-label="fig:method"}](loco-manipulation-crop.pdf){width="4.5in"}
Robot Platform
==============
We have designed and built a novel mobile rescue robot platform called ResQbot as a proof of concept for the implementation of the proposed casualty-extraction procedure. Figure 1a shows the ResQbot platform that has been developed in this project. This platform is designed to be able to perform a casualty-extraction task based on the proposed loco-manipulation approach. This platform consists of two main modules: a differential-drive mobile base and a motorised stretcher-bed conveyor module. The platform is also equipped with a range of perception devices, including an RGB-D camera and a stereoscopic camera rig; these devices provide the perception required during the operation. We used an Xbox joystick controller for the operator interface to control the robot. We also proposed an immersive teleperception interface for this mobile robot using an HTC Vive virtual reality headset. This headset will provide teleperception for the operator by displaying a real-time visual image sent from the onboard stereo camera of the mobile robot. Figure \[fig:prototype\]b shows the operator teleoperates ResQbot using the proposed teleoperation and teleperception devices.
Mobile Base Module
------------------
The mobile base module used for the ResQbot platform is a differential drive module. This platform is a customised version of a commercially available powered wheelchair—Quickie Salsa-M—manufactured by Sunrise Medical[^2]. This mobile base is chosen for its versatile design and stability owing to its original design purpose, which was to carry the disabled both indoors and outdoors. This platform has a compact turning circle while ensuring stability and safety through its all-wheel independent suspension and anti-pitch technology over rough or uneven terrain. Its mobile base is also capable of manoeuvring through narrow pathways and confined spaces, due to its compact design (only 600 mm in width).
Motorised Stretcher Bed Module
------------------------------
ResQbot is equipped with an active stretcher-bed module that enables active pulling up of the victim’s body while the mobile platform is moving. This module is mounted at the back of the mobile base via hinges allowing it to fold (for compact navigation) or unfold (for loading and transporting the victim) on demand. This stretcher bed is composed of a belt-conveyor module that is capable of transporting a maximum payload of 100 kg at its maximum power. This belt conveyor is powered by a 240 VDC motor with 500 W maximum power. The motor is controlled through a driver module powered by a 240 VAC onboard power inverter, and the pulse-width modulation (PWM) control signal is used to control the motor’s speed.
During loading of the victim’s body, the active-pulling speed of the belt conveyor has to be synchronised with the locomotion speed of the mobile platform. Therefore, this module is equipped with a closed-loop speed control system to synchronise the conveyor speed and the mobile base locomotion speed. An incremental rotary encoder connected to the conveyor’s pulley is used to provide speed measurements of the belt conveyor as feedback to the controller. Another incremental rotary encoder is connected to an omnidirectional wheel attached to the floor. This encoder provides measurements of the mobile base linear speed. The measured mobile base linear speed is used for the speed reference of the conveyor controller.
For a safe transportation process, the victim has to be safely placed onboard the stretcher bed. Thus, this stretcher-bed module is also equipped with a motorised stretcher strap to enable fastening of the victim on the bed as a safety measure. This stretcher-strap module is powered by a 24 VDC motor. The motor is controlled to fasten and unfasten the stretcher strap during the casualty-extraction procedure.
Experimental Setups and Results
===============================
**Experimental Setting.** We have conducted a number of experimental trials to evaluate our proof-of-concept mobile-rescue-robot platform, ResQbot, in terms of task accomplishment, safety and teleperception comparison. In these experiments, we conducted a mock casualty-extraction procedure using ResQbot by teleoperation with three different teleperception modalities:
- **Direct mode (baseline)**: user controls the robot while being present at the scene;
- **Conventional mode**: user receives visual feedback provided by a monocular camera through a display monitor;
- **Immersive mode**: the user receives stereoscopic vision provided by an onboard stereoscopic camera module and through a virtual reality headset.
Ninety series of trials in total were conducted, with 30 series for each teleperception modality. We conducted this number of trials to capture any possible problems encountered during the trials. For the whole experiment, we used the same setup of the victim and its relative position and orientation with respect to the ResQbot. These various victim positions are inside the area of the ResQbot perception device’s field of view.
To evaluate the safety of the proposed casualty-extraction procedure using the ResQbot platform, we conducted the trials using a sensorised dummy as the casualty. This dummy was equipped with an inertial measurement unit (IMU) sensor placed on its head. We used 3DM-GX4-25[^3] IMU sensor, with resolution $<$ 0.1 mg and bias instability $\pm$ 0.04 mg. During the trials, we recorded the data from this sensor at 100 Hz sampling frequency.
**Task Accomplishment**. In general, we achieved successful task accomplishment of the casualty-extraction procedure in every trial. Screenshot images in Figure \[fig:experiment\] demonstrate the procedure applied during the casualty-extraction operation performed by ResQbot in the experiments.
\[fig:resqbot-platform\] \[fig:resqbot-tele\] \[fig:resqbot-platform\] \[fig:resqbot-platform\]
**Safety Evaluation**. In this experiment, we observed the impact applied to the dummy’s head during the casualty-extraction procedure. This impact caused force and displacement of the dummy’s head. The observation was focused on the part during the loading phase when the robot—i.e. the stretcher-bed conveyor—made first contact with the dummy’s head. Two extreme cases were selected (i.e. roughest and smoothest operation, respectively) for this safety evaluation. These two cases represent the largest and the smallest maximum instant acceleration of each test within the whole experiment.
Figure \[fig:results\] shows the dummy’s head displacement, speed and acceleration during the loading process caused by the robot’s first contact. Two extreme cases are presented; one is the smoothest trial (in blue), and the other one is the roughest trial (in red). The dashed vertical line (in green) indicates the time-step at which the contact was initiated during the loading process.
The forces applied to the dummy’s head can be estimated based on these measured accelerations from the IMU, and it can be calculated via: $$F_i=ma+F_s$$ in which, $F_i$ corresponds to the estimated instantaneous force applied to the dummy’s head, and $F_s$ corresponds to the static friction force between the dummy’s head and the ground [@22:friction]. The estimated force was calculated based on the measured maximum instant acceleration (a) during the operation and the approximated mass of the dummy’s head (m). Table \[tab1\] summarises the observations of the two significant cases—i.e. smoothest and roughest trial—during the experiment. These cases correspond to the maximum instantaneous accelerations of the dummy’s head caused by the robot during the casualty-extraction procedure.
We compared the results presented in table \[tab1\] with several key safety thresholds—which were reported in the literature as possible causes of head or neck injuries to the casualty [@23:Engsberg:2009spinal; @24:eurailsafe:online]. Figure \[fig:summary\] illustrates the comparison between the trial results and the thresholds from the literature. It can be seen that all evaluated parameters in the experiments are relatively small and below the threshold. Even though not conclusive yet, these preliminary results show high safety promise for the proposed platform of the casualty-extraction procedure, and for further development. Thus it also motivates more elaborate safety evaluations for the practical deployment of the platform.
\[fig:resqbot-platform\] \[fig:resqbot-tele\] \[fig:resqbot-platform\]
**Smooth Trial** **Rough Trial**
------------------------------------- -- ------------------ -- -----------------
Max. instant acceleration ($m/s^2$) $\approx 0.154$ $\approx 4.042$
Max. Velocity ($m/s$)
during initial contact $\approx 0.015$ $\approx 0.16$
Victim’s head displacement ($m$) $\approx 0.004$ $\approx 0.051$
Max. impact force ($N$) $\approx 23.63$ $\approx 41.12$
: The summary of two significantly different trials—i.e. smooth and rough operation—performed during the experiment.[]{data-label="tab1"}
{width="4.75in"}
\[fig:summary\]
**Teleperception Comparison**. In this work, we proposed immersive teleperception via an HTC Vive headset to provide visual perception for the operator when operating the ResQbot platform. We evaluated this proposed teleperception modality by comparing it to the conventional teleperception setup (i.e. providing visual perception via single monitor) and the direct-observation scenario as a baseline (i.e. controlling the robot while being present at the scene).
We compared these three perception modes in terms of the smoothness of the casualty-extraction procedure, which is represented by the maximum force applied to the dummy’s head during the experiment in both cases. Figure \[fig:boxplot\] shows a box plot of the distribution of the estimated maximal forces in the three different perception modes during the experiments. According to Figure \[fig:boxplot\], in terms of the smoothness of the procedure, we observed that the VR mode (i.e. immersive) results in a higher population of smoother trials than the conventional mode. In fact, the VR mode achieves smooth operations with low maximal estimated forces under 28 N for approximately 50$\%$ of its trials, similar to the direct-observation mode, in which the operator has direct access to observe the scene during the operation.
{width="3.0in"}
\[fig:boxplot\]
Conclusion
==========
In this paper, we presented a proposed mobile rescue robot system that is capable of safely loading and transporting a casualty. We proposed an immersive stereoscopic teleperception modality via an HTC Vive headset to provide the teleoperator with more realistic and intuitive perception information during the operation. As a proof of concept of the proposed system, we designed and built a novel mobile rescue robot called ResQbot, which is controlled via teleoperation. We evaluated the proposed platform in terms of task accomplishment, safety and teleperception comparison for completing a casualty-extraction procedure. Based on the results of our experiments, the proposed platform is capable of performing a safe casualty-extraction procedure and offers great promise for further development. Moreover, the teleperception comparisons highlight that the proposed immersive teleperception can improve the performance of the teleoperator controlling the mobile robot’s performance during a casualty-extraction procedure.
Acknowledment {#acknowledment .unnumbered}
=============
Roni Permana Saputra would like to thank Indonesia Endowment Fund for Education - LPDP, for the financial support of the PhD program. The authors would also like to show our gratitude to Arash Tavakoli and Nemanja Rakicevic for helpful discussions and inputs for the present work.
[5]{} The Telegraph News: Grenfell Tower inferno a ’disaster waiting to happen’ as concerns are raised for safety of other buildings, http://www.telegraph.co.uk/news/2017/06/14/grenfell-tower-inferno-disaster-waiting-happen-concerns-raised/
Yoo, A. C., Gilbert, G. R., Broderick, T. J.: Military robotic combat casualty extraction and care. In Surgical Robotics (pp. 13-32). Springer, Boston, MA (2011)
Murphy, R. R., Tadokoro, S., Nardi, D., Jacoff, A., Fiorini, P., Choset, H., Erkmen, A. M.: Search and rescue robotics. In Springer Handbook of Robotics. pp. 1151–1173 (2008)
Shen, S., Michael, N., and Kumar, V.: Autonomous indoor 3D exploration with a micro-aerial vehicle. In Robotics and Automation (ICRA), 2012 IEEE International Conference on, pp. 9–15, IEEE (2012)
Waharte, S., and Trigoni, N.: Supporting search and rescue operations with UAVs. In Emerging Security Technologies (EST), 2010 International Conference on, pp. 142–147, IEEE (2010)
Goodrich, Michael A et al.: Supporting wilderness search and rescue using a camera-equipped mini [UAV]{}. J. Field Robot. 25, 89–110 (2008)
Gearin, M.: Remote-controlled robots. http://www.telerob.com/en/products/remote-controlled-robots
Schwarz, Max et al.: NimbRo Rescue: Solving disaster-response tasks with the mobile manipulation robot Momaro. J. Field Robot. 25, 89–110 (2017)
Wolf, A., Brown, H. B., Casciola, R., Costa, A., Schwerin, M., Shamas, E., and Choset, H.: A mobile hyper redundant mechanism for search and rescue tasks. In Intelligent Robots and Systems, 2003.(IROS 2003). Proceedings. 2003 IEEE/RSJ International Conference on, 3, pp. 2889–2895, IEEE (2003)
Theobald, D.: Mobile extraction-assist robot. U.S. Patent No. 7,719,222. Washington, DC: U.S. Patent and Trademark Office (2010)
Vecna Robotics: The BEAR Battlefield Extraction-Assist Robot. (2010)
Quick, D.: Battlefield Extraction-Assist Robot to ferry wounded to safety. https://newatlas.com/battlefield-extraction-assist-robot/17059/
Iwano, Y., Osuka, K., Amano, H.: Development of rescue support stretcher system with stair-climbing. In Safety, Security, and Rescue Robotics (SSRR), 2011 IEEE International Symposium on, pp. 245–250, IEEE (2011)
Iwano, Y., Osuka, K., Amano, H.: Development of rescue support stretcher system. In Safety, Security, and Rescue Robotics (SSRR), 2010 IEEE International Symposium on, pp. 1-6, IEEE (2010)
Iwano, Y., Osuka, K., Amano, H.: Evaluation of rescue support stretcher system. In 2012 IEEE RO-MAN: The 21st IEEE International Symposium on Robot and Human Interactive Communication, pp. 245–250, IEEE (2012)
Sahashi, T., Sahashi, A., Uchiyama, H., Fukumoto, I.: A study of operational liability of the medical rescue robot under disaster. In System Integration (SII), 2011 IEEE/SICE International Symposium on, pp. 1281–1286, IEEE (2011)
Ota, K.: RoboCue, the Tokyo Fire Department’s Rescue-Bot. Popular Science Magazine, pp. 2011–03 (2011)
Tokyo Fire Department: Rescue robot (nicknamed Robochee). http://www.tfd.metro.tokyo.jp/ts/soubi/robo/05.htm
Nosowitz, D.: Meet Japan’s Earthquake Search-and-Rescue Robots. https://www.popsci.com/technology/article/2011-03/six-robots-could-shape-future-earthquake-search-and-rescue (2011)
Saputra, R. P., Kormushev, P.: ResQbot: A Mobile Rescue Robot for Casualty Extraction. In 2018 ACM/IEEE International Conference on Human-Robot Interaction (HRI’18), pp. 239–240, ACM, New York, NY, USA (2018)
Engineers edge: Coefficient of Friction Equation and Table Chart, http://www.engineersedge.com
Engsberg, JR and Standeven, JW and Shurtleff, TL and Tricamo, JM and Landau, WM: Spinal cord and brain injury protection: [T]{}esting concept for a protective device J. Spinal Cord Med. 47, 634–639 (2009)
EURailSafe: Head Injury Criteria Tolerance Levels, www.eurailsafe.net/subsites/operas/HTML/Section3
[^1]: Virtual Reality Head Mounted Display by VIVE: https://www.vive.com/
[^2]: Quickie Salsa-M powered wheelchair by Sunrise Medical: www.sunrisemedical.co.uk
[^3]: LORD MicroStrain IMU: http://www.microstrain.com/inertial/3dm-gx4-25
|
УДК 517.9
**Р. Р. Салимов, Б. А. Клищук** (R. R. Salimov, B. A. Klishchuk), Институт математики НАН Украины, Киев.
**НИЖНИЕ ОЦЕНКИ ДЛЯ ПЛОЩАДИ ОБРАЗА КРУГА**
(**LOWER BOUNDS FOR AREAS OF IMAGES OF DISCS**).
In this article we consider $Q$-homeomorphisms with respect to the p-modulus on the complex plane with $p>2$. It is obtained a lower area estimate for image of discs under such mappings. We solved the extremal problem about minimization of the area functional of images of discs.
В работе рассматриваются $Q$-гомеоморфизмы относительно $p$-модуля на комплексной плоскости при $p>2$. Получена нижняя оценка площади образа круга при таких отображениях. Решена экстремальная проблема о минимизации функционала площади образа круга.
[**1. Введение.**]{} Задача об искажении площадей при квазиконформных отображениях берет свое начало в работе Б.Боярского, см. [@Bo]. Ряд результатов в этом направлении получен в работах [@GR], [@Ast], [@EH].
Впервые верхняя оценка площади образа круга при квазиконформных отображениях встречается в монографии М.А. Лаврентьева, см. [@Lav]. В монографии [@BGMR], см. предложение 3.7, получено уточнение неравенства Лаврентьева в терминах угловой дилатации. Также ранее в работах [@LS] и [@S1] были получены верхние оценки искажения площади круга для кольцевых и нижних $Q$-гомеоморфизмов. В данной работе получены нижние оценки площади образа круга при $Q$-гомеоморфизмах относительно $p$-модуля при $p>2$.
Для простоты изложения ограничимся только плоским случаем. Напомним некоторые определения. Пусть задано семейство $\Gamma$ кривых $\gamma$ в комплексной плоскости ${\Bbb C}$. Борелевскую функцию $\varrho:{\Bbb
C}\to[0,\infty]$ называют [*допустимой*]{} для $\Gamma$, пишут $\varrho\in{\rm adm}\,\Gamma$, если $$\label{eq1.2KR} \int\limits_{\gamma}\varrho(z)\,|dz|\
\geqslant\ 1\qquad\forall\ \gamma\in\Gamma.$$ Пусть $p\in (1,\infty)$. Тогда [*$p$–модулем*]{} семейства $\Gamma$ называется величина $$\label{eq1.3KR}\mathcal{M}_p(\Gamma)\ =\
\inf_{\varrho\in\mathrm{adm}\,\Gamma}\int\limits_{{\Bbb
C}}\varrho^p(z)\,dm(z)\, .$$
Предположим, что $D$ — область в комплексной плоскости $\mathbb{C}$, т.е. связное открытое подмножество $\mathbb{C}$ и $Q: D\rightarrow [0,\infty]$ — измеримая функция. Гомеоморфизм $f: D \rightarrow \mathbb{C}$ будем называть $Q$-гомеоморфизмом относительно $p$-модуля, если
$$\label{eq1.4KR}\mathcal{M}_p(f\Gamma)\ \leqslant\
\int\limits_{{D}}Q(z)\,\varrho^{p}(z)\,dm(z)\,$$
для любого семейства $\Gamma$ кривых в $D$ и любой допустимой функции $\varrho$ для $\Gamma$.
Исследование неравенств типа (\[eq1.4KR\]) при $p=2$ восходит к Л. Альфорсу (см., напр., теорему 3, разд. D, гл. I, [@A]), а также О. Лехто и К. Вертанену (см. неравенство (6.6), разд. 6.3, гл. V в [@LV]). В работе В.Я. Гутлянского (совместно с К. Бишопом, О. Мартио и М. Вуориненом) доказан многомерный аналог неравенства (\[eq1.4KR\]) для квазиконформных отображений (см. [@BiGMV]).
Отметим также, что если в (\[eq1.4KR\]) функцию Q считать ограниченной п.в. некоторой постоянной $K\in [1,\infty)$ и $p = 2$, то мы приходим к классическим квазиконформным отображениям, которые были впервые введены в работах Грётча, Лаврентьева и Морри.
Пусть $Q:D\to[0,\infty]$ — измеримая функция. Для любого числа $r>0$ обозначим
$$q_{z_0}(r)=
\frac{1}{2\pi\, r}\int\limits_{S(z_0, r)}Q(z)\, |dz|$$ — среднее интегральное значение функции $Q$ по окружности $S(z_0,r)=\{z\in
\mathbb{C}: \, |z-z_0| = r\}$.
[**Теорема 1.**]{}
*Пусть $D$ и $D'$ — ограниченные области в $\Bbb C$ и $f: D \rightarrow D' $ — $Q$-гомеоморфизм относительно $p$-модуля, $p>2$, $Q \in L^{1}_{\rm
loc}(D \setminus\{z_{0}\})$. Тогда при всех $r\in (0, d_0)$, $d_0 = {\rm
dist}(z_{0}, \partial D)$, имеет место оценка*
$$\label{a1*}
|fB(z_0,r)| \geqslant
\pi\,\left(\frac{p-2}{p-1}\right)^{\frac{2(p-1)}{p-2}}
\left(\int\limits_{0}^{r}\frac{dt}{t^\frac{1}{p-1}\,q_{z_0}^{\frac{1}{p-1}}(t)}\right)^{\frac{2(p-1)}{p-2}}\,,$$
где $B(z_0,r)=\{z\in \mathbb{C}: |z-z_0|\leqslant r\}\,$.
Отметим, что при $p>2$ и $Q(z)\leqslant K$ из теоремы 1 мы приходим к результату для круга из работы [@Ge], см. лемму 7.
[**3. Доказательство основной теоремы 1.**]{} Приведем некоторые вспомогательные сведения о емкости конденсатора. Следуя работе [@MRV], пару $\mathcal{E}=(A,C)$, где $A\subset\mathbb{C}$ — открытое множество и $C$ — непустое компактное множество, содержащееся в $A$, называем [*конденсатором*]{}. Конденсатор $\mathcal{E}$ называется [*кольцевым конденсатором*]{}, если $\mathfrak{R}=A\setminus C$ — кольцевая область, т.е., если $\mathfrak{R}$ — область, дополнение которой $\overline{\mathbb{C}}\setminus \mathfrak{R}$ состоит в точности из двух компонент. Конденсатор $\mathcal{E}$ называется [*ограниченным конденсатором*]{}, если множество $A$ является ограниченным. Говорят также, что конденсатор $\mathcal{E}=(A,C)$ лежит в области $D$, если $A\subset D$. Очевидно, что если $f:D\to\mathbb{C}$ — непрерывное, открытое отображение и $\mathcal{E}=(A,C)$ — конденсатор в $D$, то $(fA,fC)$ также конденсатор в $fD$. Далее $f\mathcal{E}=(fA,fC)$.
Пусть $\mathcal{E}=(A,C)$ — конденсатор. Обозначим через $\mathcal{C}_0(A)$ множество непрерывных функций $u:A\to\mathbb{R}^1$ с компактным носителем. $\mathcal{W}_0(\mathcal{E})=\mathcal{W}_0(A,C)$ — семейство неотрицательных функций $u:A\to\mathbb{R}^1$ таких, что 1) $u\in \mathcal{C}_0(A)$, 2) $u(x)\geqslant1$ для $x\in C$ и 3) $u$ принадлежит классу ${\rm
ACL}$. При $p \geqslant1$ величину $$\label{eqks2.6}{\rm cap}_p\,\mathcal{E}={\rm cap}_p\,(A,C)=\inf\limits_{u\in \mathcal{W}_0(\mathcal{E})}\,
\int\limits_{A}\,\vert\nabla u\vert^p\,dm(z)\,,$$ где $$\label{eqks2.5}\vert\nabla
u\vert= \sqrt{\left(\frac{\partial u}{\partial
x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2}\,
%\sqrt{ \left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}}\right)^2}\,.$$ называют [*$p$-ёмкостью*]{} конденсатора $\mathcal{E}$. В дальнейшем мы будем использовать установленное в работе [@Sh] равенство $$\label{EMC}
{\rm cap}_p\,\mathcal{E}=\mathcal{M}_p(\Delta(\partial A,\partial C; A\setminus
C)),$$ где для множеств $\mathcal{F}_1$, $\mathcal{F}_2$ и $\mathcal{F}$ в $\mathbb{C}$, $\Delta(\mathcal{F}_1,\mathcal{F}_2;\mathcal{F})$ обозначает семейство всех непрерывных кривых, соединяющих $\mathcal{F}_1$ и $\mathcal{F}_2$ в $\mathcal{F}$.
Известно, что при $p\geqslant1$, см. предложение 5 из [@Kru], $$\label{eqks2.8} {\rm
cap}_p\,\mathcal{E}\geqslant\frac{\left[\inf
l(\sigma)\right]^p}{|A\setminus C|^{p-1}}\,.$$ Здесь $l(\sigma)$ — длина гладкой (бесконечно дифференцируемой) кривой $\sigma$, которая является границей $\sigma=\partial U$ ограниченного открытого множества $U$, содержащего $C$ и содержащегося вместе со своим замыканием $\overline{U}$ в $A$, а точная нижняя грань берется по всем таким $\sigma$.
[*Доказательство теоремы 1.*]{} Пусть $\mathcal{E}=\left(A, C\right)$ — конденсатор, где $A=\{z\in
D: |z-z_{0}|<t+\Delta t\}$, $C=\{z\in D: |z-z_{0}|\leqslant t\}$,$t+\Delta t < d_{0}$. Тогда $f\mathcal{E} = \left(fA,fC\right)$ — кольцевой конденсатор в $D^{\prime}$ и согласно (\[EMC\]) имеем равенство
$$\label{a2} {\rm cap}_{p}\, f\mathcal{E}= \mathcal{M}_{p}\left(\Delta(\partial fA, \partial fC; f(A\setminus C)\right).$$
В силу неравенства (\[eqks2.8\]) получим $$\label{a3}{\rm cap}_{p}\, f\mathcal{E} \geqslant \frac{\left[\inf\
l (\sigma)\right]^{p}}{|fA\setminus fC|^{p-1}}\,.$$ Здесь $l(\sigma)$ — длина гладкой (бесконечно дифференцируемой) кривой $\sigma$, которая является границей $\sigma=\partial U$ ограниченного открытого множества $U$, содержащего $C$ и содержащегося вместе со своим замыканием $\overline{U}$ в $A$, а точная нижняя грань берется по всем таким $\sigma$.
С другой стороны, в силу определения $Q$-гомеоморфизма относительно $p$-модуля, имеем
$$\label{a4} {\rm cap}_{p}\, f\mathcal{E} \leqslant \int\limits
_{D}Q(z)\,\varrho ^{p}(z)\,dm(z)$$
для любой $\varrho\in{\rm adm}\ \Delta(\partial A, \partial C; A\setminus C).$
Легко проверить, что функция $$\varrho(z)= \begin{cases} \frac{1}{|z-z_{0}|\,\ln\frac{t+\Delta t}{t}},& z \in A\setminus C\\
0,& z \not\in A\setminus C \end{cases}$$ является допустимой для семейства $\Delta(\partial A, \partial C; A\setminus C)$ и поэтому
$$\label{a5} {\rm cap}_{p}\, f\mathcal{E} \leqslant
\frac{1}{\ln^{p}\left(\frac{t+\Delta t}{t}\right)}
\int\limits_{R}\frac{Q(z)}{|z-z_{0}|^{p}}\,dm(z),$$
где $R = \{z\in D: t\leqslant |z-z_{0}|\leqslant t+\Delta t\}$.
Комбинируя неравенства (\[a3\]) и (\[a5\]), получим
$$\label{a6}
%\begin{split}
\frac{\left[\inf\ l (\sigma)\right]^{p}}{|fA\setminus fC|^{p-1}}
\leqslant \frac{1}{\ln^{p}\left(\frac{t+\Delta
t}{t}\right)}\int\limits _{R}\frac{Q(z)}{|z-z_{0}|^{p}}\,dm(z).
%\end{split}$$
По теореме Фубини имеем $$\label{a7} \int\limits _{R}\frac{Q(z)}{|z-z_{0}|^{p}}\,dm(z) =
\int\limits_{t}^{t+\Delta
t}\frac{d\tau}{\tau^{p}}\int\limits_{S(z_{0},\tau)}Q(z)\,|dz| = 2\pi
\int\limits_{t}^{t+\Delta t} \tau^{1-p}\,q_{z_{0}}(\tau)\,d\tau,
%\end{center}$$ где $q_{z_{0}}(\tau) = \frac{1}{2\pi \tau}\, \int\limits_{S(z_0,
\tau)}\, Q(z)\, |dz|$ и $S(z_0,\tau)=\{z\in \mathbb{C}: |z-z_0| =
\tau\}$. Таким образом,
$$\label{a8} \inf\ l (\sigma) \leqslant
(2\pi)^{\frac{1}{p}}\,\frac{|fA\setminus
fC|^{\frac{p-1}{p}}}{\ln\left(\frac{t+\Delta
t}{t}\right)}\left[\int\limits_{t}^{t+\Delta t}
\tau^{1-p}\,q_{z_{0}}(\tau)\,d\tau\right]^{\frac{1}{p}}.$$
Далее, воcпользовавшись изопериметрическим неравенством $$\label{a9} \inf\ l(\sigma) \geqslant 2 \sqrt{\pi
|fC|},$$ получим $$\label{a10} 2 \sqrt{\pi \, |fC|}
\leqslant(2\pi)^{\frac{1}{p}}\,\frac{|fA\setminus
fC|^{\frac{p-1}{p}}}{\ln\left(\frac{t+\Delta
t}{t}\right)}\left[\int\limits_{t}^{t+\Delta t}
\tau^{1-p}\,q_{z_{0}}(\tau)\,d\tau\right]^{\frac{1}{p}}.$$
Определим функцию $\Phi(t)$ для данного гомеоморфизма $f$ следующим образом $$\label{b1}
\Phi(t) = |fB(z_{0},t)|,$$ где $B(z_{0},t) = \{z\in
\mathbb{C}: |z - z_{0}|\leqslant t\}$. Тогда из соотношения (\[a10\]) следует, что $$\label{a11} 2 \sqrt{\pi \, \Phi(t)} \leqslant
(2\pi)^{\frac{1}{p}}\,\frac{[\frac{\Phi(t+\Delta t)-\Phi(t)}{\Delta
t}]^{\frac{p-1}{p}}}{\frac{\ln(t+\Delta t)-\ln t}{\Delta
t}}\left[\frac{1}{\Delta t}\int\limits_{t}^{t+\Delta t}
\tau^{1-p}\,q_{z_{0}}(\tau)\,d\tau\right]^{\frac{1}{p}}.$$
Устремляя в неравенстве (\[a11\]) $\Delta t\to 0$, и учитывая монотонное возрастание функции $\Phi$ по $t \in (0,d_{0})$, для п.в. $t$ имеем:
$$\label{a12}
\frac{2\pi^{\frac{p-2}{2(p-1)}}}{t^\frac{1}{p-1}\,q_{z_{0}}^{\frac{1}{p-1}}(t)}
\leqslant \frac{\Phi'(t)}{\Phi^{\frac{p}{2(p-1)}}(t)}.$$
Отсюда легко вытекает следующее неравенство: $$\label{a13}
\frac{2\pi^{\frac{p-2}{2(p-1)}}}{t^\frac{1}{p-1}\,q_{z_{0}}^{\frac{1}{p-1}}(t)}\,
\leqslant
\left(\frac{\Phi^{\frac{p-2}{2(p-1)}}(t)}{\frac{p-2}{2(p-1)}}\right)^{\prime}.$$
Поскольку $p>2$, то функция $g(t)=\frac{\Phi^{\frac{p-2}{2(p-1)}}(t)}{\frac{p-2}{2(p-1)}}$ является неубывающей на $(0, d_0)$, где $d_0 ={\rm dist}(z_{0},
\partial D)$. Интегрируя обе части неравенства по $t \in
[\varepsilon,r]$ и учитывая, что $$\int\limits_{\varepsilon}^{r} \left(\frac{\Phi^{\frac{p-2}{2(p-1)}}(t)}{\frac{p-2}{2(p-1)}}\right)^{\prime} dt =\int\limits_{\varepsilon}^{r} \ g'(t)\, dt \leqslant g(r)-g(\varepsilon) \leqslant
\frac{\Phi^{\frac{p-2}{2(p-1)}}(r)-\Phi^{\frac{p-2}{2(p-1)}}(\varepsilon)}{\frac{p-2}{2(p-1)}}\,,$$ см., напр., теорему IV. 7.4 в [@Sa], получаем $$\label{a14}
2\pi^{\frac{p-2}{2(p-1)}}\,\int\limits_{\varepsilon}^{r}\frac{dt}{t^\frac{1}{p-1}q_{z_{0}}^{\frac{1}{p-1}}(t)}
\leqslant
\frac{\Phi^{\frac{p-2}{2(p-1)}}(r)-\Phi^{\frac{p-2}{2(p-1)}}(\varepsilon)}{\frac{p-2}{2(p-1)}}\,.$$ Устремляя в неравенстве (\[a14\]) $\varepsilon\to 0$, приходим к оценке $$\label{a15} \Phi(r) \geqslant \pi\,
\left(\frac{p-2}{p-1}\right)^{\frac{2(p-1)}{p-2}}
\left(\int\limits_{0}^{r}\frac{dt}{t^\frac{1}{p-1}\,q_{z_{0}}^{\frac{1}{p-1}}(t)}\right)^{\frac{2(p-1)}{p-2}}.$$ Наконец, обозначая в последнем неравенстве $\Phi(r) =
|fB(z_{0},r)|$, имеем $$\label{a16} |fB(z_{0},r)| \geqslant
\pi\,\left(\frac{p-2}{p-1}\right)^{\frac{2(p-1)}{p-2}}
\left(\int\limits_{0}^{r}\frac{dt}{t^\frac{1}{p-1}\,q_{z_{0}}^{\frac{1}{p-1}}(t)}\right)^{\frac{2(p-1)}{p-2}}$$ и тем самым завершаем доказательство теоремы 1.
[**3. Следствия из теоремы 1.**]{} Из теоремы 1 непосредственно вытекают следующие утверждения.
Воспользовавшись условием $q_{z_{0}}(t) \leqslant q_{0}\,t^{-\alpha}$, оценим правую часть неравенства (\[a1\*\]) и проведя элементарные преобразования приходим к следующему результату.
[**Следствие 1.**]{}
*Пусть $D$ и $D'$ — ограниченные области в $\Bbb C$ и $f: D \rightarrow D' $ — $Q$-гомеоморфизм относительно $p$-модуля при $p>2$. Предположим, что функция $Q$ удовлетворяет условию*
$$\label{b2J}
q_{z_{0}}(t) \leqslant q_{0}\,t^{-\alpha},\, q_{0} \in (0, \infty)\,,\, \alpha \in [0, \infty)$$
для $z_{0}\in D$ и п.в. всех $t\in (0, d_0)$, $d_0 = {\rm
dist}(z_{0}, \partial D)$. Тогда при всех $r\in (0, d_0)$ имеет место оценка
$$\label{a1p}
|fB(z_0,r)| \geqslant
\pi^{-\frac{\alpha}{p-2}}\, \left(\frac{p-2}{\alpha+p-2}\right)^{\frac{2(p-1)}{p-2}}q_{0}^{\frac{2}{2-p}} \, |B(z_0,r)|^{1+\frac{\alpha}{p-2}}\,.$$
В частности, полагая здесь $\alpha=0$, получаем следующее заключение.
[**Следствие 2.**]{} [*Пусть $D$ и $D'$ — ограниченные области в $\Bbb C$ и $f: D \rightarrow D' $ — $Q$-гомеоморфизм относительно $p$-модуля, $p>2$ и $q_{z_0}(t) \leqslant q_{0}<\infty$ для п.в. $t \in (0, \, d_0)$, $d_0 = {\rm
dist}(z_{0}, \partial D)$. Тогда имеет место оценка $$\label{a1}
|fB(z_0, r)| \geqslant q_{0}^{\frac{2}{2-p}}\,|B(z_0,r)|$$ для всех $r \in (0,d_0)$.* ]{}
Следствие 3 является частным случаем результата Геринга для $E = B(z_0,r)$, см. лемму 7 в [@Ge].
[**Следствие 4.**]{}
*Пусть $f: \mathbb{B} \rightarrow \mathbb{B}$ — $Q$-гомеоморфизм относительно $p$-модуля при $p>2$. Предположим, что функция $Q(z)$ удовлетворяет условию*
$$\label{b2}
q(t) \leqslant \frac{q_{0}}{t\,\ln^{p-1}\frac{1}{t}},\, q_{0} \in (0, \infty)\,,$$
при п.в. всех $t\in (0, 1)$, где $q(t)=\frac{1}{2\pi t}\, \int\limits_{S_{t}}\, Q(z)\, |dz|$ — среднее интегральное значение над окружностью $S_{t}=\{z\in
\mathbb{C}: \, |z| = t\}$. Тогда при всех $r\in (0, 1)$ имеет место оценка
$$\label{a1}
|fB_{r}| \geqslant
\pi\, \left(\frac{p-2}{p-1}\right)^{\frac{2(p-1)}{p-2}}\, q_{0}^{\frac{2}{2-p}} \, \left(r\ln\frac{e}{r}\right)^{\frac{2(p-1)}{p-2}},\,$$
где $B_{r}=\{z\in \mathbb{C}: |z|\leqslant r\}.\,$
Пусть $Q: \mathbb{B}\rightarrow [0,\infty]$ — измеримая функция, удовлетворяющая условию $$\label{b20}
q(t) \leqslant q_{0}\,,\, q_{0} \in (0, \infty)$$ при п.в. $t\in (0, 1)$, где $q(t)=\frac{1}{2\pi t}\, \int\limits_{S_{t}}\, Q(z)\, |dz|$ — среднее интегральное значение над окружностью $S_{t}=\{z\in
\mathbb{C}: \, |z| = t\}$.
Пусть $\mathcal{H}=\mathcal{H}(q_{0}, p, \mathbb{B})$ — множество всех $Q$-гомеоморфизмов $f: \mathbb{B} \rightarrow \mathbb{C}$ относительно $p$-модуля при $p>2$ с условием (\[b20\]). Рассмотрим на классе $\mathcal{H}$ функционал площади
$$\label{b3}
\mathbf{S}_{r}(f)= |fB_{r}|\,.$$
[*Доказательство.*]{} В силу следствия 1 немедленно вытекает оценка
$$\label{b00*} \mathbf{S}_{r}(f)\geqslant \, \pi \, q_{0}^{\frac{2}{2-p}} \, r^{2} \,.$$
Укажем гомеоморфизм $f \in \mathcal{H}$ на котором реализуется минимум функционала $\mathbf{S}_r(f)$. Пусть $f_0:\mathbb{B} \to
\mathbb{C}$, где
$$f_0(z)=q_{0}^{\frac{1}{2-p}} \, z \,$$
Очевидно, что равенство в (\[b00\*\]) достигается на отображении $f_0$. Осталось показать, что отображение, определенное таким образом, является $Q$-гомеоморфизмом относительно $p$-модуля с $Q(z)=q_0$. Действительно,
$$l(z,f_0)=L(z,f_0)= q_{0}^{\frac{1}{2-p}}\,, \, \quad J(z,f_0)=q_{0}^{\frac{2}{2-p}}$$
и $$K_{I,\,p}(z,f_0)=\frac{J(z,f_0)}{l^p(z,f_0)}=q_0 \,.$$
По теореме 1.1 из работы [@SS] отображение $f_0$ является $Q$-гомеоморфизмом относительно $p$-модуля с $Q(z)= K_{I,\,p}(z,f_0)=q_0$.
[99]{}
Гомеоморфные решения систем Бельтрами // ДАН СССР, 102 (1955), 661–664.
Area distortion under quasiconformal mappings // Ann. Acad. Sci. Fenn. Ser. A I Math. 388, 1966, 1–15.
Area dislortron of quasrconformal mapprngs // Acta Math. 173 ( 1994). 37-60.
On the area distortion by quasiconformal mappings // Proc. Amer. Math. Soc.. 123 (1995). 2793-2797.
Вариационный метод в краевых задачах для систем уравнений эллиптического типа. — Москва. — 1962. — 136 с.
Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane. EMS Tracts in Mathematics, 19. European Mathematical Society (EMS), Zьrich, 2013. x+205 pp. ISBN: 978-3-03719-122-4
, К теории экстремальных задач // Збірник праць Ін-ту математики НАНУ. – 2010. – Т. 7, №2. – С.264–269
, Нижние оценки p-модуля и отображения класса Соболева // Алгебра и анализ, 26:6 (2014), 143–171
Альфорс Л. Лекции по квазиконформным отображениям // Москва: Мир, 1969. – 133 с.
Lehto O. Quasiconformal Mappings in the Plane // New York etc.: Springer, 1973. – 258 p.
On conformal dilatation in space // Intern. J. Math. and Math. Scie. – 2003. – V. 22. – P. 1397–1420.
, Lipschitz mappings and the $p$-capacity of ring in $n$-space // Advances in the theory of Riemann surfaces (Proc. Conf. Stonybrook, N.Y., 1969), Ann. of Math. Studies. – 1971. – **66**. – P. 175–193.
, Definitions for quasiregular mappings // Ann. Acad. Sci. Fenn. Ser. A1. Math. – 1969. – **448**. – P. 1–40.
, On the equality between $p$-capacity and $p$-modulus // Sibirsk. Mat. Zh. – 1993. – **34**, no. 6. – 216-221.
, Ёмкости конденсаторов и пространственные отображения, квазиконформные в среднем // Матем. сб. – 1986. – **130**, № 2. – C. 185-206.
, Теория интеграла. – Издательство ИЛ, М., 1949. – 495 с.
The Poletskii and Vaisala inequalities for the mappings with (p,q)-distortion // Complex Variables and Elliptic Equations. - V. 59, no. 2- 2014. - P. 217 - 231.
Авторы: **Руслан Радикович Салимов, Богдан Анатольевич Клищук**
**Институт математики НАН Украины, Киев**
E-mail: **ruslan623@yandex.ru, bogdanklishchuk@mail.ru**
|
---
abstract: 'One of the most interesting puzzles in particle physics today is that new physics is expected at the TeV energy scale to solve the hierarchy problem, and stabilise the Higgs mass, but so far no unambiguous signal of new physics has been found. Strong constraints on the energy scale of new physics can be derived from precision tests of the electroweak theory and from flavour-changing or $C\!P$-violating processes in strange, charm and beauty hadron decays. Decays that proceed via flavour-changing-neutral-current processes are forbidden at the lowest perturbative order in the Standard Model and are, therefore, [*rare*]{}. Rare $b$ hadron decays are playing a central role in the understanding of the underlying patterns of Standard Model physics and in setting up new directions in model building for new physics contributions. In this article the status and prospects of this field are reviewed.'
address:
- 'Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom'
- |
Laboratori Nazionali di Frascati - INFN\
via E. Fermi 40 - 00044 Frascati (Rome) Italy
- 'Excellence Cluster Universe, TUM, Boltzmannstr. 2, 85748 Garching, Germany'
author:
- Thomas Blake
- Gaia Lanfranchi
- 'David M. Straub'
bibliography:
- 'bibliography.bib'
title: |
Rare $B$ Decays\
as\
Tests of the Standard Model
---
Flavor Physics, B Physics, Rare Decays, Standard Model, New Physics
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank W. Altmannshofer, C. Bobeth, A. Buras, T. Gershon, C. Langenbruch, and V. Vagnoni for helpful discussion and valuable feedback on the manuscript. T.B. and G.L. would like to thank the members of the LHCb collaboration for their productive collaboration that led to some of the results discussed in this article. This work is supported in part by the Royal Society (T.B.), the INFN (G.L.), and by the DFG cluster of excellence “Origin and Structure of the Universe” (D.S.).
|
---
abstract: 'We observe several interesting phenomena in a technicolor-like model of electroweak symmetry breaking based on the D4-D8-[$\overline{{\rm D}8}\ $]{}system of Sakai and Sugimoto. The benefit of holographic models based on D-brane configurations is that both sides of the holographic duality are well understood. We find that the lightest technicolor resonances contribute negatively to the Peskin-Takeuchi $S$-parameter, but heavy resonances do not decouple and lead generically to large, positive values of $S$, consistent with standard estimates in QCD-like theories. We study how the $S$ parameter and the masses and decay constants of the vector and axial-vector techni-resonances vary over a one-parameter family of D8-brane configurations. We discuss possibilities for the consistent truncation of the theory to the first few resonances and suggest some generic predictions of stringy holographic technicolor models.'
author:
- 'Christopher D. Carone'
- Joshua Erlich
- Marc Sher
date: April 2007
title: ' Holographic Electroweak Symmetry Breaking from D-branes 0.1in'
---
Introduction {#sec:intro}
============
In recent years, the holographic relationship between strongly-coupled field theories and higher-dimensional theories that include gravity (the AdS/CFT correspondence [@AdSCFT]) has allowed quantitative predictions for observables in QCD that are in surprisingly good agreement with the experimental data [@AdSQCD1; @AdSQCD2; @AdSQCD3; @Erdmenger; @Evans]. Encouraged by the success of such models, a similar approach to studying technicolor-like models of electroweak symmetry breaking (EWSB) based on the AdS/CFT correspondence has led to new possibilities for physics that may be probed at the Large Hadron Collider (LHC) [@Holo-EWSB1; @Holo-EWSB2; @CET; @piai2]. Holographic technicolor models are closely related to extra-dimensional models of electroweak symmetry breaking that have been studied in much detail recently [@EWSB-models], but their philosophy is different. In the AdS/CFT approach, the form of the 5D holographic theory is suggested by the mechanisms of chiral symmetry breaking and confinement as they appear in string theory constructions of models similar to QCD.
One benefit of holographic models rooted in D-brane configurations of string theory, as opposed to in the phenomenological models of Refs. [@Holo-EWSB1; @Holo-EWSB2; @CET; @piai2], is that the large-$N$ and curvature corrections, for example, are well defined and can, in principle, be calculated or estimated systematically. Furthermore, the gauge theories described by low-energy fluctuations of D-brane configurations are known, so that both sides of the holographic duality are well understood. The stringy model of this sort most similar to the light quark sector of QCD is the D4-D8-[$\overline{{\rm D}8}\ $]{}model of Sakai and Sugimoto [@SS1; @SS2], which we review in Section \[sec:SS\]. The Sakai-Sugimoto model is nonsupersymmetric, contains $N_f$ flavors of massless quarks transforming in the fundamental representation of the SU$(N)$ gauge group, and is confining with non-Abelian chiral symmetry breaking. (A related model based on D6-branes rather than D8-branes does not have this last feature [@Mateos].) At large $N$ and large ’t Hooft coupling $(g^2N)$ with $N_f\ll N$, the spectrum of resonances, their decay widths and couplings can be reliably calculated [@SS1; @SS2].
Minimal technicolor models are essentially scaled up versions of QCD, in which an SU(2)$\times$U(1) subgroup of the chiral symmetry is gauged. Chiral symmetry breaking due to the strong technicolor interactions then translates into electroweak symmetry breaking. A nice feature of technicolor models is that they avoid the hierarchy problem associated with scalar Higgs fields. To produce fermion masses, however, the technicolor sector must be extended, and it is challenging to accommodate a heavy top quark in such models while avoiding large flavor-changing neutral current effects. Also disappointing is that corrections to precision electroweak observables are estimated to be generically too large to be consistent with LEP data [@PT]. Walking technicolor models, with couplings that run more slowly than in QCD, provide one approach to solving these problems (see, for example, Refs. [@walking]).
If the electroweak subgroup of the chiral symmetry in the Sakai-Sugimoto model is gauged, then the model becomes a technicolor-like model of electroweak symmetry breaking. The Sakai-Sugimoto model has, in addition to $N$, $N_f$ and the ’t Hooft coupling, a free parameter $U_0$, not present in QCD, that fixes the D8-[$\overline{{\rm D}8}\ $]{}brane configuration. (The definition of $U_0$ is given in Section \[sec:SS\].) We can therefore hope that for some choice of $U_0$ the model will be consistent with precision electroweak constraints. Indeed, in phenomenological holographic technicolor models, additional parameters not present in QCD allow for corrections to precision electroweak observables that render them consistent with current bounds [@Holo-EWSB1; @Holo-EWSB2; @CET; @piai; @agashe]. The Sakai-Sugimoto model also includes a larger group of symmetries (a global SO(5) symmetry) and states (Kaluza-Klein modes around a circle of size comparable to the confining scale) not present in QCD. The existence of these additional states may be considered a prediction of the model. We find it interesting that in calculable string theory models with chiral symmetry breaking it seems to be difficult to separate additional physics from the confining scale. In our adaptation of the Sakai-Sugimoto model to EWSB, one expects technicolor-like resonances to be discovered together with Kaluza-Klein modes of a large extra dimension.
The most serious problem for old-fashioned technicolor models is the generically large value of the Peskin-Takeuchi $S$-parameter, which parameterizes a class of corrections to precision electroweak observables [@PT]. In Section \[sec:Modes\], we calculate several of the lightest vector and axial vector resonance masses, their decay constants, and their contribution to the $S$-parameter, as a function of $U_0$ (the free parameter in the model) for fixed $g^2N$ and $N$. We find that the contribution of the lightest resonances to $S$ is negative, which would seem to make such models promising candidates for a theory of EWSB. However, we also sum over all modes via a holographic sum rule and demonstrate a surprising non-decoupling of heavy resonances. The non-decoupling of contributions to the $S$-parameter is a consequence of the rapid growth of the decay constants as a function of resonance number, which offsets the suppression from the increase in resonance masses. Including the effects of the entire tower of resonances, we find that the $S$-parameter in the model is generally large and positive.
The decoupling of the heavier resonances can be recovered if we introduce an artificial ultraviolet (UV) regulator that brings the D8-brane boundary in from infinity. Although only the first few resonances are required in this limit to accurately compute $S$, we then find that their contribution typically becomes positive and large. For regulator scales lower than the weak scale, we show that it is possible to obtain positive values of $S$ consistent with current bounds. We discuss the phenomenological implications of these results, and other approaches to decreasing the $S$ parameter, in Section \[sec:Conclusions\].
The D4-D8-$\mathbf{\overline{D8}}$ System {#sec:SS}
=========================================
The D4-D8-[$\overline{{\rm D}8}\ $]{}system of Sakai and Sugimoto [@SS1; @SS2] is similar to QCD in many ways. If there are $N$ D4 branes and $N_f$ sets of D8 and [$\overline{{\rm D}8}\ $]{}branes, then the D-brane configuration describes an SU($N$) gauge theory coupled to $N_f$ flavors of fermions in the fundamental representation, which we will refer to as either quarks or techniquarks depending on the context. The D4 branes are wrapped on a circle; antiperiodic boundary conditions for the fermions break supersymmetry and lift the masses of the extraneous (adjoint) fermions. The quarks experience chiral symmetry breaking and confinement, as is qualitatively understood by properties of the D8-brane configuration and the background geometry. Properties of the hadronic bound states (masses, decay rates and couplings) in this system are calculable, as we review here.
The D4 brane geometry, in the notation of [@SS1] (except for the signature of the metric), is given by, $$ds^2=\left(\frac{U}{R}\right)^{3/2}\left(\eta_{\mu\nu}dx^\mu dx^\nu-f(U)
d\tau^2\right)- \left(\frac{R}{U}\right)^{3/2}\left(\frac{dU^2}{f(U)}+
U^2\,d\Omega_4^2\right),
\label{eq:themetric}$$ where $$f(U)=1-\frac{U_{KK}^3}{U^3} \,\,\,\, \mbox{ and } \,\,\,\, \quad R^3=\pi g_sN l_s^3.$$ The D4 brane extends in the $x^\mu, \mu=0,1,2,3$ and $\tau$ directions, where the $\tau$ direction is compactified on a circle. In the remaining dimensions, the metric is spherically symmetric: $U$ is the radial coordinate and $d\Omega_4^2$ is the metric of the unit 4-sphere. The scale $U_{KK}$ is a free parameter, but to avoid a singularity at $U=U_{KK}$ the variable $\tau$ is periodic with period $4\pi R^{3/2}/(3 U_{KK}^{1/2})$. We will refer to the geometry projected onto the $\tau$ and $U$ directions as the $U$-tube, illustrated in Fig. \[fig:U-tube\]. The D8 and [$\overline{{\rm D}8}\ $]{}branes extend in the $x^\mu$ and 4-sphere directions, and follow a minimal energy trajectory $U(\tau)$ on the $U$-tube. Physical results will depend on $R$ and $U_{KK}$ in the combination, $$\frac{R^3}{U_{KK}}\equiv \frac{9}{4}M_{KK}^{-2},$$ so that there is a single dimensionful scale, $M_{KK}$, governing the dynamics of the model. The boundary of the spacetime is at $U=\infty$, which is topologically $S^1\times S^4\times M^4$, where $M^4$ is 3+1 dimensional Minkowski space (with the Lorentz invariance of the 4D theory). The dilaton appears in the Dirac-Born-Infeld (DBI) action governing the dynamics of the D8 branes; in the D4 brane background the dilaton profile is, $$e^\phi=g_s\left(\frac{U}{R}\right)^{3/4}.$$ The RR four-form is also turned on in the D4-brane background, but we will not need its profile in what follows.
Confinement in the model is related to the termination of the geometry at $U_{KK}$, [*i.e.*]{} the restriction $U\geq U_{KK}$. One way to understand confinement in this context is to consider a string stretched from the boundary at one point in $M^4$ to another as a function of separation of the string endpoints. The minimum energy string extends away from the boundary, and for large enough separation between string endpoints, will stretch all the way to $U_{KK}$. For still larger separation, the string stretches from infinity along the $U$-tube to $U_{KK}$, moves along $M^4$, and returns to $U=\infty$. The motion along $M^4$ at roughly fixed $U\approx U_{KK}$ gives a contribution to the energy of the string proportional to the separation of the endpoints along $M^4$. This linear potential is a signature of confinement.
One striking difference between QCD and the Sakai-Sugimoto model is that the latter predicts additional Kaluza-Klein (KK) modes associated with the compact dimensions. Notably, $M_{KK}$ sets both the confinement scale and the mass scale of the KK modes; above the confining scale, the additional dimension parameterized by the coordinate $\tau$ becomes apparent. While it is common to ignore the Kaluza-Klein modes, and assume that only the 4D model of the lowest modes is realistic, we consider the existence of these Kaluza-Klein excitations as a prediction of holographic EWSB. Thus, we expect a tower of heavy gauge bosons to appear in addition to technicolor-like resonances. The D4-D8-[$\overline{{\rm D}8}\ $]{}configuration also has an SO(5) symmetry from the 4-sphere directions, which becomes an SO(5) global symmetry in the field theory, leading to additional KK modes.
In order to decouple string loop ($g_s$) and gravity ($\alpha'$) corrections, $N$ and $g_sN$ are taken to be large. In the probe brane limit, $N_f\ll N$ (first introduced in the context of the D3-D7 system by Karch and Katz [@KK]), the D4 branes generate a spacetime geometry in which stable D8 brane configurations minimize their action. This is the easiest case to study, as the backreaction on the geometry due to the D8 branes can be ignored. The D8-brane configurations in the D4-D8-[$\overline{{\rm D}8}\ $]{}system were worked out in Refs. [@SS1] and [@Aharony]. It was found that the D8 and [$\overline{{\rm D}8}\ $]{}branes (which are otherwise distinguished by their orientation) join at some $U_0\geq U_{KK}$. The D8-[$\overline{{\rm D}8}\ $]{}profile $U(\tau)$ is determined by minimizing the Dirac-Born-Infeld action with gauge fields on the branes turned off, $$S_{DBI}=-T\int d^9x\, e^{-\phi} \sqrt{{\rm det}\,g_{MN}},$$ where $g_{MN}$ is the induced metric on the D8-branes. The result of minimizing this action is [@Aharony], $$f(U)+\left(\frac{R}{U}\right)^3\frac{U'(\tau)^2}{f(U)}
=\frac{U^8 f(U)^2}{U_0^8f(U_0)} \,\,\,.
\label{eq:D8config}$$ For the solution with $U_0=U_{KK}$, the D8-[$\overline{{\rm D}8}\ $]{}branes stretch from antipodal points along the $\tau$ circle at $U=\infty$ to the tip of the $U$-tube at $U=U_{KK}$, as in Fig. \[fig:U-tube\]a. Generic D8-[$\overline{{\rm D}8}\ $]{}configurations are sketched in Fig. \[fig:U-tube\]b.
$\begin{array}{ccc}
\epsfxsize=3in
\epsffile{U-tube-a.eps} &\ \ \ &
\epsfxsize=3in
\epsffile{U-tube-b.eps} \end{array}$
For the solutions with $U_0>U_{KK}$ the asymptotic distance along the $\tau$ circle between the D8 and [$\overline{{\rm D}8}\ $]{}branes as $U\rightarrow\infty$ is found to be [@Aharony] $$L=2R^{3/2}\int_{U_0}^\infty dU\,\frac{1}{f(U)U^{3/2}\sqrt{\frac{f(U)U^8}{
f(U_0)U_0^8}-1}}.$$ The value of $L/\pi R$ is plotted as a function of $U_{0}/U_{KK}$ in Fig. \[fig:LpiR\].
![\[fig:LpiR\] The asymptotic distance between D8 and [$\overline{{\rm D}8}\ $]{}branes along the $\tau$ circle as $U\rightarrow\infty$ as a function of $U_{0}/U_{KK}$.](LpiR.eps)
The induced metric on the D8-[$\overline{{\rm D}8}\ $]{}configuration is obtained by eliminating $d\tau^2$ in favor of $dU^2$ in Eq. (\[eq:themetric\]) using Eq. (\[eq:D8config\]): $$ds^2=\left(\frac{U}{R}\right)^{3/2}\,\eta_{\mu\nu}dx^\mu dx^\nu-
\left(\frac{R}{U}\right)^{3/2}\,U^2\,d\Omega_4^2-
\left(\frac{R}{U}\right)^{3/2}\left[\frac{1}{f(U)}+\left(\frac{U}{R}\right)^3
\frac{f(U)}{U'(\tau)^2}\right]\,dU^2. \label{eq:induced}$$ The determinant of the induced metric is then $$\sqrt{\det\, g}=\frac{U^{29/4}R^{3/4}\sqrt{\det\,g_{\Omega_4}}}{\left(
U^8f(U)-U_0^8f(U_0)\right)^{1/2}}.
\label{eq:detinduced}$$ The connection of the branes at $U_0$ signals the breaking of the SU$(N_f)\times$SU$(N_f)$ chiral symmetry of the 4+1 dimensional theory (or correspondingly the gauge invariance on the D8 branes) to an SU$(N_f)$ subgroup. Quarks in this theory are massless because the D8 branes and D4 branes cannot be separated: the D8 branes are codimension 1 and the D4 branes extend in the transverse direction. Chiral symmetry breaking is associated with the formation of a quark condensate, although it is difficult to ascertain its exact value as a function of $U_0$. (If the quark mass were nonvanishing, one could vary the action with respect to it to obtain the condensate, as in Ref. [@Mateos]. For nonvanishing quark masses in D8-brane scenarios, see Ref. [@quarkmass].) On the other hand, the technipion decay constant can be estimated holographically, as we describe in the next section, and is fixed to $f_\pi=246$ GeV in order to reproduce the spectrum of electroweak gauge bosons (or to $f_\pi=92$ MeV to reproduce the pion decay constant in QCD). In QCD, the confinement and chiral symmetry breaking scales are related to one another, as is the case in the D-brane construction. However, in the D4-D8-[$\overline{{\rm D}8}\ $]{}model there is an additional parameter, $U_0/U_{KK}$, that does not correspond to any parameter in QCD. The ratio of the chiral symmetry breaking scale to the confinement scale varies as a function of the free parameter, as noticed in Ref. [@Harvey].
Holographic Technicolor from the D4-D8-$\mathbf{\overline{D8}}$ System {#sec:Modes}
======================================================================
In technicolor models, a new set of asymptotically free gauge interactions are postulated under which the Standard Model particles are neutral. Fermions charged under both the technicolor and electroweak interactions condense due to strong technicolor dynamics and break the electroweak gauge group to the U(1) gauge invariance of electromagnetism. Just as hadrons appear as bound states of confined quarks and gluons, technihadrons are predicted to exist as bound states of techniquarks and technigluons. The properties of technihadrons are difficult to calculate for the same reason that the properties of hadrons in QCD are difficult to calculate: the interactions are strong at the low energies of interest. However, in holographic models it is straightforward to estimate masses, decay rates and couplings of hadronic resonances. In QCD, the chiral symmetry of the light quarks is broken to a diagonal subgroup, yielding a spectrum of Goldstone bosons that we refer to generically as pions. In technicolor models, an SU(2)$\times$U(1) subgroup of the chiral symmetry is gauged and identified with the electroweak gauge group. With two techniquark flavors, the would-be pions are eaten by the $W$ and $Z$ boson and do not appear as physical states in the spectrum. However, techni-vector ($\rho$) resonances and techni-axial vector ($a_1$) resonances are expected to be present, and are a generic prediction of technicolor models.
In phenomenological models of EWSB based on the AdS/CFT correspondence, it has been found that corrections to precision electroweak observables can be suppressed by adjusting available parameters [@Holo-EWSB1; @Holo-EWSB2; @CET; @piai]. There are even scenarios in which the $S$-parameter is negative [@Holo-EWSB2], although there is some debate as to whether scenarios leading to negative $S$ can be physically realized [@Holo-EWSB1; @csaba]. A disadvantage to the phenomenological approach is that it is not known whether an ordinary quantum field theory interpretation of those models is always possible. On the other hand, models based on actual D-brane configurations avoid this problem: the appropriate field theory description is that of low-energy fluctuations on the given D-brane configuration. Unfortunately, the AdS/CFT correspondence is most powerful as a calculational tool in limits of the parameters that may not be physically relevant. In particular, in the supergravity limit it is assumed that $N$ and $g^2 N$ are large, so that gravity and stringy effects are decoupled from the field theory that lives on the D-branes. As a result of the large-$N$ limit, the resonances are infinitely narrow in a spectral decomposition of correlation functions, an outcome that is generic in holographic models if higher dimensional loop corrections are not taken into account.
One of the most serious problems for technicolor models is their generically large value of the Peskin-Takeuchi $S$-parameter [@PT]. The $S$ parameter can be defined in terms of matrix elements of the vector current $J_\mu^{a\,V}$ and axial-vector current $J_\mu^{a\,A}$ two-point functions [@PT]: $$\begin{aligned}
i\,\int d^4x\,e^{iq\cdot x}\,\langle J_\mu^{a\,V}(x)J_\nu^{b\,V}(0)\rangle &=&
\left(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2}\right) \,\delta^{ab}\,\Pi_V(-q^2), \nonumber \\
i\,\int d^4x\,e^{iq\cdot x}\,\langle J_\mu^{a\,A}(x)J_\nu^{b\,A}(0)\rangle &=&
\left(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2}\right)\, \delta^{ab}\,\Pi_A(-q^2)\end{aligned}$$ $$S=-4\pi \frac{d}{dQ^2}\left.\left(\Pi_V-\Pi_A\right)
\right|_{Q^2=0} \,\,,
\label{eq:S}$$ where $Q^2=-q^2$. From a dispersive representation with delta function resonances, the current two-point functions can be expressed in terms of the resonance masses and decay constants (up to constant local counterterms) yielding $$\begin{aligned}
\Pi_V(-q^2)&=&\sum_n\frac{g_{Vn}^2 \, q^2}{m_{Vn}^2(-q^2+m_{Vn}^2)} \,\,\, , \nonumber \\
\Pi_A(-q^2)&=&-f_\pi^2+\sum_n\frac{g_{An}^2 \, q^2}{m_{An}^2(-q^2+m_{An}^2)} .
\label{eq:Pi}\end{aligned}$$ It then follows that the $S$ parameter can be written as a sum over the vector and axial vector resonances as $$S=4\pi\sum_n\left(\frac{g_{Vn}^2}{m_{Vn}^4}-\frac{g_{An}^2}{m_{An}^4} \right).$$ The spectrum and decay constants of the lightest few hadronic resonances in the Sakai-Sugimoto model for the configuration $U_{0}=U_{KK}$ in which the D8-branes stretche between antipodal points on the $\tau$ circle were calculated in Ref. [@SS2]. The extension of these results to other D8-brane configurations was described in Ref. [@Aharony], although the purpose of that paper was to study finite temperature physics, and numerical solutions for the resonance masses and decay constants were not computed. The analysis is similar to that for the D4-D6-$\overline{\mbox{D6}}$ system in Ref. [@Mateos].
To calculate the decay constants $g_{Vn}$ and $g_{An}$, and masses $m_{Vn}$ and $m_{An}$, one studies the DBI action on the probe D8-branes, $$S_{D8}=-T\int_{{\rm D}8+{\overline{{\rm D}8}}} d^4x\,dU\,d\Omega_4\,e^{-\phi}
\sqrt{\det\,(g_{MN}+(2\pi\alpha')F_{MN})}\,\,\, ,$$ where $F_{MN}=F_{MN}^aT^a$ is the gauge field strength on the D8 branes, $\alpha'=l_s^2$, and $g_{MN}$ is the induced metric, Eq. (\[eq:induced\]). The determinant is over the Lorentz matrix structure, and traces over the gauge group in the expansion of the determinant are to be understood. There is a Chern-Simons term in the D8-brane action which we consistently ignore, since we restrict our attention to terms only quadratic in the gauge fields. The integral is over $U\in(U_0,\infty)$ twice: once over the D8 brane segment and once over the [$\overline{{\rm D}8}\ $]{}brane segment. We only study configurations constant along the $S^4$ and such that $F_{MN}=0$ for $M,N \neq 0,1,2,3,U$. From a 5D perspective, the remaining modes are related to these by the SO(5) symmetry of the brane configuration, plus a set of scalar fields from the decomposition of the gauge fields around the 4-sphere. Expanding the action to quadratic order in $F_{MN}$, $$\begin{aligned}
S_{D8} &\approx&
-\frac{3}{2}{\widetilde{T}}(2\pi\alpha')^2R^3 U_{KK}^{-1/2} \cdot \nonumber \\
&& \int d^4x\,dU\,{{\rm Tr}}\left[\frac{1}{2}F_{\mu\nu}
F^{\mu\nu}\,U^{-1/2}\gamma(U)^{1/2}+ F_{\mu U}F^{\mu U}\,U^{5/2}
R^{-3}\gamma(U)^{-1/2}\right],
\label{eq:quad-action}\end{aligned}$$ where ${\widetilde{T}}=\frac{2}{3} R^{3/2}U_{KK}^{1/2}T V_4 g_s^{-1}$, $V_4=8\pi^2/3$ is the area of the unit 4-sphere, and contractions of indices are with respect to the 5D Minkowski metric. We have defined the function $\gamma(U)$ as in Ref. [@Aharony]: $$\gamma(U)=\frac{U^8}{U^8 f(U)-U_0^8f(U_0)}.$$
Expanding $F_{MN}$ in modes, the vector modes are symmetric upon reflection about $U=U_0$ while the axial vector modes are antisymmetric. These modes are identified with the tower of vector and axial vector mesons in the 4+1 dimensional gauge theory, with the $\tau$ circle dimension suppressed. The normalizable modes satisfy Dirichlet boundary conditions: $A_\mu(x,\infty)=0$ on both the D8 and [$\overline{{\rm D}8}\ $]{}branches of the $U$-segment $U\in(U_0,\infty)$. The symmetric solutions also satisfy $\partial_U A_\mu|_{U=U_0}=0$, while the antisymmetric modes satisfy $A_\mu(x,U_0)=0$.
Note that in the $U$ coordinate system one must consider two branches of $U\in(U_0,\infty)$, the D8-brane branch and the [$\overline{{\rm D}8}\ $]{}-brane branch. The branes meet at $U_0$ so we sometimes refer simply to the D8-brane configuration. To simplify the discussion of boundary conditions we will sometimes find it convenient to change coordinates, for example to $s$ defined by $s^2=U-U_0$, $s\in(-\infty,\infty)$, which covers both branches of the D8-[$\overline{{\rm D}8}\ $]{}configuration. Since the equations of motion are more cumbersome in such coordinate systems, unless $U_0=
U_{KK}$, we will use the $U$ coordinate in most of what follows.
We will work in a gauge in which $A_U(x,u)=0$. As in Ref. [@SS2], we expand the vector field in normalizable modes satisfying Dirichlet boundary conditions $A_\mu^n(x,\infty)$=0:[^1] $$A_\mu(x,U)=\sum_n \left(V_\mu^n(x)\psi_{Vn}(U)+
A_\mu^n(x)\psi_{An}(U)\right).$$ The $\psi_{Vn}(U)$ are the symmetric modes, and $\psi_{An}(U)$ are antisymmetric. The equations of motion are $$\begin{aligned}
&-U^{1/2}\gamma^{-1/2}\partial_U\left(U^{5/2}\gamma^{-1/2}\partial_U\psi_{Vn}
\right)=R^3 m_{Vn}^2\psi_{Vn},& \nonumber \\
&\partial_\mu F_V^{\mu\nu}(x) = -m_{Vn}^2 V_n^\nu(x),&
\label{eq:EOM}\end{aligned}$$ and similarly for the axial vector modes with $V\rightarrow A$. The spectrum alternates between symmetric and antisymmetric modes, so that the vector and axial vector masses alternate, as in Fig. \[fig:masses\].
![\[fig:masses\] Masses of the lightest four vector (solid lines) and axial vector (dotted lines) resonances as a function of $U_0/U_{KK}$.](kkmassplot.eps)
A source for the SU(2)$_L$ or SU(2)$_R$ current corresponds to a non-normalizable solution to the equation of motion for $F_{\mu\nu}$ localized near either the D8 or [$\overline{{\rm D}8}\ $]{}brane boundary, respectively. Similarly, a source for SU(2)$_V$ corresponds to a non-normalizable solution symmetric around $U=U_0$, while a source for SU(2)$_A$ corresponds to an anti-symmetric configuration. We will call the symmetric non-normalizable solutions, Fourier transformed in the coordinates $x^\mu$, $A_\mu(q,U)={\cal V}_\mu(q)\psi_V^0(q^2,U)$, where $\psi_V^0(q^2,\infty)=1$ on both branches of $U$. We call the antisymmetric solutions $A_\mu(q,U)={\cal A}_\mu(q)\psi_A^0(q^2,U)$, where $\psi_A^0(q^2,\infty)=1$ on the D8 brane branch and $-1$ on the [$\overline{{\rm D}8}\ $]{}branch. The gauge transformation which sets $A_U(q,U)=0$ reappears in the sources ${\cal V_\mu}$ and ${\cal A_\mu}$ and gives rise to the pion kinetic terms and couplings, as discussed in [@SS2]. For our purposes it will be good enough to set these pion degrees of freedom to zero. They will get eaten after gauging of the electroweak subgroup of the chiral symmetry, in any case. The bulk-to-boundary propagators $\psi_V^0(q^2,U)$ and $\psi_A^0(q^2,U)$ satisfy the equation of motion (\[eq:EOM\]) with $m_{Vn}^2$ replaced by $q^2$. A general decomposition of the gauge fields into normalizable modes in addition to a source for the SU(2)$\times$SU(2) current is then $$A_\mu(q,U)=\sum_n\left(V_\mu^n(q)\psi_{Vn}(U)+A_\mu^n(q)\psi_{An}(U)\right)
+{\cal V}_\mu(q)\psi_V^0(q^2,U)+{\cal A}_\mu(q)\psi_A^0(q^2,U).$$
Up to gauge fixing terms, the decomposition of the action, Eq. (\[eq:quad-action\]), to quadratic order in the gauge fields is given by $$\begin{aligned}
S_{D8}&=& -\frac{3}{2}{\widetilde{T}}(2\pi\alpha')^2R^3U_{KK}^{-1/2} \nonumber \\
&& \,{\rm Tr}\int_{{\rm D8}+{\overline{{\rm D}8}}} d^4q\,dU
\left\{\frac12 \left[|F_{\mu\nu}^{V0}(q)|^2(\psi_{V}^0(q^2,U))^2+
|F_{\mu\nu}^{A0}(q)|^2(\psi_{A}^0(q^2,U))^2\right.\right. \nonumber
\\
&&+\sum_n\left.\left(|F_{\mu\nu}^{Vn}(q)|^2\psi_{Vn}^2+
|F_{\mu\nu}^{An}(q)|^2\psi_{An}^2\right)\right]
\frac{\gamma^{1/2}}{U^{1/2}} \nonumber
\\
&&-\left[(\partial_U\psi_{V}^0)^2\,|V_\mu^0(q)|^2+
(\partial_U\psi_{A}^0)^2\,|A_\mu^0(q)|^2 \right.\nonumber \\
&&+\sum_n \left.\left(
(\partial_U\psi_{Vn})^2\,|V_\mu^n(q)|^2+
(\partial_U\psi_{An})^2\,|A_\mu^n(q)|^2\right)\right]
\frac{U^{5/2}}{\gamma^{1/2}} \nonumber \\
&&+\left.\left(F_{\mu\nu}^{V0}(q)F^{\mu\nu}_{Vn}(-q)\, \psi_{Vn}\psi_V^0+
F_{\mu\nu}^{A0}(q)F^{\mu\nu}_{An}(-q)\, \psi_{An}\psi_A^0\right)\frac{\gamma^{
1/2}}{U^{1/2}}\right\}. \end{aligned}$$ If the SU$(N_f)$ generators are normalized so that Tr$T^aT^b=\delta^{ab}/2$, then the gauge fields are canonically normalized if: $$\begin{aligned}
3{\widetilde{T}}(2\pi\alpha')^2R^3U_{KK}^{-1/2}\,\int_{U_0}^\infty dU
\psi_{Vn}\psi_{Vm}\gamma^{1/2}U^{-1/2} = \delta_{mn}, \label{eq:cnormv} \\
3{\widetilde{T}}(2\pi\alpha')^2R^3U_{KK}^{-1/2}\,\int_{U_0}^\infty dU
\psi_{An}\psi_{Am}\gamma^{1/2}U^{-1/2} = \delta_{mn}.\end{aligned}$$ It is natural to define, as in Ref. [@SS2], $$\kappa\equiv{\widetilde{T}}(2\pi\alpha')^2R^3=\frac{g^2N^2}{108\pi^3},$$ where we have used the relations [@SS1]: $$R^3=\frac{g^2N^2 l_s^2}{2M_{KK}},\quad U_{KK}=\frac{2}{9}g^2NM_{KK}l_s^2,
\quad g_s=\frac{g^2}{2\pi M_{KK}l_s}.$$ Using the equations of motion Eq. (\[eq:EOM\]) we can write the action in terms of the canonically normalized modes as $$\begin{aligned}
S_{D8}&=&-{\rm Tr}\,\int d^4x\,\sum_n\left[\frac12 (F_{\mu\nu}^{Vn})^2+\frac12
(F_{\mu\nu}^{An})^2 -m_{Vn}^2\,(V_\mu^n)^2 -m_{An}^2\,(A_\mu^n)^2 \nonumber
\right.\\
&&+\left.\left(a_{Vn}F_{\mu\nu}^{V0}F_{Vn}^{\mu\nu}+
a_{An}F_{\mu\nu}^{A0}F_{An}^{\mu\nu}\right) \right] +S_{{\rm source}},
\end{aligned}$$ where $$\begin{aligned}
a_{Vn}&=&-
\left.3\kappa (m_{Vn}^2R^3)^{-1}U^{5/2}U_{KK}^{-1/2}\gamma^{-1/2}\partial_U\psi_{Vn}\right|
_{U_{{\rm D}8}=\infty}, \nonumber \\
a_{An}&=&-
\left.3\kappa (m_{An}^2R^3)^{-1}U^{5/2}U_{KK}^{-1/2}\gamma^{-1/2}\partial_U\psi_{An}\right|
_{U_{{\rm D}8}=\infty}.
\label{eq:aVaA}\end{aligned}$$ $S_{{\rm source}}$ is the kinetic term for the sources ${\cal V}$ and ${\cal A}$, and is given by, $$S_{{\rm source}}=
-{\rm Tr}\,\int d^4q\,q^2\,\left[a_{V0}\,|V_\mu^0(q)|^2+
a_{A0}\,|A_\mu(q)|^2\right]. \label{eq:Ssource}$$ The constants $a_{V0}$ and $a_{A0}$ are given by, $$\begin{aligned}
a_{V0}&=&-
\left.3\kappa (q^2R^3)^{-1}U^{5/2}U_{KK}^{-1/2}\gamma^{-1/2}\partial_U\psi_V^0
(q^2,U)\right|
_{U_{{\rm D}8}=\infty}, \nonumber \\
a_{A0}&=&-
\left.3\kappa (q^2R^3)^{-1}U^{5/2}U_{KK}^{-1/2}\gamma^{-1/2}\partial_U\psi_A^0
(q^2,U)\right|
_{U_{{\rm D}8}=\infty}.
\label{eq:aV0aA0}\end{aligned}$$
We have used the equations of motion and boundary conditions on $\psi_{Vn}, \psi_{An}$, $\psi^0_{V}$, and $\psi^0_{A}$ to eliminate vanishing surface terms, [*e.g.*]{}, $$\begin{aligned}
&&\int_{{\rm D}8+{\overline{{\rm D}8}}} dU\,U^{-1/2}\gamma^{1/2}\psi_{Vn}(U)\,\psi_V^0(q^2,U) \nonumber \\
&=&-\int_{{\rm D}8+{\overline{{\rm D}8}}} dU\, (m_{Vn}^2R^3)^{-1}\partial_U\left(
U^{5/2}\gamma^{-1/2}\,\partial_U\psi_{Vn}\right)\psi_V^0 \nonumber \\
&=&-\left.2(m_{Vn}^2R^3)^{-1}U^{5/2}\gamma^{-1/2}\partial_U\psi_{Vn}\right|
_{U_{{\rm D}8}=\infty},\end{aligned}$$ and similarly for $V\rightarrow A$.
Diagonalizing the kinetic terms, we obtain the action, $$\begin{aligned}
S_{D8}&=&-
{\rm Tr}\,\int d^4x\,\sum_n\left[\frac12 (\widetilde{F}_{\mu\nu}^{Vn})^2
-m_{Vn}^2\,(\widetilde{V}_\mu^n-a_{Vn}{\cal V}
_\mu)^2 \right.\nonumber \\
&&+\left.\frac12
(\widetilde{F}_{\mu\nu}^{An})^2 -m_{An}^2\,(\widetilde{A}_\mu^n-a_{An}{\cal A}
_\mu)^2
\right] +\widetilde{S}_{{\rm source}}, \end{aligned}$$ where $$\widetilde{V}^n_\mu=V^n_\mu+a_{Vn}{\cal V}_\mu,$$ $\widetilde{F}_{\mu\nu}^{Vn}$ is the gauge kinetic term for $\widetilde{V}^n
_\mu$, and similarly for $V\rightarrow A$. The kinetic term for the source picks up an additional contribution in this process. From the coupling between the sources and the modes we identify the decay constants, $$\begin{aligned}
g_{Vn}&=&m_{Vn}^2 a_{Vn} \nonumber \\
&=&-\left.3\kappa R^{-3}U^{5/2}U_{KK}^{-1/2}\gamma^{-1/2}\partial_U\psi_{Vn}\right|
_{U_{{\rm D}8}=\infty}, \nonumber \\
g_{An}&=&m_{An}^2 a_{An} \nonumber \\
&=&-\left.3\kappa R^{-3}U^{5/2}U_{KK}^{-1/2}\gamma^{-1/2}\partial_U\psi_{An}\right|
_{U_{{\rm D}8}=\infty}. \label{eq:thedcs}\end{aligned}$$ The decay constants grow with $U_0$ as the masses do, as demonstrated in Fig. \[fig:g\].
![\[fig:g\] Decay constants of the lightest four vector (solid lines) and axial vector (dotted lines) resonances as a function of $U_0/U_{KK}$.](kkdecayc.eps)
The spectrum of the lightest four vector and axial vector resonances, and their decay constants, were given in Ref. [@SS2] for the antipodal D8-[$\overline{{\rm D}8}\ $]{}configuration $U_0=U_{KK}$. The contribution of the first four pairs of resonances to the $S$ parameter Eq. (\[eq:S\]) using the results of Ref. [@SS2] is $S_4\approx -0.4\kappa \approx -0.006$, where in the last approximation we set $g^2N=4\pi$ and $N=4$. The lightest pair of resonances contributes negatively to the $S$-parameter. In fact, the three-digit precision to which the results in Ref. [@SS2] are quoted is not good enough for this calculation because of the propagation of errors in summing over the small differences between the large contributions of the vector and axial vector modes to $S$. We find by increasing the precision of our numerical calculation that $S_4\approx -3\kappa \approx -.05$, larger by an order of magnitude than our estimate from Sakai and Sugimoto’s results. It is clear that the contributions from individual sets of modes to the $S$ parameter are sensitive to the details of the model. Increasing the parameter $U_0/U_{KK}$ makes the contribution of the lightest modes to the $S$-parameter even more negative, as seen in Fig. \[fig:S4\].
![\[fig:S4\] Partial sums of the lightest four vector and axial vector resonance contributions to the $S$ parameter, as a function of $U_0/U_{KK}$.](svsu0plot.eps)
On the other hand, we can also calculate the $S$-parameter exactly (in the large $N$ limit) by using the bulk-to-boundary propagators $\psi_V^0$ and $\psi_A^0$, analogous to the derivation of the sum rules in Ref. [@sumrules]. It is convenient to temporarily employ a coordinate $s$ that spans both the D8 and [$\overline{{\rm D}8}\ $]{}branches, $$s^2 = U-U_0 \,\,\, ,$$ where $s>0$ ($s<0$) corresponds to the D8 brane ([$\overline{{\rm D}8}\ $]{}brane). In momentum space, the vector bulk-to-boundary propagator $\psi_V^0(q^2,s)$ satisfies $$\left[R^3 q^2 U^{-1/2} \gamma^{1/2} + \frac{1}{2 s} \partial_s (\frac{1}{2 s}
U^{5/2} \gamma^{-1/2}\partial_s)\right] \psi_V^0(q^2,s) = 0 \,\,\, ,$$ with $\psi_V^0(q^2,s)=1$ at $s=\pm\infty$. We aim to relate this to the following Green’s function $G(s,s',q^2)$ for the equation of motion operator $$\left[R^3 q^2 U^{-1/2} \gamma^{1/2} + \frac{1}{2 s} \partial_s (\frac{1}{2 s} U^{5/2} \gamma^{-1/2}\partial_s)
\right] G(s,s',q^2) = \frac{1}{|2s|} \delta(s-s') \,\,\, ,
\label{eq:greendef}$$ where we assume Dirichlet boundary conditions $G(\pm\infty,s',q^2)=0$. It follows from the normalization condition for the modes, Eq. (\[eq:cnormv\]), that we may rewrite the delta function in Eq. (\[eq:greendef\]) using the completeness relation $$\frac{3}{2} U_{KK}^{-1/2} \kappa \, U^{-1/2} \gamma^{1/2}\, \sum_n \psi_{n}(s)
\psi_{n}(s') = \frac{1}{|2s|} \delta(s-s') \,\,\,.$$ If we then apply the ansatz $$G(s,s',q^2)=\sum_{nm} c_{nm} \, \psi_{n}(s)\, \psi_{m}(s') \,\,\,,$$ one can extract the coefficients $c_{nm}$ from Eq. (\[eq:greendef\]) by application of the equations of motion. It follows that $G(s,s',q^2)$ may be expressed as a sum over modes, $$G(s,s',q^2) = \frac{3}{2} \frac{U_{KK}^{-1/2} \kappa}{R^3} \sum_n \frac{\psi_{n}(s)\psi_{n}(s')}
{q^2-m_{n}^2} \,\,\,.$$ For definiteness, let us consider the relationship between $G(s,s',q^2)$ and the bulk-to-boundary propagator of the vector modes $\psi_V^0(q^2,s)$. We begin with the identity $$\frac{s'}{|s'|} \psi_V^0(q^2,s') = \int_{-\infty}^{\infty} ds \, \delta(s-s')\,
\frac{s}{|s|} \psi_V^0(q^2,s) \,\,\,.
\label{eq:dident}$$ Replacing the delta function using Eq. (\[eq:greendef\]), one may integrate by parts twice to show that the right-hand-side of Eq. (\[eq:dident\]) is a surface term $$\frac{s'}{|s'|} \psi_V^0(q^2,s') = \frac{1}{s} \, U^{5/2} \gamma^{-1/2}\, \partial_s G_V(s,s',q^2)|^{s=\infty} \,,$$ where we have defined $G_V(s,s',q^2)=[G(s,s',q^2)+G(-s,s',q^2)]/2$. In terms of the mode decomposition, $G_V(s,s',q^2)$ contains only those modes that are even under reflection about $s=0$ ([*i.e.*]{}, $U=U_0$), which we called $\psi_{Vn}(s)$ earlier: $$\frac{s'}{|s'|} \psi_V^0(q^2,s') = 2 \, U^{5/2} \gamma^{-1/2} \, \left(\frac{3}{2}
\frac{U_{KK}^{-1/2} \kappa}{R^3} \right) \sum_n\left.
\frac{\frac{1}{2s} \partial_s \psi_{Vn}(s)\psi_{Vn}(s')}{q^2-m_{Vn}^2}
\right|^{s=\infty} \,\,\,.
\label{eq:intres1}$$ The form of this result is suggestive: $ U^{5/2}
\gamma^{-1/2} \frac{1}{2 s} \partial_s \psi_{Vn}(s)|^{s=\infty}$ is precisely the same as the quantity $U^{5/2} \gamma^{-1/2} \partial_U \psi_{Vn}(U)|_{U_{D8}=\infty}$ that appeared in our holographic expressions for the decay constants, Eqs. (\[eq:thedcs\]). We now take a derivative of each side with respect to $s'$ and take the limit $s'\rightarrow \infty$. We may re-express the derivatives of the wave functions $\psi_{Vn}$ appearing in Eq. (\[eq:intres1\]) in terms of decay constants, for $s=s'=\infty$. However, the sum is convergent only when one takes the limit $s' \rightarrow \infty$ with $s' \neq s$. The difference between the limit of the sum and its value at the limit point is a divergent quantity $\Delta_V$, such that $$\sum_n \frac{g_{Vn}^2}{q^2-m_{Vn}^2} + \Delta_V = \frac{3 U_{KK}^{-1/2} \kappa}{R^3} \left[
\frac{{U'}^{5/2} {\gamma'}^{-1/2}}{2s'} \partial_{s'} \psi_V^0(q^2,s') \right]^{s'\rightarrow \infty} \,\,\, ,$$ where primed quantities are functions of $s'$, or in original $U$ coordinates, $$\sum_n \frac{g_{Vn}^2}{q^2-m_{Vn}^2} + \Delta_V = \frac{3 U_{KK}^{-1/2} \kappa}{R^3} \left[
U^{5/2} \gamma^{-1/2} \partial_U \psi_V^0(q^2,U) \right]^{U \rightarrow \infty} \,\,\, .
\label{eq:g2m2res}$$ Sums that involve higher powers of $q^2-m_{Vn}^2$ in the denominator are convergent, which suggests that $\Delta_V$ is a constant independent of $q^2$. This can be shown by taking $q^2$ derivatives in Eq. (\[eq:intres1\]), so that the sum is convergent and there is no ambiguity in the evaluation of the subsequent limits, and comparing to the same $q^2$ derivatives of Eq. (\[eq:g2m2res\]). We fix $\Delta_V$ by noting that $\psi_V^0(0,U)$ is a constant, so that the right-hand-side of Eq. (\[eq:g2m2res\]) vanishes in the limit $q^2=0$; it follows that $$\Delta_V = \sum_ m \frac{g_{Vn}^2}{m_{Vn}^2} \,\,\,.$$ The left-hand-side of Eq. (\[eq:g2m2res\]) may now be combined to give precisely $\Pi_V(-q^2)$ defined in Eq. (\[eq:Pi\]), $$\Pi_V(-q^2) = \sum_n\frac{g_{Vn}^2 \, q^2}{m_{Vn}^2(-q^2+m_{Vn}^2)}= -\frac{3 U_{KK}^{-1/2} \kappa}{R^3} \left[
U^{5/2} \gamma^{-1/2} \partial_U \psi_V^0(q^2,U) \right]^{U \rightarrow \infty} \,\,\, .
\label{eq:bigpiVres}$$
To obtain the axial-vector self-energy, $\Pi_A(-q^2)$, one must take into account that the axial bulk-to-boundary propagator $\psi_A^0(q^2,s)$ satisfies the boundary conditions $\psi_A^0(q^2,-\infty)=-1$ and $\psi_A^0(q^2,\infty)=1$. One can then derive an expression of the same form as Eq. (\[eq:g2m2res\]) with $\Delta_V$, $g_{Vn}$, $m_{Vn}$ and $\psi_V^0(q^2,U)$ replaced by $\Delta_A$, $g_{An}$, $m_{An}$ and $\psi_A^0(q^2,U)$, respectively. The evaluation of $\Delta_A$, however, is different in this case. One can show that the right-hand-side of the axial version of Eq. (\[eq:g2m2res\]) evaluates to a non-vanishing constant when $q^2=0$, which we identify with the square of the pion decay constant, $f_\pi^2$. Indeed, we find that when $U_0=U_{KK}$ this constant agrees with the prediction $f_\pi^2=4\kappa\,M_{KK}^2/\pi $ as determined in Refs. [@SS1; @SS2] by studying the pion part of the effective action. Thus we set $$\Delta_A = \sum_ m \frac{g_{An}^2}{m_{An}^2} -f_\pi^2 \,\,\,,$$ and $$\Pi_A(-q^2) = -f_\pi^2+\sum_n\frac{g_{An}^2 \, q^2}{m_{An}^2(-q^2+m_{An}^2)} =
-\frac{3 U_{KK}^{-1/2} \kappa}{R^3} \left[
U^{5/2} \gamma^{-1/2} \partial_U \psi_A^0(q^2,U) \right]^{U \rightarrow \infty} \,\,\, .
\label{eq:bigpiAres}$$
Using the definition of the $S$ parameter given in Eq. (\[eq:S\]), it immediately follows that $$S= - 12 \pi \frac{U_{KK}^{-1/2} \, \kappa}{R^3} \left[
U^{5/2} \gamma^{-1/2} \frac{d}{dq^2}\left(\partial_U \psi_V^0 - \partial_U \psi_A^0
\right) \right]^{U=\infty,\,\,q^2=0} \,\,\,.
\label{eq:sfull}$$ This result represents a sum over all resonances. We find that Eq. (\[eq:sfull\]) leads to large, positive $S$ that increases with $U_0/U_{KK}$, a surprising result given the behavior shown in Fig. \[fig:S4\]. We explore this issue further in the next section. For the antipodal D8-[$\overline{{\rm D}8}\ $]{}configuration we obtain $$S= 58.9 \, \kappa \approx 0.9 \,\,\,,$$ where have again taken $g^2 N = 4 \pi$ and $N=4$ for the numerical evaluation. This is a factor of $2$ larger than the estimate of $S$ in a one-doublet, SU(4) technicolor models with QCD-like dynamics [@PT]. As a function of $U_0/U_{KK}$ we plot the $S$ parameter in Fig. \[fig:S\].
![\[fig:S\] Sum over all contributions to $S$ from the technicolor sector, as a function of $U_0/U_{KK}$.](svu0bbpplot.eps)
It is worth noting that the scale $M_{KK}$, with respect to which we have been measuring masses and decay constants, is determined by the electroweak scale $$f_\pi^2=-\Pi_A(0)=(246\ {\rm GeV})^2,
\label{eq:fPiPiA}$$ where $\Pi_A(-q^2)$ is given by Eq. (\[eq:bigpiAres\]). Eq. (\[eq:fPiPiA\]) is simply one of the Weinberg sum rules with the renormalization condition $\Pi_V(0)=0$, as is satisfied by Eq. (\[eq:bigpiVres\]). This expression for $f_\pi$ can also be derived by studying the two-point coupling of the axial vector source ${\cal A}_\mu$ to the pion field, as in Ref. [@SS2]. As noted earlier, we have set the pion field to zero in our analysis. In the model with $U_0=U_{KK}$ we find that $f_\pi^2=1.27 \,\kappa \,M_{KK}^2$, in agreement with Refs. [@SS1; @SS2]. Hence, in that model $M_{KK}=218 / \sqrt{\kappa}$ GeV $\approx 1.8$ TeV, where in the last approximation we set $g^2N=4\pi$ and $N=4$. $M_{KK}$ sets the scale of additional Kaluza-Klein states in the spectrum. $M_{KK}$ also sets the scale of the technihadrons: in the same approximation we find that the lightest technirho has mass $m_\rho\approx 1.5$ TeV; and the lightest techni-axial vector has mass $m_{a_1}\approx 2.2$ TeV. Note that once the electroweak scale is fixed, the only remaining freedom in the model in addition to $N$ and $g^2N$ is the one-parameter choice of D8-brane configuration. Although we have found that the $S$ parameter is too large in this model to be a consistent model of EWSB, the couplings between technicolor resonances and additional phenomenology can be calculated in the model to the same accuracy as the calculations above. We turn our attention to ways in which the $S$ parameter prediction of our model might be reduced in the next section.
It is also worth noting that we can derive the above relations for $\Pi_V(-q^2)$ and $\Pi_A(-q^2)$ by the usual AdS/CFT prescription of varying the action with respect to the sources for the currents ${\cal V}_\mu$ and ${\cal A}_\mu$. In particular, the term in $\langle J_V^\mu(q)J_V^\nu(-q)\rangle$ proportional to $g^{\mu\nu}$ is obtained by varying $iS_{{\rm source}}$, with $S_{{\rm source}}$ given in Eq. (\[eq:Ssource\]), with respect to $i{\cal V}_\mu$ twice, using the equations of motion. (The transverse tensor structure can be obtained by keeping track of gauge fixing terms, as in [@AdSQCD3].) The result is, $$\Pi_V(-q^2)=q^2 a_{V0}(q^2).$$ Comparing with Eq. (\[eq:aV0aA0\]) we see that the AdS/CFT procedure has successfully reproduced our result, Eq. (\[eq:bigpiVres\]). Similarly, we find $\Pi_A(-q^2)=q^2 a_{A0}(q^2)$.
Discussion {#sec:Conclusions}
==========
If we allow the D8-brane boundary to move in from infinity artificially, then the confining scale can be raised as in Refs. [@CET; @agashe]. In doing so, however, we are leaving the realm of string theory in favor of a bottom-up approach to holographic model building. Models of electroweak symmetry breaking constructed in this way have been discussed previously in Refs. [@Holo-EWSB1; @Holo-EWSB2; @CET; @piai2]. In our case, we may consider a 5D model motivated by the AdS/CFT correspondence, but with bulk geometry described by the induced metric on the D8-branes in the D4-D8 system. The location of the UV boundary determines the size of the radial dimension and, hence, the resonance mass scale. Since the higher resonances decouple more quickly as the size of the radial dimension is reduced, we can test the agreement of the sum over modes with the sum rule approach. Fig. \[fig:UVreg\] demonstrates the behavior of the $S$ parameter as the boundary location is varied. Here we work in the limit $U_0=U_{KK}$, and map the $U$ coordinate to a finite interval $-1+y_{reg} < y < 1-y_{reg}$, via the relations $$U = (U_{KK}^3 + U_{KK} z^2)^{1/3} \,\,\,\,\, \mbox{ and } \,\,\,\,\,
y = \frac{2}{\pi} \arctan(z/U_{KK}) \,\,\,.$$ The physical boundaries at $U=\infty$ correspond to $y=\pm 1$. It is clear from Fig. \[fig:UVreg\] that the lightest four pairs of vector and axial-vector modes provide a better approximation of the $S$ parameter as $y_{reg}$ increases.
![\[fig:UVreg\] $S$ parameter as function of UV regulator $y_{reg}$. As $y_{reg}$ increases, the $S$ parameter decreases and the higher resonances decouple more quickly.](niceplot.eps)
We also note that for a severely regulated theory, $y_{reg} \approx 0.9$, the $S$ parameter can be made consistent with the current limit $S \alt 0.09$ [@pdg]. This corresponds to a radial dimension of proper length approximately $0.5/
M_{KK}$. For the value $M_{KK}\approx 1.8$ TeV obtained in the previous section, the radial dimension has size approximately (3.6 TeV)$^{-1}$.
It is interesting to note that this version of holographic duality for chiral symmetries, with two boundary components rather than the single boundary of Anti-de Sitter space, was anticipated by Son and Stephanov in Ref. [@Son-S]. They considered a model of mesons with light-quark quantum numbers based on an extension of the notion of hidden local symmetry [@Bando]. The model is equivalent to a deconstructed extra-dimensional SU$(N)$ gauge theory with an SU$(N_f)\times$SU$(N_f)$ flavor symmetry corresponding to the global symmetries of the fields localized at the two boundaries. To calculate current correlators, the global symmetries are weakly gauged, and the gauge fields at the ends of the extra dimension act as sources for the currents. The configurations with SU$(N_f)$ gauge fields turned on at one boundary or the other are analogous to the non-normalizable modes localized at the boundary of the D8 brane or [$\overline{{\rm D}8}\ $]{}brane segment.
A deconstructed [@decons] version of the Sakai-Sugimoto model would truncate the model to a finite number of resonances. Since we have found that the light resonances in the Sakai-Sugimoto model can contribute negatively to the $S$-parameter, it would be interesting to study the behavior of the $S$-parameter as a function of the number of lattice sites of the deconstructed model. Again, such an approach deviates from string theory since there is no simple deconstruction of the radial dimension that captures all of gravitational physics of the original theory, even in the limit of a large number of lattice sites. Nevertheless, deconstruction often leads to 4D theories that possess important features of the higher-dimensional theory that they are designed to approximate. In the present case, the Sakai-Sugimoto model would simply provide a paradigm for the choice of gauge groups and symmetry breaking pattern in the deconstructed theory. Electroweak constraints would require detailed study, especially away from the large $N$ limit where radiative corrections are generally non-negligible and must be taken into account. It would also be interesting to compare these deconstructed models with existing models of electroweak symmetry breaking that were motivated in part by deconstruction of extra dimensions, such as little Higgs models [@littleHiggs].
In summary, we have studied the viability of the Sakai-Sugimoto model of intersecting D4 and D8 branes as a technicolor-like model of electroweak symmetry breaking. We determined the masses of the technicolor resonances and their decay constants in terms of the electroweak scale. We found that the contribution of the strong dynamics to the Peskin-Takeuchi $S$ parameter is larger by at least an order of magnitude than the experimentally allowed value $S\alt 0.09$, even allowing for adjustment of the D8-brane configuration. However, we observed that the lightest pair of vector and axial vector resonances contributes negatively to $S$, while the final result reflects a non-decoupling of heavy resonances that seems to be sensitive to the details of the model. The model can be made consistent with precision electroweak constraints by artificially introducing an ultraviolet regulator, as we showed in Fig. \[fig:UVreg\], or perhaps by deconstructing the model along the radial direction in order to truncate it to the lightest several technicolor resonances. In order to become a complete model of electroweak symmetry breaking, the Standard Model fermions would need to be included together with a mechanism for generation of their masses.
We thank Ofer Aharony and Veronica Sanz for useful discussions. CDC thanks the NSF for support under Grant No. PHY-0456525. M.S. thanks the NSF for support under Grant No. PHY-0554854. The work of J.E. is supported in part by the NSF under Grant No. PHY-0504442 and the Jeffress Foundation under Grant No. J-768.
[99]{}
J. M. Maldacena, Adv. Theor. Math. Phys. [**2**]{}, 231 (1998) \[Int. J. Theor. Phys. [**38**]{}, 1113 (1999)\] \[arXiv:hep-th/9711200\]; S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B [**428**]{}, 105 (1998) \[arXiv:hep-th/9802109\]; E. Witten, Adv. Theor. Math. Phys. [**2**]{}, 253 (1998) \[arXiv:hep-th/9802150\]. G. F. de Teramond and S. J. Brodsky, Phys. Rev. Lett. [**94**]{}, 201601 (2005) \[arXiv:hep-th/0501022\]. J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. Lett. [**95**]{}, 261602 (2005) \[arXiv:hep-ph/0501128\]. L. Da Rold and A. Pomarol, Nucl. Phys. B [**721**]{}, 79 (2005) \[arXiv:hep-ph/0501218\]. J. Babington, J. Erdmenger, N. J. Evans, Z. Guralnik and I. Kirsch, Phys. Rev. D [**69**]{}, 066007 (2004) \[arXiv:hep-th/0306018\]. N. Evans, J. P. Shock and T. Waterson, Phys. Lett. B [**622**]{}, 165 (2005) \[arXiv:hep-th/0505250\]. D. K. Hong and H. U. Yee, Phys. Rev. D [**74**]{}, 015011 (2006) \[arXiv:hep-ph/0602177\]. J. Hirn and V. Sanz, Phys. Rev. Lett. [**97**]{}, 121803 (2006) \[arXiv:hep-ph/0606086\]; C. D. Carone, J. Erlich and J. A. Tan, Phys. Rev. D [**75**]{}, 075005 (2007) \[arXiv:hep-ph/0612242\]. M. Piai, arXiv:0704.2205 \[hep-ph\]. L. Randall and R. Sundrum, Phys. Rev. Lett. [**83**]{}, 3370 (1999) \[arXiv:hep-ph/9905221\]; N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B [**429**]{}, 263 (1998) \[arXiv:hep-ph/9803315\]; C. Csaki, C. Grojean, H. Murayama, L. Pilo and J. Terning, Phys. Rev. D [**69**]{}, 055006 (2004) \[arXiv:hep-ph/0305237\]; C. Csaki, C. Grojean, L. Pilo and J. Terning, Phys. Rev. Lett. [**92**]{}, 101802 (2004) \[arXiv:hep-ph/0308038\]; G. Cacciapaglia, C. Csaki, G. Marandella and J. Terning, arXiv:hep-ph/0611358. T. Sakai and S. Sugimoto, Prog. Theor. Phys. [**113**]{}, 843 (2005) \[arXiv:hep-th/0412141\]. T. Sakai and S. Sugimoto, Prog. Theor. Phys. [**114**]{}, 1083 (2006) \[arXiv:hep-th/0507073\].
M. Kruczenski, D. Mateos, R. C. Myers and D. J. Winters, JHEP [**0307**]{}, 049 (2003) \[arXiv:hep-th/0304032\]; M. Kruczenski, D. Mateos, R. C. Myers and D. J. Winters, JHEP [**0405**]{}, 041 (2004) \[arXiv:hep-th/0311270\]. M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. [**65**]{}, 964 (1990); M. E. Peskin and T. Takeuchi, Phys. Rev. D [**46**]{}, 381 (1992). R. Sundrum and S. D. H. Hsu, Nucl. Phys. B [**391**]{}, 127 (1993) \[arXiv:hep-ph/9206225\]; D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D [**72**]{}, 055001 (2005) \[arXiv:hep-ph/0505059\]; D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D [**73**]{}, 037701 (2006) \[arXiv:hep-ph/0510217\]; N. Evans and F. Sannino, arXiv:hep-ph/0512080. M. Piai, arXiv:hep-ph/0609104. K. Agashe, R. Contino and A. Pomarol, Nucl. Phys. B [**719**]{}, 165 (2005) \[arXiv:hep-ph/0412089\].
A. Karch and E. Katz, JHEP [**0206**]{}, 043 (2002) \[arXiv:hep-th/0205236\]. O. Aharony, J. Sonnenschein and S. Yankielowicz, arXiv:hep-th/0604161. K. Hashimoto, T. Hirayama and A. Miwa, arXiv:hep-th/0703024. E. Antonyan, J. A. Harvey, S. Jensen and D. Kutasov, arXiv:hep-th/0604017. K. Agashe, C. Csaki, C. Grojean and M. Reece, arXiv:0704.1821 \[hep-ph\]. J. Hirn and V. Sanz, arXiv:hep-ph/0702005. W. M. Yao [*et al.*]{} \[Particle Data Group\], J. Phys. G [**33**]{}, 1 (2006). D. T. Son and M. A. Stephanov, Phys. Rev. D [**69**]{}, 065020 (2004) \[arXiv:hep-ph/0304182\]. M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. [**54**]{}, 1215 (1985); M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. [**164**]{}, 217 (1988). N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Rev. Lett. [**86**]{}, 4757 (2001) \[arXiv:hep-th/0104005\]. C. T. Hill, S. Pokorski and J. Wang, Phys. Rev. D [**64**]{}, 105005 (2001) \[arXiv:hep-th/0104035\]. H. C. Cheng, C. T. Hill, S. Pokorski and J. Wang, Phys. Rev. D [**64**]{}, 065007 (2001) \[arXiv:hep-th/0104179\]. N. Arkani-Hamed, A. G. Cohen, E. Katz, A. E. Nelson, T. Gregoire and J. G. Wacker, JHEP [**0208**]{}, 021 (2002) \[arXiv:hep-ph/0206020\]; N. Arkani-Hamed, A. G. Cohen, E. Katz and A. E. Nelson, JHEP [**0207**]{}, 034 (2002) \[arXiv:hep-ph/0206021\];
[^1]: Since we are interested in the case $N_f=2$ we should be careful not to fall prey to the Witten anomaly. To be precise, the boundary condition is that the gauge fields are pure gauge at infinity, and such configurations fall into one of two classes (because $\pi_4(SU(2))=\mathbb{Z}_2$). This subtlety will not affect any of our results, and we simply note that we must assume the techniquarks transform under an even-dimensional representation of the technicolor group to have a well-defined theory.
|
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abstract: 'Time-domain spectroscopy of the classical accreting T Tauri star, TW Hya, covering a decade and spanning the far UV to the near-infrared spectral regions can identify the radiation sources, the atmospheric structure produced by accretion, and properties of the stellar wind. On time scales from days to years, substantial changes occur in emission line profiles and line strengths. Our extensive time-domain spectroscopy suggests that the broad near-IR, optical, and far-uv emission lines, centered on the star, originate in a turbulent post-shock region and can undergo scattering by the overlying stellar wind as well as some absorption from infalling material. Stable absorption features appear in H[$\alpha$]{}, apparently caused by an accreting column silhouetted in the stellar wind. Inflow of material onto the star is revealed by the near-IR 10830Å line, and its free-fall velocity correlates inversely with the strength of the post-shock emission, consistent with a dipole accretion model. However, the predictions of hydrogen line profiles based on accretion stream models are not well-matched by these observations. Evidence of an accelerating warm to hot stellar wind is shown by the near-IR line, and emission profiles of , , , , and . The outflow of material changes substantially in both speed and opacity in the yearly sampling of the near-IR line over a decade. Terminal outflow velocities that range from 200 [km s$^{-1}$]{} to almost 400 [km s$^{-1}$]{} in appear to be directly related to the amount of post-shock emission, giving evidence for an accretion-driven stellar wind. Calculations of the emission from realistic post-shock regions are needed.'
author:
- 'A. K. Dupree, N. S. Brickhouse, S. R. Cranmer, P. Berlind, Jay Strader'
- 'Graeme H. Smith'
title: |
Structure and Dynamics of the Accretion Process and Wind\
in TW Hya [^1]
---
Introduction
============
TW Hya (CD $-$34 7151, TWA 1, HIP 53911) remains arguably the closest accreting T Tauri Star (Wichmann [et al.]{} 1998), and is oriented with its rotation axis almost along our line of sight which sets the surrounding accretion disk approximately in the plane of the sky (Krist [et al.]{} 2000; Qi [et al.]{} 2004). These characteristics make TW Hya a subject of intensive study at all wavelengths because it is bright and the accretion process is directly accessible by this polar orientation (Donati [et al.]{}2011; Johnstone [et al.]{} 2014).
TW Hya has been a frequent target of intensive study of optical emission lines along with several other bright T Tauri stars: BP Tau (Gullbring [et al.]{} 1996); DF Tau (Johns-Krull & Basri 1997); DQ Tau (Basri et al. 1997); DR Tau (Alencar [et al.]{} 2001); and RW Aur (Alencar [et al.]{} 2005) among them. These studies reveal the intrinsic variability of the optical lines. These stars have larger inclinations than TW Hya, \[20$^\circ$ for DR Tau (Schegerer [et al.]{} 2009); 39$^\circ$ for BP Tau (Guilloteau et al. 2011); the star RW Aur A is inclined by 37$^\circ$; others are not known\]. The uniqueness of TW Hya resides in the imaging of the surrounding disk in the infrared (Krist [et al.]{} 2000) and in the sub-millimeter range (Qi [et al.]{} 2004) revealing its face-on orientation. Thus the interesting polar region where accretion is ongoing can be viewed directly. This paper reports spectral sequences not only from the optical region, but the near-infrared and ultraviolet as well. The broad wavelength coverage spans accretion phenomena, the presence of winds, and the intrinsic chromosphere/corona of the star.
TW Hya itself is of low mass. Estimates range from 0.4 – 0.8 M$_\odot$ (Batalha [et al.]{} 2002; Donati [et al.]{} 2011; Vacca & Sandell 2011). The spectral type is believed to be close to a K7 dwarf (Alencar & Batalha 2002). A possible later spectral type (M2.5V, Vacca & Sandell 2011) is controversial (Andrews [et al.]{}2012; McClure [et al.]{} 2013). The inferred mass and radius for a M2.5V star lead to lower free-fall accretion velocities and consequently shock temperatures lower than measured directly (Brickhouse [et al.]{} 2010). This star has become a fiducial object in the astrophysics of accretion and low mass star formation.
Currently, the ‘standard model’ of accretion suggests (Hartmann 1998) that material from a surrounding disk is channeled by magnetic fields in an ‘accretion funnel’ towards the central star forming an accretion shock, a small post-shock cooling zone, and a hot spot or ring on or near the stellar surface. The source of optical emission is assigned to the accreting funnel flows (Hartmann [et al.]{} 1994; Muzerolle [et al.]{}2001; Kurosawa & Romanova 2013). Alencar and Batalha (2002) in a very detailed analysis of 42 optical spectra of TW Hya acquired mostly nightly over a 1.4 year period described the behavior of several emission lines: H$\alpha$, H$\beta$, ([$\lambda$]{}5876) and Na D. They noted both outflow and inflow signatures in these profiles and documented a correlation between veiling corrected equivalent widths of major lines and veiling. This suggests that increased veiling $-$ believed to arise from the continuum and lines (Gahm [et al.]{} 2008) produced by the accretion hot spot (or ring) $-$ is related to the increased equivalent width of optical emission lines. The accretion rate and its variation can be inferred directly from the lines in the X-ray spectrum (Brickhouse [et al.]{} 2012). However, recent optical spectroscopy simultaneous with X-ray measurements demonstrated the progression of accreted material in the post-shock cooling zone through the stellar atmosphere, producing optical emission, followed by the heating of the photosphere, and subsequent enhancement of the corona (Dupree [et al.]{} 2012). This sequence challenges the ‘standard model’ which attributes optical emission to the accretion funnels (Muzerolle [et al.]{} 2001, 2005; Natta [et al.]{} 2004). In addition, over this time, a stellar wind becomes established and increases in strength over several days. X-ray spectroscopy of TW Hya and the accompanying spectroscopic diagnostics of density and temperature revealed a large post-shock volume in the corona (Brickhouse [et al.]{} 2010). This newly-discovered material has roughly 300 times larger volume and 30 times more mass than the accretion shock itself, and signals the presence of an accretion-fed corona. It appears to be a large turbulent billowing structure in the corona. This process may be similar to the recently identified brightenings in the solar corona following the impact of fragments from an erupting solar filament (Reale [et al.]{} 2013). Accreting material can supply large nearby magnetic structures and also drive a stellar wind (Cranmer 2009).
While broad ultraviolet, far-UV, and optical emission lines from TW Hya are well documented (Alencar & Batalha 2002; Herczeg [et al.]{} 2002; Dupree [et al.]{} 2005a; Ardila et al., 2013) their origin has remained elusive. Intensive study of the X-ray spectrum of TW Hya with CHANDRA coupled with simultaneous optical spectra strongly suggests that the source of its broad optical and X-ray emission arises from the turbulent postshock cooling volume in the stellar atmosphere (Brickhouse [et al.]{} 2010; Dupree [et al.]{}2012), and not from ‘accretion funnels’ approaching the star. The broad emission can suffer some absorption from the accretion stream and can also be substantially modified by a stellar wind which appears to be driven by the accretion process. The structure of this wind, its characteristics as well as the amount of mass loss remain to be determined. These processes have substantial implications for angular momentum loss from the star (Matt & Pudritz 2005; Matt [et al.]{} 2012; Bouvier [et al.]{} 2014), for the presence of dust in the surrounding circumstellar material (Alexander [et al.]{} 2005), and for an understanding of the accretion process itself and its consequences.
To summarize, the recent measurements suggest a more complicated accretion process than found in the ‘standard model’ described above. We envision a magnetically channeled accretion flow from the circumstellar disk towards the star. This is defined as an accretion funnel. This flow of plasma accelerates, free-falling towards the star and forms a shock - an abrupt increase in plasma temperature and density. The post-shock plasma, a turbulent medium, appears to cool radiatively. The subsequent effects of the accretion shock are many: a turbulent cooling region that produces the broad ultraviolet emission, the broad optical emission lines, and the broad near-IR helium emission; a heated spot or ring in the stellar photosphere that creates the veiling or filling-in of photospheric lines; a large heated volume in the stellar corona detected in ; and acceleration of a stellar wind detected as absorption in H[$\alpha$]{}, and ultraviolet line profiles.
In this paper, many new sequences of optical, near infrared, and far ultraviolet spectra are presented to infer the characteristics of the accretion process and wind on several time scales. The spectra allow identification of new structures in the stellar atmosphere, give insight into the accretion parameters, and probe the wind from the star. Spectral observations in the far-UV, optical, and near-IR regions are described in Section 2; the source of the H[$\alpha$]{} line is discussed in Section 3. Spectral indications of the accretion funnels themselves are presented in Section 4. The H$\delta$ transition is compared to predictions of current MHD models in Section 5. The variable warm wind[^2] extending to greater heights in the chromosphere than H-[$\alpha$]{} is shown by the near-infrared transition of (Section 6). Section 7 presents evidence inferred from the far ultraviolet spectra bearing on the variable post-shock conditions. Section 8 evaluates a previous claim of wind presence and characteristics. Discussion and Conclusions occur in Section 9.
Observations
============
A variety of spectroscopic observations (Table 1) has been acquired to assess the structure and dynamics of the atmosphere of TW Hya. Optical spectra were obtained during many observing runs at the Magellan-CLAY 6.5m telescope at Las Campanas Observatory, Chile. MIKE, the double-echelle spectrograph (Bernstein [et al.]{} 2003) was used with a 0.75$\times$5 slit that yields a 2-pixel resolution element of $\lambda/\Delta\lambda \sim $30,000 on the red side ($\lambda\lambda$4900–9300) which contains the H$\alpha$ line and $\lambda/\Delta\lambda\sim$37,000 on the blue side ($\lambda\lambda$3200–5000) used for the higher Balmer lines. The IDL pipeline developed by S. Burles, R. Bernstein, and J. X. Prochaska[^3] was used to extract the spectra, and IRAF software[^4] was employed in the analysis. Additional optical spectra were taken at the FLWO 1.5-m telescope with the Tillinghast Reflector Echelle Spectrograph (TRES). TRES is a temperature-controlled, fiber-fed instrument with a resolving power R$\sim$44,000. The spectra were reduced with a custom IDL pipeline.[^5]
Near-infrared spectra were obtained with PHOENIX at Gemini-S during two classical observing runs. The PHOENIX setup consisted of a slit width of 4 pixels yielding a spectral resolution of $\sim$50,000. The order-sorting filter J9232 was selected which spans 1.077–1.089$\mu$m and allows access to the line at 1.083$\mu$m. Standard procedures were followed by acquiring target spectra using a nodding mode (A–B) with a spatial separation of 5 arcsec. Because standard comparison lamps have a sparse wavelength pattern in this near-infrared region, we observed a bright K giant containing many securely identified narrow photospheric absorption lines in order to determine the wavelength scale. Other near-infrared spectra were taken with NIRSPEC (McLean [et al.]{} 1998, 2000) on the Keck II telescope. The echelle cross-dispersed mode of NIRSPEC with the NIRSPEC-1 order-sorting filter was selected and the slit of 0.42$\times$12 gives a nominal spectral resolution of 23,600. The long-wavelength blocking filter was not used in order to minimize unwanted fringing. Internal flat-field lamps, NeArKr arcs, dark frames, and K0 giants were used as calibration exposures. Spectra were taken in a standard nodding mode. Data reduction was carried out with the REDSPEC package specifically written for NIRSPEC (McLean [et al.]{} 2003).
Far-UV spectra of TW Hya from FUSE were taken from the MAST archive and the extraction procedure was fully described elsewhere (Dupree [et al.]{} 2005a). Some HST spectra of TW Hya obtained with STIS were also taken from the MAST archive at STScI, and from the CoolCAT UV spectral catalogue (Ayres 2005).[^6] Other HST spectra came directly from the MAST archive at STScI. Details of these spectra are also given in Table 1.
Identifying the Source of H[$\alpha$]{}
=======================================
The H[$\alpha$]{} profile is useful for probing the accretion process. A broad H[$\alpha$]{} line has long been considered a signal of ongoing accretion in young stars (Bertout 1989; Hartmann [et al.]{} 1994; Alencar & Batalha 2002), and the cause of the broadening has frequently been ascribed to ‘turbulence’ (Alencar & Basri 2000). In this section, we present the most extensive study of TW Hya in the optical region. The H[$\alpha$]{} profile was measured on 41 nights over a decade (Figure 1), with anywhere from 2 to 300 spectra per night. Most of the observations were taken on sequential days which enables the identification of dynamic events in a stellar atmosphere. As has been noted almost two decades ago (Johns-Krull & Basri 1997), snapshot spectra do not provide a characteristic representation of a complex accreting system, and time sequences are invaluable. The H[$\alpha$]{} profiles in TW Hya show dramatic changes in the strength of the emission and the presence of absorption features at negative (outflowing) velocities.[^7] The signature of a wind in the H[$\alpha$]{} profile has been noted previously (Hartmann 1982; Calvet & Hartmann 1992; Gullbring [et al.]{} 1996; Alencar & Basri 2000; Alencar & Batalha 2002) but only recently has the hourly development of wind absorption became evident (Dupree [et al.]{} 2012). The shape of the emission profile indicates material motions of gas at chromospheric temperatures. Comparison of profiles in Figure 1 and evaluation of the flux at negative and positive velocities demonstrates the presence of roughly symmetric H[$\alpha$]{} profiles (to within 10%) centered on TW Hya for 42% of the spectra and a positive flux enhancement (‘red enhanced’) for 52% of the spectra. The full width at half maximum (FWHM) of the lines rangew from 200 to 300 [km s$^{-1}$]{}. This positional symmetry is consistent with formation on the star itself. When the profiles appear to be ‘red-enhanced’, this is caused by substantial absorption and scattering (on the “blue-side”) at negative velocities by outflowing material forming the stellar wind. A particularly instructive sequence occurred in 2007 Feb (Dupree [et al.]{} 2012). At that time, the H[$\alpha$]{} profile was roughly symmetric on the first night of observation, then the wind opacity systematically increased over the following 3 nights.
These observations demonstrate that the H[$\alpha$]{} profile from TW Hya is inherently broad and symmetrically positioned on the stellar velocity, but can be substantially absorbed by the stellar wind. In addition, changes in the long-wavelength side of the profile (cf. Figure 1) can result from a combination of enhanced post-shock emission and changing absorption by the accretion stream. The pole of TW Hya is observed directly. It is here that accretion occurs most of the time (Donati [et al.]{} 2011), and the shock and post-shock cooling region can be seen. The source of this broad emission has been long debated. With the assumption that emission arises from the accretion stream, models tend to predict pointed profiles from a star observed at low inclination. Frequently absorption on the red side that results from the infalling material emerges from the calculations (Muzerolle [et al.]{} 2001). This makes H[$\alpha$]{}appear shifted to shorter wavelengths in contrast to most of these observations. Authors have noted that the magnetospheric accretion models do not adequately represent spectral profiles found in several T Tauri stars (e.g. Alencar & Basri 2000). Changes in damping parameters and the introduction of Stark broadening, can manage to broaden the H[$\alpha$]{} profile but the observed profiles in TW Hya have flatter peaks and are frequently wind absorbed in contrast to the calculated profiles (Figure 2).
During many nights the observed H[$\alpha$]{} profile remains centered on the stellar radial velocity and appears symmetric without a distinct absorption feature. As previously suggested (Dupree [et al.]{} 2012), the broad H[$\alpha$]{} lines can result naturally from turbulent conditions in a post-shock cooling volume. This identification derived from the time study of an X-Ray accretion event which was followed by sequential changes in emission line profiles and fluxes, veiling, and subsequent coronal X-Ray enhancement. The breadth of the H[$\alpha$]{} line is slightly larger ($\sim$200$-$275 [km s$^{-1}$]{}), but comparable to that of X-ray lines arising in the shock (126 to 229 [km s$^{-1}$]{}) while slightly less than the full width of the symmetrized far-ultraviolet lines ( and ) observed to be $\sim$ 325 [km s$^{-1}$]{} in TW Hya (Dupree [et al.]{} 2005a). Moreover, the amount of material in the post-shock region inferred from the X-ray diagnostics (Brickhouse [et al.]{} 2010) as well as near-infrared measurements (Vacca & Sandell 2011) exceeds by a few orders of magnitude the material expected in an accretion column (cf. Dupree [et al.]{} 2012). These facts strongly suggest that the symmetric H[$\alpha$]{} emission line arises predominantly from the turbulent post-shock cooling region. It has been known for a long time that models of a stellar chromosphere alone can not produce strong hydrogen emission lines (Calvet [et al.]{} 1984). A symmetric H[$\alpha$]{} profile centered on the stellar radial velocity is expected from TW Hya because the postshock region is in our full view. When a wind from the star is present, the short wavelength side of the profile is weakened, and the resulting profile is stronger on the long wavelength side of the line as frequently appears in TW Hya. Absorption by infalling material can affect the profile as well.
Systems with different inclinations might be expected to produce similar turbulently broadened profiles if the post-shock region is not obscured by a disk. In fact, broad H[$\alpha$]{} has long been known (Bertout 1989; White & Basri 2003) to signal the presence of a classical accreting T Tauri star. Differences in the profile may arise from wind scattering which would be determined by the line-of-sight component of the expansion velocity. But as is evident from Figure 1, where the line-of-sight reflects the maximum outflowing velocity from TW Hya, the wind velocity and opacity can change substantially with time. The profile, discussed in Section 6 offers an interesting test of the effects of inclination.
Detection of Shadowing by Accretion Funnels
===========================================
The H[$\alpha$]{} profiles also reveal structures that occur in the stellar wind and their behavior becomes apparent in the spectral sequences lasting several days. Alencar and Batalha (2002) remarked on a broad (FWHM $\sim$125 [km s$^{-1}$]{}) absorption feature in H[$\alpha$]{} that appeared in nightly spectra suggesting variation in wind opacity. Time domain spectroscopy presented here confirms the changes in the wind, and also reveals another feature. Profiles in Figure 1 show an absorption feature that occurs at $\sim$$-$50 to $-$100 [km s$^{-1}$]{}. This feature can change in strength and velocity from night to night, but is constant in velocity over a single night. Particularly instructive are the nights following 2004 Apr 27 (Figure 3) where the absorption feature appears at $-$80 [km s$^{-1}$]{}, vanishes on Apr 28, re-appears on Apr 29 at $-$150 [km s$^{-1}$]{}, and on Apr 30 at $-$100 [km s$^{-1}$]{}. Similar behavior occurred in 2007 Feb and 2009 Jun (Figure 1) and also in 2006 (Figure 4). These absorption features in the wind remain at constant velocity during the night. The three-night sequence from 2006 April is presented in gray scale representation in Figure 5. The first night shows a narrow ($\sim$50 [km s$^{-1}$]{}) absorption feature at $-$125 [km s$^{-1}$]{} at constant velocity; this feature becomes weaker early in night 2. However, during the second night of observations, the absorption at $-$50 [km s$^{-1}$]{} at the start of the night is replaced by an absorption feature at $-200$ [km s$^{-1}$]{} by the end of the night. The right panel of Figure 5 which contains a line plot of the 8 hours of observation during night 2 illustrates this jump quite clearly. In addition, the variable infall of material at high velocities can be seen in the profile. The short rotation period of TW Hya (3.57 days; Huélamo [et al.]{} 2008) causes different viewing angles to occur during the 3 successive nights of our observations. The narrow absorptions appear constant in velocity suggesting they are located at high latitudes on the star. If the features moved outward within the wind, the speeds of $\sim$150 [km s$^{-1}$]{} would allow material to cover many ($\sim$6) stellar radii over an 8-hour span of observations and this motion would also be visible in the line profiles. But such a shift is not observed.
Such static behavior differs from high-lying stellar prominences found in rapidly rotating stars such as AB Dor (Collier Cameron & Robinson 1989) and in rapidly rotating T Tauri stars (Oliveira [et al.]{} 2000; Skelly [et al.]{} 2008; Günther [et al.]{} 2013) where the absorption feature moves rapidly in velocity from negative to positive (or vice versa) as the feature traverses the stellar disk. The absorption features in TW Hya are stable in velocity for hours, and are not at all similar to the moving Discrete Absorption Components (DACs) observed in the winds of O stars (cf. Howarth [et al.]{} 1995) which generally are very broad ($\sim$400 [km s$^{-1}$]{}) and move at thousands of [km s$^{-1}$]{} through the atmosphere (cf. Kaper [et al.]{} 1999); only absorption features in one Wolf-Rayet star (HD 50896) are broad, but static (Massa [et al.]{} 1995).
[*The static nature of these relatively narrow absorptions in TW Hya suggests they arise in a stable cool structure located in the wind itself that is silhouetted in the hydrogen line profile.*]{} These could be the signature of the accreting columns crossing the path of the wind. In an accelerating wind, features located at $-$50 to $-$150 [km s$^{-1}$]{}, would exist at about 1.2-1.3 R$_\star$ (Dupree [et al.]{} 2008), judging from the chromospheric helium line asymmetries, and above the shocked material produced by accretion (Sacco [et al.]{} 2010). The strength of the absorptions, from 0.4 to 0.6 of the local continuum provided by the H[$\alpha$]{} emission, indicates an optical depth, $\tau$ of $\sim$0.7. Here we take $\tau = \kappa \rho H$ where $\kappa (T, n_2/n_1)$ is the line absorption coefficient (incorporating a dependence on temperature and the level population ratio of hydrogen), $\rho$ is the hydrogen density, and $H$, the thickness of the line-forming region. The temperature and density structure of the ‘funnel flows’ is uncertain. However, the preshock electron density was determined (Brickhouse [et al.]{} 2010) to be 5.8$\times$10$^{11}$ cm$^{-3}$ which implies a hydrogen density of 5$\times$10$^{11}$ cm$^{-3}$ for a plasma with 10% helium. For a temperature T $\sim$7000K, the observed broadening and our non-LTE calculations[^8] for chromospheric hydrogen suggest that the n$_2$/n$_1$ level populations of the H[$\alpha$]{} line range from 6.4$\times$10$^{-7}$ to 1.8$\times$10$^{-8}$ in this temperature and density regime (Dupree [et al.]{} 2008). The scale length to produce the narrow absorption is small, less than 0.01 R$_\star$. Another model of the accreting stream (Hartmann [et al.]{} 1994) gives T = 7000K and the level 2 population of hydrogen as 8 $\times$10$^5$ cm$^{-3}$ near the star, such that the inferred thickness is much smaller, $\sim$100 km. In either case, formation in a narrow accretion funnel appears plausible to account for these absorption features.
Chromospheres, winds, and wind structures produce asymmetric lines and absorption features that may be broad (such as a central reversal) or narrow as discussed here. The uniqueness of these narrow ‘notches’ relates to their constant velocity in the H[$\alpha$]{} line profile. A blocking structure located in the wind could be seen at all angles where it is observed against a region producing emission. Many studies of emission from T Tauri stars have been published but very few have the time sampling of multiple sequential spectra that are presented here.
The H$\delta$ Line
==================
Sophisticated models of magnetospheric accretion processes have also been constructed and used to predict hydrogen line profiles. A sequence of Balmer lines calculated by Kurosawa & Romanova (2013) suggested that the H$\delta$ transition at 4101Å is an optimum line to reveal persistent red-shifted sub-continuum absorption resulting from the inflowing accretion stream in the ‘standard’ model. Absorption is expected when the hotspot is in the line of sight of the observer as is the accretion stream. TW Hya is a good test of these assumptions. Because of its pole-on orientation, the accretion stream and hot spot are in the line of sight. Our MIKE spectra capture the pivotal H$\delta$ line, as well as the complete Balmer series. Figure 6 shows four lines of the Balmer series (H$\alpha$, H$\beta$, H$\gamma$, H$\delta$) taken over 4 sequential nights in 2007 (Dupree [et al.]{} 2012). During this period the H$\alpha$ line displays a slight inflow asymmetry (blue side stronger than red side) early on the first night, followed by the subsequent onset of a wind. H$\beta$ and H$\gamma$ exhibit an absorption feature at $\sim +$50 [km s$^{-1}$]{} which signals enhanced inflow following the X-ray accretion event (Dupree [et al.]{} 2012). However the predicted sub-continuum absorption signalling infalling material is not evident in the H$\delta$ line. The period of radial velocity variation, namely 3.57 days (Huélamo et al. 2008) could cause a change in the absorption strength if there were substantial modulation in the orientation of the hot spot and the accretion column. The lack of an absorption feature in the H$\delta$ profile suggests that the H$\delta$ emission does not arise from infalling material, but must originate from another region. Another sequence of H$\delta$ profiles over 4 consecutive nights is shown in Figure 7. The first night of the sequence shows enhanced emission with shallow broad absorption between $+$100 to $+$200 [km s$^{-1}$]{}, inferred from the line asymmetry. This pattern may be the signal of an accretion event. However, subsequent nights (July 14$-$16) reveal a symmetric profile, indicating the infall has lessened, and the symmetry and FHWM ($\sim$150 [km s$^{-1}$]{}) are consistent with formation in a turbulent post-shock cooling volume. Variability of the accretion rate is also documented by the X-ray diagnostics (Brickhouse [et al.]{} 2012) which have shown a five-fold reduction in the rate from exposures separated by 2.7 days. Other studies have measured H$\delta$ in T Tauri stars (Edwards [et al.]{} 1994; Petrov [et al.]{} 1996; Alencar & Basri 2000) and the profiles in the majority of stars do not resemble the models presented in Kurosawa & Romanova (2013).
Variable Warm Wind in Near IR Helium
====================================
The broad near-IR line of at 10830Å, because of its width (FWHM $\sim$200 [km s$^{-1}$]{}), also appears formed in the post-shock material. This feature has become a useful probe of the wind and infall environment of accreting T Tauri stars (Dupree 2003; Edwards [et al.]{} 2003; Dupree [et al.]{} 2005a; Edwards [et al.]{} 2006). For TW Hya, we have acquired near-IR spectra spanning 8 years including sequences of consecutive nights. A comparison of the spectra obtained during this 8 year span is shown in Figure 8. Over this long time scale, practically every characteristic of the P Cygni profiles changes: the emission level, the wind speed and opacity, and the appearance (or not) of inflowing material. The emission strength varies by more than a factor of 2; the terminal velocity of the wind varies from $-$200 to $-$315 [km s$^{-1}$]{}; the wind opacity changes over the absorption profile; the infall terminal velocity varies from +270 to +363 [km s$^{-1}$]{} as does the opacity of the infalling material. Speaking precisely, an [*outflow*]{} of material is observed in the near-infrared helium profile of TW Hya. The escape velocity from the photosphere is 530 [km s$^{-1}$]{} for TW Hya assuming M=0.8M$_\odot$ and R=1.1R$_\odot$ (Donati [et al.]{} 2011). At a distance of 1 stellar radius above the surface, this value decreases to 400 [km s$^{-1}$]{}, which is comparable to the observed typical terminal velocities. It is highly likely then that the outflows observed in the near-IR helium line and the sequential acceleration with temperature displayed by the UV lines in Dupree [et al.]{} (2005a) form a true stellar wind.[^9]
The time-domain spectroscopy revealing these changes suggests that all T Tauri objects undoubtedly exhibit similar variability that snapshot studies (Edwards [et al.]{} 2006) can not discern. The extent of the variation also blurs the search for correlations of wind parameters with geometric inclination of the star or disk (Appenzeller & Bertout 2013).
The helium absorption profile is broad, signalling its origin in a stellar wind, through its classic P Cygni shape. There is no sign of a narrow low-velocity ($\sim$10 to 100 [km s$^{-1}$]{}) feature such as identified in several spectra of other T Tauri stars (Edwards [et al.]{} 2006) and conjectured to originate from the circumstellar disk. In the case of TW Hya, H$_2$ and \[O I\] emission arise from the disk (Herczeg [et al.]{} 2002, Pascucci [et al.]{} 2011) since their velocities coincide with that of the star itself. The infrared \[\] transition exhibits a low velocity ($-$5 [km s$^{-1}$]{}) outflow which suggests the presence of a cool (1000 K) photoevaporative molecular wind (Rigliaco [et al.]{} 2013). It is hard to see how helium which is more highly excited would be produced in such a wind.
Several sequences of the line over consecutive nights are shown in Figure 9. At these times minimal variation of the line profiles occurred on a short time scale: a small change in the emission in 2009; changes in the inflow opacity in 2010. During 2009 and 2010 the wind absorption had constant high optical depth (a flat-bottomed profile) over a 50 to 100 [km s$^{-1}$]{} span, and subtle changes in opacity would not be observable. In contrast, the H[$\alpha$]{} profile over sequential nights reveals changes in wind opacity and absorption troughs (cf. Fig. 1 A-E, I, K, L). In 2009, the absorption feature at $-$50 [km s$^{-1}$]{} is also found in the H[$\alpha$]{} profile from those nights (see Figure 1). At that time, the H[$\alpha$]{} line revealed wind scattering that extends to $-$100 [km s$^{-1}$]{}, and the helium wind extended to $-$200 [km s$^{-1}$]{}, an indication of accelerated expansion.
The emission strength exhibits substantial change. Of course, the amount of emission depends on the intrinsic line strength which is then modified by the wind scattering and some absorption from infalling material. Close inspection of the profiles suggests that the emission itself is indeed changing. On one occasion, 2002 May 20, the profile displays emission above the continuum level extending to both $-$300 and $+$300 [km s$^{-1}$]{} indicating a strong and broad intrinsic emission component.
The inflow signature in helium is a particularly useful diagnostic of the accretion process. In a standard magnetospheric accretion model (Brickhouse [et al.]{} 2012), the terminal velocity of the infalling material, determined by the free-fall distance from the circumstellar disk, controls the temperature of the shock. X-ray spectra indicate that the shock temperature, measured directly from the ratio of forbidden plus intercombination to resonance line fluxes of , varied by a factor of 1.6 over a span of 13 days between CHANDRA pointings (Brickhouse [et al.]{} 2012). The shock temperatures vary between 1.9MK and 3.1MK. In this standard model, the post-shock temperature is given by $T_{post} \sim 3mv_{ff}^2/16k$, where $v_{ff}$ is the free-fall velocity, $k$ is the Boltzmann constant and $m$ is the mean atomic mass (Brickhouse [et al.]{} 2010). The measured shock temperatures from 3 CHANDRA pointings in 2007 suggest that terminal free-fall velocities from 380 to 485 [km s$^{-1}$]{} occurred.
In the model (Brickhouse [et al.]{} 2012), when the free-fall velocity is low, the disk supplying the accreting material originates closer in to the star. The filling factor on the stellar surface increases which implies an increase in the mass accretion rate, and also in the material in the post-shock cooling zone. A range of parameters for this model is shown in Figure 10 ([*left panel*]{}). This scenario is confirmed by the behavior of the helium line. [*The amount of emission in the line appears inversely linked to the terminal velocity of the inflowing material*]{} (Figure 10, [*right panel*]{}). Lower values of the inflow terminal velocity lead to much stronger emission arising in the post-shock region, which is consistent with the higher mass accretion rate as predicted in the dipole model. The helium absorption indicates lower velocities than the terminal velocities of the models suggesting that the absorption arises from a volume above the accretion shock. It is also quite likely that the neutral helium is ionized by the X-ray emission from the shock as the helium approaches in the pre-shock stream (Lamzin 1999; Gregory [et al.]{} 2007; Brickhouse [et al.]{} 2010), and as a result does not reach a terminal free-fall velocity.
It is interesting that the helium line gives evidence of subcontinuum absorption due to inflow, when the hydrogen lines generally do not show such strong signatures of inflowing material. On 2007 March 1, we obtained (almost) simultaneous spectra of helium (Figure 8) and the hydrogen lines (shown as night 4 in Figure 6). The reversal of the negative velocity side of the Balmer lines on to the positive velocity side (Figure 6) indicates weak broad absorption near $+$200 [km s$^{-1}$]{} on the H$\beta$, H$\gamma$, and H$\delta$ transitions. However the helium line displays substantial absorption extending to $+$325 [km s$^{-1}$]{}. The temperature structure along the accretion column is not known but it appears likely that helium forms at higher temperatures where the hydrogen is ionized, and these temperatures are associated with higher infalling velocities. This could create a higher opacity in helium with the ability to probe a different, potentially more extended accreting region than represented by the hydrogen lines.
Because stronger emission signals enhanced accretion of material, this material might cause an increase in both the wind opacity and wind speed if the accretion contributes to wind acceleration. There is a hint of this in nine out of the ten near-IR spectra of (Figure 11). The one outlier with the fastest outflow speed belongs to the 2007 March 1 observation discussed above. Not only does this profile exhibit the highest outflow velocity, but also the largest subcontinuum ‘blue absorption’ of all ten nights of observations. It is difficult to quantify this further because the intrinsic profile of the helium line is unknown. The profile exhibits absorption on the ‘red’ side caused by accretion, and substantial scattering on the ‘blue’ side in the stellar wind. Figure 11 shows that increased emission in 10830Å may be linked directly to a faster outflowing terminal velocity, giving evidence for an accretion-driven stellar wind.
Variable Post-Shock Conditions from UV lines
============================================
The UV and far-UV lines also exhibit changes in flux as evidenced by HST and FUSE spectra. The expanding atmosphere of TW Hya is revealed by the asymmetric profiles of the major resonance lines (Dupree [et al.]{} 2005a), but substantial changes in the profiles are evident at different times. Figure 12 ([*left panel*]{}) contains resonance line profiles of (1548Å) and (1239Å) measured with HST/STIS on 8 May 2000. The line profiles are strikingly similar to one another, confirming formation in the same region of the atmosphere. Ardila [et al.]{} (2013) also remarked recently on this / similarity from a much larger sample of accreting stars. The temperatures of formation of and in a collisionally ionized plasma are similar at 1–2$\times$10$^5$K. The profiles exhibit both extended emission to positive velocities, and the characteristic sharp cutoff caused by wind absorption at negative velocities. More wind opacity is evident in the profile than in . This is to be expected since the strength of absorption is proportional to the value of $gf\lambda \times N_i/N_H$ where $N_i$ is taken to be $N_C$, $N_O$, or $N_N$ giving the appropriate elemental abundance. The value of $gf\lambda \times N_C/N_H$ for carbon is larger than the similar quantity for the transition. Note the extended wing at positive velocities on this date giving a half-maximum value of $+$200 [km s$^{-1}$]{}. Contrast these profiles with the resonance lines of and on 20 Feb 2003 (Figure 12, [*right panel*]{}) measured with [[*FUSE*]{}]{}. Both of these lines are also similar to one another, and their formation temperatures are typically 8$\times$10$^4$–3$\times$10$^5$K in equilibrium conditions. The profile exhibits more wind opacity than which again can be understood from the difference in the quantity $gf\lambda \times N_C/N_H$ which is larger for than for by a factor of 2.8. The relative amount of absorption in the wind is in harmony with atomic physics. Thus, [*based on these spectra, we conclude that the UV and far-UV lines are produced in the same plasma region, but conditions in this region change dramatically from time to time.*]{}
Intercomparison of two of the ultraviolet profiles (Figure 13) reveals the meaningful differences. Excess emission on the positive velocity side of the profile in 2000 does not occur in the 2003 observation. The half-maximum value of the profile in 2003 extends only to $+$125 [km s$^{-1}$]{}, whereas extended to $+$200 [km s$^{-1}$]{} in 2000. Additionally there is greater wind scattering on the negative velocity side of the profile in as compared to . While changes in the wind opacity between the two observations are unknown, the atomic physics of these lines must be a first consideration. possesses a 17% larger atomic parameter ($gf\lambda \times N_i/N_H$) than (where $N_i$=$N_C$ or $N_O$) which contributes to its greater opacity. We also do not know the distribution of material with temperature in the wind which can affect both the ionization equilibrium and the formation of the lines.
Variation of the and emission on shorter time scales can be found in Figure 14 from HST/STIS spectra. Substantial changes are apparent on a time scale of years between the HST measurements, and also over a time scale of 6 days which separates the 2010 Jan 29 and 2010 Feb 4 observations of . The emission appears to strengthen with , however the 4 simultaneous measurements show scatter. The line is not broadened as much as the emission, suggesting that the contribution from the turbulent post-shock cooling region is less. The positive velocity side of the emission ([$\lambda$]{}1335.7) resembles the positive velocity side of the H$\delta$ line (cf. Figure 7) indicating that both may be formed in a region distinct from the higher temperature species (, , etc.).
A closer look at the variations in the high temperature emissions is provided by the sequence of individual segments of the [[*FUSE*]{}]{} spectra. Under collisionally-dominated equilibrium conditions, is formed at a temperature of 3$\times$10$^5$K. However, the line continues to be formed in hotter coronal plasmas due to the extended wing of Li-like ions in the ionization balance at high temperatures. has been used consistently as a diagnostic of the dynamical properties of outflows in the solar corona (Kohl [et al.]{} 2006) and X-ray emitting shocks in OB-star winds (Lehner [et al.]{} 2003). Evidence of the dramatic changes in emission from TW Hya can be found from the individual and profiles assembled over a 32 hour span and shown in Figure 15. Each segment represents an integration time ranging from 2600 to 3700 s. The impression that varies more than is quantitatively confirmed by an extraction of the line fluxes. varies by a factor of $\sim$1.8 and by a factor of $\sim$1.5. Evaluating the negative and positive velocity sides of each line shows that the fluxes are not correlated between the sides. However the uncertainty in flux on the negative velocity side is larger because the total counts are lower. And the total counts in the line are a factor of 4 or more than in the line, making the parameters a more secure measurement. The relatively constant profile of the outflowing wind absorption at negative velocities contrasts dramatically with the changing emission at positive velocities during this time. Such profile variation provides additional evidence that the emission arises from the turbulent post-shock accretion volume, and the the line is scattered in the hot wind which is present during these observations. Similar behavior in 5876Åwas found in another accreting T Tauri star, RU Lupi (Gahm [et al.]{} 2013).
Fluxes taken from the individual spectra are shown in Figure 16 where in the first pointing the ([$\lambda$]{}977) line is stronger than ([$\lambda$]{}1031.91); this behavior is reversed in the second pointing about one day later. Here we have selected segments where the SiC detector (containing the line) is well-aligned with the LiF detector (containing the line).[The FUSE telescope contained 4 coaligned optical channels. Pointing on a target was maintained by the fine error sensor which viewed one of the channels. Because thermal changes on orbit caused rotation of the mirrors, this could lead to misalignment of the optical channels and a target could drift in and out of the aperture in the channels not used for guiding. This could affect the measurement of the line since it occurs in a channel not used for guiding. In our analysis, segments for were selected when the channels were coaligned.]{} However, the atom has a metastable level whose population is dependent on density which could contribute to the behavior of . And of course, the post-shock cooling region can change as well. During the second pointing when both lines could be measured simultaneously over $\sim$5 hours, the fluxes are correlated with a linear correlation coefficient of 0.9.
Remarks concerning a Hot Wind
=============================
The presence of a hot wind from TW Hya has been questioned by Johns-Krull and Herczeg (2007), hereafter JKH. These authors offer several arguments purporting to show that the wind is not hot. We argue below why we believe that their concerns are not valid, and discuss four relevant issues.
[**(1)**]{} As evidence for outflow, JKH require sub-continuum absorption in a line profile to be present at ‘blue-shifted’ wavelengths. While that phenomenon is well-understood from characteristic P Cygni line profiles, the signature of outflowing material can be revealed in other ways. A shift of the apparent wavelength of the centroid of an emission or absorption line can reveal mass motions. Mass motions can also be signaled by line asymmetries. Hummer and Rybicki (1968) first pointed out that apparently redshifted line profiles can arise in a differentially [*expanding*]{} atmosphere; in fact these have been widely observed in luminous stars (cf. Mallik 1986, Robinson & Carpenter 1995, Dupree [et al.]{} 2005b, Lobel & Dupree 2001) and in the Sun (Rutten & Uitenbroek 1991). The assertion by JKH that sub-continuum absorption must be present to identify outflow neglects both line-transfer physics and stellar observations themselves.
[**(2)**]{} In an attempt to demonstrate the absence of absorption in resonance doublets, JKH arbitrarily scale and overlay the long wavelength member of the doublet on to the short wavelength member of the doublet by forcing both the line peaks and blue wings to match. They claim that signs of absorption are missing in the red wing of the 1548Å line that would be caused by the wind absorption in the 1550Å line. Such arbitrary scaling is not justified. Atomic physics, supported by laboratory measurements, specifies the wavelength separation between these lines as well as the ratio of the line oscillator strengths. In an effectively optically thin doublet such as , the line flux ratio ([$\lambda$]{}1548/[$\lambda$]{}1550) is 2. Since it is preferable not to amplify the errors in the weaker line of the doublet, here we place the stronger (1548Å) line onto the weaker (1550Å) transition. The original HST data used by JKH, namely [*O58D0130*]{}, has been superseded by a more optimal spectrum reduction, namely CoolCAT (Ayres 2010) and we also consider that reduction (see Table 1). In Figure 17, we show the CoolCAT reprocessed profile as well as the original spectrum used by JKH.[^10]
When the 1548Å line is properly overlaid on the 1550Å line by shifting it according to the wavelength separation (2.577Å, Griesmann & Kling 2000), and scaling the flux of 1548Å by 2, two facts are immediately apparent from either data reduction: (1) the wind opacity in the resonance line 1548Å, indicated by the negative velocity side of the profile, is greater than in the subordinate line 1550Å, as expected from atomic physics, and (2) excess wind absorption does indeed occur in the long wavelength wing of 1548Å caused by wind absorption from the 1550Å line. Consideration of the errors in the measured fluxes at individual wavelengths in the HST/STIS spectrum indicates that the separation visible in Figure 17 between the 1550Å line and the shifted scaled 1548Å line ranges from $\sim$2.5 to 3.2$\sigma$ in the velocity region from $+$175 to $+$350 [km s$^{-1}$]{}. Moreover, in this region the dashed line in Figure 17 systematically lies below the solid curve. Thus the effects of hot wind absorption are clearly evident in the profiles.
[**(3)**]{} Trying to demonstrate the absence of wind absorption by the 1037.61Å transition affecting the 1037.02Å line, JKH resort to an [*ad hoc*]{} construction. In this case, both the activity of TW Hya and atomic physics undermine their arguments. The 1037.61Å line, has a neighboring line separated by $-$172 [km s$^{-1}$]{} which places the line in the velocity range of scattering produced by a hot wind. To attempt to show that the line is not weakened by scattering, JKH predict the intrinsic strength of a blended and feature. By taking the line profile at 1335.71Å observed in 2000 with HST/STIS, these authors scale the profile to mimic the transition at 1037.02Å, and add it to the line profile of 1031.91Å observed in 2003. The resulting scaled combination is then compared to the blended (1037.61Å) and (1037.02Å) emission feature. JKH do not present the parameters used for this scaling, shifting, and combination.
From first principles, the intrinsic relative strength of the 1335.7Å line to the 1037.02Å line is hard to predict. The ratio of $gf$[$\lambda$]{} values gives a factor of 5.3 for the $\lambda\lambda$1335.7/1037.02 ratio, and the Boltzmann factor ratio could be 2.4 at a temperature of 40000K suggesting the 1037.02Å line could be 13 times weaker than the 1335.71Å line under optically thin conditions in LTE, and not 1.5 times weaker as JKH propose. However unknown optical depths and non-LTE conditions occur in the formation of resonance lines of causing them to appear weaker and decrease the $\lambda\lambda$1335.7/1037.02 ratio. These issues confound any meaningful scaling. Moreover, the line profiles themselves change with time as demonstrated earlier. Comparison of HST spectra from 2000 to FUSE spectra from 2003 (cf. Figure 13) clearly shows the substantial changes in the emission profiles of ultraviolet lines on the ‘red’ sides ($+$100 to $+$400 [km s$^{-1}$]{}) between these three years. It is obvious from HST spectra of in 2000, 2002, and 2010 (Figure 18) that the critical region of the 1335.7Å line profile used to assess absorption, namely the region from $+$35 to $+$85 [km s$^{-1}$]{}, varies in both absolute flux as well as slope. The claim made by JKH — that this scaled, shifted, blended feature does not exhibit strong absorption on the red side caused by the hot wind — is unjustified, in particular given the substantial changes in both the emission and the wind.
[**(4)**]{} JKH suggest that two H$_2$ lines arising from the circumstellar disk exhibit behavior that suggests a cool wind is present, but not a hot wind. Their interpretation does not agree with the geometry of TW Hya and its disk, nor with the known characteristics of a wind from a dwarf star.
The H$_2$ line at 1333.85Å occurs $-$165 [km s$^{-1}$]{} from the line at 1334.5Å (at $-$430 [km s$^{-1}$]{}in Figure 18). JKH claim that this H$_2$ feature is absorbed by the wind because their model overpredicts its strength in their spectrum taken in 2000. By contrast, an H$_2$ line lying at a velocity of $-$165 [km s$^{-1}$]{} from the 1548Å feature does not suffer wind absorption when compared to their model (Herczeg [et al.]{} 2002). Thus JKH conclude that a cool wind exists, but not a hot wind.
Many spectra subsequent to the one used by JKH reveal conflicting patterns (see Fig. 18). In 2002, the wind extends to $-$251 [km s$^{-1}$]{}, which would weaken not only the line located at 1333.85Å, but should weaken another H$_2$ line located at $-$225 [km s$^{-1}$]{} (corresponding to $-$500 [km s$^{-1}$]{} in Figure 17), but it does not. Additionally, in the presence of a strong wind, another H$_2$ line emerges at 1335.2Å ($\sim -$125 [km s$^{-1}$]{} in Fig. 18; the R(2) 0-4 transition) which did not appear previously. All of these spectra indicate optically thick winds in are present and the wind opacity is variable. Furthermore, the H$_2$ lines behave in apparently unpredictable fashion – perhaps because they are produced through photoexcitation by variable Lyman-[$\alpha$]{} emission. With a more precise location of the H$_2$ emission, the behavior of the molecular lines might aid in the wind description. However, since these lines are pumped by H-Ly[$\alpha$]{}, which itself varies, trying to pin down a consistent model may prove challenging. Moreover, serious considerations discussed earlier in (3), rule out the putative effects of wind absorption.
Trying to assess potential effects of a wind upon the observed H$_2$ emission involves many factors. First, the line opacity in the wind must coincide in wavelength with the H$_2$ emission from the disk. H$_2$ is believed to be produced less than 2 AU from the star (Herczeg [et al.]{} 2002) where it originates in the gaseous component of the circumstellar disk around TW Hya. Recent ALMA observations suggest the dust-depleted cavity lies within 4 AU of the star (Rosenfeld [et al.]{} 2012). Ions in a radially-moving stellar wind passing over a distant disk will absorb at lower velocities than when observed against the nearly pole-on star itself. Rosenfeld [et al.]{} (2012) suggest the TW Hya disk might have a warp of $\sim$8 degrees, such that the effective velocity of any absorption would occur at a still lower velocity. Thus the wind may not have significant opacity at the position of the H$_2$ emission when passing across the disk at a low angle. Moreover, any channeling, flux-tube expansion, or inhomogeneities (all of which occur in the solar wind) will modify the amount of absorption and the velocity at which it occurs. A more fundamental concern appears to be the ionization stage in the wind itself. If the stellar wind at 2 AU does not contain or , then absorption will not occur. From very detailed knowledge about the wind of one star, our Sun, the low species ions are effectively absent at a distance of 1 AU (Landi [et al.]{} 2012) and the wind contains the ionization stages of the corona. Thus at a distance of 1 AU or larger, the wind of TW Hya, if similar to the solar wind, would appear to lack the ions that JKH invoke for their arguments.
The presence of a hot wind from TW Hya is not unexpected. The Sun is a low mass star, with weak magnetic fields and it possesses an outer atmosphere with temperatures of a few MK. Wave-heating mechanisms (Kohl [et al.]{} 2006) or possibly nano-flares resulting from reconnection (Tripathi [et al.]{} 2010) can heat the corona. Self-consistent models for hot winds have been constructed for accreting T Tauri stars (Cranmer 2008). These winds are turbulence driven, and the turbulence has two sources: interior convection and ‘ripples’ from nearby accretion shocks. Both observations and theory offer much evidence that low-mass cool stars can support a hot outer atmosphere.
We conclude that the stellar wind of TW Hya is a hot wind, evidenced by the ultraviolet spectra as originally suggested (Dupree [et al.]{} 2005a).
Discussion and Conclusions
==========================
The many spectra of TW Hya reported here document the source and components of emission line features and the variability of the accretion process, subsequent atmospheric heating, and wind expansion. We find:
1\. Time-domain spectroscopy reveals symmetric H[$\alpha$]{} profiles in approximately half of our spectra and indications of outflowing wind asymmetries in the remaining spectra. This spectroscopic example indicates that the source of the broad emission feature arises from the TW Hya star itself, and can be identified with the post-shock cooling zone. Broad lines ranging from X-ray through infrared transitions can be straightforwardly explained by the presence of a variable turbulent post-shock cooling volume producing the emission which can be modified by wind scattering and/or material infall. Earlier evidence of a logical progression in line shape and flux following an X-Ray accretion event also supports this interpretation (Dupree [et al.]{} 2012). Models are now needed for the temperature structure and emergent emission from this post-shock region. A recent MHD simulation (Matsakos [et al.]{} 2013) of a two-dimensional accretion shock impacting a chromosphere suggests that substantial chaotic motions can result in the low chromosphere. These need to be followed up with calculation of the radiation emitted from the cooling plasma.
2\. Absorption features at negative velocities in the H[$\alpha$]{} line noted by Alencar et al (2002), are demonstrated with the frequent continuous spectral coverage reported in this paper, to be stable in velocity during a night’s observation. Moreover, their stable behavior and opacity characteristics suggest they appear to arise from the silhouette of an accretion stream high in the stellar chromosphere. At positive velocities, absorption in H[$\alpha$]{} mimics the infall speeds detected in .
3\. Current models (Muzerolle [et al.]{} 2001; Kurosawa & Romanova 2012, 2013) that consider the optical emission from accretion ‘funnels’ do not agree with the observed line profiles in TW Hya. The observed profiles lack the shift, the asymmetries and/or absorption predicted from accretion stream models. In addition, many models do not include a stellar wind, which is clearly indicated in the observed line profiles, nor emission from a post-shock cooling region.
4\. A hot wind, and its variable speed and opacity are revealed by the UV and far-UV resonance lines with spectra spanning a decade in time. Previous arguments offered (JKH) against the presence of a hot wind are shown to be unpersuasive. Broad line profiles and variability support both formation in a turbulent medium and the presence of a hot wind. This hot wind is a natural extension of the wind indicated by the near-IR helium line profile.
5\. The near-IR transition at 10830Å shows less night-to-night absorption variability than H[$\alpha$]{}, suggesting a large-scale wind outflow from the star. The emission in the helium line correlates inversely with the velocity of the accreting inflowing material consistent with a simple model of dipole accretion, and formation of the emission in the turbulent post-shock cooling zone. The stellar wind velocity appears related to the strength of helium emission providing direct spectroscopic evidence that the accretion process leads to wind acceleration.
We thank the anonymous referee for his/her comments that improved the manuscript. The authors gratefully acknowledge the helpful support from Gemini-S astronomers, the staff at Magellan, and KECK II while acquiring these spectra. Observers at FLWO were a great help on short notice. This research has made use of NASA’s Astrophysics Data System Bibliographic Services. Some of the data presented here was obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. We wish to extend special thanks to those of Hawaiian ancestry from whose sacred mountain of Mauna Kea we are privileged to conduct observations. Without their generous hospitality, the Keck results presented in this paper would not have been possible.
[*Facilities:*]{} , , ,
Alencar, S. H. P., & Basri, G. 2000, , 119, 1881
Alencar, S. H. P., Basri, G., Hartmann, L, & Calvet, N. 2005, , 440, 595
Alencar, S. H. P., & Batalha, C. 2002, , 571, 378
Alencar, S. H. P., Johns-Krull, C. M., & Basri, G. 2001, , 122, 3335
Alexander, R., D., Clarke, C. J., & Pringle, J. E. 2005, MNRAS, 358, 283
Andrews, S. M., Wilner, D. J., Hughes, A. M., et al. 2012, , 744, 162
Appenzeller, I., & Bertout, C. 2013, , 558, A83
Ardila, D. R., Herczeg, G. J., Gregory, S. G., et al. 2013, ApJS, 207, 1
Avrett, E. H., & Loeser, R. 2008, , 175, 229
Ayres, T. R. 2005, Proc. 13 Cambridge Workshop on Cool Stars, Stellar Systems and the Sun (CoolStars 13), ed. F. Favata, G. A. J. Hussain, & B. Battrick, ESA SP-560, p. 419
Ayres, T. R. 2010, , 187, 149
Basri, G., Johns-Krull, C. M., & Mathieu, R. D. 1997, , 114, 781
Batalha, C., Batalha, N. M., Alencar, S. H. P., Lopes, D. F., & Duarte, E. S. 2002, , 580, 343
Bernstein, R. A., Shectman, S. A., Gunnels, S. M., Mochnacki, S., & Athey, A. E. 2003, SPIE, 4841, 1694
Bertout, C. 1989, , 27, 351
Bouvier, J., Matt, S. P., Mohanty, S., [et al.]{} 2014, [*Protostars and Planets VI*]{}, in press (arXiv: 1309.7851)
Brickhouse, N. S., Cranmer, S. R., Dupree, A. K., Luna, G. J. M., & Wolk, S. 2010, , 710, 1835
Brickhouse, N. S., Cranmer, S. R., Dupree, A. K., [et al.]{} 2012, ApJ, 760, L21
Calvet, N., Basri, G., & Kuhi, L. V. 1984, , 277, 725
Calvet, N., & Hartmann, L. 1992, , 386, 239
Collier Cameron, A., & Robinson, R. D. 1989, , 236, 57
Cranmer, S. R. 2008, , 680, 316
Cranmer, S. R. 2009, , 706, 824
Donati, J.-F., Gregory, S. G., Alencar, S. H. P., [et al.]{} 2011, , 417, 472
Dupree, A. K. 2003, [*Stars as Suns: Activity, Evolution, and Planets*]{}, IAU Symp. 219, A. K. Dupree & A. O. Benz, eds, p.623
Dupree, A. K., Avrett, E. H., Brickhouse, N. S., Cranmer, S. R., & Szalai, T. 2008, in Cambridge Workshop on Cool Stars, Stellar Systems and the Sun (CoolStars 14), ed. G. T. VanBelle, ASPCS 384, CD (arXiv:astro-ph/0702395)
Dupree, A. K., Brickhouse, N. S., Cranmer, S. R., et al., 2012, , 750, 73
Dupree, A. K., Brickhouse, N. S., Smith, G. H., & Strader, J. 2005a, , 625, L131
Dupree, A. K., Lobel, A., Young, P. R., [et al.]{} 2005b, , 622, 629
Edwards, S., Fischer, W., Hillenbrand, L. & Kwan, J. 2006, , 646, 319.
Edwards, S., Fischer, W., Kwan, J., Hillenbrand, L., & Dupree, A. K. 2003, , 599, L41
Edwards, S., Hartigan, P., Ghandour, L., & Andrulis, C. 1994, , 108, 1056
Gahm, G. F., Stempels, H. C., Walter, F. W., Petrov, P. P., & Herczeg, G. J. 2013, , 560, A57
Gahm, G. F., Walter, F. M., Stempels, H. C., Petrov, P. P., & Herczeg, G. J. 2008, , 482, L35
Gregory, S. G., Wood, K., & Jardine, M. 2007, , 379, L35
Griesmann, U., & Kling, R. 2000, , 536, L113
Guilloteau, S., Dutrey, A., Piétu, V., & Boehler, Y. 2011, , 529, A105
Gullbring, E., Petrov, P. P., Ilyin, I., Tuominen, I., Gahm, G. F., & Loden, K. 1996, , 314, 835
Günther, H. M., Wolter, U., Robrade, J., & Wolk, S. J. 2013, , 771, 70
Hartmann, L. 1982, , 48, 109
Hartmann, L. 1998, Accretion Processes in Star Formation, (New York, NY: Cambridge Univ. Press)
Hartmann, L., Hewett, R., & Calvet, N. 1994, , 426, 669
Herczeg, G. J., Linsky, J. L., Valenti, J. A., Johns-Krull, C. M., & Wood, B. E. 2002, , 572, 310
Howarth, I. D., Prinja, R. K., & Massa, D. 1995, , 452, L65
Huélomo, N., Figueira, P., Bonfils, X., [et al.]{} 2008, , 489, L9
Hummer, D. G., & Rybicki, G. B. 1968, , 153, L107
Johns-Krull, C., & Basri, G. 1997, , 474, 433
Johns-Krull, C., & Herczeg, G. J. 2007, , 655, 345 (JKH) Johnstone, C. P., Jardine, M., Gregory, S. G., Donati, J.-F., & Hussain, G. 2014, , 437, 3202
Kaper, L., Henrichs, H. F., Nichols, J. S., & Telting, J. H. 1999, , 344, 231
Kohl, J. L., Noci, G., Cranmer, S. R., & Raymond, J. C. 2006, A&AR, 13, 31
Krist, J. E., Stapelfeldt, K. R., Ménard, F., Padgett, D. L., & Burrows, C. J. 2000, , 538, 793
Kurosawa, R., & Romanova, M. M. 2012, MNRAS, 426, 2901
Kurosawa, R., & Romanova, M. M. 2013, MNRAS, 431, 2673
Lamzin, S. A. 1999, Astron. Lett., 25, 430
Landi, E., Gruesbeck, J. R., Lepri, S. T., Zurbuchen, T. H., & Fisk, L. A. 2012, , 761, 48
Lehner, N., Fullerton, A. W., Massa, D., Sembach, K. R., & Zsargó, J. 2003, , 589, 526
Lobel, A., & Dupree, A. K. 2001, , 558, 815
Mallik, S. V. 1986, MNRAS, 222, 307
Massa, D., Fullerton, A. W., Nichols, J. S., [et al.]{} 1995, , 452, L53
Matsakos, T., Chièze, J.-P., Stehlé, C., [et al.]{} 2013, , 557, A69
Matt, S., & Pudritz, R. E. 2005, ApJ, 632, L135
Matt, S. P., Pinzón, G., Greene, T. P., & Pudritz, R. E. 2012, ApJ, 745, 101
McClure, M. K., Calvet, N., Espaillat, C., [et al.]{} 2013, , 769, 73
McLean, I. S., Becklin, E. E., Bendiksen, O., [et al.]{}1998, SPIE, 3354, 566
McLean, I. S., McGovern, M. R., Burgasser, A. J., [et al.]{} 2000, SPIE, 4008, 1048
McLean, I. S., McGovern, M. R., Burgasser, A. J., [et al.]{} 2003, , 596, 561
Muzerolle, J., Calvet, N., & Hartmann, L. 2001, , 550, 944 Muzerolle, J., Luhman, K. L., Briceño, C., Hartmann, L, & Calvet, N. 2005, , 625, 906
Natta, A., Testi, L., Muzerolle, J., Randich, S., Comerón, F., & Persi, P. 2004, , 424, 603
Oliveira, J. M., Foing, B. H., vanLoon, J. Th., & Unruh, Y. C. 2000, , 362, 615
Pascucci, I., Sterzik, M., Alexander, R. D., [et al.]{} 2011, , 736, 13
Petrov, P. P., Gullbring, E., Ilyin, I., Gahm, G. F., Tuominen, I., Hackman, T., & Loden, K. 1996, , 314, 821
Qi, C., Ho, P. T. P., Wilner, D. J., [et al.]{} 2004, , 616, l11
Reale, F., Orlando, S., Testa, P., [et al.]{} 2013, Sci, 341, 251
Rigliaco, E., Pascucci, I., Gorti, U., Edwards, S., & Hollenbach, D. 2013, , 772:60
Robinson, R. D., & Carpenter, K. G. 1995, , 442, 328
Rosenfeld, K. A., Qi, C., Andrews, S. M., [et al.]{} 2012, , 757, 129
Rutten, R. J., & Uitenbroek, H. 1991, Sol. Phys., 134, 15
Sacco, G. G., Orlando, S., Argiroffi, C., [et al.]{} 2010, , 522, A55
Schegerer, A. A., Wolf, S., Hummel, C. A., Quanz, S. P., & Richichi, A. 2009, , 502, 367
Skelly, M. B., Unruh, Y. C., Collier Cameron, A., [et al.]{} 2008, , 385, 708
Tripathi, D., Mason, H. E., & Klimchuk, J. A. 2010, , 723, 713
Vacca, W. D. & Sandell, G. 2011, , 732, 8
White, R. J., & Basri, G. 2003, , 582, 1109
Wichmann, R., Bastian, U., Krautter, J., Jankovics, I., & Rucinski, S. M. 1998, , 301, L39
Yang, J., Herczeg, G. J., Linsky, J. L., [et al.]{} 2012, , 744, 121
![Single H[$\alpha$]{} spectra of TW Hya selected from ten years of measurements from 2004 to 2013. Note that there are two different scales on the y-axis. Observations in 2005 Jul, 2006 Apr, and 2011 Apr indicate stronger H[$\alpha$]{}. The narrow absorption feature near $+$50 [km s$^{-1}$]{} is due to water vapor. All of these spectra suggest that H[$\alpha$]{} is centered on the stellar radial velocity, and the profile is substantially modified by wind absorption causing the apparent ‘red’ enhancement.](fig1g.eps)
![Single H[$\alpha$]{} spectra of TW Hya selected from ten years of measurements from 2004 to 2013. Note that there are two different scales on the y-axis. Observations in 2005 Jul, 2006 Apr, and 2011 Apr indicate stronger H[$\alpha$]{}. The narrow absorption feature near $+$50 [km s$^{-1}$]{} is due to water vapor. All of these spectra suggest that H[$\alpha$]{} is centered on the stellar radial velocity, and the profile is substantially modified by wind absorption causing the apparent ‘red’ enhancement.](fig1k.eps)
![The observed H[$\alpha$]{} spectrum from 2010 Jun 30 ([*solid line*]{}) with two models overlaid from Muzzerolle [et al.]{} (2001) that assume either no line-damping or an arbitrary damping parameter. These models assume formation of H[$\alpha$]{} in an accretion stream which can be seen by the weaker positive velocity side of the profile as compared with the negative side, and slight subcontinuum absorption when damping is absent. A stellar wind is not included in the models accounting in part for the excess emission of the model on the negative velocity side. Also the model profiles appear ‘pointed’ which is not found in the observed profiles (see also Figure 1).](fig2.eps)
![Magellan/MIKE spectra over 4 nights in 2004 showing the appearance of wind structures stable in velocity. The time interval between the first and last spectrum is shown. Each spectrum has been offset for display. The broken vertical lines mark the constant velocity of the absorption feature during each night. The notch in the H[$\alpha$]{} emission at $+$50 [km s$^{-1}$]{} results from water vapor absorption.](fig3.eps)
![Magellan/MIKE spectra of the H[$\alpha$]{} line over 3 consecutive nights in April 2006. The narrow absorption near $+$50 [km s$^{-1}$]{} is caused by water vapor. More spectra were obtained than displayed here and a gray scale representation of all spectra is shown in Figure 5.](fig4.eps)
![[*Left panel:*]{} Gray scale representation of the H[$\alpha$]{} profiles of TW Hya over 3 consecutive nights in April 2006 where each spectrum has been divided by the [*maximum*]{} flux at each wavelength in order to display absorption features. During Night 2, absorption at $-$50 [km s$^{-1}$]{} is replaced by a discontinuous jump to a new absorption feature at $-$200 [km s$^{-1}$]{} (at $\sim$29 hrs). [*Right panel:*]{} Profiles during night 2 of the gray scale representation above. Note the weakening of the absorption at $-$50 [km s$^{-1}$]{} during the night, and the abrupt appearance of another absorption feature at $-$200 [km s$^{-1}$]{} in the middle of the night which becomes prominent at the end of the night. Broad absorption extending from $\sim +$150 to $\sim +$400 [km s$^{-1}$]{} arising from infalling material is variable, and the velocities are commensurate with the velocities measured in .](fig5left.ps)
![[*Left panel:*]{} Gray scale representation of the H[$\alpha$]{} profiles of TW Hya over 3 consecutive nights in April 2006 where each spectrum has been divided by the [*maximum*]{} flux at each wavelength in order to display absorption features. During Night 2, absorption at $-$50 [km s$^{-1}$]{} is replaced by a discontinuous jump to a new absorption feature at $-$200 [km s$^{-1}$]{} (at $\sim$29 hrs). [*Right panel:*]{} Profiles during night 2 of the gray scale representation above. Note the weakening of the absorption at $-$50 [km s$^{-1}$]{} during the night, and the abrupt appearance of another absorption feature at $-$200 [km s$^{-1}$]{} in the middle of the night which becomes prominent at the end of the night. Broad absorption extending from $\sim +$150 to $\sim +$400 [km s$^{-1}$]{} arising from infalling material is variable, and the velocities are commensurate with the velocities measured in .](fig5right.eps)
![Balmer series over 4 consecutive nights in 2007 (Feb. 26 – Mar. 1). The y-axis represents a continuum normalized flux. Profiles from night 4 (2007 March 1) have the negative velocity segments overlaid on the positive velocity side and marked by a broken ([*red*]{}) line. The effects of wind absorption can be seen when the broken line lies below the solid line; effects of absorption by the accretion stream can be seen at $\sim +$200 [km s$^{-1}$]{} where the broken line lies slightly above the solid line. ](fig6.eps)
![Typical H$\delta$ line profile ($\lambda$4101.73) over 4 consecutive nights in 2008. The broken line represents the negative velocity wing of the first night’s observation that has been reflected around the axis at zero velocity. This illustrates the slight broad absorption present at velocities +50 to +250 [km s$^{-1}$]{}. Profiles from the following 3 nights are symmetric. Subcontinuum absorption predicted by magnetospheric accretion models is missing in the positive velocity wing of the line.](fig7.eps)
  
![[*Left panel:*]{} The magnetic dipole accretion model predicts the relationship between the accretion rate and the terminal free-fall velocity. A slower infall velocity results from a smaller radius of the inner circumstellar disk, and a larger filling factor in the stellar atmosphere produces a higher mass accretion rate. Parameter ranges constrained by CHANDRA diagnostics are shown here. The $\times$ symbols denote the average CHANDRA spectrum, while the other symbols mark 3 individual CHANDRA pointings (cf. Brickhouse [et al.]{} 2012). [*Right panel:*]{} Equivalent width of the emission component of the line ($\lambda$10830) as a function of the terminal velocity indicated by the inflowing subcontinuum absorption. These observations display general agreement with the magnetic dipole accretion model. The emission in the post-shock cooling zone increases because of increased accretion at lower terminal free-fall velocities.](fig10left.eps)
![[*Left panel:*]{} The magnetic dipole accretion model predicts the relationship between the accretion rate and the terminal free-fall velocity. A slower infall velocity results from a smaller radius of the inner circumstellar disk, and a larger filling factor in the stellar atmosphere produces a higher mass accretion rate. Parameter ranges constrained by CHANDRA diagnostics are shown here. The $\times$ symbols denote the average CHANDRA spectrum, while the other symbols mark 3 individual CHANDRA pointings (cf. Brickhouse [et al.]{} 2012). [*Right panel:*]{} Equivalent width of the emission component of the line ($\lambda$10830) as a function of the terminal velocity indicated by the inflowing subcontinuum absorption. These observations display general agreement with the magnetic dipole accretion model. The emission in the post-shock cooling zone increases because of increased accretion at lower terminal free-fall velocities.](fig10right.eps)
![UV and far-UV spectra of major resonance lines. Peak fluxes are scaled to match to illustrate the shape of the profiles. [*Left panel:*]{} [[*HST*]{}]{}:STIS spectra of ([$\lambda$]{}1239) and ([$\lambda$]{}1548) taken in the year 2000. [*Right panel:*]{} [[*FUSE*]{}]{} spectra of ([$\lambda$]{}977) and taken three years later in 2003 (Dupree [et al.]{} 2005a). Note that the profiles observed at the same time are similar one to another with the exception of different absorption levels on the negative velocity side (see text for explanation).](fig12left.eps)
![UV and far-UV spectra of major resonance lines. Peak fluxes are scaled to match to illustrate the shape of the profiles. [*Left panel:*]{} [[*HST*]{}]{}:STIS spectra of ([$\lambda$]{}1239) and ([$\lambda$]{}1548) taken in the year 2000. [*Right panel:*]{} [[*FUSE*]{}]{} spectra of ([$\lambda$]{}977) and taken three years later in 2003 (Dupree [et al.]{} 2005a). Note that the profiles observed at the same time are similar one to another with the exception of different absorption levels on the negative velocity side (see text for explanation).](fig12right.eps)
![Comparison of resonance lines at different times showing the dramatic change in emission at positive velocities. The second member of the doublet appears at $+$500 [km s$^{-1}$]{}. See text for discussion of these profiles. Peak fluxes are scaled in this figure to illustrate the shape of the line profiles.](fig13.eps)
![ ([*1335Å*]{}) and ([*1548Å*]{}) multiplets showing the changes in both intrinsic line strength and wind opacity, even on short time scales, such as the 6 days separating 2010 Jan 29 and 2010 Feb 4. The line is generally narrower on the positive velocity side than the which is similar to the high temperature profile. The broken vertical lines mark the rest velocities of the doublets.](fig14left.eps)
![ ([*1335Å*]{}) and ([*1548Å*]{}) multiplets showing the changes in both intrinsic line strength and wind opacity, even on short time scales, such as the 6 days separating 2010 Jan 29 and 2010 Feb 4. The line is generally narrower on the positive velocity side than the which is similar to the high temperature profile. The broken vertical lines mark the rest velocities of the doublets.](fig14right.eps)
![Fluxes measured from the [*FUSE*]{} line profiles shown in the previous figure display similar behavior in the and emission during several hours of the second pointing. During the second pointing, the absolute flux values are reversed from the first pointing, and the line fluxes are correlated.](fig16.eps)
![The solid lines show the ‘CoolCAT’ optimum reduction ([ *upper panel*]{}) of the same STIS spectrum as that used by JKH ([*lower panel*]{}, O59D01030). The broken lines (red in online version) represent the profile of the 1548Å multiplet multiplied by 0.5 and shifted by the velocity equivalent of the wavelength separation of the doublet. Wind absorption by the 1550Å component is indicated by the weaker extended positive velocity wing of the 1548Å transition ([*broken line*]{}). Wind absorption occurs systematically between $+$175 and $+$375 [km s$^{-1}$]{} of the scaled 1548Å line in the velocity scale of the figure. Additionally the effect of increased opacity in the 1548Å line is evident in the region $-$125 to $+$25 [km s$^{-1}$]{}, where the broken line lies systematically below the solid line. Thus, two signatures of wind absorption can be noted in these doublet profiles: absorption of the red wing of 1548Å by the blue wing of 1550Å, and the increased scattering of the 1548Å line in the wind as compared with 1550Å.](fig17top.eps)
![The solid lines show the ‘CoolCAT’ optimum reduction ([ *upper panel*]{}) of the same STIS spectrum as that used by JKH ([*lower panel*]{}, O59D01030). The broken lines (red in online version) represent the profile of the 1548Å multiplet multiplied by 0.5 and shifted by the velocity equivalent of the wavelength separation of the doublet. Wind absorption by the 1550Å component is indicated by the weaker extended positive velocity wing of the 1548Å transition ([*broken line*]{}). Wind absorption occurs systematically between $+$175 and $+$375 [km s$^{-1}$]{} of the scaled 1548Å line in the velocity scale of the figure. Additionally the effect of increased opacity in the 1548Å line is evident in the region $-$125 to $+$25 [km s$^{-1}$]{}, where the broken line lies systematically below the solid line. Thus, two signatures of wind absorption can be noted in these doublet profiles: absorption of the red wing of 1548Å by the blue wing of 1550Å, and the increased scattering of the 1548Å line in the wind as compared with 1550Å.](fig17bottom.eps)
![HST/STIS spectra of TW Hya in the region of the 1335Å doublet in 2000, 2002, and 2010 showing the variabilities in emission, H$_2$ emission and wind speed and opacity. The short wavelength component ($-$275 km s$^{-1}$) of the doublet is bifurcated by interstellar absorption. The broken vertical lines mark the region ($+$35 to $+$85 [km s$^{-1}$]{}) where JKH postulated the lack of absorption. This section of the profile is obviously substantially variable in both flux and slope of the emission line. The arrows indicate the extent of wind absorption for one member of the doublet. Note that the $H_2$ lines vary, but not in any systematic way with wind speed or opacity. In 2002, when the wind opacity is substantial between $-$100 and $-$250 [km s$^{-1}$]{}, another $H_2$ emission line appears (R(2)0-4 at 1335.2Å at $\sim -$125 [km s$^{-1}$]{}). ](fig18.eps)
[^1]: Data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. Infrared spectra were taken at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), formerly the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência e Tecnologia (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). This paper also includes spectra gathered with the 6.5-meter Magellan Telescope/CLAY located at Las Campanas Observatory, Chile. Additional spectra were obtained at the 1.5m Tillinghast Telescope at the Fred Lawrence Whipple Observatory of the Smithsonian Astrophysical Observatory.
[^2]: Here we take a warm wind to have a temperature of $\sim$15,000K indicative of the formation region of ; a hot wind is considered to have a temperature of $\sim$10$^5$ K or higher. A stellar wind, if like the solar wind, exhibits a progression in temperature from ‘warm’ to ‘hot’.
[^3]: See http://web.mit.edu/ burles/www/MIKE/
[^4]: See [*iraf.noao.edu*]{}
[^5]: See [*www.sao.arizona.edu/FLWO/60/tres.html*]{}
[^6]: See [ *casa.colorado.edu/ ayres/CoolCAT/*]{}.
[^7]: Water vapor produces an absorption feature on the positive velocity side near $+$50 [km s$^{-1}$]{}.
[^8]: The PANDORA code (Avrett & Loeser 2008) was used in a spherical expanding semi-empirical model constructed to match wind scattering profiles of H[$\alpha$]{}, , and the chromospheric density inferred from the ultraviolet lines.
[^9]: Other studies (Batalha [et al.]{} 2002) suggest M=0.7M$_\odot$ and R=0.8R$_\odot$ which would indicate a comparable escape velocity (410 [km s$^{-1}$]{}) at 1 stellar radius above the photosphere.
[^10]: More recent papers (Yang [et al.]{} 2012) than Herczeg [et al.]{} (2002) estimating the UV line fluxes from TW Hya using revised calibrations report a substantially increased value of the flux by a factor of $\sim$5.
|
---
abstract: 'A crucial component to maximizing the science gain from the multi-messenger follow-up of gravitational-wave (GW) signals from compact binary mergers is the prompt discovery of the electromagnetic counterpart. Ideally, the GW detection and localization must be reported early enough to allow for telescopes to slew to the location of the GW-event before the onset of the counterpart. However, the time available for early warning is limited by the short duration spent by the dominant ($\ell = m = 2$) mode within the detector’s frequency band, before the binary merges. This can be circumvented if one could exploit the fact that GWs also contain contributions from higher modes that oscillate at higher harmonics of the orbital frequency, which enter the detector band well before the dominant mode. In this *letter*, we show that these higher modes, although smaller in amplitude, will enable us to significantly improve the early warning time for compact binaries with asymmetric masses (such as neutron-star-black-hole binaries). We investigate the gain in the early-warning time when the $\ell = m = 3$ and $\ell = m = 4$ modes are included in addition to the dominant mode. This is done by using a fiducial threshold of 1000 sq. deg. on the localization sky-area for electromagnetic follow-ups. We find that, in LIGO’s projected “O5-like” network of five GW detectors, for neutron-star-black-hole mergers expected to produce counterparts, we get early-warning gains of up to $\sim 25$ s, assuming the source at a distance of $40$ Mpc. These gains increase to $\sim 40$ s in the same five-detector network with three LIGO detectors upgraded to “Voyager” sensitivity, and $\sim 5$ min. in a third-generation network when the source is placed at $100$ Mpc.'
author:
- 'Shasvath J. Kapadia$^1$'
- Mukesh Kumar Singh$^1$
- Md Arif Shaikh$^1$
- Deep Chatterjee$^2$
- 'Parameswaran Ajith$^{1,3}$'
title: |
Of Harbingers and Higher Modes:\
Improved gravitational-wave early-warning of compact binary mergers
---
Introduction {#sec:introduction}
============
The first gravitational-wave (GW) detection of a binary neutron star merger, GW170817 [@GW170817-DETECTION], also produced an electromagnetic counterpart that was followed up extensively by various telescopes worldwide observing different bands of the electromagnetic spectrum [@GW170817-MMA]. This event became a watershed in multimessenger astronomy, as it demonstrated the immense science gain in observing the same transient in multiple observational windows. GW170817 verified the previously conjectured engine of short gamma-ray bursts (GRBs) as the merger of binary neutron stars (BNS) [see @NakarGRB for a review]. In addition, it enabled an unparalleled study of a new class of optical transients called kilonovae [@MetzgerKN], which revealed an important environment in which heavy elements get synthesized [@GW170817-HEAVY-ELEMENTS]. The multimessenger observations also provided stringent constraints on the speed of GWs [@GW170817-TGR], gave important clues to the nuclear equation of state at high densities [@GW170817-SOURCE-PROPERTIES; @GW170817-EOS], and an independent estimation of the Hubble constant [@GW170817-HUBBLE].
An early warning of the merger from the GW data would allow many additional science benefits: For example, it would enable the observations of possible precursors [@2012PhRvL.108a1102T], a better understanding of the kilonova physics and the formation of heavy elements by identifying the peak of kilonova lightcurves [@Drout2017; @Cowperthwaite2017], and possible signatures of any intermediate merger product (e.g: hypermassive NS [@HotokezakaHNS]) that might have been formed [^1].
BNS mergers are traditionally expected to produce EM counterparts, and therefore it is not surprising that the first efforts towards GW early-warning focused on such events. The inspiral of BNSs lasts for several minutes within the frequency band of ground based GW detectors. If sufficient signal-to-noise ratio (SNR) could be accumulated during this time, ideally tens of seconds to a minute before merger, it could allow for a tight enough sky map for telescopes, enabling them to point at the binary before it merges [@CannonEW].
Early warning for heavier binaries, like neutron-star-black-hole (NSBH) binaries or binary black holes (BBHs), is more challenging, given that they spend significantly smaller durations in the band of ground based detectors (for e.g., GW150914 spent $\sim0.1$s in the LIGO detectors’ frequency band [@GW150914-DETECTION]) [^2]. A possible way to achieve early warning is to detect these systems early in the inspiral, although that would require ground-based detectors to be sensitive at very low frequencies. Seismic noise being the dominant impediment to such low-frequency detections, a “multi-band” detection strategy has been proposed, where the upcoming space-based detector LISA would detect the binary early in its inspiral, potentially years before it reaches the frequency band of ground based detectors [@SesanaMultiband].
In this *letter*, we describe an alternative method for early-warning targeted at unequal-mass compact binaries (especially NSBHs), that could be applied to the upcoming second and third generation (2G and 3G) network of ground-based detectors [@LIGOProspects; @CE; @ET]. The method essentially relies on the fact that the detected GW signal from asymmetric binary inspirals, within a range of inclination angles, could contain contributions from several higher modes in addition to the dominant, quadrupole ($\ell = m = 2$) mode [@Varma:2014jxa]. Since the majority of the higher modes (with $m > 2$) oscillate at larger multiples of the orbital frequency than the dominant mode, we expect these higher modes to enter the frequency band of the detector well before the dominant mode. Thus, using GW templates including higher modes in online GW searches [e.g: @gstlal; @mbta; @pycbc; @spiir] would enable us to detect and localize the binary earlier than analyses that only use the dominant mode, potentially allowing significant early-warning times.
We investigate the early-warning time gained by including the higher modes $\ell = m = 3$ and $\ell = m = 4$ in addition to the dominant mode, for binaries with secondary masses spanning the range $m_2 = 1-3 M_{\odot}$ and mass-ratios spanning $q := m_1/m_2 = 4-20$. We find that, for a network of five detectors with projected sensitivities pertaining to the 5th observing run (O5) [@observer_summary], we get time gains of up to $\sim 1$ minute for trigger-selection (achieving a network SNR of 4) and $\sim 25$ seconds for localization (achieving a sky area of 1000 sq. deg.), for binaries located at a distance of GW170817 ($d_L \simeq 40$ Mpc). These gains increase by about a factor of $1.5-2$ for the same detector-network with the three LIGO detectors, including LIGO-India [@LIGO-INDIA], upgraded to “Voyager” sensitivity [@Adhikari:2019zpy]. They further increase to $\sim 50$ minutes for detection and $\sim 4-5$ minutes for localization, in a 3G network consisting of two Cosmic Explorer detectors and one Einstein telescope detector, even when the source is placed at 100 Mpc. We also investigate the effect of varying the sky location, orientation and luminosity distance of the source.
The *letter* is organized as follows. Section \[sec:ew\_hm\] elaborates on the early-warning method, while also describing higher modes and giving quantitative arguments as to why one should expect them to enhance early-warning times. Section \[sec:results\] describes the results, in particular the time gained in detecting and localizing the sources (described above) by including higher modes, for various upcoming observing scenarios involving ground-based interferometric detectors. The *letter* ends with Section \[sec:conclusion\], which gives a summary and an assessment of the benefits of the proposed method.
Early Warning with Higher Modes {#sec:ew_hm}
===============================
The gravitational waveform, conveniently expressed as a complex combination $h(t) := h_{+}(t) - ih_{\times}(t)$ of two polarizations $h_{+}(t)$ and $h_{\times}(t)$, can be expanded in the basis of spin $-2$ weighted spherical harmonics $\Ylm$ [@NewmanPenrose] $$h(t; \iota, \varphi_{o}) = \ffrac{1}{d_{L}} \sum_{\ell = 2}^{\infty}\sum_{m = -\ell}^{\ell} {h_{\ell m}(t, \blambda)} \, \Ylm.$$ Here, $d_L$ is the luminosity distance, and $h_{\ell m}$ are the multipoles of the waveform that depend exclusively on the intrinsic parameters of the system $\blambda$ (component masses, spins, etc.) and time $t$. On the other hand, the dependence of the waveform on the orientation of the source with respect to the line-of-sight of the detector is captured by the basis functions $\Ylm$ of the spin $-2$ weighted spherical harmonics, where $\iota, \varphi_{o}$ are the polar and azimuthal angles in the source-centered frame, that define the line of sight of the observer with respect to the total angular momentum of the binary. For non-precessing binaries, due to symmetry, modes with negative $m$ are related to the ones with corresponding positive $m$ by $h_{\ell-m} = (-1)^\ell h_{\ell m}^*$. In this *letter*, we consider only non-precessing binaries. Hence, even when we mention only modes with positive $m$, it is implied that the corresponding $-m$ modes are also considered.
The dominant multipole corresponds to $\ell= m =2$, which is the quadrupole mode. The next two subdominant multipoles are $\ell= m = 3$ and $\ell= m = 4$. The contribution of subdominant modes relative to the quadrupole mode depends on the asymmetries of the system — for e.g., relative contribution of higher modes is larger for binaries with large mass ratios. Also, due to the nature of the spin $-2$ weighted spherical harmonics, the higher-mode contribution to the observed signal is the largest for binaries with large inclination angles (say, $\iota = 60^\circ$).
The instantaneous frequency of each spherical harmonic mode is related to the orbital frequency in the following way (assuming a non-precessing orbit): $$F_{\ell m}(t) \simeq m \, F_{\mathrm{orb}}(t).$$ Thus, higher modes (with $m > 2$) enter the frequency band of the detector (say, 10 Hz) before the dominant mode (see Fig. \[fig:hm\_illustrative\] for a qualitative illustration) .
![Schematic illustration of how different modes appear in the detector band. We show the real part of the *whitened* modes $h_{\ell m}$ (with $\ell = m = \{2, 3, 4\}$) of a compact binary coalescence waveform, plotted as a function of time. The modes are whitened by the noise PSD of Advanced LIGO to show their expected contribution to the SNR. The higher the $m$, the earlier it enters the frequency band of the detector. This can be seen by the appearance of the non-zero amplitudes of the higher modes at a time $\Delta \tau$ before the merger (dashed black vertical line), where $\Delta \tau$ increases with increasing $m$.[]{data-label="fig:hm_illustrative"}](hlm_whitened_onecol.pdf){width="1.0\columnwidth"}
The time taken by the binary to merge, once it has reached an orbital frequency of $F_{\mathrm{orb}}$, is approximately given by @Sathyaprakash:1994nj $$\tau \simeq \frac{5}{256}\,\mathcal{M}^{-5/3}\, (2\pi F_\mathrm{orb})^{-8/3} \propto (F_{\ell m}/m)^{-8/3},
\label{eq:chirptime}$$ where $\mathcal{M} := (m_1 m_2)^{3/5}/(m_1 + m_2)^{1/5}$ is the chirp mass of the binary. Thus, the in-band duration of a higher mode $h_{\ell m}$ is a factor $(m/2)^{8/3}$ larger than the corresponding $\ell = m = 2$ mode. For the $\ell = m = 3$ mode, this amounts to $\sim 3$ fold increase in the observable duration as compared to the $\ell = m = 2$ mode, and for the $\ell = m = 4$ mode a $\sim 6$ fold increase
However, the time gained in reaching a fiducial threshold-SNR and localizing a source to a fiducial sky area depends on two competing factors: On the one hand, higher modes are excited only for binaries with large mass ratios (and hence larger chirp masses, when we fix a lower limit on $m_2 \simeq 1 M_\odot$). On the other hand, according to Eq., heavier binaries will merger quicker in the detector band. Thus, the region of the $m_1-m_2$ plane that maximizes the time gains corresponds to regions where the masses are sufficiently asymmetric to excite the higher modes significantly, while not too heavy to make the system hurry through the frequency band of the detector; this region will change depending on the sensitivities of the detectors in a given observing scenario.
For stationary Gaussian noise (which we assume throughout this paper), the power-spectral-density (PSD) of the noise completely determines its statistical properties. Based on this assumption, assessing a trigger to be worthy of follow-up can be reduced to setting a threshold on the SNR, corresponding to a given false alarm probability. The localization area, at a given confidence, is completely determined by the separation of the detectors, their individual effective bandwidths, and the SNRs. In this *letter* we use the method proposed by @Fairhurst1 [@Fairhurst2] to estimate the sky area from the times of arrival of the signal at the detectors, and timing uncertainties. In this method, the localization sky area of a source at a given right ascension (RA) and declination (Dec) can be computed from the pair-wise separation of the detectors, as well as each detectors’ timing errors. Note that if the detectors are approximately co-planar, then the mirror degeneracy with respect to the plane of the detector needs to be broken by additional waveform consistency tests between detectors.
Results {#sec:results}
=======
![Gains in the early warning time of compact binary mergers due to the inclusion of higher modes, as compared to the same using the dominant mode only. These plots correspond to the intrinsic parameter ranges $m_2 = 1-3 M_{\odot}$ and $q = 4-20$ (extrinsic parameters set to their optimal values). The binary components are assumed to be non-spinning; however, even including a primary spin as large $\chi_1 = 0.9$ does not alter the time gains significantly. We also plot the contours that demarcate the region corresponding to binaries that would produce a non-zero ejecta mass and therefore an EM counterpart, for various $\chi_1$ values [@FoucartEMB]. Two sets of contours, corresponding to two different nuclear EOS, 2H (solid, black contours) and SLy (dotted, black contours) are plotted. As expected, systems that are both asymmetric, while relatively lower in total mass, tend to maximize the time gains. These gains can be as much as $\sim 25$ seconds in the O5 scenario, $\sim 40$ seconds in the Voyager scenario, and $\sim 5$ minutes in the 3G scenario. A significant fraction of the systems with gains close to the maximum values are expected to be electromagnetically bright, even for moderately spinning ($\chi_1 \sim 0.6$) primary masses, assuming the two EOS considered here. []{data-label="fig:results-delta-t-em-bright"}](skyarea-delta-t-publication.pdf){width="1.0\columnwidth"}
We generate two sets of GW signals — one set containing just the $\ell = m = 2$ mode, while the other includes the $\ell = m = 3$ and $\ell = m = 4$ modes in addition to the dominant mode. These are generated using the <span style="font-variant:small-caps;">IMRPhenomHM</span> model [@imrphenomhm], as implemented in the <span style="font-variant:small-caps;">LALSuite</span> software package [@lalsuite]. We consider three observing scenarios. The first is the “O5” scenario, consisting of LIGO-Hanford, LIGO-Livingston, Virgo, KAGRA and LIGO-India. We assume the most optimistic projected sensitivities, from the document [@observer_summary]. Since KAGRA’s projected sensitivity for O5 only has a lower limit, we assume that KAGRA’s sensitivity will equal Virgo’s. The second is the “Voyager” scenario [noise PSD taken from @Voyager_PSD], where we assume that all three LIGO detectors, including LIGO-India, will be upgraded to Voyager sensitivity, while Virgo and KAGRA will operate at their O5 sensitivities. The third is the 3G scenario, where the assumed network consists of two Cosmic Explorer detectors, and one Einstein Telescope. [The projected PSD for the Einstein Telescope is taken from [@ET_PSD], and that for Cosmic Explorer is taken from [@CE_PSD]]{}.
In Fig. \[fig:results-delta-t-em-bright\], we summarize the early-warning time gained by the inclusion of higher modes, for unequal-mass binary systems, with the secondary mass spanning [$m_2 = 1-3 M_{\odot}$]{} and the mass ratio spanning [$q = 4-20$]{} [^3].
We only show results for the case of non-spinning binaries, since time gains do not change appreciably with spin. Furthermore, we focus on the mass range that corresponds to NSBHs. This is done for two reasons: the first is that an EM counterpart for binaries detectable by ground based detectors are expected to require a NS; the second is that this region contains the maximum gain in the early warning time, for the mass ratios we consider. We also include contours that demarcate the region of the $m_1-m_2$ plane that are expected to produce EM-counterparts, based on the spin of the primary and the NS equation of state (EOS) of the secondary [@FoucartEMB]. For this purpose, we consider three values of the spin ($0, 0.6, 0.9$), and two EOS: 2H [@2H] and SLy [@SLy]. The former is a “stiff” EOS, predicting a relatively broad region of the component-mass space to produce counterparts, while the latter is a more “realistic” EOS, as indicated by the GW-based investigations of the source properties of the GW170817 merger event [@GW170817-SOURCE-PROPERTIES].
We set the luminosity distance to $40$ Mpc – similar to that of GW170817 [@GW170817-DETECTION]. We then sample the extrinsic parameter space of RA, Dec, polarization, and inclination, to determine the optimal localization sky area [^4]. For the O5 and Voyager scenarios, we set the lower limit of the detector bandwidths to be $10$ Hz. We then determine the upper frequency (and the time to coalescence from that frequency) for which the $90\%$ localization sky area reduces to 1000 $\mathrm{deg}^2$, for waveforms with and without the higher modes. The difference in the times to coalescence gives us the time gained $\Delta \tau_c(\Omega_{90})$. For the 3G scenario, we estimate the times gained by setting the luminosity distance to 100 Mpc and the lower cutoff frequency to 5Hz. Using a near-identical process to the one described above, we also determine the times gained in reaching a threshold network SNR of $4$, which we denote as $\Delta\tau_c(\rho_4)$. We get localization time gains of up to $\sim 25$ seconds, $\sim 40$ seconds and $\sim 5$ minutes, for each of the observing scenarios (O5, Voyager and 3G) respectively, and trigger-selection time gains of up to $\sim 60$ seconds, $\sim 100$ seconds and $\sim 50$ minutes. A significant fraction of the systems with time gains close to the maximum values, are expected to produce electromagnetic counterparts for the assumed EOS (2H and SLy), for moderate to highly spinning primary masses.
We further investigate the variation of the localization time gains with extrinsic parameters. We start by focusing on those determining the antenna pattern functions of the GW detectors. We draw 1000 samples, at random, from uniform distributions in RA $\in [0, 2\pi]$, $\cos\left(\mathrm{Dec}\right) \in [-1, 1]$ and polarization $\in [0, 2\pi]$, while fixing the masses to $m_1 = 15 M_{\odot}, m_2 = 1.5 M_{\odot}$ [^5], and fixing the inclination to $60$ degrees. We then evaluate the localization time gains for the O5 scenario, which we represent as cumulative histograms in Fig. \[fig:results-skyarea-percent-hist\]. We find that the median time gains are $\sim 8$ seconds corresponding to $\sim 40\%$ improvements over the time to coalescence for the dominant mode. These gains increase to about $\sim 15$ seconds ($\sim 60 \%$ improvement) for the Voyager scenario and $\sim 120$ seconds ($\sim 60 \%$ improvement) for the 3G scenario.
We then vary the inclination, and distance, individually, while fixing the other extrinsic parameters to their optimal values, and the masses to the same values as above (Fig. \[fig:results-skyarea-delta-t-var-dist\]). We find that for inclination, within a $\sim 60$ degree-window centered on the optimal value of $60$ degrees, the time gains don’t decrease by more than $\sim 50\%$ of the maximum value. Additionally, the time gains have an (approximate) inverse scaling with the distance.
Summary and Outlook {#sec:conclusion}
===================
Current and upcoming transient surveys plan to participate in the follow-up of GW transients [see, e.g., @Graham_2019], especially if they are expected to produce EM counterparts via the tidal disruption of one or both of the binary’s components. Real-time dissemination of sky localization from GW data is useful for fast and wide-field surveys (e.g., [@Bellm_2018; @Ivezic_2019]) to begin follow-up observations, the slew times of which can be $\sim 30-60$s.
Given these slew times, we show that the inclusion of higher modes in GW low-latency searches [such as @gstlal; @mbta; @pycbc; @spiir] will improve the early-warning time. This is especially true for asymmetric mass compact binaries with inclined orbits, where higher multipoles of the gravitational radiation are expected to make appreciable contributions to the SNRs. The existence of some of these higher multipoles, and their conformity to General Relativity, was recently confirmed with the detection of an asymmetric BBH merger GW910412 observed by LIGO and Virgo during O3 [@GW190412].
In particular we have shown that, for the parameter space we have explored ($m_2 \in [1, 3] M_{\odot}$, $q \in [4-20]$), and assuming a luminosity distance similar to that of GW170817, it is possible to achieve a localization of $\sim 1000~\sqdeg$ with time gains of up to [$\Delta \tau_c(\Omega_{90}) \sim 25$ seconds]{} for O5 sensitivities, as compared to when only the dominant mode is considered. For 3G detectors, the improvement in early warning time can be $\Delta\tau_c(\Omega_{90}) \sim 5$ minutes when we keep the source at 100 Mpc. These correspond to improvements of over a $70\%$ (with respect to using waveforms with just the $\ell = m = 2$ mode) for some of the binary configurations we have explored.
In addition, we have investigated the variation of these time gains with the sky location and orientation of the binary while fixing the masses to $m_1 = 15 M_{\odot}, m_2 = 1.5 M_{\odot}$. Such a binary is expected to produce a counterpart upon merger, even for moderate spins of the primary. We find that for a significant region of this parameter space, the gains don’t decrease by more than $50\%$ of the maximum values. We also study the variation of the gain in early-warning time, for a range of luminosity distances spanning $10-100$Mpc [^6], and get time gains spanning $\sim 50 - 6$ seconds in the O5 scenario. These gains would increase by over an order of magnitude for 3G detectors. We also estimated time gains for trigger-selection, where we set the threshold network SNR to 4. These times gains are greater than those for localization: For O5, we get gains of $\sim 1$ minute, and for the next-generation detectors we get gains of up to $\sim 50$ minutes. However, from a EM follow-up perspective, the early-warning times for localization are likely more useful.
The choice of $1000\,\sqdeg$ as the fiducial threshold for sky area is similar to some of those considered by [@CannonEW], and can be justified if a catalog of galaxies is available, allowing EM telescopes to adopt some form of slewing strategy [see, e.g., @SlewingStrategy] to efficiently probe the estimated localization region.
One might ask how often would we expect to see NSBH mergers with counterparts, within $100$ Mpc? This question is difficult to answer given that no confirmed NSBH merger has so far been observed. That said, upper limits on the rate of mergers of these transients do exist [@GWTC-1], which we could use to do a more elaborate population study to estimate a distribution of time gains. We leave this for future work.
#### Acknowledgments:
We are grateful to Shaon Ghosh for reviewing our manuscript and providing useful comments. We also thank Stephen Fairhurst for clarifications on [@Fairhurst1; @Fairhurst2], and Srashti Goyal for help with the <span style="font-variant:small-caps;">PyCBC</span> implementation of the antenna pattern functions. SJK’s, MKS’s, MAS’s and PA’s research was supported by the Department of Atomic Energy, Government of India. In addition, SJK’s research was funded by the Simons Foundation through a Targeted Grant to the International Centre for Theoretical Sciences, Tata Institute of Fundamental Research (ICTS-TIFR). PA’s research was funded by the Max Planck Society through a Max Planck Partner Group at ICTS-TIFR and by the Canadian Institute for Advanced Research through the CIFAR Azrieli Global Scholars program. DC would like to thank the ICTS-TIFR for their generous hospitality; a part of this work was done during his visit to ICTS.
natexlab\#1[\#1]{}\[1\][[\#1](#1)]{} \[1\][doi: [](http://doi.org/#1)]{} \[1\][[](http://ascl.net/#1)]{} \[1\][[](https://arxiv.org/abs/#1)]{}
, B. P., [Abbott]{}, R., [Abbott]{}, T. D., [et al.]{} 2018, , 121, 161101,
—. 2018, Living Reviews in Relativity, 21, 3,
Abbott, B. P., [et al.]{} 2016, Phys. Rev. Lett., 116, 061102,
—. 2017, Phys. Rev. Lett., 119, 161101,
—. 2017, Astrophys. J., 848, L12,
, B. P., [et al.]{} 2017, , 551, 85,
Abbott, B. P., [et al.]{} 2017, Classical and Quantum Gravity, 34, 044001,
—. 2018.
—. 2019, Physical Review X, 9, 011001,
—. 2019, Physical Review X, 9, 031040,
Adams, T., Buskulic, D., Germain, V., [et al.]{} 2016, Classical and Quantum Gravity, 33, 175012,
Adhikari, R. X., [et al.]{} 2019, Class. Quant. Grav., 36, 245010,
Bellm, E. C., Kulkarni, S. R., Graham, M. J., [et al.]{} 2018, Publications of the Astronomical Society of the Pacific, 131, 018002,
, K., [Cariou]{}, R., [Chapman]{}, A., [et al.]{} 2012, , 748, 136,
Chu, Q. 2017, PhD thesis, The University of Western Australia
Connaughton, V., Burns, E., Goldstein, A., [et al.]{} 2016, The Astrophysical Journal, 826, L6,
Coughlin, M. W., Tao, D., Chan, M. L., [et al.]{} 2018, Monthly Notices of the Royal Astronomical Society, 478, 692,
Cowperthwaite, P. S., Berger, E., Villar, V. A., [et al.]{} 2017, ApJ, 848, L17,
, F., & [Haensel]{}, P. 2001, , 380, 151,
, M. R., [Piro]{}, A. L., [Shappee]{}, B. J., [et al.]{} 2017, Science, 358, 1570,
, S. 2009, New Journal of Physics, 11, 123006,
—. 2011, Classical and Quantum Gravity, 28, 105021,
Foucart, F. 2012, Phys. Rev. D, 86, 124007,
Graham, M. J., Kulkarni, S. R., Bellm, E. C., [et al.]{} 2019, Publications of the Astronomical Society of the Pacific, 131, 078001,
Hild, S. 2012, Classical and Quantum Gravity, 29, 124006,
, K., [Kiuchi]{}, K., [Kyutoku]{}, K., [et al.]{} 2013, , 88, 044026,
Ivezi[ć]{}, [Ž]{}., Kahn, S. M., Tyson, J. A., [et al.]{} 2019, The Astrophysical Journal, 873, 111,
, [LIGO Scientific Collaboration]{}, & [Virgo Collaboration]{}. 2019, Advanced LIGO, Advanced Virgo and KAGRA observing run plans. <https://dcc.ligo.org/public/0161/P1900218/002/SummaryForObservers.pdf>
, D., [Metzger]{}, B., [Barnes]{}, J., [Quataert]{}, E., & [Ramirez-Ruiz]{}, E. 2017, , 551, 80,
Kyutoku, K., Shibata, M., & Taniguchi, K. 2010, Phys. Rev. D, 82, 044049,
. 2015, Instrument Science White Paper. <https://dcc.ligo.org/public/0120/T1500290/002/T1500290.pdf>
—. 2020, [LIGO]{} [A]{}lgorithm [L]{}ibrary - [LALS]{}uite, free software (GPL),
Loeb, A. 2016, The Astrophysical Journal, 819, L21,
London, L., Khan, S., Fauchon-Jones, E., [et al.]{} 2018, Phys. Rev. Lett., 120, 161102,
Messick, C., Blackburn, K., Brady, P., [et al.]{} 2017, Phys. Rev. D, 95, 042001,
, B. D. 2017, Living Reviews in Relativity, 20, 3,
, E. 2007, , 442, 166,
Newman, E. T., & Penrose, R. 1966, Journal of Mathematical Physics, 7, 863,
Nitz, A. H., Dal Canton, T., Davis, D., & Reyes, S. 2018, Phys. Rev. D, 98, 024050,
Punturo, M., Abernathy, M., Acernese, F., [et al.]{} 2010, Classical and Quantum Gravity, 27, 194002,
, D., [Adhikari]{}, R. X., [Ballmer]{}, S., [et al.]{} 2019, in , Vol. 51, 35.
Sathyaprakash, B. 1994, Phys. Rev. D, 50, 7111,
Sesana, A. 2016, Phys. Rev. Lett., 116, 231102,
, [the Virgo Collaboration]{}, [Abbott]{}, R., [et al.]{} 2020, arXiv e-prints, arXiv:2004.08342.
, D., [Read]{}, J. S., [Hinderer]{}, T., [Piro]{}, A. L., & [Bondarescu]{}, R. 2012, , 108, 011102,
Unnikrishnan, C. S. 2013, International Journal of Modern Physics D, 22, 1341010,
Varma, V., Ajith, P., Husa, S., [et al.]{} 2014, Phys. Rev. D, 90, 124004,
[^1]: To put things in context, for GW170817 the GCN (Gamma-ray Coordinate Network) circular was released only 30 minutes after merger [@GW170817-MMA]
[^2]: One might argue that stellar mass BBH mergers are not likely to produce electromagnetic emissions under standard scenarios. Nevertheless, there are proposals of possible counterparts to stellar-mass BBH mergers (e.g., [@Loeb_2016]). The Fermi satellite had also announced a candidate gamma-ray counterpart coincident with GW150914 [@Connaughton_2016].
[^3]: We do not consider mass ratios $q > 20$ because the waveforms that we use are calibrated to numerical relativity results only for binaries with $q \lesssim 20$ [@imrphenomhm]. We also do not show the results for $q < 4$, as the improvements are not significant.
[^4]: The antenna pattern functions were computed using the <span style="font-variant:small-caps;">PyCBC</span> software package [@pycbc]
[^5]: This choice of masses does not correspond to the optimal mass-combination for time gain, within the mass-space we consider. Nevertheless, it does represent a system that could potentially have an EM counterpart for a moderately $\chi_1 \sim 0.6$ spinning primary assuming a 2H equation of state.
[^6]: Beyond this distance, observing counterparts with electromagnetic telescopes becomes difficult since capturing the image would require significant increases in exposure times.
|
---
abstract: 'Let $M$ be an irreducible Riemannian symmetric space. The index $i(M)$ of $M$ is the minimal codimension of a totally geodesic submanifold of $M$. In [@BO] we proved that $i(M)$ is bounded from below by the rank ${\operatorname{rk}}(M)$ of $M$, that is, ${\operatorname{rk}}(M) \leq i(M)$. In this paper we classify all irreducible Riemannian symmetric spaces $M$ for which the equality holds, that is, ${\operatorname{rk}}(M) = i(M)$. In this context we also obtain an explicit classification of all non-semisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric R-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with $i(M) \in \{4,5,6\}$.'
address:
- 'King’s College London, Department of Mathematics, London WC2R 2LS, United Kingdom'
- 'Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina'
author:
- Jürgen Berndt
- Carlos Olmos
title: |
Maximal totally geodesic submanifolds\
and index of symmetric spaces
---
Introduction
============
Let $M$ be a connected Riemannian manifold and denote by ${\mathcal S}$ the set of all connected totally geodesic submanifolds $\Sigma$ of $M$ with $\dim(\Sigma) < \dim(M)$. The index $i(M)$ of $M$ is defined by $$i(M) = \min\{ \dim(M) - \dim(\Sigma) \mid \Sigma \in {\mathcal S}\} = \min\{ {\operatorname{codim}}(\Sigma) \mid \Sigma \in {\mathcal S}\}.$$ This notion was introduced by Onishchik in [@On] who also classified the irreducible simply connected Riemannian symmetric spaces $M$ with $i(M) \leq 2$.
In [@BO] we investigated $i(M)$ for irreducible Riemannian symmetric spaces $M$. We proved that the rank ${\operatorname{rk}}(M)$ of $M$ is always less or equal than the index of $M$ and classified all irreducible Riemannian symmetric spaces $M$ with $i(M) \leq 3$. The motivation for this paper was to understand better the equality case ${\operatorname{rk}}(M) = i(M)$. The main result of this paper is the classification of all irreducible Riemannian symmetric spaces $M$ with ${\operatorname{rk}}(M) = i(M)$.
\[main\] Let $M$ be an irreducible Riemannian symmetric space of noncompact type. The equality ${\operatorname{rk}}(M) = i(M)$ holds if and only if $M$ is isometric to one of the following symmetric spaces:
- $SL_{r+1}({\mathbb R})/SO_{r+1}$, $r \geq 1$;
- $SO^o_{r,r+k}/SO_rSO_{r+k}$, $r \geq 1$, $k \geq 0$, $(r,k) \notin \{(1,0),(2,0)\}$.
Duality between Riemannian symmetric spaces of noncompact type and of compact type preserves totally geodesic submanifolds, and if $M$ is an irreducible Riemannian symmetric space of compact type and $\hat{M}$ is its Riemannian universal covering space (which is also a Riemannian symmetric space of compact type), then $i(M) = i(\hat{M})$. Therefore Theorem \[main\] leads, via duality and covering maps, to the classification of irreducible Riemannian symmetric spaces of compact type with ${\operatorname{rk}}(M) = i(M)$.
In order to compute the index explicitly we need to have a good understanding of maximal totally geodesic submanifolds. Every maximal totally geodesic submanifold $\Sigma$ in an irreducible Riemannian symmetric space $M$ of noncompact type is either semisimple or non-semisimple. As part of our investigation we obtain an explicit classification for the non-semisimple case and a conceptual characterization of such submanifolds in terms of symmetric R-spaces. Denote by $r$ the rank of $M$ and write $M = G/K$, where $G$ is the connected identity component of the isometry group $I(M)$ of $M$ and $K = G_p$ is the isotropy group of $G$ at $p \in M$. Consider a set of simple roots $\Lambda = \{\alpha_1,\ldots,\alpha_r\}$ of a restricted root space decomposition of the Lie algebra ${\mathfrak{g}}$ of $G$ and denote by $\delta = \delta_1 \alpha_1 + \ldots + \delta_r \alpha_r$ the highest root. Let ${\mathfrak{q}}_i$ be the parabolic subalgebra of ${\mathfrak{g}}$ which is determined by the root subsystem $\Phi_i = \Lambda \setminus \{\alpha_i\}$ and consider the Chevalley decomposition ${\mathfrak{q}}_i = {\mathfrak{l}}_i \oplus {\mathfrak{n}}_i$ of ${\mathfrak{q}}_i$ into a reductive subalgebra ${\mathfrak{l}}_i$ and a nilpotent subalgebra ${\mathfrak{n}}_i$. Let $L_i$ be the connected closed subgroup of $G$ with Lie algebra ${\mathfrak{l}}_i$ and denote by $F_i$ the orbit of $L_i$ containing $p$. Then $F_i$ is a non-semisimple totally geodesic submanifold of $M$ which decomposes into $F_i = {\mathbb{R}}\times B_i$, where $B_i$ is a semisimple Riemannian symmetric space of noncompact type. The classification and characterization of non-semisimple maximal totally geodesic submanifolds in $M$ is as follows:
\[main2\] Let $M = G/K$ be an irreducible Riemannian symmetric space of noncompact type and let $\Sigma$ be a non-semisimple connected complete totally geodesic submanifold of $M$. Then the following statements are equivalent:
- $\Sigma$ is a maximal totally geodesic submanifold of $M$;
- $\Sigma$ is isometrically congruent to $F_i = {\mathbb{R}}\times B_i$ and $\delta_i = 1$;
- The normal space $\nu_p\Sigma$ of $\Sigma$ at $p$ is the tangent space of a symmetric R-space in $T_pM$;
- The pair $(M,\Sigma)$ is as in Table [\[totgeodnsstable\]]{}.
An R-space is a real flag manifold and a symmetric R-space is a real flag manifold which is also a symmetric space. R-spaces are projective varieties and symmetric R-spaces were classified and investigated by Kobayashi and Nagano in [@KN]. They arise as certain orbits of the isotropy representation of semisimple Riemannian symmetric spaces.
This paper is organized as follows. In Section \[preliminaries\] we summarize basic material about Riemannian symmetric spaces of noncompact type, their restricted root space decompositions and Dynkin diagrams, parabolic subalgebras, and their boundary components with respect to the maximal Satake compactification.
In Section \[reflective\] we obtain some sufficient criteria for totally geodesic submanifolds in Riemannian symmetric spaces of noncompact type to be reflective. As is well-known, totally geodesic submanifold are in one-to-one correspondence with Lie triple system. If the orthogonal complement of a Lie triple system is also a Lie triple system, then the Lie triple system and the corresponding totally geodesic submanifold are said to be reflective. Geometrically, reflective submanifolds arise as connected components of fixed point sets of isometric involutions. Reflective submanifolds in irreducible simply connected Riemannian symmetric spaces of compact type were classified by Leung in [@L1] and [@L2]. The concept of reflectivity turns out to be very useful in our context. One of our main criteria is Proposition \[ref3\] which states that if the kernel of the slice representation of a semisimple totally geodesic submanifold $\Sigma$ in an irreducible Riemannian symmetric space of noncompact type has positive dimension, then $\Sigma$ is reflective. This criterion then provides a lower bound for the codimension of $\Sigma$ which we will use in index calculations.
In Section \[nonsemisimple\] we will prove Theorem \[main2\]. The first step is to show that any non-semisimple maximal totally geodesic submanifold in $M$ is congruent to one of the orbits $F_i$ introduced above. The coefficient $\delta_i$ of $\alpha_i$ in the highest root $\delta$ then plays a crucial role for the next step. If $\delta_i \geq 2$, we construct explicitly a larger Lie triple system containing the Lie triple system corresponding to $F_i$. The situation for $\delta_1 = 1$ is much more involved. With delicate arguments using Killing fields, Jacobi fields, reflections and transvections we can show that $F_i$ is maximal when $\delta_i = 1$. As an application of Theorem \[main2\] we obtain that every maximal totally geodesic submanifold of an irreducible Riemannian symmetric space of noncompact type whose root system is of type $(BC_r)$, $(E_8)$, $(F_4)$ or $(G_2)$ must be semisimple. Another application states that every non-semisimple maximal totally geodesic submanifold of an irreducible Riemannian symmetric space of noncompact type must be reflective. As a third application we obtain that the index of $SL_{r+1}({\mathbb{R}})/SO_{r+1}$ is equal to its rank $r$.
In Section \[ex\] we prove that the two classes of symmetric spaces listed in Theorem \[main\] satisfy the equality ${\operatorname{rk}}(M) = i(M)$. For this we explicitly construct totally geodesic submanifolds $\Sigma$ of $M$ with ${\operatorname{codim}}(\Sigma) = {\operatorname{rk}}(M)$ using standard algebraic theory of symmetric spaces.
In Section \[proof\] we prove Theorem \[main\]. A crucial step is Proposition \[boundary\_reduction\] which states that if $M$ satisfies the equality ${\operatorname{rk}}(M) = i(M)$, then every irreducible boundary component $B$ of the maximal Satake compactification of $M$ satisfies ${\operatorname{rk}}(B) = i(B)$. As an application we obtain that with the possible exception of $E_6^6/Sp_4$, $E_7^7/SU_8$ and $E_8^8/SO_{16}$ there are no other irreducible Riemannian symmetric spaces $M$ of noncompact type with ${\operatorname{rk}}(M) = i(M)$ than those discussed in Section \[ex\]. The exceptional symmetric space $E_6^6/Sp_4$ has the interesting property that each of its irreducible boundary components $B$ satisfies ${\operatorname{rk}}(B) = i(B)$. In order to come to a conclusion for this exceptional symmetric space we developed the criteria about reflective submanifolds in Section \[reflective\]. Using these criteria we can show that $E_6^6/Sp_4$ does not satisfy the equality ${\operatorname{rk}}(M) = i(M)$. Since $E_6^6/Sp_4$ arises as a boundary component of $E_7^7/SU_8$ and of $E_8^8/SO_{16}$ we can then conclude that these two symmetric spaces do not satisfy the equality ${\operatorname{rk}}(M) = i(M)$ either.
In Section \[applications\] we apply some of the results in Sections \[reflective\] and \[nonsemisimple\] to calculate explicitly the index of some other symmetric spaces. We also classify the irreducible Riemannian symmetric spaces of noncompact type with $i(M) \in \{4,5,6\}$.
Riemannian symmetric spaces of noncompact type {#preliminaries}
==============================================
We assume that the reader is familiar with the general theory of Riemannian symmetric spaces as in [@H] and summarize below some basic facts and notations which are used in this paper.
Let $M = G/K$ be an irreducible Riemannian symmetric space of noncompact type, where $G = I^o(M)$ is the connected component of the isometry group $I(M)$ of $M$ containing the identity transformation, $p \in M$ and $K = G_p$ is the isotropy group of $G$ at $p$. Then $G$ is a noncompact real semisimple Lie group and $K$ is a maximal compact subgroup of $G$. Let ${\mathfrak{g}}= {\mathfrak{k}}\oplus {\mathfrak{p}}$ be the corresponding Cartan decomposition of ${\mathfrak{g}}$ and denote by $\theta$ the corresponding Cartan involution on ${\mathfrak{g}}$. Let $B$ be the Killing form of ${\mathfrak{g}}$. Then $\langle X,Y \rangle = -B(X,\theta Y)$ is a positive definite inner product on ${\mathfrak{g}}$. The vector space ${\mathfrak{p}}$ can be identified canonically with the tangent space $T_pM$ of $M$ a $p$. Since the Riemannian metric on $M$ is unique up to homothety, we can assume that the Riemannian metric on $M$ coincides with the $G$-invariant Riemannian metric induced by $\langle \cdot , \cdot \rangle$.
We denote by $r = {\operatorname{rk}}(M)$ the rank of $M$. Let ${\mathfrak{a}}$ be a maximal abelian subspace of ${\mathfrak{p}}$ and denote by ${\mathfrak{a}}^\ast$ the dual space of ${\mathfrak{a}}$. Note that $\dim({\mathfrak{a}}) = r$. For each $\alpha \in {\mathfrak{a}}^\ast$ we define ${\mathfrak{g}}_{\alpha} = \{X \in {\mathfrak{g}}\mid [H,X] = \alpha(H)X\ {\rm for\ all\ }H \in {\mathfrak{a}}\}$. If $\alpha \neq 0$ and ${\mathfrak{g}}_\alpha \neq \{0\}$, then $\alpha$ is a restricted root and ${\mathfrak{g}}_\alpha$ a restricted root space of ${\mathfrak{g}}$ with respect to ${\mathfrak{a}}$. The positive integer $m_\alpha = \dim({\mathfrak{g}}_\alpha)$ is called the multiplicity of the root $\alpha$. We denote by $\Psi$ the set of restricted roots with respect to ${\mathfrak{a}}$. The direct sum decomposition $${\mathfrak{g}}= {\mathfrak{g}}_0 \oplus \left(\bigoplus_{\alpha \in \Psi} {\mathfrak{g}}_{\alpha}\right)$$ is the restricted root space decomposition of ${\mathfrak{g}}$ with respect to ${\mathfrak{a}}$. The eigenspace ${\mathfrak{g}}_0$ decomposes into ${\mathfrak{g}}_0 = {\mathfrak{k}}_0 \oplus {\mathfrak{a}}$, where ${\mathfrak{k}}_0 = Z_{{\mathfrak{k}}}({\mathfrak{a}})$ is the centralizer of ${\mathfrak{a}}$ in ${\mathfrak{k}}$.
Let $\{\alpha_1,\ldots,\alpha_r\} = \Lambda \subset \Psi$ be a set of simple roots of $\Psi$. We denote by $H^1,\ldots,H^r \in {\mathfrak{a}}$ the dual basis of $\alpha_1,\ldots,\alpha_r \in {\mathfrak{a}}^*$ defined by $\alpha_i(H^j) = \delta_{ij}$ for all $i,j \in \{1,\ldots,r\}$, where $\delta_{ij} = 0$ for $i \neq j$ and $\delta_{ij} = 1$ for $i = j$. Riemannian symmetric spaces of noncompact type are uniquely determined by the Dynkin diagram of their restricted root system together with the multiplicities of the simple roots. In Table \[dynkin\] we list the Dynkin diagrams and root multiplicities for all irreducible Riemannian symmetric spaces of noncompact type.
[ | p[4.5cm]{} p[2.6cm]{} p[2.4cm]{} p[1.6cm]{} | ]{}
------------------------------------------------------------------------
Dynkin diagram & $M$ & Multiplicities & Comments\
------------------------------------------------------------------------
& $SO^o_{1,1+k}/SO_{1+k}$ & $k$ & $k \geq 1$\
& $SL_{r+1}({\mathbb R})/SO_{r+1}$ & $1,1,\ldots,1,1$ & $r \geq 2$\
& $SL_{r+1}({\mathbb C})/SU_{r+1}$ & $2,2,\ldots,2,2$ & $r \geq 2$\
& $SU^*_{2r+2}/Sp_{r+1}$ & $4,4,\ldots,4,4$ & $r \geq 2$\
& $E_6^{-26}/F_4$ & $8,8$ &\
------------------------------------------------------------------------
& $SO^o_{r,r+k}/SO_{r}SO_{r+k}$ & $1,1,\ldots,1,1,k$ & $r \geq 2, k \geq 1$\
& $SO_{2r+1}({\mathbb C})/SO_{2r+1}$ & $2,2,\ldots,2,2,2$ & $r \geq 2$\
------------------------------------------------------------------------
& $Sp_r({\mathbb R})/U_r$ & $1,1,\ldots,1,1,1$ & $r \geq 3$\
& $SU_{r,r}/S(U_rU_r)$ & $2,2,\ldots,2,2,1$ & $r \geq 3$\
& $Sp_r({\mathbb C})/Sp_r$ & $2,2,\ldots,2,2,2$ & $r \geq 3$\
& $SO^*_{4r}/U_{2r}$ & $4,4,\ldots,4,4,1$ & $r \geq 3$\
& $Sp_{r,r}/Sp_rSp_r$ & $4,4,\ldots,4,4,3$ & $r \geq 2$\
& $E_7^{-25}/E_6U_1$ & $8,8,1$ &\
------------------------------------------------------------------------
& $SO^o_{r,r}/SO_{r}SO_{r}$ & $1,1,\ldots,1,1,1,1$ & $r \geq 4$\
& $SO_{2r}({\mathbb C})/SO_{2r}$ & $2,2,\ldots,2,2,2,2$ & $r \geq 4$\
------------------------------------------------------------------------
& $SU_{r,r+k}/S(U_rU_{r+k})$ & $2,2,\ldots,2,2,(2k,1)$ & $r \geq 1, k \geq 1$\
& $Sp_{r,r+k}/Sp_rSp_{r+k}$ & $4,4,\ldots,4,4,(4k,3)$ & $r \geq 1, k \geq 1$\
& $SO^*_{4r+2}/U_{2r+1}$ & $4,4,\ldots,4,4,(4,1)$ & $r \geq 2$\
& $E_6^{-14}/Spin_{10}U_1$ & $6,(8,1)$ &\
& $F_4^{-20}/Spin_9$ & $(8,7)$ &\
------------------------------------------------------------------------
& $E_6^6/Sp_4$ & $1,1,1,1,1,1$ &\
& $E_6({\mathbb C})/E_6$ & $2,2,2,2,2,2$ &\
------------------------------------------------------------------------
& $E_7^7/SU_8$ & $1,1,1,1,1,1,1$ &\
& $E_7({\mathbb C})/E_7$ & $2,2,2,2,2,2,2$ &\
------------------------------------------------------------------------
& $E_8^8/SO_{16}$ & $1,1,1,1,1,1,1,1$ &\
& $E_8({\mathbb C})/E_8$ & $2,2,2,2,2,2,2,2$ &\
------------------------------------------------------------------------
& $F_4^4/Sp_3Sp_1$ & $1,1,1,1$ &\
& $E_6^2/SU_6Sp_1$ & $1,1,2,2$&\
& $E_7^{-5}/SO_{12}Sp_1$ & $1,1,4,4$ &\
& $E_8^{-24}/E_7Sp_1$ & $1,1,8,8$ &\
& $F_4({\mathbb C})/F_4$ & $2,2,2,2$&\
------------------------------------------------------------------------
& $G_2^2/SO_4$ & $1,1$ &\
& $G_2({\mathbb C})/G_2$ & $2,2$ &\
Parabolic subalgebras (resp. subgroups) of real semisimple Lie algebras (resp. Lie groups) play an important role for the geometry of Riemannian symmetric spaces of noncompact type for which their is no analogue in the compact case. We will now describe how to construct all parabolic subalgebras of ${\mathfrak{g}}$. We denote by $\Psi^+$ the set of positive roots in $\Psi$ with respect to the set $\Lambda$ of simple roots. Let $\Phi$ be a subset of $\Lambda$. We denote by $\Psi_\Phi$ the root subsystem of $\Psi$ generated by $\Phi$, that is, $\Psi_\Phi$ is the intersection of $\Psi$ and the linear span of $\Phi$. We define a reductive subalgebra ${\mathfrak{l}}_\Phi$ and a nilpotent subalgebra ${\mathfrak{n}}_\Phi$ of ${\mathfrak g}$ by $${\mathfrak{l}}_\Phi = {\mathfrak{g}}_0 \oplus \left(\bigoplus_{\alpha \in \Psi_\Phi} {\mathfrak{g}}_{\alpha}\right)
\ \ \textrm{and}\ \
{\mathfrak{n}}_\Phi = \bigoplus_{\alpha \in \Psi^+\setminus \Psi_\Phi^+} {\mathfrak g}_{\alpha}.$$ It follows from properties of root spaces that $[{\mathfrak{l}}_\Phi,{\mathfrak{n}}_\Phi] \subset {\mathfrak{n}}_\Phi$ and therefore $${\mathfrak{q}}_\Phi = {\mathfrak{l}}_\Phi \oplus {\mathfrak{n}}_\Phi$$ is a subalgebra of ${\mathfrak{g}}$, the so-called parabolic subalgebra of ${\mathfrak{g}}$ associated with the subsystem $\Phi$ of $\Psi$. The decomposition ${\mathfrak{q}}_\Phi = {\mathfrak{l}}_\Phi \oplus {\mathfrak{n}}_\Phi$ is the Chevalley decomposition of the parabolic subalgebra ${\mathfrak{q}}_\Phi$.
Every parabolic subalgebra of ${\mathfrak{g}}$ is conjugate in ${\mathfrak{g}}$ to ${\mathfrak{q}}_\Phi$ for some subset $\Phi$ of $\Lambda$. The set of conjugacy classes of parabolic subalgebras of ${\mathfrak{g}}$ therefore has $2^r$ elements. Two parabolic subalgebras ${\mathfrak{q}}_{\Phi_1}$ and ${\mathfrak{q}}_{\Phi_2}$ of ${\mathfrak{g}}$ are conjugate in the full automorphism group ${\operatorname{Aut}}({\mathfrak{g}})$ of ${\mathfrak{g}}$ if and only if there exists an automorphism $F$ of the Dynkin diagram associated to $\Lambda$ with $F(\Phi_1) = \Phi_2$. If $|\Phi| = r-1$ then ${\mathfrak{q}}_\Phi$ is said to be a maximal parabolic subalgebra of ${\mathfrak{g}}$.
Let $${\mathfrak{a}}_\Phi = \bigcap_{\alpha \in \Phi} \ker(\alpha) \subset {\mathfrak{a}}$$ be the split component of ${\mathfrak{l}}_\Phi$ and denote by ${\mathfrak{a}}^\Phi = {\mathfrak{a}}\ominus {\mathfrak{a}}_\Phi$ the orthogonal complement of ${\mathfrak{a}}_\Phi$ in ${\mathfrak{a}}$. The reductive subalgebra ${\mathfrak{l}}_\Phi$ is the centralizer (and the normalizer) of ${\mathfrak{a}}_\Phi$ in ${\mathfrak{g}}$. The orthogonal complement ${\mathfrak{m}}_\Phi = {\mathfrak{l}}_\Phi \ominus {\mathfrak{a}}_\Phi$ of ${\mathfrak{a}}_\Phi$ in ${\mathfrak{l}}_\Phi$ is a reductive subalgebra of ${\mathfrak{g}}$. The decomposition $${\mathfrak{q}}_\Phi = {\mathfrak{m}}_\Phi \oplus {\mathfrak{a}}_\Phi \oplus {\mathfrak{n}}_\Phi$$ is the Langlands decomposition of the parabolic subalgebra ${\mathfrak{q}}_\Phi$. We have $[{\mathfrak{m}}_\Phi,{\mathfrak{a}}_\Phi] = 0$ and $[{\mathfrak{m}}_\Phi,{\mathfrak{n}}_\Phi] \subset {\mathfrak{n}}_\Phi$. Moreover, ${\mathfrak{g}}_\Phi = [{\mathfrak{m}}_\Phi,{\mathfrak{m}}_\Phi] =[{\mathfrak{l}}_\Phi,{\mathfrak{l}}_\Phi]$ is a semisimple subalgebra of ${\mathfrak{g}}$. The center ${\mathfrak{z}}_\Phi$ of ${\mathfrak{m}}_\Phi$ is contained in ${\mathfrak{k}}_0$ and induces the direct sum decomposition ${\mathfrak{m}}_\Phi = {\mathfrak{z}}_\Phi \oplus {\mathfrak{g}}_\Phi$ and therefore, since ${\mathfrak{z}}_\Phi \subset {\mathfrak{k}}_0$, we see that ${\mathfrak{g}}_\Phi \cap {\mathfrak{k}}_0 = {\mathfrak{k}}_0 \ominus{\mathfrak{z}}_\Phi$.
For each $\alpha \in \Psi$ we define ${\mathfrak{k}}_\alpha = {\mathfrak{k}}\cap ({\mathfrak{g}}_{-\alpha} \oplus {\mathfrak{g}}_\alpha)$ and ${\mathfrak{p}}_\alpha = {\mathfrak{p}}\cap ({\mathfrak{g}}_{-\alpha} \oplus {\mathfrak{g}}_\alpha)$. Then we have ${\mathfrak{k}}_{-\alpha} = {\mathfrak{k}}_\alpha$, ${\mathfrak{p}}_{-\alpha} = {\mathfrak{p}}_\alpha$ and ${\mathfrak{k}}_\alpha \oplus {\mathfrak{p}}_\alpha = {\mathfrak{g}}_{-\alpha} \oplus {\mathfrak{g}}_\alpha$ for all $\alpha \in \Psi$. From general root space properties it follows that $${\mathfrak{f}}_\Phi = {\mathfrak{l}}_\Phi \cap {\mathfrak{p}}= {\mathfrak{a}}\oplus \left( \bigoplus_{\alpha \in \Psi_{\Phi}} {\mathfrak{p}}_\alpha \right)
\ \textrm{and}\
{\mathfrak{b}}_\Phi = {\mathfrak{m}}_\Phi \cap {\mathfrak{p}}= {\mathfrak{g}}_\Phi \cap {\mathfrak{p}}= {\mathfrak{a}}^\Phi \oplus \left( \bigoplus_{\alpha \in \Psi_{\Phi}} {\mathfrak{p}}_\alpha \right)$$ are Lie triple systems in ${\mathfrak{p}}$. We define a subalgebra ${\mathfrak{k}}_\Phi$ of ${\mathfrak{k}}$ by $${\mathfrak{k}}_\Phi = {\mathfrak{q}}_\Phi \cap {\mathfrak{k}}= {\mathfrak{l}}_\Phi \cap {\mathfrak{k}}= {\mathfrak{m}}_\Phi \cap{\mathfrak{k}}= {\mathfrak{k}}_0 \oplus \left( \bigoplus_{\alpha \in \Psi_{\Phi}}
{\mathfrak{k}}_\alpha \right).$$ Then we have $[{\mathfrak{k}}_\Phi , {\mathfrak{m}}_\Phi ] \subset {\mathfrak{m}}_\Phi$, $[{\mathfrak{k}}_\Phi , {\mathfrak{a}}_\Phi ] = \{0\}$ and $[{\mathfrak{k}}_\Phi , {\mathfrak{n}}_\Phi ] \subset {\mathfrak{n}}_\Phi$. Moreover, ${\mathfrak{g}}_\Phi = ({\mathfrak{g}}_\Phi \cap {\mathfrak{k}}_\Phi) \oplus {\mathfrak{b}}_\Phi$ is a Cartan decomposition of the semisimple subalgebra ${\mathfrak{g}}_\Phi$ of ${\mathfrak{g}}$ and ${\mathfrak{a}}^\Phi$ is a maximal abelian subspace of ${\mathfrak{b}}_\Phi$. If we define $({\mathfrak{g}}_\Phi)_0 = ({\mathfrak{g}}_\Phi \cap {\mathfrak{k}}_0) \oplus {\mathfrak{a}}^\Phi$, then ${\mathfrak{g}}_\Phi = ({\mathfrak{g}}_\Phi)_0 \oplus \left(\bigoplus_{\alpha \in \Psi_\Phi} {\mathfrak{g}}_{\alpha}\right)$ is the restricted root space decomposition of ${\mathfrak{g}}_\Phi$ with respect to ${\mathfrak a}^\Phi$ and $\Phi$ is the corresponding set of simple roots.
Let $F_\Phi$ and $B_\Phi$ be the connected complete totally geodesic submanifold of $M$ corresponding to the Lie triple systems ${\mathfrak{f}}_\Phi$ and ${\mathfrak{b}}_\Phi$, respectively. Then $B_\Phi$ is a Riemannian symmetric space of noncompact type with ${\operatorname{rk}}(B_\Phi) = |\Phi|$, also known as a boundary component in the maximal Satake compactification of $M$ (see [@BJ]). Note that $B_\Phi$ is irreducible if and only if the Dynkin diagram corresponding to $\Phi$ is connected. The totally geodesic submanifold $F_\Phi$ is isometric to the Riemannian product $B_\Phi \times {\mathbb{R}}^{r-|\phi|}$, where ${\mathbb{R}}^{r-|\phi|}$ is the totally geodesic Euclidean space in $M$ corresponding to the abelian Lie triple system ${\mathfrak{a}}_\Phi$. For $i \in \{1,\ldots,r\}$ we define $\Phi_i = \Lambda \setminus \{\alpha_i\}$, ${\mathfrak{l}}_i = {\mathfrak{l}}_{\Phi_i}$, $F_i = F_{\Phi_i}$, $B_i = B_{\Phi_i}$, etcetera. Then we have $F_i = {\mathbb{R}}\times B_i$.
Reflective submanifolds {#reflective}
=======================
Let $\Sigma'$ be a connected totally geodesic submanifold of $M$. Since $M$ is homogeneous we can assume that $p \in \Sigma'$. Moreover, since every connected totally geodesic submanifold of a Riemannian symmetric space is contained in a connected complete totally geodesic submanifold, we can also assume that $\Sigma'$ is complete. Since $M$ is of noncompact type, $\Sigma'$ is the Riemannian product of a (possibly $0$-dimensional) Euclidean space and a (possibly $0$-dimensional) Riemannian symmetric space of noncompact type. This implies in particular that $\Sigma'$ is simply connected.
The tangent space ${\mathfrak{m}}' = T_p\Sigma' \subset T_pM = {\mathfrak{p}}$ is a Lie triple system in ${\mathfrak{p}}$ and thus ${\mathfrak{g}}' = [{\mathfrak{m}}',{\mathfrak{m}}'] \oplus {\mathfrak{m}}' \subset {\mathfrak{k}}\oplus {\mathfrak{p}}= {\mathfrak{g}}$ is a subalgebra of ${\mathfrak{g}}$. Let $G'$ be the connected closed subgroup of $G$ with Lie algebra ${\mathfrak{g}}'$. Then $\Sigma'$ is the orbit $G' \cdot p$ of the $G'$-action on $M$ containing $p$. Thus we can write $\Sigma' = G'/K'$, where $K' = G'_p$ is the isotropy group of $G'$ at $p$. Since $\Sigma'$ is simply connected, the isotropy group $K'$ is connected. The Lie algebra ${\mathfrak{k}}'$ of $K'$ is given by ${\mathfrak{k}}' = [{\mathfrak{m}}',{\mathfrak{m}}']$. Note that $G'$ is a normal subgroup of $G^{\Sigma'} = \{g\in G \mid g(\Sigma') = \Sigma'\}$ and $K'$ is a normal subgroup of $(G^{\Sigma'})_p$.
The following Slice Lemma was proved in [@BO] and will be used later. We formulate it here for the noncompact case, but it is valid also for the compact case.
[(Slice Lemma)]{}\[Simons\] Let $M=G/K$ be an irreducible Riemannian symmetric space of noncompact type with ${\operatorname{rk}}(M)\geq 2$, where $G= I^o(M)$ and $K = G_p$ is the isotropy group of $G$ at $p \in M$. Let $\mathfrak g = \mathfrak k \oplus \mathfrak p$ be the corresponding Cartan decomposition. Let $\Sigma'$ be a nonflat totally geodesic submanifold of $M$ such that $p\in \Sigma'$. Let $G'$ be the connected closed subgroup of $G$ with Lie algebra $[{\mathfrak{m}}',{\mathfrak{m}}'] \oplus {\mathfrak{m}}' $, where $T_p\Sigma' = {\mathfrak{m}}' \subset \mathfrak p = T_pM$, and $K' = G'_p$. Then the slice representation of $K'$ on $\nu _p \Sigma'$ is nontrivial.
In general, the orthogonal complement ${\mathfrak{m}}''$ of a Lie triple system ${\mathfrak{m}}'$ in ${\mathfrak{p}}$ is not a Lie triple system. If ${\mathfrak{m}}''$ is a Lie triple system, then ${\mathfrak{m}}'$ is said to be a reflective Lie triple system and $\Sigma'$ is said to be a reflective submanifold of $M$. The notion comes from the fact that the geodesic reflection of $M$ in $\Sigma'$ is a well-defined global isometry of $M$ if and only if both ${\mathfrak{m}}'$ and ${\mathfrak{m}}''$ are Lie triple systems. Reflective submanifolds therefore always come in pairs $\Sigma'$ and $\Sigma''$ corresponding to the two reflective Lie triple systems ${\mathfrak{m}}'$ and ${\mathfrak{m}}''$. In this situation we write $\Sigma'' = G''/K''$, where $G''$ is the connected closed subgroup of $G$ with Lie algebra ${\mathfrak{g}}'' = [{\mathfrak{m}}'',{\mathfrak{m}}''] \oplus {\mathfrak{m}}''$ and $K'' = G''_p$ is the connected closed subgroup of $K$ with Lie algebra ${\mathfrak{k}}'' = [{\mathfrak{m}}'',{\mathfrak{m}}'']$. The reflective submanifolds of irreducible simply connected Riemannian symmetric spaces of compact type were classified by Leung ([@L1],[@L2]). Using duality one obtains the classification of reflective submanifolds in irreducible Riemannian symmetric spaces of noncompact type.
Let $R$ denote the Riemannian curvature tensor of $M$. As $\Sigma'$ is totally geodesic in $M$, the restriction of $R$ to $\Sigma'$ coincides with the Riemannian curvature tensor of $\Sigma'$. We will regard, via the isotropy representation at $p$, $K' \subset K \subset SO(T_pM)$. Note that ${\mathfrak{k}}$ and ${\mathfrak{k}}'$ are generated by the curvature transformations $R_{x,y} \in {\mathfrak{so}}(T_pM)$ with $x,y \in T_pM$ and $x,y \in T_p\Sigma'$, respectively. The curvature operator $\tilde R : \Lambda ^2 (T_pM) \to \Lambda ^2 (T_pM)$ is negative semidefinite. This implies, as is well-known, that $K'$ acts almost effectively on $T_p\Sigma'$.
Let $\rho : K' \to SO (\nu _p\Sigma'),\ k \mapsto d_pk_{\vert \nu _p\Sigma'}$ be the slice representation of $K'$ on the normal space $\nu _p\Sigma'$ of $\Sigma'$ at $p$ and denote by $\ker(\rho)$ the kernel of $\rho$. Let $\chi : K'' \to SO (T _p\Sigma''),\ k \mapsto d_pk_{\vert T _p\Sigma''}$ be the isotropy representation of $K''$ on the tangent space $T_p\Sigma''$.
\[ref1\] Let $M = G/K$ be an irreducible Riemannian symmetric space of noncompact type, $\Sigma' = G'/K'$ be a reflective submanifold of $M$ and $\Sigma'' = G''/K''$ be the reflective submanifold of $M$ with $T_p\Sigma'' = \nu_p\Sigma'$. Then:
- $\rho (K')$ is a normal subgroup of $\chi(K'')$.
- The subspace $(\nu_p\Sigma')_o = \{ \xi \in \nu_p\Sigma' \mid \rho(k')\xi = \xi\ \textrm{for\ all}\ k' \in K'\}$ of $\nu_p\Sigma' = T_p\Sigma''$ is $\chi(K'')$-invariant and $\Sigma'' = \Sigma''_o \times \Sigma''_1$ (Riemannian product), where $\Sigma''_o$ is the totally geodesic submanifold of $\Sigma''$ with $T_p\Sigma''_o = (\nu_p\Sigma')_o$. Moreover, if ${\operatorname{rk}}(M) \geq 2$, then $\Sigma''_o$ is flat.
As previously observed, $K'$ is a normal subgroup of $(G^{\Sigma'})_p$. Observe also that $K'' \subset (G^{\Sigma '})_p$ and that $\rho (K') \subset \chi (K'')$ (for the last inclusion see the paragraph below Lemma 2.1 in [@BO]). Then $\rho (K') = \rho (k'' K ' (k'')^{-1}) = \chi (k'') \rho(K') (\chi (k''))^{-1}$ for all $k''\in K''$ and thus $\rho (K')$ is a normal subgroup of $\chi (K'')$. Thus the subspace $(\nu_p\Sigma')_o$ of $T_p\Sigma''$ is $\chi(K'')$-invariant and hence also invariant under the holonomy group of $\Sigma''$ at $p$. Since $\Sigma''$ is simply connected, the de Rham decomposition theorem for Riemannian manifolds implies that $\Sigma''$ decomposes as a Riemannian product into $\Sigma'' = \Sigma''_o \times \Sigma''_1$, where $\Sigma''_o$ is the totally geodesic submanifold of $\Sigma''$ with $T_p\Sigma''_o = (\nu_p\Sigma')_o$.
We write $\Sigma''_o = G''_o/K''_o$, where $G''_o$ is the connected closed subgroup of $G$ with Lie algebra ${\mathfrak{g}}''_o = [T_p\Sigma''_o,T_p\Sigma''_o] \oplus T_p\Sigma''_o$ and $K''_o$ is the isotropy group of $G''_o$ at $p$. Let $x_1,x_2 \in T_p\Sigma''_o = (\nu_p\Sigma')_o$. For all $y\in T_p\Sigma''_1$ we have $R_{x_1,x_2}y = 0$ since $\Sigma'' = \Sigma''_o \times \Sigma''_1$ is a Riemannian product and totally geodesic in $M$. Clearly, $T_p\Sigma''_1$ is $K''_o$-invariant and hence $T_p\Sigma'$ is also $K''_o$-invariant. If $x', y' \in T_p\Sigma'$, then $\langle R_{x_1,x_2}x', y'\rangle = \langle R_{x', y'}x_1,x_2\rangle = 0$, since $x_1, x_2 \in (\nu_p\Sigma')_o$ are fixed under the slice representation of $K'$. Since $\nu _p \Sigma''_o = T_p\Sigma''_1 \oplus T_p\Sigma'$ and ${\mathfrak{k}}''_o$ is linearly spanned by the curvature endomorphisms of pairs of elements in $T_p\Sigma''_o$, we conclude that the slice representation of $K''_o$ on $\nu _p \Sigma''_o$ is trivial. It follows from the Slice Lemma \[Simons\] that $\Sigma''_o$ is flat if ${\operatorname{rk}}(M) \geq 2$. This finishes the proof of part (ii).
\[ref2\] Let $M = G/K$ be an irreducible Riemannian symmetric space of noncompact type with ${\operatorname{rk}}(M) \geq 2$ and let $\Sigma$ be a totally geodesic submanifold of $M$ which decomposes into a Riemannian product $\Sigma = \Sigma_0 \times \Sigma_1$ with a Euclidean factor $\Sigma_0$ and a semisimple factor $\Sigma_1$ with $\dim(\Sigma_0) > 0$ and $\dim(\Sigma_1) > 0$. Then $\Sigma_1$ is not a reflective submanifold of $M$.
Assume that $\Sigma_1$ is a reflective submanifold of $M$. We will apply Lemma \[ref1\] with $\Sigma' = \Sigma_1$. In the notation of Lemma \[ref1\], we have $T_p\Sigma_0 \subset (\nu_p\Sigma')_o$ and therefore $\Sigma_0$ is contained in the flat factor $\Sigma''_o$ of $\Sigma''$. This implies that $R_{x_0,x''} = 0$ for all $x_0\in T_p\Sigma _0$ and $x'' \in T_p\Sigma''$. We obviously also have $R_{x_0,x_1} = 0$ for all $x_0 \in T_p\Sigma_0$ and $x_1 \in T_p\Sigma_1 = T_p\Sigma'$. Since $T_pM = T_p\Sigma' \oplus T_p\Sigma''$ this implies $R_{x_0, \cdot } = 0$ for all $x_0 \in T_p\Sigma_0$, which is a contradiction.
The next result provides a useful sufficient criterion for a semisimple totally geodesic submanifold of an irreducible Riemannian symmetric space to be reflective.
\[ref3\] Let $M = G/K$ be an irreducible Riemannian symmetric space of noncompact type with ${\operatorname{rk}}(M) \geq 2$ and let $\Sigma = G'/K'$ be a semisimple totally geodesic submanifold of $M$. Assume that the kernel $\ker(\rho)$ of the slice representation $\rho : K' \to SO (\nu _p \Sigma)$ has positive dimension. Then we have $$\nu _p \Sigma = \{\xi \in T_pM \mid \rho(k)\xi = \xi\ \textrm{for\ all}\ k \in \ker(\rho)^o\}$$ and, in particular, $\Sigma$ is a reflective submanifold of $M$.
The subspace ${\mathbb{V}}= \{\xi \in T_p\Sigma \mid d_pk(\xi) = \xi\ \textrm{for\ all}\ k \in \ker(\rho)^o\}$ of $T_p\Sigma$ is $K'$-invariant since $\ker(\rho)^o$ is a normal subgroup of $K'$.
We first assume that $\mathbb V = T_p \Sigma$. Since $\ker(\rho)^o$ acts trivially on $\nu _p\Sigma$ we conclude that $\ker(\rho)^o$ and hence $K'$ acts trivially on $T_pM$, which is a contradiction.
Next, we assume that ${\mathbb{V}}$ is a nontrivial proper $K'$-invariant subspace of $T_p\Sigma$. Then $\Sigma$ decomposes as a Riemannian product $\Sigma = \Sigma_1 \times \Sigma_2$, where ${\mathbb{V}}= T_p\Sigma_1$. If we write, as usual, $\Sigma_i = G_i/K_i$, then $K' = K_1 \times K_2$ (almost direct product). Let ${\mathfrak{h}}_i$ be the orthogonal projection of the Lie algebra of $\ker(\rho)$ into ${\mathfrak{k}}_i$ and let $H_i$ be the connected subgroup of $K_i$ with Lie algebra ${\mathfrak{h}}_i$. Then $H_1$ acts trivially on ${\mathbb{V}}= T_p\Sigma_1$ since both $\ker(\rho)^o$ and $H_2$ act trivially on ${\mathbb{V}}$. Since $K_1$ acts almost effectively on $T_p\Sigma_1$ and $H_1$ is connected, it follows that $H_1$ is trivial. Thus we have shown that $\ker(\rho)^o \subset K_2$.
Note that $\{\xi \in T_pM \mid \rho(k)\xi = \xi\ \textrm{for\ all}\ k \in \ker(\rho)^o\} = \mathbb V \oplus \nu _p \Sigma = \nu _p \Sigma_2$. This shows that $\Sigma _2$ is a reflective submanifold of $M$. Let $\Sigma _3 = G_3/K_3$ be the reflective submanifold of $M$ with $T_p\Sigma_3 = \nu _p\Sigma _2$. We denote by $\rho _i : K_i \to SO (\nu _p \Sigma _i)$ the slice representation of $K_i$ on the normal space $\nu_p\Sigma_i$ and by $\chi_i : K_i \to SO(T_p\Sigma_i)$ the isotropy representation of $K_i$, $i \in \{1,2,3\}$.
From Lemma \[ref1\](i) we see that $\rho_2 (K_2)$ is a normal subgroup of $\chi_3(K_3)$. Let $\mathbb W$ be the set of fixed vectors of $\rho _2 (K_2)$ in $\nu _p \Sigma _2 = T_p \Sigma _3 = \mathbb V \oplus \nu _p \Sigma = T_p\Sigma _1 \oplus \nu _p \Sigma$. Since $K_2$ acts trivially on $T_p\Sigma _1$ one has that $T_p\Sigma _1 \subset \mathbb W$. From Lemma \[ref1\](ii) we know that $\mathbb W$ is the tangent space of a Euclidean factor of $\Sigma _3$. This is a contradiction since $\Sigma _1$ is contained in this Euclidean factor, however, $\Sigma _1$ is not flat as $\Sigma $ is semisimple. It follows that $\mathbb V =\{ 0 \}$, which proves the assertion.
The following consequence of Proposition \[ref3\] states that totally geodesic submanifolds of sufficiently small codimension in irreducible Riemannian symmetric spaces are reflective.
\[ref4\] Let $M$ be an $n$-dimensional irreducible Riemannian symmetric space of noncompact type with $r = {\operatorname{rk}}(M) \geq 2$ and let $\Sigma$ be a semisimple connected complete totally geodesic submanifold of $M$ with ${\operatorname{codim}}(\Sigma) = d$. If $$\frac{1}{2}d(d+1) +{\operatorname{rk}}(\Sigma) < n,$$ then $\Sigma$ is a reflective submanifold of $M$. In particular, if $$d(d+1) < 2(n-r),$$ then $\Sigma$ is a reflective submanifold of $M$.
As usual, we write $\Sigma = G'/K'$. If $\dim(K') > \dim(SO(\nu_p\Sigma)) = \frac{1}{2}d(d-1)$, then the kernel of the slice representation $\rho : K' \to SO(\nu_p\Sigma)$ must have positive dimension and therefore $\Sigma$ is a reflective submanifold of $M$ by Proposition \[ref3\]. A principal $K'$-orbit on $\Sigma$ has dimension $n-d-{\operatorname{rk}}(\Sigma)$ and thus $\dim(K') \geq n - d - {\operatorname{rk}}(\Sigma)$. Consequently, if $\frac{1}{2}d(d-1) < n - d - {\operatorname{rk}}(\Sigma)$, then $\Sigma$ is a reflective submanifold of $M$. The inequality $\frac{1}{2}d(d-1) < n - d - {\operatorname{rk}}(\Sigma)$ is equivalent to $\frac{1}{2}d(d+1) + {\operatorname{rk}}(\Sigma) < n$. The last statement follows from the fact that ${\operatorname{rk}}(\Sigma) \leq {\operatorname{rk}}(M) = r$.
Non-semisimple maximal totally geodesic submanifolds {#nonsemisimple}
====================================================
Let $\Sigma$ be a connected totally geodesic submanifold of $M$. We may assume that $\Sigma$ is complete and $p \in \Sigma$. Every connected complete totally geodesic submanifold of a Riemannian symmetric space is again a Riemannian symmetric space. In this paper, when we consider a totally geodesic submanifold $\Sigma$ of $M$, we always assume that $p \in \Sigma$ and that $\Sigma$ is connected and complete. Since $M$ is of noncompact type, it follows that $\Sigma$ is isometric to the Riemannian product $\Sigma_0 \times \Sigma_1$, where $\Sigma_0$ is a (possibly $0$-dimensional) Euclidean space and $\Sigma_1$ is a (possibly $0$-dimensional) Riemannian symmetric space of noncompact type.
The next result relates non-semisimple maximal totally geodesic submanifolds of $M$ to the reductive factors in the Chevalley decompositions of the maximal parabolic subalgebras of ${\mathfrak{g}}$.
\[totgeodnss\] Let $M = G/K$ be an irreducible Riemannian symmetric space of noncompact type and let $\Sigma$ be a non-semisimple maximal totally geodesic submanifold of $M$. Then $\Sigma$ is congruent to $F_i = {\mathbb{R}}\times B_i$ for some $i \in \{i,\ldots,r\}$.
Let ${\mathfrak{a}}$ be a maximal abelian subspace of ${\mathfrak{p}}$ with $T_p\Sigma_0 \subset {\mathfrak{a}}$ and consider the restricted root space decomposition of ${\mathfrak{g}}$ induced by ${\mathfrak{a}}$. We define $\Upsilon = \{ \alpha_i \in \Lambda \mid \alpha_i(T_p\Sigma_0) = 0\} \subset \Lambda$. Assume that $\Upsilon = \Lambda$, which means that $\alpha_i(T_p\Sigma_0) = 0$ for all $\alpha_i \in \Lambda$. This implies $T_p\Sigma_0 = \{0\}$ and therefore $\Sigma = \Sigma_1$ is semisimple, which is a contradiction. Thus we have $|\Upsilon| < |\Lambda| = r$ and therefore there exists $i \in \{1,\ldots,r\}$ such that $\Upsilon \subset \Phi_i$. Then we get $$\begin{aligned}
T_p\Sigma
& \subset & Z_{\mathfrak{p}}(T_p\Sigma_0) = \{ X \in {\mathfrak{p}}\mid [X,Y] = 0\ \textrm{for\ all}\ Y \in T_p\Sigma_0\} \\
& \subset & Z_{\mathfrak{g}}(T_p\Sigma_0) = \{ X \in {\mathfrak{g}}\mid [X,Y] = 0\ \textrm{for\ all}\ Y \in T_p\Sigma_0\} \\
& = & {\mathfrak{g}}_0 \oplus \left( \bigoplus_{\alpha \in \Psi,\alpha(T_p\Sigma_0) = \{0\}} {\mathfrak{g}}_\alpha \right)
= {\mathfrak{l}}_\Upsilon \subset {\mathfrak{l}}_i,\end{aligned}$$ which implies $T_p\Sigma \subset {\mathfrak{l}}_i \cap {\mathfrak{p}}= {\mathfrak{f}}_i$ and therefore $\Sigma \subset F_i = {\mathbb{R}}\times B_i$. If $\Sigma$ is a maximal totally geodesic submanifold of $M$ we must have $\Sigma = F_i$, since $F_i$ is a totally geodesic submanifold of $M$ which is strictly contained in $M$.
The remaining problem is to clarify which of the totally geodesic submanifolds $F_i$ are maximal. The solution of this problem is related to symmetric R-spaces. Let $M = G/K$ be an irreducible Riemannian symmetric space of noncompact type and consider the isotropy representation $$\chi : K \to T_pM = {\mathfrak{p}},\ v \mapsto d_pk(v) = {\operatorname{Ad}}(k)v.$$ For every $0 \neq v \in {\mathfrak{p}}$ the orbit $$K\cdot v = \{{\operatorname{Ad}}(k)v \mid k \in K\} \subset {\mathfrak{p}}$$ is called an R-space (or real flag manifold). One can show that the normal space $\nu_v(K \cdot v)$ of $K \cdot v$ at $v$ is equal to $$\nu_v(K \cdot v) = Z_{\mathfrak{p}}(v) = \{ w \in {\mathfrak{p}}\mid [v,w] = 0 \},$$ where $Z_{\mathfrak{p}}(v)$ is the centralizer of $v$ in ${\mathfrak{p}}$. It follows from the Jacobi identity that $Z_{\mathfrak{p}}(v)$ is a Lie triple system. Thus, for every $0 \neq v \in {\mathfrak{p}}$, there exists a connected complete totally geodesic submanifold $\Sigma^v$ of $M$ with $T_p\Sigma^v = \nu_v(K \cdot v)$. Since every $v \in {\mathfrak{p}}$ is contained in a maximal abelian subspace of ${\mathfrak{p}}$ we can assume that $v \in {\mathfrak{a}}$. Then we have ${\mathfrak{l}}_\Phi = Z_{\mathfrak{g}}(v)$ with $\Phi = \{\alpha_i \in \Lambda \mid \alpha_i(v) = 0\}$, which implies ${\mathfrak{f}}_\Phi = Z_{\mathfrak{p}}(v) = \nu_v(K \cdot v)$ and therefore $F_\Phi = \Sigma^v$.
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$(R)$ & Highest root $\delta = \delta_1\alpha_1 + \ldots + \delta_r\alpha_r$ & Comments\
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$(A_r)$ & $\alpha_1 + \ldots + \alpha_r$ & $r \geq 1$\
$(B_r)$ & $\alpha_1 + 2\alpha_2 + \ldots + 2\alpha_r$ & $r \geq 2$\
$(C_r)$ & $2\alpha_1 + \ldots + 2\alpha_{r-1} + \alpha_r$ & $r \geq 3$\
$(D_r)$ & $\alpha_1 + 2\alpha_2 + \ldots + 2\alpha_{r-2} + \alpha_{r-1} + \alpha_r$ & $r \geq 4$\
$(BC_r)$ & $2\alpha_1 + \ldots + 2\alpha_r$ & $r \geq 1$\
$(E_6)$ & $\alpha_1 + 2\alpha_2 + 2\alpha_3 + 3\alpha_4 + 2\alpha_5 + \alpha_6$ &\
$(E_7)$ & $2\alpha_1 + 2\alpha_2 + 3\alpha_3 + 4\alpha_4 + 3\alpha_5 + 2\alpha_6 + \alpha_7$ &\
$(E_8)$ & $2\alpha_1 + 3\alpha_2 + 4\alpha_3 + 6\alpha_4 + 5\alpha_5 + 4\alpha_6 + 3\alpha_7 + 2\alpha_8$ &\
$(F_4)$ & $2\alpha_1 + 3\alpha_2 + 4\alpha_3 + 2\alpha_4$ &\
$(G_2)$ & $3\alpha_1 + 2\alpha_2$ &\
A special situation occurs when the orbit $K \cdot v$ is a symmetric space. In this situation the orbit $K \cdot v$ is called an irreducible symmetric R-space. The irreducibility here refers to the irreducibility of the symmetric space $G/K$ and not to the orbit. An irreducible symmetric R-space can be reducible as a Riemannian manifold. The irreducible symmetric R-spaces were classified by Kobayashi and Nagano in [@KN]. Their classification can be read off from the Dynkin diagram and highest root of the symmetric spaces $G/K$. In Table \[dynkin\] we already listed the Dynkin diagrams. In Table \[highestroot\] we list the corresponding highest roots $\delta = \delta_1\alpha_1 + \ldots + \delta_r\alpha_r$.
Kobayashi and Nagano proved that an R-space $K\cdot v$ is symmetric if and only if $v = H^i$ and $\delta_i = 1$. From Tables \[dynkin\] and \[highestroot\] one can easily get the classification of irreducible symmetric R-spaces. We can now state the main result of this section:
\[rspace\] Let $M = G/K$ be an irreducible Riemannian symmetric space of noncompact type and let $\Sigma$ be a non-semisimple connected complete totally geodesic submanifold of $M$. Then the following statements are equivalent:
- $\Sigma$ is a maximal totally geodesic submanifold of $M$;
- $\Sigma$ is isometrically congruent to $F_i = {\mathbb{R}}\times B_i$ and $\delta_i = 1$;
- $\nu_p\Sigma$ is the tangent space of a symmetric R-space in $T_pM$;
- The pair $(M,\Sigma)$ is as in Table [\[totgeodnsstable\]]{}.
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$M$ & $B$ & ${\operatorname{codim}}(\Sigma)$ & Comments\
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$SL_{r+1}({\mathbb{R}})/SO_{r+1}$ & $SL_i({\mathbb{R}})/SO_i \times SL_{r+1-i}({\mathbb{R}})/SO_{r+1-i}$ & $i(r+1-i)$ & $r \geq 2$, $1 \leq i \leq [r/2]$\
$SL_{r+1}({\mathbb{C}})/SU_{r+1}$ & $SL_i({\mathbb{C}})/SU_i \times SU_{r+1-i}({\mathbb{C}})/SU_{r+1-i}$ & $2i(r+1-i)$ & $r \geq 2$, $1 \leq i \leq [r/2]$\
$SU^*_{2r+2}/Sp_{r+1}$ & $SU^*_{2i}/Sp_i \times SU^*_{2(r+1-i)}/Sp_{r+1-i}$ & $4i(r+1-i)$ & $r \geq 2$, $1 \leq i \leq [r/2]$\
$E_6^{-26}/F_4$ & ${\mathbb{R}}H^9$ & $16$ &\
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$SO^o_{r,r+k}/SO_{r}SO_{r+k}$ & $SO^o_{r-1,r-1+k}/SO_{r-1}SO_{r-1+k}$ & $2r-2+k$ & $r \geq 2, k \geq 1$\
$SO_{2r+1}({\mathbb{C}})/SO_{2r+1}$ & $SO_{2r-1}({\mathbb{C}})/SO_{2r-1}$& $4r-2$ & $r \geq 2$\
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$Sp_r({\mathbb{R}})/U_r$ & $SL_r({\mathbb{R}})/SO_r$ & $\frac{1}{2}r(r+1)$ & $r \geq 3$\
$SU_{r,r}/S(U_rU_r)$ & $SL_r({\mathbb{C}})/SU_r $ & $r^2$ & $r \geq 3$\
$Sp_r({\mathbb{C}})/Sp_r$ & $SL_r({\mathbb{C}})/SU_r$ & $r(r+1)$ & $r \geq 3$\
$SO^*_{4r}/U_{2r}$ & $SU^*_{2r}/Sp_r$ & $r(2r-1)$ & $r \geq 3$\
$Sp_{r,r}/Sp_rSp_r$ & $SU^*_{2r}/Sp_r$ & $r(2r+1)$ & $r \geq 2$\
$E_7^{-25}/E_6U_1$ & $E_6^{-26}/F_4$ & $27$ &\
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$SO^o_{r,r}/SO_{r}SO_{r}$ & $SO^o_{r-1,r-1}/SO_{r-1}SO_{r-1}$ & $2(r-1)$ & $r \geq 4$\
& $SL_r({\mathbb{R}})/SO_r$ & $\frac{1}{2}r(r-1)$ & $r \geq 4$\
$SO_{2r}({\mathbb{C}})/SO_{2r}$ & $SO_{2(r-1)}({\mathbb{C}})/SO_{2(r-1)}$& $4(r-1)$ & $r \geq 4$\
& $SL_r({\mathbb{C}})/SU_r$ & $r(r-1)$ & $r \geq 4$\
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$E_6^6/Sp_4$ & $SO^o_{5,5}/SO_5SO_5$ & $16$ &\
$E_7^7/SU_8$ & $E_6^6/Sp_4$ & $27$ &\
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$E_6({\mathbb C})/E_6$ & $SO_{10}({\mathbb{C}})/SO_{10}$ & $32$ &\
$E_7({\mathbb C})/E_7$ & $E_6({\mathbb{C}})/E_6$ & $54$ &\
The equivalence of (ii) and (iv) is a straightforward computation using Tables \[dynkin\] and \[highestroot\]. Kobayashi and Nagano proved that an R-space $K \cdot v$ is symmetric if and only if $v = H^i$ and $\delta_i = 1$. In this situation we have $\nu_{H^i}(K \cdot H^i) = Z_{\mathfrak{p}}(H^i) = {\mathfrak{f}}_i = T_pF_i$ and hence $\nu_pF_i = T_{H^i}(K \cdot H^i)$. This gives the equivalence of (ii) and (iii). We shall now prove the equivalence of (i) and (ii).
We first assume that $\Sigma$ is a maximal totally geodesic submanifold of $M$. From Proposition \[totgeodnss\] we know that, up to conjugacy, $\Sigma = F_i$ for some $i \in \{1,\ldots,r\}$. Assume that $\delta_i \geq 2$ and let $t$ be a prime number with $t \leq \delta_i$. Then define the semisimple subalgebra ${\mathfrak{h}}_{i,t}$ of ${\mathfrak{g}}$ by $${\mathfrak{h}}_{i,t} = {\mathfrak{g}}_0 \oplus \left( \bigoplus_{\alpha \in \Psi,\alpha(H^i) \equiv 0(\textrm{mod}\,t)} {\mathfrak{g}}_\alpha \right) .$$ Since $${\mathfrak{l}}_i = {\mathfrak{g}}_0 \oplus \left( \bigoplus_{\alpha \in \Psi,\alpha(H^i) = 0} {\mathfrak{g}}_\alpha \right)$$ and $\delta_i \geq t$ we see that ${\mathfrak{l}}_i$ is strictly contained in ${\mathfrak{h}}_{i,t}$. It follows that the Lie triple system ${\mathfrak{l}}_i \cap {\mathfrak{p}}= {\mathfrak{f}}_i$ is strictly contained in the Lie triple system ${\mathfrak{h}}_{i,t} \cap {\mathfrak{p}}$. This is a contradiction since, by assumption, $T_p\Sigma = {\mathfrak{f}}_i$ is a maximal Lie triple system. Consequently we must have $\delta_i = 1$. This finishes the proof for “(i) $\Rightarrow$ (ii)”.
Conversely, let us assume that $\Sigma = F_i$ for some $i \in \{1,\ldots,r\}$ and that $\delta_i = 1$. We denote by $S_i$ the symmetric R-space $K \cdot H^i \subset {\mathfrak{p}}= T_pM$. Then we have $\alpha(H^i) \in \{-1,0,+1\}$ for all $\alpha \in \Psi$ and therefore ${\operatorname{ad}}(H^i)^2$ induces a vector space decomposition ${\mathfrak{g}}= {\mathfrak{g}}^0 \oplus {\mathfrak{g}}^1$ of ${\mathfrak{g}}$, where $${\mathfrak{g}}^0 = {\mathfrak{l}}_i = {\mathfrak{g}}_0 \oplus \left( \bigoplus_{\alpha \in \Psi,\alpha(H^i) = 0} {\mathfrak{g}}_\alpha \right)\ \textrm{and}\
{\mathfrak{g}}^1 = \bigoplus_{\alpha \in \Psi,\alpha(H^i) = \pm 1} {\mathfrak{g}}_\alpha .$$ The map $X_0 + X_1 \to X_0 - X_1$ defines an involutive automorphism of ${\mathfrak{g}}= {\mathfrak{g}}^0 \oplus {\mathfrak{g}}^1$. We denote by $s_i : {\mathfrak{p}}\to {\mathfrak{p}}$ the induced isomorphism on ${\mathfrak{p}}$. Then we have $s_i(X) = -X$ for all $X \in T_{H^i}S_i = {\mathfrak{g}}^1 \cap {\mathfrak{p}}= \oplus_{\alpha \in \Psi,\alpha(H^i) = 1} {\mathfrak{p}}_\alpha $ and $s_i(X) = X$ for all $X \in \nu_{H^i}S_i = {\mathfrak{g}}^0 \cap {\mathfrak{p}}= {\mathfrak{f}}_i = {\mathbb{R}}\times {\mathfrak{b}}_i = Z_{\mathfrak{p}}(H^i)$. The isomorphism $s_i$ is the orthogonal reflection of ${\mathfrak{p}}$ in the normal space $\nu_{H^i}S_i$ and its restriction to $S_i$ leaves $S_i$ invariant and hence induces an involutive isometry on $S_i$ for which $H^i$ is an isolated fixed point. This shows that $S_i$ is a symmetric R-space and that $S_i$ is an extrinsically symmetric submanifold of the Euclidean space $T_pM = {\mathfrak{p}}$ with $s_i$ as the extrinsic symmetry.
Since $[{\mathfrak{g}}^\nu,{\mathfrak{g}}^\mu] \subset {\mathfrak{g}}^{(\nu+\mu)({\rm mod}\,2)}$, we see that $\nu_{H^i}S_i = {\mathfrak{g}}^0 \cap {\mathfrak{p}}$ and $T_{H^i}S_i = {\mathfrak{g}}^1 \cap {\mathfrak{p}}$ are Lie triple systems. It follows that both the tangent space and the normal space of the symmetric R-space $S_i$ at $H^i$ are reflective Lie triple systems.
Let ${\mathbb{V}}\neq {\mathfrak{p}}$ be a Lie triple system in ${\mathfrak{p}}$ with ${\mathfrak{f}}_i \subset {\mathbb{V}}$ and let $\Sigma'$ be the connected complete totally geodesic submanifold of $M$ with $T_p\Sigma' = {\mathbb{V}}$. Then we have $\Sigma' = G'/K'$, where $G'$ and $K'$ is the connected closed subgroup of $G$ with Lie algebra ${\mathfrak{g}}' = [{\mathbb{V}},{\mathbb{V}}] \oplus {\mathbb{V}}$ and ${\mathfrak{k}}' = [{\mathbb{V}},{\mathbb{V}}]$, respectively.
Since ${\operatorname{rk}}(M) \geq 2$, the semisimple factor ${\mathfrak{b}}_i$ of ${\mathfrak{f}}_i$ is non-trivial and therefore ${\mathbb{V}}$ is a non-abelian subspace of ${\mathfrak{p}}$. Since $T_{H^i}S_i$ is a Lie triple system, ${\mathbb{V}}\cap T_{H^i}S_i$ is a Lie triple system as well. Let $N$ be the connected component containing $H^i$ of the intersection $\Sigma' \cap S_i$. It is clear from the construction that $N$ is a smooth submanifold of $S_i$ in an open neighborhood of $H^i$.
We identify $X\in {\mathfrak{g}}$ with the Killing field $q\mapsto X.q = \frac{d}{dt}_{|t = 0}(t \mapsto {\operatorname{Exp}}(tX)(q))$ on $M$. The orthogonal projection $\bar X$ of $X_{\vert \Sigma'}$ to $T\Sigma'$ is a Killing field on the totally geodesic submanifold $\Sigma'$ which lies in the transvection algebra of $\Sigma'$ (see the paragraph below Lemma 2.1 in [@BO]). Note that $\bar X.p=0$ if $X\in {\mathfrak{k}}$. Then, if $X\in {\mathfrak{k}}$, there exists $X' \in {\mathfrak{k}}'$ such that $Z_{\vert \Sigma'}$ is always perpendicular to $\Sigma'$, where $Z=X-X'$. This implies that $Z.{\mathbb{V}}\subset {\mathbb{V}}^\perp$. In fact, let $\gamma _u$ be the geodesic in $M$ with initial condition $\gamma_u'(0) = u \in {\mathbb{V}}$. The Jacobi field $Z.\gamma _u(t)$ is perpendicular to $T_{\gamma_u(t)} \Sigma'$ and therefore its covariant derivative $(Z.\gamma _u)' (t)$ must be so. Hence $(Z.\gamma _u)'(0)= Z.u \in {\mathbb{V}}^\perp$. So, if $X\in {\mathfrak{k}}$, we have $X.u = X'.u + Z.u$ and thus $T_u(K\cdot u) \subset T_u(K'\cdot u) \oplus {\mathbb{V}}^\perp$ for all $u \in {\mathbb{V}}$. This implies $T_u(K\cdot u) = T_u(K'\cdot u)\oplus {\mathbb{V}}^\perp$ (orthogonal direct sum) and $\nu _u (K'\cdot u) = \nu _u (K\cdot u)$ for all $u\in S_i$, since $\nu _{H^i}S_i \subset {\mathbb{V}}$.
As we have previously observed, $N$ is a submanifold of $S_i$ in an open neighborhood of $H^i$. Since $K'\cdot H^i \subset {\mathbb{V}}$ and $K'\subset K$ we obtain $K'\cdot H^i \subset N$ and $K'\cdot N = N$. From the previous paragraph we conclude that $N$ coincides with $K'\cdot H^i$ around $H^i$, since both submanifolds of $S_i$ have the same dimension. Moreover, since ${\mathbb{V}}$ is $K'$-invariant, ${\mathbb{V}}$ contains the normal space $\nu _w S_i= \nu _w (K\cdot w)$ for all $w \in K'\cdot H^i$. This implies in particular that $N$ is totally geodesic in $S_i$ at all points $w \in K'\cdot H^i$. Thus $N$ is a submanifold around any $w\in K'\cdot H^i$ and $N$ coincides, around $w$, with this orbit. Therefore $K'\cdot H^i$ is an open subset of $N$. Since $K'\cdot H^i$ is compact and $N$ is Hausdorff, the orbit $K'\cdot H^i$ is a closed subset of $N$. Since $N$ is connected this implies that $N = K'\cdot H^i$ is a totally geodesic submanifold of $S_i$.
Let us consider the extrinsic symmetry $s_i$ at $H^i$ of the extrinsically symmetric submanifold $S_i$ of $T_pM$. Since $s_i$ leaves $S_i$, ${\mathbb{V}}$ and $\{H^i\}$ invariant, it also leaves $N = K'\cdot H^i$, the connected component of $S_i \cap {\mathbb{V}}$ containing $H^i$, invariant. Hence $s_i$ restricted to ${\mathbb{V}}$ is an extrinsic symmetry of $N\subset {\mathbb{V}}$ at $H^i$. This proves that $N$ is an extrinsically symmetric submanifold of $\mathbb V$.
Note that the extrinsic symmetry $s_i$ has the property $s_i ({\mathbb{V}}) = {\mathbb{V}}$ and therefore $s_i K's_i^{-1} = K'$.
We want to prove that $N = \{ H^i\}$, or equivalently, that ${\mathbb{V}}= \nu_{H^i}S_i$. Assume that this is not true. Let ${\mathbb{W}}\subset {\mathbb{V}}$ be the linear span of $N = K'\cdot H^i$. Then ${\mathbb{W}}$ is the tangent space to a (symmetric) Riemannian factor of $\Sigma'$, since it is $K'$-invariant. The subspace ${\mathbb{W}}$ cannot have an abelian part since $N = K'\cdot H^i$ is full in ${\mathbb{W}}$. Also, since $K$ acts irreducibly on $T_pM$, $K$ must act effectively on the symmetric orbit $S_i$. The group $K$ is generated by the so-called geometric transvections $\{ s_x\circ s_y\} $, where $x, y \in S_i$ and $s_x$ denotes the extrinsic symmetry at $x$. In fact, the connected group $K$ cannot be bigger than the group of transvections of the symmetric space $S_i$ since $S_i$ is compact, and so any Killing field on $S_i$ is bounded and hence belongs to the Lie algebra of the transvection group.
Let $K''$ be the connected closed subgroup of $K'$ with Lie algebra ${\mathfrak{k}}'' = [{\mathbb{W}},{\mathbb{W}}] \oplus {\mathbb{W}}$. Note that $K'\cdot H^i = K''\cdot H^i$. Moreover, $K''$ acts almost effectively on $N$. In fact, $K''$ acts almost effectively on ${\mathbb{W}}$ (see Section 2 of [@BO]) and if $k''\in K''$ acts trivially on $N$ it must act trivially on its linear span. We also have $K ' = K'' \times \bar{K}$ (almost direct product), where $\bar{K}$ is the connected closed subgroup of $K'$ with Lie algebra $\bar{{\mathfrak{k}}} = [{\mathbb{W}}^\perp \cap {\mathbb{V}},{\mathbb{W}}^\perp \cap {\mathbb{V}}] \oplus ({\mathbb{W}}^\perp \cap {\mathbb{V}})$.
As we have seen above, $N = K''\cdot H^i$ is a symmetric submanifold of $S_i$ and thus $K''$ must be generated by $\{ s_{x '}\circ s_{y'}\}$ with $x', y' \in K'\cdot H^i$. The following observation is crucial: $\{ s_{x '}\circ s_{y'}\} $ is the identity on the orthogonal complement of ${\mathbb{W}}$. In fact, $s_{x '}$ is the identity on ${\mathbb{W}}^\perp \cap {\mathbb{V}}$, since this subspace is contained in $\nu _{x'} S_i$. Moreover, $s_{x '}$ is minus the identity on ${\mathbb{V}}^\perp$, which is tangent to $S_i$ at $x'$. The same is true if one replaces $x'$ by $y'$ and so $s_{x '}\circ s_{y'}$ is the identity on ${\mathbb{V}}^\perp \oplus ({\mathbb{W}}^\perp \cap {\mathbb{V}}) = {\mathbb{W}}^\perp $. This implies that $K''$ acts trivially on ${\mathbb{W}}^\perp$, which contradicts the Slice Lemma \[Simons\]. Then ${\mathbb{V}}= \nu_{H^i}S_i$ which implies that $\nu_{H^i} S_i = T_p\Sigma$ is maximal. Thus we have proved that $\Sigma$ is a maximal totally geodesic submanifold of $M$. This finishes the proof of “(ii) $\Rightarrow$ (i)”.
From Theorem \[rspace\] and Table \[highestroot\] we obtain
\[nonss\] Let $M = G/K$ be an irreducible Riemannian symmetric space of noncompact type. If the restricted root system of $M$ is of type $(BC_r)$, $(E_8)$, $(F_4)$ or $(G_2)$, then every maximal totally geodesic submanifold of $M$ is semisimple.
We have seen in the proof of Theorem \[rspace\] that $\nu_pF_i$ is a Lie triple system when $\delta_i = 1$, which implies that $T_pF_i$ is a reflective Lie triple system when $\delta_i = 1$. From Theorem \[rspace\] we can therefore conclude:
\[nssrefl\] Let $M = G/K$ be an irreducible Riemannian symmetric space of noncompact type and let $\Sigma$ be a non-semisimple maximal totally geodesic submanifold of $M$. Then $\Sigma$ is a reflective submanifold of $M$.
We remark that the analogous statement for the semisimple case does not hold. For example, $SL_3({\mathbb{R}})/SO_3$ is a semisimple maximal totally geodesic submanifold of $G_2^2/SO_4$ which is not reflective.
We recall from [@BO] the following result:
\[bound\] Let $M$ be an irreducible Riemannian symmetric space. Then $${\operatorname{rk}}(M) \leq i(M).$$
From Table \[totgeodnsstable\] we obtain that the codimension of the totally geodesic submanifold $\Sigma = {\mathbb{R}}\times SL_{r}({\mathbb{R}})/SO_{r}$ in $M = SL_{r+1}({\mathbb{R}})/SO_{r+1}$ is equal to $r = {\operatorname{rk}}(M)$, which implies $i(M) \leq {\operatorname{rk}}(M)$. Using Theorem \[bound\] we thus conclude:
\[SLSO\] For $M = SL_{r+1}({\mathbb{R}})/SO_{r+1}$ we have ${\operatorname{rk}}(M) = r = i(M)$.
Examples of symmetric spaces with ${\operatorname{rk}}(M) = i(M)$ {#ex}
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We first consider the symmetric space $M = SL_{r+1}({\mathbb R})/SO_{r+1}$ for $r \geq 1$ and present a more explicit version of Corollary \[SLSO\]. This symmetric space has ${\operatorname{rk}}(M) = r$ and $\dim(M) = \frac{1}{2}r(r+3)$. For $r = 1$ we get the real hyperbolic plane ${\mathbb{R}}H^2$. Thus, if $\Sigma$ is a geodesic in $M$, we have ${\operatorname{codim}}(\Sigma) = 1 = {\operatorname{rk}}(M)$. For $r \geq 2$ we consider the Cartan decomposition ${\mathfrak{g}}= {\mathfrak{k}}\oplus {\mathfrak{p}}$ of the Lie algebra ${\mathfrak{g}}= {\mathfrak{sl}}_{r+1}({\mathbb R})$ of $G = SL_{r+1}({\mathbb R})$ which is induced by the Lie algebra ${\mathfrak k} = {\mathfrak{so}}_{r+1}$ of $K = SO_{r+1}$. The vector space ${\mathfrak{p}}$ is given by $${\mathfrak{p}}= \{A \in {\mathfrak{sl}}_{r+1}({\mathbb R}) \mid A^T = A\}.$$ We now define $${\mathfrak{m}}= \left\{ \left. A = \begin{pmatrix} -{\mathfrak{tr}}(B) & 0 \\ 0 & B \end{pmatrix} \in {\mathfrak{p}}\ \right| \ B \in {\mathfrak{gl}}_r({\mathbb{R}}),\ B^T = B \right\} .$$ Then we have $$[[{\mathfrak{m}},{\mathfrak{m}}],{\mathfrak{m}}] = \left\{ \left. A = \begin{pmatrix} 0 & 0 \\ 0 & \ B \end{pmatrix} \in {\mathfrak{p}}\ \right| \ B \in {\mathfrak{sl}}_r({\mathbb{R}}),\ B^T = B \right\} \subset {\mathfrak{m}},$$ which shows that ${\mathfrak{m}}$ is a Lie triple system in ${\mathfrak{p}}$. We have $\dim({\mathfrak{m}}) = \frac{1}{2}r(r+1)$ and hence $\dim({\mathfrak{p}}) - \dim({\mathfrak{m}}) = \frac{1}{2}r(r+3) - \frac{1}{2}r(r+1) = r$. Thus the connected complete totally geodesic submanifold $\Sigma$ of $M$ corresponding to the Lie triple system ${\mathfrak m}$, which is isometric to ${\mathbb{R}}\times SL_r({\mathbb R})/SO_r$, satisfies ${\operatorname{codim}}(\Sigma) = r = {\operatorname{rk}}(M)$. From Theorem \[bound\] we can therefore conclude that the index of $SL_{r+1}({\mathbb R})/SO_{r+1}$ is equal to the rank of $SL_{r+1}({\mathbb R})/SO_{r+1}$. We remark that ${\mathbb{R}}\times SL_r({\mathbb R})/SO_r$ is tangent to the normal space of a Veronese embedding of the real projective space ${\mathbb{R}}P^r$ into ${\mathfrak{p}}$ (see e.g. Lemma 8.1 in [@ORi]).
Next, we consider the symmetric space $M = SO^o_{r,r+k}/SO_rSO_{r+k}$ with $r \geq 1$, $k \geq 0$ and $(r,k) \notin \{(1,0),(2,0)\}$. This symmetric space has ${\operatorname{rk}}(M) = r$ and $\dim(M) = r(r+k)$. For $(r,k) = (1,0)$ we have $\dim(M) = 1$ and so $M$ is not of noncompact type. For $(r,k) = (2,0)$ we have the symmetric space $M = SO^o_{2,2}/SO_2SO_2$ which is isometric to the Riemannian product of two real hyperbolic planes and therefore not irreducible. Note that $SO^o_{1,2}/SO_2 = SL_2({\mathbb R})/SO_2$ and $SO^o_{3,3}/SO_3SO_3 = SL_4({\mathbb R})/SO_4$.
For $r = 1$ we get the $(k+1)$-dimensional real hyperbolic space $M = {\mathbb{R}}H^{k+1} = SO^o_{1,1+k}/SO_{1+k}$. This space contains a totally geodesic hypersurface $\Sigma = {\mathbb{R}}H^k$ and therefore ${\operatorname{rk}}(M) = 1 = i(M)$.
Now assume that $r \geq 2$ and consider the Cartan decomposition ${\mathfrak{g}}= {\mathfrak{k}}\oplus {\mathfrak{p}}$ of the Lie algebra ${\mathfrak{g}}= {\mathfrak{so}}_{r,r+k}$ of $G = SO^o_{r,r+k}$ which is induced by the Lie algebra ${\mathfrak k} = {\mathfrak{so}}_r \oplus {\mathfrak{so}}_{r+k}$ of $K = SO_rSO_{r+k}$. The vector space ${\mathfrak{p}}$ is given by $${\mathfrak{p}}= \left\{A \in {\mathfrak{so}}_{r,r+k} \ \left| \ A = \begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix},\ B \in M_{r,r+k}({\mathbb{R}}) \right. \right\},$$ where $M_{r,r+k}({\mathbb{R}})$ denotes the vector space of $r \times (r+k)$-matrices with real coefficients. We define a linear subspace ${\mathfrak{m}}$ of ${\mathfrak{p}}$ by $${\mathfrak{m}}= \left\{ \left. A = \begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix} \in {\mathfrak{p}}\ \right| \ B = \begin{pmatrix} C & 0 \end{pmatrix},\ C \in M_{r,r+k-1}({\mathbb{R}}) \right\}.$$ A straightforward calculation shows that $[[{\mathfrak{m}},{\mathfrak{m}}],{\mathfrak{m}}] \subset {\mathfrak{m}}$, that is, ${\mathfrak{m}}$ is a Lie triple system in ${\mathfrak{p}}$. We have $\dim({\mathfrak{m}}) = r(r+k-1)$ and hence $\dim({\mathfrak{p}}) - \dim({\mathfrak{m}}) = r(r+k) - r(r+k-1) = r$. Thus the connected complete totally geodesic submanifold $\Sigma$ of $M$ corresponding to the Lie triple system ${\mathfrak m}$, which is isometric to $SO^o_{r,r+k-1}/SO_rSO_{r+k-1}$, satisfies ${\operatorname{codim}}(\Sigma) = r = {\operatorname{rk}}(M)$. From Theorem \[bound\] it follows that the index of $SO^o_{r,r+k}/SO_rSO_{r+k}$ is equal to $r$.
Altogether we have now proved the “if”-part of Theorem \[main\]:
\[examples\] Let $M$ be one of the following Riemannian symmetric spaces of noncompact type:
- $SL_{r+1}({\mathbb R})/SO_{r+1}$, $r \geq 1$;
- $SO^o_{r,r+k}/SO_rSO_{r+k}$, $r \geq 1$, $k \geq 0$, $(r,k) \notin \{(1,0),(2,0)\}$.
Then ${\operatorname{rk}}(M) = r = i(M)$.
The classification {#proof}
==================
The following result was proved in [@BO] and will be used later.
\[flat\] Let $M$ be an irreducible Riemannian symmetric space, $\Sigma$ a connected totally geodesic submanifold of $M$ and $p \in \Sigma$. Then there exists a maximal abelian subspace ${\mathfrak{a}}$ of ${\mathfrak{p}}$ such that ${\mathfrak{a}}$ is transversal to $T_p\Sigma$, that is, ${\mathfrak{a}}\cap T_p\Sigma = \{0\}$.
Let $M = G/K$ be an irreducible Riemannian symmetric space of noncompact type and assume that $i(M) = r = {\operatorname{rk}}(M)$. Then there exists a connected complete totally geodesic submanifold $\Sigma$ of $M$ with $p \in \Sigma$ such that ${\operatorname{codim}}(\Sigma) = r$. According to Theorem \[flat\] there exists a maximal abelian subspace ${\mathfrak{a}}$ of ${\mathfrak{p}}$ such that ${\mathfrak{a}}$ is transversal to $T_p\Sigma$. Let $\Psi$ be the set of restricted roots with respect to ${\mathfrak{a}}$ and $\Lambda = \{\alpha_1,\ldots,\alpha_r\}$ be a set of simple roots for $\Psi$. The next result provides a necessary criterion for an irreducible Riemannian symmetric space $M$ with ${\operatorname{rk}}(M) \geq 2$ to satisfy the equality ${\operatorname{rk}}(M) = i(M)$.
\[boundary\_reduction\] [(Boundary Reduction)]{} Let $M$ be an irreducible Riemannian symmetric space of noncompact type with ${\operatorname{rk}}(M) \geq 2$ and assume that the equality ${\operatorname{rk}}(M) = i(M)$ holds. Then every irreducible boundary component $B_\Phi$ of $M$ satisfies ${\operatorname{rk}}(B_\Phi) = i(B_\Phi)$.
Let $\Sigma_\Phi$ be the connected complete totally geodesic submanifold of $F_\Phi$ corresponding to the Lie triple system $T_p\Sigma \cap T_pF_\Phi$. Since $T_pM = T_p\Sigma \oplus {\mathfrak{a}}$ (direct sum) and ${\mathfrak{a}}\subset T_pF_\Phi$, we have $T_pF_\Phi = T_p\Sigma_\Phi \oplus {\mathfrak{a}}$ (direct sum). Thus the codimension of $\Sigma_\Phi$ in $F_\Phi$ is equal to $\dim{{\mathfrak{a}}} = r = {\operatorname{rk}}(M)= {\operatorname{rk}}(F_\Phi)$.
The orthogonal projection $(T_p\Sigma_\Phi)_{T_pB_\Phi}$ of the Lie triple system $T_p\Sigma_\Phi$ onto $T_pB_\Phi$ is a Lie triple system. Let $\Sigma'_\Phi$ be the connected complete totally geodesic submanifold of $B_\Phi$ corresponding to the Lie triple system $(T_p\Sigma_\Phi)_{T_pB_\Phi}$. Since $T_pF_\Phi = T_p\Sigma_\Phi \oplus {\mathfrak{a}}= T_pB_\Phi \oplus {\mathfrak{a}}_\Phi$ (direct sum) and ${\mathfrak{a}}= {\mathfrak{a}}^\Phi \oplus {\mathfrak{a}}_\Phi$, we have $T_pB_\Phi = T_p\Sigma'_\Phi \oplus {\mathfrak{a}}^\Phi$ (direct sum), which implies that the codimension of $\Sigma'_\Phi$ in $B_\Phi$ is equal to $\dim({\mathfrak{a}}^\Phi) = \dim({\mathfrak{a}}) - \dim({\mathfrak{a}}_\Phi) = r - (r-|\Phi|) = |\Phi| = {\operatorname{rk}}(B_\Phi)$. This implies $i(B_\Phi) \leq {\operatorname{rk}}(B_\Phi)$. However, since $B_\Phi$ is irreducible, we also have ${\operatorname{rk}}(B_\Phi) \leq i(B_\Phi)$ by Theorem \[bound\]. Altogether this implies ${\operatorname{rk}}(B_\Phi) = i(B_\Phi)$.
We recall the following result from [@BO]:
\[classification\] [(Symmetric spaces with index $\leq 3$)]{} Let $M$ be an irreducible Riemannian symmetric space of noncompact type.
- $i(M) = 1$ if and only if $M$ is isometric to
- the real hyperbolic space ${\mathbb R}H^{k+1} = SO^o_{1,1+k}/SO_{1+k}$, $k \geq 1$.
- $i(M) = 2$ if and only if $M$ is isometric to one of the following spaces:
- the complex hyperbolic space ${\mathbb C}H^{k+1} = SU_{1,1+k}/S(U_1U_{1+k})$, $k \geq 1$;
- the Grassmannian $SO^o_{2,2+k}/SO_2SO_{2+k}$, $k \geq 1$;
- the symmetric space $SL_3({\mathbb R})/SO_3$.
- $i(M) = 3$ if and only if $M$ is isometric to one of the following spaces:
- the Grassmannian $SO^o_{3,3+k}/SO_3SO_{3+k}$, $k \geq 1$;
- the symmetric space $G^2_2/SO_4$;
- the symmetric space $SL_3({\mathbb C})/SU_3$;
- the symmetric space $SL_4({\mathbb R})/SO_4$.
The Riemannian symmetric spaces of noncompact type with ${\operatorname{rk}}(M) = 1 = i(M)$ are precisely the real hyperbolic spaces $SO^o_{1,1+k}/SO_{1+k}$, $k \geq 1$. The irreducible Riemannian symmetric spaces of noncompact type with ${\operatorname{rk}}(M) \geq 2$ whose rank one boundary components are all real hyperbolic spaces are precisely those for which the restricted root system is reduced, that is, is not of type ($BC_r$). From Proposition \[boundary\_reduction\] we therefore obtain:
[(Rank One Boundary Reduction)]{} \[rank\_one\_reduction\] Let $M$ be an irreducible Riemannian symmetric space of noncompact type with ${\operatorname{rk}}(M) \geq 2$ and assume that ${\operatorname{rk}}(M) = i(M)$. Then the restricted root system of $M$ is not of type ($BC_r$).
According to Theorem \[classification\], the Riemannian symmetric spaces of noncompact type with ${\operatorname{rk}}(M) = 2 = i(M)$ are precisely $SO^o_{2,2+k}/SO_2SO_{2+k}$, $k \geq 1$, and $SL_3({\mathbb R})/SO_3$. The corresponding Dynkin diagrams with multiplicities are $$\xy
\POS (0,0) *\cir<2pt>{} ="a",
(10,0) *\cir<2pt>{}="b",
(0,-3) *{1},
(10,-3) *{k},
\ar @2{->} "a";"b"
\endxy
\ \ \textrm{and}\ \
\xy
\POS (0,0) *\cir<2pt>{} ="a",
(10,0) *\cir<2pt>{}="b",
(0,-3) *{1},
(10,-3) *{1},
\ar @{-} "a";"b"
\endxy\ \ .$$ We can easily extract from Table \[dynkin\] the Dynkin diagrams of rank $\geq 3$ with multiplicities for which every connected subdiagram of rank $2$ is one of the above: $$\xy
\POS (0,0) *\cir<2pt>{} ="a",
(7,0) *\cir<2pt>{}="b",
(14,0) *\cir<2pt>{}="c",
(21,0) *\cir<2pt>{}="d",
(0,-3) *{1},
(7,-3) *{1},
(14,-3) *{1},
(21,-3) *{1},
\ar @{-} "a";"b",
\ar @{.} "b";"c",
\ar @{-} "c";"d"
\endxy
\ \ ,\ \
\xy
\POS (0,0) *\cir<2pt>{} ="a",
(7,0) *\cir<2pt>{}="b",
(14,0) *\cir<2pt>{}="c",
(21,0) *\cir<2pt>{}="d",
(28,0) *\cir<2pt>{}="e",
(0,-3) *{1},
(7,-3) *{1},
(14,-3) *{1},
(21,-3) *{1},
(28,-3) *{k},
\ar @{-} "a";"b",
\ar @{.} "b";"c",
\ar @{-} "c";"d",
\ar @2{->} "d";"e"
\endxy
\ \ ,\ \
\xy
\POS (0,0) *\cir<2pt>{} ="a",
(7,0) *\cir<2pt>{}="b",
(14,0) *\cir<2pt>{}="c",
(21,0) *\cir<2pt>{}="d",
(28,2) *\cir<2pt>{}="e",
(28,-2) *\cir<2pt>{}="f",
(0,-3) *{1},
(7,-3) *{1},
(14,-3) *{1},
(21,-3) *{1},
(31,3) *{1},
(31,-3) *{1},
\ar @{-} "a";"b",
\ar @{.} "b";"c",
\ar @{-} "c";"d",
\ar @{-} "d";"e",
\ar @{-} "d";"f"
\endxy\ \ ,$$ $$\xy
\POS (0,0) *\cir<2pt>{} ="a",
(14,5) *\cir<2pt>{} = "b",
(7,0) *\cir<2pt>{}="c",
(14,0) *\cir<2pt>{}="d",
(21,0) *\cir<2pt>{}="e",
(28,0) *\cir<2pt>{}="f",
(0,-3) *{1},
(12,5) *{1},
(7,-3) *{1},
(14,-3) *{1},
(21,-3) *{1},
(28,-3) *{1},
\ar @{-} "a";"c",
\ar @{-} "c";"d",
\ar @{-} "b";"d",
\ar @{-} "d";"e",
\ar @{-} "e";"f",
\endxy
\ \ ,\ \
\xy
\POS (0,0) *\cir<2pt>{} ="a",
(14,5) *\cir<2pt>{} = "b",
(7,0) *\cir<2pt>{}="c",
(14,0) *\cir<2pt>{}="d",
(21,0) *\cir<2pt>{}="e",
(28,0) *\cir<2pt>{}="f",
(35,0) *\cir<2pt>{}="g",
(0,-3) *{1},
(12,5) *{1},
(7,-3) *{1},
(14,-3) *{1},
(21,-3) *{1},
(28,-3) *{1},
(35,-3) *{1},
\ar @{-} "a";"c",
\ar @{-} "c";"d",
\ar @{-} "b";"d",
\ar @{-} "d";"e",
\ar @{-} "e";"f",
\ar @{-} "f";"g",
\endxy\ \ ,$$ $$\xy
\POS (0,0) *\cir<2pt>{} ="a",
(14,5) *\cir<2pt>{} = "b",
(7,0) *\cir<2pt>{}="c",
(14,0) *\cir<2pt>{}="d",
(21,0) *\cir<2pt>{}="e",
(28,0) *\cir<2pt>{}="f",
(35,0) *\cir<2pt>{}="g",
(42,0) *\cir<2pt>{}="h",
(0,-3) *{1},
(12,5) *{1},
(7,-3) *{1},
(14,-3) *{1},
(21,-3) *{1},
(28,-3) *{1},
(35,-3) *{1},
(42,-3) *{1},
\ar @{-} "a";"c",
\ar @{-} "c";"d",
\ar @{-} "b";"d",
\ar @{-} "d";"e",
\ar @{-} "e";"f",
\ar @{-} "f";"g",
\ar @{-} "g";"h"
\endxy
\ \ ,\ \
\xy
\POS (0,0) *\cir<2pt>{} ="a",
(7,0) *\cir<2pt>{}="b",
(14,0) *\cir<2pt>{}="c",
(21,0) *\cir<2pt>{}="d",
(0,-3) *{1},
(7,-3) *{1},
(14,-3) *{1},
(21,-3) *{1},
\ar @{-} "a";"b",
\ar @2{->} "b";"c",
\ar @{-} "c";"d"
\endxy
\ \ .$$
From Proposition \[boundary\_reduction\] we thus obtain:
[(Rank Two Boundary Reduction)]{} \[rank\_two\_reduction\] Let $M$ be an irreducible Riemannian symmetric space of noncompact type with ${\operatorname{rk}}(M) \geq 3$ and assume that ${\operatorname{rk}}(M) = i(M)$. Then $M$ must be among the following spaces:
- $SL_{r+1}({{\mathbb{R}}})/SO_{r+1}$, $r \geq 3$;
- $SO^o_{r,r+k}/SO_{r}SO_{r+k}$, $r \geq 3$, $k \geq 0$;
- $E_6^6/Sp_4$;
- $E_7^7/SU_8$;
- $E_8^8/SO_{16}$;
- $F_4^4/Sp_3Sp_1$.
We know from Proposition \[examples\] that the symmetric spaces in (1) and (2) satisfy the equality ${\operatorname{rk}}(M) = i(M)$. In order to prove Theorem \[main\] it remains to show that the four exceptional spaces in Corollary \[rank\_two\_reduction\] do not satisfy the equality ${\operatorname{rk}}(M) = i(M)$. For $M = F_4^4/Sp_3Sp_1$ we can apply rank three boundary reduction:
\[f4sp3sp1\] The symmetric space $M = F_4^4/Sp_3Sp_1$ does not satisfy the equality ${\operatorname{rk}}(M) = i(M)$.
The Dynkin diagram with multiplicities for $F_4^4/Sp_3Sp_1$ is $$\xy
\POS (0,0) *\cir<2pt>{} ="a",
(7,0) *\cir<2pt>{}="b",
(14,0) *\cir<2pt>{}="c",
(21,0) *\cir<2pt>{}="d",
(0,-3) *{1},
(7,-3) *{1},
(14,-3) *{1},
(21,-3) *{1},
\ar @{-} "a";"b",
\ar @2{->} "b";"c",
\ar @{-} "c";"d"
\endxy\ .$$ We see from Theorem \[classification\] that the boundary component $B_\Phi = Sp_3({{\mathbb{R}}})/U_3$ corresponding to the rank three subdiagram $$\xy
\POS
(7,0) *\cir<2pt>{}="b",
(14,0) *\cir<2pt>{}="c",
(21,0) *\cir<2pt>{}="d",
(7,-3) *{1},
(14,-3) *{1},
(21,-3) *{1},
\ar @2{->} "b";"c",
\ar @{-} "c";"d"
\endxy$$ does not satisfy the equality ${\operatorname{rk}}(B_\Phi) = i(B_\Phi)$. The statement thus follows from Proposition \[boundary\_reduction\].
The situation for the exceptional symmetric space $E_6^6/Sp_4$ is quite interesting as the following result shows.
\[e6sp4\] Every irreducible boundary component $B_\Phi$ of $M = E_6^6/Sp_4$ satisfies ${\operatorname{rk}}(B_\Phi) = i(B_\Phi)$. However, $M$ does not satify the equality ${\operatorname{rk}}(M) = i(M)$.
We list the different types of irreducible boundary components of $M$ by cardinality of $\Phi$.
- $|\Phi| = 1$: $B_\Phi = SL_2({\mathbb{R}})/SO_2$;
- $|\Phi| = 2$: $B_\Phi = SL_3({\mathbb{R}})/SO_3$;
- $|\Phi| = 3$: $B_\Phi = SL_4({\mathbb{R}})/SO_4$;
- $|\Phi| = 4$: $B_\Phi = SL_5({\mathbb{R}})/SO_5$ and $B_\Phi = SO^o_{4,4}/SO_4SO_4$;
- $|\Phi| = 5$: $B_\Phi = SL_6({\mathbb{R}})/SO_6$ and $B_\Phi = SO^o_{5,5}/SO_5SO_5$.
As we have shown in Proposition \[examples\], each of these boundary components satisfies ${\operatorname{rk}}(B_\Phi) = i(B_\Phi)$.
We have $n = \dim(M) = 42$ and $r = {\operatorname{rk}}(M) = 6$. Assume that there exists a maximal totally geodesic submanifold $\Sigma$ of $M$ with $d = {\operatorname{codim}}(\Sigma) = 6$. We first assume that $\Sigma$ is semisimple. Then the inequality in Corollary \[ref4\] is satisfied and thus $\Sigma$ is a reflective submanifold of $M$. As usual, we write $\Sigma = G'/K'$, where $G'$ is the connected closed subgroup of $E_6^6$ with Lie algebra ${\mathfrak{g}}' = [T_p\Sigma , T_p\Sigma] \oplus T_p\Sigma$ and $K' = G'_p$. Note that $K'$ is connected since $\Sigma$ is simply connected. Let $s \in I(M)$ be the geodesic reflection of $M$ in $\Sigma$ and define $\tau : E_6^6 \to E_6^6,\ g \mapsto sgs^{-1}$. It is clear that $G'$, and hence also $K'$, is contained in the fixed point set of $\tau$. Since $s$ commutes with the geodesic symmetry of $M$ at $p$, we have $\tau(Sp_4) = Sp_4$. Let $H$ be the connected component of the fixed point set of $\tau_{|Sp_4}$ containing the identity transformation of $Sp_4$. Note that $K' \subset H$. Then $Sp_4/H$ is a (simply connected) Riemannian symmetric space of compact type. However, as we observed in the proof of Corollary \[ref4\], we have $\dim(K ') \geq \dim(\Sigma) - {\operatorname{rk}}(M) = 30$ and therefore $\dim(Sp_4/H) \leq \dim(Sp_4/K') \leq 6$. Since there is no Riemannian symmetric space of $Sp_4$ of dimension $\leq 6$ we conclude that there is no reflective submanifold $\Sigma$ of $M$ with ${\operatorname{codim}}(\Sigma) = 6$. \[Note: This fact can also be seen directly from Leung’s classification of reflective submanifolds. However, we prefer to give a conceptual proof here.\] Therefore $\Sigma$ cannot be semisimple. If $\Sigma$ is non-semisimple, then $\Sigma = {\mathbb{R}}\times SO^o_{5,5}/SO_5SO_5$ by Table \[totgeodnsstable\] and hence ${\operatorname{codim}}(\Sigma) = 16$, which is a contradiction. Altogether we can now conclude that there is no totally geodesic submanifold in $M$ with ${\operatorname{codim}}(M) = 6$. This implies $rk(M) < i(M)$.
As a consequence of Proposition \[e6sp4\] we can now settle the two remaining cases.
\[e7e8\] The symmetric spaces $M = E_7^7/SU_8$ and $M = E^8_8/SO_{16}$ do not satisfy the equality ${\operatorname{rk}}(M) = i(M)$.
We see from Table \[dynkin\] that the Dynkin diagram with multiplicities for $E_6^6/Sp_4$ can be embedded into the Dynkin diagrams with multiplicities for $E_7^7/SU_8$ and $E^8_8/SO_{16}$. This means that $E_6^6/Sp_4$ is an irreducible boundary component of both $E_7^7/SU_8$ and $E^8_8/SO_{16}$. From Proposition \[boundary\_reduction\] and Proposition \[e6sp4\] we can conclude that both $M = E_7^7/SU_8$ and $M = E^8_8/SO_{16}$ do not satisfy the equality ${\operatorname{rk}}(M) = i(M)$.
Theorem \[main\] now follows from Proposition \[examples\], Corollary \[rank\_two\_reduction\], Corollary \[f4sp3sp1\], Proposition \[e6sp4\] and Corollary \[e7e8\]. We also obtain the following interesting characterization of the exceptional symmetric space $E_6^6/Sp_4$:
The exceptional symmetric space $E^6_6/Sp_4$ is the only irreducible Riemannian symmetric space $M$ of noncompact type with $rk(M) \geq 3$ for which every irreducible boundary component $B$ satisfies ${\operatorname{rk}}(B) = i(B)$ but the manifold itself does not satisfy ${\operatorname{rk}}(M) = i(M)$.
Further applications {#applications}
====================
In this section we will calculate $i(M)$ for a few irreducible Riemannian symmetric spaces $M$ of noncompact type using the methods we developed in this paper and Leung’s classification of reflective submanifolds. We first recall some known results to put our results into context.
The totally geodesic submanifolds of Riemannian symmetric spaces $M$ of noncompact type with ${\operatorname{rk}}(M) = 1$ were classified by Wolf in [@Wo]. We use the following notations: ${\mathbb{R}}H^{k+1} = SO^o_{1,1+k}/SO_{1+k}$ denotes the $(k+1)$-dimensional real hyperbolic space, ${\mathbb{C}}H^{k+1} = SU_{1,1+k}/S(U_1U_{1+k})$ denotes the $(k+1)$-dimensional complex hyperbolic space, ${\mathbb{H}}H^{k+1} = Sp_{1,1+k}/Sp_1Sp_{1+k}$ denotes the $(k+1)$-dimensional quaternionic hyperbolic space, and ${\mathbb{O}}H^2 = F_4^{-20}/Spin_9$ denotes the Cayley hyperbolic plane. Here, $k \geq 1$. The totally geodesic submanifolds of irreducible Riemannian symmetric spaces $M$ of noncompact type with ${\operatorname{rk}}(M) = 2$ were classified by Klein in [@K1], [@K2], [@K3] and [@K4]. From Wolf’s and Klein’s classifications we obtain $i(M)$ for all irreducible Riemannian symmetric spaces $M$ of noncompact type with ${\operatorname{rk}}(M) \leq 2$. Some of the indices for ${\operatorname{rk}}(M) = 2$ were calculated by Onishchik in [@On]. We summarize all this in Table \[iMforrkMleq2\].
[ | p[2.7cm]{} p[4cm]{} p[1.2cm]{} p[0.8cm]{} p[1.5cm]{} | ]{}
------------------------------------------------------------------------
$M$ & $\Sigma$ & $\dim(M)$ & $i(M)$ & Comments\
------------------------------------------------------------------------
${\mathbb{R}}H^{k+1}$ & ${\mathbb{R}}H^k$ & $k+1$ & $1$ & $k \geq 1$\
${\mathbb{C}}H^{k+1}$ & ${\mathbb{C}}H^k$ (and ${\mathbb{R}}H^2$ for $k=1$) & $2(k+1)$ & $2$ & $k \geq 1$\
${\mathbb{H}}H^{k+1}$ & ${\mathbb{H}}H^k$ (and ${\mathbb{C}}H^2$ for $k=1$)& $4(k+1)$ & $4$ & $k \geq 1$\
${\mathbb{O}}H^2$ & ${\mathbb{O}}H^1$, ${\mathbb{H}}H^2$ & $16$ & $8$ &\
------------------------------------------------------------------------
$SL_3({\mathbb{R}})/SO_3$ & ${\mathbb{R}}\times {\mathbb{R}}H^2$ & $5$ & $2$ &\
$SO^o_{2,2+k}/SO_2SO_{2+k}$ & $SO^o_{2,1+k}/SO_2SO_{1+k}$ & $2(k+2)$ & $2$& $k \geq 1$\
$SL_3({\mathbb{C}})/SU_3$ & $SL_3({\mathbb{R}})/SO_3$ & $8$ & $3$ &\
$G_2^2/SO_4$ & $SL_3({\mathbb{R}})/SO_3$ & $8$ & $3$ &\
$SO_5({\mathbb{C}})/SO_5$ & $SO_4({\mathbb{C}})/SO_4$, $SO^o_{2,3}/SO_2SO_3$ & $10$ & $4$ &\
$SU_{2,2+k}/S(U_2U_{2+k})$ & $SU_{2,1+k}/S(U_2U_{1+k})$ & $4(k+2)$ & $4$ & $k \geq 1$\
$SU^*_6/Sp_3$ & $SL_3({\mathbb{C}})/SU_3$, ${\mathbb{H}}H^2$ & $14$ & $6$ &\
$G_2({\mathbb{C}})/G_2$ & $G_2^2/SO_4$, $SL_3({\mathbb{C}})/SU_3$ & $14$ & $6$ &\
$Sp_{2,2}/Sp_2Sp_2$ & $Sp_2({\mathbb{C}})/Sp_2$ & $16$ & $6$ &\
$SO^*_{10}/U_5$ &$SO^*_8/U_4$, $SU_{2,3}/S(U_2U_3)$ & $20$ & $8$ &\
$Sp_{2,2+k}/Sp_2Sp_{2+k}$ & $Sp_{2,1+k}/Sp_2Sp_{1+k}$ & $8(k+2)$ & $8$ & $k \geq 1$\
$E_6^{-26}/F_4$ & ${\mathbb{O}}H^2$ & $26$ & $10$ &\
$E_6^{-14}/Spin_{10}U_1$ & $SO^*_{10}/U_5$ & $32$ & $12$ &\
Let $M$ be a connected Riemannian manifold and denote by ${\mathcal S}_r$ the set of all connected reflective submanifolds $\Sigma$ of $M$ with $\dim(\Sigma) < \dim(M)$. The reflective index $i_r(M)$ of $M$ is defined by $$i_r(M) = \min\{ \dim(M) - \dim(\Sigma) \mid \Sigma \in {\mathcal S}_r\} = \min\{ {\operatorname{codim}}(\Sigma) \mid \Sigma \in {\mathcal S}_r\}.$$ It is clear that $i(M) \leq i_r(M)$ and thus $i_r(M)$ is an upper bound for $i(M)$. Leung classified in [@L1] and [@L2] the reflective submanifolds of irreducible simply connected Riemannian symmetric spaces of compact type. Using duality this allows us to calculate $i_r(M)$ explicitly for all irreducible Riemannian symmetric spaces $M$ of noncompact type. We list the reflective indices $i_r(M)$ for all $M$ with ${\operatorname{rk}}(M) \geq 3$ in Table \[summary\].
[ | p[2.7cm]{} p[3.7cm]{} p[1.2cm]{} p[0.8cm]{} p[1.6cm]{} p[2.2cm]{} |]{}
------------------------------------------------------------------------
$M$ & $\Sigma$ & $\dim(M)$ & $i_r(M)$ & Comments & $i(M) = i_r(M)$?\
------------------------------------------------------------------------
$SL_{r+1}({\mathbb{R}})/SO_{r+1}$ & ${\mathbb{R}}\times SL_r({\mathbb{R}})/SO_r$ & $\frac{1}{2}r(r+3)$ & $r$ & $r \geq 3$ & yes\
$SL_4({\mathbb{C}})/SU_4$ & $Sp_2({\mathbb{C}})/Sp_2$ & $15$ & $5$ & & yes\
$SL_{r+1}({\mathbb{C}})/SU_{r+1}$ & ${\mathbb{R}}\times SL_r({\mathbb{C}})/SU_r$ & $r(r+2)$ & $2r$ & $r \geq 4$ & ?\
$SU^*_{2r+2}/Sp_{r+1}$ & ${\mathbb{R}}\times SU^*_{2r}/Sp_r$ & $r(2r+3)$ & $4r$ & $r \geq 3$ & ?\
------------------------------------------------------------------------
$SO^o_{r,r+k}/SO_{r}SO_{r+k}$ & $SO^o_{r,r+k-1}/SO_{r}SO_{r+k-1}$ & $r(r+k)$ & $r$& $r \geq 3, k \geq 1$ & yes\
$SO_{2r+1}({\mathbb{C}})/SO_{2r+1}$ & $SO_{2r}({\mathbb{C}})/SO_{2r}$ & $r(2r+1)$ & $2r$ & $r \geq 3$ & yes\
------------------------------------------------------------------------
$Sp_r({\mathbb{R}})/U_r$ & ${\mathbb{R}}H^2 \times Sp_{r-1}({\mathbb{R}})/U_{r-1}$ & $r(r+1)$ & $2r-2$ & $r \geq 3$ & yes for $r\leq 5$, otherwise ?\
$SU_{r,r}/S(U_rU_r)$ & $SU_{r-1,r}/S(U_{r-1}U_r)$ & $2r^2$ & $2r$ & $r \geq 3$ & yes\
$Sp_r({\mathbb{C}})/Sp_r$ & ${\mathbb{R}}H^3 \times Sp_{r-1}({\mathbb{C}})/Sp_{r-1}$ & $r(2r+1)$ & $4r-4$ & $r \geq 3$ & ?\
$SO^*_{4r}/U_{2r}$ & $SO^*_{4r-2}/U_{2r-1}$ & $2r(2r-1)$ & $4r-2$ & $r \geq 3$ & ?\
$Sp_{r,r}/Sp_rSp_r$ & $Sp_{r-1,r}/Sp_{r-1}Sp_r$ & $4r^2$ & $4r$ & $r \geq 3$ & ?\
$E_7^{-25}/E_6U_1$ & $E_6^{-14}/Spin_{10}U_1$ & $54$ & $22$ & & ?\
------------------------------------------------------------------------
$SO^o_{r,r}/SO_{r}SO_{r}$ & $SO^o_{r-1,r}/SO_{r-1}SO_{r}$ & $r^2$ & $r$ & $r \geq 4$ & yes\
$SO_{2r}({\mathbb{C}})/SO_{2r}$ & $SO_{2r-1}({\mathbb{C}})/SO_{2r-1}$ & $r(2r-1)$ & $2r-1$ & $r \geq 4$ & yes\
------------------------------------------------------------------------
$SU_{r,r+k}/S(U_rU_{r+k})$ & $SU_{r,r+k-1}/S(U_rU_{r+k-1})$ & $2r(r+k)$ & $2r$ & $r \geq 3, k \geq 1$ & yes\
$Sp_{r,r+k}/Sp_rSp_{r+k}$ & $Sp_{r,r+k-1}/Sp_rSp_{r+k-1}$ & $4r(r+k)$ & $4r$ & $r \geq 3, k \geq 1$ & yes for $r-1 \leq k$, otherwise ?\
$SO^*_{4r+2}/U_{2r+1}$ &$SO^*_{4r}/U_{2r}$ & $2r(2r+1)$ & $4r$ & $r \geq 3$ & ?\
------------------------------------------------------------------------
$E_6^6/Sp_4$ & $F_4^4/Sp_3Sp_1$ & $42$ & $14$ & & ?\
$E_6({\mathbb{C}})/E_6$ & $F_4({\mathbb{C}})/F_4$ & $78$ & $26$ & & ?\
------------------------------------------------------------------------
$E_7^7/SU_8$ & ${\mathbb{R}}\times E^6_6/Sp_4$ & $70$ & $27$ & & ?\
$E_7({\mathbb C})/E_7$ & ${\mathbb{R}}\times E_6({\mathbb{C}})/E_6$ & $133$ & $54$ & & ?\
------------------------------------------------------------------------
$E_8^8/SO_{16}$ & ${\mathbb{R}}H^2 \times E_7^7/SU_8$ & $128$ & $56$ & & ?\
$E_8({\mathbb{C}})/E_8$ & ${\mathbb{R}}H^3 \times E_7({\mathbb{C}})/E_7$ & $248$ & $112$ & & ?\
------------------------------------------------------------------------
$F_4^4/Sp_3Sp_1$ & $SO^o_{4,5}/SO_4SO_5$ & $28$ & $8$ & & yes\
$E_6^2/SU_6Sp_1$ & $F_4^4/Sp_3Sp_1$ & $40$ & $12$ & & ?\
$E_7^{-5}/SO_{12}Sp_1$ & $E_6^2/SU_6Sp_1$ & $64$ & $24$ & & ?\
$E_8^{-24}/E_7Sp_1$ & $E_7^{-5}/SO_{12}Sp_1$ & $112$ & $48$ & & ?\
$F_4({\mathbb{C}})/F_4$ & $SO_9({\mathbb{C}})/SO_9$ & $52$ & $16$& & ?\
As an application of Corollaries \[ref4\] and \[nssrefl\] we will now calculate the index of a few symmetric spaces. Let $\Sigma$ be a maximal totally geodesic submanifold of an $n$-dimensional irreducible Riemannian symmetric space $M$ of noncompact type with $r = {\operatorname{rk}}(M) \geq 2$ such that $i(M) = {\operatorname{codim}}(\Sigma)$. If $\Sigma$ is non-semisimple, then $\Sigma$ is a reflective submanifold by Corollary \[nssrefl\]. If $\Sigma$ is semisimple and $d = {\operatorname{codim}}(\Sigma)$ satisfies $d(d+1) < 2(n - r)$, then $\Sigma$ is a reflective submanifold of $M$ by Corollary \[ref4\]. It follows that if ${\operatorname{codim}}(\Sigma) \leq i_r(M) - 1$ and $(i_r(M)-1)i_r(M) < 2(n-r)$, then $\Sigma$ is a reflective submanifold. Altogether this implies the following
\[irM=iM\] Let $M$ be an irreducible Riemannian symmetric space of noncompact type with ${\operatorname{rk}}(M) \geq 2$. If $$(i_r(M)-1)i_r(M) < 2(\dim(M) - {\operatorname{rk}}(M)),$$ then $i(M) = i_r(M)$.
The inequality in Proposition \[irM=iM\] can be checked explicitly for each symmetric space $M$ in Table \[summary\]:
\[moreexamples\] The following Riemannian symmetric spaces $M$ of noncompact type with ${\operatorname{rk}}(M) \geq 3$ satisfy the inequality in Proposition \[irM=iM\] and therefore satisfy the equality $i(M) = i_r(M)$:
- $SL_{r+1}({\mathbb{R}})/SO_{r+1}$, $r \geq 3$;
- $SL_4({\mathbb{C}})/SU_4$;
- $SO^o_{r,r+k}/SO_rSO_{r+k}$, $r \geq 3$, $k \geq 1$;
- $SO_{2r+1}({\mathbb{C}})/SO_{2r+1}$, $r \geq 3$;
- $Sp_r({\mathbb{R}})/U_r$, $3 \leq r \leq 4$;
- $SO^o_{r,r}/SO_rSO_r$, $r \geq 4$;
- $SO_{2r}({\mathbb{C}})/SO_{2r}$, $r \geq 4$;
- $SU_{r,r+k}/S(U_rU_{r+k})$, $r \geq 3$, $k \geq 1$;
- $Sp_{r,r+k}/Sp_rSp_{r+k}$, $3 \leq r \leq k$.
We inserted this result into the last column of Table \[summary\].
We can also use these methods to determine all irreducible Riemannian symmetric spaces $M$ of noncompact type with $i(M) = 4$.
\[iM=4\] [(Symmetric spaces with index four)]{} Let $M$ be an irreducible Riemannian symmetric space of noncompact type. Then $i(M) = 4$ if and only if $M$ is isometric to one of the following symmetric spaces:
- ${\mathbb{H}}H^{k+1} = Sp_{1,1+k}/Sp_1Sp_k$, $k \geq 1$;
- $SU_{2,2+k}/S(U_2U_{2+k})$, $k \geq 1$;
- $SO^o_{4,4+k}/SO_4SO_{4+k}$, $k \geq 0$;
- $SO_5({\mathbb{C}})/SO_5$;
- $Sp_3({\mathbb{R}})/U_3$;
- $SL_5({\mathbb{R}})/SO_5$.
From Tables \[iMforrkMleq2\] and \[summary\] and Corollary \[moreexamples\] we see that every symmetric space listed in Theorem \[iM=4\] satisfies $i(M) = 4$. Conversely, let $M$ be an irreducible Riemannian symmetric space of noncompact type with $i(M) = 4$ and let $\Sigma$ be a maximal totally geodesic submanifold of $M$ with $d = {\operatorname{codim}}(\Sigma) = 4$. If ${\operatorname{rk}}(M) \leq 2$ we obtain from Table \[iMforrkMleq2\] that $M$ is one of the spaces in (i), (ii) and (iv). Assume that ${\operatorname{rk}}(M) \geq 3$. If $\Sigma$ is non-semisimple, then $\Sigma$ is reflective by Corollary \[nssrefl\]. If $\Sigma$ is semisimple and $\dim(M) - {\operatorname{rk}}(M) \geq 11$, then $\Sigma$ is reflective by Corollary \[ref4\]. Thus we have $i_r(M) = i(M) = 4$ if $\dim(M) - {\operatorname{rk}}(M) \geq 11$ and we can use Table \[summary\] to see that $M$ is isometric to a space in (iii). The symmetric spaces $M$ with ${\operatorname{rk}}(M) \geq 3$ and $\dim(M) - {\operatorname{rk}}(M) < 11$ are $SL_4({\mathbb{R}})/SO_4$ and $SO^o_{3,4}/SO_3SO_4$ (which both have index $3$ by Theorem \[classification\]), $Sp_3({\mathbb{R}})/U_3$ and $SL_5({\mathbb{R}})/SO_5$ (which both have index $4$ by Corollary \[moreexamples\]). This concludes the proof of Theorem \[iM=4\]
The analogous argument does not work for index five. For example, $M = SU_{3,3}/S(U_3U_3)$ has $i_r(M) = 6$, but for $d= 5$ the inequality $d(d+1) < 2(\dim(M) - {\operatorname{rk}}(M))$ is not satisfied, so we can only conclude $i(M) \in \{5,6\}$ with our results so far. However, using the classification in [@BT] of cohomogeneity one actions on irreducible Riemannian symmetric spaces of noncompact type, we can improve the inequality in Corollary \[ref4\] when ${\operatorname{codim}}(\Sigma) \geq 5$:
\[ref4plus\] Let $M$ be an $n$-dimensional irreducible Riemannian symmetric space of noncompact type with $r = {\operatorname{rk}}(M) \geq 2$ and let $\Sigma$ be a semisimple connected complete totally geodesic submanifold of $M$ with ${\operatorname{codim}}(\Sigma) = d \geq 5$. If $$d(d-1) < 2(n - r - 1),$$then $\Sigma$ is a reflective submanifold of $M$.
As usual, we write $\Sigma = G'/K'$ and identify $SO_d$ with $SO(\nu_p\Sigma)$. Since $d \geq 5$ and any connected subgroup of $SO_d$ is totally geodesic in $SO_d$, we see from Corollary \[moreexamples\] that the minimal codimension of a connected subgroup of $SO_d$ is equal to $d-1$, which is exactly the codimension of $SO_{d-1}$. A principal $K'$-orbit on $\Sigma$ has dimension $n-d-{\operatorname{rk}}(\Sigma)$, which implies $\dim(K') \geq n - d - {\operatorname{rk}}(\Sigma) \geq n - d - r$. Consequently, if $\frac{1}{2}(d-1)(d-2) < n - d - r$, then $\dim(K') > \frac{1}{2}(d-1)(d-2) = \dim(SO_{d-1}) $. The inequality $\frac{1}{2}(d-1)(d-2) < n - d - r$ is equivalent to $d(d-1) < 2(n - r - 1)$. If the kernel $\ker(\rho)$ of the slice representation $\rho : K' \to SO (\nu_p \Sigma)$ has positive dimension, then $\Sigma$ is a reflective submanifold by Proposition \[ref3\]. If $\dim(\ker(\rho)) = 0$, then we must have ${\mathfrak{k}}' = {\mathfrak{so}}_d$ and the action of $K'$ on the unit sphere in $\nu_p\Sigma$ is transitive. This implies that $\Sigma$ is a totally geodesic singular orbit of a cohomogeneity one action on $M$. It was proved in [@BT] that with five exceptions any such orbit is reflective. Three of the five exceptions do not satisfy the assumption $d \geq 5$. The remaining two exceptions are $\Sigma = G_2^{\mathbb{C}}/G_2$ in $M = SO_7({\mathbb{C}})/SO_7$ and $\Sigma = SL_3({\mathbb{C}})/SU_3$ in $M = G_2^{\mathbb{C}}/G_2$, and both do not satisfy the inequality $d(d-1) < 2(n - r - 1)$. It follows that $\Sigma$ is reflective.
Note that the assumption $d \geq 5$ in Proposition \[ref4plus\] is essential. For example, $\Sigma = G_2^2/SO_4$ is a semisimple totally geodesic submanifold of $M = SO^o_{3,4}/SO_3SO_4$ with $d = {\operatorname{codim}}(\Sigma) = 4$. The inequality in Proposition \[ref4plus\] is satisfied, but $\Sigma$ is non-reflective. For $d=3$ the totally geodesic submanifold $\Sigma = SL_3({\mathbb{R}})/SO_3$ in $M = G_2^2/SO_4$ provides a counterexample.
From Proposition \[ref4plus\] we obtain:
\[rankreflective\] Let $M$ be an irreducible Riemannian symmetric space of noncompact type and let $\Sigma$ be a semisimple connected complete totally geodesic submanifold of $M$ with ${\operatorname{codim}}(\Sigma) \geq 5$. If ${\operatorname{codim}}(\Sigma) = {\operatorname{rk}}(M)$, then $\Sigma$ is a reflective submanifold of $M$.
For $d = {\operatorname{codim}}(\Sigma) = {\operatorname{rk}}(M) = r$ the inequality in Proposition \[ref4plus\] becomes $r^2 + r < 2n-2$. It is clear that $n = \dim(M) \geq \frac{1}{2}\#(R) + r$, where $(R)$ denotes the restricted root system of $M$. For every root system occuring here we have $r^2 + r \leq \#(R)$, with equality if and only if $(R) = (A_r)$. Altogether this implies $r^2 + r \leq \#(R) \leq 2n - 2r < 2n-2$ and hence $\Sigma$ is reflective by Proposition \[ref4plus\].
From Proposition \[ref4plus\] we also obtain:
\[betterestimate\] Let $M$ be an irreducible Riemannian symmetric space of noncompact type with ${\operatorname{rk}}(M) \geq 2$, $i(M) \geq 5$ and $i_r(M) \geq 6$. If $$(i_r(M)-2)(i_r(M)-1) < 2(\dim(M) - {\operatorname{rk}}(M) - 1),$$ then $i(M) = i_r(M)$.
Let $\Sigma$ be a maximal totally geodesic submanifold of $M$ such that $d = {\operatorname{codim}}(\Sigma) = i(M) \geq 5$. We put $n = \dim(M)$ and $r = {\operatorname{rk}}(M)$. If $\Sigma$ is non-semisimple, then $\Sigma$ is a reflective submanifold by Corollary \[nssrefl\] and hence $d \geq i_r(M)$. If $\Sigma$ is semisimple and $d < i_r(M)$, then $d(d-1) < 2(n - r - 1)$ by assumption and thus $\Sigma$ is a reflective submanifold by Corollary \[ref4plus\], which is a contradiction to $d < i_r(M)$. It follows that $d \geq i_r(M)$ and therefore $i(M) = i_r(M)$.
We can use Corollary \[betterestimate\] to calculate a few more indices for symmetric spaces which cannot be obtained via the inequality in Proposition \[irM=iM\] and are therefore not listed in Corollary \[moreexamples\]:
\[moreindices\] The following symmetric spaces satisfy $i(M) = i_r(M)$:
- $Sp_5({\mathbb{R}})/U_5$;
- $SU_{r,r}/S(U_rU_r)$, $r \geq 3$;
- $Sp_{r,r+k}/Sp_rSp_{r+k}$, $k + 1 = r \geq 3$;
- $F_4^4/Sp_3Sp_1$.
Let $M$ be one of the symmetric spaces in (i)-(iv). It is clear that $rk(M) \geq 2$. From Theorems \[classification\] and \[iM=4\] we see that $i(M) \geq 5$ and from Table \[summary\] we see that $i_r(M) \geq 6$. It is a straightforward calculation to show that $M$ satisfies the inequality in Corollary \[betterestimate\], which then implies $i(M) = i_r(M)$.
We inserted this result into the last column of Table \[summary\].
We can now also settle the classifications for $i(M) = 5$ and $i(M) = 6$.
\[iM=5\] [(Symmetric spaces with index five)]{} Let $M$ be an irreducible Riemannian symmetric space of noncompact type. Then $i(M) = 5$ if and only if $M$ is isometric to one of the following symmetric spaces:
- $SO^o_{5,5+k}/SO_5SO_{5+k}$, $k \geq 0$;
- $SL_4({\mathbb{C}})/SU_4$;
- $SL_6({\mathbb{R}})/SO_6$.
From Corollary \[moreexamples\] and Table \[summary\] we see that every symmetric space listed in Theorem \[iM=5\] satisfies $i(M) = 5$. Conversely, let $M$ be an irreducible Riemannian symmetric space of noncompact type with $i(M) = 5$ and let $\Sigma$ be a maximal totally geodesic submanifold of $M$ with $d = {\operatorname{codim}}(\Sigma) = 5$. From Table \[iMforrkMleq2\] we obtain ${\operatorname{rk}}(M) \geq 3$. If $\Sigma$ is non-semisimple, then $\Sigma$ is reflective by Corollary \[nssrefl\]. If $\Sigma$ is semisimple and $\dim(M) - {\operatorname{rk}}(M) > 11$, then $\Sigma$ is reflective by Proposition \[ref4plus\]. Thus we have $i_r(M) = i(M) = 5$ if $\dim(M) - {\operatorname{rk}}(M) > 11$ and we can use Table \[summary\] to see that $M$ is isometric to one of the spaces in (i)-(iii). If $\dim(M) - {\operatorname{rk}}(M) < 11$ we saw in the proof of Theorem \[iM=4\] that $i(M) \in \{3,4\}$. There is no symmetric space $M$ with ${\operatorname{rk}}(M) \geq 3$ and $\dim(M) - {\operatorname{rk}}(M) = 11$. This concludes the proof of Theorem \[iM=5\]
\[iM=6\] [(Symmetric spaces with index six)]{} Let $M$ be an irreducible Riemannian symmetric space of noncompact type. Then $i(M) = 6$ if and only if $M$ is isometric to one of the following symmetric spaces:
- $SO^o_{6,6+k}/SO_6SO_{6+k}$, $k \geq 0$;
- $SU_{3,3+k}/S(U_3U_{3+k})$, $k \geq 0$;
- $SU^*_6/Sp_3$;
- $G_2^{\mathbb{C}}/G_2$;
- $Sp_{2,2}/Sp_2Sp_2$;
- $Sp_4({\mathbb{R}})/U_4$;
- $SO_7({\mathbb{C}})/SO_7$;
- $SL_7({\mathbb{R}})/SO_7$.
From Tables \[iMforrkMleq2\] and \[summary\] we see that every symmetric space listed in Theorem \[iM=6\] satisfies $i(M) = 6$. Conversely, let $M$ be an irreducible Riemannian symmetric space of noncompact type with $i(M) = 6$ and let $\Sigma$ be a maximal totally geodesic submanifold of $M$ with $d = {\operatorname{codim}}(\Sigma) = 6$. If ${\operatorname{rk}}(M) \in \{1,2\}$ we see from Table \[iMforrkMleq2\] that $M$ is one of the spaces in (iii)-(v). We assume that ${\operatorname{rk}}(M) \geq 3$. If $\Sigma$ is non-semisimple, then $\Sigma$ is reflective by Corollary \[nssrefl\]. If $\Sigma$ is semisimple and $\dim(M) - {\operatorname{rk}}(M) > 16$, then $\Sigma$ is reflective by Proposition \[ref4plus\]. Thus we have $i_r(M) = i(M) = 6$ if $\dim(M) - {\operatorname{rk}}(M) > 16$ and we can use Table \[summary\] to see that $M$ is isometric to one of the spaces in (i), (ii), (vii) and (viii). If $\dim(M) - {\operatorname{rk}}(M) < 12$ we saw in the proof of Theorem \[iM=5\] that $i(M) \in \{3,4\}$. The symmetric spaces $M$ with ${\operatorname{rk}}(M) \geq 3$ and $12 \leq \dim(M) - rk(M) \leq 16$ are $SO^o_{3,5}/SO_3SO_5$ and $SO^o_{3,6}/SO_3SO_6$ (which both have index $3$ by Theorem \[classification\]), $SO^o_{4,4}/SO_4SO_4$ and $SO^o_{4,5}/SO_4SO_5$ (which both have index $4$ by Theorem \[iM=4\]), $SL_6({\mathbb{R}})/SO_6$ and $SL_4({\mathbb{C}})/SU_4$ (which both have index $5$ by Theorem \[iM=5\]), $Sp_4({\mathbb{R}})/U_4$ (which has index $6$ by Corollary \[moreexamples\] and Table \[summary\]), $SU_{3,3}/S(U_3U_3)$ (which has index $6$ by Corollary \[moreindices\] and Table \[summary\]). This concludes the proof of Theorem \[iM=5\]
We cannot continue beyond $i(M) = 6$ with our methods. For example, the symmetric space $M = Sp_3({\mathbb{C}})/Sp_3$ satisfies $\dim(M) = 21$ and ${\operatorname{rk}}(M) = 3$. Thus the inequality $d(d-1) < 2(\dim(M) - {\operatorname{rk}}(M) - 1) = 34$ in Proposition \[ref4plus\] is satisfied if and only if $d \leq 6$. However, from Table \[summary\] we know that $i_r(M) = 8$. Thus we must have $i(M) \in \{7,8\}$. We cannot exclude the possiblity $i(M) = 7$ here.
It is worthwhile to point out that the only irreducible Riemannian symmetric space $M$ with $i(M) < i_r(M)$ known to us is $M = G_2^2/SO_4$. This leads us to the
Let $M$ be an irreducible Riemannian symmetric space of noncompact type and $M \neq G_2^2/SO_4$. Then $i(M) = i_r(M)$.
We verified the conjecture in this paper for several symmetric spaces and for all symmetric spaces with $i(M) \leq 6$ or $\dim(M) \leq 20$. In the last column of Table \[summary\] we summarize the current status of this conjecture.
[\[10\]]{}
J. Berndt and C. Olmos, On the index of symmetric spaces, preprint arXiv:1401.3585 \[math.DG\].
J. Berndt and H. Tamaru, Cohomogeneity one actions on noncompact symmetric spaces with a totally geodesic singular orbit, *Tohoku Math. J. (2)* **56** (2004), 163–177.
A. Borel and L. Ji, *Compactifications of symmetric and locally symmetric spaces*, Birkhäuser, Boston, 2006.
S. Helgason, *Differential geometry, Lie groups, and symmetric spaces*, Corrected reprint of the 1978 original. Graduate Studies in Mathematics, 34. American Mathematical Society, Prividence, RI, 2001.
S. Klein, Totally geodesic submanifolds of the complex quadric, *Differential Geom. Appl.* **26** (2008), 79–96.
S. Klein, Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram, *Geom. Dedicata* **138** (2009), 25–50.
S. Klein, Totally geodesic submanifolds of the complex and the quaternionic $2$-Grassmannians, *Trans. Amer. Math. Soc.* **361** (2010), 4927–4967.
S. Klein, Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank $2$, *Osaka J. Math.* **47** (2010), 1077–1157.
S. Kobayashi and T. Nagano, On filtered Lie algebras and geometric structures I, *J. Math. Mech.* **13** (1964), 875–907.
D. S. P. Leung, On the classification of reflective submanifolds of Riemannian symmetric spaces, *Indiana Univ. Math. J.* **24** (1974), 327–339, Errata: *Indiana Univ. Math. J.* **24** (1975), 1199.
D. S. P. Leung, Reflective submanifolds. III. Congruency of isometric reflective submanifolds and corrigenda to the classification of reflective submanifolds, *J. Differential Geom.* **14** (1979), 167–177.
C. Olmos and R. Riaño-Riaño, Normal holonomy of orbits and Veronese submanifolds, *J. Math. Soc. Japan* (to appear), preprint arXiv:1306.2225 \[math.DG\].
**2** (1980), 64–85.
J.A. Wolf, Elliptic spaces in Grassmann manifolds, *Illinois J. Math.* **7** (1963), 447–462.
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---
abstract: 'We study the gravitational collapse in modified gravitational theories. In particular, we analyze a general $f(R)$ model with uniformly collapsing cloud of self-gravitating dust particles. This analysis shares analogies with the formation of large-scale structures in the early Universe and with the formation of stars in a molecular cloud experiencing gravitational collapse. In the same way, this investigation can be used as a first approximation to the modification that stellar objects can suffer in these modified theories of gravity. We study concrete examples, and find that the analysis of gravitational collapse is an important tool to constrain models that present late-time cosmological acceleration.'
author:
- 'J.A.R.Cembranos$^{(a)}$[^1], A.de la Cruz-Dombriz$\,^{(b, c)}$[^2] and B.Montes Núñez$\,^{(d)}$[^3]'
title: 'Gravitational collapse in $f(R)$ theories'
---
Introduction
============
In the general study of astrophysical weak gravitational fields, relativistic effects tend to be ignored. However, there are clear examples of stellar objects in which these effects may have important consequences, such as neutron stars, white dwarfs, supermassive stars or black holes. Indeed, it becomes necessary to consider observationally consistent gravitational theories to study these objects. General Relativity (GR) has been the most widely used theory but other gravitational theories may be studied for a better understanding of the features and properties of such objects and to compare their predictions with experimental results.
The gravitational collapse for a spherically symmetric stellar object has been extensively studied in the GR framework (see [@Weinberg] and references therein). By assuming the metric of the space-time to be spherically symmetric and that the collapsing fluid is pressureless, the found metric interior to the object turns to be Robertson-Walker type with a parameter playing the role of spatial curvature and proportional to initial density. The time lapse and the size of the object are given by a cycloid parametric equation with an angle parameter $\psi$. Further results are that the time when the object gets zero size is finite and inversely proportional to the square root of the initial density. Finally, the redshift seen by an external observer is nevertheless infinite when time approaches the collapse time.
In spite of the fact that GR has been one of the most successful theories of the twentieth century, it does not give a satisfactory explanation to some of the latest cosmological and astrophysical observations with usual matter sources. In the first place, a dark energy contribution needs to be considered to provide cosmological acceleration whereas the baryonic matter content has to be supplemented by a dark matter (DM) component to give a satisfactory description of large scale structures, rotational speeds of galaxies, orbital velocities of galaxies in clusters, gravitational lensing of background objects by galaxy clusters, such as the Bullet Cluster, and the temperature distribution of hot gas in galaxies and clusters of galaxies. All these evidences have revealed the interest to study alternative cosmological theories. This extra DM component is required to account for about $20\%$ of the energy content of our Universe. Although there are many possible origins for this component [@DM], DM is usually assumed to be in the form of thermal relics that naturally freeze-out with the right abundance in many extensions of the standard model of particles [@WIMPs]. Future experiments will be able to discriminate among the large number of candidates and models, such as direct and indirect detection designed explicitly for their search [@isearches], or even at high energy colliders, where they could be produced [@Coll].
A larger number of possibilities can be found in the literature to generating the present accelerated expansion of the Universe [@DE]. One of these methods consists of modifying Einstein’s gravity itself [@otras; @Tsujikawareview2010] without invoking the presence of any exotic dark energy among the cosmological components. In this context, functions of the scalar curvature when included in the gravitational action give rise to the so-called $f(R)$ theories of modified gravity [@Tsujikawa_Felice_fR_theories]. They amount to modifying the l.h.s. of the corresponding equations of motion and provide a geometrical origin to the accelerated cosmological expansion. Although such theories are able to describe the accelerated expansion on cosmological scales correctly [@delaCruzDombriz:2006fj], they typically give rise to strong effects on smaller scales. In any case, viable models can be constructed to be compatible with local gravity tests and other cosmological constraints [@varia].
The study of alternative theories of gravitation requires establishing methods able to confirm or discard their validity by studying the cosmological evolution, the growing of cosmological perturbation and, at astrophysical level, the existence of objects predicted by GR such as black holes [@delaCruzDombriz:2009et] or dust clouds forming compact structures. It is well-known that $f(R)$ gravity theories may mimic any cosmological evolution by choosing adequate $f(R)$ models, in particular that of $\Lambda\text{CDM}$ [@delaCruzDombriz:2006fj]. This is the so-called [*degeneracy problem*]{} that some modified gravity theories present: accordingly, the exclusive use of observations such as high-redshift Hubble diagrams from SNIa [@Riess], baryon acoustic oscillations [@BAO] or CMB shift factor [@Spergel], based on different distance measurements which are sensitive only to the expansion history, cannot settle the question of the nature of dark energy [@Linder] since identical results may be explained by several theories. Nevertheless, it has been proved that $f(R)$ theories - even mimicking the standard cosmological expansion - provide different results from $\Lambda\text{CDM}$ if the scalar cosmological perturbations are studied [@delaCruzDombriz:2008cp]. Consequently, the power spectra would be distinguishable from that predicted by $\Lambda\text{CDM}$ [@PRL_Dombriz].
It is therefore of particular interest, to establish the predictions of $f(R)$ theories concerning the gravitational collapse, and in particular collapse times, for different astrophysical objects. Collapse properties may be either exclusive for Einstein’s gravity or intrinsic to any covariant gravitation theory. On the other hand, obtained results may be shed some light about the models viability and be useful to discard models in disagreement with expected physical results. In [@Sharif:2010um] the authors studied gravitational collapse of a spherically symmetric perfect fluid in $f(R)$ gravity. By proceeding in a similar way to [@Weinberg], the object mass was deduced from the junction conditions for interior and exterior metric tensors. Finally, they concluded that $f(R_0)$ (constant scalar curvature term) slows down the collapse of matter and plays the role of a cosmological constant. Authors in [@Bamba:2011sm] paid attention to the curvature singularity appearing in the star collapse process in $f(R)$ theories. This singularity was claimed to be generated in viable $f(R)$ gravity and can be avoided by adding a $R^{\alpha}$ term. They also studied exponential gravity and the time scale of the singularity appearance in that model. It was shown that in case of star collapse, this time scale is much shorter than the age of the universe. Analogous studies were carried on by [@Arbuzova:2010iu] claiming that in this class of theories, explosive phenomena in a finite time may appear in systems with time dependent increasing mass density.
Reference [@Santos:2011ye] includes a complete study of neutron stars in $f(R)$ theories. The most relevant result in this investigation suggests that $f(R)$ theory allows stars in equilibrium with arbitrary baryon number, no matter how large they are. Very recently authors in [@Hwang] studied collapse of charged black holes by using the double-null formalism. Charged black holes in f(R) gravity can have a new type of singularity due to higher curvature corrections, the so-called f(R)-induced singularity, although it is highly model-dependent.
The present work has been arranged as follows: in section II, $f(R)$ modified gravity theories will be introduced. Gravitational collapse in $f(R)$ theories will be presented in section III. After performing some calculations, the evolution equation for the object scale factor will be obtained. This equation will be used throughout the following sections. Section IV is then dedicated to achieve solutions for the modified equations in three qualitatively different $f(R)$ models, which try to illustrate the broad phenomenology of the subject. This is therefore the aim of this section: to study gravitational collapse by calculating the evolution of the object scale factor in particular $f(R)$ models. Finally, the conclusions based upon the presented results will be analyzed in detail in section V.
II. $f(R)$ theories of gravity
==============================
With the aim of proposing and alternative theory to GR, a possible modification consists of adding a function of the scalar curvature, $f(R)$, to the Einstein-Hilbert (EH) Lagrangian. Therefore the gravitational action becomes [^4] $$S_G=\frac{1}{16\pi G}\int \text{d}^4x \sqrt{\vert g \vert}\left(R+f(R)\right).
\label{Modified action}$$ By performing variations with respect to the metric, the modified Einstein equations turn out to be $$(1+f_R)R_{\mu\nu}-\frac{1}{2}(R+f(R))g_{\mu\nu}+{\cal D}_{\mu\nu}f_R\,=\,-8\pi G \,T_{\mu\nu},
\label{fieldtensorialequation}$$ where $T_{\mu\nu}$ is the energy-momentum tensor of the matter content, $f_R\,\equiv\,\text{d}f(R)/\text{d}R$ and ${\cal D}_{\mu\nu}\equiv \nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\square$ with $\square\,\equiv\,\nabla_{\alpha}\nabla^{\alpha}$ and $\nabla$ is the usual covariant derivative.
These equations may be written *à la Einstein* by isolating on the l.h.s. the Einstein tensor and the $f(R)$ contribution on the r.h.s. as follows $$\begin{aligned}
R_{\mu\nu}&-&\frac{1}{2}R g_{\mu\nu}\,=\,\frac{1}{(1+f_R)}\left[\vphantom{\frac{1}{2}}-8\pi G T_{\mu\nu}
\right. \nonumber\\ &-& \left.
{\cal D}_{\mu\nu}f_R+\frac{1}{2}\left(f(R)-Rf_R\right)g_{\mu\nu}\right]
$$ We can also find the expression for the scalar curvature by contracting with $g^{\mu\nu}$ which gives: $$\begin{aligned}
(1-f_R)R+2f(R)+3\square f_R\,=8\pi G \,T.
\label{}\end{aligned}$$ Note that, unlike GR where $R$ and $T$ are related algebraically, for a general $f(R)$ those two quantities are dynamically related. In the homogeneous and isotropic case, the scalar curvature in $f(R)$ theories becomes $$\begin{aligned}
R=\frac{8\pi G \,T-2f(R)-3 \ddot{f}_R\,}{(1-f_R)}
\label{Scalar curvature in f(R)}\end{aligned}$$ where dot means the derivative with respect to cosmic time.
III. Gravitational collapse in $f(R)$
=====================================
In the case of our investigation, we introduce the spherically symmetric metric $$\text{d}s^2 =\text{d}t^2-U(r,t)\text{d}r^2-V(r,t)(\text{d}\theta^2+\text{sin}^2\theta \text{d}\phi^2)
\label{Intervalo colapso}$$ If the collapsing object is approximated to be pressureless $p\simeq 0$, the components of the energy-momentum tensor can be expressed as follows $$\begin{aligned}
T_{\mu\nu}=\rho u_{\mu}u_{\nu}\;;\;
T^{t}_{\;\;t}=\rho\;;\;
T^{i}_{\;\;i}=0 \;\; \text{if}\;\; i=r,\theta,\,\phi.
\label{pressureless_fluid}\end{aligned}$$ We may further simplify the collapse model by considering $\rho$ independent from the position. Therefore, we can search -as is actually the usual approach in the GR case- a separable solution for this metric as follows $$U(r,t)=A_1^2(t)h(r),\quad V(r,t)=A_2^2(t)r^2,
\label{Funciones U y V en f(R)}$$ where a previous reparametrization of the radial coordinate is required. When $f(R)$ modified tensorial equations are studied in the homogeneous and isotropic case - in which $f(R)$ does not depend on the position-, the trace component provides $$\begin{aligned}
\left(\frac{\dot{A}_2}{A_2}-\frac{\dot{A}_1}{A_1}\right)\frac{g'}{g}=0\Rightarrow \frac{\dot{A}_2}{A_2}=\frac{\dot{A}_1}{A_1}.
\label{A1_vs_A2}\end{aligned}$$ From , we deduce that $A_1$ and $A_2$ are proportional, in other words, $A_1(t)=C(r)A_2(t)$. So, if we choose $A_1(t)=A_2(t)\equiv A(t)$, the dependence in the radial coordinate is reabsorbed by $h(r)$. Hence: $$U(r,t)=A^2(t)h(r),\quad V(r,t)=A^2(t)r^2.
\label{U and V general functions}$$ Components $tt$, $rr$ and $\theta\theta$ for the modified tensorial equations may be written respectively in terms of the functions $A(t)$ and $h(r)$ as follows $$\begin{aligned}
&&3\frac{\ddot{A}}{A}=\frac{1}{(1+f_R)}\left[-8\pi G\rho+
3\frac{\dot{A}}{A}\dot{f}_{R}+
\frac{1}{2}\left(R+f(R)\right)\right],\nonumber\\
&&
\label{Rtt with U and V simplified}\end{aligned}$$ $$\begin{aligned}
&&A\ddot{A}+2\dot{A}^2+\frac{h'}{r h^2}
=\frac{A^2}{(1+f_R)}\left[\ddot{f}_{R}\,
+2\frac{\dot{A}}{A}\dot{f}_{R}
\right. \nonumber\\ && \left.
+\frac{1}{2}\left(R+f(R)\right)\right],
\label{Rrr with U and V simplified}\end{aligned}$$ $$\begin{aligned}
&&A\ddot{A}+2\dot{A}^2+\frac{1}{r^2}-\frac{1}{hr^2}
+\frac{h'}{2r h^2}\nonumber\\
&=&\frac{A^2}{(1+f_R)}\left[\ddot{f}_{R}
+2\frac{\dot{A}}{A}\dot{f}_{R}
+\frac{1}{2}\,(R+f(R))\right].\nonumber\\
\label{Rthth with U and V simplified}\end{aligned}$$ Let us point out two important aspects of equations and : firstly, terms on the r.h.s of both equations are equal. Secondly, the term on the l.h.s. exclusively depends on $r$, whereas the term on the r.h.s. only depends on $t$ in both equations, so that they must be constants[^5]. Therefore we may equal l.h.s. of both equations to provide $$\begin{aligned}
\frac{1}{r}\frac{h'}{h^2}=\frac{1}{r^2}-\frac{1}{hr^2}+\frac{1}{2r}\frac{h'}{h^2}\equiv 2k\,,
\label{Eq para constante k in modified case}\end{aligned}$$ where we have equaled both equations (multiplied by a factor $A^2$) to a constant $-2k$. The resulting solution is $h(r)=(1-kr^2)^{-1}$. Once we have calculated $h(r)$, the resulting metric can be expressed as follows: $$\text{d}s^2 =\text{d}t^2-A^2(t)\left[\frac{\text{d}r^2}{1-kr^2}+r^2(\text{d}\theta^2+\text{sin}^2\theta \text{d}\phi^2)\right]\,,
\label{M�trica homog�nea y is�tropa in modified case}$$ which is formally the same as the one obtained in the GR case [@Weinberg]. Expression for $k$ may be substituted in either expression or yielding: $$\begin{aligned}
&-&\frac{\ddot{A}}{A}-2\left(\frac{\dot{A}}{A}\right)^2-\frac{2k}{A^2}=\frac{1}{(1+f_R)}\left[-\ddot{f}_{R}
-2\frac{\dot{A}}{A}\dot{f}_{R}
\right. \nonumber\\ &-& \left.
\frac{1}{2}\left(R+f(R)\right)\right].
\label{eq casi casi}\end{aligned}$$ Taking into account $\rho\,(t)=\rho\,(t=0)/A(t)^3$ (given by the energy motion equation for dust matter) and the results in (\[eq casi casi\]), equation becomes: $$\begin{aligned}
\dot{A}^2&=&-k+\frac{1}{(1+f_R)}\left[\frac{4}{3}\pi G \rho(0)A^{-1}+ \frac{1}{2}A^2\ddot{f}_{R}
\right. \nonumber\\ &+& \left.
\frac{1}{2}A\dot{A}\dot{f}_{R}
+\frac{A^2}{6}\left(R+f(R)\right)\right].
\label{Eq for dotA}\end{aligned}$$ Furthermore, provided that the fluid is assumed to be at rest for $t=0$, initial conditions $\dot{A}(t=0)=0$ and $A(t=0)=1$ hold. This last condition means that the scale factor of the object at initial time is normalized to unity. In order to simplify the notation we define $R(t=0)\equiv R_0$ and $\rho\,(t=0)\equiv \rho_0$. Therefore, evaluation of at $t=0$ allows to recast $k$ as follows $$\begin{aligned}
k&=&\frac{1}{(1+f_R(R_0))}\left[\frac{4\pi G}{3}\rho_0+\frac{1}{2}\ddot{f}_{R}(R_0)
+\frac{1}{6}(R_0+f(R_0))\right]
\nonumber\\
&&\label{Curvature in terms of R(0)}\end{aligned}$$ Once $k$ has been expressed in terms of different quantities initial values, equations (\[Scalar curvature in f(R)\]) and (\[Curvature in terms of R(0)\]) may be inserted in (\[Eq for dotA\]) to provide
$$\begin{aligned}
\dot{A}^2&=&-\frac{1}{6(1-f_R^2(R_0))}\left[\vphantom{\ddot{f}_R(R_0)}\,8\pi G\rho_0\left(2-f_R(R_0)\right)
- f(R_0)(1+f_R(R_0))-3\ddot{f}_R(R_0)f_R(R_0)\,\right]
+\frac{1}{(1-f_R^2)}\,\frac{8\pi G}{3}\rho_0\,A^{-1}\nonumber\\
&&-\frac{1}{6(1-f_R^2)}\left[8\pi G\rho_0\,A^{-1}f_R+3A^2\ddot{f}_{R}f_R-
3A\dot{A}\dot{f}_{R}(1-f_R)+A^2f(R)(1+f_R)\right].
\label{Simplified Eq for dotA}\end{aligned}$$
The previous expression will be solved perturbatively to first order in perturbations for different $f(R)$ models. Let us remind at this stage that the zeroth order solution of GR is given by the parametric equations of a cycloid [@Weinberg]: $$t=\frac{\psi+\text{sin}\,\psi}{2\sqrt{k}},\;\;\; A_G=\frac{1}{2}(1+\text{cos}\,\psi).
\label{Eqs cicloide}$$ Expression clearly implies that a sphere with initial density $\rho_0$ and negligible pressure will collapse from rest to a state of infinite proper energy density in a finite time that we will denote $T_G$. This time is obtained for the first value of $\psi$ such as $A_G=0$, i.e. for $\psi=\pi$. It means $$\begin{aligned}
T_G=\left(\frac{\pi+\text{sin}\pi}{2\sqrt{k}}\right)=\frac{\pi}{2\sqrt{k}}=\frac{\pi}{2}\left(\frac{3}{8\pi G\rho_0}\right)^{1/2}.\label{Valor de t para colapso}\end{aligned}$$
In order to study the modification to the gravitational collapse in $f(R)$ theories, we will expand $A$ around $A_G$ and $f(R)$ around the scalar curvature in GR ($R=R_G$): $$A=A_G+g(\psi)\,,
\label{Series expansion for A}$$ $$f(R)\simeq f(R_G)+f'(R_G)(R-R_G).
\label{Series expansion for f(R)}$$ The presence of a function $f(R)$ in the gravitational Lagrangian will represent a correction of first order with respect to the usual EH Lagrangian. Hence, $g(\psi)$ as defined in (\[Series expansion for A\]) will be also first order at least. By substituting the series expansions and in expression until first order in $\varepsilon$, we find that equation (\[Simplified Eq for dotA\]) becomes
$$\begin{aligned}
\text{tg}\left(\frac{\psi}{2}\right)g'\,&=&\,-\frac{1}{2}\text{cos}^{-2}\left(\frac{\psi}{2}\right) g
+\frac{1}{12k}\text{cos}^{2}\left(\frac{\psi}{2}\right)\left(f(R_{G0})+3 k f_{R}(R_{G0})\right)
-\frac{1}{4}f_R(R_G)\nonumber\\
&&+\frac{1}{4\sqrt{k}}\,\text{sin}^{}\left(\frac{\psi}{2}\right)\text{cos}^{3}\left(\frac{\psi}{2}\right)\dot{f}_{R}(R_G)
-\frac{1}{12k}\,\text{cos}^{6}\left(\frac{\psi}{2}\right) f(R_G)\,,
\label{Eq g' with generic Rs}\end{aligned}$$
where we have cancelled out the GR exact solution and only kept first order perturbed terms. Equation will provide $g(\psi)$ evolution for different $f(R)$ models to be considered in the next section.
IV. $f(R)$ theory results
=========================
In this section we shall consider three illustrative $f(R)$ models and study the gravitational collapse process for collapsing dust. The models under consideration are
Model 1: $f(R)=\varepsilon R^2$
-------------------------------
This function has been proposed both as a viable inflation candidate [@Starobinsky:1980te] and as a dark matter model [@R2DM]. In this last reference, the $\varepsilon$ parameter definition reads $$\begin{aligned}
\varepsilon=\frac{1}{6m_0^2}\, ,
\label{Eq epsilon1}\end{aligned}$$ and the minimum value allowed for $m_0$ is computed as $m_0=2.7\times10^{-12}$ GeV at 95 $\%$ confidence level, i.e. $\varepsilon\leq 2.3\times10^{22}\,\text{GeV}^{-2}$. On the other hand, $\varepsilon>0$ is needed to ensure the stability of the model, since in the opposite case, a tachyon is present in the theory. These constraints are in agreement with [@Berry].
After some algebra, for this model, equation (\[Eq g’ with generic Rs\]) can be written as follows: $$\begin{aligned}
&&g'(\psi )+g(\psi ) \csc (\psi )+\frac{9}{8}k\varepsilon\left(\sin (\psi )+2 \tan
\left(\frac{\psi }{2}\right)
\right. \nonumber\\ &-& \left.
4 \tan ^4\left(\frac{\psi }{2}\right)
\csc (\psi )\right)=0.
\label{Simplified Final eq for g' model 1}\end{aligned}$$ The homogeneous equation associated to (\[Simplified Final eq for g’ model 1\]) presents the solution $g_{hom}(\psi )\propto\cot\left(\psi/2\right)$, which diverges at $\psi=0$. Therefore, its contribution will be ignored in the upcoming analysis. The analytical full solution of becomes
$$\begin{aligned}
&&g(\psi )= c_1 \cot \left(\frac{\psi }{2}\right)-\frac{9}{128}k\varepsilon \cot
\left(\frac{\psi }{2}\right) \left\{\vphantom{\frac{\psi}{2}}
-
16\left(\psi+ \sin (\psi)\right)
+\frac{64}{5} \tan \left(\frac{\psi }{2}\right)\left[4-\sec ^2\left(\frac{\psi}{2}\right)\left(\sec^2\left(\frac{\psi}{2}\right) - 2\right)\right]
\right\}\,,\nonumber\\
&&
\label{Analytical solution for g model 1}\end{aligned}$$
This analytical solution given by can be compared with the GR one by plotting them together as shown in Figure \[Figure\_Intersection\_1\].
[![Comparison between the solution given by setting $c_1=0$ and the GR case for the model 1: $f(R)= \varepsilon R^2$. The plotted $k$ value is fixed by Eq. (\[Curvature\]), where the density is $\rho_{\text{SF}}\simeq1.5 \times 10^{-38}$ GeV$^4 \simeq 3.5\times10^{-18}\,{\text{kg}}/{\text{m}^3}$, i.e., the matter density in the early Universe at redshift $z\simeq 1100$ marking the decoupling of matter and radiation and the beginning of structure formation (SF). The modification is extraordinarily small and has been increased 52 orders of magnitude to make it observable: $\hat{g}(\psi)=10^{52} g(\psi)$.[]{data-label="Figure_Intersection_1"}](Model1-3pi.eps "fig:")]{}
As we see in this Figure, in the first stage of the collapse, the correction is negative, what implies that we have a larger contraction. On the contrary, very close to $\psi=\pi$, where the solution can be approximated as $$\begin{aligned}
g(\psi )\simeq\frac{72k\varepsilon}{5(\psi-\pi)^4}.
\label{Series expansion of analytical solution for g model 1}\end{aligned}$$ the sign of the modification changes and the total collapse is avoided. Exactly at this moment, the perturbation leaves the linear regime and a more complete analysis is required. It is interesting to estimate when the linear approach fails and an important modification is expected.
[![Validity of the perturbative regime for model 1, showing different relevant regions: In blue we show the region where our linear approach loses its validity. The excluded region is depicted in yellow and determined by the condition $\varepsilon\leq 2.3\times10^{22}\,\text{GeV}^{-2}$. Finally, the density marking the beginning of structure formation (SF) and the dark energy (DE) density ($\rho_{\text{DE}}\simeq2.8 \times 10^{-47}$ GeV$^4$) have also been plotted for reference.[]{data-label="Validity regime 1"}](Validityregimemodel1.eps "fig:")]{}
By using the collapse time parametrization (\[Eqs cicloide\]) in the $\psi\rightarrow \pi$ limit, one gets $$\begin{aligned}
t&=&\frac{\psi+\sin(\psi)}{2\sqrt{k}}\simeq \frac{\pi+1/6(\Delta\psi)^3}{2\sqrt{k}},
\label{Collapsing time series expansion model 1}\end{aligned}$$ or written in terms of the relative variation: $$\begin{aligned}
\frac{\Delta t}{t_{GR}}&\simeq& \frac{(\Delta\psi)^3}{6\pi}\, .
\label{Collapsing time series expansion model 1 variation}\end{aligned}$$
We are interested in estimating the region of the parameter space of the model, where the modified collapse, and the result for $\psi_{C}$ ($\psi$ value for the collapse) is significantly different from the one predicted by GR ($\psi_{C,GR}\equiv \pi$). With this purpose, we can estimate the values for which $A_G$ is of the same order of its correction. As one can see in Figure \[Figure\_Intersection\_1\], this deviation is more important close to the final stage of the collapse, for $\psi \sim \pi$. In this region, $A_G$ can be approximated by: $$\begin{aligned}
A_G=\frac{1}{2}\left(1+\text{cos}\, \psi \right)\simeq\frac{\left(\psi-\pi\right)^2}{4}\,.
\label{A_G series expansion}\end{aligned}$$ We can use these approximations in the limit $\psi\rightarrow\pi$ to determine the intersection between the particular solution of and the GR solution: $A_G$. This calculation will help establishing the validity regime of the perturbative approach. Therefore, by imposing $|g(\psi)|=|A_G|$, with $g(\psi)$ given by equation (\[Series expansion of analytical solution for g model 1\]), we obtain: $$\begin{aligned}
\psi=\pi-\left(\frac{288|k\varepsilon|}{5}\right)^{1/6}\,.
\label{Regime validity psi value}\end{aligned}$$
At this point, it is necessary to clarify the physical value for $k$ in order to discuss if the departure from linearity is important. $k$ is the initial condition given by equation that depends on the matter density, the initial curvature and the particular $f(R)$ model. In our analysis we are interested in studying the modification to the gravitational collapse in GR and for this reason we will assume the same value of $k$ than as given in GR. This implies that the entire modification has a dynamical origin and it does not come from a change in the initial conditions. Therefore, we will assume that $k$ only depends on the matter density: $$\begin{aligned}
k=\frac{8\pi G}{3}\rho_0.
\label{Curvature}\end{aligned}$$
For the most physically interesting values of $k$ and $\varepsilon$, for which we have studied gravitational collapse of a dust matter cloud, the value of $\psi$ is quite close to $\psi=\pi$ and therefore the asymptotic approach to obtain (\[Regime validity psi value\]) is fully justified. The results are summarized in Figure \[Validity regime 1\], were the non-linear regime is shown for different values of $\epsilon$ and initial densities. For example, it is interesting to check the behavior for the matter density in the early Universe at redshift $z\simeq 1100$, which marks the decoupling of matter and radiation and the beginning of structure formation (SF): $\rho_{\text{SF}}\simeq1.5 \times 10^{-38}$ GeV$^4 \simeq 3.5\times10^{-18}\,{\text{kg}}/{\text{m}^3}$. In this particular case, the calculated root presents a slight difference with respect to the solution for GR. In particular, $\psi_{C}$ differs from $\psi_{C,\,\text{GR}}\equiv \pi$ in the ninth significant figure.
Although this modification is not detectable for this first model as we deduce from the last considerations, it is interesting to stress that the relative modification is higher for denser media since the correction increases with $\rho_0$ as $\Delta t/t_{GR} \propto \sqrt{\rho_0}$. This behavior is significantly different with respect to other models as we will see in the following sections. In Figure \[Validity regime 1\], we have represented some relevant density values as well as the non linear regime region. The loss of the linear regime takes place at high densities since the correction is directly proportional to $\rho$.
Model 2: $f(R)=\varepsilon R^{-1}$
----------------------------------
We will continue our analysis with the $f(R)$ model proposed in reference [@Carroll:2003wy] as a dark energy candidate. This possibility is currently excluded, but this model is a simple example that help to understand the gravitational collapse modifications in models that provide late-time acceleration.
For this model, equation becomes: $$\begin{aligned}
g'(\psi )+g(\psi ) \csc (\psi )=\frac{\varepsilon}{6k^2}\sin \left(\frac{\psi }{2}\right) \cos ^{13}\left(\frac{\psi }{2}\right)\,,
\label{Simplified Final eq for g' model 2}\end{aligned}$$ whose full solution is
$$\begin{aligned}
g(\psi )&=& c_1 \cot \left(\frac{\psi }{2}\right)+\frac{1}{6} \frac{\varepsilon}{k^2}\left(\frac{33 \psi }{2048}+\frac{165 \sin (\psi)}{8192}
-
\frac{11 \sin (2 \psi )}{8192}-\frac{121\sin (3 \psi )}{24576}-\frac{25 \sin (4 \psi )}{8192}
\right. \nonumber\\ &-& \left.
\frac{43 \sin (5 \psi )}{40960}-\frac{5 \sin (6 \psi )}{24576}-\frac{\sin (7 \psi )}{57344}\right) \cot
\left(\frac{\psi }{2}\right).\nonumber\\
\label{Analitycal solution for g model 2}\end{aligned}$$
In Figure \[Figure\_Intersection\_2\], it is possible to see the behavior of the modification for $\varepsilon=-\mu^4$, and $\mu=10^{-42}$ GeV as it was the value originally proposed in reference [@Carroll:2003wy].
[![Analogous representation to the one shown in Fig. \[Figure\_Intersection\_1\], which includes the GR solution, the modification given by and the sum of the two. In this figure $\hat{g}(\psi)=10^{19} g(\psi)$ in order to make the modification observable.[]{data-label="Figure_Intersection_2"}](Model2-3pi.eps "fig:")]{}
The series expansion of around $\psi=\pi$ reads, in this case: $$\begin{aligned}
g(\psi )\simeq-\frac{11\pi\varepsilon(\psi-\pi)}{8192k^2}\;.
\label{Series expansion of analytical solution for g model 2}\end{aligned}$$ Once again the intersection between the particular solution of and the GR solution $A_G$ can be determined in the $\psi\rightarrow \pi$ limit, with help of equation (\[A\_G series expansion\]). $|g(\psi)|=|A_G|$ implies $$\begin{aligned}
\psi\simeq\pi-\frac{11\pi}{2048}\frac{|\varepsilon|}{k^2}.
\label{Regime validity psi value model 2}\end{aligned}$$
[![Analogous to Figure \[Validity regime 1\] for model 2. The limit of the region depicted in yellow shows the proposed value $\varepsilon=-\mu^4$, and $\mu=10^{-42}$ GeV according to [@Carroll:2003wy].[]{data-label="Validity model 2"}](Validityregimemodel2.eps "fig:")]{}
As it can be seen in Figure \[Figure\_Intersection\_2\], the difference between the modified $\psi_{C}$ and $\psi_{C,\,\text{GR}}$ is not distinguishable for $\varepsilon=-\mu^4$ if density is higher than the standard dark energy density. The same result is found for $\varepsilon>-\mu^4$ ($|\varepsilon|<\mu^4$ and negative). The situation changes for $\varepsilon<-\mu^4$ ($|\varepsilon|>\mu^4$ and positive). This behavior can be observed in Figure \[Validity model 2\], where the validity of the linear regime is shown to decrease for higher values of $|\varepsilon|$ and lower densities. As we will see in the following example, this is a general property of $f(R)$ models that provide accelerated cosmologies, at least, for densities higher than the vacuum energy. Results in Figure \[Validity model 2\] can be understood by estimating the correction of the collapsing time in the linear regime as it is determined by equation (\[Collapsing time series expansion model 1 variation\]). This $f(R)$ model provides a relative difference for the collapsing time in GR value given by $$\begin{aligned}
\frac{\Delta t}{t_{GR}}&\simeq&-
\frac{11^3 \varepsilon^3 \pi^2}{3 k^6 2^{34}}\,.
\label{Modification time collapse model 2}\end{aligned}$$ We observe that the correction is more negligible for denser objects. This unexpected fact can be understood since GR modification to the scale factor is proportional to $g \propto \varepsilon/k^2$ whereas $k\propto\rho_0$. According to this dependence, a stellar object with a higher density will suffer a less important modification and vice versa. The relative time modification is lower for denser media since the correction decreases with $\rho_0$ as $\Delta t/t_{GR} \propto \rho_0^{-6}$.
Model 3: $f(R)=\lambda R_{0}\left[\left(1+\frac{R^2}{R_{0}^2}\right)^{-n}-1\right]$
-----------------------------------------------------------------------------------
The last $f(R)$ model to be analyzed in the present work is the well-known Starobinsky model proposed in Reference [@Starobinsky:2007hu]. For this model, $n,\lambda>0$ and $R_{0}$ is considered to be of the order of the presently observed effective cosmological constant[^6]. With such parameter choice, this model is a viable dark energy candidate. The relation between $\lambda$ and $R_{0}$ in vacuum is given by $H_0^2=\lambda R_0/6$ according to [@Starobinsky:2007hu], where $H_0$ is the present Hubble parameter (see [@WMAP] for recent WMAP data) and the proposed value for $\lambda=0.69$.
For the sake of simplicity, let us choose $n=1$. In this case, the equation (\[Eq g’ with generic Rs\]) may be rewritten as follows:
$$\begin{aligned}
&&-g'(\psi )-g(\psi ) \csc (\psi )-\frac{9k\lambda R_{0} \sin ^{3}(\psi)\csc^{4}\left(\frac{\psi}{2}\right)}{32\left(9k^2+R_0^2\right)^2}\left(R_0^2+3k^2\right)
+\frac{72k\lambda R_0 \sin^4\left(\frac{\psi}{2}\right)\csc(\psi)}{\left(R_0^2+9k^2\sec^{12}\left(\frac{\psi}{2}\right) \right)^2}\left(R_0^2+3k^2\sec^{12}\left(\frac{\psi}{2}\right) \right)
\nonumber\\
&&-\frac{9 k \lambda R_0^3\tan \left(\frac{\psi }{2}\right) \sec ^4\left(\frac{\psi}{2}\right) \left(R_0^2-27 k^2 \sec^{12}\left(\frac{\psi }{2}\right)\right)}{2 \left(9 k^2
\sec ^{12}\left(\frac{\psi}{2}\right)+R_0^2\right)^3}
=0.
\label{Final eq for g' model 3}\end{aligned}$$
Unlike the other two cases, we are not able to find analytical solution for equation . Thus, specific values for $k$ and $\lambda$ parameters and $R_{0}$ are required to find a numerical solution. This solution is plotted in Figure \[Figure\_Intersection\_3\].
[![The plotted lines are analogous to the ones in Fig. \[Figure\_Intersection\_1\] and Fig. \[Figure\_Intersection\_2\] but with the solution given by . In this figure $\hat{g}(\psi)=10^{9} g(\psi)$ in order to make the modification observable.[]{data-label="Figure_Intersection_3"}](Model3Alv.eps "fig:")]{}
In any case, equation (\[Final eq for g’ model 3\]) can be studied in the asymptotic limits $\psi\rightarrow0$ and $\psi\rightarrow\pi$. Thus, the corresponding series expansion of (\[Final eq for g’ model 3\]) in the $\psi\rightarrow0$ becomes $$\begin{aligned}
&-&g'(\psi )-\frac{g(\psi )}{\psi }
-\frac{9 k \lambda R_0 \left(3 k^2-R_0^2 \right)\psi}{4 \left(9 k^2
+R_0^2\right)^2}=0\,,\nonumber\\
\label{Final Eq Psi 0 series expansion}\end{aligned}$$ whose analytical solution is $$\begin{aligned}
g(\psi )&=& \frac{c_1}{\psi }+\frac{3 k \lambda R_0 \left(R_0^2-3 k^2 \right)\psi ^2}{4 \left(9 k^2+R_0^2\right)^2}\,.
\label{Solution Psi 0 series expansion}\end{aligned}$$ Since the homogeneous equation does not depend on the $f(R)$ model, the condition $c_1=0$ is also necessary in order to have a finite solution. When the considered asymptotic limit is $\psi\rightarrow\pi$, equation (\[Final eq for g’ model 3\]) approximately becomes $$\begin{aligned}
-g'(\psi )+\frac{g(\psi )}{\psi-\pi }+\frac{9k \lambda R_0 (\psi-\pi)^3 \left(3 k^2+ R_0^2\right)}{32 \left(9 k^2+ R_0^2\right)^2}=0\,,\nonumber\\
\label{Eq Psi pi dominant terms simplified}\end{aligned}$$ whose analytical solution is $$\begin{aligned}
g(\psi )= c_1 (\psi -\pi )+\frac{3k \lambda R_0 (\psi-\pi)^4
\left(3 k^2+2 R_0^2\right)}{32 \left(9 k^2+ R_0^2\right)^2}.\nonumber\\
\label{Solution Psi pi series expansion}\end{aligned}$$ This asymptotic limit of the linear correction depends on a higher power of $(\psi-\pi)$ than the GR solution given by eq. (\[A\_G series expansion\]). This fact implies that we cannot estimate the validity of the linear regime by using the $\psi\rightarrow\pi$ as in the previous cases. The modification is more important at intermediates values of $\psi$, as it can be observed in Figure \[Figure\_Intersection\_3\]. The numerical results are showed in Figure 6. In a similar way to the second model, for $\lambda<\lambda_0$ the modification of the collapse is always linear and not important. The situation is different for $\lambda>\lambda_0$, where the collapse is severely modified at densities closer to the vacuum one. We have checked numerically that denser environments are less affected by this gravitational model.
[![Analogous to Figure \[Validity regime 1\] for studied model 3. The limit of the region depicted in yellow shows the proposed value $\lambda=0.69$ [@Starobinsky:2007hu].[]{data-label="Validity model 3"}](Newvalidityregimemodel3.eps "fig:")]{}
V. Conclusions
==============
In this work we have studied the gravitational collapse in $f(R)$ gravity theories. These theories provide corrections to the field equations that modify the evolution of gravitational collapse with respect to the usual General Relativity results. In this context, viable $f(R)$ models must provide similar results for the collapse times to the values obtained in General Relativity. In addition, collapse times must be much shorter than the age of the universe and long enough to allow matter cluster.
The analyzed $f(R)$ models present both important different quantitative and qualitative behaviors when compared with General Relativity collapses. In fact, all of them show a collapsing initial epoch with higher contraction than in General Relativity. This result is expected since $f(R)$ theories modify the gravitational interaction by the addition of a new scalar mediator. It is well-known that a scalar force is always attractive and can only reduce the time of gravitational collapse. This result is interesting since observations of structures at high redshift introduce some tension with the standard $\Lambda$CDM model [@clusters], and the tendency of $f(R)$ models to increase the gravitational attraction at early times can alleviate this problem.
Although this general behavior is shared by the three models analyzed throughout this investigation, they present significant differences when the modifications to the General Relativity collapse leave the linear regime. On the one hand, the $R^2$ model has a modification that increases with the density of the collapse object: $(\Delta t_c/t_{GR}) \propto \sqrt{\rho}$. The opposite behavior is found for the $R^{-1}$ model, where this modification decreases with density as $(\Delta t_c/t_{GR}) \propto \rho^{-6}$. Finally, a similar situation is reproduced numerically in the Starobinsky model. The departure from the linear collapse is able to exclude interesting parameters regions of these models that support late-time acceleration as seen in Figures 2, 4 and 6.
Another relevant question is related to the physics of stellar objects when analyzed in the $f(R)$ modified gravity theories frame. Although we cannot use straightforwardly the results of this analysis due to the fundamental role that pressure plays in the stability of these objects, we may get an idea of the importance of the corrections. Inside these objects, pressure is the same order of magnitude as density, and it is expected to introduce an important modification into star evolution and dynamics. Therefore, it is enough to take into account the typical value of the density of a neutron star, approximately $10^{-3}\, \text{GeV}^4$ to estimate if correction will be important. Although this value is 35 orders of magnitude larger than the dust density used above, the results do not change dramatically. A direct extrapolation suggests that we may expect even more negligible modifications for $f(R)$ models that present dark energy scenarios (models 2 and 3). In models with higher powers of the scalar curvature, the correction to General Relativity will be more important but still negligible (as for model 1).
[**Acknowledgments.**]{} This work has been supported by MICINN (Spain) project number FPA 2008-00592 and Consolider-Ingenio MULTIDARK CSD2009-00064. AdlCD also acknowledges financial support from NRF and URC research fellowships (South Africa) and kind hospitality of UCM, Madrid while elaborating part of the manuscript.
[99]{}
S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, (1972).
L. Covi, J. E. Kim and L. Roszkowski, Phys. Rev. Lett. [**82**]{}, 4180 (1999); J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. D [**68**]{}, 085018 (2003); J. L. Feng, A. Rajaraman and F. Takayama, Int. J. Mod. Phys. D [**13**]{}, 2355 (2004); J. A. R. Cembranos, J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. Lett. [**95**]{}, 181301 (2005); J. A. R. Cembranos, J. L. Feng, L E. Strigari, Phys. Rev. D [**75**]{}, 036004 (2007); J. A. R. Cembranos, J. H. Montes de Oca Y., L. Prado, J. Phys. Conf. Ser. [**315**]{}, 012012 (2011); J. A. R. Cembranos, J. L. Diaz-Cruz and L. Prado, Phys. Rev. D [**84**]{}, 083522 (2011). H. Goldberg, Phys. Rev. Lett. [**50**]{}, 1419 (1983); J. R. Ellis [*et al.*]{}, Nucl. Phys. B [**238**]{}, 453 (1984); K. Griest and M. Kamionkowski, Phys. Rep. **333**, 167 (2000); J. A. R. Cembranos, A. Dobado and A. L. Maroto, Phys. Rev. Lett. [**90**]{}, 241301 (2003); Phys. Rev. D [**68**]{}, 103505 (2003); AIP Conf. Proc. [**670**]{}, 235 (2003); Phys. Rev. D [**73**]{}, 035008 (2006); Phys. Rev. D [**73**]{}, 057303 (2006); Int. J. Mod. Phys. [**D13**]{}, 2275 (2004); A. L. Maroto, Phys. Rev. D [**69**]{}, 043509 (2004); Phys. Rev. D [**69**]{}, 101304 (2004); A. Dobado and A. L. Maroto, Nucl. Phys. B **592**, 203 (2001); J. A. R. Cembranos [*et al.*]{}, JCAP [**0810**]{}, 039 (2008). J. A. R. Cembranos and L. E. Strigari, Phys. Rev. D [**77**]{}, 123519 (2008); J. A. R. Cembranos, J. L. Feng and L. E. Strigari, Phys. Rev. Lett. [**99**]{}, 191301 (2007); J. A. R. Cembranos [*et al.*]{}, Phys. Rev. D [**83**]{}, 083507 (2011); J. Phys. Conf. Ser. [**314**]{}, 012063 (2011); AIP Conf. Proc. [**1343**]{}, 595 (2011); arXiv:1111.4448 \[astro-ph.CO\]. J. Alcaraz [*et al.*]{}, Phys. Rev.[**D67**]{}, 075010 (2003); P. Achard [*et al.*]{}, Phys. Lett. [**B597**]{}, 145 (2004); J. A. R. Cembranos, A. Rajaraman and F. Takayama, Europhys. Lett. [**82**]{}, 21001 (2008); J. A. R. Cembranos, A. Dobado and A. L. Maroto, Phys. Rev. [**D65**]{} 026005 (2002); J. Phys. A [**40**]{}, 6631 (2007); Phys. Rev. [**D70**]{}, 096001 (2004); J. A. R. Cembranos [*et al.*]{}, AIP Conf. Proc. [**903**]{}, 591 (2007). S. Weinberg, Rev. Mod. Phys., [**61**]{}, 1-23, (1989); T. Biswas [*et al.*]{}, Phys. Rev. Lett. [**104**]{}, 021601 (2010); JHEP [**1010**]{}, 048 (2010); Phys. Rev. D [**82**]{}, 085028 (2010); J. A. R. Cembranos, AIP Conf. Proc. [**1182**]{}, 288 (2009); Phys. Rev. D [**73**]{}, 064029 (2006); AIP Conf. Proc. [**1343**]{}, 604 (2011); J. A. R. Cembranos, K. A. Olive, M. Peloso and J. P. Uzan, JCAP [**0907**]{}, 025 (2009); S. Nojiri and S. D. Odintsov, Int. J. Geom. Meth. Mod. Phys. [**4**]{} 115, (2007); J. Beltrán and A. L. Maroto, Phys. Rev. D [**78**]{}, 063005 (2008); JCAP 0903, 016 (2009); Phys. Rev. D [**80**]{}, 063512 (2009); Int. J. Mod. Phys. D [**18**]{}, 2243-2248 (2009).
A. Dobado and A. L. Maroto Phys. Rev. [**D52**]{}, 1895, (1995); G. Dvali, G. Gabadadze and M. Porrati, [*Phys. Lett.*]{} [**B485**]{}, 208, (2000); S. M. Carroll et al., [*Phys. Rev.*]{} [**D71**]{} 063513, (2005); J. A. R. Cembranos, Phys. Rev. [**D73**]{} 064029, (2006); S. Nojiri and S. D. Odintsov, Int. J. Geom. Meth. Mod. Phys. [**4**]{} 115, (2007); S. Tsujikawa, Lect. Notes Phys. [**800**]{} 99, (2010). A. De Felice, S. Tsujikawa, . Living Rev.Rel.13:3, (2010). A. de la Cruz-Dombriz, A. Dobado, Phys. Rev. [**D74**]{} 087501, (2006). T. P. Sotiriou, Gen. Rel. Grav. [**38**]{} 1407, (2006); O. Mena, J. Santiago and J. Weller, Phys. Rev. Lett. [**96**]{} 041103, (2006); V. Faraoni, Phys. Rev. [**D74**]{} 023529, (2006); S. Nojiri and S. D. Odintsov, Phys. Rev. [**D74**]{} 086005, (2006); A. de la Cruz-Dombriz and D. Saez-Gomez, arXiv:1112.4481 \[gr-qc\]; I. Sawicki and W. Hu, Phys. Rev. [**D75**]{} 127502, (2007). A. de la Cruz Dombriz, [*Some cosmological and astrophysical aspects of modified gravity theories*]{}, PhD. thesis (2010), \[arXiv:1004.5052 \[gr-qc\]\]. ISBN 978-84-693-7628-7; P. K. S. Dunsby, E. Elizalde, R. Goswami, S. Odintsov and D. S. Gomez, Phys. Rev. D [**82**]{}, 023519 (2010); A. M. Nzioki, S. Carloni, R. Goswami and P. K. S. Dunsby, Phys. Rev. D [**81**]{}, 084028 (2010); N. Goheer, J. Larena and P. K. S. Dunsby, Phys. Rev. D [**80**]{}, 061301 (2009). A. de la Cruz-Dombriz, A. Dobado and A. L. Maroto, Phys. Rev. D [**80**]{} 124011, (2009) \[Erratum-ibid. D [**83**]{} 029903, (2011)\], J. Phys. Conf. Ser. [**229**]{}, 012033 (2010); J. A. R. Cembranos, A. de la Cruz-Dombriz and P. J. Romero, arXiv:1109.4519 \[gr-qc\]. A. G. Riess et al. \[Supernova Search Team Collaboration\], Astron. J. 116, 1009, (1998); S. Perlmutter et al.\[Supernova Cosmology Project Collaboration\], Astro- phys. J. 517, 565, (1999).
D. J. Eisenstein et al. Astrophys. J. 633: 560-574, (2005).
D. N. Spergel et al. \[WMAP Collaboration\], Astrophys. J. Suppl. 170 377, (2007).
E. Linder. Phys. Rev. D 72 : 043529, (2005).
A. de la Cruz-Dombriz, A. Dobado, A. L. Maroto, Phys. Rev. D[**77** ]{} 123515, (2008); A. Abebe [*et al.*]{}, arXiv:1110.1191 \[gr-qc\]; S. Carloni, P. K. S. Dunsby and A. Troisi, Phys. Rev. D [**77**]{}, 024024 (2008). A. de la Cruz-Dombriz, A. Dobado and A. L. Maroto, Phys. Rev. Lett. [**103**]{} 179001 (2009). M. Sharif and H. R. Kausar, Astrophys. Space Sci. [**331**]{} 281, (2011). K. Bamba, S. Nojiri and S. D. Odintsov, Phys. Lett. [**B698** ]{} 451-456, (2011). E. V. Arbuzova and A. D. Dolgov, Phys. Lett. B [**700**]{} 289, (2011). E. Santos, \[arXiv:1104.2140 \[gr-qc\]\].
D. -i. Hwang, B. -H. Lee and D. -h. Yeom, JCAP [**1112**]{} 006 (2011). A. A. Starobinsky, Phys. Lett. B [**91**]{} 99, (1980).
J. A. R. Cembranos, Phys. Rev. Lett. [**102**]{}, 141301 (2009); J. Phys. Conf. Ser. [**315**]{}, 012004 (2011). C. P. L. Berry and J. R. Gair, Phys. Rev. D [**83**]{} 104022 (2011). S. M. Carroll, V. Duvvuri, M. Trodden, M. S. Turner, Phys. Rev. D [**70** ]{} 043528, (2004). A. A. Starobinsky, JETP Lett. [**86**]{} 157, (2007). http://lambda.gsfc.nasa.gov/product/map/current\
/params/olcdm\_sz\_lens\_wmap7\_bao\_h0.cfm
R. Foley [*et al.*]{}, ApJ, 731, 86, (2011); M. Brodwin [*et al.*]{}, ApJ, 721, 90, (2010); M. Jee [*et al.*]{}, ApJ, 704, 672, (2009); M. Baldi, V. Pettorino, MNRAS, 412, L1, (2011); M. J. Mortonson, W. Hu, D. Huterer, Phys. Rev. D [**83**]{}, 023015, (2011).
[^1]: E-mail: cembranos@physics.umn.edu
[^2]: E-mail: alvaro.delacruzdombriz@uct.ac.za
[^3]: E-mail: barbara.montes@ciemat.es
[^4]: In the present work we employ the natural units system in which $\hbar=c=1$. Note also that our definition for the Riemann tensor is $R_{\mu\nu\kappa}^{\sigma}=\partial_{\kappa}\Gamma_{\mu\nu}^{\sigma}-\partial_{\nu}\Gamma_{\mu\kappa}^{\sigma}+
\Gamma_{\kappa\lambda}^{\sigma}\Gamma_{\mu\nu}^{\lambda}-\Gamma_{\nu\lambda}^{\sigma}\Gamma_{\mu\kappa}^{\lambda}$.
[^5]: This fact is also satisfied in the GR case and allows the simplification of the calculus.
[^6]: It is important to remark that $R_{0}$ in this context is a parameter of the model and not the initial scalar curvature.
|
---
author:
- |
Gonzalo E. Reyes\
Département de mathématiques\
Antoine Royer\
École Polytechnique\
Université de Montréal
date: February 2003
title: On the law of motion in Special Relativity
---
\[theorem\][Proposition]{} \[theorem\][Lemma]{} \[theorem\][Remark]{} \[theorem\][Corollary]{} \[theorem\][Definition]{} \[subsection\]
\#1 =bbold12 =bbold8 \#1 \#1 **\#1**
Abstract {#abstract .unnumbered}
========
Newton’s law of motion for a particle of mass $m$ subject to a force $\bf f$ acting at time $t$ may be formulated either as $$\bf f=d/dt\; (m\bf u(t))$$ or, since $m=m_0$ is a constant, as $$\bf f=m\bf a(t)$$ where $\bf u(t)$ and $\bf a(t)$ are the velocity and the acceleration, respectively, of the particle at time $t$ relative to an inertial frame $S,\;$ ‘the laboratory’. This law may be interpreted in either of two ways:
1. The force **f acting on the particle at time $t$ during an infinitesimal time $\delta t$ imparts to the laboratory a boost $ \delta \bf u=(1/m)\bf f \delta t,$ while the particle maintains the velocity $\bf u(t)$ relative to the new frame $S'$.**
2. The force **f acting on the particle at time $t$ during an infinitesimal time $\delta t$ imparts to the particle a boost $\delta\bf w=(1/m)\bf f \delta t$ relative to its proper frame $S_0$ which moves with velocity $\bf u(t)$ relative to $S$.**
We show that the relativistic law of motion admits both interpretations, the first of which is in fact equivalent to the law of motion. As a consequence, we show that the relativistic law of motion may also be reformulated as $$\bf f=m\bf a$$ in analogy with Newton’s law, but with a [*relativistic*]{} mass and a [*relativistic*]{} acceleration defined in terms of the [*relativistic*]{} addition law of velocities, rather than ordinary mass and ordinary vectorial addition of velocities that lead to the classical acceleration and to Newton’s law.
Introduction {#introduction .unnumbered}
============
It is well-known that the Special Theory of Relativity is based on two postulates: the principle of relativity and the constancy of the speed of light. From these postulates, one can deduce the Lorentz transformations that connect the space-time coordinates of a particular event in two inertial frames. From these transformations, in turn, lenght contraction, time dilation, the ‘addition’ formula for velocities, etc., follow straightforwardly (see e.g [@ein], [@mo], [@pau], [@rin]). Thus, these postulates are sufficient for the development of the theory as far as [*kinematics*]{} is concerned.
On the other hand, this is not so for [*dynamics*]{}. In the earlier papers, the law of motion was obtained from electrodynamical considerations. After a tortuous path, Planck and independently Tolman, finally arrived at the familiar formulation of today connecting force, mass and velocity at a given time: $$\left \{ \begin{array}{c}
\bf f=d/dt(m\bf u(t))\\
m=m_0\gamma (u)\\
\gamma (u)=1/\sqrt{(1-u^2/c^2)}
\end{array}
\right.$$ The time $t$, the mass $m$, the force **f and the velocity $u(t)$ are relative to an inertial frame $S$. (See [@dug], [@ein], [@aim] and [@pau] for historical references).**
In his lecture on ‘Space and Time’ (see [@ein]), delivered in 1908 and published a year later, Minkowski introduced an [*invariant*]{} reformulation of the usual law of motion: $$\bf F=m_0\bf A$$ where $\bf F$ is the 4-force acting on the particle and $\bf A$ the 4-acceleration.
It should be noticed, however, that $\bf F$ is a 4-force and thus differs from the 3-force $\bf f$. Even the spatial component $(F_1, F_2, F_3)$ of Minkowski 4-force differs from the previously introduced 3-forces. In fact $$(F_1, F_2, F_3)={\gamma}(u) \bf f$$ The corresponding spatial component of the 4-acceleration is therefore $$(A_1, A_2, A_3) ={\gamma}(u)d/dt({\gamma}(u)\bf u)$$ The time components of the Minkowski force and the acceleration are given by $ F_4=(\gamma (u)/c) \bf f.\bf u$ and $ A_4=c\gamma (u)d/dt(\gamma (u)).$ The equation $ F_4=m_0 A_4$ follows from the corresponding equation for the spatial component and is in fact equivalent to $\bf f.d\bf r=c^2dm,$ as shown by an easy calculation.
The arguments generally given in favor of the usual law of motion are (see [@rin2]):
1. it reduces to the law of Newton $\bf f= m\bf a$ in the classical limit
2. it leads to the conservation of momentum in simple collisions, provided that we assume the law of equality of action and reaction at contact
3. it leads to $\delta E=c^2\delta m,$ the infinitesimal version of Einstein’s famous law $E=mc^2$ by adopting the classical definition of work as $force\times distance,$ as we will see later on.
4. it is consistent with the well-established Lorentz law of force in electrodynamics
These are strong arguments not only in favor of the usual law of motion, but (iii) and (iv) provide good reasons to consider also the 3-force (along with the Minkowski 4-force ).
The aim of this note is to show that this law of motion admits the two interpretations of Newton’s law stated in the Abstract. To formulate them in a concise manner, we shall use the diagram
as a shorthand to describe the following situation: the particle $P$ moves with velocity $\bf u'$ relative to $S',$ which itself moves with velocity $\bf v$ relative to $S$ and $\bf u$ is the velocity of the particle relative to $S.$ It will be particularly handy when we deal with relativity and Lorentz transformations.
To simplify the notation, we shall use either $\partial_t$ or $\dot{(...)}$ for $d/dt$. i.e., the derivative with respect to time.
Newtonian Dynamics {#newtonian-dynamics .unnumbered}
==================
Newton’s law of motion for a particle of mass $m$ subject to a force $\bf f$ acting at time $t$ is $$\bf f=\partial_t (m\bf u(t))$$ where $\bf u(t)$ is the velocity of the particle at time $t$ relative to an inertial system $S,$ to be referred to as ‘the laboratory’.
This law can be rewritten as $$\bf u(t+\delta t)=\bf u(t)+(1/m)\bf f\delta t$$ by letting $\delta t$ an infinitesimal time increment and using $$\bf u(t+\delta t)=\bf u(t)+ \dot{\bf u}(t)\delta t$$ Now, the point is that this law may be interpreted in either of two equivalent ways:
1. The force **f acting on the particle at time $t$ during an infinitesimal lapse of time $\delta t$ imparts to the laboratory $S$ , relative to which the particle has velocity **u(t), a boost $ \delta \bf u=(1/m)\bf f \delta t$ while the particle maintains the velocity $\bf u(t)$ relative to the new frame $S'$. Thus, the new velocity of the particle relative to $S$ at time $t+\delta t$ is $$\bf u(t+\delta t)=(1/m)\bf f \delta t + \bf u(t)$$ Using our diagram,****
2. The force **f acting on the particle at time $t$ during an infinitesimal lapse of time $\delta t$ imparts to the particle a boost $\delta\bf w=(1/m)\bf f \delta t$ relative to its proper frame $S_0$ which moves with velocity $\bf u(t)$ relative to $S$. Then the new velocity of the particle relative to $S$ at time $t+\delta t$ is $$\bf u(t+\delta t)=\bf u(t)+(1/m)\bf f\delta t$$ Diagrammatically,**
Relativistic Dynamics {#relativistic-dynamics .unnumbered}
=====================
As we mentioned already, in Special Relativity Newton’s law of motion is replaced by $$\left \{ \begin{array}{c}
\bf f=\partial_t(m\bf u(t))\\
m=m_0\gamma (u)
\end{array}
\right.$$ From now on, we choose a system of unities such that $c=1$. Thus $\gamma (u)=1/\sqrt{1-u^2}$.
We wish here to point out a suggestive interpretation of the law of motion which does not seem to have been noticed before. In fact, we wish to point out that this law admits precisely the two previous interpretations, [*provided*]{} that we take care to refer all quantities to the corresponding frames, since force, mass and time are frame-dependent, contrary to the classical case, and take into account the fact that relativistic composition of velocities $\oplus$ is [*not commutative*]{}. Unlike the Newtonian case, however, only the first interpretation is equivalent to the law of motion. The second, although a consequence of this law, does not seem to be equivalent to it.
Thus,
1. The force **f acting on the particle at time $t$ during an infinitesimal lapse of time $\delta t$ imparts to the laboratory $S$ , relative to which the particle has velocity **u(t), a boost $ \delta \bf u=(1/m)\bf f \delta t$ (where **f, $\delta t$ and m [*are measured in $S$*]{}), while the particle maintains the velocity $\bf u(t)$ relative to the new frame $S'$. Thus, the new velocity of the particle relative to $S$ at time $t+\delta t$ is $$\bf u(t+\delta t)=(1/m)\bf f\delta t\oplus \bf u(t)$$ Using our diagram once again, we obtain******
2. The force **f acting on the particle at time $t$ during an infinitesimal lapse of time $\delta t$ imparts to the particle a boost $\delta w_0=(1/m_0)\bf f_0(\delta t)_0$ relative to $S_0,$ its rest frame which moves with velocity $\bf u(t)$ relative to $S$. (Here **f, $(\delta t)_0$ and $m_0$ are measured in $S_0$). Then the new velocity of the particle relative to $S$ at time $t+\delta t$ (measured in $S$) is $$\bf u(t+\delta t)=\bf u(t)\oplus (1/m_0)\bf f_0(\delta t)_0$$ Diagrammatically,****
To prove our claim, we use some basic facts about relativistic kinematics that can be found in the Appendix.
When the velocity $ \bf v$ is infinitesimal, $\bf v=\delta \bf w,$ we get the infinitesimal Lorentz transformation $$\begin{array}{lll}
\delta \bf v\oplus \bf u &=& (\bf u+\delta \bf v)/(1+\delta \bf v.\bf u) \\
&=& (\bf u+\delta \bf v)(1-\delta \bf v.\bf u) \\
&=& \bf u+\delta \bf v-\bf u(\bf u.\delta \bf v))
\end{array}$$ since $\gamma(\delta \bf v)=1$
Then $$\bf u.\partial_t\gamma u = \bf u.(\dot{\gamma}\bf u+\gamma \dot{\bf u})
= \dot{\gamma}u^2+\gamma \bf u.\dot{\bf u}
= \dot{\gamma}u^2+\dot{\gamma}/\gamma^2
= \dot{\gamma} \;\;\;\;\;(*)$$ since $\dot{\gamma}=\gamma^3\bf u.\dot{\bf u}.$
Assuming the law of motion $\bf f=m_0\partial_t(\gamma \bf u)$, $$\delta w = \delta t\bf f/m
=\delta t\gamma^{-1}\partial_t\gamma \bf u
= \dot{\bf u}\delta t+\delta t(\dot{\gamma}/\gamma)\bf u$$ and $$\bf u.\delta w = \bf u.\delta t\gamma^{-1}\partial_t\gamma \bf u
=\delta t\gamma^{-1}\bf u.\partial_t\gamma \bf u
= \delta t(\dot{\gamma}/\gamma)$$ Thus, $$\begin{array}{lll}
\delta \bf w\oplus \bf u&=& \bf u+\delta \bf w -\bf u(\bf u.\delta w) \\
&=& \bf u+\delta \bf w -\bf u\delta t(\dot{\gamma}/\gamma) \\
&=& \bf u+\dot{\bf u}\delta t \\
&=& \bf u (t+\delta t)
\end{array}$$ We have obtained thus the first interpretation.
Notice that from (\*) we also have $$\bf u.\partial_t(m\bf u)= m_0\dot{\gamma}
= \dot{m}$$ so that $$0= \delta t\bf u.(\bf f-\partial_tm\bf u)
= \delta E-\delta m$$ the infinitesimal version of Einstein’s law. (Recall that $c=1$).
To obtain the second interpretation from the relativistic law of motion, we define left and right accelerations by the formulas $$\left \{ \begin{array}{c}
\bf a_l\delta t \oplus \bf u(t)=\bf u(t+\delta t)\\
\bf u(t)\oplus \bf a_r\delta t=\bf u(t+\delta t)
\end{array}
\right.$$
From these equations we obtain unique solutions $$\left \{ \begin{array}{l}
\bf a_l=\bf a+\bf u\gamma^2( \bf a.\bf u) \\
\bf a_r=\gamma \bf a+\bf u(1/\bf u^2)(\gamma (\gamma -1)\;\bf a.\bf u)
\end{array}
\right.$$ Furthermore, it is easily checked that $$\left \{ \begin{array}{l}
\bf a_l=(1/\gamma)[\bf a_r+\bf u(1/\bf u^2)\bf a_r.\bf u(\gamma -1)] \\
\bf a_r=\gamma(\bf a_l+\bf u[(1/\bf u)^2\bf a_l.\bf u(\gamma -1)-\bf a_l.\bf u\gamma])
\end{array}
\right.$$ Now, $(\delta t)_0=(1/\gamma)\delta t$ (time dilation) and thus, $$(1/\gamma m_0)\bf f_0=\bf a_r\;\; iff \;\;\bf f_0=\gamma \{\bf f+\bf u[(1/\bf u)^2(\bf f.\bf u (\gamma -1))-\bf f.\bf u\gamma]\}$$ But from the law of motion $\bf f=\partial_t(m\bf u)$ (and $m=m_0\gamma$), it follows easily that $\bf f=m\bf a_l$ which in turn, implies the right hand side. But the right hand side is true, since it is precisely the Lorentz transformation of the force, as deduced from the law of motion (see the Appendix)
To show that, conversely, the first interpretation (together with $m=m_0\gamma$) implies the relativistic law of motion, notice that from the first interpretation, $(1/m)\bf f\delta t=\bf a_l\delta t,$ which in turn implies $\bf f=m\bf a_l, $ a formula which is obviously equivalent (by taking derivatives) to $\bf f=\partial_t(m\bf u(t),$ provided that $m=m_0\gamma.$
Thus, we may reformulate the relativistic law of motion as $$\bf f=m\bf a_l$$ where $m=m_0\gamma$ is the [*relativistic*]{} mass of the particle and $\bf a_l$ is [*the relativistic*]{} “left acceleration”.
The second interpretation, $(1/\gamma m_0)\bf f_0= a_r,$ can be written in a way that is analogous to Newton’s law, namely $$\bf f_0=m_0\delta \bf w_0/(\delta t)_0$$ (since $\delta \bf w_0=1/m_0\bf f_0(\delta t)_0$), but with the relativistic “acceleration” $\delta \bf w_0/(\delta t)_0=\gamma \bf a_r$. This expression has the following physical interpretation: assume that an observer attached to the frame $S_0$ moving with the particle let a test body ‘fall’ freely relative to $S$. Then $-\delta \bf w_0/(\delta t)_0$ is the acceleration at the moment of the take-off, as measured by this observer in his frame (i.e. $S_0$). In classical mechanics, this acceleration is implicit in the studies of Huygens on centrifugal force ([*De vi centrifuga*]{}). In fact, Huygens imagines a man attached to a turning wheel and holding a thread tied to a ball of lead in his hand. The thread is suddenly cut and Huygens studies the motion of the ball at the instant when the thread is cut. This “take-off” acceleration plays an important role in some historico-critical studies such as those of the ‘Ecole du fil’ of of F.Reech and J.Andrade (see [@dug]).
Appendix:Lorentz transformations and composition of velocities {#appendixlorentz-transformations-and-composition-of-velocities .unnumbered}
==============================================================
We recall the Lorentz transformation that connects the position of an event in two inertial frames $S$ and $S'$ such that $S'$ (or rather its origin) moves with uniform velocity $\bf v$ with respect to $S.$ Indeed, if $(\bf r, t)$ describes the event in $S,$ and $(\bf r', t')$ describes the same event in $S',$ then (by mapping these quantities in an independent Euclidean 3-space): $$\left \{ \begin{array}{c}
\bf r'=\bf r+\bf v\gamma (v)\{(1/c^2)\gamma (v)/(1+\gamma (v)) \bf v.\bf r- t\} \\
t'=\gamma (v) (t-\bf v.\bf r/c^2)
\end{array}
\right.$$ Similarly, if a particle moves with velocity **u with respect to $S$ and velocity $\bf u'$ with respect to $S',$ then $$\bf u'= \{\bf u/\gamma (v) +\bf v [(1/c^2)\gamma (v)/ (1+\gamma (v)) \bf u.\bf v-1]\}/(1-\bf u.\bf v/c^2)$$ Finally, if the particle is subject to a force **f relative to $S,$ then relative to $S'$ the force is given by $$\bf f'=\{\bf f/\gamma (v)+\bf v[(1/c^2)\gamma (v)/(1+\gamma (v)) \bf f.\bf v-\bf f.\bf u/c^2]\}/(1-\bf u.\bf v/c^2)$$ These formulas may be found in [@rin] (pages 23, 40 and 97), although the expression $(1-1/\gamma (v))/v^2$ has been replaced by $(1/c^2)\gamma (v)/(1+\gamma (v))$ used by Ungar [@un]. This last formula makes sense for every $\bf v$ such that $v<c.$****
The Lorentz transformation of velocities may be expressed in terms of composition or addition of velocities.
In fact, let $V=\{\vec{v}\in {\bb{R}}^3|v^2<c^2\}.$ Define an operation on the open domain $V\times V$ of ${\bb{R}}^3\times {\bb{R}}^3$ with values in $V$ by the formula $$\bf v\oplus \bf u=\{\bf u/\gamma (v)+\bf v[(1/v^2)(1-1/\gamma (v))\bf u.\bf v +1]\}/(1+\bf u.\bf v/c^2)$$ Although this operation is neither commutative nor associative, it is [*gyrocommutative*]{} and [*gyroassociative*]{}. (See [@un] for these notions as well as for further properties of this operation).
In terms of this operation, we can express the Lorentz transformation of velocities of the beginning of this Appendix as $$\bf u'=(-\bf v)\oplus \bf u$$ or, equivalently, as $$\bf u=\bf v\oplus \bf u'$$ Notice that even if $\bf u$ and $\bf v$ are velocities, $\bf v\oplus \bf u$ is not interpretable as a velocity. To make contact with actual composition of velocities we need to devise physical set-ups to realize each vector as a velocity (relative to a suitable frame) and the operation as actual composition of velocities according to our diagram
Acknowledgments {#acknowledgments .unnumbered}
===============
The first author owes a great debt of gratitude to Nicanor Parra. He formulated clearly the problem of the status of the 3-force and the law of motion in the Theory of Special Relativity and told him about the take-off acceleration in classical mechanics, suggesting that it could be used in Special Relativity. He had countless conversations on and off on ‘natural and supernatural’ mechanics, since he followed Parra’s course in 1957 at the Instituto Pedagógico of the Universidad de Chile in Santiago. His encouragement is greatly appreciated. He is also in debt to Jorge Krausse. Besides discussions of a general nature, he helped him to find his way in the literature on Relativity Theory.
[99]{} Dugas, R., Histoire de la Mécanique \[1950\], Neuchatel
Eintein, A. [*et al*]{}, The principle of Relativity \[1952, Republication of the 1923 translation published by Meuthen and Company, Ltd. of the 4th edition of Das Relativitätprinzip, Teubner 1922\], Dover
Miller, A.I., Albert Einstein’s Special Theory of Relativity \[1998\], Springer
M$\o$ller, C., The Theory of Relativity \[1969, Corrected sheets of the first edition\], Oxford at the Clarendon Press
Pauli, W., Theory of Relativity \[1967, Second Reprint\], Pergamon Press
Rindler, W., Special Relativity \[1965, Second edition\], Oliver and Boyd
Rindler, W., Essential Relativity \[1969\], Van Ostrand
Ungar, A.A., Thomas precession: Its Underlying Gyrogroup Axioms and Their Use in Hyperbolic Geometry and Relativistic Physics, Foundations of Physics, Vol. 27, No. 6, 1997
|
Tsemo Aristide
College Boreal, 1 Yonge Street,
M5E 1E5, Toronto, ON
Canada
tsemo58@yahoo.ca
[**Closed models, strongly connected components and Euler graphs.**]{}
**Abstract.**
[*In this paper, we continue our study of closed models defined in categories of graphs. We construct a closed model defined in the category of directed graphs which characterizes the strongly connected components. This last notion has many applications, and it plays an important role in the web search algorithm of Brin and Page, the foundation of the search engine Google. We also show that for this closed model, Euler graphs are particular examples of cofibrant objects. This enables us to interpret in this setting the classical result of Euler which states that a directed graph is Euleurian if and only if the in degree and the out degree of every of its nodes are equal. We also provide a cohomological proof of this last result.*]{}
[**1. Introduction.**]{}
In this paper, we pursue our investigation of closed models defined in the category $Gph$ of directed graphs. Recall that in \[2\] and \[3\], that we have published in collaboration with Terrence Bisson, we have introduced two closed models: the first is related to the zeta function of directed graphs and the second to dynamical systems. These constructions have been generalized in \[10\] where we have defined the notion of closed models defined by counting and study the existence of such closed models in the category of undirected graphs. For the closed model defined in \[2\], a morphism of $Gph$ $f:X\rightarrow Y$ is a weak equivalence if and only if for every cycle $c_n, n>0$, the morphism of sets $Hom(c_n,X)\rightarrow Hom(c_n,Y)$ induced by $f$ is a bijection.
In this paper, we modify this condition by allowing $n$ to be equal to zero, otherwise said, we are counting also the nodes of $X$. This new closed model defined in $Gph$ enables to study other interesting properties of this category in particular it enlightens the important notion of strongly connected component of a directed graph, which has many applications in web search engines: the well known search engine Google designed by Brin and Page \[5\] uses the notion of pagerank to construct an hierarchy of the web which can be calculated by using strongly connected components and Markov matrices. More precisely, we show that a morphism $f:X\rightarrow Y$ is a weak equivalence for this closed model if and only if it induces a bijection between the respective sets of strongly connected components of $X$ and $Y$ and its restriction to each strongly connected component of $X$ is an isomorphism onto a strongly connected component of $Y$. The cofibrant objects obtained here enable us also to study Eulerian graphs and to interpret the famous Euler theorem which states that a finite directed graph $X$ is Eulerian if and only if for every node $x$ of $X$ the inner and the outer degree of $x$ are equal. We also provide a construction of new closed models from a closed model defined by counting. This enables us to give a conceptual formulation of the closed model defined in \[2\].
We also introduce an homology theory in the category $Gph$ and show that the positive cycles of the first homology group of a directed graph is the set of cycles; this also enables us to give an homological interpretation of the Euler’s theorem that we have just quoted and to establish a link between the notions studied in this paper and simplicial sets. In this regard, we show that there exists a closed model defined in the category of $1$-simplicial sets also called the category of reflexive graphs which has many similarities which the closed model studied earlier in this paper.
[**2. Some basic properties of the category of directed graphs.**]{}
Let $C$ be the category which has two objects that we denote by $0$ and $1$; the morphisms of $C$ which are not identities are $s,t\in Hom(0,1)$.
[**Definitions 2.1.**]{}
The category $Gph$ of presheaves over $C$ is the category of directed graphs. Thus, a directed graph $X$ is defined by two sets $X(0)$ and $X(1)$, and two maps $X(s),X(t):X(1)\rightarrow X(0)$. The elements of $X(0)$ are called the nodes of $X$ and the elements of $X(1)$ the arcs of $X$. For every arc $a\in X(1)$, $X(s)(a)$ is the source of $a$ and $X(t)(a)$ is the target of $a$. We will also often say that $a$ is an arc between $X(s)(a)$ and $X(t)(a)$ or that $a$ connects $X(s)(a)$ and $X(t)(a)$.
A morphism $f:X\rightarrow Y$ between two directed graphs is a morphism of presheaves: it is defined by two maps $f(0):X(0)\rightarrow Y(0)$ and $f(1):X(1)\rightarrow Y(1)$ such that $Y(s)\circ f(1) = f(0)\circ X(s)$ and $Y(t)\circ f(1) = f(0)\circ X(t)$.
Let $X$ be a finite directed graph, suppose that the cardinality of $X(0)$ is $n$, the adjacency matrix $A_X$ of $X$ is the $n\times n$ matrix whose entry $(i,j)$ is the cardinal of $X(x_i,x_j)$, the set of arcs between $x_i$ and $x_j$.
[**Definitions 2.2.**]{}
Let $X$ be a graph, and $x$ a node of $X$. We denote by $X(x,*)$ the set of arcs of $X$ whose source is $x$, and by $X(*,x)$ the set of arcs of $X$ whose target is $x$. If $X$ is finite, the inner degree of $x$ is the cardinality of $X(*,x)$ and the outer degree of $x$ is the cardinality of $X(x,*)$.
Examples of directed graphs are:
The directed dot graph $D$; $D(0)$ is a singleton and $D(1)$ is empty. Geometrically it is represented by a point.
The directed arc $A$. The set of nodes of $A$ contains two elements $x,y$, and $A$ has a unique arc $a$ such $A(s)(a)=x$ and $A(t)(a)=y$. Geometrically, it is represented by an arc between $x$ and $y$ as follows: $x \longrightarrow y$.
The directed cycle $c_n, n\geq 1$ of length $n$; $c_n(0)$ is a set which contains $n$ elements that we denote by $x^n_0,...,x^n_{n-1}$. For $i<n-1$, there is a unique arc $a^n_i$ whose source is $x^n_i$ and whose target is $x^n_{i+1}$; there is an arc $a^n_{n-1}$ whose source is $x^n_{n-1}$ and whose target is $x^n_0$. Often, we will say that $D$ is the cycle $c_0$ of length $0$.
The directed line $L$ is the graph such that $L(0)$ is the set of integers $Z$, and for every integer $n$, there exists a unique arc $a_n$ such that $L(s)(a_n) = n$ and $L(t)(a_n) = n+1$.
The directed path $P_n$ of length $n$; the set of nodes $P_n(0)$ has $n$ elements $x^n_0,....,x^n_{n-1}$ and for $i<n-1$, there exists an arc $a^n_i$ between $x^n_i$ and $x^n_{i+1}$; $x^n_0$ is the source of the path and $x^n_{n-1}$ is its end.
[**Definitions 2.3.**]{}
Let $X$ be an object of $Gph$ and $x,y$ two nodes of $X$. A path between $x$ and $y$ is a morphism $f:P_n\rightarrow X$ such that $f(0)(x^n_0) = x$ and $f(0)(x^n_{n-1}) = y$. We say that $X$ is connected if and only if for every nodes $x$ and $y$ of $X$, there exists a finite set of nodes $(x_i)_{i=1,...,l}$ such that $x_1=x, x_l=y$ and for $i<l$, there exists a path between $x_i$ and $x_{i+1}$ or a path between $x_{i+1}$ and $x_i$.
The graph $X$ is strongly connected if and only if for every nodes $x,y$ of $X$ there exists a path between $x$ and $y$ and a path between $y$ and $x$. This is equivalent to saying that there exists a cycle which contains $x$ and $y$.
Let $X$ be a directed graph, consider the equivalent relation $R$ defined on the space of nodes of $X$ such that $x Rx$ for every $x\in X(0)$, if $x$ is distinct of $y$ then $x R y$ if and only if there exists a cycle which contains $x$ and $y$.
We denote by $U_1,...,U_p,...$ the set of equivalent classes of this relation. We denote by $X_{U_i}$ the subgraph of $X$ whose set of nodes is $U_i$. An arc $a\in X(1)$ is an arc of $X_{U_i}$ if and only if $X(s)(a)$ and $X(t)(a)$ are elements of $U_i$. The graphs $X_{U_i}$ are the strongly connected components of $X$.
[**3. Closed models in $Gph$.**]{}
We recall now the notion of closed model category:
[**Definition 3.1.**]{}
Let $C$ be a category, we say that the morphism $f:X\rightarrow Y$ has the left lifting property with respect to the morphism $g:A\rightarrow B$ (resp., $g$ has the right lifting property with respect to $f$) if and only if for each commutative square
$$\dot{}
\begin{CD}
X @> l>> A\\
@VV f V @VV g V\\
Y @>m>> B
\end{CD}$$ there exists a morphism $n:Y\rightarrow A$ such that $ l = n\circ f$ and $m = g\circ n$. Let $I$ be a class of maps of $C$, we denote by $inj(I)$ the class of morphisms of $C$ such that for every $f$ in $I$ and every $g\in inj(I)$, $g$ has the right lifting property with respect to $f$. We denote $cell(I)$ the subclass of maps of $C$ which are retracts of transfinite composition of pushouts of elements $I$.
Two class of maps $L$ and $R$ define a weak factorization system $(L,R)$ of $C$ if and only if: for every morphism $f$ of $C$, there exists $g\in R$ and $h\in L$ such that $f = g\circ h$ and $L$ is the class of morphisms which have the left lifting property with respect to every morphism $R$ and $R$ is the class of morphisms which have the right lifting property with respect to every morphism of $L$.
[**Definition 3.2.**]{}
A closed model category is a category $M$ which has projective limits and inductive limits endowed with three subclasses of morphisms $W,F,C$ called respectively the weak equivalences, the fibrations and the cofibrations. We denote by $F'$ (resp., $C'$) the intersection $F\cap W$ (resp., $C\cap W$). The subclass $F'$ is called the class of weak fibrations and $C'$ the class of weak cofibrations. The following two axioms are also satisfied:
M1. $(C,F')$ and $(C',F)$ are weak factorization systems.
M2 Let $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ be two maps in $M$, if two maps of the triple $\{ f,g,g\circ f\}$ is a weak equivalence so is the third.
In this paper, we are only going to consider locally presentable categories. This has the virtue to avoid set theoretical difficulties when one tries to find weak factorizations systems. We are going to use Proposition 1.3 of Beke \[1\] which asserts that if $I$ is a class of morphisms of a locally presentable category, $(cell(I), inj(I))$ is a weak factorization system. The categories of graphs used here are locally presentable categories since they are isomorphic to categories of presheaves defined on a small category.
[**Definition 3.3.**]{}
A closed model structure defined on $C$ is cofibrantly generated if and only if there exists a set of morphisms $I$ (resp., $J$) such that $inj(I)$ (resp., $inj(J)$) is the class of weak fibrations (resp., the class of fibrations).
In \[10\] we have introduced the notion of a closed model category defined by counting which we outline: it is a closed model category $C$, whose class of weak equivalences $W$ is defined as follows:
Firstly, we consider a set of objects of $C$, $(X_l)_{l\in L}$. Let $\phi$ be the initial object of $C$, we can define the morphisms $i_l:\phi\rightarrow X_l$ and the folding morphism $j_l:X_l+X_l\rightarrow X_l$ which is the sum of two copies of $Id_{X_l}:X_l\rightarrow X_l$. The class $W$ is $inj(I)$ where $I=\{i_j,j_l; {l\in L}\}$. Thus a morphism $f:X\rightarrow Y$ is a weak equivalence if and only if for every $l\in L$, the map $Hom(X_l,X)\rightarrow Hom(X_l,Y)$ which sends $g:X_l\rightarrow X$ to $f\circ g$ is bijective, and $(cell(I),W)$ is a weak factorization system. We can define a closed model on $C$ whose class of weak equivalences is $W$, the class of fibrations is the class of morphisms of $C$ and the class of cofibrations is $cell(I)$. Remark that such a closed model is cofibrantly generated since its class of fibrations is $inj(\phi)$ where $\phi$ is the initial object.
[**Proposition 3.1.**]{}
[*Let $W$ be the class of weak equivalences of the closed model defined by counting the objects $(X_l)_{l\in L}$, $J$ a set of morphisms $(f_j)_{j\in P}$ such that $cell(J) \subset W$. Denote by $F$ the class $inj(J)$ and by $Cof$ the class of morphisms $cell(I\bigcup J)$. Then $(W,F,Cof)$ defines a closed cofibrantly generated closed model on $C$.*]{}
[**Proof.**]{}
We are going to apply the result of D. Kan quoted by Hirschhorn \[9\] p. 213, Theorem 11.3.1 that shows that the sets of morphisms $I\bigcup J$ and $J$ define a cofibrantly generated closed model on $C$ where $I=\{i_l,j_l; {l\in L}\}$.
A morphism $f$ of $cell(J)$ is an element of $W$ by assumption, and is obviously contained in $cell(I\bigcup J)$. A morphism $f$ of $C$ which is right orthogonal to $I\bigcup J$ is a weak equivalence since it is right orthogonal to $I$ and is obviously right orthogonal to $J$. This verifies the conditions 2 and 3 of the theorem of Kan. A morphism $f$ which is right orthogonal to $J$ and is in $W$ is a morphism right orthogonal to $I\bigcup J$. This verifies the condition $4 (b)$.
[**Examples.**]{}
We present now the following closed model defined by counting the cycle graphs $(c_n)_{n>0}$ in the category $Gph$. A morphism $f:X\rightarrow Y$ is contained in the class $W'$ of weak equivalences of this closed model if for every $n>0$, the map $Hom(c_n,X)\rightarrow Hom(c_n,Y)$ is bijective. We have a closed model $(W',Fib',Cof')$ for which $Fib'$ is the class of all the maps and $Cof'$ is $cell(i_n,j_n,n>0)$, where $i_n:\phi\rightarrow c_n$ and $j_n:c_n+c_n\rightarrow c_n$.
We can apply the Proposition 3.1, to obtain other closed models with the same class of weak equivalences. On this purpose, consider a non empty graph $X$ such that for every integer $n>0$, $Hom(c_n,X)$ is empty. Such a graph is called acyclic. Let $x$ be any node of $X$, consider the morphism $s^x:D\rightarrow X$ such that the image of $s^x(0)$ is $x$. An element of $cell(s^x)$ is a composition of morphisms $f:Y\rightarrow Z$, where $f$ is the canonical embedding of $Y$ into a graph $Z$ obtained by attaching an acyclic graph to a node of $Y$. See also \[4\] Proposition 4. We deduce that the class $cell(s^x)$ is contained in $W'$. We can thus apply the Proposition 3.1 to obtain the closed model $(W',F_X,Cof_X)$ such that $F_X$ is $inj(s^x)$, and $Cof_X = cell(i_n,j_n,s^x,n>0)$. In particular, if $s:D\rightarrow A$ is the morphism between the dot graph and the arc graph such that $s(0)$ is the source of $A$, we obtain the closed model presented in \[2\] for which the class of fibrations is $inj(s)$ and the cofibrations are $cell(i_n,j_n,s)$. Other examples may rise some interest. We can define $t:D\rightarrow A$ such that the image of $t(0)$ is the target of $A$ and obtain a closed model whose weak equivalences are $W'$, the class of fibrations is $inj(t)$ and the cofibrations are $cell(i_n,j_n,t,n>0)$. We can also defined the closed model whose weak equivalences are $W'$, the class of fibrations is $inj(s,t)$ and the cofibrations are $cell(i_n,j_n,s,t)$.
[**4. Closed models and strongly connected components.**]{}
One of the main purposes of this paper is to study a closed model defined by counting on $Gph$ related to $(W',Fib',Cof')$. This time, we count the cycles $(c_n)_{n\geq 0}$. That is, we are also counting nodes. Thus a weak equivalence $W$ for this closed model is a morphism $f:X\rightarrow Y$ such that for every $n\geq 0$, the map $Hom(c_n,X)\rightarrow Hom(c_n,Y)$ which associates $f\circ g$ to each element $g\in Hom(c_n,X)$ is bijective. We obtain a closed model $(W,Fib,Cof)$ for which $Fib$ is the class of all the morphisms of $Gph$ and $Cof$ is $cell(i_n,j_n, n\geq 0)$. This closed model is related to strongly connected components of directed graphs, a notion which is intensively used in computer science and in particular in web search as shows the work of Brin and Page \[5\], the conceptual foundation of the search engine Google. Given a network (a directed graph), it is important for a web search engine to recommend pages to an user, on this purpose, a weight is assigned to each page (vertex) called the pagerank which depends on the number of important links that the page receives (the weight of the source of the incoming arcs). If $A$ is the adjacency matrix of the network, to obtain the pagerank, one has to define a new matrice $P$ by replacing the non zero coefficients of $A$ by numbers which quantify the importance of the link, and the pagerank of the page $i$ is just the sum of the entries of the $i$-row of $P$. It is also reasonable to normalize the columns of the matrix $P$ to minimize the importance of outgoing links from a page, so surfing online is assimilated to a random walk described by the Markov matrix $P$. Linear algebra shows thus the pagerank is an eigenvalue of $P$. If $P$ is irreducible, the Perron theorem shows the existence of a unique maximal positive eigenvalue which defines the pagerank. The fact that $P$ is irreducible means also that the graph is strongly connected. In practice this is not true, but research shows that 90 percent of the world wide web is connected and contains a giant strongly connected component. To cope of the general situation, google uses transition probabilities.
We have the following result:
[**Theorem 4.1.**]{}
[*A morphism $f:X\rightarrow Y$ of $Gph$ is an element of $W$ if and only if it induces a bijection between the sets of strongly connected components of $X$ and $Y$ and the restriction of $f$ to a strongly component of $X$ is an isomorphism onto a strongly connected component of $Y$.*]{}
[**Proof.**]{}
Firstly, we show that the image of a strongly connected component $U$ of $X$ is a strongly connected component. The restriction $f_{\mid U}$ of $f$ to $U$ is injective on nodes, since $f$ induces a bijection on the set of nodes. Let $a$ and $b$ be two arcs of $U$ such that $f(1)(a) = f(1)(b)$. Since $f$ is injective on nodes, $s(a) = s(b)$ and $t(a) = t(b)$. Consider a path $p$ in $U$ between $t(a)$ and $s(a)$. We can construct two cycles $c$ and $c'$ obtained respectively by the concatenation of $a$ and $p$ and the concatenation of $b$ and $p$ The images of $c$ and $c'$ by $f$ coincide. This implies that $c=c'$ since $f$ is injective on cycles, thus $a=b$. The image of $U$ is thus imbedded in a strongly connected component $V$ of $Y$.
Suppose that there exists a node $y$ in $V$ which is not in the image of $U$. Let $y'=f(0)(x)$, $x\in U$. Since $V$ is strongly connected, there exists a cycle $c$ of $V$ whose set of nodes contains $y$ and $y'$. Consider the cycle $c'$ of $X$ whose image by $f$ is $c$; $c'$ contains $x$ since $f(0)$ is injective. This implies that $c'$ is in $U$, and $c$ is contained in the image of $U$. This is a contradiction with the fact that $y$ is not in the image of $U$. Consider an arc $b$ of $V$ which is not in the image of $U$. There exists a cycle $c$ of $V$ that contains $b$. Since $f$ induces a bijection on cycles, there exists a cycle $c'$ of $X$ whose image by $f$ is $c$. Let $a$ be the arc of $c'$ whose image by $f$ is $b$; $s(a)$ and $t(a)$ are contained in $U$ since their image are contained in $f(0)(U)$. This implies that $a$ is in $U$ since $U$ is a strongly connected component and henceforth $b$ is in the image of $U$. Thus the restriction of $f$ to $U$ is surjective on arcs. Since the restriction of $f$ to $U$ is injective, we deduce that $f$ induces an isomorphism of $U$ onto its image $V$.
Let $V$ be a strongly connected component of $Y$, and $y$ a node of $V$. There exists a node $x\in X(0)$ such that $f(0)(x) = y$. The image of the strongly connected component which contains $x$ is $V$. This implies that $f$ induces a bijection on strongly connected components.
Conversely, suppose that $f$ induces a bijection between the set of on strongly connected components of $X$ and $Y$ and the restriction of $f$ to a strongly connected component of $X$ is an isomorphism. Let $c$ and $c'$ two $n$-cycles ($n$ eventually $0$) of $X$ whose image by $f$ coincide. This implies that that $c$ and $c'$ are in the same strongly connected component $U$, and are equal since the restriction of $f$ to $U$ is an imbedding. Let $c$ be a cycle of $Y$, $c$ is an element of a strongly connected component $V$. The strongly connected component $U$ of $X$ whose image maps isomorphically to $V$ contains a cycle whose image is $c$. We deduce that $f$ is a weak equivalence.
[**Corollary 4.1.**]{}
[*A morphism $f:X\rightarrow Y$ between two strongly connected directed graphs is a weak equivalence if and only if it is an isomorphism.*]{}
[**Cofibrant replacement.**]{}
We are going to study in this section the notion of cofibrant replacement for the closed model defined in this section 4 on $Gph$ by $(W,Fib=Hom(Gph),Cof)$. Recall that an object $X$ is cofibrant if and only if the map $\phi\rightarrow X$ is a cofibration where $\phi$ is the initial object. The object $Y$ is a cofibrant replacement of $X$ if and only if $Y$ is a cofibrant object and there exists a weak equivalence $f:Y\rightarrow X$. We know that $Cof = cell(i_n,j_n,n\geq 0)$. This implies that the $n$-cycles $n\geq 0$ are cofibrant. We deduce also that the sum of cycles are cofibrant objects.
[**Some cofibrant maps: Gluing nodes and paths.**]{}
Let $f:c_0\rightarrow c_m$ and $g:c_0\rightarrow c_n$ two morphisms of graphs. Consider the pushout diagram:
$$\dot{}
\begin{CD}
c_0 @> f>> c_m\\
@VV g V @VV V\\
c_n @>m>> X
\end{CD}$$
The graph $X$ is obtained by identifying a node of $c_m$ with a node of $c_n$. We say also that $X$ is obtained by attaching $c_m$ and $c_n$ by a node. The graph $X$ is cofibrant. We can iterate this operation to create more cofibrant objects: for example we can attach more cycles or identify paths as follows:
Consider the graph $X$ defined as follows: there exist two cycles $c_m$ and $c_n$, nodes $x,y$ of $c_m$ and nodes $x',y'$ of $c_n$ such that there exist a path $p_1\in c_m$ between $y$ and $x$ and a path $p_2$ in $c_n$ between $y'$ and $x'$ which have the same length. We can construct the graph $X$ obtained by attaching $c_m$ and $c_n$ by identifying $x, x'$ and $y,y'$. We denote by $[x]$ $(resp., [y])$ the node of $X$ corresponding to $x$ (resp., $y$). In $X$, we have paths $l_1,l_2$ between $[y]$ and $[x]$ and obtained respectively from $p_1$ and $p_2$ and which have the same length. There exist also another $l_3$ between $[x]$ and $[y]$ in $X$. We can construct the cycles $c = l_1l_3$ and $l_2l_3$ which have the same length $p$. Let $f:c_{p}\rightarrow X$ whose image is $l_1l_3$ and $g:c_{p}\rightarrow X$ whose image is $l_2l_3$. We can construct the pushout of $f+g:c_p+c_p\rightarrow X$ by $j_{p}:c_{p}+c_{p}\rightarrow c_{p}$. It is a morphism $h:X\rightarrow Y$ and $Y$ is obtained from $X$ by identifying $l_1$ and $l_2$. We say that $Y$ is obtained by gluing the paths $l_1$ and $l_2$.
[**Theorem 4.2.**]{}
[*A strongly connected graph is a cofibrant object.*]{}
[**Proof.**]{}
Let $X$ be a strongly connected graph. There exists a family of cycles $(c_{n_i},i\in I)$ and a morphism $f:\sum_ic_{n_i}\rightarrow X$ surjective on nodes and arcs. We can write $f = h\circ g$ where $g$ is a cofibration and $h$ a weak fibration. Write $h:Y\rightarrow X$, without restricting the generality, we can suppose that the image $Y$ of $g$ is connected. Thus $Y$ can be constructed from a cycle $c_p$ by repeating the following operations: attach a cycle to a point, identifying two nodes or two arcs. This implies that $Y$ is strongly connected. The Corollary 4.1 implies that $h$ is an isomorphism, we deduce that $f$ is a cofibration and $X$ is cofibrant.
The previous construction yields to the following:
[**Corollary 4.2.**]{}
[*Let $X$ be a directed graph, consider the subgraph $c(X)$ of $X$ which has the same nodes of $X$, an arc of $X$ is an arc of $c(X)$ if and only if it is contained in a strongly connected component of $X$, the canonical embedding $c_X:c(X)\rightarrow X$ is a cofibrant replacement of $X$.*]{}
[**Proof.**]{}
The graph $c(X)$ is the disjoint union of the strongly connected components of $X$. The Theorem 4.2 implies that $c(X)$ is a cofibrant object, and the Theorem 4.1 implies that the canonical embedding $c(X)\rightarrow X$ is a weak equivalence.
[**Application to Eulerian graphs.**]{}
We are going to apply these results to Eulerian cycles. Remark that:
[**Proposition 4.1.**]{}
[*Let $X$ be a finite strongly connected directed graph, there exists an integer $n(X)$, and a morphism $f:c_{n(X)}\rightarrow X$ surjective on arcs.*]{}
[**Proof.**]{}
We fix a node $x_0$ of $X$. We can index the arcs of $X$ by $a_1,...,a_l$. Since $X$ is strongly connected, there exists a path $p_i$ from $x_0$ to $s(a_i)$ and a path $p_i'$ from $t(a_i)$ to $x_0$ $i=1,...l$. We can construct the cycle $p'_la_lp_l...p'_ia_ip_i...p'_1a_1p_1$ which contains all the arcs of $X$.
This leads to to the following definition:
[**Definition 4.1.**]{}
An Eulerian cycle in a directed graph $X$ is a cycle $f:c_n\rightarrow X$ such that $f(1)$ is a bijection.
We have the following proposition:
[**Proposition 4.2.**]{}
[*A finite directed graph $X$ is Eulerian if and only if it is cofibrant and obtained from a cycle by identifying nodes.*]{}
[**Proof.**]{}
Let $X$ be an Eulerian graph. There exist an integer $n$ and a morphism $f:c_n\rightarrow X$ surjective on nodes and bijective on arcs; $f$ is a cofibration since it is the composition of morphisms which identify nodes and henceforth, we deduce that $X$ is cofibrant since $c_n$ is cofibrant. Conversely, a cofibrant graph $X$ obtained from a cycle $c_n$ by identifying some of its nodes is Eulerian and the canonical morphism $f:c_n\rightarrow X$ is an Eulerian cycle.
[**Proposition 4.3.**]{}
[*Consider a graph $X$ constructed recursively as follows: $X_0$ is a cycle $c_n$, to construct $X_1$, identify two nodes of $c_n$ or attach a cycle to a node of $c_n$. Suppose defined $X_n$, to obtain $X_{n+1}$, identify two nodes of $X_n$ or attach a cycle to a node of $X_n$. Each graph $X_n$ is Eulerian.*]{}
[**Proof.**]{}
The graph $X_0=c_n$ is Eulerian. Suppose that $X_n$ is Eulerian. Let $f:c_p\rightarrow X_n$ be an Eulerian cycle. If $X_{n+1}$ is obtained from $X_n$ by identifying two nodes, let $g:X_n\rightarrow X_{n+1}$ be the identifying morphism, $g\circ f$ is an Eulerian cycle of $X$. Suppose that $X_{n+1}$ is obtained from $X_n$ by attaching a cycle $c_m$. The concatenation of the cycles $f$ and $c_m$ is an Eulerian cycle of $X_{n+1}$.
[**Theorem 4.3.**]{}
[*A finite directed connected graph $X$ is obtained by the processus described in Proposition 4.3 if and only if for every node $x$ of $X$, the in and out degree of $x$ are equal.*]{}
[**Proof.**]{}
Suppose that $X$ is an Eulerian graph, then Proposition 4.2 shows that there exists a sequence of graphs $X_0=c_n,...,X_n=X$ such that $X_{i+1}$ is obtained from $X_i$ by identifying two nodes of $X_i$. The identification of two nodes of an Eulerian graph increases the in degree and the out degree of a node by the same number, we deduce that if $X_i$ is Eulerian, then $X_{i+1}$ is Eulerian. Since $c_n$ is Eulerian, we deduce recursively that $X$ is Eulerian.
Conversely, suppose that $X$ is a connected directed finite graph such that the in degree and the out degree of every node of $X$ coincide, we are going to show that $X$ is constructed by the process described at Proposition 4.3. Let $x$ be any node of $X$ and $a_0\in X(x,*)$, then $X(t(a_0),*)$ is not empty since its in degree is equal to its out degree, we consider $a_1\in X(t(a_0),*)$, if $t(a_1) = x$ we stop otherwise there exists $a_2\in X(t(a_1),*)$ by continuing this process we obtain a cycle $f_1:c_{n_1}\rightarrow X$ injective on arcs. We can consider the subgraph $X_1$ of $X$ which is the image of $f_1$; $X_1$ is obtained from $c_{n_1}$ by identifying nodes. If $X_1$ is not $X$, since $X$ is connected, we have $x_2\in X_1$ such that $X(x_2,*)$ contains an arc $a^2_1$ which is not in $X_1$, since the in degree and the out degree of $t(a^2_1)$ are equal, if $t(a^2_1)$ is distinct of $x_2$ there exists an arc $a^2_2\in X(t(a^2_2),*)$ as above, we conclude the existence of an injective morphism $f_2:c_{n_2}\rightarrow X$ whose image is a cycle through $x_2$. We can construct the subgraph of $X$ which is the union of $X_1$ and the image of $f_2$. Remark that $X_2$ is obtained from $X_1$ by attaching a cycle and identifying nodes. We can repeat the process to obtain an increasing sequence of graphs $X_1\subset X_2\subset...X_i\subset X_{i+1}\subset...$ such that $X_{i+1}$ is obtained from $X_i$ by attaching a cycle and identifying nodes of this cycle. Since $X$ is finite, we deduce the existence of $n$ such that $X_n= X$. The Theorem 4.3 shows that $X$ is Eulerian.
[**Corollary. 4.3. (Euler).**]{}
[*A finite directed graph $X$ is Eulerian if and only if for every node $x$ of $X$, the in and out degree of $x$ are equal.*]{}
[**5. Cohomological interpretation.**]{}
Let $X$ be a directed graph. We denote by $Z(X(0))$ (resp., $Z(X(1))$ the free commutative group generated by the set $X(0)$ (resp., by the arcs of $X$). The elements of $Z(X(0))$ are called the $0$-chains. A $1$-chain $u$ of $X$ is the linear sum $\sum_{i=1}^{i=l}d_if_{n_i}$ where $d_i$ is an integer and $f_{n_i}:P_{n_i}\rightarrow X$ is a morphism between the path of length $n_i$ and $X$. We denote by $Z(ch(X))$ the space of $1$-chains of $X$. To each $1$-chain $u$, we associate $u'$ the element of $Z(X(1))$ defined by $\sum_{i=1}^{i=l} d_i\sum_{m=0}^{m=n_i-1}f_{n_i}(1)(a^{n_i}_m)$, we will often call $u'$ the image of $u$.
We say that $u$ is positive if and only if $d_i\geq 0, i=1,...,l$.
The length $l_X(u)$ of $u$ is $\sum_in_i\mid d_i\mid$.
Suppose that $X$ is finite, for each arc $a\in X(1)$, we define the morphism $f_a:P_1\rightarrow X$ whose image is $a$; the fundamental chain $[X]$ of $X$ is $\sum_{a\in X(1)}f_a$.
We define the linear map $d^X_1:Z(ch(X))\rightarrow Z(X(0))$ such that for every chain $f:P_n\rightarrow X$ of $X$, $d^X_1(f) = t(f)-s(f)$. Remark that $d_1^X(f) =\sum_{i=0}^{i=n-1} t(f(1)(a^n_i))-s(f(1)(a^n_i))$.
We also define the linear map $d^X_0:Z(X(0))\rightarrow Z$ such that for every node $x$ of $X$, $d^X_0(x) = 1$. We have the relation $d^X_0\circ d^X_1 = 0$. We denote by $H_1(X)$ the kernel of $d_1$, and by $H_0(X)$ the quotient of the kernel of $d_0$ by the image of $d_1$.
Each morphism $f:X\rightarrow Y$ between directed graphs induces natural morphisms $f^*_0:Z(X(0))\rightarrow Z(Y(0))$ and $f_1^*:Z(ch(X))\rightarrow Z(ch(Y))$.
Remark that if $f:c_n\rightarrow X$ is an $n$-cycle of $X$, the composition of $f\circ p_n$ of $f$ with the canonical morphism $p_n:P_{n+1}\rightarrow c_n$ is a chain such that $d^X_1(f\circ p_n)=0$.
[**Proposition 5.1.**]{}
[*Let $X$ be a finite directed graph, $u=\sum_{i\in I} d_if_{n_i}$ a positive $1$-chain, $d_1^X(u)=0$ if and only if there exists a finite set of cycles $g_j:c_{n_j}\rightarrow X$ such that the images of $\sum_i d_if_{n_i}$ and $\sum_j g_j\circ p_{n_j}$ coincide.*]{}
[**Proof.**]{}
Without restricticting the generality, we can assume that $d_i=1, i\in I$ since the chain is positive. We are going to give a recursive proof depending of the cardinality of $I$. Suppose that $I$ is a singleton, then $u=f$ where $f:P_n\rightarrow X$. The fact that $d_1^X(f)=0$ is equivalent to say that $f$ factors by a morphism $c_n\rightarrow X$.
Suppose that the result is true if the cardinality of $I$ is $l$. Assume now that the cardinality of $I$ is $l+1$. Remark that $d^X_1(u)=\sum_if_{n_i}(0)(t(P_{n_i}))-f_{n_i}(0)(s(P_{n_i})) = 0$. This implies the existence of $i_p$ such that $f_{n_{i_p}}(0)(s(P_{n_{i_p}}))=f_{n_0}(0)(t(P_{n_0}))$ we can thus define the concantenation $f_{n_{i_p}}f_{n_0}$ of $f_{n_p}$ which is an $n_0+n_p$-chain. We consider the family $L=\{f_{n_i}, f_{n_{i_p}}f_{n_0}, i\in I\}-\{f_{n_0},f_{n_{i_p}}\}$ whose cardinal is strictly inferior to the cardinal of $I$ and such that $\sum_{i\neq 0,p }f_{n_i}+f_{n_{i_p}}f_{n_0}$ has the same image than $u$. We can apply the recursive hypothesis to it and obtain a family of cycles $g_j:c_{n_j}\rightarrow X$ such that the images of $\sum_i d_if_{n_i}$ and $\sum_j g_j\circ p_{n_j}$ coincide.
This enables to give another proof of the theorem of Euler:
[**Corollary. 5.1. (Euler).**]{}
[*Let $X$ be a finite connected directed graph, there exists a morphism $f:c_n\rightarrow X$ bijective on arcs if and only if for every node $x$ of $X$ the in and the out degrees of $x$ coincide.*]{}
[**Proof.**]{}
Suppose that for every node $x$ of $X$, the in degree $in(x)$ and the out degree $out(x)$ of $X$ coincide, we have $d^X_1([X]) =\sum_{x\in X(0)}(out(x)-in(x)) =0$. The Proposition 5.1 implies the existence of morphism $f_{n_1}:c_{n_1}\rightarrow X,..., f_{n_l}:c_{n_l}\rightarrow X$ such that $\sum_{i=1}^{i=l}f_{n_i}$ and $[X]$ have the same image. We also deduce that $\sum_{i=1}^{i=l}f_{n_i}$ is bijective on arcs since the coefficients of its image are $1$. Since $X$ is connected, we deduce the existence of a morphism $f:c_n\rightarrow X$ bijective on arcs by making a concatenation of $f_{n_i},i=1,...l$.
[**Remark.**]{}
Let $X$ be a finite graph, $H_0(X)=0$ if and only if $X$ is connected, and $H_1(X)=0$ if and only if $X$ is acyclic: this is equivalent to saying that for every integer $n>0$, $Hom(c_n,X)$ is empty. In fact, there exists a bijection between the set of cycles of $X$ and positive elements of $H_1(X)$. This allows to give another description of the class of weak equivalences $W'$ studied in: a morphism $f:X\rightarrow Y$ is an element of $W'$ if and only if $f_1^*:H_1(X)\rightarrow H_1(Y)$ is bijective on positive chains.
Let $X$ be a finite strongly connected finite directed graph. We have seen that there exists a morphism $f:c_n\rightarrow X$ surjective on nodes and arcs. A good question is to find the lower bound $n(X)$ of $n$. We know that if $X$ is Eulerian, $n(X)$ is the cardinal of the number of arcs of $X$. The Proposition 5.1 shows that to find $n(X)$, it is sufficient to find a positive chain $c$ such that $d_1([X]+c)=0$ and the length of $l([X]+c)$ is minimal.
[**6. Closed models on $RGph$.**]{}
The cohomological interpretation of the proof ot the Euler theorem suggests that this theory is related to simplicial sets. In fact, $1$-simplicial sets are often called reflexive graphs, in this part, we are going to study a closed model in the category $RGph$ of reflexive graphs related to the closed model that we have just studied in $Gph$. Consider the category $C_R$ which has two objects $0_R$ and $1_R$, the morphisms of $C_R$ different of the identities are $s_R,t_R\in Hom_{C_R}(0_R,1_R)$ and a morphism $j_R\in Hom_{C_R}(1_R,0_R)$ such that $j_R\circ s_R = j_R\circ t_R = id_{0_R}$.
The category of presheaves over $C_R$ is called the category of reflexive graphs.
An object $X$ of the category $RGph$ is defined by two sets $X(0_R)$ and $X(1_R)$, two morphisms $X(s_R),X(t_R):X(1_R)\rightarrow X(0_R)$ and a morphism $X(j_R):X(0_R)\rightarrow X(1_R)$ such that $X(s_R)\circ X(j_R) = X(t_R)\circ X(j_R)=Id_{X(0_R)}$. Let $x$ be an element of $X(0_R)$, we will often denote $X(j_R)(x)$ by $[x]$. Geometrically, a node $x\in X(0_R)$ is represented by a point; we do not represent geometrically $X(j_R)(X(0_R))$. If $a\in X(1_R)$ is an arc which is not an element of $X(j_R)(X(0_R))$, it is represented by a directed arrow between $X(s_R)(a)$ and $X(t_R)(a)$.
Examples of reflexive graphs are:
The reflexive dot graph $D_R$; $D_R(0_R)$ and $D_R(1_R)$ are singletons.
The reflexive arc $A_R$; $A_R(0_R)$ contains two elements $x,y$; $A_R(1_R)$ contains three elements $[x], [y]$ and $a$ such that $A_R(j_R)(x) =[x], A_R(j_R)(y) = [y]$, $A_R(s_R)(a) = x$ and $A_R(t_R)(a) = y$.
The reflexive cycle of length $n$, $c_n^R$; $c_n^R(0_R)$ contains $n$ elements that we denote by $x^n_0,...,x^n_{n-1}$, For $i<n-1$, there is a unique arc $a^n_i$ whose source is $x^n_i$ and whose target is $x^n_{i+1}$; there is an arc $a^n_{n-1}$ whose source is $x^n_{n-1}$ and whose target is $x^n_0$. There exists arcs $[x^n_0],...,[x^n_{n-1}]$ such that $c_n^R(j_R)(x^n_i) = [x^n_i]$.
We are going to transport the closed models defined on $Gph$ to $RGph$.
We recall the transport theorem due to Crans, see Cisinski \[6\] 1.4.23.
[**Theorem 6.1.**]{}
*Let, $C$, $D$ be categories such that:*
\(i) $C$ and $D$ are complete and cocomplete and $L:C\rightarrow D$ a functor which has a right adjoint $R$.
Suppose that $C$ is endowed with a closed model structure $(W_C,Fib_C,Cof_C)$ cofibrantly generated by $I$ and $J$ such that:
\(ii) $L(I)$ and $L(J)$ allow the small element argument
\(iii) for every arrow $d$ of $D$ which is the transfinite composition of pushouts of arrows $L(c)$ where $c$ is an element of $W_C\cap Cof_C$, the arrow $R(d)$ is a weak equivalence in $C$.
Then there exists a closed model structure $(W_D,Cof_D,Fib_D)$ on $D$ such that:
T1 An arrow $d$ of $D$ is in $W_D$ if and only if $R(d)$ is in $W_C$
T2 An arrow $f$ of $C$ is in $Fib_D$ if and only if $R(f)$ is in $Fib_C$.
T3 An arrow of $D$ is in $Cof_D$ if and only it has the left lifting property with respect to all elements of $W_D\cap Fib_D$.
We thus deduce the following result:
[**Proposition 6.1.**]{}
[*Let $C,D$ be categories of presheaves defined on a set. Let $(W_C,Fib_C,Cof_C)$ be a closed model defined by counting the set of objects $(X_l)_{l\in L}$ of $C$. We suppose that $Fib_C$ is the class of all maps of $C$. Let $F:C\rightarrow D$ be a functor which has a right adjoint $G$. Suppose that $D$ is complete and cocomplete, then we can transfer $(W_C,Fib_C,Cof_C)$ to $D$ to obtain a closed model $(W_D,Fib_D,Cof_D)$ whose class of weak equivalences is defined by counting the set $(F(X_l))_{l\in L}$.*]{}
[**Proof.**]{}
The condition $(i)$ and $(ii)$ are satisfied since $C$ and $D$ are categories of presheaves defined on a set. Since the weak cofibrations are isomorphisms, the condition $(iii)$ is also satisfied. We deduce the class of weak equivalences of the closed model $(W_D,Fib_D,Cof_D)$ transfered to $D$ are morphisms $f:U\rightarrow V$ such that $G(f)$ is a weak equivalence. This is equivalent to saying that for every $l\in L$, the morphism of sets $Hom(X_l,G(U))\rightarrow Hom(X_l,G(V))$ which sends $h$ to $G(f)\circ h$ is an isomorphism. Since $G$ is the right adjoint of $F$, we deduce that this last condition is equivalent to saying that the morphism $Hom(F(X_l),U)\rightarrow Hom(F(X_l),V)$ which sends $h$ to $h\circ f$ is an isomorphism. Thus $(W_D,Fib_D,Cof_D)$ is obtained by counting the family $(F(X_l))_{l\in L}$.
We are going to apply the previous proposition to the following situation: consider the functor $f_R:C\rightarrow C_R$ defined on objects by $f_R(0) = 0_R$ and $f_R(1) = 1_R$. On morphisms, it is defined by $f_R(s) = s_R$ and $f_R(t) = t_R$.
Recall that if $S$ is a presheaf defined on a category $D$, and $F:D'\rightarrow D$ a functor, the inverse image $F^*S$ of $S$ is the presheaf defined on $D'$ such that for every object $X$ of $D'$, $F^*S(X) = S(F(X))$. When applying this construction to the functor $f_R$, we obtain that: if $X$ is a reflexive graph, $f_R^*(X)(0)=X(0_R)$ and $f_R^*(X)(1)=X(1_R)$. In particular, $f^*_R(D_R) = c_1$ and $f^*_R(A_R)$ is the directed graph which has two nodes $x$ and $y$, there exists an arc $a$ whose source is $x$ and whose target is $y$, there exists two loops $a_x$ such that $s(a_x)=x$ and $a_y$ such that $s(a_y)=y$. The Proposition 5.1 p.23 of \[8\] insures that the functor $f^*_R$ has a left adjoint ${f_R}_*$ and a right adjoint ${f_R}_!$.
[**Proposition 6.2.**]{}
[*The closed models of $RGph$ obtained by transferring the closed models $(W,Cof,Fib)$ which counts the cycles $(c_n)_{n\geq 0}$ and $(W',Cof',Fib')$ which counts the cycles $(c_n)_{n>0}$ to $RGph$ by the adjunction pair $({f_R}_*,f_R^*)$ are identic.*]{}
[**Proof.**]{}
The Proposition 6.1 implies that the transfer of $(W,Cof,Fib)$ (resp., $(W',Cof',Fib')$) on $RGph$ is the closed model defined by counting $(c_n^R)_{n\geq 0}$ (resp., $(c_n^R)_{n\geq 1})$. Thus we have to show that a morphism $f:X\rightarrow Y$ is right orthogonal to $i_n^R,j_n^R,n\geq 0$ if and only if it is right orthogonal to $i_n^R,j_n^R, n\geq 1$. On this purpose, it is enough to show that if $f$ is right orthogonal to $i_n^R,j_n^R, n\geq 1$, then it is right orthogonal to $i_0^R$ and $j_0^R$. Suppose that such an $f$ is not right orthogonal to $i_0^R$ or $j_0^R$. This equivalent to saying that $f$ does not induces a bijection between the nodes of $X$ and $Y$. If $f(0):X(0_R)\rightarrow Y(0_R)$ is not injective, let $x,y\in X(0_R)$ such that $f(0)(x)=f(0)(y)$. There exist morphisms $u,v:c_1^R\rightarrow X$ such that $u(0)(x^1_0) = x, v(0)(x^1_0) = y$, and $u(1)(a^1_0)=[x]$ and $v(1)(a^1_0) = [y]$. Consider the morphism $w:c_1^R\rightarrow Y$ such that $w(0)(x^1_0) = f(0)(x)$ and $w(1)(a^1_0) = [x]$. The following diagram does not have a filler. $$\begin{CD}
c_1^R+c_1^R @>u+v>> X\\
@VV j_1^R V @VV f V\\
c_1^R @>w>> Y
\end{CD}$$
This is a contradiction with the fact that $f$ is right orthogonal to $j_1^R$; thus $f(0)$ is injective.
Suppose that $f(0)$ is not surjective. Then there exists a node $y$ of $Y$ which is not in the image of $f(0)$. Let $u:c_1^R\rightarrow Y$ defined by $u(0)(x^1_0) = y$ and $u(1)(a^1_0) = [y]$. The following diagram does not have a filler:
$$\begin{CD}
\phi @> >> X\\
@VV i_1 V @VV f V\\
c_1^R @>u>> Y
\end{CD}$$
This is in contradiction with the fact that $f$ is right orthogonal to $i_1^R$. We deduce that $f(0)$ is surjective.
[**Definitions 6.1.**]{}
Let $X$ be a reflexive graph, the cycle $f:c_n^R\rightarrow X$ is degenerated if there exists $i$ such that $f(1)(a_i^n)=[y]$ where $y$ is a node of $Y$. A cycle is nondegenerated if it is not degenerated.
The following proposition shows that a morphism of $W_R$ preserves the nondegenerated cycles.
[**Proposition 6.3.**]{}
[*A weak equivalence $f:X\rightarrow Y$ of $RGph$ induces a bijection on nondegenerated cycles.*]{}
[**Proof.**]{}
Suppose that the image of a cycle $u:c_n^R\rightarrow X$ is degenerated. This implies that there exists a cycle $v:c_{n-1}^R\rightarrow Y$ which has the same image than $u$ and such that there exists a commutative diagram:
$$\begin{CD}
\phi @> >> X\\
@VV V @VV f V\\
c_{n-1}^R @>v>> Y
\end{CD}$$
which has a filler $w:c_{n-1}^R\rightarrow X$, and there exists a degenerated morphism $h:c_n^R\rightarrow c^R_{n-1}$ such that $ f\circ w\circ h =f\circ u$. Since the image of $w$ and the image of $u$ are different, we deduce that $f$ does not induces an injection on $n$-cycles. This is a contradiction with the fact that $f$ is a weak equivalence.
There exist morphisms which induces bijection on nondegenerated cycles, but which are not weak equivalences an example is the canonical morphism $f:A_R\rightarrow D_R$. The following result can be compared to \[10\] Theorem 4.9:
[**Proposition 6.4.**]{}
[*Let $W'_R$ be the class of morphisms of $RGph$ which induce a bijection on nondegenerated cycles. There does not exist a closed model whose class of weak equivalences is $W'_R$.*]{}
[**Proof.**]{}
Suppose that such a closed model exists. Consider the canonical morphism $f:A_R\rightarrow D_R$, we can write $f=g\circ h$ where $g$ is a weak fibration and $h$ a cofibration, the $2$-$3$ property implies that $h$ is a weak cofibration. Write $g:X\rightarrow D_R$, suppose that the cardinality of $X(0_R)$ is superior or equal to $2$. Let $l:c_1^R\rightarrow D_R$, the pullback of $l$ by $g$ is not a weak equivalence since its domain contains at least two distinct subgraphs isomorphic to $c_1^R$, this implies that the cardinal of $X(0_R)$ is $1$ and henceforth the cardinal of $X(1_R)$ is $1$ since $g$ is a weak equivalence; thus $g$ is the identity. We deduce that $f=h$ is a weak cofibration.
Let $Y$ be the reflexive graph such that $Y(0_R)$ contains two elements $u$ and $v$, $Y(1_R)$ contains $[u], [v]$ and two elements $c,d$ such that $X(s_R)(c)=X(s_R)(d) =u$ and $X(t_R)(c) = X(t_R)(d) = v$. Consider the morphism $k:A_R\rightarrow Y$ such that $k(0)(x)=u, k(0)(y) = v$ and $k(1)(a) =c$. The image of the pushout $m$ of $f$ by $k$ is $c_1^R$. This implies that $m$ is not weak equivalence. This is a contradiction with the fact that the pushout of a weak cofibration is a weak cofibration.
We will show now that some properties of the closed model defined on $RGph$ similar to the properties of the closed model $(W,Fib,Cof)$ defined on $Gph$. A reflexive graph $X$ is strongly connected if and only if for every nodes $x$ and $y$ of $X$, there exists a reflexive cycle $f:c_n^R\rightarrow X$ such that the image of $f(0)$ contains $x$ and $y$.
[**Proposition 6.5.**]{}
[*A strongly connected reflexive graph $X$ is cofibrant.*]{}
[**Proof.**]{}
Let $X$ be a strongly connected reflexive graph $X$. There exists a graph $X'$ in $Gph$ such that ${f_R}_*(X')=X$; $X'(0) = X(0_R)$ and $X'(1)$ is $X(1_R)-\{ [x], x\in X(0)\}$. The graph $X'$ is also strongly connected, thus it is a cofibrant object of $(W,Cof,Fib)$. Since the map $c_{X'}:\phi\rightarrow X'$ is a cofibration, this implies that $c_{X'}$ is an element of $cell(i_n,j_n,n\geq 0)$. We deduce that $c_X:\phi\rightarrow X$ is an element of $cell(i_n^R,j_n^R,n\geq 0)$ since $X={f_R}_*(X')$ and left adjoint preserve colimits and henceforth that $X$ is a cofibrant object.
[**Proposition 6.6.**]{}
[*A morphism $f:X\rightarrow Y$ between two reflexive graphs is a weak equivalence if and only if it induces a bijection between strongly connected components and its restriction to each strongly connected component is an isomorphism onto a strongly connected component of $Y$.*]{}
[**Proof.**]{}
Let $f:X\rightarrow Y$ be a weak equivalence of the closed model defined on $RGph$. The morphism ${f_R}^*(f)$ is also a weak equivalence. The Theorem 4.1 implies that it induces a bijection between the strongly connected components of ${f_R}^*(X)$ and ${f_R}^*(Y)$ and the restriction of ${f_R}^*(f)$ to each connected component of ${f_R}^*(X)$ is an isomorphism. Remark that ${f_R}^*(X)(0)=X(0_R)$ and ${f_R}^*(X)(1)=X(1_R)$, since ${f_R}^*$ is just the forgetful functor. This implies that the strongly connected components of ${f_R}^*(X)$ are of the form $V={f_R}^*(U)$ where $U$ is a strongly connected component of $X$ and that $f$ induces a bijection between strongly connected components and its restriction to each strongly connected component is an isomorphism onto a strongly connected component of $Y$.
[**References.**]{}
\[1\] Beke, T. (2000). Sheafifiable homotopy model categories. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 129, No. 03, pp. 447-475). Cambridge University Press.
\[2\] Bisson, T., Tsemo, A. (2009). A homotopical algebra of graphs related to zeta series. Homology, Homotopy and Applications, 11(1), 171-184.
\[3\] Bisson, T., Tsemo, A. (2011). Symbolic dynamics and the category of graphs. Theory and Applications of Categories, 25(22), 614-640.
\[4\] Bisson, T., Tsemo, A. (2011). Homotopy equivalence of isospectral graphs. New York J. Math, 17, 295-320.
\[5\] Brin, S., Page, L. (2012). Reprint of: The anatomy of a large-scale hypertextual web search engine. Computer networks, 56(18), 3825-3833.
\[6\] Cisinski D.C.,(2006) Les préfaisceaux comme type d’homotopie, Astérisque, Volume 308, Soc. Math. France.
\[7\] Euler, L. (1741). Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae, 8, 128-140.
\[8\] Artin, M., Grothendieck, A., Verdier, J. L. (1972). Théorie des topos et cohomologie étale des schémas. Tome 1. Lecture notes in mathematics, 269.
\[9\] Hirschhorn, P. S. (2009). Model categories and their localizations (No. 99). American Mathematical Soc.
\[10\] Tsemo, A. (2013). Applications of closed models defined by counting to graph theory and topology. arXiv preprint arXiv:1308.3983.
|
---
abstract: 'We report experimental demonstration of the feasibility of reaching temperatures below 1 mK using cryogen-free technology. Our prototype system comprises an adiabatic nuclear demagnetisation stage, based on hyperfine-enhanced nuclear magnetic cooling, integrated with a commercial cryogen-free dilution refrigerator and 8 T superconducting magnet. Thermometry was provided by a current-sensing noise thermometer. The minimum temperature achieved at the experimental platform was 600 $\mu$K. The platform remained below 1 mK for over 24 hours, indicating a total residual heat-leak into the experimental stage of 5 nW. We discuss straightforward improvements to the design of the current prototype that are expected to lead to enhanced performance. This opens the way to widening the accessibility of temperatures in the microkelvin regime, of potential importance in the application of strongly correlated electron states in nanodevices to quantum computing.'
address:
- '$^1$Oxford Instruments Omicron NanoScience, Tubney Woods, Abingdon, Oxfordshire, OX13 5QX, UK'
- '$^2$Department of Physics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK'
author:
- 'G Batey$^1$, A Casey$^2$, M N Cuthbert$^1$, A J Matthews$^1$, J Saunders$^2$, A Shibahara$^2$'
bibliography:
- '1mK.bib'
title: 'A microkelvin cryogen-free experimental platform with integrated noise thermometry'
---
\[intro\]Introduction
=====================
Quantum effects in condensed matter physics can be revealed when systems are cooled to a temperature below which the thermal energy is less than their characteristic energy scale. The long tradition of cooling condensed matter systems to progressively lower temperatures has uncovered a wealth of new phenomena. However most of the emphasis on cooling systems to below 1 mK has focussed on quantum fluids and solid. The widely agreed imperative now is to open the microkelvin range to experiments on nanoelectronic devices and strongly correlated electron systems. Examples are: fractional quantum Hall effect with potential applications in topological quantum computing [@Sarma2005; @Marcus2008]; new strongly correlated quantum states in semiconductor nanodevices [@Hanson2007]; charge pumps [@Pepper2010]; possible nuclear spin ordering in two dimensional quantum wells [@Loss2007]; brute force cooling of nanomechanical resonators into the quantum regime [@Schwab2004]. In addition the decoherence times of superconducting qubits may ultimately depend upon the intrinsic coherence of Josephson junctions [@Paik2011] which should be improved at lower temperatures [@Catelani2011].
In order to cool these systems into the microkelvin range, the requirements are: an appropriate low-temperature platform and thermometry; appropriate sample thermalisation. This paper considers the first of these.
Continuous cooling to the low mK regime is usually achieved using a dilution refrigerator. Whilst very powerful dilution refrigerators can operate at temperatures down to $\sim$ 2 mK [@Vermeulen1987; @Cousins1999], and the cold liquid in dilution refrigerators can be coupled to a condensed matter experiment [@Samkharadze2011], the dramatic decrease in availability (and associated increase in cost) of $^3$He in recent years [@Kouzes2009] has made these very large systems uneconomical for all but the most specialised low-temperature facilities.
On the other hand, the temperature regime down to $\sim$ 10 mK has recently been opened up by the availability of cryogen-free systems. Here a dilution refrigerator with a relatively small charge of helium mixture is operated with a pulse tube refrigerator pre-cooling stage [@uhlig2002]. These “dry” cryogen-free systems can be installed without the need for associated complex research infrastructure, such as a helium liquefaction plant, or in remote locations. Such systems can be automated to a higher degree than their “wet” counterparts, and are simple to operate. In conventional systems, with liquid helium precooling, the overall diameter is constrained by the neck of the helium dewar, which influences the rate of liquid helium “boil-off”. In cryogen-free systems this restriction does not apply. It allows experimental platforms typically several hundred mm in diameter. This has enabled a range of more complex services to be installed onto such refrigerators; for example bulky signal-conditioning elements such as cryogenic amplifiers, microwave components (bias-tees, circulators, switches etc.) and filtering (such as metal powder filters) which have proved invaluable for experiments aimed at quantum information processing.
Nuclear cooling [@Andres1982] offers the possibility of extending experimental temperatures into the $\mu$K regime, well below those accessible to even the most powerful dilution refrigerators. A variety of strategies have been developed [@Pickett1988; @Pobell2007]. The most popular is the adiabatic nuclear demagnetization of copper, precooled in a (large) magnetic field (required to generate the initial entropy reduction in the nuclear refrigerant) by a thermal link connected to a dilution refrigerator with an unloaded base temperature of < 10 mK *via* a superconducting heat switch. In this case the demagnetisation cooling process is“single-shot”, as only a finite amount of energy can be absorbed by the nuclear stage as the system warms. These restrictions have not proved to be a problem in practice as the duty cycle between the pre-cooling and demagnetisation stages of a typical experimental run are short compared to the low-temperature hold time, provided the heat leak into the system is sufficiently small. Recently, semiconductor nano-structures have been cooled using an array of nuclear refrigerators attached to measurement leads [@Clark2010]. However, it is not obvious that this approach could be implemented for a wide range of condensed matter physics experiments, especially those requiring complicated wiring arrangements.
The approach adopted here is the integration of cryogen-free dilution refrigerators and superconducting magnets [@Batey2009], with the entire system running from a single pulse tube cooler, with a “bolt-on” nuclear refrigerator to extend the accessible temperature range to below 1 mK, thus maintaining compatibility with a wide range of experimental applications. Given the extreme precautions taken with nuclear adiabatic demagnetization cryostats to create ultra-low mechanical noise environments in order to minimize heat leaks due to vibration of the nuclear stage, the feasibility of realising this goal was not apparent; this has been addressed in the work reported here.
\[fridge\]The cryogen-free system
=================================
The pulse tube coolers used on cryogen-free dilution refrigerators are known to be a source of mechanical noise, particularly on systems where the room temperature, first ($\sim$ 50 K) and second stage ($\sim$ 3 K) components of the pulse tube coldhead (the volume in which the gas expands) are not vibrationally decoupled from the dilution refrigerator [@pelliccione2013]. Initial measurements of cryogen-free refrigerator and magnet systems [@Batey2009] did not show a significant increase in the base temperature of the refrigerator when the installed magnet was persistent at its full field of 12 T. However in those tests the mixing chamber plate only experienced the fringing field of the magnet, not the full field to which any nuclear refrigerant would be exposed. To our knowledge it has not been demonstrated previously that this potential obstacle to the implementation of nuclear demagnetisation techniques in a cryogen-free environment could be overcome.
The choice of refrigerant for nuclear cooling has been reviewed [@Pickett1988; @Pobell2007]. Here we consider two options: copper or PrNi$_5$. Copper is recognised as being the refrigerant of choice for the attainment of the lowest possible temperature, but in order to generate a significant entropy reduction in the refrigerant the starting conditions are demanding: a 9 % entropy reduction can be achieved by cooling to a temperature of $\sim$ 10 mK in a magnetic field of $\sim$ 8 T. The high bulk electrical conductivity of copper at low temperatures coupled with the large magnetic field and the possibility of mechanical vibrations in a cryogen-free system led to copper being rejected as the refrigerant in this work. A large entropy reduction can be achieved in PrNi$_5$ at higher temperatures and in lower applied fields due to the hyperfine-enhancement of the field experienced by the nuclei; in a field of 6 T a 70 % entropy reduction is reached at 25 mK. Additionally, the poor bulk conductivity of the material (comparable to that of brass) should minimise eddy current heating resulting from any motion of the refrigerant in the field. The nuclear spin ordering in PrNi$_5$ also allows the possibility to demagnetize to zero magnetic field; subsequent warming of the stage is governed by the Schottky heat capacity maximum which occurs at $\sim$ 0.5 mK [@Kubota1980], this contrasts with the use of copper, where nuclear ordering is only observed at the much lower temperature of 58 nK [@Huiku1982], and so in the temperature regime of the present application the system is a nuclear paramagnet and the nuclear heat capacity is proportional to the square of the final magnetic field.
In view of the concerns about the possible effects of vibrational heating, both in the pre-cool phase and the warm-up following demagnetization, we opted for PrNi$_5$ as our coolant. The dilution refrigerator used in this work, an Oxford Instruments Triton 200, has been described previously [@Batey2009]. The typical performance parameters are a cooling power in excess of 200 $\mu$W at 100 mK, and a base temperature below 10 mK. We used a standard Triton 200, with the only anti-vibration measures being the decoupling of the pulse tube cooler first and second stages from the refrigerator plates using flexible copper braids.
![\[fig:fridge\] A cross-section view of the layout of the demagnetisation stage below the mixing chamber. The 4 K shield extension allows the superconducting magnet to be mounted in a lower position with respect to the mixing chamber. The superconducting heat switch and noise thermometer are positioned just below the mixing chamber flange. The low thermal conductivity support structure of the demagnetisation stage is not shown for clarity and nor are the thermal links, which are described in the main text.](Fig_1.eps){width="95.00000%"}
The layout of the demagnetisation stage below the mixing chamber is shown in figure \[fig:fridge\]. The superconducting magnet available for this work was of standard solenoid design, with a 77 mm cold bore and a maximum field of 8 T when operated at 5 K. The field compensation of this solenoid was not optimised; however the stray field at the mixing chamber and heat switch was lowered by the simple expedient of increasing the distance between the mixing chamber plate and the centre of the solenoid. This increase in cryostat length was also necessary to accommodate the nuclear refrigeration stage. The overall system height was < 2.5 m, compatible with standard laboratory space.
The nuclear stage consists of 128 g of the inter-metallic compound PrNi$_5$ in the form of nine 6 mm diameter $\times$ 50 mm long rods, which were pre-tinned with 99.99 % cadmium. Each rod was connected to the upper and lower experimental plates (figure \[fig:fridge\]) with 1 mm diameter copper wire with a residual resistance ratio $\sim$ 1000. One wire per rod extends to the upper plate and eight wires per rod to the lower plate. The stage used in this work was adapted from a pre-existing stage made at Cornell [@Parpia1985]. The upper experimental plate was connected to the mixing chamber through a superconducting aluminium heat switch [@Lawson1982]. The switch had six 1 mm diameter silver wires bonded to either end, which in turn were diffusion bonded to copper blocks which made mechanical connections to the upper experimental plate and the mixing chamber. This link was particularly long, $\sim$ 30 cm, to mitigate the effects of the non-optimal stray field, however it was still possible to precool the stage to below 20 mK in a 6.2 T field within 24 hours. The heat load on the mixing chamber during the precool was a combination of eddy current heating of the nuclear stage and components of the mixing chamber exposed to the large stray fields of the magnet, in particular the brass mixing chamber radiation shield which experienced the full field of the magnet. With the heat switch open, it was possible to cool the mixing chamber to $\sim$ 15 mK.
\[thermo\]The current-sensing noise thermometer
===============================================
Attaining temperatures significantly below dilution refrigerator temperatures poses an experimental challenge. Measuring the temperatures so-attained poses another. The base temperature of dilution refrigerators can be established with nuclear orientation thermometry [@Marshak1983], however this slow technique is not suitable as a practical thermometer for regular use, and is not usable at $\mu$K temperatures. Other convenient methods of temperature measurement, such as carbon resistance sensors [@Samkharadze2010], shot noise in tunnel junctions [@Spietz2006] or Coulomb blockade [@Pekola1994] also prove difficult to implement below a few mK. The most common approach is to rely on the Curie-law paramagnetism of nuclear spins [@buchal1978] or the magnetism of dilute electronic paramagnets [@Paulson1979] calibrated with a fixed point device. Other methods are possible if a sample of superfluid $^3$He is part of the set-up [@Todoschenko2002].
Here we have deployed a current sensing noise thermometer [@Casey2003]. This thermometer exploits the low intrinsic noise and high sensitivity of a dc Superconducting Quantum Interference Device, SQUID, configured as a current amplifier to detect the Johnson noise produced by a resistive element. The resistive sensor is mounted on a copper platform linked to the demagnetisation stage. The sensor is connected to the SQUID input inductance *via* a superconducting twisted pair. The mean square noise current flowing in the SQUID input coil per unit bandwidth, arising from the thermal noise in the resistor, is then given by $$\label{eq:noise}
\left\langle I^{2}_{N} \right\rangle = \frac{4k_{B}T}{R}\left(\frac{1}{1+\omega^{2}\tau^{2}}\right)$$ where $\omega = 2 \pi f$ and the time constant $\tau = L/R$. Here $L$ is dominated by the input coil inductance of the SQUID $L_{i}$; the resistance $R$ can be chosen either to optimise speed of measurement or reduce the noise temperature of the measurement system. In this work a sensor resistance of 0.24 m$\Omega$ was chosen, when coupled to an extremely sensitive 2-stage SQUID [@Drung2007] this results in a current noise power due to the SQUID of $2.5\times10^{-25}$ A$^{2}$/Hz or equivalently an amplifier with a noise temperature of $T_{N}=1$ $\mu$K. For comparison we note that a thermometer prepared in a similar way mounted on a traditional copper nuclear demagnetisation cryostat has been cooled to below 200 $\mu$K in our laboratory. We also remark that noise thermometers optimised for operation in the dilution refrigerator temperature range can achieve 1 % precision in 100 ms of measurement time.
Typical examples of the current noise measured over a range of temperatures are shown in figure \[fig:thermom\]. For these measurements we have mounted the noise thermometer in a region of low stray field near to the heat switch; see figure \[fig:fridge\]. It is thermally connected to the lower nuclear stage plate through a link consisting of 37 $\times$ 0.7 mm diameter, annealed copper wires. Extraneous noise peaks arising from environmental noise have been filtered. Scatter in the measured frequency-dependent noise power is higher than that observed on a conventional (non cryogen-free) refrigerator, which may be a quieter environment due to the absence of microphonic and / or triboelectric noise arising from vibrations caused by the pulse tube cooler. Nevertheless we obtain a precision of 2 % in a measurement time of 200 s from fits to the power spectrum.
\[results\]Results
==================
Here we present the results from this prototype system, which demonstrate the feasibility of the approach, despite a number of non-optimal conditions, discussed in the conclusion. The precool field was limited to < 6.2 T by the tolerable upper limit to the stray field at the superconducting heat switch. Typically a 24 hour precool was sufficient to cool the stage to 20 mK, achieving an entropy reduction of 80 % of the free spin value [@Folle1981]. The superconducting heat switch was then opened and the demagnetisation carried out in a series of steps. The field was halved at each step and the rate was adjusted so that each step took the same amount of time, until a final step from 100 mT to zero applied field. Hence demagnetising to zero field could be achieved in around 6 hours.
Figure \[fig:results1\] shows the results of such a demagnetisation on two separate cryostat cool-downs. In one case an intrinsic heat leak to the stage of 20 nW was observed. The heat leak was determined by applying additional electrical power to the stage and monitoring the time required to warm it over the temperature range from 2 to 12 mK, figure \[fig:results2\]. We found the heat leak was sensitive to the precise arrangement of the flexible lines around the pulse tube refrigerator cold-head and to the addition of lead shot damping masses ($\sim$ 10 kg) to the top of the cold-head itself. Whilst the intrinsic heat leak could be reduced in this way no correlation could be observed with the vibrational spectra measured with accelerometers [@HSJ], sensitive in the frequency range 1-1000 Hz, installed on the refrigerator 3 K stage. The background heat leak was improved to 5 nW, comparable to that achieved in many conventional copper nuclear demagnetisation systems. Heat leaks below the nW level can only be achieved by taking the utmost care in vibration isolation and choice of materials used in construction [@Buck1990]. In the run where a 20 nW heat leak was observed the thermometer was cooled to 600 $\mu$K. After reaching this temperature it stayed below 1 mK for 16 hours. An increased hold time can be achieved at the expense of a higher minimum temperature by stopping the demagnetization at a higher final field. In figure \[fig:results1\] we show the results of a demagnetisation to 210 mT with a lowest temperature of 1.2 mK. In this case the stage remained below the base temperature of the dilution refrigerator for over two days. Finally, with an improved heat leak of 5 nW, a hold-time below 1 mK of over 24 hours was achieved.
\[conc\]Conclusions
===================
We have presented initial results obtained on a cryogen-free dilution refrigerator with a bolt-on PrNi$_5$ nuclear demagnetisation stage. The results: a base temperature of 600 $\mu$K and a residual heat leak 5 nW, indicate that cryogen-free systems are indeed suitable environments for microkelvin experiments despite the concerns over the vibrational stability of such systems. Pre-cooling the nuclear refrigerant to temperatures < 20 mK in a field of $\sim$ 6.2 T is accomplished in around 24 hours, with the demagnetisation taking a further 6 hours. The hold time below 1 mK can be > 24 hours. This represents a reasonable duty cycle for many experiments in this regime.
Such a simple demagnetization stage could, in principle, be added to any cryogen-free dilution refrigerator system that is suitable for the operation of a superconducting magnet. The temperatures were determined using a current-sensing dc SQUID noise thermometer that allowed us to measure directly sub-mK temperatures.
The present set-up is subject to a number of relatively straightforward modifications that are anticipated to lead to improvements in performance. These include optimisation of magnet design to provide appropriate field cancellation regions. In this work the nuclear stage was added to an existing cryogen-free dilution refrigerator for which there was no particular attention devoted to vibration isolation. Remotely mounting the turbo-molecular pump, used for the $^3$He circulation, and decoupling the pulse tube cooler from the system top-plate will reduce the levels of vibrational noise, together with improvements in the rigidity of the refrigerator mounting frame.
Cryogen-free sub-mK platforms do therefore seem be a realistic prospect, dramatically improving the accessibility of the ultra-low temperature frontier for exploration and discovery. The addition of a research magnet, to enable the combination of a high magnetic fields and microkelvin temperatures remains a future technical challenge to be addressed.
References {#references .unnumbered}
==========
|
---
abstract: 'The study of the interaction between solid objects and magnetohydrodynamic (MHD) fluids is of great importance in physics as consequence of the significant phenomena generated, such as planets interacting with stellar wind produced by their host stars. There are several computational tools created to simulate hydrodynamic and MHD fluids, such as the FLASH code. In this code there is a feature which permits the placement of rigid bodies in the domain to be simulated. However, it is available and tested for pure hydrodynamic cases only. Our aim here is to adapt the existing resources of FLASH to enable the placement of a rigid body in MHD scenarios and, with such a scheme, to produce the simulation of a non-magnetized planet interacting with the stellar wind produced by a sun-like star. Besides, we consider that the planet has no significant atmosphere. We focus our analysis on the patterns of the density, magnetic field and velocity around the planet, as well as the influence of the viscosity on such patterns. At last, an improved methodological approach is available to other interested users.'
author:
- 'Edgard F. D. Evangelista'
- 'Oswaldo D. Miranda'
- Odim Mendes
- 'Margarete O. Domingues'
date: 'Received: date / Accepted: date'
title: 'Simulating the interaction of a non-magnetized planet with the stellar wind produced by a sun-like star using the FLASH Code'
---
Introduction
============
The simulation of rigid bodies interacting with fluids is a problem of great interest in physics as consequence of the significant phenomena generated. As examples of an application of such a problem, one may cite aerodynamic studies of mechanical structures such as airfoils and planets interacting with stellar winds. In the literature, one may find examples of approaches to the problem in question: in [@takahashi:2002], the authors describe the modeling of the interaction of a fluid with a rigid body, where they use the Cubic Interpolated Propagation (CIP) to simulate the fluid itself and the Volume of Solid (VoS) to handle the interaction of the body with the fluid; in [@takashi:1992], it is shown a computational approach to solve problems of rigid objects in contact with viscous incompressible fluids, in which the authors used the arbitrary Lagrangian-Eulerian method and the streamline-upwind/Petrov-Garlerkin finite element volume scheme.
It is worth bearing in mind that the examples such as the ones discussed above involved pure hydrodynamic scenarios only. However, when dealing with electrically conducting fluids, including plasmas undergoing the effects of electromagnetic fields, the hydrodynamic model should be replaced by appropriate physical-mathematical frameworks. Of these, one of the simplest is MHD, which describes the behavior of plasmas under the influence of magnetic fields[@powell:1999]. For example, in [@grigoriadis:2010] the authors use the immersed boundary method to address the case of a MHD fluid interacting with a circular cilinder.
In Astrophysics one may cite as examples of MHD studies of interactions of fluids and bodies the paper [@johnstone:2015], where the authors use the 3D code Nurgush to simulate the shocks between the winds from two low-mass stars forming a binary system, and [@vernisse:2013], in which it is used the code A.I.K.E.F. (Adaptive Ion-Kinetic-Electron-Fluid) as a tool to treat lunar type plasma interactions. Other pertinent examples include the study of exoplanets under the influence of the environment produced by their host stars: [@cohen:2015] addresses to Venus-like, non-magnetized exoplanets interacting with the wind from a M-dwarf star; [@nichols:2016] discusses exoplanets with magnetospheres undergoing Earth-like magnetospheric interaction with the solar wind; and in [@bourrier:2016] the authors analyze observations of the “warm Neptune” $\mbox{GJ}436\mbox{b}$ and the interaction between its exhosfere with the stellar wind. Further, it is worth mentioning [@spreiter:1970], which focuses on the interaction of the solar wind with non-magnetized planets, while [@dryer:1973] studies the flow of the solar wind around Jupiter, Saturn, Uranus, Neptune and Pluto.
There are several computational schemes created to handle hydrodynamic and MHD problems. In this paper we use the FLASH code of the University of Chicago. However, it is important to point out that the tool which permits the placement of bodies in the simulations are, until this time, implemented and tested in such a code for pure hydrodynamic cases only.
Our aim here is to simulate the MHD interaction of the wind produced by a sun-like star with a non-magnetized planet, which has the approximate size of Earth and is placed at an orbital distance equal to the mean radius from the sun to Mercury. Furthermore, in our model the planet has no significant atmosphere. We achieve this by adapting the existing tools for simulating solid objects in pure hydrodynamic scenarios present in FLASH.
We investigate the influence of the viscosity on the regions around the planet, particularly its effects on the recirculation patterns and the behavior of the wake. Besides, in order to analyze the consistence of our scheme, we pay special attention to the magnetic field profiles and the mesh refinement in the MHD scenarios. For the sake of comparison, we perform a similar simulation in a pure hydrodynamic scenario.
The scheme presented here is interesting once it creates new perspectives for using the FLASH code, concerning the simulations of interactions of MHD fluids with rigid bodies. In addition, with the exponential growth of interest in research associated with exoplanets in the last two decades, both in observational and theoretical aspects, there is now strong interest in the studies of orbital evolution of planets due to their interaction with the protoplanetary disc, the central star and other planets (see, e.g., the recent work [@alvarado-gomez:2016]). Such studies are situated in a step that can be immediately extended from the work presented here.
This paper is organized as follows: in Section \[MHD\] we show the basic formalism of MHD; in Section \[numerical\] we discuss the numerical details of the simulations, concerning both the computational and the physical aspects; in Section \[results\] the results and their respective discussions are presented, while the conclusions are given in Section \[conclusions\].
Basic formalism of MHD {#MHD}
======================
Magnetohydrodynamics is one of the simplest frameworks for modelling the interaction between a conducting fluid and a magnetic field[@bateman:1978] and describes the macroscopic behavior of electrically conducting fluids, of which the most common is the plasma[@biskamp:2003]. Roughly speaking, MHD consists in the combination of the equations governing the fluid dynamics with Maxwell’s equations of the electromagnetism.
Though the resulting system of equations can be presented in different ways, it is usually written in conservative form such that, in a fixed frame of reference (or Eulerian coordinate system), it assumes the form for the case where the viscosity is non-negligible:[@goedbloed:2004; @lifschitz:1989]
$$\begin{aligned}
&\frac{\partial}{\partial t}(\rho\mathbf{v})= \nonumber \\
&\nabla\cdot\left[-\rho\mathbf{v}\mathbf{v}+\frac{1}{\mu}\mathbf{B}\mathbf{B}-\mathbb{I}\left(p+\frac{B^{2}}{2\mu}\right)\right] + \rho\nu\nabla^{2}\mathbf{v}, \label{mhd:1} \\[8pt]
&\frac{\partial\mathbf{B}}{\partial t}=\nabla\cdot(\mathbf{v}\mathbf{B}-\mathbf{B}\mathbf{v}), \label{mhd:2} \\[8pt]
&\frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho\mathbf{v}), \label{mhd:3} \\[8pt]
&\frac{\partial\epsilon}{\partial t}=\nabla\cdot\left[-\left(\epsilon+p+\frac{\mathbf{B}^{2}}{2\mu}\right)\mathbf{v}+\frac{1}{\mu}(\mathbf{B}\cdot\mathbf{v})\mathbf{B}\right], \label{mhd:4} \\[8pt]
& \nabla\cdot\mathbf{B}=0, \label{mhd:5}\end{aligned}$$
where: $\epsilon=\rho\mathbf{v}^{2}/2+p/(\gamma-1)+\mathbf{B}^{2}/2\mu$ is the total energy density of the fluid; $\mu$, $\mathbf{v}$, $\mathbf{B}$, $\rho$, $p$ are the magnetic permeability, the velocity, the magnetic field, the density and the pressure of the plasma; $\mathbb{I}$ is the $3\times 3$ identity matrix and $\nu$ is the kinematic viscosity[@goedbloed:2004]. Besides, it is considered a equation of state in the form $p=(\gamma-1)\epsilon$ where $\gamma$ is the adiabatic index.
From the form of such equations one may note that, with the exception of the term proportional to $\nu$ in Eq. (\[mhd:1\]), their right-hand sides are the divergent of the fluxes through the boundaries of the volume considered[@bateman:1978] and they represent, from top to bottom, the time evolution of the momentum, magnetic field, mass density and total energy, while Eq. (\[mhd:5\]) is the zero-divergence constraint on the magnetic field.
Computational codes use the conservative equations of MHD as shown here, once that particular form make them suitable for working out finite difference schemes. On the other hand, properties of the analytic equations can be used to validate the performance of numerical schemes[@bateman:1978].
Numerical Aspects {#numerical}
=================
The FLASH code has been originally developed for simulating astrophysical phenomena involving MHD and is distributed by the Center for Astrophysical Thermonuclear Flashes (FLASH Center) of the University of Chicago[^1]. The default package used by this code for handling the adaptive-mesh refinement grid is PARAMESH[@macneice:2000], which employs a refinement criteria adapted from Löhner’s error estimator[@lohner:1987] with a threshold $10^{-2}$ in order to trigger the mesh refinement process. Besides, FLASH uses the Message-Passing Interface (MPI) library and HDF5 to allow portability on a variety of computers when dealing with parallel computation[@fryxell:2000]. Its modular architecture is such that it permits customization of the codes in order to simulate particular cases by means of changes in the algorithms and creation of new physics modules.
We consider five, six and seven levels of refinement in our simulations. However, we focus our analysis on the scenarios with five and seven levels; a result with six levels was generated just in order to investigate the convergence of the solutions and it is briefly mentioned in Subsection \[BxDiffZero\] (with a panel shown in Subsection \[comparison\]). Increasing the level of refinement by one duplicates the number of blocks in each coordinate and, as we start with a domain of $3\times 3\times 3$ blocks, we obtain $48\times 48\times 48$, $96\times 96\times 96$ and $192\times 192\times 192$ blocks with five, six and seven levels, respectively. Each block has $8\times 8\times 8$ cells.
In our MHD simulation we use the unsplit staggered mesh (USM) algorithm in order to solve Eqs. (\[mhd:1\])-(\[mhd:4\]). It is based on the Godunov method, basically consisting in a conservative finite-volume scheme using spatial discretisation to solve the partial differential equations. For the pure hydrodynamic scenario it is used the unsplit hydro solver (UHS) which, in the present context, can be treated as a simplified version of USM where a fundamental difference is the presence of magnetic and electric fields in the latter. It is worth recalling that UHS uses the zone-edge data-extrapolated method as a specific predictor-corrector formulation.
The FLASH code employs as default the Roe approximate Riemann solver[@roe:1981], which has been applied to a wide range of physical problems. However, despite the sucess of that solver, it can fail in regions of very low densities, producing unphysical states near strong rarefaction regions. Such a characteristic can represent a critical disadvantage in MHD scenarios, where in general the gas pressure is much less than the magnetic pressure[@li:2005]. Besides, due to the fact that the Roe solver demands eigen decomposition, it may become computationally costly in MHD problems. In order to overcome the mentioned limitations we use the HLL (Harten-Lax-van Leer-Contact) solver in both MHD and pure hydro scenarios, once this scheme satisfies the integral form of the conservation laws and it is computationally more robust[@einfeldt:1991].
The time advancement of the equations in USM and UHS is based on a MUSCL-Hancock[@vanleer:1984] type algorithm and the code uses the constrained transport method to assure numerically the physical constraint given by Eq. (\[mhd:5\]) (see [@lee:2009]). On the other hand, all the simulations use a Courant Friedrichs Lewy (CFL) condition[@courant:1967] of $0.8$ and have an adiabatic index $\gamma=5/3$.
Physical parameters of the problem {#physicalparam}
----------------------------------
We created a scenario representing a planet with the approximate size of Earth orbiting a sun-like star and placed at an orbital distance equal to the mean radius from the sun to Mercury, namely, $\unit[0.39]{AU}\approx \unit[6.0\times 10^{12}]{cm}$. Such a planet is inserted as a sphere of radius $\unit[6\times 10^{8}]{cm}$ and center at $\unit[(6.0\times 10^{9},0,0)]{cm}$ in a rectangular box whose dimensions are: $x\in [0.0,18.0] \unit[\times 10^{9}]{cm}$, $y\in [-9.0,9.0] \unit[\times 10^{9}]{cm}$, $z\in [-9.0,9.0] \unit[\times 10^{9}]{cm}$ for the scenarios presented in Subsection \[PureHydro\] and \[BxEqualZero\]; and $x\in [0.0,18.0] \unit[\times 10^{9}]{cm}$, $y\in [-12.0,6.0] \unit[\times 10^{9}]{cm}$, $z\in [-12.0,6.0] \unit[\times 10^{9}]{cm}$ for the case shown in Subsection \[BxDiffZero\].
The magnetic field $\mathbf{B}$ to be used as an of the initial parameters in our simulations is determinated by means of Parker’s model for the solar wind, such that its components are written in spherical coordinates as[@parker:1958]
$$\begin{aligned}
&B_r = B_0\left(\frac{b}{r}\right)^{2}, \label{parkerB:1} \\
&B_\theta = 0, \label{parkerB:2} \\
&B_\phi = B_0\left(\frac{\Omega}{v_{\mbox{\scriptsize{sw}}}}\right)(r-b)\left(\frac{b}{r}\right)^{2}\sin\theta, \label{parkerB:3}\end{aligned}$$
where $B_0$ and $b$ are constants, $\Omega$ is the angular velocity of the sun, $v_{\mbox{\scriptsize{sw}}}$ is the radial velocity of the solar wind and $r$ is the heliocentric distance. As we are assuming that the orbit of our hypothetical planet is in the ecliptic plane ($\theta=\pi/2$) and given the fact that Eqs. (\[parkerB:1\])-(\[parkerB:3\]) do not depend on $\phi$, we can, for the sake of convenience, consider that $B_r$ and $B_\phi$ have the directions of the x-axis and y-axis in our domain, respectively.
Since, according to [@kivelson:1995], the components of the magnetic field at (written as $B^e_r$ and $B^e_\phi$) have values such that $\sqrt{(B^e_r)^2+(B^e_\phi)^2}=\unit[7]{nT}$ and $B^e_\phi/B^e_r \approx 1$, we can use Eqs. (\[parkerB:1\])-(\[parkerB:3\]) to evaluate such components at (writting them as $B^m_r$ and $B^m_\phi$ in this case). With effect, from Eq. (\[parkerB:1\]) we deduce that $B^m_r(0.39)^2=B^e_r(1)^2$, giving $B^m_r=\unit[32.5]{nT}$. Now, dividing Eq. (\[parkerB:3\]) by Eq. (\[parkerB:1\]) and using $v_{\mbox{\scriptsize{sw}}}=\unit[400]{km~s^{-1}}$, $\Omega=\unit[2.7\times 10^{-6}]{\mbox{rad}~s^{-1}}$ and $b=\unit[4.6\times 10^{-2}]{AU}$ [@tautz:2011], we have $B_\phi/B_r\approx r$; for $r=\unit[0.39]{AU}$ we obtain finally $B^m_\phi=\unit[12.7]{nT}$.
From Eqs. (\[parkerB:1\])-(\[parkerB:3\]) we note that Parker’s model does not define a component perpendicular to $B_r$ and $B_\phi$. On the other hand, the presence of a $B_\theta$ different from zero is justified, for example, by the transport of magnetic fields on the solar surface and turbulence[@korth:2011], making interesting the inclusion of such a component in our scenarios. According to [@korth:2011], measurements of $B_\theta$ taken between $\unit[0.31]{AU}$ and $\unit[0.47]{AU}$ by spacecrafts such as MESSENGER and Helios present large flutuations around zero, making difficult in principle to choose a “typical” value to be used here. However, as we can deduce from the histograms shown in [@korth:2011], more than $\approx 90 \%$ of the pertinent observational data lie in the interval $\approx [-15,15]\unit{nT}$, suggesting us that it would be reasonable to consider an initial $B_\theta$ (written as $B_z$ hereafter) of $\sim\unit[10]{nT}$ in our simulations.
The remaining initial parameters, namely, $\rho$ (obtained from the proton density $n_p$ and the electron density $n_e$) and $p$ at $r=\unit[0.39]{AU}$ can be obtained by a similar procedure to the one used in the evaluation of $B^m_r$ and $B^m_\phi$. With effect, from Parker’s model, we may consider that $n_e$ and $n_p$ has a dependence on $r$ in the form $n_{e,p} \propto r^{-2}$[@parker:1958]. Besides, let us assume that the proton temperature $T_p$ and the electron temperature $T_e$ vary with $r$ as $T_p \propto r^{-1}$ and $T_e \propto r^{-1/2}$[@kivelson:1995]. Now, from the fact that at $r=\unit[1]{AU}$ we have $n_p=n_e=\unit[7]{cm^{-3}}$, $T_p=\unit[1.2\times 10^{5}]{K}$ and $T_e=\unit[1.4\times 10^{5}]{K}$[@kivelson:1995], we are able to deduce that such variables have the values $n=\unit[46]{cm^{-3}}$ (dropping the subscripts), $T_p=\unit[3.08\times 10^{5}]{K}$ and $T_e=\unit[2.24\times 10^{5}]{K}$ at $r=\unit[0.39]{AU}$.
The pressure is calculated by $p=nk_B(T_p+T_e)$ where $k_B$ is the Boltzmann constant, giving $p=\unit[3.38\times 10^{-9}]{dyn~cm^{-2}}$; besides, $\rho=n(m_p+m_e)$ with $m_p$ and $m_e$ representing the proton and electron masses, yielding $\rho=\unit[1.17\times 10^{-23}]{g~cm^{-3}}$. The values of $\rho$, $p$ and $\mathbf{B}$ calculated above are used as initial conditions of the domain (inside the planet we use different conditions, as explained subsequently). On the other hand, the initial $\mathbf{v}$ of the domain is given by $(v_{\mbox{\scriptsize{sw}}},v_{\phi},0)$. From Parker’s model we have $v_{\phi}=\Omega(r-b)\sin\theta$ which, in our case, yields the value $v_{\phi}=\unit[140]{km~s^{-1}}$. Table \[tab:1\] summarizes the initial $\rho$, $p$, $\mathbf{v}$ and $\mathbf{B}$ to be used in the simulations.
[cccc]{} $\rho$ & $p$ & $\mathbf{v}$ & $\mathbf{B}$\
($\times 10^{-23}$) & ($\times 10^{-9}$) & ($\times 10^{7}$) &\
$\unit{g~cm^{-3}}$ & $\unit{dyn~cm^{-2}}$ & $\unit{cm~s^{-1}}$ & $\unit{nT}$\
1.17 & 3.38 & (4.0,1.4,0) & (32.5,12.7,10.0)\
The stellar wind is represented as flowing from the border at $x=0$ of the domain with the velocity given by Table \[tab:1\]. In order to do so we employ the user defined boundary condition, defining at such a border the values for $\rho$, $p$, $\mathbf{v}$ and $\mathbf{B}$ given in Table \[tab:1\]. The outflow boundary condition, which stands for a zero normal gradient at the region being considered, is applied to the remaining edges. As a particular case shown in Appendix 1, we performed a simulation where we consider the user defined condition at the left ($x=0$), top and bottom boundaries, whereas at the right one we maintain the outflow condition.
The physical initial conditions inside the solid body are defined in the following way: $\mathbf{v}_{\mbox{\scriptsize{body}}}=0$, $\mathbf{B}_{\mbox{\scriptsize{body}}}=0$, $\rho_{\mbox{\scriptsize{body}}}=\unit[1.17\times 10^{-22}]{g~cm^{-3}}$ and $p_{\mbox{\scriptsize{body}}}=\unit[3.38\times 10^{-9}]{dyn~cm^{-2}}$. Actually, in preliminar simulations we tested different values for $\rho_{\mbox{\scriptsize{body}}}$ and $p_{\mbox{\scriptsize{body}}}$ and we verified that the results are not noticeably affected by the exact numerical choice of such parameters in the cases where they are greater than or equal to, respectively, $\rho$ and $p$ in Table \[tab:1\]. Despite the fact that in a typical planet $\rho\sim\unit[1]{g~cm^{-3}}$, we consider the mentioned value of $\rho_{\mbox{\scriptsize{body}}}$ for the sake of convenience in the treatment and visualization of the results. It is worth noting that inside rigid bodies the MHD equations do not evolve; further, the code applies the reflecting boundary condition at the surface of such objects.
Values of the viscosity to be used in the simulations {#viscosity}
-----------------------------------------------------
According to the model for the kinematic viscosity $\nu$ of the solar wind discussed in [@subramanian:2012], we have $\nu=\unit[300]{km^2~s^{-1}}$ at $r=\unit[0.39]{AU}$, giving us one of the values of $\nu$ to be employed in our scenarios. On the other hand, [@tejada:2005] presents a estimate of $\nu\sim\unit[1000]{km^2~s^{-1}}$ at $r=\unit[0.72]{AU}$ (the heliocentric distance of Venus), while the model by [@subramanian:2012] yields $\nu\approx\unit[600]{km^2~s^{-1}}$ at the same $r$. Such a fact suggests us that, according to the literature, there may be discordance about the evaluations of $\nu$ corresponding to each $r$. Therefore, besides considering $\nu=\unit[300]{km^2~s^{-1}}$, it would be interesting to simulate additional cases with different values of $\nu$. For this purpose we use $\nu=\unit[1000]{km^2~s^{-1}}$ and $\nu=\unit[5000]{km^2~s^{-1}}$. The latter is artificially high and was included in order to analyze the effects of the viscosity on the processes being simulated.
It is useful to define the Reynolds number $Re$, which may be written in function of $\nu$ as[@grigoriadis:2010]
$$\label{reynolds}
Re=\frac{uD}{\nu},$$
where $u$ is the velocity of the fluid ($\sqrt{v^{2}_{\mbox{\scriptsize{sw}}}+v^{2}_{\phi}}$ in our case) and $D$ represents a characteristic linear dimension of the body. Here, $D$ is considered as the diameter of the planet.
The model used in our simulations is essentially collisionless, once in such a formalism the viscosity is considered as totally caused by protons being scattered by “kinks” in the magnetic fields, while the proton-proton collisions are neglected in the deductions. Though such a model is suitable for our purposes, the solar wind may in fact be weakly collisional for the scales used here; with effect, strictly speaking, the wind is considered collisionless up to $\sim\unit[10]{R_{\odot}}$. See [@marsch:2006] for a detailed discussion.
It is worth mentioning that in the regions where the solar wind is collisional, we have the predominance of Coulomb collisions. Such processes have influence on the physical characteristics of the plasma, such as affecting the ion velocity distributions. See [@livi:1986] for details.
Results
=======
In this section we present the simulations for three cases: purely hydrodynamic, MHD with the initial $\mathbf{B}$ given by Table \[tab:1\] and MHD considering an initial $\mathbf{B}$ in the form $(0,12.7,10.0)~\unit{nT}$.
Purely hydrodynamic case {#PureHydro}
------------------------
Figure \[figs:1\] shows the density profiles in the xy-plane and at the instant $t=\unit[1200]{s}$ (after the vanishing of the transients present at the initial instants of the simulation) for the purely hydrodynamic case. We considered the values of $\rho$, $p$ and $\mathbf{v}$ in Table \[tab:1\] as initial parameters of the domain; besides, we used five levels of refinement. The dimensions of the box are in $\unit[10^{9}]{cm}$ and $\rho$ is in units of $\log(\rho/\unit[10^{-24}]{g~cm^{-3}})$.
The left profile in Fig. \[figs:1\] represents the case where we neglect the viscosity; the right one corresponds to $\nu=\unit[300]{km^2~s^{-1}}$ ($Re=17000$). We may note the formation of vortices past the planet, which tend to the right top region of the domain due to the presence of $v_{\phi}$. Comparing both panels we note that the differences between their correspondent patterns caused by the viscosity are very small for that particular value of $\nu$. Also, both scenarios are characterized by $\rho\approx\unit[3.0\times10^{-23}]{g~cm^{-3}}$ and $p\approx\unit[2.0\times 10^{-8}]{dyn~cm^{-2}}$ at the left side of the body and $\rho\approx\unit[1.0\times10^{-23}]{g~cm^{-3}}$ and $p\approx\unit[3.0\times 10^{-9}]{dyn~cm^{-2}}$ at right (in the wake between $x=\unit[7\times 10^{9}]{cm}$ and $x=\unit[9\times 10^{9}]{cm}$).
It is worth noting the shock seen between the mentioned wake and the vortices, where its left side is characterized by $\rho=\unit[1.3\times 10^{-23}]{g~cm^{-3}}$, $p=\unit[4\times 10^{-9}]{dyn~cm^{-2}}$ and $|\mathbf{v}|=\unit[3.5\times 10^{7}]{cm~s^{-1}}$, while the right one has $\rho=\unit[2.0\times 10^{-23}]{g~cm^{-3}}$, $p=\unit[1.5\times 10^{-8}]{dyn~cm^{-2}}$ and $|\mathbf{v}|=\unit[2.0\times 10^{7}]{cm~s^{-1}}$.
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Figure \[figs:2\] presents the velocity vector field and the vorticity profiles for the purely hydrodynamic simulations at $t=\unit[1200]{s}$ in the xy-plane. Note that we focus on the regions around the planet. The vorticity $\omega_z$ is calculated from $\mathbf{v}$ by means of
$$\label{vort}
\omega_z=\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}.$$
Four scenarios are considered: using $\nu=\unit[5000]{km^2~s^{-1}}$ ($Re=1020$), $\nu=\unit[1000]{km^2~s^{-1}}$ ($Re=5100$), $\nu=\unit[300]{km^2~s^{-1}}$ ($Re=17000$) and with no viscosity. The maximum value of $|\mathbf{v}|$ in the four profiles of Fig. \[figs:2\] are of $\approx\unit[5.0\times 10^7]{cm~s^{-1}}$. The length $L$ of the reciculation region is calculated from the surface of the planet (point $x=\unit[6.6\times 10^{9}]{cm},y=0$) until the edge of the circulation pattern seen at the right top of panels of Fig. \[figs:2\]. For the range of values considered here, $L$ slightly decreases as $Re$ increases: for $Re=1020$, $Re=5100$ and $Re=17000$ we have $L=\unit[2.3\times 10^{9}]{cm}$, $L=\unit[2.1\times 10^{9}]{cm}$ and $L=\unit[2.0\times 10^{9}]{cm}$, respectively (see Fig. \[figs:9\]); besides, $L=\unit[1.8\times 10^{9}]{cm}$ with no viscosity. On the other hand, the maximum value of $|\omega_z|$ increases as $Re$ increases (see the values of $|\omega_z|$ for each case in Fig. \[figs:2\] and the diagrams in Fig. \[figs:9\].) Note that, for convenience, $|\omega_z|$ for this case is shown multiplied by four in Fig. \[figs:9\].
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Our simulations might be compared to other results found in the literature. In fact, the particular case of the solar wind interacting with the Moon presented in [@spreiter:1970] has a characteristic in common with our hydrodynamic simulations: in both cases there is no formation of bow shock, once the authors of such a paper considered that the Moon has no magnetic field and ionosphere to deflect the solar wind.
Observing Fig. \[figs:2\] we note that the fluid is deflected as it pass around the planet once that in FLASH the surface of rigid bodies is treated as a reflecting boundary. However, the velocities of the fluid drop nearly to zero at the region where it reaches radially the surface of the planet. On the other hand, we should bear in mind that in [@spreiter:1970] the particles of the solar plasma which hit the lunar surface are stopped and removed from the flow. Both scenarios are intrinsically different once in [@spreiter:1970] the fluid is absorbed by the surface of the body.
In [@grigoriadis:2010] it is shown, among other results, the influence of the viscosity on the size of the recirculation regions for a hydrodynamic fluid interacting with a cylinder. The authors found that, for $Re\gtrapprox50$, higher values of $Re$ are related to smaller $L$. We observed a similar behavior in our hydrodynamic simulations, though the geometry of the body in [@grigoriadis:2010] is not the same as the one used here (see Fig. \[figs:9\]).
MHD scenario with initial $B_x=\unit[32.5]{nT}$ {#BxDiffZero}
-----------------------------------------------
Figure \[figs:3\] presents the density profiles of the MHD simulations at $t=\unit[1400]{s}$ in the xy-plane (left panels) and xz-plane (righ panels) with the initial conditions of the domain given in Table \[tab:1\] and using five levels of refinement; the upper and lower panels correspond respectively to the scenarios with no viscosity and considering $\nu=\unit[300]{km^2~s^{-1}}$ ($Re=17000$). Generally speaking, there is the formation of a thick, distinct bow shock with $\rho\approx\unit[3\times 10^{-23}]{g~cm^{-3}}$ and $p\sim\unit[1\times 10^{-8}]{dyn~cm^{-2}}$. The low-density tails are characterized by $\rho\sim\unit[10^{-24}]{g~cm^{-3}}$ and $p\sim\unit[1\times 10^{-7}]{dyn~cm^{-2}}$ at their central regions in both scenarios. Further, note that, for five levels of refinement, the viscosity has no noticeable effects on the density profiles.
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Besides, for the sake of testing the convergence of the solutions (bearing in mind the numerical dissipation effects), the simulation for $Re=17000$ was obtained using seven levels of refinement (shown in Fig. \[figs:3b\]).
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Figures \[figs:3\] and \[figs:3b\] show the outlines of the mesh refinement. The most refined areas are along the shocks, as well as around the object and where $\rho$ and $\mathbf{B}$ present variations (see Fig. \[figs:4\] too), indicating us that PARAMESH remains stable in MHD simulations with solid objects under the present conditions.
Comparing Figs. \[figs:3\] and \[figs:3b\] we may note that the wake is thinner for seven levels when compared to the other scenarios. Such a behavior is related to the refinement of the solutions: the higher the refinement, the thinner the wakes, with their dimensions converging to a particular value for sufficiently high refinements. With effect, concerning the dimensions of the wake, the simulation with six levels (left panel of Fig. \[figs:8b\]) represents an intermediate case between the less and the more refined ones.
Note the structures in the wake of the right profile of Fig. \[figs:3b\]. A closer view of these structures is shown in Fig. \[figs:4b\], which shows the density in colors (same scale as in Fig. \[figs:3\] and \[figs:3b\]) and $\mathbf{B}_{xz}=\{B_{x},B_{z}\}$ as a vector field. The vectors of $\mathbf{B}_{xz}$ are not scaled by magnitude for a better visualization but $|\mathbf{B}_{xz}|$ has a maximum value of $\sim\unit[100]{nT}$ in the region. The behavior of $\mathbf{B}_{xz}$ in Fig. \[figs:4b\] is suggestive of a magnetic reconnection process possibly happening in such a region.
It is interesting to observe that, though the stellar wind is parallel to the x-axis, there is no symmetry around $y=0$ in Fig. \[figs:3\] and \[figs:3b\]. We explain this behavior as follows: as the simulation evolves, the plasma starting with velocity $\mathbf{v}=v_{\mbox{\scriptsize{sw}}} \boldsymbol{\hat{\textbf{i}}}$ undergoes magnetic forces due to $B_y$ and $B_z$, causing the emergence of $v_y$ and $v_z$ components in the fluid velocities (though some of $v_y$ and $v_z$ arises from the interaction with the rigid body). Then $B_x$ exerts forces transverse to the x-axis on the portions of the fluid where $v_y$ and $v_z$ are different from zero.
We plot the magnetic field at $t=\unit[1400]{s}$, shown in Fig. \[figs:4\]. The perspective is from the xy-plane, with the components $B_x$ and $B_y$ represented as a vector field and $B_z$ in color plot. The left and right panels correspond to the scenarios with no viscosity and with $\nu=\unit[300]{km^2~s^{-1}}$, respectively. In both cases we have $\sqrt{B^2_x+B^2_y}\approx\unit[300]{nT}$ around the planet and $B_z\approx\unit[13]{nT}$ along the bow shock; besides, in the wake $|\mathbf{B}|$ has the lowest values.
Analyzing the initial $\mathbf{B}$ given in Table \[tab:1\], we deduce that the interaction of the wind with the body increases $\sqrt{B^2_x+B^2_y}$ by a factor of $\approx 8.5$, while the values of $B_z$ remains of the same order of magnitude.
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![Zoom of the structures in the wake of the right bottom panel of Fig. \[figs:3\]. The density is shown in colors (same scale as in Fig. \[figs:3\]) and the vector field (not scaled by magnitude) represents $\mathbf{B}_{xz}=\{B_{x},B_{z}\}$.[]{data-label="figs:4b"}](Fig15.pdf){width="0.7\linewidth"}
Figure \[figs:5\] shows the velocity vector field and the vorticity profiles for the MHD simulations at $t=\unit[1400]{s}$ in the xy-plane. As in the previous case, four scenarios are considered: with $\nu=\unit[5000]{km^2~s^{-1}}$ (Re=$1020$), $\nu=\unit[1000]{km^2~s^{-1}}$ (Re=$5100$), $\nu=\unit[300]{km^2~s^{-1}}$ (Re=$17000$) and with no viscosity. The four scenarios were generated with five levels of refinement. The maximum value of the velocity in the four scenarios of Fig. \[figs:5\] are of $\approx\unit[3.0\times 10^8]{cm~s^{-1}}$. Here we have $L\approx\unit[1.7\times 10^{9}]{cm}$ for the four scenarios; the maximum $|\omega_z|$ slightly increases with $Re$ and its values may be observed in Fig. \[figs:5\] and Fig. \[figs:9\].
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MHD scenario with initial $B_x=0$ {#BxEqualZero}
---------------------------------
As an extra result, we performed simulations using the same parameters as the ones shown in Subsection \[BxDiffZero\] but considering $B_x=0$ in the initial conditions. Though this scenario is not realistic, once from Parker’s model $B_r/B_{\phi} \ll 1$ only for large heliocentric distances, it will help us to observe the influence of the transversal components of $\mathbf{B}$ on the interaction of the wind with the planet. The densities and mesh refinement with five levels at $t=\unit[1400]{s}$ are shown in Fig. \[figs:6\]. We note that there is symmetry about $y=0$ and, as in the previous MHD case, it is formed a discernible bow shock. The bow shocks in Fig. \[figs:6\] are characterized by $\rho=\unit[1\times 10^{-23}]{g~cm^{-3}}$ and $p=\unit[2.0\times 10^{-8}]{dyn~cm^{-2}}$, while the wakes have $\rho\sim \unit[10^{-24}]{g~cm^{-3}}$ and $p=\unit[7.0\times 10^{-8}]{dyn~cm^{-2}}$ at their central regions.
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Concerning the mesh refinement, we see that the most refined areas are around the shocks, object and wakes, following the variations of $\rho$ and $\mathbf{B}$. As in the previous MHD scenario, we point out the stability of the numerical schemes that are integrated in the PARAMESH structure in this case.
Figure \[figs:7\] for $\mathbf{B}$ follows the same scheme of Fig. \[figs:4\]. We have $\sqrt{B^2_x+B^2_y}=\unit[143]{nT}$ (left panel) and $\sqrt{B^2_x+B^2_y}=\unit[149]{nT}$ (right panel) around the planet and $|B_z|$ has maximun values of $\approx\unit[19]{nT}$. We see that $\sqrt{B^2_x+B^2_y}$ is increased by the factors $\approx 4.1$ (left panel) and $\approx 4.3$ (right panel) when compared to its initial value; $B_z$ reachs values which are $1.9$ higher than the initial one. In Figs. \[figs:4\] and \[figs:7\] we observe a pattern of circulation of $\mathbf{B}$ in the xy-plane. Particularly, in the inner regions of the wake we have magnetic field lines which are oppositely directed and are close to each other. Under certain circumstances, such a behavior could potentially create suitable conditions for the onset of magnetic reconnection.
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The velocity vector field and the vorticity profiles at $t=\unit[1400]{s}$ are given in Fig. \[figs:8\]. The scheme is similar to Fig. \[figs:5\]. However, note that here, in order to better observe the recirculation zones, the arrows of the velocity fields are not scaled by magnitude. We have $L=\unit[5.8\times 10^{8}]{cm}$, $L=\unit[9.6\times 10^{8}]{cm}$ and $L=\unit[1.7\times 10^{9}]{cm}$ for increasing values of $Re$ ($L=\unit[3.3\times 10^{9}]{cm}$ with no viscosity); $|\omega_z|$ increases between $Re=1020$ and $Re=5100$ and its values are shown in Fig. \[figs:8\] and Fig. \[figs:9\].
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In the MHD simulations there is the formation of a low-density layer between the object and the interacting stellar wind, which has a minimum thickness of, for example, $\approx \unit[1.3\times 10^{8}]{cm}$ in the upper panels of Fig. \[figs:3\] and $\approx \unit[2.0\times 10^{8}]{cm}$ in Fig. \[figs:6\]. As this phenomenon is not present in the hydrodynamic case we deduce that it is mainly due to the action of $\mathbf{B}$ which, moreover, reach its maximum values in the areas adjacent to the object.
In [@spreiter:1970], the scenario of the interaction of the solar wind with Venus (considered as having no significant magnetic field) presents a bow shock similar to the ones in our MHD simulations; besides, in such a scenario there is a low-density layer of thickness $\unit[5\times 10^7]{km}$ between the shock and Venus. According to the authors, that layer is formed when the ionosphere of the planet deflects the solar wind, preventing it to hit the surface. Though in our model the planet has no atmosphere, the action of $|\mathbf{B}|$ around the body produced a similar effect, as explained in the previous paragraph.
The influence of the viscosity on the length of the recirculation zone in MHD simulations may be found, for example, in [@grigoriadis:2010]. In this paper, the MHD scenarios (with streamwise and transverse magnetic fields) for $Re=100$ have, generally speaking, higher $L$ when compared to the cases where $Re=40$.
Influence of the boundaries on the simulations {#comparison}
----------------------------------------------
Though we are using outflow boundary conditions, it would in principle be possible that some interaction at the borders could propagate back to the domain and influence the results of the simulations. In order to investigate the influence of the boundaries on our results, we performed the simulation of the MHD scenario with initial $B_x=\unit[32.5]{nT}$, $\nu=\unit[300]{km^2~s^{-1}}$ and five levels of refinement using domains with two sizes: $x\in [0.0,18.0] \unit[\times 10^{9}]{cm}$, $y\in [-12.0,6.0] \unit[\times 10^{9}]{cm}$, $z\in [-12.0,6.0] \unit[\times 10^{9}]{cm}$ and $x\in [0.0,12.0] \unit[\times 10^{9}]{cm}$, $y\in [-6.0,6.0] \unit[\times 10^{9}]{cm}$, $z\in [-6.0,6.0] \unit[\times 10^{9}]{cm}$. Figure \[figs:8b\] presents the density panels at $t=\unit[1400]{s}$ for the bigger (center) and smaller (right) domains. Besides, for the sake of comparison, we show a simulation with the same domain and conditions than the one of the center panel but using six levels of refinement (left.) Note that the center profile is the same as the one presented in Fig. \[figs:3\] (bottom left panel). It was shown here again to facilitate a visual comparison.
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{width="0.32\linewidth"} {width="0.32\linewidth"} {width="0.32\linewidth"}
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From Fig. \[figs:8b\] we note that the center and right profiles have essentially the same characteristics; we do not observe patterns which would potentially be caused by “back reactions” of the boundaries. The patterns in the form of shocks in the right bottom of the panels are created near the planet at the first instants of the simulation and propagate from the left. In Fig. \[figs:6\] the reader may observe similar patterns above the planet in the left panels.
Though the center and right panels in Fig. \[figs:8b\] have similar characteristics, we may note that they are not equal. We explain the difference between that two cases as follows: in both scenarios we started with the same number of blocks, that is, $3\times3\times3$ in the first level and reaching to $48\times 48\times 48$ in the fifth (see Section \[numerical\]). So, the domain at right in Fig. \[figs:8b\] is smaller than the other but has the same number of blocks, such that it seems “more refined” (see the discussion in Subsection \[BxDiffZero\]). With effect, we may compare the center and right profiles in Fig. \[figs:8b\] to the left one and to the cases with seven levels of refinement in Fig. \[figs:3\]. Particularly, note the similarity between the right and left profiles in Fig. \[figs:8b\]. We conclude that the size of the domain do has influence on the results in the sense of refinement, as explained above.
Conclusions
===========
In this paper we simulated the interaction between the wind produced by a sun-like star and a non-magnetized planet. Such a planet has the approximate size of Earth and an orbital radius of $\unit[0.39]{AU}$, which corresponds to the mean distance between the sun and Mercury. We used the FLASH code to simulate hydrodynamic and MHD scenarios, having as purpose to implement and test the inclusion of a solid and stationary object in MHD simulations in this code. The results presented here are new and interesting once the tool for simulating solid bodies in FLASH is currently implemented and tested for hydrodynamic cases only.
The hydrodynamic simulation used as initial parameters of the domain the values of $\mathbf{v}$, $\rho$ and $p$ shown in Table \[tab:1\], besides a maximum time of $\unit[1200]{s}$ and five levels of refinement. We presented the profiles of $\rho$ in the xy-plane for the scenarios with no viscosity and with $\nu=\unit[300]{km^2~s^{-1}}$; besides, we plotted the velocity fields and the vorticity for $Re=1020$, $Re=5100$, $Re=17000$ and no viscous. The differences in the profiles of $\rho$ between the two cases are very small, while the velocity fields indicated us that $L$ slightly decreases (and $|\omega_z|$ increases) as $Re$ increases. See Fig. \[figs:9\].
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{width="0.47\linewidth"} {width="0.47\linewidth"}
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For the MHD scenario we considered the same initial parameters of the domain as in the hydrodynamic simulation besides adding $\mathbf{B}$. We used Parker’s model to define the initial $B_x$ and $B_y$, whereas the values of $B_z$ was estimated by means of observations from the spacecrafts MESSENGER and Helios[@korth:2011], giving $B_x=\unit[32.5]{nT}$, $B_y=\unit[12.7]{nT}$ and $B_z=\unit[10]{nT}$. As an extra MHD result, we simulated the case where we have initially $B_x=0$ which, though is not realistic, helped us to investigate the influence of the transversal components of $\mathbf{B}$ on the simulations.
For the MHD simulations with initial $B_x=\unit[32.5]{nT}$ we shown the profiles of $\rho$ and the outlines of the mesh refinement in the xy and xz-plane: using five levels for the cases with $Re=17000$ and with no viscosity; and seven levels for $Re=17000$. Besides, we present $\mathbf{B}$ and the velocity (with vorticity) fields in the xy-plane. The velocity fields corresponded to $Re=1020$, $Re=5100$, $Re=17000$ and no viscous scenarios. We observed the formation of a bow shock with $\rho\approx\unit[3.0\times 10^{-23}]{g~cm^{-3}}$ and $p\sim\unit[1.0\times 10^{-8}]{dyn~cm^{-2}}$ and a wake with $\rho\sim\unit[10^{-24}]{g~cm^{-3}}$ and $p=\unit[1.0\times 10^{-7}]{dyn~cm^{-2}}$ at its central line. We observed that, in our simulations with five levels of refinement, the viscosity has no noticeable effects on the density profiles. Besides, we briefly discussed the simulation with six levels of refinement (left panel of Fig. \[figs:8b\]) and we concluded that the solutions converge for $Re=17000$. The interaction of the wind with the planet causes the increase in $|\mathbf{B}|$ around the body when compared to its initial values: $\sqrt{B_x^2+B_y^2}$ is higher by a factor $8.5$ and $|B_z|$ remains of the same order of magnitude when compared to the initial conditions. We investigated the possible occurrence of magnetic reconnection in a case where $\nu=\unit[300]{km^2~s^{-1}}$ (right panel of Fig. \[figs:3b\]). The velocity and vorticity fields of Fig. \[figs:5\], as well as Fig. \[figs:9\], show that $\omega_z$ slightly increases as $Re$ increases while $L$ remains approximately with the same size in the four cases.
In the case where we have and initial $B_x=0$ and using five levels of refinement, we observed the characteristics: the bow shock has $\rho=\unit[1.0\times 10^{-23}]{g~cm^{-3}}$ and $p=\unit[2.0\times 10^{-8}]{dyn~cm^{-2}}$; the wake is characterized by $\rho\sim\unit[10^{-24}]{g~cm^{-3}}$ and $p=\unit[7.0\times 10^{-8}]{dyn~cm^{-2}}$ at its inner regions. Contrary to the previous MHD case, these results present symmetry around $y=0$. Besides, $\sqrt{B_x^2+B_y^2}$ around the body is higher by a factor $4.1-4.3$ than the value calculated from the initial conditions, while $|B_z|$ is higher by a factor $1.9$.
Figure \[figs:8\] and Fig. \[figs:9\] show us that $L$ increases from $Re=1020$ to $Re=17000$, while $\omega_z$ increases between $Re=1020$ and $Re=5100$. For the scenario with no viscosity, $L=\unit[3.3\times 10^{9}]{cm}$. As in the previous MHD case, we observed higher refinement along the shocks, around the object and other regions where $\rho$ and $\mathbf{B}$ present variations, indicating us that PARAMESH remained stable in those cases.
The presence of an initial $B_x$ different from zero in the MHD simulations causes the loss of symmetry around the x-axis both in y and z-directions. We explained such a behavior as the action of the component $B_x$ on the portions of the fluid with $v_y\neq 0$ and $v_z\neq 0$, generating a dominant force in the $y$ and $z$-direction. The absence of a bow shock in our purely hydrodynamic simulations is a characteristic observed in the interaction of the solar wind with the Moon found in [@spreiter:1970]. Still in [@spreiter:1970], the interaction of the wind with Venus has some features in common with our MHD scenarios: the presence of a bow shock and the formation of a low-density layer between the shock and the object. In our case, this phenomenon is mainly due to the action of $|\mathbf{B}|$ around the body, while in [@spreiter:1970] it is caused by the ionosphere of the planet.
The influence of the viscosity on $L$ shown in [@grigoriadis:2010] for the hydrodynamic case is similar to the one deduced from Fig. \[figs:2\]: for $Re\gtrapprox50$, higher $Re$ are related to smaller $L$; in [@grigoriadis:2010], the MHD case with $Re=100$ has, generally speaking, higher $L$ when compared to the scenario with $Re=40$. In our scenario with initial $B_x=0$, $L$ increases with $Re$.
We investigated the potential influence of the size and borders of the domain on the simulations. In the case used as example, we did not observe patterns which would be caused by the influence of the boundaries; however, we deduced that the size of the domain has effect on the refinement of the solutions.
From all the discussions presented here, we concluded that, under the conditions considered in this paper, our scheme generated promising results and it creates new perspectives for using the FLASH code in realistic simulations of planets interacting with stellar winds. For example, it is known that Mercury has a tenuous exosphere which undergoes strong variations between the perihelion and the aphelion, making interesting the inclusion of objects with atmospheres in future works in order to study such scenarios. We will investigate in more details the effects of higher levels of refinement on the simulations, as well as the influence of the sizes of the domain on the results. Further, we will consider scenarios with different boundary conditions and investigate how their choice affect the results.
The authors acknowledge INPE for providing the necessary computer resources. FLASH code was in part developed by the DOE NNSA-ASC OASCR Flash Center at the University of Chicago. **Funding information** EFDE acknowledges Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, PCI/INPE program, grant 300887/2017-5); OM, MOD and ODM acknowledge Financiadora de Estudos e Projetos (FINEP, under agreement 01.12.0527.00), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP, grant 2015/25624-2), CNPq (grants 424352/2018-4 and 307083/2017-9) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).
Appendix 1 {#Ap1 .unnumbered}
==========
Figure \[figs:10\] show the xy-plane of the simulation for a scenario similar to the one of the top panels of Fig. \[figs:3\] but considering the user defined condition at the left ($x=0$), top and bottom boundaries; at the right one we maintain the outflow condition.
We note that the lower region of the wake in Fig. \[figs:10\] is slightly wider than the one observed in the upper left panel of Fig. \[figs:3\]. This feature, probably, is due to the stellar wind flowing from the lower boundary, once we are considering that $v_y\neq 0$.
![Same as in the top left panel of Fig. \[figs:3\] but considering the user defined condition at the left ($x=0$), top and bottom boundaries; at the right one we use the outflow condition.[]{data-label="figs:10"}](Fig35.pdf){width="1.0\linewidth"}
Appendix 2 {#appendix-2 .unnumbered}
==========
This Appendix yields further computational details of the simulations shown in this paper and it would be of special interest for those readers which have some familiarity with the FLASH code.
All the necessary files to the MHD simulation are placed in the folder . Such files are:
- `Makefile.h`: contains auxiliary instructions used to compile the particular problem being treated.
- `Config`: in this file we specify the required units and define the default runtime parameters. Particularly, we used the units and .
- `flash.par`: in this file we define the initial runtime parameters such as the initial values of the physical quantities, boundary conditions, maximum level of refinement and the Riemann solver being used. See Section \[numerical\] for such parameters.
- `Simulation_data.F90`: this module stores data specific to the problem being simulated.
- `Simulation_init.F90`: this routine gets the necessary parameters and initialize other variables in the module.
- `Simulation_initBlock.F90`: it applies the initial conditions, as well as rigid bodies and other desired particularities, to the domain of the problem. Here we insert a body in the form of a sphere of radius $R$ in the simulation by means of the algorithm:
r$=\sqrt{(x_i-x_{c})^{2}+(y_i-y_{c})^{2}+(z_i-z_{c})^{2}}$ $\mbox{VAR(BDRY)}=-1$ $\mbox{VAR(BDRY)}=1$
where $(x_i,y_i,z_i)$ and $(x_c,y_c,z_c)$ are the coordinates of the i-th cell of the domain and of the center of the sphere, respectively. The variable BDRY is defined in such a way that it has the value $+1$ in the cells inside the object; in the rest of the domain we have $\mbox{VAR(BDRY)}=-1$. Besides, inside the sphere the physical parameters have the particular values discussed in Section \[numerical\].
- `Grid_bcApplyToRegionSpecialized.F90`: a default version of this module is found in the folder . We use it to define specific boundary conditions at the left edge of the domain, describing the stellar wind flowing toward the body, as explained in Section \[numerical\].
The files `Makefile.h`, `Simulation_data.F90` and `Simulation_init.F90` have the standard form used in many of the supplied test problems implemented in FLASH4, which are placed in the folder .
In order to compile and run our MHD simulation, we use the following commands:
`.\setup -auto -<n>d magnetoHD/StarPlanetInt +usm` `cd object` `make` `mpirun -np N flash4`
where $<$n$>$ is the number of dimensions of the simulation and N is the number of processors being used.
The files used in the pure hydrodynamic scenario are placed in . They are similar to the ones of the MHD case, but with the following modifications:
- we exclude from the files all the variables related to the magnetic field, including `killdivb`.
- in `Config` we use the unit instead of .
To compile and run the pure hydrodynamic simulation, we use:
`.\setup -auto -<n>d StarPlanetInt +uhd` `cd object` `make` `mpirun -np N flash4`
T. Takahashi, H. Ueki, A. Kunimatsu, H. Fujii, in *ACM SIGGRAPH 2002 Conference Abstracts and Applications* (ACM, New York-USA, 2002), p.266. <https://doi.org/10.1145/1242073.1242279>
N. Takashi, T.J.R. Hughes, Comput. Method. Appl. M. **95**, 115–138 (1992). <https://doi.org/10.1016/0045-7825(92)90085-X>
K.G. Powell, P.L. Roe, T.J. Linde, T.I. Gombosi, D.L. [De Zeeuw]{}, J. Comp. Phys. **154**, 284–309 (1999). <https://doi.org/10.1006/jcph.1999.6299>
D.G.E. Grigoriadis, I.E. Sarris, S.C. Kassinos, Comput. Fluids **39**, 345–258 (2010). <https://doi.org/10.1016/j.compfluid.2009.09.012>
C.P. Johnstone, et al., Astron. Astrophys. **577**, A122 (2015). <https://doi.org/10.1051/0004-6361/201425134>
Y. Vernisse, et al., Planet. Space Sci. **84**, 37–47 (2013). <https://doi.org/10.1016/j.pss.2013.04.004>
O. Cohen, et al., Astrophys. J. **806**(1), 41 (2015). <https://doi.org/10.1088/0004-637X/806/1/41>
J.D. Nichols, S.E. Milan, Mon. Not. R. Astron. Soc. **461**, 2353–2366 (2016). <https://doi.org/10.1093/mnras/stw1430>
V. Bourrier, et al., Astron. Astrophys. **591**, A121 (2016). <https://doi.org/10.1051/0004-6361/201628362>
J.R. Spreiter, A.L. Summers, A.W. Rizzi, Planet. Space Sci. **18**, 1281–1299 (1970). <https://doi.org/10.1016/0032-0633(70)90139-X>
M. Dryer, A.W. Rizzi, Wen-Wu Shen, Astrophys. Space Sci. **22**, 329–351 (1973). <https://doi.org/10.1007/BF00647431>
J.D. Alvarado-G[ó]{}mez, et al., Astron. Astrophys. **594**, A95 (2016). <https://doi.org/10.1051/0004-6361/201628988>
G. Bateman, *MHD Instabilities* (The MIT Press, Massachusetts-USA, 1978).
D. Biskamp, *Magnetohydrodynamic Turbulence* (Cambridge University Press, Cambridge-UK, 2003).
J.P. Goedbloed, S. Poedts, *Principles of Magnetohydrodynamics* (Cambridge University Press, Cambridge-UK,2004).
A.E. Lifschitz, *Developments in Electromagnetic Theory and Applications: Magnetohydrodynamics and Spectral Theory* (Kluwer Academic Publishers, Dordrecht-The Netherlands, 1989).
P. MacNeice, et al., Comput. Phys. Commun. **126**(3), 330–354 (2000). <https://doi.org/10.1016/S0010-4655(99)00501-9>
R. L[ö]{}hner, Comp. Meth. App. Mech. Eng. **61**(3), 323–338 (1987). <https://doi.org/10.1016/0045-7825(87)90098-3>
B. Fryxell, et al., Astrophys. J. Suppl. S. **131**(1), 273–334 (2000). <https://doi.org/10.1086/317361>
P.L. Roe, J. Comp. Phys. **43**(2), 357–372 (1981). <https://doi.org/10.1016/0021-9991(81)90128-5>
S. Li, J. Comput. Phys. **203**(1), 344–357 (2005). <https://doi.org/10.1016/j.jcp.2004.08.020>
B. Einfeldt, C.D. Munz, P.L. Roe, B. Sjögreen, J. Comput. Phys. **92**(2), 273–295 (1991). <https://doi.org/10.1016/0021-9991(91)90211-3>
B. van Leer, SIAM J. Sci. Stat. Comp. **5**(1), 1–20 (1984). <https://doi.org/10.1137/0905001>
D. Lee, A.E. Deane, J. Comput. Phys. **228**(4), 952–975 (2009). <https://doi.org/10.1016/j.jcp.2008.08.026>
R. Courant, K. Friedrichs, H. Lewy, IBM J. Res. Dev. **11**(2), 215–234 (1967). <https://doi.org/10.1147/rd.112.0215>
E.N. Parker, Astrophys. J. **128**, 664–676 (1958). <https://doi.org/10.1086/146579>
M.G. Kivelson, C.T. Russell (ed.), *Introduction to Space Physics* (Cambridge University Press, Cambridge-UK,1995).
R.C. Tautz, A. Shalchi, A. Dosch, J. Geophys. Res.: Space Phys. **116**(A2), A02102 (2011). <https://doi.org/10.1029/2010JA015936>
H. Korth, et al., Planet. Space Sci. **59**, 2075–2085 (2011). <https://doi.org/10.1016/j.pss.2010.10.014>
P. Subramanian, A. Lara, A. Borgazzi, Geophys. Res. Lett. **39**(19), L19107 (2012). <https://doi.org/10.1029/2012GL053625>
H. P[é]{}rez-de-Tejada, Astrophys. J. Lett. **618**(2), L145–L148 (2005). <https://doi.org/10.1086/425864>
E. Marsch, Living Rev. Solar Phys. **3**(1), 1 (2006). <https://doi.org/10.12942/lrsp-2006-1>
S. Livi, E. Marsch, H. Rosenbauer, J. Geophys. Res.: Space Phys. **91**(A7), 8045–8050 (1986). <https://doi.org/10.1029/JA091iA07p08045>
[^1]: <http://flash.uchicago.edu/site/flashcode>
|
---
abstract: 'We propose a multi-stage learning approach for pruning the search space of maximum clique enumeration, a fundamental computationally difficult problem arising in various network analysis tasks. In each stage, our approach learns the characteristics of vertices in terms of various neighborhood features and leverage them to prune the set of vertices that are likely *not* contained in any maximum clique. Furthermore, we demonstrate that our approach is domain independent – the same small set of features works well on graph instances from different domain. Compared to the state-of-the-art heuristics and preprocessing strategies, the advantages of our approach are that (i) it does not require any estimate on the maximum clique size at runtime and (ii) we demonstrate it to be effective also for dense graphs. In particular, for dense graphs, we typically prune around 30 % of the vertices resulting in speedups of up to 53 times for state-of-the-art solvers while generally preserving the size of the maximum clique (though some maximum cliques may be lost). For large real-world sparse graphs, we routinely prune over 99 % of the vertices resulting in several tenfold speedups at best, typically with no impact on solution quality.'
author:
- Marco Grassia$^1$
- Juho Lauri$^2$
- |
Sourav Dutta$^3$Deepak Ajwani$^4$ $^1$University of Catania, Italy\
$^2$Nokia Bell Labs, Ireland\
$^3$Eaton Corp., Ireland\
$^4$University College Dublin, Ireland marco.grassia@studium.unict.it, juho.lauri@gmail.com, souravdutta@eaton.com, deepak.ajwani@ucd.ie
bibliography:
- 'ijcai19.bib'
title: 'Learning Multi-Stage Sparsification for Maximum Clique Enumeration[^1]'
---
Introduction
============
A large number of optimization problems in diverse domains such as data mining, decision-making, planning, routing and scheduling are computationally hard (i.e., ${\textsc{NP}}$-hard). No efficient polynomial-time algorithms are known for these problems that can solve every instance of the problem to optimality and many researchers consider that such algorithms may not even exist. A common way to deal with such optimization problems is to design heuristics that leverage the structure in real-world instance classes for these problems. This is a time-consuming process where algorithm engineers and domain experts have to identify the key characteristics of the instance classes and carefully design algorithm for optimality on instances with those characteristics.
In recent years, researchers have started exploring if machine learning techniques can be used to (i) automatically identify characteristics of the instance classes and (ii) learn algorithms specifically leveraging those characteristics. In particular, recent advances in deep learning and graph convolutional networks have been used in an attempt to directly *learn* the output of an optimization algorithm based on small training examples (see e.g., [@Vinyals2015; @Bello2016; @Nowak2017]). These approaches have shown promising early results on some optimization problems such as the Travelling Salesman Problem (TSP). However, there are two fundamental challenges that limit the widespread adoption of these techniques: (i) requirement of large amounts of training data whose generation requires solving the ${\textsc{NP}}$-hard optimization problem on numerous instances and (ii) the resultant lack of scalability (most of the reported results are on small test instances).
Recently, [@our-nips] proposed a probabilistic preprocessing framework to address the above challenges. Instead of directly learning the output of the ${\textsc{NP}}$-hard optimization problem, their approach learns to prune away a part of the input. The reduced problem instance can then be solved with exact algorithms or constraint solvers. Because their approach merely needs to learn the elements of the input it can confidently prune away, it needs significantly less training. This also enables it to scale to larger test instances. They considered the problem of maximum clique enumeration and showed that on sparse real-world instances, their approach pruned 75-98% of the vertices. Despite the conceptual novelty, the approach still suffered from (i) poor pruning on dense instances, (ii) poor accuracy on larger synthetic instances and (iii) non-transferability of training models across domains. In this paper, we build upon their work and show that we can achieve a significantly better accuracy-pruning trade-off, both on sparse and dense graphs, as well as cross-domain generalizability using a multi-stage learning methodology.
**Maximum clique enumeration** We consider the maximum clique enumeration (MCE) problem, where the goal is to list all *maximum* (as opposed to maximal) cliques in a given graph. The maximum clique problem is one of the most heavily-studied combinatorial problems arising in various domains such as in the analysis of social networks [@soc; @Fortunato2010; @Palla2005; @Papadopoulos2012], behavioral networks [@beha], and financial networks [@finan]. It is also relevant in clustering [@dynamic; @Yang2016] and cloud computing [@Wang2014; @Yao2013]. The listing variant of the problem, MCE, is encountered in computational biology [@bio; @Eblen2012; @Yeger2004; @mce] in problems like the detection of protein-protein interaction complex, clustering protein sequences, and searching for common cis-regulatory elements [@protein].
It is ${\textsc{NP}}$-hard to even approximate the maximum clique problem within $n^{1-\epsilon}$ for any $\epsilon > 0$ [@Zuckerman2006]. Furthermore, unless an unlikely collapse occurs in complexity theory, the problem of identifying if a graph of $n$ vertices has a clique of size $k$ is not solvable in time $f(k) n^{o(k)}$ for any function $f$ [@Chen2006]. As such, even small instances of this problem can be non-trivial to solve. Further, under reasonable complexity-theoretic assumptions, there is no polynomial-time algorithm that preprocesses an instance of $k$-clique to have only $f(k)$ vertices, where $f$ is any computable function depending solely on $k$ (see e.g., [@fpt-book]). These results indicate that it is unlikely that an efficient preprocessing method for MCE exists that can reduce the size of input instance drastically while guaranteeing to preserve all the maximum cliques. In particular, it is unlikely that polynomial-time sparsification methods (see e.g., [@Batson2013]) would be applicable to MCE. This has led researchers to focus on heuristic pruning approaches.
A typical preprocessing step in a state-of-the-art solver is the following: (i) quickly find a large clique (say of size $k$), (ii) compute the core number of each vertex of the input graph $G$, and (iii) delete every vertex of $G$ with core number less than $k-1$. This can be equivalently achieved by repeatedly removing all vertices with degree less than $k$. For example, the solver `pmc` [@Rossi2015b] – which is regarded as “*the* leading reference solver” [@San2016] – use this as the only preprocessing method. However, there are two major downsides to this preprocessing step. First, it is crucially dependant on $k$, the size of a large clique found. Since the maximum clique size is ${\textsc{NP}}$-hard to approximate within a factor of $n^{1-\epsilon}$, maximum clique estimates with no formal guarantees are used. Second and more important, it is typical that even if the estimate $k$ was equal to the size of a maximum clique in $G$, the core number of most vertices could be considerably higher than $k-1$. This is particularly true in the case of dense graphs and it results in little or *no* pruning of the search space. Similarly, other preprocessing strategies (see e.g., [@Eblen2010] for more discussion) depend on ${\textsc{NP}}$-hard estimates of specific graph properties and are not useful for pruning dense graphs.
#### Our Results
We demonstrate 30 % vertex pruning rates on average for dense networks, for which exact state-of-the-art methods are not able to prune anything, while typically only compromising the number of maximum cliques and not their size. For sparse networks, our preprocessor typically prunes well over 99 % of the vertices without compromising the solution quality. In both cases, these prunings result in speedups as high as several tenfold for state-of-the-art MCE solvers. For example, after the execution of our multi-stage preprocessor, we correctly list all the 196 maximum cliques (of size 24) in a real-world social network (socfb-B-anon) with 3 M vertices and 21 M edges in only 7 seconds of solver time, compared with 40 minutes of solver time with current state-of-the-art preprocessor (see Table \[tbl:pruning\]).
Preliminaries and Related Work
==============================
Let $G=(V,E)$ be an undirected simple graph. A *clique* is a subset $S \subseteq V$ such that every two distinct vertices of $S$ are adjacent. We say that the vertices of $S$ form a $k$-clique when $|S| = k$. The *clique number* of $G$, denoted by $\omega(G)$, is the size of a maximum clique in $G$. A *$k$-coloring* of $G$ is a function $c: V \to \{1, \ldots, k\}$. A *coloring* is a $k$-coloring for some $k \le |V|$. A coloring $c$ is *proper* if $c(u)\neq c(v)$ for every edge $\{u,v\} \in E$. The *chromatic number* of $G$, denoted by $\chi(G)$, is the smallest $k$ such that $G$ has a proper $k$-coloring. It is easy to see that $\chi(G) \geq \omega(G)$ as at least $k$ colors are needed to color a $k$-clique. Finally, a *$k$-core* of a graph $G$ is a maximal subgraph of $G$ where every vertex in the subgraph has degree at least $k$ in the subgraph. The *core number* of a vertex $v$ is the largest $k$ for which a $k$-core containing $v$ exists.
#### Machine learning and ${\textsc{NP}}$-hard problems
There has been work on using machine learning to help tackle hard problems with different approaches. Some solve a problem by augmenting existing solvers [@Liang2016], predicting a suitable solver to run for a given instance [@Fitzgerald2015; @Loreggia2016], or attempting to discover new algorithms [@Khalil2017]. In contrast, some methods address the problems more directly. Examples include approaches to TSP [@Hopfield1985; @Fort1988; @Durbin1987], with recent work in [@Vinyals2015; @Bello2016; @Nowak2017].
#### Maximal clique enumeration
We note that there are algorithms [@Eppstein2010; @Cheng2011] for *maximal* clique enumeration, in contrast to our problem of *maximum* clique enumeration. The two set of algorithms are required in very different applications, and the runtime of maximal clique enumeration is generally significantly higher.
#### Probabilistic preprocessing
Recently, [@our-nips] proposed a probabilistic preprocessing framework for fine-grained search space classification. It treats individual vertices of $G=(V,E)$ as classification problems and the problem of learning a preprocessor reduces to that of learning a mapping $\gamma : V \to \{0,1\} $ from a set of $L$ training examples $T = \{ \langle f(v_i), y_i \rangle \}^L_{i=1}$, where $v_i \in V$ is a vertex, $y_i \in \{0,1\}$ a class label, and $f : V \to \mathbb{R}^d$ a mapping from a vertex to a $d$-dimensional feature space. To learn the mapping $\gamma$ from $T$, a probabilistic classifier $P$ is used which outputs a probability distribution over $\{0,1\}$ for a given $f(u)$ for $u \in V$. Then, on input graph $G$, all vertices from $G$ that are predicted by $P$ to not be in a solution with probability at least $q$ (for some *confidence threshold* $q$) are pruned away. Here, $q$ trades-off the pruning rate with the accuracy of the pruning.
This framework showed that there is potential for learning a heuristic preprocessor for instance size pruning. However, the speedups obtained were limited and the training models were not transferable across domains. We build upon this work and show that we can achieve cross-domain generalizability and considerable speedups, both on sparse and dense graphs, using a multi-stage learning methodology.
Proposed framework {#sec:framework}
==================
In this section, we introduce our multi-stage preprocessing approach and then give the features that we use for pruning.
#### Multi-stage sparsification
A major difficulty with the probabilistic preprocessing described above is that when training on sparse graphs, the learnt model focused too heavily on pruning out the easy cases, such as low-degree vertices and not on the difficult cases like vertices with high degree and high core number. To improve the accuracy on difficult vertices, we propose a multi-stage sparsification approach. In each stage, the approach focuses on gradually harder cases that were difficult to prune by the classifier in earlier stages.
Let $\mathcal{G}_1$ be the input set of networks. Consider a graph $G \in \mathcal{G}_1$. Let $\mathcal{M}$ be the set of all maximum cliques of $G$, and denote by $V(\mathcal{M})$ the set of all vertices in $\mathcal{M}$. The positive examples in the training set $T_1$ consist of all vertices that are in some maximum clique ($V(\mathcal{M})$) and the negative examples are the ones in the set $V \setminus V(\mathcal{M})$. Since the training dataset can be highly skewed, we under-sample the larger class to achieve a balanced training data. A probabilistic classifier $P_1$ is trained on the balanced training data in stage $1$. Then, in the next stage, we remove all vertices that were predicted by $P_1$ to be in the negative class with a probability above a predefined threshold $q$. We focus on the set $\mathcal{G}_2$ of subgraphs (of graphs in $\mathcal{G}_1$) induced on the remaining vertices and repeat the above process. The positive examples in the training set $T_2$ consists of all vertices in some maximum clique ($V(\mathcal{M})$) and the negative examples are the ones in the set $V \setminus V(\mathcal{M})$, training dataset is balanced by under-sampling and we use that balanced dataset to learn the probabilistic classifier $P_2$. We repeat the process for $\ell$ stages.
As we show later, the multi-stage sparsification results in significantly more pruning compared to a single-stage probabilistic classifier.
![While the shown proper 3-coloring is optimal, we can swap the non-white colors in either triangle to see that $\chi_d(v) = 1/3$.[]{data-label="fig:chrom-density"}](chrom-density){width="30.00000%"}
#### Graph-theoretic features
We use the following graph-theoretic features: **(F1)** number of vertices, **(F2)** number of edges, **(F3)** vertex degree, **(F4)** local clustering coefficient (LCC), and **(F5)** eigencentrality.
The crude information captured by features (F1)-(F3) provide a reference for the classifier for generalizing to different distributions from which the graph might have been generated. Feature (F4), the LCC of a vertex is the fraction of its neighbors with which the vertex forms a triangle, encapsulating the well-known small world phenomenon. Feature (F5) eigencentrality represents a high degree of connectivity of a vertex to other vertices, which in turn have high degrees as well. The *eigenvector centrality* $\vec{v}$ is the eigenvector of the adjacency matrix $A$ of $G$ with the largest eigenvalue $\lambda$, i.e., it is the solution of $\vec{A}\vec{v} = \lambda\vec{v}$. The $i$th entry of $\vec{v}$ is the *eigencentrality* of vertex $v$. In other words, this feature provides a measure of local “denseness”. A vertex in a dense region shows higher probability of being part of a large clique.
#### Statistical features
In addition, we use the following statistical features: **(F6)** the $\chi^2$ value over vertex degree, **(F7)** average $\chi^2$ value over neighbor degrees, **(F8)** $\chi^2$ value over LCC, and **(F9)** average $\chi^2$ value over neighbor LCCs.
The intuition behind (F6)-(F9) is that for a vertex $v$ present in a large clique, its degree and LCC would deviate from the underlying expected distribution characterizing the graph. Further, the neighbors of $v$ also present in the clique would demonstrate such behaviour. Indeed, statistical features have been shown to be robust in approximately capturing local structural patterns [@graph].
Statistical significance is captured by the notion of p-value [@fitStatistics], and well-estimated [@pear] by the [*Pearson’s chi-square statistic*]{}, $\chi^2$, computed as $\chi^2 = \sum_{\forall i}\left[\left(O_i - E_i\right)^2 / E_i\right]$, where $O_i$ and $E_i$ are the observed and expected number of occurrences of the possible outcomes $i$.
#### Local chromatic density
Let $G=(V,E)$ be a graph. We define the *local chromatic density* of a vertex $v \in V$, denoted by $\chi_d(v)$, as the minimum ratio of the number of distinct colors appearing in $N(v)$ and any optimal proper coloring of $G$. Put differently, the local chromatic density of $v$ is the minimum possible number of colors in the immediate neighborhood of $v$ in any optimal proper coloring of $G$ (see Figure \[fig:chrom-density\]).
We use the local chromatic density as the feature **(F10)**. A vertex $v$ with high $\chi_d(v)$ means that the neighborhood of $v$ is dense, as it captures the adjacency relations between the vertices in $N(v)$. Thus, a vertex in such a dense region has a higher chance of belonging to a large clique.
However, the problem of computing $\chi_d(v)$ is computationally difficult. In the decision variant of the problem, we are given a graph $G=(V,E)$, a vertex $v \in V$, and a ratio $q \in (0,1)$. The task is to decide whether there is proper $k$-coloring $c$ of $V$ witnessing $\chi_d(v) \geq q$. The omitted proof is by a polynomial-time reduction from graph coloring.
Given a graph $G=(V,E)$, $v \in V$, and $q \in (0,1)$, it is ${\textsc{NP}}$-hard to decide whether $\chi_d(v) \leq q$.
Despite its computational hardness, we can in practice compute $\chi_d(v)$ by a heuristic. Indeed, to compute $\chi_d(v)$ for every $v \in V$, we first compute a proper coloring for $G$ using e.g., the well-known linear-time greedy heuristic of [@Welsh1967]. After a proper coloring has been computed, we compute the described ratio for every vertex from that.
[\*[111]{}[l]{}]{} & & & &\
(lr)[1-2]{} (lr)[3-4]{} (lr)[5-6]{} (lr)[7-8]{} (lr)[9-10]{} **W/o** & **With** & **W/o** & **With** & **W/o** & **With** & **W/o** & **With** & **W/o** & **With**\
0.95 & 0.98 & 0.89 & 0.99 & 0.90 & 0.95 & 0.96 & 0.99 & 0.87 & 0.96\
#### Learning over edges
Instead of individual vertices, we can view the framework also over *individual edges*. In this case, the goal is to find a mapping $\gamma' : E \to \{0,1\}$, and the training set $L'$ contains feature vectors corresponding to edges instead of vertices. We also briefly explore this direction in this work.
#### Edge features
We use the following features (E1)-(E9) for an edge $\{u,v\}$. **(E1)** Jaccard similarity is the number of common neighbors of $u$ and $v$ divided by the number of vertices that are neighbors of at least one of $u$ and $v$. **(E2)** Dice similarity is twice the number of common neighbors of $u$ and $v$, divided by the sum of their degrees. **(E3)** Inverse log-weighted similarity is as the number of common neighbors of $u$ and $v$ weighted by the inverse logarithm of their degrees. **(E4)** Cosine similarity is the number of common neighbors of $u$ and $v$ divided by the geometric mean of their degrees. The next three features are inspired by the vertex features: **(E5)** average LCC over $u$ and $v$, **(E6)** average degree over $u$ and $v$, and **(E7)** average eigencentrality over $u$ and $v$. **(E8)** is the number of length-two paths between $u$ and $v$. Finally, we use **(E9)** *local edge-chromatic density*, i.e., the number of distinct colors on the common neighbors of $u$ and $v$ divided by the total number of colors used in any optimal proper coloring.
The intuition behind (E1)-(E4) is well-established for community detection; see e.g., [@Harenberg2014] for more. For (E8), observe that the number of length-two paths is high when the edge is part of a large clique, and at most $n-2$ when $\{u,v\}$ is an edge of a complete graph on $n$ vertices. Notice that (E9) could be converted into a deterministic rule: the edge $\{u,v\}$ can be safely deleted if the common neighbors of $u$ and $v$ see less than $k-2$ colors in any proper coloring of the input graph $G$, where $k$ is an estimate for $\omega(G)$. To our best knowledge, such a rule has not been considered previously in the literature. Further, notice that there are situations in which this rule *can* be applied whereas the similar vertex rule uncovered from (F10) cannot. To see this, let $G$ be a graph consisting of two triangles $\{a,b,c\}$ and $\{x,y,z\}$, connected by an edge $\{a,x\}$, and let $k = 3$. The vertex rule cannot delete $a$ nor $x$, but the described edge rule removes $\{a,x\}$.
Experimental results
====================
In this section, we describe how multi-stage sparsification is applied to the MCE problem and our computational results. To allow for a clear comparison, we follow closely the definitions and practices specified in [@our-nips]. Thus, unless otherwise mentioned and to save space, we refer the reader to that work for additional details.
All experiments ran on a machine with Intel Core i7-4770K CPU (3.5 GHz), 8 GB of RAM, running Ubuntu 16.04.
#### Training and test data
All our datasets are obtained from Network Repository [@Rossi2015] (available at <http://networkrepository.com/>). For dense networks, we choose a total of 30 networks from various categories with the criteria that the edge density is at least 0.5 in each. We name this category “dense”. The test instances are in Table \[tbl:dense\], chosen based on empirical hardness (i.e., they are solvable in reasonable amount of time).
For sparse networks, we choose our training data from four different categories: 31 biological networks (“bio”), 32 social networks (“soc”), 107 Facebook networks (“socfb”), and 13 web networks (“web”). In addition, we build a fifth category “all” that comprises all networks from the mentioned four categories. The test instances are in Table \[tbl:pruning\].
#### Feature computation
We implement the feature computation in C++, relying on the `igraph` [@igraph] C graph library. In particular, our feature computation is single-threaded with further optimization possible.
#### Domain oblivious training via local chromatic density
In [@our-nips], it was assumed that the classifier should be trained with networks coming from the same domain, and that testing should be performed on networks from that domain. However, we demonstrate in Table \[tbl:chrom-feat\] that a classifier can be trained with networks from various domains, yet predictions remain accurate across domains (see column “all”). The accuracy is boosted considerably by the introduction of the local chromatic density (F10) into the feature set (see Table \[tbl:chrom-feat\]). In particular, when generalizing across various domains, the impact on accuracy is almost 10 %. For this reason, rather than focusing on network categories, we only consider networks by edge density (at least 0.5 or not).
#### State-of-the-art solvers for MCE
To our best knowledge, the only publicly available solvers able to list all maximum cliques[^2] are `cliquer` [@Ostergard2002], based on a branch-and-bound strategy; and `MoMC` [@Li2017], introducing incremental maximum satisfiability reasoning to a branch-and-bound strategy. We use these solvers in our experiments.
Dense networks {#subs:dense}
--------------
In this subsection, we show results for probabilistic preprocessing on dense networks (i.e., edge density at least 0.5).
\[tbl:dense\]
Instance $|V|$ $|E|$ $\omega$ n. $\omega$ Pruning `cliquer` `MoMC`
--------------- -------- --------- ----------- ------------- --------- --------- ---------- ----------- --------- --------------------- -----------------------
brock200-1 200 14.8 K 21 (20) 2 (16) — — **0.34** **0.55** $<$0.01 **0.39 (53.07)** 0.04 (44.57)
keller4 171 9.4 K **11\*** 2304 (37) — — **0.30** **0.50** $<$0.01 **$<$0.01 (38.11)** 0.02 (5.68)
keller5 776 226 K **27\*** 1000 (5) — — **0.28** **0.48** 0.19 `t/o` **1421.24 ($>$2.53)**
p-hat300-3 300 33.4 K **36\*** **10\*** — — **0.38** **0.58** 0.02 **87.1 (9.12)** 0.05 (6.00)
p-hat500-3 500 93.8 K **50\*** 62 (40) — — **0.34** **0.52** 0.07 `t/o` **2.51 (5.98)**
p-hat700-1 700 61 K **11\*** **2\*** — — **0.36** **0.47** 0.03 0.08 (1.22) **0.05 (1.30)**
p-hat700-2 700 121.7 K **44\*** **138\*** — — **0.36** **0.45** 0.11 `t/o` 1.35 (—)
p-hat1000-1 1 K 122.3 K **10\*** 276 (165) — — **0.36** **0.47** 0.08 **0.86 (2.22)** 0.71 (1.67)
p-hat1500-1 1.5 K 284.9 K 12 (11) 1 (376) — — **0.33** **0.43** 0.25 13.18 (—) **3.2 (1.54)**
fp 7.5 K 841 K **10\*** **1001\*** — — **0.06** **0.29** 0.36 0.65 (—) **5.19 (1.13)**
nd3k 9 K 1.64 M **70\*** **720\*** — — **0.23** **0.28** 1.28 `t/o` **7.05 (1.09)**
raefsky1 3.2 K 291 K **32\*** 613 (362) — — **0.33** **0.38** 0.11 2.80 (—) **0.31 (1.36)**
HFE18\_96\_in 4 K 993.3 K **20\*** **2\*** $<$1e-4 $<$1e-4 **0.26** **0.27** 0.49 58.88 (1.05) **4.30 (1.18)**
heart1 3.6 K 1.4 M **200\*** 45 (26) $<$1e-4 $<$1e-4 **0.19** **0.25** 0.66 `t/o` 19.37 (—)
cegb2802 2.8 K 137.3 K **60\*** 101 (38) 0.09 0.04 **0.39** **0.46** 0.09 0.05 (—) **0.15 (1.61)**
movielens-1m 6 K 1 M **31\*** **147\*** 0.05 0.007 **0.22** **0.23** 0.98 31.31 (—) **2.85 (1.14)**
ex7 1.6 K 52.9 K **18\*** 199 (127) 0.02 0.01 **0.26** **0.28** 0.04 0.01 (—) **0.1 (1.29)**
Trec14 15.9 K 2.87 M **16\*** **99\*** 0.16 0.009 **0.34** **0.15** 2.19 3.62 (—) 0.35 (—)
#### Classification framework for dense networks
For training, we get 4762 feature vectors from our “dense” category. As a baseline, a 4-fold cross validation over this using logistic regression from [@our-nips] results in an accuracy of **0.73**. We improve on this by obtaining an accuracy of **0.81** with gradient boosted trees (further details omitted), found with the help of `auto-sklearn` [@autosklearn].
#### Search strategies
Given the empirical hardness of dense instances, one should not expect a very high accuracy with polynomial-time computable features such as (F1)-(F10). For this reason, we set the confidence threshold $q=0.98$ here.
#### The failure of $k$-core decomposition on dense graphs
It is common that widely-adopted preprocessing methods like the $k$-core decomposition cannot prune any vertices on a dense network $G$, even if they had the computationally expensive knowledge of $\omega(G)$. This is so because the degree of each vertex is higher than than the maximum clique size $\omega(G)$.
We showcase precisely this poor behaviour in Table \[tbl:dense\]. For most of the instances, the $k$-core decomposition with the exact knowledge of $\omega(G)$ cannot prune any vertices. In contrast, the probabilistic preprocessor prunes typically around 30 % of the vertices and around 40 % of the edges.
#### Accuracy
Given that around 30 % of the vertices are removed, how many mistakes do we make? For almost all instances we retain the clique number, i.e., $\omega(G') = \omega(G)$, where $G'$ is the instance obtained by preprocessing $G$ (see column “$\omega$” in Table \[tbl:dense\]). In fact, the only exceptions are and , for which $\omega(G') = \omega(G) - 1$ still holds. Importantly, for about half of the instances, we retain *all* optimal solutions.
#### Speedups
We show speedups for the solvers after executing our pruning strategy in Table \[tbl:dense\] (last two columns). We obtain speedups as large as 53x and for 38x and , respectively. This might not be surprising, since in both cases we lose some maximum cliques (but note that for , the size of a maximum clique is still retained). For , the preprocessor makes no mistakes, resulting in speedups of upto 9x. The speedup for is *at least* 2.5x, since the original instance was not solved within 3600 seconds, but the preprocessed instances was solved in roughly 1421 seconds.
Most speedups are less than 2x, explained by the relative simplicity of instances. Indeed, it seems challenging to locate dense instances of MCE that are (i) structured and (ii) solvable within a reasonable time.
Sparse networks {#subs:sparse}
---------------
In this subsection, we show results for probabilistic preprocessing on sparse networks (i.e., edge density below 0.5).
#### Classification framework for sparse networks
We use logistic regression trained with stochastic gradient descent.
\[tbl:pruning\]
Instance $|V|$ $|E|$ $\omega$ n. $\omega$ Pruning `cliquer` `MoMC`
------------------------- ------- ------- ----------- ------------- ------- ------- ----------- ----------- ---------- ------------------- ---------------------
bio-WormNet-v3 16 K 763 K **121\*** **18\*** 0.868 0.602 **0.987** **0.975** 0.36 0.37 (—) **0.40 (3.94)**
ia-wiki-user-edits-page 2 M 9 M **15\*** **15\*** 0.958 0.641 **0.997** **0.946** 1.12 **1.16 (29.94)** `s`
rt-retweet-crawl 1 M 2 M **13\*** **26\*** 0.979 0.863 **0.997** **0.989** 0.38 **0.41 (5.66)** `s`
soc-digg 771 K 6 M **50\*** **192\*** 0.969 0.496 **0.998** **0.964** 4.80 **4.91 (1.78)** `s`
soc-flixster 3 M 8 M **31\*** **752\*** 0.986 0.834 **0.999** **0.989** 1.32 **1.41 (3.86)** `s`
soc-google-plus 211 K 2 M **66\*** **24\*** 0.986 0.785 **0.998** **0.972** 0.35 0.35 (—) **0.41 (3.98)**
soc-lastfm 1 M 5 M **14\*** 330 (324) 0.933 0.625 **0.993** **0.938** 2.24 **2.57 (10.56)** `s`
soc-pokec 2 M 22 M **29\*** **6\*** 0.824 0.595 **0.975** **0.940** 17.59 **24.40 (45.80)** `s`
soc-themarker 69 K 2 M **22\*** **40\*** 0.713 0.151 **0.972** **0.842** 2.03 **4.95 (—)** `s`
soc-twitter-higgs 457 K 15 M **71\*** **14\*** 0.852 0.540 **0.986** **0.943** 9.52 **9.85 (1.92)** `s`
soc-wiki-Talk-dir 2 M 5 M **26\*** **141\*** 0.993 0.830 **0.999** **0.970** 1.09 **3.47 (1.25)** `s`
socfb-A-anon 3 M 24 M **25\*** **35\*** 0.879 0.403 **0.984** **0.907** 28.49 **38.05 (55.95)** `s`
socfb-B-anon 3 M 21 M **24\*** **196\*** 0.884 0.378 **0.986** **0.920** 28.33 **35.49 (67.46)** `s`
socfb-Texas84 36 K 2 M **51\*** **34\*** 0.540 0.322 **0.957** **0.941** 1.04 **1.07 (1.32)** `s`
tech-as-skitter 2 M 11 M **67\*** **4\*** 0.997 0.971 **1.000** **0.998** 0.28 0.28 (—) **0.36 (4.31)**
web-baidu-baike 2 M 18 M **31\*** **4\*** 0.933 0.618 **0.992** **0.934** 9.67 **11.00 (7.48)** `s`
web-google-dir 876 K 5 M **44\*** **8\*** 1.000 0.999 **1.000** **1.000** $<$ 0.00 $<$ 0.00 (—) **$<$ 0.00 (2.06)**
web-hudong 2 M 15 M 267 (266) 59 (1) 1.000 0.996 **1.000** **0.997** 0.09 0.10 (—) **0.1 (9.99)**
web-wikipedia2009 2 M 5 M **31\*** **3\*** 0.999 0.988 **1.000** **1.000** 0.03 0.03 (—) **0.03 (4.28)**
#### Implementing the $k$-core decomposition
Recall the exact state-of-the-art preprocessor: (i) use a heuristic to find a large clique (say of size $k$) and (ii) delete every vertex of $G$ of core number less than $k-1$. For sparse graphs, a state-of-the-art solver `pmc` has been reported to find large cliques, i.e., typically $k$ is at most a small additive constant away from $\omega(G)$ (a table of results seen at <http://ryanrossi.com/pmc/download.php>). Further, given that some real-world sparse networks are scale-free (many vertices have low degree) the $k$-core decomposition can be effective in practice.
To ensure highest possible prune ratios for the $k$-core decomposition method, we supply it with the number $\omega(G)$ instead of an estimate provided by any real-world implementation. This ensures *ideal conditions*: (i) the method always prunes as aggressively as possible, and (ii) we further assume its execution has zero cost.We call this method the *$\omega$-oracle*.
#### Test instance pruning
Before applying our preprocessor on the sparse test instances, we prune them using the $\omega$-oracle. This ensures that the pruning we report is highly non-trivial, while also speeding up feature computation.
#### Search strategies
We experiment with the following two multi-stage search strategies:
- *Constant confidence (CC):* at every stage, perform probabilistic preprocessing with confidence threshold $q$.
- *Increasing confidence (IC):* at the first stage, perform probabilistic preprocessing with confidence threshold $q$, progressing $q$ by $d$ for every later stage.
Our goal is two-fold: to find (i) a number of stages $\ell$ and (ii) parameters $q$ and $d$, such that the strategy never errs while pruning as aggressively as possible. We do a systematic search over parameters $\ell$, $q$, and $d$. For the CC strategy, we let $\ell \in \{1,2,\ldots,8\}$ and $q \in \{ 0.55, 0.6, \ldots, 0.95 \}$. For the IC strategy, we try $q \in \{ 0.55, 0.60, 0.65 \}$, $d = 0.05$, and set $\ell$ so that in the last stage the confidence is 0.95.
We find the CC strategy with $q = 0.95$ to prune the highest while still retaining all optimal solutions. Thus, for the remaining experiments, we use a CC strategy with $q=0.95$.
Our 5-stage strategy outperforms, almost always safely, the $\omega$-oracle (see Table \[tbl:pruning\]). In particular, note that even if the difference between the vertex pruning ratios is small, the impact for the number of edges removed can be considerable (see e.g., all instances of the “soc” category).
#### Speedups
We show speedups for the solvers in Table \[tbl:pruning\]. We use as a baseline the solver executed on an instance pruned by the $\omega$-oracle, which renders many of the instances easy already. Most notably, this is *not* the case for , , and , all requiring at least 5 minutes of solver time. The largest speedup is for , where we go from requiring 40 minutes to only 7 seconds of solver time. For `MoMC`, most instances report a segmentation fault for an unknown reason.
#### Comparison against Lauri and Dutta
The results in Table \[tbl:pruning\] are not directly comparable to those in [@our-nips Table 1]. First, the authors only give vertex pruning ratios. While the difference in vertex pruning ratios might sometimes seem underwhelming, even small increases can translate to large decrements in the number of edges. On the other hand, the difference is often clear in our favor as in and (i.e., 0.76 vs. 0.96 and 0.90 vs. 0.99). Second, the authors use estimates on $\omega(G)$ – almost always less than the exact value – whereas we use the exact value provided by the $\omega$-oracle. Thus, the speedups we report are *as conservative as possible* unlike theirs.
Edge-based classification
-------------------------
For edges, we do a similar training as that described for vertices. For the category “dense”, we obtain 79472 feature vectors. Further, for this category, the edge classification accuracy is **0.83**, which is 1 % higher than the vertex classification accuracy using the same classifier as in Subsection \[subs:dense\]. However, we note that the edge feature computation is noticeably slower than that for vertex features.
Model analysis
--------------
Gradient boosted trees (used with dense networks in Subsection \[subs:dense\]) naturally output feature importances. We apply the same classifier for the sparse case to allow for a comparison of feature importance. In both cases, the importance values are distributed among the ten features and sum up to one.
Unsurprisingly, for sparse networks, the local chromatic density (F10) dominates (importance 0.22). In contrast, (F10) is ineffective for dense networks (importance 0.08), since the chromatic number tends to be much higher than the maximum clique size. In both cases, (F5) eigencentrality has relatively high importance, justifying its expensive computation.
For dense networks, (F7) average $\chi^2$ over neighbor degrees has the highest importance (importance 0.23), whereas in the sparse case it is least important feature (importance 0.03). This is so because all degrees in a dense graph are high and the degree distribution tends to be tightly bound or coupled. Hence, even slight deviations from the expected (e.g., vertices in large cliques) depict high statistical significance scores.
Discussion and conclusions {#sec:disc}
==========================
We proposed a multi-stage learning approach for pruning the search space of MCE, generalizing an earlier framework of [@our-nips]. In contrast to known exact preprocessing methods, our approach requires no estimate for the maximum clique size at runtime – a task ${\textsc{NP}}$-hard to even approximate – and particularly challenging on dense networks. We provide an extensive empirical study to show that our approach can routinely prune over 99 % of vertices in sparse graphs. More importantly, our approach can typically prune around 30 % of the vertices on dense graphs, which is considerably more than the existing methods based on $k$-cores.
#### Future improvements
To achieve even larger speedups, one can consider parallelization of the feature computation (indeed, our current program is single-threaded). In addition, at every stage, we recompute all features from scratch. There are two obvious ways to speed this part: (i) it is unnecessary to recompute a local feature (e.g., degree or local clustering coefficient) for vertex $v$ if none of its neighbors were removed, and (ii) more generally, there is considerable work in the area of dynamic graph algorithms under vertex deletions. Another improvement could be to switch more accurate but expensive methods for feature computation (e.g., (F10) which is ${\textsc{NP}}$-hard) when the graph gets small enough.
#### Dynamic stopping criteria
We refrained from multiple stages of preprocessing for dense networks due to the practical hardness of the task. However, for sparse networks, it was practically always safe to perform five (or even more) of stages of preprocessing with no effect on solution quality. An intriguing open problem is to propose a dynamic strategy for choosing a suitable number of stages $\ell$. There are several possibilities, such as stopping when pruning less than some specified threshold, or always pruning aggressively (say up to $\ell=10$), computing a solution, and then backtracking by restoring the vertices deleted in the previous stage, halting when the solution does not improve anymore.
#### Classification over edges
To speed up our current implementation for edge feature computation, a first step could be a well-engineered neighborhood intersection to speed up (E1)-(E4). Luckily, this is a core operation in database systems with many high-performance implementations available [@Lemire2016; @Inoue2014; @Lemire2015]. Further path-based features are also possible, like the number of length-$d$ paths for $d=3$, which is still computed cheaply via e.g., matrix multiplication. For larger $d$, one could rely on estimates based on random walk sampling. In addition, it is possible to leave edge classification for the later stages, once the vertex classifier has reduced the size of the input graph enough. We believe that there is further potential in exploring this direction.
[^1]: Part of this work was done while the authors were at Nokia Bell Labs, Ireland
[^2]: For instance, `pmc` [@Rossi2015b] does not have this feature.
|
---
abstract: 'The existence of macroscopic regions with antibaryon excess in the matter – dominated Universe is a possible consequence of the evolution of baryon charged, pseudo – Nambu – Goldstone field with lepton number violating couplings. Such regions can survive the annihilation with surrounding matter only in the case if their sizes exceeds the critical surviving size. The evolution of survived antimatter – regions with high original antibaryon density inside results in the formation of globular clusters, which is made out from antimatter stars. The origin of antimatter regions in the chosen scenario is accompanied the formation of closed domain walls, which can collapse into massive black holes deposed inside the high density antimatter regions. This fact can give us an additional hint, that an anti – stars globular cluster could be one of the collapsed – core star clusters, which populate our galaxy.'
address: |
$^*$Labor für Höchenergiephysik, ETH-Hönggerberg, HPK–Gebäude, CH–8093 Zürich\
$^{\dagger}$ Center for CosmoParticle Physics “Cosmion”, 4 Miusskaya pl., 125047 Moscow, Russia\
$^{\ddagger}$ Moscow Engineering Physics Institute, Kashirskoe shosse 31, 115409 Moscow, Russia
author:
- 'Alexander Sakharov$^*$, Maxim Khlopov${^\dagger}$$^{\ddagger}$ and Sergei Rubin$^{\dagger}$$^{\ddagger}$'
title: |
Macroscopically large antimatter regions\
in the baryon asymmetric universe
---
Introduction {#introduction .unnumbered}
============
Since a long time the generally accepted motivation for baryon asymmetric Universe were the direct observations, which claim to exclude the macroscopic amount of antimatter within the distance up to 20Mpc from the solar system [@exl]. Moreover, if larger than 20Mpc regions of matter and antimatter coexist, then it would be impossible to keep them out of the close contact during an early time, because the uniformity of CMBR excludes the existence of any significant voids. The annihilation, which would take place at the border between matter and antimatter region, during the period $1100>z>20$, would disturb the diffuse $\gamma$ – ray background [@crg], if the size of matter or antimatter regions does not exceed $10^3$Mpc. Thus the baryon symmetric Universe is practically excluded. However, such arguments cannot exclude the case when the Universe is composed almost entirely of matter with relatively small insertions of primordial antimatter. Thus we could expect the existence of macroscopically large antimatter regions in the baryon asymmetric universe as a whole. We call such a region the local antimatter area (LAA).
Any primordial LAA having initial size up to $\simeq 1$pc or more at the end of radiation dominated (RD) stage is survived the boundary annihilation with surrounding matter until the contemporary epoch [@we] and in the case of successive homogeneous expansion has the critical surviving size $l_c\simeq 1kpc$ or more. The smaller LAA’s will be eaten up by the annihilation. This fact makes problematically to apply any model with usual thermal phase transition to modulate primordial matter antimatter distribution over the size exceeding $\l_c$ [@dolg; @zil]. We could think about a possible inflational blow upping of the correlation length of a usual phase transition [@dolsil], but one should also take care about some unwanted topological defects, which could accompany phase transitions and significantly contribute to the energy density of the universe. Mostly, to get rid from unwanted topological defects some mechanisms of symmetry restoration should be invoked [@dolsil]. Here we present the issue for inhomogeneous baryogenesis [@zil], which is free from the difficulties connected with usual phase transition approach and able to generate a considerable number of above – critical LAAs, what makes reasonable to discuss the existence of primordial LAA in our galactic volume.
Formation and evolution of LAA’s {#formation-and-evolution-of-laas .unnumbered}
================================
#### The Formation scenario. {#the-formation-scenario. .unnumbered}
Our antimatter generation scenario [@zil] is based on the spontaneous baryogenesis mechanism [@sb], which implies the existence of complex scalar field $\chi =(f/\sqrt{2})\exp{(\theta )}$ carrying baryonic charge with explicitly broken $U(1)$ symmetry. The explicit breakdown of $U(1)$ symmetry is coming from the phase depended term, which tilts the bottom of the Nambu – Goldstone (NG) potential. We suppose [@zil] that the radial mass $m_{\chi}$ of field $\chi$ is larger then the Hubble constant $H$ during inflation, while for the angular mass of $\chi$ just the opposite condition $m_{\theta}\ll H$ is satisfied at that period. It makes sure that $U(1)$ symmetry is already broken spontaneously at the beginning of inflation, but the background vacuum energy is still so high, that the tilt of the potential is vanished. This implies that the phase $\theta$ behaves as ordinary massless NG boson and the radius of NG potential is firmly established by the scale $f$ of spontaneous $U(1)$ symmetry breaking. Owing to quantum fluctuations of effectively massless angular component $\theta$ at the de Sitter background [@linde] the phase $\theta$ is varied in different regions of the Universe. Actually, such fluctuations can be interpreted as the one – dimensional Brownian motion [@linde] along the circle valley corresponding to the bottom of the NG potential. When the vacuum energy decreases the tilt of potential becomes topical, and pseudo NG (PNG) field starts oscillate. Let us assume that the phase value $\theta
=0$ corresponds to South Pole of NG field circle valley, and $\theta
=\pi$ corresponds to the opposite pole. The positive gradient of phase in this picture is routed as counterclockwise direction, and the dish of PNG potential would locate at the South Pole of circle. The possible interaction of field $\chi$ that violates the lepton number can have such a structure [@zil; @dolgmain], that as the $\theta$ rolls down in clockwise direction during the first oscillation, it preferentially creates baryons over antibaryons, while the opposite is true as it rolls down in the opposite direction during the first oscillation. Thus to have the globally baryon dominated Universe one must have the phase sited in the range $[\pi ,0 ]$, just at the beginning of inflation (when the size of the modern Universe crosses the horizon). Then subsequent quantum fluctuations move the phase to some points $\bar\theta_i$ at the range $[0,\pi ]$ causing the antibaryon excess production. If it takes place not later then after 15 e – folds from the beginning of inflation [@zil], the size of LAA’s will exceed the critical surviving size $l_c$. Let set the phase at the point $\theta_{60}$ in the range $[\pi
,0 ]$, where for simplicity we suggest that the total number of inflational e – folds is 60. The phase makes Brownian step $\delta\theta
=H/(2\pi f)$ at each e–fold. Because the typical wavelength of the fluctuation $\delta\theta$ generated during such timescale is equal to $H^{-1}$, the whole domain $H^{-1}$, containing $\theta_{60}$, after one e–fold effectively becomes divided into $e^3$ separate, causal disconnected domains of radius $H^{-1}$. Each domain contains almost homogeneous phase value $\theta_{60-1}=\theta_{60}\pm\delta\theta$. In half of these domains the phase evolves towards $\pi$ (the North Pole) and in the other domains it moves towards zero (the South Pole). To have LAA’s with appropriate sizes to avoid full annihilation one should require that the phase value crosses $\pi$ or zero not later then after $15$ steps. The numerical calculations [@zil] of the domain size distribution filled with appropriate phase values $\bar\theta_i$ show that a volume box corresponding to each galaxy can contains 1–10 above – critical regions with appropriate phase $\bar\theta_i$ at the condition that the fraction of the universe containing $\bar\theta_i$ is many orders of magnitude less then 1 [@zil]. The last conclusion makes sure that the universe will become baryon asymmetric as a whole. At the some moment after the end of inflation deeply at the Friedman epoch the condition $m_{\theta}\ll H$ is violated and the oscillations of $\theta$ around the minima of PNG potential are started. Then the stored energy density $\rho_{\theta}\simeq
\theta^2m_{\theta}^2f^2$ will convert into baryons and antibaryons. All domains where the phase starts to oscillate from the values $\bar\theta_i$ will contain antimatter. The density of antimatter depends on the initial value $\bar\theta_i$ and can be different in the different domains [@zil]. The average number density of surrounding matter should be normalised on the observable one $n_B/s\simeq 3\cdot
10^{-10}$. This normalisation sets the condition $f/m_{\theta}\ge 10^{10}$ for the PNG potential [@zil].
#### Anti – star globular cluster formation. {#anti-star-globular-cluster-formation. .unnumbered}
At the condition $f\ge H\simeq 10^{13}$GeV [@zil] we can have a considerable number of high density above – critical LAA’s that makes sense to discuss the possible evolution of such a LAA in our galaxy. It is well known [@glob] that clouds, which have temperature near $10^4$K and densities several ten times that of the surrounding hot gas, are gravitationally unstable if their masses are of the order of $10^5M_{\odot}-10^6M_{\odot}$. These objects are identified as the progenitor of globular cluster (GC) and reflect the Jeans mass at the recombination epoch. From the other side the typical size of that mass is close to the $l_c$ at the end of RD epoch. Thus if the primordial antibaryon density inside a LAA was one order of magnitude higher then surrounding matter density, that LAA can evolve into antimatter GC [@khl]. Moreover, to imprint a characteristic Jeans mass the proto – GC must cool slowly [@glob] after the recombination, so the heating of dense antimatter might be supported by annihilation with surrounding matter. Thereby GC at the large galactocentric distance is the ideal astrophysical objects which could be made out of antimatter, because GC’s are the oldest galactic system to form in the universe, and contain stars of the first population. The existence of one of such anti – star GC with the mass $10^3M_{\odot}-10^5M_{\odot}$ will not disturb observable $\gamma$ – ray background [@khl], but the expected fluxes of $\overline{^4He}$ and $\overline{^3He}$ from such an antimatter object [@bgk] are only factor two below the limit of AMS–01 (STS–91) experiment [@ams] and definitely accessible for the sensitivity of coming up AMS–02 experiment.
#### Topological defects and black holes (BH). {#topological-defects-and-black-holes-bh. .unnumbered}
The angular term of $\chi$ potential $m_{\theta}^2f^2(1-\cos\theta)$, which breaks $U(1)$ symmetry explicitly has a number of discrete degenerate minima [@bh]. The equation of motion with such a potential admits a kink – like, domain wall (DW) solution, which interpolates between two adjacent vacua, for example between $\theta =0$ and $\theta =2\pi$. From the other side our scenario [@zil] deals with the situation when at the beginning of inflation the universe contains the uniform $\theta$ in the interval $[\pi ,0]$ and hence the final vacuum state of baryon asymmetric part is $\theta =2\pi$. On the contrary, there will be the island with $\theta$ in the range $[0,\pi ]$ where the phase came trough the North Pole due to the fluctuations. The phase inside that islands will produce preferentially antimatter and come to the vacuum state $\theta =0$. Thus both states $\theta =2\pi$ and $\theta =0$ are separated by closed DW’s. The collapse of such a DW is unavoidable [@bh], and DW’s which are generated before 20 inflation e – folds will form BH’s. The density profile, concentration and observable central cusp in the stars velocity dispersion of the collapsed – core clusters GC NGC 7078 (M15) consist with the hypothesis of the massive $\approx 10^3M_{\odot}$ central BH existence [@glob]. This mass corresponds to the 33 inflation e – folds. It means that the DW was originally already encompassed the size $\l_c$ giving rise the central BH formation, which could induce the collapsed – core properties of M15.
#### Acknowledgments. {#acknowledgments. .unnumbered}
S.R. and M.K. work at the project “Cosmoparticle Physics” and acknowledge support from Cosmion – ETHZ collaboration.
Steigman, G. A., [*Ann. Rev. Astron. Astrophys.*]{} [**14**]{}, 339 (1976). Cohen, A. G., De Rujula, A., and Glashow, S. L., [*Astrophys. J.*]{} [**495**]{}, 539 (1998). Khlopov, M. Yu., et al, [*Astropart. Phys.*]{} [**12**]{}, 367 (2000). Dolgov, A. D., hep – ph/9605280; [*Phys. Rep.*]{} [**222**]{}, 309 (1992). Khlopov, M. Yu., Rubin, S. G., and Sakharov, A. S., [*Phys. Rev.*]{} [**D62**]{}, 083505 (2000). Kuzmin, V., Tkachev, I., and Shaposhnikov, M., [*Phys. Lett.*]{} [**105B**]{}, 167 (1981). Cohen, A. G., and Kaplan, D. B., [*Phys. Lett.*]{} [**B199**]{}, 251 (1987); [*Nucl. Phys.*]{} [**B308**]{}, 913 (1988). Dolgov, A. D., et al, [*Phys. Rev.*]{} [**D56**]{}, 6155 (1997). Linde, A., [*Particle Physics and Inflationary Cosmology*]{}: Harwood, 1990. Meylan, G., and Heggie, D. C., [*Astron. Astrophys. Rev.*]{} [**8**]{}, 1 (1997). Khlopov. M. Yu., [*Gravitation & Cosmology*]{} [**4**]{}, 1 (1998). Belotsky, K. M., et al, [*Phys. Atom. Nucl.*]{} [**63**]{}, 233 (2000); [*astro–ph/9807027*]{}. AMS Collaboration, Alcaraz, J., et al, [*Phys. Lett.*]{} [**461B**]{}, 387 (1999). Rubin, S. G., Khlopov, M. Yu., and Sakharov A, S., [*hep-ph/0005271*]{}.
|
---
abstract: 'Multiply constant-weight codes (MCWCs) were introduced recently to improve the reliability of certain physically unclonable function response. In this paper, the bounds of MCWCs and the constructions of optimal MCWCs are studied. Firstly, we derive three different types of upper bounds which improve the Johnson-type bounds given by Chee [*et al.*]{} in some parameters. The asymptotic lower bound of MCWCs is also examined. Then we obtain the asymptotic existence of two classes of optimal MCWCs, which shows that the Johnson-type bounds for MCWCs with distances $2\sum_{i=1}^mw_i-2$ or $2mw-w$ are asymptotically exact. Finally, we construct a class of optimal MCWCs with total weight four and distance six by establishing the connection between such MCWCs and a new kind of combinatorial structures. As a consequence, the maximum sizes of MCWCs with total weight less than or equal to four are determined almost completely.'
author:
- 'Xin Wang, Hengjia Wei, Chong Shangguan, and Gennian Ge [^1] [^2] [^3] [^4] [^5]'
bibliography:
- 'REF.bib'
title: 'New bounds and constructions for multiply constant-weight codes'
---
Multiply constant weight codes, spherical codes, Plotkin bound, Johnson bound, linear programming bound, Gilbert-Varshamov bound, concatenation, graph decompositions, skew almost-resolvable squares
Introduction
============
Modern cryptographic practice rests on the use of one-way functions, which are easy to evaluate but difficult to invert. Unfortunately, commonly used one-way functions are either based on unproven conjectures or have known vulnerabilities. Physically unclonable functions (PUFs), introduced by Pappu [*et al.*]{} [@phy], provide innovative low-cost authentication methods and robust structures against physical attacks. Recently, PUFs have become a trend to provide security in low cost devices such as Radio Frequency Identifications (RFIDs) and smart cards [@loop; @prf; @phy; @puf]. Multiply constant-weight codes (MCWCs) establish the connection between the design of the Loop PUFs [@loop] and coding theory, thus were put forward in [@MCWC]. In an MCWC, each codeword is a binary word of length $mn$ which is partitioned into $m$ equal parts and has weight exactly $w$ in each part [@MCWC]. The more general definition of MCWCs with different lengths and weights in different parts can be found in [@zhang]. This definition generalizes the classic definitions of constant-weight codes (CWCs) (where $m=1$) and doubly constant-weight codes (where $m=2$) [@jon; @lev].
The theory of MCWCs is at a rudimentary stage. In [@zhang] Chee [*et al.*]{} extended techniques of Johnson [@jon] and established certain preliminary upper and lower bounds for possible sizes of MCWCs. They also showed that these bounds are asymptotically tight up to a constant factor. In [@Cheeoptimal], Chee [*et al.*]{} gave some combinatorial constructions for MCWCs which yield several new infinite families of optimal MCWCs. In particular, by establishing the connection between MCWCs and combinatorial designs and using some existing results in design theory, they determined the maximum sizes of MCWCs with total weight less than or equal to four, leaving an infinite class open. In the same paper, they also showed that the Johnson-type bounds are asymptotically tight for fixed weights and distances by applying Kahn’s Theorem [@Kahn] on the size of the matching in hypergraphs. Furthermore, in [@Cheegraph], they demonstrated that one of the Johnson-type bounds is asymptotically exact for the distance $2mw-2$. This was achieved by applying the theory of edge-colored digraph-decompositions [@LW].
In this paper, we continue the study on the bounds of MCWCs and the constructions of optimal MCWCs. Our main contributions are as follows:
- We extend the techniques of Agrell [*et al.*]{} [@var] and improve the Johnson-type bounds derived in [@zhang]. We also show that the generalised Gilbert-Varshmov (GV) bound [@gil; @varsh] is better than the asymptotic lower bounds derived in [@zhang], where the concatenation techniques are employed.
- We obtain the asymptotic existence of two classes of optimal MCWCs. One of them generalizes the known result of [@Cheegraph] for MCWCs with different weights in different parts. The other shows that another Johnson-type bound is asymptotically exact for distance $2mw-w$.
- We consider the open case of optimal MCWCs in [@Cheeoptimal], i.e., doubly constant-weight codes with weight two in each part and distance six. We establish an equivalence relation between such MCWCs and certain kind of combinatorial structures, which are called skew almost-resolvable squares. Accordingly, several new constructions are proposed. As a consequence, the maximum sizes of MCWCs with total weight less than or equal to four are determined almost completely, leaving a very small number of lengths open.
The rest of this article is organized as follows. Section 2 collects the necessary definitions and notations. Section 3 gives three forms of upper bounds, which can improve the previous Johnson-type bounds. Section 4 studies the asymptotic lower bounds of MCWCs. Section 5 presents the asymptotic existence of two classes of optimal MCWCs. Section 6 handles the optimal MCWCs with total weight four. A conclusion is made in Section 7.
Definitions and Notations
=========================
Multiply Constant-weight Codes
------------------------------
All sets considered in this paper are finite if not obviously infinite. We use $[n]$ to denote the set $\{1,2,\ldots,n\}$. If $X$ and $R$ are finite sets, $R^X$ denotes the set of vectors of length $|X|$. Each component of a vector ${\mathbf{u}}\in
R^X$ takes value in $R$ and is indexed by an element of $X$, that is, ${\mathbf{u}}=({\mathbf{u}}_x)_{x\in X}$, and ${\mathbf{u}}_x\in R$ for each $x\in X$. A [*$q$-ary code of length $n$*]{} is a set ${{\mathcal C}}\subseteq {{\mathbb{Z}}}_q^X$ for some $X$ with size $n$. The elements of ${{\mathcal C}}$ are called [*codewords*]{}. The [*support*]{} of a vector ${\mathbf{u}}\in {{\mathbb{Z}}}_q^X$, denoted ${{\rm supp}}({\mathbf{u}})$, is the set $\{x \in X : {\mathbf{u}}_x \not= 0\}$. The [*Hamming norm*]{} or the [*Hamming weight*]{} of a vector ${\mathbf{u}}\in{{\mathbb{Z}}}_q^X$ is defined as $\|{\mathbf{u}}\|=| {{\rm supp}}({\mathbf{u}})|$. The distance induced by this norm is called the [*Hamming distance*]{}, denoted $d_H$, so that $d_H({\mathbf{u}},{\mathbf{v}})=\| {\mathbf{u}}-{\mathbf{v}}\|$, for ${\mathbf{u}},{\mathbf{v}}\in{{\mathbb{Z}}}_q^X$. A code ${{\mathcal C}}$ is said to [*have distance $d$*]{} if the Hamming distance between any two distinct codewords of ${{\mathcal C}}$ is at least $d$. A $q$-ary code of length $n$ and distance $d$ is called an $(n,d)_q$ code. When $q=2$, an $(n,d)_2$ code is simply called an $(n,d)$ code.
Let $m$, $N$ be positive integers and $X$ be a set of size $N$. Suppose that $X$ can be partitioned as $X=X_1\cup X_2\cup\cdots \cup X_m$ with $|X_i|=n_i$, $i=1,2,\ldots,m$. An $(N,d)$ code ${{\mathcal C}}\subseteq {{\mathbb{Z}}}_2^X$ is said to be of [*multiply constant-weight*]{} and denoted by MCWC$(w_1,n_1;w_2,n_2;\cdots;w_m,n_m;d)$, if each codeword has the weight $w_1$ in the coordinates indexed by $X_1$, weight $w_2$ in the coordinates indexed by $X_2$, and so on and so forth. When $w_1=w_2=\cdots=w_m=w$ and $n_1=n_2=\cdots=n_m=n$, we simply denote this multiply constant-weight code of length $N=mn$ by MCWC$(m,n,d,w)$.
The largest size of an $(n,d)_q$ code is denoted by $A_q(n,d)$. When $q=2$, the size is simply denoted by $A(n,d)$. The largest size of an MCWC$(w_1, n_1;w_2, n_2; \ldots;w_m, n_m; d)$ is denoted by $T(w_1, n_1;w_2, n_2;\ldots;w_m, n_m; d)$; the largest size of an MCWC$(m,n,d,w)$ is denoted by $M(m,n,d,w)$; and the largest size of a CWC$(n,d,w)$ is denoted by $A(n,d,w)$. The code achieving the largest size is said to be [*optimal*]{}.
Next, we will restate the known results about MCWCs without proof, more details can be found in [@zhang]. The authors of [@zhang] first use the concatenation technique to construct MCWCs from the classic $q$-ary codes.
\[concatenation\]([@zhang]) Let $q\leqslant A(n,d_1,w)$, we have $$M(m,n,d_1d_2,w)\geq A_q(m,d_2).$$ Specially, $M(m,qw,2d,w)$ $\geq$ $A_q(mw,d)$.
As MCWC is a generalization of CWC, the techniques of Johnson for CWC [@jon] can be naturally extended to give the recursive bounds as follows:
([@zhang])\[jonp\] $$\label{jon1}
T(w_1,n_1;w_2,n_2;\ldots;w_m,n_m;d)\leq \lfloor \frac{n_i}{w_i}T(w_1,n_1;\ldots;w_i-1,n_i-1;\ldots;w_m,n_m;d)\rfloor,$$ $$\label{jon2}
T(w_1,n_1;w_2,n_2;\ldots;w_m,n_m;d)\leq \lfloor \frac{n_i}{n_i-w_i}T(w_1,n_1;\ldots;w_i,n_i-1;\ldots;w_m,n_m;d)\rfloor,$$ $$\label{jon3}
T(w_1,n_1;w_2,n_2;\ldots;w_m,n_m;d)\leq \lfloor \frac{u}{w_1^2/n_1+w_2^2/n_2+\cdots+w_m^2/n_m-\lambda}\rfloor,$$ where $d=2u$ and $\lambda=w_1+w_2+\cdots+w_m-u$.
([@zhang])\[mjonp\] $$\label{mjon1}
M(m,n,d,w)\leq \lfloor \frac{n^m}{w^m}M(m,n-1,d,w-1)\rfloor,$$ $$M(m,n,d,w)\leq \lfloor \frac{n^m}{(n-w)^m}M(m,n-1,d,w)\rfloor,$$ $$\label{mjon3}
M(m,n,d,w)\leq \lfloor \frac{d/2}{d/2+mw^2/n-mw}\rfloor.$$
Association Schemes
-------------------
Let $X$ be a finite set with at least two elements and, for any integer $n\geq 1$, let ${\cal R}=\{R_0,R_1,\ldots,R_n\}$ be a family of $n+1$ relations $R_i$ on $X$. The pair $(X,{\cal R})$ will be called an [*association scheme with $n$ classes*]{} if the following three conditions are satisfied:
- The set $\cal R$ is a partition of $X^2$ and $R_0$ is the diagonal relation, i.e., $R_0=\{(x,x)|x\in X\}$.
- For $i=0,1,\ldots,n$, the inverse $R_{i}^{-1}=\{(y,x)|(x,y)\in R_i\}$ of the relation $R_i$ also belongs to $\cal R$.
- For any triple of integers $i,j,k=0,1,\ldots,n$, there exists a number $p^{(k)}_{i,j}=p^{(k)}_{j,i}$ such that, for all $(x,y)\in R_k$: $$|\{z\in X|(x,z)\in R_i, (z,y)\in R_j\}|=p^{(k)}_{i,j}.$$ The $p^{(k)}_{i,j}$’s are called the [*intersection numbers*]{} of the scheme $(X,{\cal R})$.
Any relation $R_i$ can be described by its [*adjacency matrix*]{} $D_i\in {\mathbb{C}}(X,X)$, defined as follows: $$D_i(x,y)=\left\{
\begin{array}{ll}
1, & (x,y)\in R_i,\\
0, & (x,y)\not\in R_i.
\end{array}
\right.$$
We call the linear space $$A=\{ \sum_{i=0}^{n}\alpha_i D_i|\alpha_i\in \mathbb{C} \}$$ the [*Bose-Mesner algebra*]{} of the association scheme $(X,\cal R)$. There is a set of pairwise orthogonal idempotent matrices $J_0,J_1,\ldots,J_n$, which forms another basis of this Bose-Mesner algebra.
Given two bases $\{D_k\}$ and $\{J_k\}$ of the Bose-Mesner algebra of a scheme, let us consider the linear transformations from one into the other: $$D_k=\sum_{i=0}^{n}P_k(i)J_i,~~~~k=0,1,\ldots,n.$$
From these we construct a square matrix $P$ of order $n+1$ whose $(i,k)$-entry is $P_k(i)$: $$P=[P_k(i):0\leq i,k\leq n].$$
Since $P$ is nonsingular, there exists a unique square matrix $Q$ of order $n+1$ over $\mathbb{C}$ such that $$PQ=QP=|X|I.$$ The matrices $P$ and $Q$ are called the [*eigenmatrices*]{} of the association scheme.
Let ${\cal R}=\{R_0,R_1,\ldots,R_n\}$ be a set of $n+1$ relations on $X$ of an association scheme. For a nonempty subset $Y$ of $X$, let us define the [*inner distribution*]{} of $Y$ with respect to $\cal R$ to be the $(n+1)$-tuple $\alpha=(\alpha_0,\alpha_1,\ldots,\alpha_n)$ of nonnegative rational numbers $\alpha_i$ given by $$\alpha_i=|Y|^{-1}|R_i\cap Y^2|.$$
In [@del], Delsarte gave a key observation about the inner distribution and the eigenmatrix Q.
\[lp\]([@del]) The components $\alpha Q_k$ of the row vector $\alpha Q$ are nonnegative.
Let $w$ and $n$ be integers, with $1\leq w\leq n$. In the Hamming space of dimension $n$ over $\mathbb{F}=\{0,1\}$, we consider the subset $X$ of $\mathbb{F}^n$ as follows: $$X=\{x\in \mathbb{F}^n|w_H(x)=w\},$$ and we define the distance relations $R_0,R_1,\ldots,R_w$: $$R_i=\{(x,y)\in X^2|d(x,y)=2i\}.$$ For given $n$ and $w$, with $1\leq w\leq n/2$, we call $(X,\cal R)$ the [*Johnson scheme*]{} $J(w,n)$, i.e., binary codes with length $n$ and constant weight $w$.
Given an integer $k$, with $0\leq k\leq w$, we define [*Eberlein polynomial*]{} $E_k(u)$, in the indeterminate $u$, as follows: $$E_k(u)=\sum_{i=0}^{k}(-1)^i{u \choose i}{w-u \choose k-i}{n-w-u \choose k-i}.$$
([@del]) The eigenmatrices $P$ and $Q$ of the Johnson scheme $J(w,n)$ are given by $$P_k(i)=E_k(i),$$ $$Q_i(k)=\frac{\mu_i E_k(i)}{{w \choose i}{n-w \choose i}},$$ where $\mu_i=\frac{n-2i+1}{n-i+1}{n \choose i}$.
Design Theory
-------------
To give our constructions of optimal MCWCs, we need the following notations and results in design theory.
Let $K$ be a subset of positive integers and $\lambda$ be a positive integer. A [*pairwise balanced design*]{} ($(v, K, \lambda)$-PBD or $(K, \lambda)$-PBD of order $v$) is a pair ($X,{{\mathcal B}}$), where $X$ is a finite set ([*the point set*]{}) of cardinality $v$ and ${{\mathcal B}}$ is a family of subsets ([*blocks*]{}) of $X$ that satisfy (1) if $B\in {{\mathcal B}}$, then $|B|\in K$ and (2) every pair of distinct elements of $X$ occurs in exactly $\lambda$ blocks of ${{\mathcal B}}$. The integer $\lambda$ is the [*index*]{} of the PBD. When $K=\{k\}$, a $(v, \{k\}, \lambda)$-PBD is also known as a [*balanced incomplete block design*]{} (BIBD), which is denoted by BIBD$(v,k,\lambda)$.
\[PBD579\] \[PBD5-9\] For any odd integer $v\geq 5$, a $(v,\{5,7,9\},1)$-PBD exists with exceptions $v\in [11,19]\cup\{23\}\cup[27,33]\cup\{39\}$, and possible exceptions $v\in \{43,51,59,71,75,83,87,95,99,107,111,113,115,119,139,179\}$.
An [*$\alpha$-parallel class*]{} of blocks in a BIBD $(X,{\cal B})$ is a subset ${\cal B}' \subset {\cal B}$ such that each point $x \in X$ is contained in exactly $\alpha$ blocks in ${\cal B}'$. When $\alpha =1$, we simply call it a [*parallel class*]{}, as usual. If the block set $\cal B$ can be partitioned into $\alpha$-parallel classes, then the BIBD is called [*$\alpha$-resolvable*]{} (or just [*resolvable*]{} if $\alpha=1$). We will use $\alpha$-resolvable BIBDs to construct optimal MCWCs.
A [*group divisible design*]{} (GDD) is a triple $(X,{\cal G},{\cal B})$ where $X$ is a set of points, ${\cal G}$ is a partition of $X$ into [*groups*]{}, and ${\cal B}$ is a collection of subsets of $X$ called [*blocks*]{} such that any pair of distinct points from $X$ occurs either in some group or in exactly one block, but not both. A $K$-GDD of type $g^{u_1}_1 g^{u_2}_2\ldots g^{u_s}_s$ is a GDD in which every block has size from the set $K$ and in which there are $u_i$ groups of size $g_i, i=1,2,\ldots,s$. When $K=\{k\}$, we simply write $k$ for $K$. A $k$-GDD of type $m^k$ is also called a [*transversal design*]{} and denoted by TD$(k,m)$.
\[TD\] Let $m$ be a positive integer. Then:
1. a TD$(4,m)$ exists if $m\not\in \{2,6\}$;
2. a TD$(5,m)$ exists if $m\not\in \{2,3,6,10\}$;
3. a TD$(6,m)$ exists if $m\not\in\{2,3,4,6,10,22\}$;
4. a TD$(m+1,m)$ exists if $m$ is a prime power.
Decomposition of Edge-colored Complete Digraphs
-----------------------------------------------
Denote the set of all ordered pairs of a finite set $X$ with distinct components by $\overline{X \choose 2}$. An [*edge-colored digraph*]{} is a triple $G =(V,C,E)$, where $V$ is a finite set of [*vertices*]{}, $C$ is a finite set of [*colors*]{} and $E$ is a subset of $\overline{X \choose 2}\times C$. Members of $E$ are called [*edges*]{}. The [*complete edge-colored digraph*]{} on $n$ vertices with $r$ colors, denoted by $K_n^{(r)}$, is the edge-colored digraph $(V,C,E)$, where $|V|=n$, $|C| = r$ and $E = \overline{X \choose 2}\times C$.
A family $\cal F$ of edge-colored subgraphs of an edge-colored digraph $K$ is a [*decomposition*]{} of $K$ if every edge of $K$ belongs to exactly one member of $\cal F$. Given a family of edge-colored digraphs $\cal G$, a decomposition $\cal F$ of $K$ is a [*$\cal G$-decomposition of $K$*]{} if each edge-colored digraph in $\cal F$ is isomorphic to some $G \in \cal G$. In [@LW], Lamken and Wilson exhibited the asymptotic existence of decompositions of $K_n^{(r)}$ for a fixed family of digraphs. To state their result, we require more concepts.
Consider an edge-colored digraph $G = (V,C,E)$ with $|C| = r$. Let $((u,v), c) \in E$ denote a directed edge from $u$ to $v$, colored by $c$. For any vertex $u$ and color $c$, define the [*indegree*]{} and [*outdegree*]{} of $u$ with respect to $c$, to be the number of directed edges of color $c$ entering and leaving $u$, respectively. Then for vertex $u$, we define the [*degree vector*]{} of $u$ in $G$, denoted by $\tau(u,G)$, to be the vector of length $2r$, $\tau(u,G) =(\textup{in}_1(u,G), \textup{out}_1(u,G),\ldots,\textup{in}_r(u,G),\textup{out}_r(u,G))$. Define $\alpha(G)$ to be the greatest common divisor of the integers $t$ such that the $2r$-vector $(t, t, \ldots, t)$ is a nonnegative integral linear combination of the degree vectors $\tau(u,G)$ as $u$ ranges over all vertices of all digraphs $G \in \cal G$.
For each $G = (V,C,E) \in \cal G$, let $\mu(G)$ be the [*edge vector*]{} of length $r$ given by $\mu(G)=(m_1(G),m_2(G),\ldots,m_r(G))$ where $m_i(G)$ is the number of edges with color $i$ in $G$. We denote by $\beta(G)$ the greatest common divisor of the integers $m$ such that $(m, m, \ldots, m)$ is a nonnegative integral linear combination of the vectors $\mu(G)$, $G \in \cal G$. Then $\cal G$ is said to be [*admissible*]{} if $(1,1,\ldots,1)$ can be expressed as a positive rational combination of the vectors $\mu(G)$, $G \in \cal G$.
\[graphdecom\] Let $\cal G$ be an admissible family of edge-colored digraphs with $r$ colors. Then there exists a constant $n_0=n_0(\cal G)$ such that a $\cal G$-decomposition of $K_n^{(r)}$ exists for every $n\geq n_0$ satisfying $n(n-1)\equiv 0 \pmod{\beta(\cal G)}$ and $n-1\pmod{\alpha(\cal G)}$.
In the same paper, the above theorem had also been extended to the multiplicity case. Consider the problem of finding a family $\cal F$ of subgraphs of $K_n^{(r)}$ each of which is isomorphic to a member of $\cal G$, so that each edge of $K_n^{(r)}$ of color $i$ occurs in exactly $\lambda_i$ of the members of $\cal F$. We can think of this as a $\cal G$-decomposition of $K_n^{[\lambda_1, \lambda_2 , \ldots, \lambda_r]}$, which denotes the digraph on $n$ vertices where there are exactly $\lambda_i$ edges of color $i$ joining $x$ to $y$ for any ordered pair $(x, y)$ of distinct vertices.
Let $\bm{\lambda}$ $=(\lambda_1, \lambda_2 , \ldots, \lambda_r)$ be a vector of positive integers. Let $\alpha(\cal{G}; \bm{\lambda})$ denote the least positive integer $t$ such that the constant vector $t\bm{\lambda}$ is an integral linear combination of $\tau(u, G)$ as $u$ ranges over all vertices of all digraphs $G \in \cal G$. Let $\beta(\cal{G}; \bm{\lambda})$ denote the least positive integer $m$ such that the constant vector $m\bm{\lambda}$ is an integral linear combination of $\mu(G)$, $G\in \cal G$. We say $\cal G$ is [*$\bm{\lambda}$-admissible*]{} when the vector $\bm{\lambda}$ is a positive rational linear combination of $\mu(G)$, $G\in \cal G$.
\[mgraphdecom\] Let $\cal G$ be a $\bm{\lambda}$-admissible family of edge-$r$-colored digraphs, where $\bm{\lambda}=(\lambda_1, \lambda_2 , \ldots,$ $\lambda_r)$. Then there exists a constant $n_0=n_0({\cal G}, \bm{\lambda})$ such that a $\cal G$-decomposition of $K_n^{[\lambda_1, \lambda_2 , \ldots, \lambda_r]}$ exists for every $n\geq n_0$ satisfying: $n(n-1)\equiv 0 \pmod{\beta(\cal G; \bm{\lambda})}$ and $n-1\pmod{\alpha(\cal G; \bm{\lambda})}$.
Upper Bounds
============
For the simplicity of illustration, when handling the general bounds of MCWCs, we only consider the special case of MCWC$(m,n,d,w)$. However, it is easy to see that our methods used can also be applied to the general case.
Bounds from Spherical Codes
---------------------------
We start with the definition of a spherical code. Different from the classic code, the spherical code is defined on the Euclidean space. A [*spherical code*]{} is a finite subset of $S(n)$, where $S(n):=\{{\mathbf{x}}\in R^n: \|{\mathbf{x}}\|=1\}$. Here $\|*\|$ is the Euclidean norm. The [*distance*]{} between two codewords is defined by $d_E({\mathbf{c}}_1,{\mathbf{c}}_2):=\|{\mathbf{c}}_1-{\mathbf{c}}_2\|$. However, to characterize the codeword separation in a spherical code, the [*minimum angle $\phi$*]{} or [*the maximum cosine $s$*]{} is often used instead of the Euclidean distance. The relation between these three parameters is $$s:=\cos \phi=1-\frac{d_{E}^2}{2}.$$ We will generally use $s$ as the separation parameter. The largest size of an $n$-dimensional spherical code with maximum cosine $s$ is defined by $A_S(n,s)$.
When $s\leq 0$, the value of $A_S(n,s)$ has been determined completely. [@acz; @dav; @erd; @ran; @sar]: $$\begin{array}{ll}
A_S(n,s)=\lfloor1-\frac{1}{s}\rfloor, & if~s\leq-\frac{1}{n};\\
A_S(n,s)=n+1, & if~-\frac{1}{n}\leq s<0;\\
A_S(n,0)=2n. &
\end{array}$$
Before proceeding further, let us remark that, under a suitable mapping, a binary code can be viewed as a spherical code. Thus an upper bound on the cardinality of the spherical code serves as an upper bound for the binary code. This observation can improve previous upper bounds in some cases.
Define $${{\mathcal H}}(n)=\{0,1\}^n,$$ $${{\mathcal M}}(m,n,w)=\{{\mathbf{x}}\in{{\mathcal H}}(n):{\mathbf{x}}\cdot {\mathbf{u}}_i=w\},$$ where ${\mathbf{u}}_i={\mathbf{e}}_i\otimes {\mathbf{j}}_n$, ${\mathbf{e}}_i$ is the standard $m$-dimensional unit vector and ${\mathbf{j}}_n$ is the $n$-dimensional all-one vector. Then any subset of ${{\mathcal H}}(n)=\{0,1\}^n$ is a binary code of length $n$ and any subset of ${{\mathcal M}}(m,n,w)$ is an MCWC$(m,n,d,w)$ for some distance $d$.
Let $\Omega(*)$ denote the mapping $0\rightarrow1$ and $1\rightarrow-1$ from binary Hamming space to Euclidean space. Then $$\Omega({{\mathcal M}}(m,n,w))=\{{\mathbf{x}}\in \Omega({{\mathcal H}}(n)):{\mathbf{x}}\cdot {\mathbf{u}}_i=n-2w ~~for ~~1\leq i\leq m \}.$$
For any point ${\mathbf{x}}\in {{\mathcal M}}(m,n,w)$, ${\mathbf{x}}$ satisfies $(\Omega({\mathbf{x}})-{\mathbf{x}}_0)\cdot {\mathbf{u}}_i =0$ and $\|\Omega({\mathbf{x}})-{\mathbf{x}}_0\|=r$, where $${\mathbf{x}}_0=(1-\frac{2w}{n}) {\mathbf{j}}_{mn},$$ and $$r=2\sqrt{\frac{mw(n-w)}{n}}.$$
Hence $\Omega({{\mathcal M}}(m,n,w))$ is a subset of the $(nm-m)$-dimensional hypersphere of radius $r$ centered at ${\mathbf{x}}_0$.
From the above analysis, we can get the following bound:
\[31\] $$\begin{array}{ll}
M(m,n,2d,w) \leq \lfloor \frac{d}{b}\rfloor, & if ~b\geq\frac{d}{nm-m+1},\\
M(m,n,2d,w) \leq m(n-1)+1, & if ~0<b<\frac{d}{n},
\end{array}$$ where $$b=d-\frac{mw(n-w)}{n}.$$
Let ${{\mathcal C}}$ be an MCWC$(m,n,2d,w)$. Translating $\Omega({{\mathcal C}})$ by ${\mathbf{x}}_0$ and scaling the radius by $1/r$, in accordance with the above analysis, yields an $(nm-m)$-dimensional spherical code with the maximum cosine $s=1-\frac{dn}{mw(n-w)}$. Thus $$\begin{array}{ll}
M(m,n,2d,w) \leq A_S(m(n-1),s), & if ~s \geq -1;\\
M(m,n,2d,w)=1, & if ~s < -1.
\end{array}$$
Using $A_S(mn-m,s)$ as an upper bound for $|\Omega({{\mathcal C}})|$ completes the proof.
The first bound in Theorem \[31\] is equivalent to the last Johnson-type bound (\[jon3\]) and the second bound improves the Johnson-type bound of Proposition \[jonp\] when $0<b<\frac{d}{n}$.
Plotkin-type Bounds
-------------------
The following proposition is well-known, while we provide a sketch of the proof for the sake of completeness.
([@var])\[plo\] Let ${{\mathcal C}}$ be an $(n,d)$ code, then $$|{{\mathcal C}}|\leq \frac{d/2}{d/2-\sum_{i=1}^{n}f_{i}(1-f_{i})}$$ provided that the denominator is positive, where $f_i$ denotes the proportion of codewords that have a $1$ in position $i$.
The proof follows from the technique of double counting. On one hand, $$d_{av}=\frac{1}{M(M-1)}\sum_{c_1,c_2\in C}d(c_1,c_2)\geq d,$$ where $M=|{{\mathcal C}}|$. On the other hand, $$d_{av}=\frac{2M}{M-1}\sum_{i=1}^{n}f_i(1-f_i).$$ By the double counting principle, $$\frac{2M}{M-1}\sum_{i=1}^{n}f_i(1-f_i)\geq d.$$
For MCWCs, we will have more restrictions concerning $f_i$, so we expect to get a better bound.
$$\label{pplo}
M(m,n,2d,w)\leq \max\{\frac{d}{d-\sum_{i=1}^{mn}f_i(1-f_i)}\}$$
where the maximum is taken over all $f_i $ $(1\le i \le mn)$ that satisfy the constraints below: $$\begin{aligned}
f_1+f_2+\cdots+f_n=w\\
f_{n+1}+f_{n+2}+\cdots+f_{2n}=w\\
\vdots\\
f_{(m-1)n+1}+f_{(m-1)n+2}+\cdots+f_{mn}=w.\end{aligned}$$
The proof follows from the definition of MCWCs and Proposition \[plo\].
\[cor\] $$\label{jon}
M(m,n,2d,w)\leq \lfloor\frac{d}{b}\rfloor,$$ where $$b=d-\frac{mw(n-w)}{n}.$$
To get an upper bound of MCWCs, we only need to determine the minimum value of $\sum_{i=1}^{n}f_{i}^{2}$, when $f_1+f_2+\cdots+f_n=w$. We use the method of Lagrange Multiplier. Let $\gamma$ be an auxiliary variable. We consider the following function: $$g(f_1,f_2,\ldots,f_n,\gamma)=\sum_{i=1}^{n}f_{i}^{2}+\gamma(f_1+f_2+\cdots+f_n-w).$$ Then $$\frac{\partial g}{\partial f_i}=2f_i+\gamma=0,$$ $$\frac{\partial g}{\partial \gamma}=\sum_{i=1}^{n}f_i-w=0.$$ Thus when $f_i=\frac{w}{n}$, the original function will achieve the minimum value. Substituting $f_i$ with $\frac{w}{n}$ in the sum of (\[pplo\]), we obtain (\[jon\]).
The bound (\[jon\]) is equivalent to the Johnson-type bound (\[mjon3\]) of Proposition \[mjonp\], however when we impose the additional constraint that $f_i$ must be multiples of $1/M$, the problem will be set in the discrete domain $\{0,1/M,2/M,\ldots,1\}$ instead of the continuous domain $[0,1]$. Similar with the above discussion of Corollary \[cor\], we will get an implicit expression of the upper bound.
If $b>0$, then $$M(m,n,2d,w)\leq \lfloor d/b\rfloor,$$ where $$\begin{array}{l}
b=d-\frac{mw(n-w)}{n}+\frac{nm}{M^2}\{Mw/n\}\{M(n-w)/n\},\\
M=M(m,n,2d,w),\\
\{x\}=x-\lfloor x\rfloor.
\end{array}$$
Linear Programming Bounds
-------------------------
Let ${{\mathcal C}}$ be an MCWC$(m,n,2d,w)$. The distance distribution of ${{\mathcal C}}$ can be defined as follows: $$A_{2i_1,2i_2,\ldots,2i_m}:=\frac{1}{|{{\mathcal C}}|}\sum_{{\mathbf{c}}\in {{\mathcal C}}}A_{2i_1,2i_2,\ldots,2i_m}({\mathbf{c}}),$$ where $A_{2i_1,2i_2,\ldots,2i_m}({\mathbf{c}}):=|\{{\mathbf{c}}_1\in {{\mathcal C}}:({\mathbf{c}}_1\oplus{\mathbf{c}})\cdot {\mathbf{u}}_j=2i_j\}|$, ${\mathbf{u}}_j:={\mathbf{e}}_j\otimes {\mathbf{j}}_n$, ${\mathbf{e}}_j$ is the standard $m$-dimensional unit vector and ${\mathbf{j}}_n$ is the $n$-dimensional all-one vector.
\[con\] Let ${{\mathcal C}}$ be an MCWC$(m,n,2d,w)$, then $$\sum_{i_1=0}^{w}\sum_{i_2=0}^{w}\cdots\sum_{i_m=0}^{w}Q_{k_1}(i_1)Q_{k_2}(i_2)\cdots Q_{k_m}(i_m)A_{2i_1,2i_2,\ldots,2i_m}\geq 0.$$
For $v=1,2,\ldots,m$, suppose $(X^{(v)};R_{0}^{(v)},\cdots,R_{w}^{(v)})$ is an association scheme with intersection numbers $p_{ijk}^{(v)}$, incidence matrices $D_i^{(v)}$, idempotents $J_i^{(v)}$, and eigenvalues $P_k^{(v)}(i)$, $Q_k^{(v)}(i)$. Then the Cartesian product $(X^{(1)}\times X^{(2)}\times \cdots\times X^{(m)}; R_{i_1\ldots i_m}=R_{i_1}^{(1)}\times\cdots\times R_{i_m}^{(m)},0\leq i_j\leq m$ for $1\leq j\leq m)$ is an association scheme with eigenmatrice $Q_{k_1}^{(1)}(i_1)Q_{k_2}^{(2)}(i_2)\cdots Q_{k_m}^{(m)}(i_m)$. Hence ${{\mathcal C}}$ is a code in the product of $m$ Johnson schemes. The result follows from Theorem \[lp\].
$$M(m,n,2d,w)\leq 1+\lfloor \max\sum_{i_1=0}^{w}\sum_{i_2=0}^{w}\cdots\sum_{i_m=0}^{w}A_{2i_1,\ldots,2i_m}\rfloor,$$ where
$$A_{2i_1,\ldots,2i_m}\geq 0,$$ $$A_{2i_1,\ldots,2i_m}=0,~~for~\sum_{j=1}^{m}i_j<d;$$ and $$\sum_{i_1=0}^{w}\sum_{i_2=0}^{w}\cdots\sum_{i_m=0}^{w}Q_{k_1}(i_1)Q_{k_2}(i_2)\cdots Q_{k_m}(i_m)A_{2i_1,2i_2,\ldots,2i_m}\geq 0.$$
Asymptotic Lower Bounds
=======================
In this section, we consider the asymptotic rate of $M(m,n,d,w)$ when $m$ is large, $n$ is a function of $m$, $d=\lfloor \delta mn\rfloor$ and $w=\lfloor \omega n\rfloor$ for $0<\delta, \omega<1$. Define the value $\mu(\delta,\omega)$ as follows: $$\mu(\delta,\omega):=\limsup_{m \rightarrow \infty}\frac{\log_2 M(m,n,\lfloor\delta mn\rfloor,\lfloor\omega n\rfloor)}{mn}.$$
In [@zhang], Chee [*et al.*]{} used the concatenation technique to give the following asymptotic lower bound.
([@zhang])\[propalb\] For $\delta\leq 1/2$, we have $$\mu(\delta,1/2)\geq 1-H(\delta),$$ where $H(x)$ denotes the binary entropy function defined by $$H(x):=-x\log_{2}x-(1-x)\log_2(1-x),$$ for all $0\leq x\leq 1$.
In this section, we will generalise Proposition \[propalb\] and give a general form of the asymptotic lower bound. After that, we will give a generalised Gilbert-Varshamov bound for MCWCs and show that this classic method can provide a better bound.
The first bound follows from Proposition \[concatenation\]. We choose the $q$-ary code that can achieve the Gilbert-Varshamov bound as outer codes. For convenience, we assume $\frac{1}{\omega}$ and $\delta mn$ are integers.
For $\omega\leq 1/2$ and $\delta\leq \max\{1/2,2\omega\}$, we have $$\mu_{c}(\delta,\omega)\geq\omega\log_2(\frac{1}{\omega})(1-H_{\frac{1}{\omega}}(\frac{\delta}{2\omega})),$$ where $H_{q}(x):=x\log_q(q-1)-x\log_q x-(1-x)\log_q(1-x)$ for $0<x\leq\frac{q-1}{q}$.
Applying Proposition \[concatenation\], we get $M(m,n,\delta mn,\omega n)\geq A_{\frac{1}{\omega}}(mwn,\frac{\delta mn}{2})$. Since $A_q(n,d)\geq q^{(1-H_q(d/n))n}$, then $$M(m,n,\delta mn,\omega n)\geq (\frac{1}{\omega})^{(1-H_{\frac{1}{\omega}}(\frac{\delta}{2\omega}))mwn},$$ thus $$\mu_c(\delta,\omega)\geq \omega\log_2(\frac{1}{\omega})(1-H_{\frac{1}{\omega}}(\frac{\delta}{2\omega})).$$
Actually, there exist algebraic geometric codes leading to an asymptotic improvement upon Gilbert-Varshamov bound when the alphabet size $q\geq 49$ [@TVZ; @X]. Since the improvement is slight, we still use the Gilbert-Varshamov bound for the sake of simplicity.
The Gilbert-Varshamov bound is one of the most well-known and fundamental results in coding theory. In fact, it can be easily applied to various kinds of codes. For MCWC$(m,n,2d,w)$, the volume of the Hamming ball of radius $2d-1$ is $$\sum_{i_1+i_2+\ldots+i_m\leq d-1}{w \choose i_1}{n-w \choose i_1}\cdots{w \choose i_m}{n-w \choose i_m}.$$
For $\omega\leq 1/2$ and $\delta\leq \max\{1/2,2\omega\}$, we have $$\mu_{GV}(\delta,\omega)\geq H_2(\omega)-\omega H_2(\frac{\delta}{2\omega})-(1-\omega)H_2(\frac{\delta}{2(1-\omega)}).$$
Since $$M(m,n,\delta mn,\omega n)\geq \frac{{n \choose \omega n}^m}{\sum_{i_1+i_2+\ldots+i_m\leq \frac{\delta mn}{2}-1}{\omega n \choose i_1}{(1-\omega)n \choose i_1}\cdots{\omega n \choose i_m}{(1-\omega)n \choose i_m}}$$ $$\geq \frac{{n \choose \omega n}^m}{\sum_{0\leq i\leq \frac{\delta mn}{2}}{\omega mn \choose i}{(1-\omega)mn \choose i}},$$ we have $$\mu_{GV}(\delta,\omega)\geq\frac{\log_2\frac{2^{nmH_2(\omega)}}{2^{\omega nmH_2(\frac{\delta}{2\omega})}2^{(1-\omega)mnH_2(\frac{\delta}{2(1-\omega)})}}}{mn}$$ $$\geq H_2(\omega)-\omega H_2(\frac{\delta}{2\omega})-(1-\omega)H_2(\frac{\delta}{2(1-\omega)}).$$
At the end of this section, we compare the two bounds given above and show that the generalised Gilbert-Varshamov bound offers a better one.
[\[com\]]{} $$\mu_{GV}(\delta,\omega)\geq\mu_{c}(\delta,\omega),$$ equality holds only when $w=\frac{1}{2}$ or $\delta=2(\omega-\omega^2)$.
Let $$f(\delta,\omega)=\mu_{GV}(\delta,\omega)-\mu_{c}(\delta,\omega)$$ $$=H_2(\omega)-(1-\omega)H_2(\frac{\delta}{2(1-\omega)})+\frac{\delta}{2}\log_2(\frac{1}{\omega}-1)-(1-\omega)\log_2(1-\omega).$$ For simplicity, letting $x=\frac{\delta}{2}$, we get $$f(x,\omega)=-(2-2\omega-x)\log_2(1-\omega)+x\log_2(\frac{x}{\omega})+(1-\omega-x)\log_2(1-\omega-x).$$ We will derive the proof by considering two cases of $\omega\leq\frac{1}{4},x\leq\omega$ and $\frac{1}{4}<\omega\leq \frac{1}{2},x\leq\frac{1}{4}$ separately.
1. $\omega\leq\frac{1}{4},x\leq\omega$.
When $x=0$, $f(0,\omega)=-(1-\omega)\log_2(1-\omega)>0$.
When $x=\omega$, $f(\omega,\omega)=(3\omega-2)\log_2(1-\omega)-(2\omega-1)\log_2(1-2\omega)$. We want to show $f(\omega,\omega)\geq 0$. Since $f(0,0)=0$ and $f(\frac{1}{4},\frac{1}{4})=2-\frac{5}{4}\log_2 3>0$, we need to show that $g(\omega)=f(\omega,\omega)$ is monotonely increasing. $$g^{'}(\omega)=3\log_2(1-\omega)-2\log_2(1-2\omega)+\frac{\omega}{\omega-1},$$ $$g^{''}(\omega)=\frac{\omega(3-2\omega)}{(\omega-1)^2(1-2\omega)}>0.$$ Since $g^{'}(0)=0$ and $g^{'}(\frac{1}{4})=\frac{5}{3}+3\log_2(\frac{3}{4})>0$, we get $g^{'}(\omega)\geq 0$, thus $f(\omega,\omega)\geq 0$.
Moreover $\frac{\partial f(x,\omega)}{\partial x}=\log_2\frac{x(1-\omega)}{\omega(1-\omega-x)}=0$, we get $x=\omega-\omega^2$. Since $f(\omega-\omega^2,\omega)=0$, with the above analysis, we get $f(\delta,\omega)\geq0$.
2. $\frac{1}{4}<\omega\leq \frac{1}{2},x\leq\frac{1}{4}$.
When $x=0$, $f(0,\omega)=-(1-\omega)\log_2(1-\omega)>0$.
When $x=\frac{1}{4}$, $f(\frac{1}{4},\omega)=-(\frac{7}{4}-2\omega)\log_2(1-\omega)+\frac{1}{4}\log_2(\frac{1}{4\omega})+(\frac{3}{4}-\omega)\log_2(\frac{3}{4}-\omega).$ We want to show $f(\frac{1}{4},\omega)\geq 0$. Since $f(\frac{1}{4},\frac{1}{4})=-\frac{5}{4}\log_2(\frac{3}{4})-\frac{1}{2}>0$ and $f(\frac{1}{4},\frac{1}{2})=0$, we show the function $f(\frac{1}{4},\omega)$ is monotonely decreasing. $$f^{'}(\frac{1}{4},\omega)=\frac{1}{\ln2}(2\ln(1-\omega)-\ln(\frac{3}{4}-\omega)+1-\frac{1}{4(1-\omega)}-\frac{1}{4\omega}),$$ $$f^{''}(\frac{1}{4},\omega)=\frac{1}{\ln2}(\frac{}{}+\frac{1-2\omega}{4\omega^2(1-\omega)^2})\geq0.$$ Since $f^{'}(\frac{1}{4},\frac{1}{4})=\frac{1}{\ln2}(\ln(\frac{9}{8})-\frac{1}{3})<0$ and $f^{'}(\frac{1}{4},\frac{1}{2})=0$, we get $f^{'}(\frac{1}{4},\omega)\leq0$, thus $f(\frac{1}{4},\omega)\geq0$.
The remainder of the proof is the same as the first case. Then, we have already proven this theorem.
Two Infinite Classes of Optimal Codes
=====================================
In [@Cheegraph], Chee [*et al.*]{} demonstrated that certain Johnson-type bounds are asymptotically exact for constant-composition codes, nonbinary constant-weight codes and MCWCs by constructing several infinite classes of optimal codes achieving these bounds. Especially, for MCWCs they showed that the bound (\[jon1\]) is asymptotically exact for distance $2mw-2$.
\[Cheegraphdecom\] Fix $m$ and $w$. There exits an integer $n_0$ such that $$M(m,n,2mw-2,w)=\frac{n(n-1)}{w^2}$$ for all $n\geq n_0$ satisfying $n-1\equiv 0 \pmod{w^2}$.
In this section, we will generalize Theorem \[Cheegraphdecom\] to the case where the weight $w_i$ may not be equal. We determine the value of $T(w_1,n;w_2,n;\ldots;w_m,n;2\sum_{i=1}^m w_i-2)$ for some modulo classes of $n$ when $n$ is sufficiently large. We also establish the connection between $\alpha$-resolvable BIBDs and MCWCs and employ Theorem \[mgraphdecom\] to establish the asymptotic existence of a class of $\alpha$-resolvable BIBDs. As a consequence, we prove that the bound (\[jon3\]) is asymptotically exact for distance $2mw-w$.
Optimal MCWCs with Distance $2\sum_{i=1}^m w_i-2$
-------------------------------------------------
Let $w_1\geq w_2\geq \cdots \geq w_m$ be nonnegative integers. Let $w=\sum_{i=1}^m w_i$. The Johnson-type bound (\[jon1\]) shows that $$\begin{split}
T(w_1,n;w_2,n;\ldots;w_m,n;2w-2) \leq & \begin{cases} \frac{n(n-1)}{w_1(w_1-1)}, \textup{\ \ if $w_1>w_2$;} \\
\frac{n(n-1)}{w_1^2}, \textup{\ \ if $w_1=w_2$.}\\
\end{cases}
\end{split}$$ We will show that this bound is asymptotically tight. To apply Theorem \[graphdecom\], we first define the family of edge-colored digraphs $\cal G$. We use the $m^2$ ordered pairs from $[m]$ as colors. Define $\overline{w}=[w_1,w_2\ldots,w_m]$. Let $G(\overline{w})$ be the digraph with vertex set
$$\label{partition0}
V(G(\overline{w}))= W_1 \cup W_2 \cup \cdots \cup W_{m}$$
where $W_i$’s are disjoint vertex sets with $|W_i|=w_i$. Here, for all distinct $x, y \in V(G(\overline{w}))$, there is an edge from $x$ to $y$ of color $(i, j)$ where $i$ and $j$ are such that $x \in W_i$ and $y \in W_j$. Then in the graph $G(\overline{w})$, there are $w_iw_j$ edges colored $(i,j)$ with $i\not = j$, and $w_i(w_i-1)$ edges colored $(i,i)$. For $i, j \in [m]$, let $G_{ij}$ be a digraph with two vertices and one directed edge of color $(i,j)$. To define ${\cal G}(\overline{w})$, we consider the following two cases depending on whether $w_1=w_2$:
1. When $w_1 > w_2$, we have $w_1(w_1-1)\geq w_1w_2$. Let $r$ be the largest integer such that $w_1-1 = w_2 = \cdots = w_r$. Then set ${\cal G}(\overline{w}) = \{G(\overline{w})\}\cup \{G_{ij}: (i,j)\in ([m]\times [m]) \backslash \{(1,i),(i,1): 1\leq i\leq r\}\}$.
2. When $w_1 = w_2$, we have $w_1w_2>w_1(w_1-1)$. Let $r$ be the largest integer such that $w_1 = \cdots = w_r$. Then set ${\cal G}(\overline{w}) = \{G(\overline{w})\}\cup \{G_{ij}: (i,j)\in ([m]\times [m]) \backslash \overline{[r] \choose 2}\}$.
Suppose that a $G(\overline{w})$-decomposition of $K_n^{(m^2)}$ exists. Then $$\begin{split}
T(w_1,n;\ldots;w_m,n;2w-2) = & \begin{cases} \frac{n(n-1)}{w_1(w_1-1)}, \textup{\ \ if $w_1>w_2$;} \\
\frac{n(n-1)}{w_1^2}, \textup{\ \ if $w_1=w_2$.}\\
\end{cases}
\end{split}$$
Let $V$ be the vertex set of $K_n^{(m^2)}$ and $\cal F$ be the $G(\overline{w})$-decomposition. Let $X=\{1,2,\ldots,m\}\times V$. The code is constructed in $2^X$. For each $F\in \cal F$ isomorphic to $G(\overline{w})$, there is a unique partition of the vertex set $V(F)=\cup_{i=1}^m S_{i}$ so that the edge from $x$ to $y$ in $F$ has color $(i,j)$ if $x \in S_i$ and $y \in S_j$. Construct a codeword $\mathbf{u}$ such that $\mathbf{u}_{(i,x)} = 1$ if $x \in S_i$, and $\mathbf{u}_{(i,x)} = 0$ otherwise. Since $|S_i|=w_i$, this code is an MCWC$(w_1,n;\ldots;w_m,n;d)$ with some distance $d$. Noting that every colored edge appears at most once in the member of $\cal F$ isomorphic to $G(\overline{w})$, we have $|\rm{supp}(\mathbf{u})\cap \rm{supp}(\mathbf{v})|\leq 1$ for any two codewords $\mathbf{u}$ and $\mathbf{v}$. Thus this code has distance $2w-2$.
Finally, let $m$ be the number of digraphs in $\cal F$ isomorphic to $G(\overline{w})$. It is easy to see that $m=\frac{n(n-1)}{w_1(w_1-1)}$ if $w_1>w_2$ and $m=\frac{n(n-1)}{w_1^2}$ otherwise.
Noting that $m_{(i,j)}(G(\overline{w}))=w_iw_j$, $i\not=j$, $m_{(i,i)}(G(\overline{w}))=w_i(w_i-1)$ and $m_{(i,j)}(G_{ij})=1$, we have $$\begin{split}
\beta({\cal G}(\overline{w})) = & \begin{cases} w_1(w_1-1), \textup{\ \ if $w_1>w_2$;} \\
w_1^2, \textup{\ \ if $w_1=w_2$.}\\
\end{cases}
\end{split}$$
Since $\textup{in}_{(i,j)}(G(\overline{w}))=w_j$, $\textup{out}_{(i,j)}(G(\overline{w}))=w_i$ for any $i \not =j$, $\textup{in}_{(i,i)}(G(\overline{w}))=\textup{out}_{(i,i)}(G(\overline{w}))=w_i-1$, it is easy to check that $$\begin{split}
\alpha({\cal G}(\overline{w})) = & \begin{cases} w_1(w_1-1), \textup{\ \ if $w_1>w_2$;} \\
w_1, \textup{\ \ if $w_1=w_2$.}\\
\end{cases}
\end{split}$$
Then applying Theorem \[graphdecom\], we can obtain the following result.
Let $w_1\geq w_2\geq \cdots \geq w_m$ be nonnegative integers and $w=\sum_{i=1}^m w_i$. There exits an integer $n_0$ such that $$\begin{split}
T(w_1,n;\ldots;w_m,n;2w-2) = & \begin{cases} \frac{n(n-1)}{w_1(w_1-1)}, \textup{\ \ if $w_1>w_2$;} \\
\frac{n(n-1)}{w_1^2}, \textup{\ \ if $w_1=w_2$.}\\
\end{cases}
\end{split}$$ for all $n\geq n_0$ satisfying $n-1\equiv 0 \pmod{w_1(w_1-1)}$ if $w_1 > w_2$, or $n-1\equiv 0\pmod{w_1^2}$ otherwise.
Optimal MCWCs with Distance $2mw-w$
-----------------------------------
We first establish a connection between $\alpha$-resolvable BIBDs and optimal MCWCs.
\[alphar2MCWC\] If there exits an $\alpha$-resolvable BIBD$(v,k,\lambda)$, then $M(m,n,d,w)=v$, where $m=\frac{\lambda(v-1)}{\alpha(k-1)}$, $n=\frac{\alpha v}{k}$, $d=2(\frac{\lambda(v-1)}{k-1}-\lambda)$, and $w=\alpha$.
The Johnson-type bound (\[jon3\]) shows that $M(m,n,d,w)\leq v$ where $m=\frac{\lambda(v-1)}{\alpha(k-1)}$, $n=\frac{\alpha v}{k}$, $d=2(\frac{\lambda(v-1)}{k-1}-\lambda)$, and $w=\alpha$.
Let $(X, \cal B)$ be an $\alpha$-resolvable BIBD$(v,k,\lambda)$. Since there are $\frac{\lambda(v-1)}{\alpha(k-1)}$ $\alpha$-parallel classes in $\cal B$, each of which consists of $\frac{\alpha v}{k}$ blocks, we can arrange all the blocks in an $m\times n$ array with $m=\frac{\lambda(v-1)}{\alpha(k-1)}$ and $n=\frac{\alpha v}{k}$, such that the blocks in each row form an $\alpha$-parallel class. Now, for each point $x\in X$, construct a codeword $\mathbf{u}$ with $\mathbf{u}_{(i,j)}=1$ if the block in the entry $(i,j)$ contains $x$, and $\mathbf{u}_{(i,j)}=0$ otherwise. Since each point appears in $\alpha$ times in each row, the code constructed above is an MCWC$(m,n,d,\alpha)$ of size $v$ for some distance $d$. Since any two distinct points of $X$ appear together in exactly $\lambda$ blocks, the supports of any two codewords intersect in exactly $\lambda$ points. Thus the code has distance $d=2(mw-\lambda)=2(\frac{\lambda(v-1)}{k-1}-\lambda)$.
In the remaining of this subsection, we employ Theorem \[mgraphdecom\] to show that when $\alpha=\lambda$ and $k\mid \alpha$, an $\alpha$-resolvable BIBD$(v,k,\lambda)$ exists for all sufficient $v$ with $v\equiv 1\pmod{k-1}$.
We first define the family of edge-$r$-colored digraphs $\cal G$ with $r=k^2-k$. We use the $(k-1)^2$ ordered pairs from $[k-1]$ and the $k-1$ singletons $(i)$, $i=1, 2, ..., k-1$ as colors. Let $\bm{\lambda}$ be a vector of length $k^2-k$ with each entry being $\lambda$. For each $(k-1)$-tuple $\mathbf{t}=(t_1,t_2\ldots,t_{k-1})$ of nonnegative integers summing to $k$, let $G(\mathbf{t})$ be the digraph with $k+1$ vertices
$$\label{partition}
V(G(\mathbf{t}))=\{w\} \cup T_1 \cup T_2 \cup \cdots \cup T_{k-1}$$
where $T_i$’s are disjoint vertex sets with $|T_i|=t_i$ and $w$ is another vertex not in any $T_i$. Here, for all distinct $x, y \in V(G(\mathbf{t}))$, there is an edge from $x$ to $y$ of color $(i, j)$ where $i$ and $j$ are such that $x \in T_i$ and $y \in T_j$, and an edge of color $(i)$ from the special vertex $w$ to each $x$ in $T_i$. Let $\cal G$ be the collection of all such $G(\mathbf{t})$.
If there exits a $\cal G$-decomposition of the edge-$r$-colored $K_m^{[\lambda, \lambda, \ldots, \lambda]}$ with $r=k^2-k$ and $m=\frac{v-1}{k-1}$, then a $\lambda$-resolvable BIBD$(m(k-1)+1,k,\lambda)$ exists.
Let $V$ be the vertex set of $K_m^{[\lambda, \lambda, \ldots, \lambda]}$ and let $X=\{\infty\} \cup (V\times [k-1])$. Let $B_x=\{\infty\}\cup (\{x\}\times \{1,2, \ldots, k-1\})$, ${\cal B}=\{B_x: x \in V\}$. The elements $V$ will be used to index the $\lambda$-parallel classes, which are denoted as ${\cal P}_x$, $x \in V$; $B_x$ will be in ${\cal P}_x$. For each $F \in \cal F$, there will be a unique partition of the $k+1$ vertices $V(F)\subset V$ as $$V(F)=\{w\} \cup S_1 \cup S_2 \cup \cdots \cup S_{k-1}$$ as in (\[partition\]). Let $$A_F=\cup_{i=1}^{k-1} S_i \times \{i\};$$ we take $\lambda$ copies of this block in the parallel class ${\cal P}_w$. Let ${\cal A}=\{A_F : F \in {\cal F}\}$ and let ${\cal B}^{\lambda}$ be a multi-set containing each member of $\cal B$ $\lambda$ times. It is easy to check that $(X, {\cal A} \cup {\cal B}^{\lambda})$ is a $((k-1)m+1, k, \lambda)$-BIBD, and that each ${\cal P}_w$ is a $\lambda$-parallel class. For example, the $\lambda$ blocks in ${\cal P}_w$ that contains a point $(y,i), y\not=w$ are $A_F$’s where $F$’s are the graphs in $\cal F$ that contain the edge of color $(i)$ from $w$ to $y$.
With the same argument as that in the proof of [@LW Therem 10.1], one can show that $m(m-1)(\lambda, \lambda, \ldots, \lambda)$ is an integral linear combination of the vectors $\mu(G(\mathbf{t}))$, $G(\mathbf{t})\in \cal G$, and $(m-1)(\lambda, \lambda, \ldots, \lambda)$ is an integral linear combination of the vectors $\tau(x, G(\mathbf{t}))$ as $x$ ranges over all vertices of all digraphs $G(\mathbf{t})\in \cal G$. Thus, the two conditions of Theorem \[mgraphdecom\] are satisfied. Applying this theorem we can obtain the following result.
\[alpharBIBD\] Given positive integers $k$ and $\lambda$ with $k \mid \lambda$, there exits a constant $m_0=m_0(k,\lambda)$ such that a $\lambda$-resolvable BIBD$(m(k-1)+1,k,\lambda)$ exists for all $m\geq m_0$.
Combining Proposition \[alphar2MCWC\] and Theorem \[alpharBIBD\], we can get the following result.
Given positive integers $k$ and $w$ with $k \mid w$, there exits a constant $m_0=m_0(k,w)$ such that $$M(m,n,2(mw-w),w)=m(k-1)+1$$ with $n=w(m(k-1)+1)/k$ for all $m\geq m_0$.
Optimal MCWCs with Weight Four
==============================
In [@Cheeoptimal], the authors determined the maximum size of MCWCs for total weight less than or equal to four, except when $m=2$, $w_1=w_2=2$, $d = 6$ and $n_1 \leq n_2 \leq 2n_1-1$, with both $n_1$ and $n_2$ being odd. We consider this open class in this section. The Johnson-type bound (\[jon1\]) yields that:
Let $n_1,n_2$ be two odd integers with $0< n_1 \leq n_2 \leq 2n_1-1$. Then T$(2,n_1;2,n_2;6) \leq \lfloor \frac{n_2(n_1-1)}{4}\rfloor$.
We will show the above bound can be achieved for most cases. Firstly, we introduce a new combinatorial structure and establish the connection between such a structure and the optimal MCWC$(2,n_1;2,n_2;6)$.
Skew Almost-resolvable Squares
------------------------------
Let $V$ be a set of $v$ points and $S$ be a set of $s$ points. A [*skew almost-resolvable square*]{}, denoted SAS$(s,v)$, is an $s\times s$ array, where the rows and the columns are indexed by the elements of $S$, and each cell is either empty or contains a pair of points from $V$, such that:
1. for every two cells $(i,j)$ and $(j,i)$ with $i \not =j $ at most one is filled;
2. the cells on the diagonal are all empty;
3. no pair of points from $V$ appears in more than one cell;
4. for each $i\in S$, the pairs in row $i$ together with those in column $i$ form a partition of $V\backslash\{x\}$ for some $x\in V$.
\[MCWC2SAS\] Let $v\equiv 1\pmod {4}$ and $s\equiv 1\pmod {2}$ with $v \leq s \leq 2v-1$. There exists an MCWC$(2,v;2,s;6)$ of size $\frac{s(v-1)}{4}$ if and only if an SAS$(s,v)$ exists.
Let $A$ be an SAS$(s,v)$ on $V$ with rows and columns indexed by $S$. We may assume that $V$ and $S$ are distinct. Let $X=V\cup S$. The code is constructed in $2^X$. For each filled cell $(i,j)$ of $A$ with $A(i,j)=\{a,b\}$, construct a codeword ${\mathbf{u}}$ where ${\mathbf{u}}_x=1$ if $x\in\{a,b,i,j\}$, and ${\mathbf{u}}_x=0$ otherwise. Then we get an MCWC$(2,v;2,s;d)$ for some distance $d$. Note that Properties 1), 3) and 4) guarantee that any pair of points of $X$ appear in at most one codeword’s support. The supports of any two distinct codewords ${\mathbf{u}}$ and ${\mathbf{v}}$ intersect in at most one point and then the code has distance $6$. According to Property 4), for each $i \in S$, there are $\frac{v-1}{2}$ cells filled in row $i$ and column $i$. Thus we have $\frac{s(v-1)}{4}$ cells filled in total and the code has size $\frac{s(v-1)}{4}$.
Conversely, let $X=X_1\cup X_2$ with $|X_1|=v$ and $|X_2|=s$. Let ${{\mathcal C}}$ be an MCWC$(2,v;2,s;6)$ of size $\frac{s(v-1)}{4}$ in $2^X$. Construct an $s\times s$ array with rows and columns indexed by the elements of $S$. For each codeword ${\mathbf{u}}\in \cal C$ with ${\rm supp}({\mathbf{u}})=\{a,b,i,j\}$, $a,b\in X_1$ and $i,j\in X_2$, fill in the cell $(i,j)$ with the pair $\{a,b\}$. It is easy to check that this array is an SAS$(s,v)$.
In the above definition of SASs, if we replace the condition 4) by the following one, we get the definition of SAS$^*(s,v)$s.
1. there exits an $i_0\in S$ such that for each $i\in S\backslash \{i_0\}$, the pairs in row $i$ and column $i$ form a partition of $V\backslash\{x\}$ for some $x\in V$; the pairs in row $i_0$ and column $i_0$ form a partition of $V\backslash\{x,y,z\}$ for some distinct $x,y,z\in V$.
Similarly, we have the following result, the proof of which is exactly the same as that of Proposition \[MCWC2SAS\] and we omit it here.
\[MCWC2SAS\*\] Let $v\equiv 3\pmod {4}$ and $s\equiv 1\pmod {2}$ with $v \leq s \leq 2v-1$. There exists an MCWC$(2,v;2,s;6)$ of size $\lfloor\frac{s(v-1)}{4}\rfloor$ if and only if an SAS$^*(s,v)$ exists.
In the following, we will discuss a useful construction method, i.e., frame construction, which will allow us to construct infinite families of SASs and SAS$^*$s.
Let $V$ be a set of $v$ points and $S$ be a set of $s$ points. Let $\{H_1,H_2,\ldots,H_n\}$ be a partition of $V$ with $|H_i|=h_i$ and $\{S_1,S_2,\ldots,S_n\}$ be a partition of $S$ with $|S_i|=s_i$. A [*skew frame-resolvable square*]{} (SFS) of type $\{(s_i,h_i):1\leq i \leq n\}$ is an $s\times s$ array, where the rows and the columns are indexed by the elements of $S$, and each cell is either empty or contains a pair of points from $V$, such that:
1. for every two cells $(i,j)$ and $(j,i)$ with $i \not =j $ at most one is filled;
2. the subarray indexed by $S_i\times S_i$ is empty, and it is called [*hole*]{};
3. no pair of points from $V$ appears in more than one cell;
4. no pair of points from $H_i$ appears in any cell;
5. for each $l\in S_i$, the pairs in row $l$ together with those in column $l$ form a partition of $V\backslash H_i$.
We will use an exponential notation $(s_1,g_1)^{n_1} \cdots (s_n,g_n)^{n_t}$ to indicate that there are $n_i$ occurrences of $(s_i, g_i)$ in the partitions. We can use GDDs to give the recursive construction of SFSs.
\[WFC\] Let $(X,{\cal G},{\cal B})$ be a GDD, and let $s,v : X \rightarrow {{\mathbb{Z}}}^{+} \cup \{0\}$ be two weight functions on $X$. Suppose that for each block $B \in \cal B$, there exists an SFS of type $\{(s(x),v(x)) : x \in B\}$. Then there is an SFS of type $\{
(\sum_{x \in G} s(x), \sum_{x \in G} v(x)) : G \in \cal G \}$.
For each $x\in X$, let $S(x)$ be an index set of $s(x)$ elements, where $S(x)$ and $S(y)$ are disjoint for any $x\not =y \in X$. For each $B \in {{\mathcal B}}$, we construct an SFS of type $\{(s(x),v(x)) : x \in B\}$ ${\cal A}_B$ on $\cup_{x \in B} (\{x\}\times \{1, 2,\ldots,v(x)\})$ and index its rows and columns using the elements of the set $\cup_{x\in{{\mathcal B}}}S(x)$.
Denote $S=\cup_{x\in X} S(x)$ and $V=\cup_{x \in X} (\{x\}\times \{1, 2,\ldots,v(x)\})$. We construct the requisite SFS $\cal A$ on $V$ and index its rows and columns by $S$ as follows: for each cell of $\cal A$ indexed by $(\alpha,\beta)$, if $\alpha \in S(x)$, $\beta \in S(y)$ with $x\not =y$ and there exists a block $B \in {{\mathcal B}}$ containing $x,y$, then we place the entry from ${\cal A}_B$ indexed by $(\alpha, \beta)$ in the cell of $\cal A$; otherwise the cell is empty.
For each $G_i\in \cal G$, denote $S_i=\cup_{x\in G_i} S(x)$ and $H_i=\cup_{x\in G_i} (\{x\}\times \{1, 2,\ldots,v(x)\})$. It is easy to check that Properties 1) – 4) in the definition of SFSs are satisfied. Now, for each $\alpha \in S_i$, we consider the pairs in row $\alpha$ and column $\alpha$. Assume that $\alpha \in S(x)$ for some $x\in G_i$. Since for each $y\not \in G_i$, there exits a unique block containing both $x$ and $y$, the set $\{B\backslash\{x\}: x\in B \in \cal B\}$ forms a partition of $X\backslash G_i$. Note that for each ${\cal A}_B$ with $x\in B$, the pairs in row $\alpha$ and column $\alpha$ of ${\cal A}_B$ form a partition of $\cup_{y\in B, y\not= x} (y\times \{1,2,\ldots, v(y)\})$. Then the pairs in row $\alpha$ and column $\alpha$ in $\cal A$ form a partition of $$\bigcup_{x\in B, B\in \cal B}\left( \bigcup_{y\in B, y\not= x} \left(y\times \{1,2,\ldots, v(y)\}\right) \right)=\bigcup_{y\in X\backslash G_i} \left(y\times \{1,2,\ldots, v(y)\}\right)=V\backslash H_i.$$
Thus we have proved that $\cal A$ is an SFS of type $\{ (\sum_{x \in G} s(x), \sum_{x \in G} v(x)) : G \in \cal G \}$.
Let $V$ be a set of $v$ points and $S$ be a set of $s$ points. Let $W$ be a subset of $V$ with $|W|=w$ and $T$ be a subset of $S$ with $|T|=t$. A [*holey skew almost-resolvable square*]{}, denoted HSAS$(s,v;t,w)$, is an $s\times s$ array, where the rows and the columns are indexed by the elements of $S$, and each cell is either empty or contains a pair of points from $V$, such that:
1. for every two cells $(i,j)$ and $(j,i)$ with $i \not =j $ at most one is filled;
2. the subarray indexed by $T\times T$ is empty, and it is called [*hole*]{};
3. no pair of points from $V$ appears in more than one cell;
4. no pair of points from $W$ appears in any cell;
5. for each $t\in T$, the pairs in row $t$ together with those in column $t$ form a partition of $V\backslash W$;
6. for each $l\in S\backslash T$, the pairs in row $l$ and column $l$ form a partition of $V\backslash \{x\}$ for some $x\in V$.
The following result is simple but useful in our constructions.
Suppose that there exist both an HSAS$(s,v;t,w)$ and an SAS$(t,w)$. Then an SAS$(s,v)$ exists.
In the following, we show how to construct SASs from SFSs.
\[BFC\]\[Basic Frame Construction\] Suppose that there exists an SFS of type $\{(s_i,h_i): 1\leq i \leq n\}$. Let $s=\sum_{i=1}^n s_i$ and $v=\sum_{i=1}^n h_i$. If for each $1\leq i \leq n-1$ there exists an HSAS$(s_i+e, h_i+w; e,w)$, furthermore,
1. if there exits an HSAS$(s_n+e, h_n+w;e,w)$, then an HSAS$(s + e, v + w;e,w)$ exists;
2. if there exits an SAS$(s_n+e, h_n+w)$, then an SAS$(s + e, v + w)$ exists;
3. if there exits an SAS$^*(s_n+e, h_n+w)$, then an SAS$^*(s + e, v + w)$ exists.
Let $A$ be an SFS of type $\{(s_i,h_i): 1\leq i \leq n\}$ on $V=\cup_{i=1}^s H_i$ with rows and columns indexed by $S$. Let $W$ be a set of size $w$, disjoint from $V$, and take our new point set to be $V\cup W$. Now, add $e$ new rows and columns. For each $1\leq i \leq n-1$, fill the $s_i \times s_i$ subsquare together with the $e$ new rows and columns with a copy of the HSAS$(s_i+e, h_i+w; e,w)$ on $H_i \cup W$, such that the intersection of the new rows and columns forms a hole. Then, fill the $s_n \times s_n$ subsquare together with the $e$ new rows and columns with a copy of the HSAS$(s_n+e, h_n+w;e,w)$ (SAS$(s_n+e, h_n+w; e,w)$, SAS$^*(s_n+e, h_n+w; e,w)$). It is routine to check that the resultant square is an HSAS$(s + e, v + w;e,w)$ (SAS$(s + e, v + w)$, SAS$^*(s + e, v + w)$).
Determining the Value of T$(2,n_1;2,n_2;6)$
-------------------------------------------
Suppose ${\mathbf{u}}\in{{\mathbb{Z}}}_2^X$ is a codeword of an MCWC$(w_1,n_1;w_2,n_2;d)$. We can represent ${\mathbf{u}}$ equivalently as a $4$-tuple $\langle
a_1, a_2, a_3,a_4\rangle\in X^4$, where ${\mathbf{u}}_{a_1}={\mathbf{u}}_{a_2}={\mathbf{u}}_{a_3}={\mathbf{u}}_{a_{4}} =1$. Throughout this section, we shall often represent codewords of MCWCs in this form.
\[MCWCsmall\] Let $n_1\in\{3,5,7,9,11,13,15,17,19,21,25,29,33,37\}$, $n_1\leq n_2\leq 2n_1-1$ and $n_2$ be odd. Then
1. T$(2,n_1;2,n_2;6)=\lfloor \frac{n_2(n_1-1)}{4}\rfloor$, except for $(n_1,n_2)=(5,7)$; furthermore
2. T$(2,5;2,7;6) = 6$.
The upper bound T$(2,5;2,7;6) \leq 6$ can be found in [@table]. Codes achieving the upper bounds are constructed as follows.
For $3\leq n_1 \leq 9$, let $X=\{0,1,2,\ldots,n_1+n_2-1\}$. $X$ can be partitioned as $X=X_1\cup X_2$ with $X_1=\{0,1,\ldots,n_1-1\}$ and $X_2=\{n_1,n_1+1,\ldots,n_1+n_2-1\}$. The desired codes are constructed on $X$ and the codewords are listed in Table \[tab.smallMCWC\].
For $n_1\in\{13,17,21,25,29,33,37\}$, the codes are constructed in the Appendix.
For $n_1\in\{11,15,19\}$ and $n_1\leq n_2\leq 2n_1-3$, take an HSAS$(n_2,n_1;3,3)$ from the Appendix and fill in the hole with an SAS$^*(3,3)$ (which is equivalent to an MCWC$(2,3;2,3;6)$ and has been constructed above) to obtain an SAS$^*(n_2,n_1)$. According to Proposition \[MCWC2SAS\*\], that is equivalent to an MCWC$(2,n_1;2,n_2;6)$ of size $\lfloor \frac{n_2(n_1-1)}{4}\rfloor$, as desired. For $n_1\in\{11,15,19\}$ and $n_2= 2n_1-1$, we proceed similarly; take an HSAS$(n_2,n_1;5,3)$ from the Appendix and fill in the hole with an SAS$^*(5,3)$ (which is equivalent to an MCWC$(2,5;2,3;6)$ and has been constructed above).
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$(n_1,n_2)$ [Codewords ]{}
------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$(3,3)$ $\begin{array}{lllllll}
\langle 0, 1, 3, 4 \rangle\\
\end{array}$
$(3,5)$ $\begin{array}{lllllll}
\langle 0, 1, 3, 4 \rangle & \langle 1, 2, 5, 6 \rangle \\
\end{array}$
$(5,5)$ $\begin{array}{lllllll}
\langle 0, 1, 5, 6 \rangle & \langle 0, 2, 7, 8 \rangle & \langle 1, 3, 7, 9 \rangle & \langle 2, 4, 5, 9 \rangle & \langle 3, 4, 6, 8 \rangle \\
\end{array}$
$(5,7)$ $\begin{array}{lllllll}
\langle 0, 1, 5, 6 \rangle & \langle 0, 2, 7, 8 \rangle & \langle 0, 3, 9, 10 \rangle & \langle 1, 2, 9, 11 \rangle & \langle 1, 4, 7, 10 \rangle & \langle 3, 4, 5, 8 \rangle\\
\end{array}$
$(5,9)$ $\begin{array}{lllllllll}
\langle 0, 3, 10, 9 \rangle & \langle 2, 3, 5, 13 \rangle & \langle 0, 2, 8, 7 \rangle & \langle 0, 4, 11, 12 \rangle & \langle 1, 2, 9, 11 \rangle & \langle 1, 3, 7, 12 \rangle & \langle 0, 1, 6, 5 \rangle & \langle 1, 4, 8, 13 \rangle & \langle 2, 4, 6, 10 \rangle \\
\end{array}$
$(7,7)$ $\begin{array}{llllllllll}
\langle 0, 1, 7, 8 \rangle &
\langle 0, 2, 9, 10 \rangle &
\langle 0, 3, 11, 12 \rangle &
\langle 1, 2, 11, 13 \rangle &
\langle 1, 4, 9, 12 \rangle &
\langle 2, 5, 7, 12 \rangle &
\langle 3, 4, 7, 10 \rangle &
\langle 3, 5, 8, 9 \rangle &
\langle 4, 6, 8, 11 \rangle &
\langle 5, 6, 10, 13 \rangle \\
\end{array}$
$(7,9)$ $\begin{array}{llllllllll}
\langle 0, 1, 7, 8 \rangle &
\langle 0, 2, 9, 10 \rangle &
\langle 0, 3, 11, 12 \rangle &
\langle 0, 4, 13, 14 \rangle &
\langle 1, 2, 11, 13 \rangle &
\langle 1, 3, 9, 14 \rangle &
\langle 1, 4, 10, 12 \rangle &
\langle 2, 3, 7, 15 \rangle &
\langle 2, 5, 8, 12 \rangle \\
\langle 3, 5, 10, 13 \rangle &
\langle 4, 5, 7, 9 \rangle &
\langle 4, 6, 8, 11 \rangle &
\langle 5, 6, 14, 15 \rangle &
\end{array}$
$(7,11)$ $\begin{array}{llllllllll}
\langle 0, 1, 7, 8 \rangle &
\langle 0, 2, 9, 10 \rangle &
\langle 1, 5, 11, 13 \rangle &
\langle 0, 4, 13, 14 \rangle &
\langle 0, 5, 15, 16 \rangle &
\langle 3, 6, 16, 13 \rangle &
\langle 1, 3, 9, 14 \rangle &
\langle 0, 3, 11, 17 \rangle &
\langle 1, 6, 15, 17 \rangle \\
\langle 2, 3, 7, 15 \rangle &
\langle 2, 4, 8, 16 \rangle &
\langle 2, 5, 12, 14 \rangle &
\langle 3, 5, 8, 10 \rangle &
\langle 1, 4, 12, 10 \rangle &
\langle 4, 5, 7, 17 \rangle &
\langle 4, 6, 9, 11 \rangle &
\end{array}$
$(7,13)$ $\begin{array}{llllllllll}
\langle 0, 1, 7, 8 \rangle &
\langle 0, 2, 9, 10 \rangle &
\langle 0, 3, 11, 12 \rangle &
\langle 1, 4, 12, 10 \rangle &
\langle 5, 4, 7, 9 \rangle &
\langle 0, 6, 17, 18 \rangle &
\langle 1, 2, 11, 13 \rangle &
\langle 1, 3, 9, 14 \rangle &
\langle 3, 5, 10, 8 \rangle \\
\langle 1, 5, 17, 19 \rangle &
\langle 3, 4, 19, 18 \rangle &
\langle 2, 4, 8, 16 \rangle &
\langle 5, 0, 16, 15 \rangle &
\langle 0, 4, 14, 13 \rangle &
\langle 2, 3, 7, 17 \rangle &
\langle 3, 6, 13, 16 \rangle &
\langle 4, 6, 11, 15 \rangle &
\langle 2, 5, 12, 18 \rangle \\
\langle 2, 6, 14, 19 \rangle &
\end{array}$
$(9,9)$ $\begin{array}{llllllllll}
\langle 7, 3, 15, 12 \rangle &
\langle 2, 1, 16, 11 \rangle &
\langle 4, 8, 9, 15 \rangle &
\langle 0, 3, 11, 10 \rangle &
\langle 2, 8, 10, 13 \rangle &
\langle 2, 6, 9, 12 \rangle &
\langle 4, 5, 11, 12 \rangle &
\langle 1, 0, 12, 17 \rangle &
\langle 6, 7, 13, 17 \rangle \\
\langle 6, 0, 14, 15 \rangle &
\langle 6, 4, 10, 16 \rangle &
\langle 1, 4, 14, 13 \rangle &
\langle 7, 1, 10, 9 \rangle &
\langle 7, 8, 14, 11 \rangle &
\langle 3, 8, 17, 16 \rangle &
\langle 5, 2, 15, 17 \rangle &
\langle 3, 5, 14, 9 \rangle &
\langle 0, 5, 13, 16 \rangle \\
\end{array}$
$(9,11)$ $\begin{array}{llllllllll}
\langle 4, 8, 13, 11 \rangle &
\langle 3, 0, 14, 10 \rangle &
\langle 6, 5, 11, 19 \rangle &
\langle 3, 1, 16, 11 \rangle &
\langle 0, 8, 16, 15 \rangle &
\langle 8, 2, 9, 17 \rangle &
\langle 6, 2, 14, 13 \rangle &
\langle 6, 8, 12, 10 \rangle &
\langle 4, 3, 17, 15 \rangle \\
\langle 6, 3, 9, 18 \rangle &
\langle 4, 1, 12, 18 \rangle &
\langle 3, 5, 13, 12 \rangle &
\langle 8, 1, 19, 14 \rangle &
\langle 7, 0, 11, 12 \rangle &
\langle 1, 7, 9, 13 \rangle &
\langle 0, 5, 17, 18 \rangle &
\langle 6, 7, 17, 16 \rangle &
\langle 2, 7, 19, 18 \rangle \\
\langle 7, 5, 14, 15 \rangle &
\langle 0, 4, 9, 19 \rangle &
\langle 1, 2, 10, 15 \rangle &
\langle 4, 5, 10, 16 \rangle &
\end{array}$
$(9,13)$ $\begin{array}{llllllllll}
\langle 6, 4, 13, 17 \rangle &
\langle 3, 2, 17, 9 \rangle &
\langle 0, 1, 9, 10 \rangle &
\langle 6, 3, 18, 15 \rangle &
\langle 5, 0, 16, 17 \rangle &
\langle 2, 5, 18, 13 \rangle &
\langle 7, 1, 18, 20 \rangle &
\langle 2, 1, 12, 15 \rangle &
\langle 2, 4, 16, 14 \rangle \\
\langle 1, 5, 19, 21 \rangle &
\langle 6, 7, 9, 12 \rangle &
\langle 8, 7, 13, 16 \rangle &
\langle 8, 2, 20, 19 \rangle &
\langle 0, 3, 19, 13 \rangle &
\langle 4, 5, 20, 9 \rangle &
\langle 8, 0, 18, 12 \rangle &
\langle 7, 3, 14, 21 \rangle &
\langle 7, 5, 15, 10 \rangle \\
\langle 7, 4, 19, 11 \rangle &
\langle 1, 3, 16, 11 \rangle &
\langle 6, 8, 10, 11 \rangle &
\langle 6, 0, 20, 14 \rangle &
\langle 3, 4, 12, 10 \rangle &
\langle 1, 8, 14, 17 \rangle &
\langle 0, 2, 11, 21 \rangle &
\langle 4, 8, 15, 21 \rangle &
\end{array}$
$(9,15)$ $\begin{array}{llllllllll}
\langle 0, 1, 9, 10 \rangle &
\langle 0, 2, 11, 12 \rangle &
\langle 1, 2, 15, 13 \rangle &
\langle 6, 2, 22, 23 \rangle &
\langle 0, 5, 17, 18 \rangle &
\langle 0, 6, 19, 20 \rangle &
\langle 0, 4, 16, 15 \rangle &
\langle 2, 3, 9, 21 \rangle &
\langle 8, 1, 17, 23 \rangle \\
\langle 4, 7, 17, 13 \rangle &
\langle 3, 4, 23, 19 \rangle &
\langle 5, 8, 16, 22 \rangle &
\langle 2, 5, 14, 20 \rangle &
\langle 2, 7, 16, 19 \rangle &
\langle 2, 4, 10, 18 \rangle &
\langle 3, 6, 17, 15 \rangle &
\langle 4, 8, 20, 21 \rangle &
\langle 1, 3, 11, 22 \rangle \\
\langle 0, 3, 14, 13 \rangle &
\langle 3, 5, 10, 12 \rangle &
\langle 1, 5, 19, 21 \rangle &
\langle 1, 4, 14, 12 \rangle &
\langle 4, 5, 9, 11 \rangle &
\langle 7, 8, 14, 11 \rangle &
\langle 7, 0, 21, 22 \rangle &
\langle 6, 7, 9, 12 \rangle &
\langle 6, 8, 10, 13 \rangle \\
\langle 5, 7, 23, 15 \rangle &
\langle 3, 7, 18, 20 \rangle &
\langle 1, 6, 16, 18 \rangle &
\end{array}$
$(9,17)$ $\begin{array}{llllllllll}
\langle 8, 7, 15, 17 \rangle &
\langle 2, 6, 18, 9 \rangle &
\langle 0, 5, 14, 17 \rangle &
\langle 4, 0, 16, 19 \rangle &
\langle 1, 8, 11, 25 \rangle &
\langle 1, 0, 18, 24 \rangle &
\langle 1, 5, 16, 9 \rangle &
\langle 2, 3, 16, 24 \rangle &
\langle 7, 0, 21, 10 \rangle \\
\langle 4, 8, 14, 9 \rangle &
\langle 5, 6, 19, 10 \rangle &
\langle 8, 5, 12, 18 \rangle &
\langle 3, 4, 15, 13 \rangle &
\langle 3, 7, 19, 25 \rangle &
\langle 0, 6, 25, 15 \rangle &
\langle 3, 1, 17, 20 \rangle &
\langle 5, 7, 24, 13 \rangle &
\langle 8, 0, 13, 20 \rangle \\
\langle 3, 6, 11, 14 \rangle &
\langle 2, 0, 11, 12 \rangle &
\langle 4, 6, 22, 17 \rangle &
\langle 8, 2, 10, 22 \rangle &
\langle 8, 6, 24, 23 \rangle &
\langle 1, 2, 15, 19 \rangle &
\langle 4, 1, 12, 10 \rangle &
\langle 7, 6, 12, 16 \rangle &
\langle 3, 0, 23, 9 \rangle \\
\langle 7, 4, 18, 11 \rangle &
\langle 3, 5, 22, 21 \rangle &
\langle 4, 5, 20, 25 \rangle &
\langle 6, 1, 21, 13 \rangle &
\langle 2, 4, 23, 21 \rangle &
\langle 7, 1, 22, 23 \rangle &
\langle 2, 7, 14, 20 \rangle &
\end{array}$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\[PBD2MCWC\] Let $t$ be a positive integer with $2t+1\geq 21$ and $2t+1\not \in\{23,27,29,33,39,43,51,59,75,83,87,95,99,107,$ $139,179\}$. Let $n_1=4t+1$ or $4t+3$, $n_1\leq n_2\leq 2n_1-1$ and $n_2$ be odd. Then T$(2,n_1;2,n_2;6)=\lfloor \frac{n_2(n_1-1)}{4}\rfloor$.
According to Propositions \[MCWC2SAS\] and \[MCWC2SAS\*\], we only need to construct the corresponding SAS$(n_2,n_1)$ when $n_1\equiv 1\pmod{4}$ or SAS$^*(n_2,n_1)$ when $n_1\equiv 3\pmod{4}$.
For each given $t$ and $2t+1\not \in\{71,111,113,115,119\}$, take a $(2t+1,\{5,7,9\},1)$-PBD from Theorem \[PBD579\], and remove one point to obtain a $\{5,7,9\}$-GDD of type $4^i 6^j 8^k$ with $4i+6j+8k=2t$. Assign each point with weights $(4,2)$ or $(2,2)$ and apply Construction \[WFC\]; the input SFSs of type $(4,2)^a(2,2)^b$ with $a+b\in\{5,7,9\}$ are constructed in the Appendix. Then we can get an SFS of type $$(8,8)^{i_8}(10,8)^{i_{10}}\cdots(16,8)^{i_{16}}(12,12)^{j_{12}}\cdots(24,12)^{j_{24}}(16,16)^{k_{16}}\cdots(32,16)^{k_{32}},$$ for any nonnegative integers $i_8, i_{10},\ldots,i_{16},j_{12},\ldots,k_{32}$ with $$\begin{aligned}
i_8+ i_{10}+\cdots+i_{16}&=i\\
j_{12}+ j_{14}+\cdots+j_{24}&=j\\
k_{16}+ k_{18}+\cdots+k_{32}&=k.\end{aligned}$$ Now, we can fill the holes of the SFS in three ways:
1. Add a new row and a new column and apply Construction \[BFC\] (2) with ‘$e=1$’ and ‘$w=1$’; the input HSAS$(r,v;1,1)$ (i.e. SAS$(r,v)$) with $v\in\{9,13,17\}$ and $v\leq r\leq 2v-1$ come from Lemma \[MCWCsmall\]. Then we get an SAS$(s,4t+1;1,1)$ with $4t+1\leq s \leq 8t+1$, as desired;
2. Add three new rows and three new columns and apply Construction \[BFC\] (1) with ‘$e=3$’ and ‘$w=3$’; the input HSAS$(r,v;3,3)$ with $v\in\{11,15,19\}$ and $v\leq r\leq 2v-3$ are constructed in the Appendix. We get an HSAS$(s,4t+3;3,3)$ with $4t+3\leq s \leq 8t+3$. Then fill in the hole with an SAS$(3,3)$ constructed in Lemma \[MCWCsmall\] to obtain the desired SAS$^*(s,4t+3)$ with $4t+3\leq s \leq 8t+3$.
3. When the SFS has type $(16,8)^i(24,12)^j(32,16)^k$, add five new rows and five new columns and apply Construction \[BFC\] (1) with ‘$e=5$’ and ‘$w=3$’; the input HSAS$(r,v;5,3)$ with $(r,v)\in\{(21,11),$ $(29,15),(37,19)\}$ are constructed in the Appendix. We get an HSAS$(8t+5,4t+3;5,3)$. Then fill in the hole with an SAS$^*(5,3)$ constructed in Lemma \[MCWCsmall\] to obtain the desired SAS$^*(8t+5,4t+3)$.
For $2t+1=71$, take a TD$(9,8)$ from Theorem \[TD\] and truncate one of its group to six points to obtain an $\{8,9\}$-GDD of type $8^8 6^1$, noting that $8\times 8+6=70=2t$. Then proceed similarly as above, we can obtain the desired SAS$(s,4t+1)$ and SAS$^*(s,4t+3)$. Here the additional input SFSs of type $(4,2)^a(2,2)^{8-a}$ with $0\leq a \leq 8$ are constructed in the Appendix.
For $2t+1\in\{111,113,115,119\}$, take a $\{7,9\}$-GDD of type $8^{15}$ from [@CD Part 4, Corollary 2.44] and truncate the last two groups to obtain $\{5,6,7,8,9\}$-GDDs of types $8^{13} 6^1$, $8^{14}$, $8^{13} 6^1 4^1$ and $8^{14} 6^1$, respectively. Then proceed similarly as above, we can obtain the desired SASs and SAS$^*$s. Here the additional input SFSs of type $(4,2)^a(2,2)^{6-a}$ with $0\leq a \leq 6$ are constructed in the Appendix.
In the proof of Lemma \[PBD2MCWC\], we have constructed HSAS$(s,4t+3;3,3)$ with $4t+3\leq s \leq 8t+3$ and HSAS$(8t+5,4t+3;5,3)$. These HSASs will be used in later constructions.
Let $t$ be a positive integer with $2t+1 \in\{39,43,51,59,75,99\}$. Let $n_1=4t+1$ or $4t+3$, $n_1\leq n_2\leq 2n_1-1$ and $n_2$ be odd. Then T$(2,n_1;2,n_2;6)=\lfloor \frac{n_2(n_1-1)}{4}\rfloor$.
For $2t+1=39$, take a $\{5,7\}$-GDD of type $6^6 2^1$ from [@CD Part 4, Example 2.51], noting that $6\times 6+2=2t$. Assign each point with weights $(4,2)$ or $(2,2)$ and apply Construction \[WFC\]. Then we can get an SFS of type $$(12,12)^{i_{12}}(14,12)^{i_{14}}\cdots(24,12)^{i_{24}}(4,4)^{j_4}(8,4)^{j_8},$$ for any nonnegative integers $i_{12}, i_{14},\ldots,i_{24},j_4,j_8$ with $i_{12}+ i_{14}+\cdots+i_{24}=6$ and $j_4+j_8=1$. Now, we can fill the holes of the SFS in three ways:
1. Add a new row and a new column and apply Construction \[BFC\] (2) with ‘$e=1$’ and ‘$w=1$’; the input HSAS$(r,13;1,1)$ (i.e. SAS$(r,13)$) with $13\leq r\leq 25$, SAS$(5,5)$ and SAS$(9,5)$ come from Lemma \[MCWCsmall\]. Then we get an SAS$(s,77)$ with $77\leq s \leq 153$, as desired.
2. Add three new rows and three new columns and apply Construction \[BFC\] (3) with ‘$e=3$’ and ‘$w=3$’; the input HSAS$(r,15;3,3)$ with $15\leq r\leq 27$ are constructed in the Appendix and the input SAS$^*(7,7)$ and SAS$^*(11,7)$ come from Lemma \[MCWCsmall\]. Then we get an SAS$^*(s,79)$ with $79 \leq s \leq 155$.
3. When the SFS has type $(24,12)^6 (8,4)^1$, add five new rows and five new columns and apply Construction \[BFC\] (3) with ‘$e=5$’ and ‘$w=3$’; the input HSAS$(29,15;5,3)$ is constructed in the Appendix and the input SAS$^*(13,7)$ comes from Lemma \[MCWCsmall\]. Then we get an SAS$^*(157,79)$, as desired.
For $2t+1\in\{43,51,59,75,99\}$, we start with $\{5,6,7,8,9\}$-GDDs of types $8^5 2^1$, $8^6 2^1$, $8^7 2^1$, $8^9 2^1$, and $8^{12} 2^1$, respectively, which will be constructed below. Proceed as above to obtain the desired SASs and SAS$^*$s; here we fill in the holes of the SFS with SAS$(r,17)$ (see Lemma \[MCWCsmall\]), HSAS$(r,19;3,3)$ (see the Appendix) and HSAS$(37,19;5,3)$ (see the Appendix). The $\{5,6,7,8,9\}$-GDDs are constructed as follows. For the types $8^5 2^1$, $8^6 2^1$ and $8^7 2^1$, take a TD$(9,8)$ from Theorem \[TD\] and truncate the last four groups. For the type $8^9 2^1$, take a TD$(9,9)$ from Theorem \[TD\] and remove one point to redefine the groups to obtain a $\{9\}$-GDD of type $8^{10}$. Then truncate the last group. For the type $8^{12} 2^1$, take a $\{9\}$-GDD of type $8^{15} 16^1$ from [@CD Part 4, Corollary 2.44] and truncate the last four groups.
Let $t$ be a positive integer. If $2t+1 \in\{107,139,179\}$. Let $n_1=4t+1$ or $4t+3$, $n_1\leq n_2\leq 2n_1-1$ and $n_2$ be odd. Then T$(2,n_1;2,n_2;6)=\lfloor \frac{n_2(n_1-1)}{4}\rfloor$.
For $2t+1=107$, take a TD$(6,20)$ from Theorem \[TD\] and truncate the last group to six points to obtain a $\{5,6\}$-GDD of type $20^5 6^1$. Assign each point with weights $(4,2)$ or $(2,2)$ and apply Construction \[WFC\]. Then we can get an SFS of type $$(40,40)^{i_{40}}(42,40)^{i_{42}}\cdots(80,40)^{i_{80}}(12,12)^{j_{12}} (14,12)^{j_{14}}\cdots(24,12)^{j_{24}},$$ for any nonnegative integers $i_{40}, i_{42},\ldots,i_{80},j_{12},\ldots,j_{24}$ with $i_{40}+ i_{42}+\cdots+i_{80}=5$ and $j_{12}+j_{14}+\ldots+j_{24}=1$. Now, we can fill the holes of the SFS in three ways:
1. Add a new row and a new column and apply Construction \[BFC\] (2) with ‘$e=1$’ and ‘$w=1$’; the input HSASs and SASs come from Lemmas \[MCWCsmall\]–\[PBD2MCWC\]. Then we get an SAS$(s,213)$ with $213\leq s \leq 425$, as desired.
2. Add three new rows and three new columns and apply Construction \[BFC\] (3) with ‘$e=3$’ and ‘$w=3$’; the input HSAS$(r,43;3,3)$ with $43\leq r\leq 83$ are constructed in the proof of Lemma \[PBD2MCWC\] and the input SAS$^*(r,15)$ comes from Lemma \[MCWCsmall\]. Then we get an SAS$^*(s,215)$ with $215 \leq s \leq 427$.
3. When the SFS has type $(80,40)^5 (24,12)^1$, add five new rows and five new columns and apply Construction \[BFC\] (3) with ‘$e=5$’ and ‘$w=3$’; the input HSAS$(85,43;5,3)$ and SAS$^*(29,15)$ come from Lemmas \[MCWCsmall\]–\[PBD2MCWC\]. Then we get an SAS$^*(429,215)$, as desired.
For $2t+1=139$ or $179$, take a TD$(8,24)$ from Theorem \[TD\] and truncate the last three groups to obtain $\{5,6,7,8,9\}$-GDDs of types $24^5 6^3$ or $24^7 6^1 4^1$. Then proceed similarly as above to obtain the desired SASs and SAS$^*$s; the input HSASs, SASs and SAS$^*$s all come from Lemma \[MCWCsmall\].
Let $t$ be a positive integer with $2t+1 \in\{83,87,95\}$. Let $n_1=4t+1$, $n_1\leq n_2\leq 2n_1-1$ and $n_2$ be odd. Then T$(2,n_1;2,n_2;6)=\frac{n_2(n_1-1)}{4}$.
Take a TD$(6,16)$ from Theorem \[TD\] and truncate the last group to obtain $\{5,6\}$-GDDs of types $16^5 2^1$, $16^5 6^1$ or $16^5 14^1$, respectively. Then proceed similarly as above to obtain the desired SASs; the input SAS$(s,v)$ with $s\in\{5,13,29,33\}$ all come from Lemma \[MCWCsmall\].
Combining the above lemmas, we get the following result.
Let $n_1,n_2$ be two odd integers with $0< n_1 \leq n_2 \leq 2n_1-1$. Then T$(2,n_1;2,n_2;6) = \lfloor \frac{n_2(n_1-1)}{4}\rfloor$, except for $(n_1,n_2)=(5,7)$, and except possibly for $n_1\in\{23,27,31,35,39,45,47,53,55,57,59,65,67,165,175,191\}$.
Conclusions
===========
In this paper, we consider the bounds and constructions of MCWCs. For the upper bound, we use three different approaches to improve the generalised Johnson bounds mentioned in [@zhang]. For the lower bound, we derive two asymptotic lower bounds, the first is from the technique of concatenation and the second is from the Gilbert-Varshamov type bound. A comparison between these two bounds is also given. For the constructions, by establishing the connections between some combinatorial structures and MCWCs, several new combinatorial constructions for MCWCs are given. We obtain the asymptotic existence result of two classes of optimal MCWCs and construct a class of optimal MWCWs which are open in [@Cheeoptimal]. As consequences, the Johnson-type bounds are shown to be asymptotically exact for MCWCs with distances $2\sum_{i=1}^mw_i-2$ or $2mw-w$. The maximum sizes of MCWCs with total weight less than or equal to four are determined almost completely.
Appendix
========
Small MCWC$(2,n_1;2,n_2;6)$ for $n_1 \equiv 1\pmod{4}$ and $13\leq n_1 \leq 37$
-------------------------------------------------------------------------------
T$(2,13;2,n;6) =3n$ for each odd $n$ and $13 \leq n \leq 25$.
Let $X_1=({{\mathbb{Z}}}_3\times \{0,1,2,3\})\cup \{\infty\}$. For $13 \leq n \leq 17$, let $X_2= ({{\mathbb{Z}}}_3\times \{4,5,6,7\})\cup (\{a\}\times \{1,\ldots,n-12\})$; for $19 \leq n \leq 23$, let $X_2= ({{\mathbb{Z}}}_3\times \{4,5,\ldots,9\})\cup (\{a\}\times \{1,\ldots,n-18\})$; for $n = 25$, let $X_2= ({{\mathbb{Z}}}_3\times \{4,5,\ldots,11\})\cup (\{a\}\times \{1\})$. Denote $X=X_1\cup X_2$. The desired codes of size $3n$ are constructed on ${{\mathbb{Z}}}_2^X$. The codewords are obtained by developing the following base codewords under the action of the cyclic group ${{\mathbb{Z}}}_3$, where the points $\infty$ and $a_i$ are fixed.
$n=13$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{0}, 0_{5}, 0_{4}\rangle &
\langle\infty, 0_{3}, 2_{6}, 2_{7}\rangle &
\langle1_{1}, 1_{0}, 2_{4}, a_{1}\rangle &
\langle2_{2}, 2_{3}, 2_{5}, a_{1}\rangle &
\langle1_{0}, 2_{2}, 0_{7}, 2_{6}\rangle &
\langle1_{1}, 2_{2}, 1_{5}, 1_{6}\rangle &
\langle0_{0}, 2_{1}, 0_{7}, 2_{4}\rangle &
\langle0_{0}, 2_{3}, 0_{6}, 1_{5}\rangle \\
\langle2_{1}, 2_{2}, 1_{4}, 0_{5}\rangle &
\langle1_{2}, 2_{3}, 2_{4}, 2_{6}\rangle &
\langle2_{0}, 0_{1}, 0_{7}, 1_{6}\rangle &
\langle1_{2}, 0_{3}, 0_{7}, 1_{4}\rangle &
\langle0_{1}, 1_{3}, 2_{5}, 2_{7}\rangle &
\end{array}$$ ]{}
$n=15$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{1}, 2_{7}, 2_{6}\rangle &
\langle\infty, 0_{3}, 0_{5}, 2_{4}\rangle &
\langle2_{3}, 2_{2}, 0_{5}, a_{1}\rangle &
\langle2_{1}, 2_{0}, 1_{7}, a_{1}\rangle &
\langle2_{2}, 0_{3}, 0_{7}, a_{2}\rangle &
\langle1_{0}, 2_{1}, 0_{4}, a_{2}\rangle &
\langle0_{0}, 2_{1}, 0_{5}, a_{3}\rangle &
\langle0_{3}, 1_{2}, 1_{4}, a_{3}\rangle \\
\langle0_{1}, 2_{3}, 2_{6}, 2_{4}\rangle &
\langle2_{1}, 0_{2}, 2_{4}, 0_{7}\rangle &
\langle1_{0}, 1_{3}, 0_{6}, 2_{7}\rangle &
\langle0_{1}, 0_{3}, 1_{6}, 2_{5}\rangle &
\langle1_{2}, 2_{0}, 2_{4}, 0_{6}\rangle &
\langle0_{2}, 0_{0}, 2_{5}, 0_{6}\rangle &
\langle0_{0}, 1_{2}, 1_{5}, 0_{7}\rangle &
\end{array}$$ ]{}
$n=17$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{1}, 1_{7}, 1_{6}\rangle &
\langle\infty, 0_{3}, 1_{4}, 2_{5}\rangle &
\langle1_{2}, 1_{3}, 0_{6}, a_{1}\rangle &
\langle0_{0}, 0_{1}, 1_{4}, a_{1}\rangle &
\langle1_{0}, 0_{1}, 1_{6}, a_{2}\rangle &
\langle1_{3}, 0_{2}, 2_{5}, a_{2}\rangle &
\langle0_{1}, 2_{0}, 2_{5}, a_{3}\rangle &
\langle2_{3}, 0_{2}, 1_{7}, a_{3}\rangle \\
\langle1_{1}, 0_{3}, 0_{4}, a_{4}\rangle &
\langle0_{0}, 1_{2}, 2_{6}, a_{4}\rangle &
\langle2_{1}, 0_{3}, 2_{4}, a_{5}\rangle &
\langle0_{2}, 1_{0}, 2_{7}, a_{5}\rangle &
\langle1_{3}, 0_{0}, 1_{5}, 2_{7}\rangle &
\langle1_{2}, 0_{1}, 1_{5}, 1_{7}\rangle &
\langle0_{0}, 0_{3}, 0_{7}, 1_{6}\rangle &
\langle1_{1}, 0_{2}, 1_{5}, 0_{6}\rangle \\
\langle0_{0}, 0_{2}, 0_{4}, 2_{4}\rangle &
\end{array}$$ ]{}
$n=19$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{0}, 0_{4}, 0_{5}\rangle &
\langle\infty, 0_{1}, 0_{6}, 0_{7}\rangle &
\langle\infty, 0_{2}, 0_{8}, 0_{9}\rangle &
\langle0_{0}, 0_{1}, 1_{4}, a_{1}\rangle &
\langle0_{2}, 0_{3}, 0_{5}, a_{1}\rangle &
\langle0_{0}, 1_{0}, 2_{5}, 0_{6}\rangle &
\langle0_{3}, 1_{3}, 0_{4}, 0_{8}\rangle &
\langle0_{0}, 1_{3}, 2_{7}, 2_{9}\rangle \\
\langle0_{0}, 0_{2}, 1_{7}, 1_{8}\rangle &
\langle0_{1}, 1_{1}, 0_{5}, 0_{8}\rangle &
\langle0_{1}, 0_{2}, 2_{4}, 1_{5}\rangle &
\langle0_{1}, 1_{2}, 1_{6}, 0_{9}\rangle &
\langle0_{2}, 1_{2}, 2_{6}, 1_{4}\rangle &
\langle0_{2}, 1_{3}, 2_{5}, 0_{7}\rangle &
\langle0_{2}, 2_{3}, 2_{7}, 1_{9}\rangle &
\langle0_{0}, 1_{1}, 0_{7}, 2_{8}\rangle \\
\langle0_{0}, 2_{1}, 2_{4}, 0_{9}\rangle &
\langle0_{0}, 2_{3}, 1_{6}, 0_{8}\rangle &
\langle0_{1}, 2_{3}, 2_{6}, 2_{9}\rangle &
\end{array}$$ ]{}
$n=21$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{1}, 1_{7}, 1_{6}\rangle &
\langle\infty, 0_{2}, 1_{8}, 0_{9}\rangle &
\langle\infty, 1_{0}, 1_{4}, 1_{5}\rangle &
\langle2_{1}, 2_{0}, 0_{7}, a_{1}\rangle &
\langle0_{2}, 0_{3}, 0_{5}, a_{1}\rangle &
\langle1_{3}, 0_{1}, 1_{4}, a_{2}\rangle &
\langle0_{0}, 0_{2}, 1_{5}, a_{2}\rangle &
\langle2_{0}, 1_{1}, 1_{4}, a_{3}\rangle \\
\langle2_{3}, 1_{2}, 0_{7}, a_{3}\rangle &
\langle0_{2}, 1_{2}, 2_{6}, 0_{4}\rangle &
\langle0_{0}, 1_{2}, 2_{9}, 2_{7}\rangle &
\langle1_{0}, 1_{3}, 1_{7}, 2_{9}\rangle &
\langle0_{1}, 0_{3}, 2_{7}, 1_{8}\rangle &
\langle0_{0}, 1_{0}, 2_{6}, 0_{8}\rangle &
\langle0_{3}, 2_{3}, 1_{4}, 2_{8}\rangle &
\langle1_{1}, 2_{2}, 1_{5}, 0_{4}\rangle \\
\langle2_{1}, 1_{3}, 1_{6}, 0_{6}\rangle &
\langle0_{0}, 1_{3}, 0_{9}, 2_{5}\rangle &
\langle1_{1}, 1_{2}, 1_{8}, 0_{8}\rangle &
\langle1_{1}, 2_{1}, 0_{9}, 0_{5}\rangle &
\langle0_{2}, 2_{3}, 0_{6}, 2_{9}\rangle &
\end{array}$$ ]{}
$n=23$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{0}, 0_{5}, 0_{4}\rangle &
\langle\infty, 1_{2}, 0_{8}, 0_{9}\rangle &
\langle\infty, 1_{1}, 2_{6}, 0_{7}\rangle &
\langle0_{3}, 0_{2}, 0_{5}, a_{1}\rangle &
\langle1_{0}, 1_{1}, 2_{4}, a_{1}\rangle &
\langle2_{1}, 1_{0}, 2_{5}, a_{2}\rangle &
\langle0_{3}, 2_{2}, 2_{4}, a_{2}\rangle &
\langle2_{0}, 1_{1}, 2_{7}, a_{3}\rangle \\
\langle2_{3}, 0_{2}, 0_{6}, a_{3}\rangle &
\langle0_{0}, 0_{2}, 2_{5}, a_{4}\rangle &
\langle0_{3}, 0_{1}, 0_{6}, a_{4}\rangle &
\langle1_{1}, 2_{3}, 0_{4}, a_{5}\rangle &
\langle0_{2}, 2_{0}, 2_{6}, a_{5}\rangle &
\langle0_{0}, 2_{2}, 0_{8}, 1_{7}\rangle &
\langle0_{0}, 2_{0}, 1_{6}, 1_{8}\rangle &
\langle0_{0}, 2_{3}, 0_{9}, 2_{4}\rangle \\
\langle0_{3}, 1_{3}, 2_{8}, 2_{5}\rangle &
\langle0_{1}, 2_{3}, 2_{9}, 0_{7}\rangle &
\langle0_{1}, 1_{1}, 2_{5}, 1_{8}\rangle &
\langle0_{1}, 1_{2}, 2_{6}, 1_{9}\rangle &
\langle0_{0}, 0_{3}, 2_{7}, 2_{9}\rangle &
\langle1_{2}, 2_{1}, 1_{8}, 2_{9}\rangle &
\langle0_{2}, 1_{2}, 2_{4}, 1_{7}\rangle &
\end{array}$$ ]{}
$n=25$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{3}, 0_{11}, 0_{7}\rangle &
\langle\infty, 2_{0}, 2_{5}, 1_{9}\rangle &
\langle\infty, 0_{2}, 0_{6}, 1_{8}\rangle &
\langle\infty, 0_{1}, 1_{10}, 1_{4}\rangle &
\langle0_{0}, 0_{2}, 0_{7}, a_{1}\rangle &
\langle0_{1}, 1_{3}, 0_{6}, a_{1}\rangle &
\langle2_{2}, 0_{1}, 0_{5}, 1_{7}\rangle &
\langle1_{3}, 1_{0}, 0_{8}, 2_{5}\rangle \\
\langle2_{0}, 0_{3}, 0_{8}, 1_{7}\rangle &
\langle0_{3}, 1_{2}, 1_{10}, 0_{10}\rangle &
\langle0_{0}, 2_{1}, 1_{6}, 0_{9}\rangle &
\langle0_{0}, 1_{2}, 2_{6}, 0_{6}\rangle &
\langle2_{0}, 1_{0}, 0_{4}, 2_{11}\rangle &
\langle0_{0}, 0_{1}, 0_{4}, 2_{10}\rangle &
\langle0_{2}, 1_{2}, 1_{9}, 2_{11}\rangle &
\langle2_{1}, 0_{2}, 0_{8}, 0_{11}\rangle \\
\langle1_{2}, 2_{0}, 0_{9}, 2_{10}\rangle &
\langle1_{0}, 0_{3}, 0_{5}, 2_{7}\rangle &
\langle0_{1}, 2_{1}, 2_{7}, 2_{9}\rangle &
\langle2_{3}, 0_{3}, 1_{4}, 1_{9}\rangle &
\langle2_{2}, 2_{1}, 1_{4}, 1_{5}\rangle &
\langle0_{0}, 1_{1}, 1_{10}, 0_{8}\rangle &
\langle1_{3}, 1_{1}, 2_{6}, 0_{11}\rangle &
\langle2_{1}, 1_{3}, 0_{5}, 2_{11}\rangle \\
\langle0_{2}, 1_{3}, 1_{4}, 2_{8}\rangle &
\end{array}$$ ]{}
T$(2,17;2,n;6) =4n$ for each odd $n$ and $17 \leq n \leq 33$.
Let $X_1=({{\mathbb{Z}}}_4\times \{0,1,2,3\})\cup \{\infty\}$. For $17 \leq n \leq 23$, let $X_2= ({{\mathbb{Z}}}_4\times \{4,5,6,7\})\cup (\{a\}\times \{1,\ldots,n-16\})$; for $25 \leq n \leq 31$, let $X_2= ({{\mathbb{Z}}}_4\times \{4,5,\ldots,9\})\cup (\{a\}\times \{1,\ldots,n-24\})$; for $n = 33$, let $X_2= ({{\mathbb{Z}}}_4\times \{4,5,\ldots,11\})\cup (\{a\}\times \{1\})$. Denote $X=X_1\cup X_2$. The codes of size $4n$ are constructed on ${{\mathbb{Z}}}_2^X$ and the base codewords are listed as follows.
$n=17$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{0}, 1_{4}, 1_{5}\rangle &
\langle\infty, 2_{1}, 1_{7}, 2_{6}\rangle &
\langle3_{0}, 3_{1}, 0_{4}, a_{1}\rangle &
\langle3_{2}, 2_{3}, 3_{5}, a_{1}\rangle &
\langle2_{0}, 2_{3}, 1_{7}, 1_{5}\rangle &
\langle0_{2}, 2_{3}, 0_{6}, 2_{5}\rangle &
\langle0_{1}, 1_{1}, 1_{5}, 0_{4}\rangle &
\langle0_{2}, 1_{2}, 2_{4}, 1_{7}\rangle \\
\langle1_{1}, 1_{3}, 3_{5}, 0_{6}\rangle &
\langle1_{1}, 0_{2}, 3_{6}, 3_{4}\rangle &
\langle1_{0}, 3_{1}, 1_{7}, 0_{6}\rangle &
\langle0_{3}, 1_{3}, 1_{6}, 2_{4}\rangle &
\langle0_{0}, 1_{0}, 2_{5}, 3_{4}\rangle &
\langle2_{0}, 3_{3}, 0_{7}, 3_{7}\rangle &
\langle0_{1}, 2_{2}, 1_{7}, 3_{5}\rangle &
\langle0_{0}, 3_{2}, 0_{6}, 1_{6}\rangle \\
\langle0_{2}, 0_{3}, 0_{4}, 2_{7}\rangle &
\end{array}$$ ]{}
$n=19$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{0}, 1_{4}, 2_{5}\rangle &
\langle\infty, 3_{1}, 2_{6}, 0_{7}\rangle &
\langle1_{0}, 2_{1}, 1_{7}, a_{1}\rangle &
\langle0_{3}, 1_{2}, 0_{6}, a_{1}\rangle &
\langle2_{0}, 3_{3}, 3_{5}, a_{2}\rangle &
\langle0_{1}, 1_{2}, 0_{4}, a_{2}\rangle &
\langle0_{2}, 0_{3}, 1_{4}, a_{3}\rangle &
\langle1_{1}, 2_{0}, 0_{5}, a_{3}\rangle \\
\langle0_{0}, 2_{3}, 0_{5}, 3_{7}\rangle &
\langle3_{3}, 2_{2}, 3_{7}, 2_{5}\rangle &
\langle0_{0}, 0_{3}, 1_{6}, 0_{4}\rangle &
\langle2_{0}, 3_{2}, 3_{4}, 1_{6}\rangle &
\langle3_{2}, 3_{1}, 1_{4}, 3_{7}\rangle &
\langle1_{1}, 0_{2}, 3_{7}, 3_{5}\rangle &
\langle0_{1}, 3_{1}, 0_{6}, 0_{5}\rangle &
\langle2_{3}, 3_{3}, 1_{4}, 1_{6}\rangle \\
\langle2_{1}, 2_{0}, 0_{6}, 1_{4}\rangle &
\langle1_{2}, 0_{2}, 2_{5}, 1_{6}\rangle &
\langle0_{0}, 3_{3}, 1_{7}, 2_{7}\rangle &
\end{array}$$ ]{}
$n=21$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{1}, 2_{6}, 2_{7}\rangle &
\langle\infty, 2_{0}, 0_{5}, 2_{4}\rangle &
\langle3_{2}, 3_{3}, 3_{5}, a_{1}\rangle &
\langle0_{0}, 0_{1}, 1_{6}, a_{1}\rangle &
\langle0_{3}, 3_{2}, 1_{5}, a_{2}\rangle &
\langle3_{1}, 1_{0}, 3_{4}, a_{2}\rangle &
\langle3_{2}, 1_{3}, 3_{6}, a_{3}\rangle &
\langle0_{0}, 1_{1}, 3_{4}, a_{3}\rangle \\
\langle2_{3}, 3_{2}, 0_{4}, a_{4}\rangle &
\langle1_{0}, 0_{1}, 2_{5}, a_{4}\rangle &
\langle2_{3}, 0_{1}, 3_{7}, a_{5}\rangle &
\langle1_{0}, 1_{2}, 0_{5}, a_{5}\rangle &
\langle2_{1}, 1_{3}, 3_{7}, 3_{5}\rangle &
\langle0_{2}, 0_{1}, 3_{6}, 3_{4}\rangle &
\langle0_{1}, 1_{3}, 2_{6}, 0_{5}\rangle &
\langle1_{2}, 0_{1}, 2_{7}, 1_{4}\rangle \\
\langle1_{0}, 3_{2}, 1_{6}, 3_{7}\rangle &
\langle3_{0}, 2_{2}, 3_{5}, 0_{4}\rangle &
\langle2_{0}, 3_{2}, 1_{7}, 0_{6}\rangle &
\langle1_{3}, 2_{3}, 1_{6}, 2_{4}\rangle &
\langle0_{0}, 1_{3}, 0_{7}, 1_{7}\rangle &
\end{array}$$ ]{}
$n=23$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{0}, 3_{5}, 3_{4}\rangle &
\langle\infty, 3_{1}, 1_{7}, 0_{6}\rangle &
\langle1_{2}, 1_{3}, 2_{6}, a_{1}\rangle &
\langle2_{0}, 2_{1}, 3_{4}, a_{1}\rangle &
\langle2_{1}, 1_{0}, 0_{4}, a_{2}\rangle &
\langle3_{3}, 2_{2}, 2_{5}, a_{2}\rangle &
\langle0_{0}, 1_{3}, 2_{5}, a_{3}\rangle &
\langle0_{1}, 2_{2}, 0_{7}, a_{3}\rangle \\
\langle1_{0}, 0_{1}, 2_{5}, a_{4}\rangle &
\langle2_{3}, 3_{2}, 3_{7}, a_{4}\rangle &
\langle2_{3}, 2_{1}, 1_{4}, a_{5}\rangle &
\langle2_{2}, 1_{0}, 1_{6}, a_{5}\rangle &
\langle0_{1}, 1_{3}, 3_{5}, a_{6}\rangle &
\langle0_{0}, 2_{2}, 2_{6}, a_{6}\rangle &
\langle3_{0}, 3_{2}, 2_{5}, a_{7}\rangle &
\langle2_{3}, 0_{1}, 1_{7}, a_{7}\rangle \\
\langle0_{2}, 0_{1}, 3_{7}, 0_{4}\rangle &
\langle0_{3}, 3_{3}, 1_{4}, 2_{6}\rangle &
\langle3_{0}, 2_{2}, 0_{6}, 3_{7}\rangle &
\langle1_{1}, 0_{1}, 3_{6}, 1_{5}\rangle &
\langle1_{3}, 2_{0}, 3_{7}, 1_{6}\rangle &
\langle2_{3}, 0_{0}, 2_{7}, 2_{4}\rangle &
\langle0_{2}, 1_{2}, 3_{4}, 2_{5}\rangle &
\end{array}$$ ]{}
$n=25$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{1}, 3_{7}, 1_{6}\rangle &
\langle\infty, 2_{0}, 2_{4}, 2_{5}\rangle &
\langle\infty, 0_{2}, 0_{9}, 0_{8}\rangle &
\langle3_{3}, 0_{2}, 3_{6}, a_{1}\rangle &
\langle0_{0}, 0_{1}, 1_{4}, a_{1}\rangle &
\langle0_{3}, 1_{3}, 2_{4}, 2_{9}\rangle &
\langle2_{2}, 0_{1}, 0_{6}, 3_{9}\rangle &
\langle0_{1}, 1_{1}, 1_{5}, 0_{4}\rangle \\
\langle1_{2}, 0_{0}, 2_{8}, 2_{6}\rangle &
\langle2_{0}, 1_{0}, 3_{5}, 0_{4}\rangle &
\langle0_{3}, 1_{0}, 1_{7}, 0_{8}\rangle &
\langle1_{2}, 1_{3}, 0_{4}, 3_{8}\rangle &
\langle2_{1}, 2_{2}, 0_{4}, 1_{7}\rangle &
\langle1_{1}, 0_{3}, 1_{8}, 0_{5}\rangle &
\langle3_{1}, 0_{3}, 1_{5}, 2_{6}\rangle &
\langle2_{0}, 0_{2}, 3_{9}, 0_{7}\rangle \\
\langle2_{0}, 0_{1}, 1_{6}, 3_{8}\rangle &
\langle0_{0}, 3_{1}, 0_{8}, 3_{9}\rangle &
\langle2_{1}, 0_{3}, 0_{9}, 0_{7}\rangle &
\langle0_{0}, 1_{3}, 3_{7}, 0_{9}\rangle &
\langle0_{0}, 0_{3}, 3_{5}, 1_{6}\rangle &
\langle0_{1}, 3_{2}, 1_{7}, 2_{8}\rangle &
\langle2_{2}, 1_{2}, 2_{4}, 0_{5}\rangle &
\langle2_{0}, 2_{2}, 2_{6}, 0_{9}\rangle \\
\langle0_{2}, 2_{3}, 0_{5}, 1_{7}\rangle &
\end{array}$$ ]{}
$n=27$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{0}, 2_{4}, 2_{5}\rangle &
\langle\infty, 2_{2}, 0_{8}, 2_{9}\rangle &
\langle\infty, 0_{1}, 1_{6}, 1_{7}\rangle &
\langle2_{3}, 2_{2}, 0_{7}, a_{1}\rangle &
\langle2_{1}, 2_{0}, 3_{4}, a_{1}\rangle &
\langle1_{2}, 2_{3}, 3_{5}, a_{2}\rangle &
\langle2_{1}, 1_{0}, 0_{6}, a_{2}\rangle &
\langle3_{0}, 1_{1}, 1_{4}, a_{3}\rangle \\
\langle2_{3}, 0_{2}, 1_{5}, a_{3}\rangle &
\langle1_{1}, 3_{2}, 1_{6}, 2_{9}\rangle &
\langle1_{1}, 2_{1}, 2_{8}, 3_{5}\rangle &
\langle0_{0}, 0_{2}, 1_{6}, 0_{7}\rangle &
\langle3_{0}, 2_{3}, 2_{5}, 1_{6}\rangle &
\langle0_{0}, 2_{2}, 1_{8}, 1_{7}\rangle &
\langle1_{0}, 3_{3}, 1_{6}, 0_{9}\rangle &
\langle0_{2}, 3_{3}, 2_{9}, 2_{4}\rangle \\
\langle0_{2}, 1_{2}, 0_{6}, 1_{4}\rangle &
\langle1_{1}, 2_{0}, 0_{7}, 3_{9}\rangle &
\langle0_{1}, 3_{3}, 3_{6}, 3_{4}\rangle &
\langle1_{0}, 0_{0}, 2_{5}, 0_{8}\rangle &
\langle0_{3}, 1_{3}, 0_{8}, 2_{4}\rangle &
\langle1_{0}, 1_{3}, 1_{9}, 0_{7}\rangle &
\langle2_{1}, 2_{3}, 0_{8}, 2_{7}\rangle &
\langle2_{0}, 3_{2}, 0_{9}, 0_{8}\rangle \\
\langle1_{1}, 3_{3}, 1_{9}, 0_{8}\rangle &
\langle0_{1}, 1_{2}, 2_{7}, 0_{5}\rangle &
\langle0_{1}, 3_{2}, 2_{4}, 3_{5}\rangle &
\end{array}$$ ]{}
$n=29$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{0}, 0_{4}, 0_{5}\rangle &
\langle\infty, 0_{1}, 0_{7}, 0_{6}\rangle &
\langle\infty, 2_{2}, 2_{9}, 2_{8}\rangle &
\langle2_{3}, 2_{2}, 2_{5}, a_{1}\rangle &
\langle0_{1}, 0_{0}, 1_{4}, a_{1}\rangle &
\langle2_{1}, 1_{0}, 0_{4}, a_{2}\rangle &
\langle2_{3}, 1_{2}, 3_{5}, a_{2}\rangle &
\langle2_{3}, 0_{2}, 2_{6}, a_{3}\rangle \\
\langle3_{1}, 1_{0}, 3_{4}, a_{3}\rangle &
\langle3_{3}, 0_{2}, 3_{4}, a_{4}\rangle &
\langle1_{1}, 2_{0}, 3_{5}, a_{4}\rangle &
\langle0_{2}, 0_{0}, 3_{5}, a_{5}\rangle &
\langle1_{1}, 1_{3}, 0_{4}, a_{5}\rangle &
\langle0_{0}, 3_{0}, 2_{8}, 3_{9}\rangle &
\langle0_{0}, 2_{3}, 1_{9}, 3_{7}\rangle &
\langle0_{3}, 3_{0}, 3_{7}, 1_{6}\rangle \\
\langle0_{2}, 0_{1}, 1_{8}, 3_{6}\rangle &
\langle3_{1}, 0_{3}, 2_{7}, 2_{9}\rangle &
\langle3_{1}, 2_{3}, 1_{8}, 0_{5}\rangle &
\langle3_{1}, 0_{2}, 1_{6}, 0_{6}\rangle &
\langle3_{1}, 1_{3}, 2_{8}, 1_{7}\rangle &
\langle1_{3}, 0_{3}, 1_{9}, 2_{4}\rangle &
\langle0_{1}, 3_{1}, 1_{9}, 3_{5}\rangle &
\langle0_{2}, 1_{0}, 2_{7}, 3_{9}\rangle \\
\langle1_{2}, 3_{1}, 0_{7}, 3_{8}\rangle &
\langle2_{0}, 1_{3}, 3_{8}, 3_{6}\rangle &
\langle0_{2}, 3_{2}, 1_{9}, 1_{4}\rangle &
\langle3_{0}, 3_{3}, 2_{6}, 3_{8}\rangle &
\langle0_{0}, 1_{2}, 2_{5}, 2_{7}\rangle &
\end{array}$$ ]{}
$n=31$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{3}, 1_{9}, 1_{5}\rangle &
\langle\infty, 2_{0}, 0_{6}, 1_{7}\rangle &
\langle\infty, 3_{2}, 0_{8}, 3_{4}\rangle &
\langle3_{3}, 3_{1}, 0_{4}, a_{1}\rangle &
\langle3_{0}, 0_{2}, 3_{5}, a_{1}\rangle &
\langle3_{1}, 2_{0}, 2_{4}, a_{2}\rangle &
\langle1_{3}, 3_{2}, 1_{6}, a_{2}\rangle &
\langle3_{1}, 1_{0}, 3_{4}, a_{3}\rangle \\
\langle1_{3}, 0_{2}, 0_{6}, a_{3}\rangle &
\langle1_{0}, 0_{1}, 2_{6}, a_{4}\rangle &
\langle0_{2}, 0_{3}, 0_{7}, a_{4}\rangle &
\langle3_{0}, 0_{3}, 0_{4}, a_{5}\rangle &
\langle2_{1}, 0_{2}, 0_{5}, a_{5}\rangle &
\langle1_{2}, 1_{1}, 2_{5}, a_{6}\rangle &
\langle3_{0}, 2_{3}, 3_{6}, a_{6}\rangle &
\langle1_{1}, 3_{3}, 1_{5}, a_{7}\rangle \\
\langle1_{2}, 3_{0}, 3_{7}, a_{7}\rangle &
\langle3_{0}, 0_{0}, 0_{8}, 2_{5}\rangle &
\langle1_{1}, 0_{1}, 3_{7}, 3_{8}\rangle &
\langle3_{3}, 0_{2}, 3_{9}, 1_{7}\rangle &
\langle1_{0}, 1_{2}, 1_{9}, 0_{6}\rangle &
\langle0_{1}, 3_{2}, 0_{9}, 0_{6}\rangle &
\langle0_{3}, 0_{0}, 2_{9}, 1_{5}\rangle &
\langle2_{0}, 1_{2}, 0_{7}, 1_{8}\rangle \\
\langle3_{3}, 2_{3}, 0_{8}, 1_{4}\rangle &
\langle2_{3}, 1_{1}, 0_{6}, 1_{8}\rangle &
\langle1_{2}, 2_{2}, 0_{8}, 0_{4}\rangle &
\langle0_{3}, 2_{0}, 0_{8}, 1_{9}\rangle &
\langle3_{0}, 3_{1}, 0_{7}, 0_{9}\rangle &
\langle2_{2}, 1_{1}, 3_{4}, 0_{9}\rangle &
\langle0_{1}, 3_{3}, 3_{5}, 0_{7}\rangle &
\end{array}$$ ]{}
$n=33$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{0}, 0_{8}, 0_{9}\rangle &
\langle\infty, 2_{1}, 3_{4}, 3_{6}\rangle &
\langle\infty, 3_{3}, 2_{11}, 3_{5}\rangle &
\langle\infty, 3_{2}, 1_{10}, 3_{7}\rangle &
\langle2_{2}, 3_{3}, 0_{5}, a_{1}\rangle &
\langle3_{1}, 1_{0}, 0_{7}, a_{1}\rangle &
\langle2_{1}, 1_{3}, 1_{11}, 3_{8}\rangle &
\langle2_{2}, 1_{2}, 1_{11}, 1_{5}\rangle \\
\langle3_{1}, 0_{1}, 3_{9}, 2_{7}\rangle &
\langle1_{2}, 1_{1}, 0_{4}, 2_{10}\rangle &
\langle0_{1}, 1_{2}, 2_{4}, 3_{8}\rangle &
\langle0_{3}, 0_{0}, 3_{9}, 1_{7}\rangle &
\langle1_{3}, 3_{2}, 2_{6}, 2_{9}\rangle &
\langle3_{0}, 0_{0}, 3_{6}, 1_{11}\rangle &
\langle2_{2}, 1_{0}, 0_{4}, 1_{10}\rangle &
\langle3_{0}, 1_{2}, 3_{11}, 0_{8}\rangle \\
\langle1_{0}, 0_{2}, 1_{9}, 1_{5}\rangle &
\langle1_{0}, 1_{1}, 2_{5}, 3_{6}\rangle &
\langle1_{0}, 2_{1}, 0_{8}, 0_{11}\rangle &
\langle0_{1}, 2_{2}, 3_{6}, 2_{10}\rangle &
\langle0_{2}, 0_{3}, 2_{9}, 3_{7}\rangle &
\langle3_{0}, 1_{3}, 2_{10}, 0_{6}\rangle &
\langle2_{0}, 2_{2}, 2_{8}, 0_{7}\rangle &
\langle3_{1}, 3_{3}, 3_{6}, 0_{11}\rangle \\
\langle0_{3}, 1_{3}, 1_{8}, 3_{4}\rangle &
\langle0_{0}, 3_{1}, 1_{9}, 2_{10}\rangle &
\langle0_{0}, 3_{3}, 1_{10}, 2_{5}\rangle &
\langle3_{2}, 0_{1}, 0_{7}, 0_{8}\rangle &
\langle2_{2}, 1_{3}, 3_{11}, 2_{4}\rangle &
\langle0_{1}, 1_{3}, 0_{10}, 1_{9}\rangle &
\langle0_{0}, 1_{3}, 1_{4}, 3_{5}\rangle &
\langle0_0, 2_0, 0_4, 2_4\rangle^s \\
\langle0_1, 2_1, 0_5, 2_5\rangle^s &
\langle0_2, 2_2, 0_6, 2_6\rangle^s &
\langle0_3, 2_3, 0_7, 2_7\rangle^s &
\end{array}$$ ]{} Note that each of the codewords marked $s$ only generates two codewords.
T$(2,21;2,n;6) =5n$ for each odd $n$ and $21 \leq n \leq 41$.
Let $X_1=({{\mathbb{Z}}}_5\times \{0,1,2,3\})\cup \{\infty\}$. For $21 \leq n \leq 29$, let $X_2= ({{\mathbb{Z}}}_5\times \{4,5,6,7\})\cup (\{a\}\times \{1,\ldots,n-20\})$; for $31 \leq n \leq 39$, let $X_2= ({{\mathbb{Z}}}_5\times \{4,5,\ldots,9\})\cup (\{a\}\times \{1,\ldots,n-30\})$; for $n = 41$, let $X_2= ({{\mathbb{Z}}}_5\times \{4,5,\ldots,11\})\cup (\{a\}\times \{1\})$. Denote $X=X_1\cup X_2$. The desired codes of size $5n$ are constructed on ${{\mathbb{Z}}}_2^X$ and the base codewords are listed as follows.
$n=21$: [$$\begin{array}{llllllllll}
\langle\infty, 2_0, 4_5, 4_4\rangle &
\langle\infty, 0_1, 0_7, 3_6\rangle &
\langle1_1, 2_0, 2_4, a_1\rangle &
\langle2_2, 0_3, 3_6, a_1\rangle &
\langle2_2, 2_3, 3_5, 3_7\rangle &
\langle4_1, 4_2, 1_4, 3_5\rangle &
\langle1_2, 4_2, 4_6, 1_5\rangle &
\langle1_2, 4_0, 4_5, 0_4\rangle \\
\langle1_3, 4_3, 4_5, 3_6\rangle &
\langle1_2, 0_3, 0_6, 1_4\rangle &
\langle0_3, 4_2, 2_7, 1_6\rangle &
\langle2_0, 0_1, 3_7, 2_7\rangle &
\langle0_1, 4_1, 4_6, 4_4\rangle &
\langle0_3, 4_3, 2_4, 4_4\rangle &
\langle1_0, 2_0, 0_6, 2_6\rangle &
\langle1_0, 3_1, 4_5, 0_5\rangle \\
\langle0_0, 4_3, 1_5, 3_7\rangle &
\langle3_0, 4_1, 2_4, 0_6\rangle &
\langle0_1, 3_1, 4_7, 3_5\rangle &
\langle2_0, 2_2, 0_4, 1_7\rangle &
\langle0_2, 2_3, 0_7, 2_7\rangle &
\end{array}$$ ]{}
$n=23$: [$$\begin{array}{llllllllll}
\langle\infty, 3_0, 3_4, 3_5\rangle &
\langle\infty, 2_1, 2_6, 2_7\rangle &
\langle2_0, 2_1, 3_4, a_1\rangle &
\langle1_2, 3_3, 3_5, a_1\rangle &
\langle2_0, 3_1, 0_4, a_2\rangle &
\langle1_2, 2_3, 4_6, a_2\rangle &
\langle0_2, 4_3, 3_5, a_3\rangle &
\langle3_0, 0_1, 0_4, a_3\rangle \\
\langle1_2, 1_3, 1_6, 3_7\rangle &
\langle0_0, 4_1, 3_7, 2_7\rangle &
\langle0_1, 1_2, 1_7, 2_6\rangle &
\langle0_3, 1_3, 0_4, 0_7\rangle &
\langle1_2, 4_3, 0_5, 0_7\rangle &
\langle1_2, 0_2, 2_4, 1_5\rangle &
\langle0_1, 2_1, 3_5, 3_6\rangle &
\langle0_0, 4_0, 1_5, 3_5\rangle \\
\langle0_1, 1_1, 0_5, 4_4\rangle &
\langle0_0, 2_0, 1_6, 0_6\rangle &
\langle0_2, 3_2, 3_4, 2_6\rangle &
\langle0_3, 2_3, 3_4, 3_6\rangle &
\langle0_0, 3_2, 4_7, 1_7\rangle &
\langle1_0, 3_3, 0_4, 1_7\rangle &
\langle0_1, 0_3, 2_5, 4_6\rangle &
\end{array}$$ ]{}
$n=25$: [$$\begin{array}{llllllllll}
\langle\infty, 2_1, 2_7, 0_6\rangle &
\langle\infty, 3_3, 2_5, 4_4\rangle &
\langle3_3, 2_2, 1_4, a_1\rangle &
\langle3_1, 0_0, 4_7, a_1\rangle &
\langle0_0, 0_1, 3_4, a_2\rangle &
\langle3_2, 0_3, 0_5, a_2\rangle &
\langle1_2, 1_0, 1_7, a_3\rangle &
\langle3_3, 1_1, 2_4, a_3\rangle \\
\langle3_0, 2_1, 3_5, a_4\rangle &
\langle3_3, 4_2, 0_4, a_4\rangle &
\langle1_1, 4_3, 0_7, a_5\rangle &
\langle1_2, 2_0, 1_6, a_5\rangle &
\langle2_1, 3_2, 1_5, 1_4\rangle &
\langle3_3, 3_2, 2_6, 2_7\rangle &
\langle1_1, 3_2, 3_4, 4_5\rangle &
\langle1_0, 4_0, 0_5, 3_4\rangle \\
\langle3_0, 3_3, 3_6, 3_4\rangle &
\langle4_2, 3_2, 1_6, 0_7\rangle &
\langle1_3, 1_1, 2_6, 3_5\rangle &
\langle4_0, 1_1, 0_6, 1_6\rangle &
\langle1_1, 0_0, 3_6, 1_4\rangle &
\langle1_2, 1_1, 1_5, 4_7\rangle &
\langle0_3, 2_2, 1_5, 3_6\rangle &
\langle2_0, 1_3, 4_5, 4_7\rangle \\
\langle0_0, 1_3, 1_7, 3_7\rangle &
\end{array}$$ ]{}
$n=27$: [$$\begin{array}{llllllllll}
\langle\infty, 0_2, 0_5, 0_4\rangle &
\langle\infty, 0_1, 3_7, 0_6\rangle &
\langle3_3, 3_2, 4_6, a_1\rangle &
\langle2_1, 2_0, 0_4, a_1\rangle &
\langle0_3, 2_2, 3_7, a_2\rangle &
\langle2_0, 0_1, 4_4, a_2\rangle &
\langle3_1, 1_0, 4_5, a_3\rangle &
\langle0_2, 2_3, 2_6, a_3\rangle \\
\langle4_0, 3_1, 2_6, a_4\rangle &
\langle0_2, 4_3, 4_7, a_4\rangle &
\langle4_3, 3_0, 2_4, a_5\rangle &
\langle1_1, 3_2, 3_6, a_5\rangle &
\langle1_3, 2_1, 2_7, a_6\rangle &
\langle2_2, 0_0, 0_4, a_6\rangle &
\langle1_0, 4_2, 2_5, a_7\rangle &
\langle0_1, 0_3, 2_7, a_7\rangle \\
\langle1_3, 2_0, 2_5, 4_5\rangle &
\langle0_1, 1_3, 3_6, 1_4\rangle &
\langle4_1, 1_1, 1_5, 0_7\rangle &
\langle4_2, 0_3, 3_6, 4_7\rangle &
\langle3_0, 4_0, 0_7, 3_6\rangle &
\langle3_2, 0_2, 4_5, 2_4\rangle &
\langle0_0, 1_1, 2_6, 1_4\rangle &
\langle2_3, 1_3, 1_5, 3_4\rangle \\
\langle2_2, 1_1, 3_4, 4_5\rangle &
\langle0_2, 4_0, 2_7, 3_7\rangle &
\langle0_0, 2_3, 4_5, 1_6\rangle &
\end{array}$$ ]{}
$n=29$: [$$\begin{array}{llllllllll}
\langle\infty, 0_0, 2_5, 1_4\rangle &
\langle\infty, 0_1, 4_7, 0_6\rangle &
\langle4_1, 0_2, 1_7, a_1\rangle &
\langle3_0, 0_3, 0_4, a_1\rangle &
\langle2_3, 2_2, 1_5, a_2\rangle &
\langle4_0, 0_1, 3_4, a_2\rangle &
\langle0_3, 4_2, 2_5, a_3\rangle &
\langle4_1, 2_0, 0_6, a_3\rangle \\
\langle1_1, 3_0, 4_5, a_4\rangle &
\langle1_3, 3_2, 0_4, a_4\rangle &
\langle0_0, 4_1, 3_5, a_5\rangle &
\langle1_3, 2_2, 0_6, a_5\rangle &
\langle0_0, 0_2, 2_7, a_6\rangle &
\langle4_1, 3_3, 4_4, a_6\rangle &
\langle1_3, 4_1, 3_6, a_7\rangle &
\langle1_2, 0_0, 0_7, a_7\rangle \\
\langle2_0, 0_3, 2_4, a_8\rangle &
\langle3_2, 4_1, 4_5, a_8\rangle &
\langle2_1, 2_3, 2_7, a_9\rangle &
\langle2_0, 1_2, 1_5, a_9\rangle &
\langle3_0, 0_0, 4_6, 4_7\rangle &
\langle2_3, 3_3, 4_7, 3_5\rangle &
\langle1_2, 1_1, 0_4, 3_5\rangle &
\langle0_3, 0_0, 3_4, 3_7\rangle \\
\langle2_1, 0_3, 0_6, 3_5\rangle &
\langle3_3, 4_0, 4_6, 1_6\rangle &
\langle1_2, 4_2, 2_4, 0_6\rangle &
\langle2_1, 0_2, 3_7, 0_7\rangle &
\langle0_1, 2_2, 2_4, 2_6\rangle &
\end{array}$$ ]{}
$n=31$: [$$\begin{array}{llllllllll}
\langle\infty, 3_1, 3_6, 3_7\rangle &
\langle\infty, 0_0, 2_4, 4_5\rangle &
\langle\infty, 1_2, 0_8, 1_9\rangle &
\langle4_0, 4_1, 0_4, a_1\rangle &
\langle1_2, 0_3, 0_5, a_1\rangle &
\langle1_0, 1_3, 4_8, 2_9\rangle &
\langle1_2, 2_3, 3_6, 3_5\rangle &
\langle0_0, 1_0, 4_9, 2_8\rangle \\
\langle0_2, 3_3, 3_8, 3_9\rangle &
\langle0_1, 3_1, 2_5, 3_8\rangle &
\langle0_2, 2_2, 0_6, 3_4\rangle &
\langle0_0, 2_3, 0_5, 4_7\rangle &
\langle0_1, 4_1, 4_4, 3_6\rangle &
\langle1_1, 1_2, 0_9, 3_4\rangle &
\langle0_1, 3_2, 3_5, 0_9\rangle &
\langle2_0, 3_1, 3_5, 4_9\rangle \\
\langle1_0, 0_1, 4_7, 1_8\rangle &
\langle0_2, 1_2, 0_4, 1_7\rangle &
\langle1_1, 2_3, 2_7, 3_8\rangle &
\langle1_1, 3_3, 2_6, 0_8\rangle &
\langle0_0, 4_2, 2_5, 2_7\rangle &
\langle0_3, 1_3, 3_6, 2_4\rangle &
\langle0_3, 3_3, 1_7, 2_9\rangle &
\langle1_2, 3_3, 2_5, 2_8\rangle \\
\langle1_1, 1_3, 4_9, 4_4\rangle &
\langle1_0, 4_3, 4_4, 4_6\rangle &
\langle0_0, 2_2, 1_7, 4_8\rangle &
\langle0_0, 3_0, 0_6, 4_6\rangle &
\langle1_0, 2_3, 1_4, 4_5\rangle &
\langle0_1, 1_2, 3_7, 2_6\rangle &
\langle0_0, 3_1, 0_7, 0_9\rangle &
\end{array}$$ ]{}
$n=33$: [$$\begin{array}{llllllllll}
\langle\infty, 4_2, 1_8, 4_9\rangle &
\langle\infty, 3_0, 3_4, 3_5\rangle &
\langle\infty, 1_1, 1_6, 3_7\rangle &
\langle3_1, 3_0, 4_4, a_1\rangle &
\langle3_3, 3_2, 3_5, a_1\rangle &
\langle1_3, 0_2, 2_5, a_2\rangle &
\langle3_0, 4_1, 1_4, a_2\rangle &
\langle3_2, 2_3, 1_5, a_3\rangle \\
\langle3_0, 0_1, 4_7, a_3\rangle &
\langle1_3, 2_3, 3_7, 1_4\rangle &
\langle2_1, 4_3, 2_7, 3_6\rangle &
\langle0_0, 1_3, 0_9, 3_9\rangle &
\langle0_2, 2_2, 3_4, 2_4\rangle &
\langle1_0, 2_2, 1_7, 0_8\rangle &
\langle1_1, 3_1, 2_9, 0_8\rangle &
\langle0_3, 3_3, 4_8, 1_4\rangle \\
\langle1_0, 4_2, 0_9, 3_5\rangle &
\langle2_1, 0_2, 0_6, 4_9\rangle &
\langle1_1, 2_3, 4_8, 4_5\rangle &
\langle1_1, 1_2, 2_8, 3_6\rangle &
\langle1_2, 3_3, 1_8, 4_6\rangle &
\langle4_1, 0_2, 0_7, 1_5\rangle &
\langle0_0, 2_2, 4_7, 3_7\rangle &
\langle2_0, 0_0, 1_6, 0_6\rangle \\
\langle1_1, 0_3, 1_9, 0_6\rangle &
\langle1_0, 4_3, 2_5, 2_9\rangle &
\langle0_1, 4_1, 0_5, 3_4\rangle &
\langle0_0, 0_3, 2_6, 0_8\rangle &
\langle4_0, 3_2, 2_8, 1_7\rangle &
\langle2_2, 0_3, 0_9, 3_6\rangle &
\langle1_0, 0_0, 4_5, 2_8\rangle &
\langle0_0, 0_2, 2_9, 4_4\rangle \\
\langle0_1, 3_3, 0_4, 3_7\rangle &
\end{array}$$ ]{}
$n=35$: [$$\begin{array}{llllllllll}
\langle\infty, 0_3, 3_7, 2_9\rangle &
\langle\infty, 0_0, 0_5, 1_4\rangle &
\langle\infty, 0_2, 0_8, 4_6\rangle &
\langle4_1, 4_0, 0_7, a_1\rangle &
\langle3_3, 0_2, 3_5, a_1\rangle &
\langle2_0, 0_1, 2_4, a_2\rangle &
\langle4_3, 3_2, 4_6, a_2\rangle &
\langle2_3, 0_2, 4_7, a_3\rangle \\
\langle1_1, 2_0, 1_5, a_3\rangle &
\langle0_3, 3_0, 4_6, a_4\rangle &
\langle1_1, 1_2, 4_7, a_4\rangle &
\langle3_1, 1_0, 0_7, a_5\rangle &
\langle1_2, 0_3, 1_4, a_5\rangle &
\langle3_2, 2_2, 4_9, 3_5\rangle &
\langle4_3, 4_2, 1_6, 2_9\rangle &
\langle1_3, 0_1, 1_9, 2_9\rangle \\
\langle1_1, 4_3, 1_4, 4_4\rangle &
\langle0_1, 4_3, 0_7, 0_6\rangle &
\langle0_1, 2_2, 1_4, 4_5\rangle &
\langle2_0, 0_3, 4_9, 4_7\rangle &
\langle3_2, 2_0, 0_7, 3_9\rangle &
\langle1_3, 1_1, 0_4, 0_8\rangle &
\langle4_1, 2_2, 3_7, 4_8\rangle &
\langle2_0, 4_0, 2_9, 0_5\rangle \\
\langle4_2, 2_0, 1_4, 0_4\rangle &
\langle4_1, 1_1, 0_6, 2_8\rangle &
\langle0_0, 1_1, 3_6, 4_6\rangle &
\langle2_2, 4_0, 3_8, 1_8\rangle &
\langle3_1, 4_1, 0_5, 3_9\rangle &
\langle2_2, 2_0, 1_9, 2_6\rangle &
\langle4_2, 0_0, 2_4, 2_6\rangle &
\langle0_0, 0_3, 0_7, 3_8\rangle \\
\langle2_3, 4_3, 4_8, 3_5\rangle &
\langle0_0, 4_3, 2_5, 0_8\rangle &
\langle0_1, 4_2, 3_5, 2_8\rangle &
\end{array}$$ ]{}
$n=37$: [$$\begin{array}{llllllllll}
\langle\infty, 3_0, 2_5, 2_4\rangle &
\langle\infty, 3_3, 0_8, 3_9\rangle &
\langle\infty, 0_2, 2_6, 1_7\rangle &
\langle3_1, 4_2, 4_7, a_1\rangle &
\langle2_3, 2_0, 2_5, a_1\rangle &
\langle2_1, 0_2, 4_4, a_2\rangle &
\langle3_3, 4_0, 1_6, a_2\rangle &
\langle0_2, 3_3, 2_4, a_3\rangle \\
\langle0_0, 3_1, 0_6, a_3\rangle &
\langle1_0, 4_3, 4_4, a_4\rangle &
\langle4_1, 4_2, 4_5, a_4\rangle &
\langle1_3, 1_2, 0_6, a_5\rangle &
\langle0_1, 0_0, 3_5, a_5\rangle &
\langle0_2, 2_0, 3_4, a_6\rangle &
\langle0_1, 0_3, 4_5, a_6\rangle &
\langle3_2, 1_0, 0_7, a_7\rangle \\
\langle4_1, 3_3, 0_4, a_7\rangle &
\langle3_1, 1_1, 0_9, 1_4\rangle &
\langle4_3, 3_2, 4_6, 1_7\rangle &
\langle0_2, 1_1, 0_4, 2_8\rangle &
\langle0_0, 4_2, 3_8, 4_6\rangle &
\langle2_3, 1_3, 0_7, 4_5\rangle &
\langle0_0, 2_0, 4_9, 0_7\rangle &
\langle2_0, 4_3, 3_9, 4_8\rangle \\
\langle3_0, 4_3, 1_9, 0_7\rangle &
\langle3_1, 0_3, 1_6, 1_9\rangle &
\langle0_0, 4_1, 4_8, 1_5\rangle &
\langle1_0, 2_1, 2_7, 1_8\rangle &
\langle2_1, 4_2, 3_5, 0_8\rangle &
\langle0_1, 3_0, 1_6, 4_6\rangle &
\langle0_0, 1_2, 2_4, 1_8\rangle &
\langle1_1, 2_1, 0_7, 2_9\rangle \\
\langle4_2, 3_3, 3_7, 1_9\rangle &
\langle2_1, 0_3, 4_8, 2_6\rangle &
\langle0_2, 0_0, 2_5, 0_9\rangle &
\langle3_3, 0_3, 1_4, 1_8\rangle &
\langle0_2, 2_2, 3_5, 3_9\rangle &
\end{array}$$ ]{}
$n=39$: [$$\begin{array}{llllllllll}
\langle\infty, 3_1, 4_9, 4_8\rangle &
\langle\infty, 0_2, 1_4, 1_5\rangle &
\langle\infty, 2_3, 3_6, 3_7\rangle &
\langle2_2, 1_3, 1_5, a_1\rangle &
\langle2_0, 3_1, 1_7, a_1\rangle &
\langle3_3, 2_0, 3_6, a_2\rangle &
\langle3_1, 2_2, 1_4, a_2\rangle &
\langle0_0, 2_1, 2_4, a_3\rangle \\
\langle2_2, 4_3, 0_5, a_3\rangle &
\langle3_0, 3_1, 3_5, a_4\rangle &
\langle2_3, 4_2, 1_4, a_4\rangle &
\langle1_3, 4_0, 0_5, a_5\rangle &
\langle2_2, 4_1, 3_6, a_5\rangle &
\langle1_2, 2_0, 3_7, a_6\rangle &
\langle4_3, 1_1, 0_4, a_6\rangle &
\langle2_0, 1_3, 3_4, a_7\rangle \\
\langle2_1, 4_2, 3_7, a_7\rangle &
\langle2_2, 4_0, 2_6, a_8\rangle &
\langle4_3, 4_1, 2_5, a_8\rangle &
\langle3_0, 0_2, 3_6, a_9\rangle &
\langle2_1, 3_3, 2_7, a_9\rangle &
\langle4_1, 2_1, 1_5, 4_8\rangle &
\langle0_1, 4_1, 0_6, 2_9\rangle &
\langle4_0, 0_0, 4_4, 0_9\rangle \\
\langle4_3, 2_3, 4_8, 4_7\rangle &
\langle2_1, 1_3, 1_9, 4_7\rangle &
\langle1_3, 0_2, 0_9, 2_8\rangle &
\langle0_3, 2_0, 3_8, 0_4\rangle &
\langle0_0, 3_1, 4_5, 2_7\rangle &
\langle4_0, 4_3, 2_9, 1_5\rangle &
\langle3_2, 4_2, 4_8, 0_9\rangle &
\langle1_0, 3_0, 0_9, 1_8\rangle \\
\langle0_0, 4_1, 2_6, 2_8\rangle &
\langle0_3, 1_3, 2_9, 3_6\rangle &
\langle1_1, 1_2, 0_8, 3_6\rangle &
\langle4_0, 4_2, 2_7, 4_7\rangle &
\langle2_0, 3_2, 0_5, 1_8\rangle &
\langle4_2, 3_1, 4_4, 3_9\rangle &
\langle0_2, 0_3, 3_4, 4_6\rangle &
\end{array}$$ ]{}
$n=41$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{0}, 3_{10}, 3_{6}\rangle &
\langle\infty, 0_{3}, 1_{11}, 3_{5}\rangle &
\langle\infty, 0_{2}, 3_{4}, 4_{7}\rangle &
\langle\infty, 3_{1}, 0_{8}, 3_{9}\rangle &
\langle0_{1}, 3_{0}, 0_{7}, a_1\rangle &
\langle3_{2}, 3_{3}, 3_{6}, a_1\rangle &
\langle2_{1}, 4_{0}, 3_{6}, 4_{5}\rangle &
\langle0_{1}, 1_{1}, 4_{8}, 2_{10}\rangle \\
\langle2_{1}, 3_{2}, 4_{9}, 1_{9}\rangle &
\langle1_{1}, 3_{1}, 0_{7}, 3_{11}\rangle &
\langle2_{2}, 0_{3}, 1_{6}, 1_{8}\rangle &
\langle0_{2}, 2_{3}, 0_{8}, 4_{10}\rangle &
\langle3_{2}, 3_{1}, 4_{4}, 4_{8}\rangle &
\langle4_{0}, 1_{2}, 4_{11}, 3_{4}\rangle &
\langle0_{1}, 1_{3}, 0_{6}, 3_{9}\rangle &
\langle1_{3}, 2_{0}, 3_{7}, 0_{5}\rangle \\
\langle0_{0}, 1_{1}, 1_{10}, 2_{5}\rangle &
\langle2_{3}, 3_{3}, 1_{9}, 2_{8}\rangle &
\langle0_{0}, 3_{3}, 0_{6}, 4_{9}\rangle &
\langle2_{1}, 0_{2}, 4_{4}, 0_{10}\rangle &
\langle4_{0}, 1_{3}, 4_{7}, 2_{7}\rangle &
\langle2_{0}, 0_{2}, 1_{10}, 3_{6}\rangle &
\langle0_{3}, 2_{1}, 1_{10}, 0_{5}\rangle &
\langle0_{0}, 0_{3}, 0_{9}, 3_{6}\rangle \\
\langle0_{1}, 2_{3}, 1_{11}, 4_{4}\rangle &
\langle2_{0}, 4_{0}, 4_{8}, 0_{4}\rangle &
\langle2_{3}, 2_{1}, 2_{4}, 0_{11}\rangle &
\langle0_{0}, 1_{0}, 3_{11}, 3_{9}\rangle &
\langle1_{3}, 2_{2}, 2_{5}, 3_{11}\rangle &
\langle3_{1}, 4_{0}, 1_{4}, 3_{8}\rangle &
\langle1_{1}, 0_{2}, 2_{7}, 1_{5}\rangle &
\langle2_{0}, 2_{2}, 1_{11}, 0_{10}\rangle \\
\langle0_{2}, 1_{2}, 4_{5}, 0_{9}\rangle &
\langle0_{1}, 2_{2}, 4_{6}, 3_{6}\rangle &
\langle1_{1}, 1_{0}, 0_{5}, 2_{9}\rangle &
\langle0_{2}, 3_{2}, 0_{11}, 3_{7}\rangle &
\langle3_{1}, 2_{3}, 1_{7}, 2_{11}\rangle &
\langle0_{2}, 1_{0}, 2_{8}, 2_{5}\rangle &
\langle1_{2}, 2_{3}, 2_{7}, 4_{8}\rangle &
\langle2_{3}, 0_{3}, 3_{4}, 0_{10}\rangle \\
\langle0_{0}, 1_{3}, 0_{4}, 0_{10}\rangle &
\end{array}$$ ]{}
T$(2,25;2,n;6) =6n$ for each odd $n$ and $25 \leq n \leq 49$.
Let $X_1=({{\mathbb{Z}}}_6\times \{0,1,2,3\})\cup \{\infty\}$. For $25 \leq n \leq 35$, let $X_2= ({{\mathbb{Z}}}_6\times \{4,5,6,7\})\cup (\{a\}\times \{1,\ldots,n-24\})$; for $37 \leq n \leq 47$, let $X_2= ({{\mathbb{Z}}}_6\times \{4,5,\ldots,9\})\cup (\{a\}\times \{1,\ldots,n-36\})$; for $n = 49$, let $X_2= ({{\mathbb{Z}}}_6\times \{4,5,\ldots,11\})\cup (\{a\}\times \{1\})$. Denote $X=X_1\cup X_2$. The desired codes of size $6n$ are constructed on ${{\mathbb{Z}}}_2^X$ and the base codewords are listed as follows.
$n=25$: [$$\begin{array}{llllllllll}
\langle\infty, 5_{0}, 0_{5}, 5_{4}\rangle &
\langle\infty, 5_{1}, 5_{7}, 5_{6}\rangle &
\langle1_{2}, 3_{3}, 1_{5}, a_{1}\rangle &
\langle1_{0}, 1_{1}, 2_{4}, a_{1}\rangle &
\langle5_{0}, 3_{3}, 4_{6}, 5_{7}\rangle &
\langle0_{0}, 5_{0}, 4_{4}, 2_{4}\rangle &
\langle0_{3}, 3_{0}, 3_{5}, 3_{6}\rangle &
\langle5_{0}, 4_{2}, 4_{7}, 2_{5}\rangle \\
\langle0_{2}, 0_{0}, 2_{7}, 5_{5}\rangle &
\langle3_{1}, 4_{2}, 1_{4}, 0_{6}\rangle &
\langle4_{0}, 0_{1}, 5_{6}, 1_{6}\rangle &
\langle5_{2}, 5_{3}, 4_{4}, 0_{5}\rangle &
\langle4_{0}, 0_{3}, 0_{5}, 2_{6}\rangle &
\langle4_{1}, 5_{3}, 5_{7}, 3_{4}\rangle &
\langle4_{0}, 2_{1}, 2_{5}, 0_{6}\rangle &
\langle0_{1}, 1_{0}, 4_{7}, 2_{7}\rangle \\
\langle2_{1}, 3_{1}, 0_{5}, 4_{5}\rangle &
\langle1_{3}, 0_{2}, 1_{4}, 1_{6}\rangle &
\langle3_{3}, 4_{3}, 5_{4}, 2_{7}\rangle &
\langle3_{2}, 5_{2}, 3_{6}, 5_{4}\rangle &
\langle0_{3}, 2_{1}, 4_{6}, 1_{7}\rangle &
\langle3_{2}, 5_{1}, 1_{4}, 2_{7}\rangle &
\langle4_{3}, 1_{1}, 1_{4}, 0_{5}\rangle &
\langle5_{2}, 3_{3}, 2_{6}, 0_{7}\rangle \\
\langle0_{2}, 1_{2}, 3_{5}, 4_{7}\rangle &
\end{array}$$ ]{}
$n=27$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{1}, 1_{7}, 1_{6}\rangle &
\langle\infty, 2_{0}, 2_{4}, 2_{5}\rangle &
\langle1_{2}, 1_{3}, 1_{5}, a_{1}\rangle &
\langle2_{0}, 2_{1}, 3_{4}, a_{1}\rangle &
\langle0_{0}, 1_{1}, 3_{4}, a_{2}\rangle &
\langle2_{2}, 1_{3}, 0_{5}, a_{2}\rangle &
\langle5_{0}, 1_{1}, 1_{4}, a_{3}\rangle &
\langle1_{2}, 2_{3}, 2_{6}, a_{3}\rangle \\
\langle0_{0}, 2_{3}, 4_{6}, 3_{5}\rangle &
\langle0_{0}, 4_{3}, 2_{6}, 0_{7}\rangle &
\langle0_{1}, 5_{1}, 3_{4}, 5_{5}\rangle &
\langle0_{0}, 0_{2}, 2_{7}, 1_{7}\rangle &
\langle0_{2}, 4_{3}, 5_{7}, 1_{5}\rangle &
\langle0_{3}, 1_{3}, 0_{4}, 5_{7}\rangle &
\langle1_{1}, 1_{2}, 4_{5}, 0_{4}\rangle &
\langle1_{0}, 5_{1}, 0_{6}, 4_{6}\rangle \\
\langle0_{1}, 2_{1}, 4_{5}, 4_{7}\rangle &
\langle0_{3}, 2_{3}, 4_{4}, 5_{6}\rangle &
\langle0_{1}, 2_{2}, 2_{6}, 5_{7}\rangle &
\langle1_{0}, 0_{2}, 4_{7}, 0_{7}\rangle &
\langle0_{2}, 2_{2}, 5_{6}, 4_{6}\rangle &
\langle0_{2}, 3_{3}, 4_{4}, 5_{5}\rangle &
\langle0_{2}, 5_{2}, 0_{4}, 2_{4}\rangle &
\langle0_{0}, 1_{0}, 2_{5}, 5_{4}\rangle \\
\langle0_{0}, 1_{3}, 5_{5}, 4_{7}\rangle &
\langle0_{0}, 3_{1}, 0_{6}, 4_{5}\rangle &
\langle0_{1}, 3_{3}, 4_{6}, 3_{7}\rangle &
\end{array}$$ ]{}
$n=29$: [$$\begin{array}{llllllllll}
\langle\infty, 4_{1}, 4_{7}, 0_{6}\rangle &
\langle\infty, 1_{0}, 0_{5}, 1_{4}\rangle &
\langle4_{1}, 4_{0}, 5_{4}, a_{1}\rangle &
\langle1_{3}, 0_{2}, 5_{6}, a_{1}\rangle &
\langle3_{3}, 5_{2}, 2_{4}, a_{2}\rangle &
\langle0_{0}, 2_{1}, 5_{7}, a_{2}\rangle &
\langle4_{1}, 1_{0}, 0_{4}, a_{3}\rangle &
\langle2_{3}, 3_{2}, 4_{5}, a_{3}\rangle \\
\langle2_{1}, 3_{0}, 3_{6}, a_{4}\rangle &
\langle0_{2}, 3_{3}, 5_{4}, a_{4}\rangle &
\langle4_{3}, 2_{2}, 4_{7}, a_{5}\rangle &
\langle1_{0}, 5_{1}, 5_{6}, a_{5}\rangle &
\langle3_{0}, 0_{3}, 0_{4}, 3_{7}\rangle &
\langle2_{3}, 3_{3}, 3_{6}, 3_{5}\rangle &
\langle4_{1}, 0_{2}, 0_{7}, 2_{4}\rangle &
\langle4_{0}, 3_{2}, 5_{5}, 1_{5}\rangle \\
\langle2_{3}, 4_{3}, 3_{7}, 1_{6}\rangle &
\langle1_{2}, 2_{2}, 4_{6}, 1_{5}\rangle &
\langle4_{0}, 0_{2}, 5_{7}, 1_{7}\rangle &
\langle3_{1}, 0_{2}, 0_{4}, 0_{6}\rangle &
\langle2_{1}, 3_{2}, 1_{7}, 0_{7}\rangle &
\langle5_{1}, 2_{3}, 0_{7}, 1_{5}\rangle &
\langle2_{1}, 0_{2}, 3_{5}, 1_{4}\rangle &
\langle5_{0}, 5_{3}, 1_{7}, 1_{6}\rangle \\
\langle5_{1}, 0_{1}, 5_{5}, 4_{6}\rangle &
\langle4_{0}, 0_{0}, 1_{6}, 0_{5}\rangle &
\langle2_{2}, 4_{0}, 0_{4}, 3_{6}\rangle &
\langle4_{1}, 3_{3}, 4_{4}, 1_{5}\rangle &
\langle0_{0}, 1_{3}, 4_{4}, 4_{5}\rangle &
\end{array}$$ ]{}
$n=31$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{1}, 2_{7}, 0_{6}\rangle &
\langle\infty, 5_{0}, 5_{4}, 3_{5}\rangle &
\langle1_{1}, 1_{0}, 5_{7}, a_{1}\rangle &
\langle1_{2}, 1_{3}, 2_{6}, a_{1}\rangle &
\langle0_{2}, 5_{3}, 3_{6}, a_{2}\rangle &
\langle5_{1}, 4_{0}, 1_{4}, a_{2}\rangle &
\langle1_{1}, 5_{0}, 1_{4}, a_{3}\rangle &
\langle0_{2}, 3_{3}, 1_{5}, a_{3}\rangle \\
\langle2_{1}, 5_{0}, 0_{5}, a_{4}\rangle &
\langle5_{3}, 3_{2}, 5_{6}, a_{4}\rangle &
\langle3_{0}, 1_{1}, 1_{6}, a_{5}\rangle &
\langle4_{3}, 0_{2}, 0_{4}, a_{5}\rangle &
\langle0_{2}, 1_{3}, 4_{6}, a_{6}\rangle &
\langle3_{0}, 2_{1}, 1_{4}, a_{6}\rangle &
\langle2_{3}, 4_{1}, 1_{6}, a_{7}\rangle &
\langle5_{0}, 1_{2}, 4_{7}, a_{7}\rangle \\
\langle1_{3}, 1_{1}, 0_{5}, 1_{5}\rangle &
\langle0_{2}, 4_{1}, 5_{5}, 0_{6}\rangle &
\langle2_{3}, 1_{3}, 2_{4}, 3_{5}\rangle &
\langle3_{1}, 2_{2}, 4_{7}, 5_{5}\rangle &
\langle1_{1}, 0_{3}, 0_{7}, 4_{7}\rangle &
\langle1_{2}, 0_{2}, 5_{4}, 1_{7}\rangle &
\langle1_{3}, 2_{0}, 2_{7}, 3_{7}\rangle &
\langle3_{1}, 0_{2}, 5_{7}, 1_{4}\rangle \\
\langle0_{0}, 3_{2}, 2_{6}, 3_{5}\rangle &
\langle2_{3}, 0_{3}, 5_{4}, 5_{7}\rangle &
\langle4_{0}, 2_{2}, 0_{7}, 0_{5}\rangle &
\langle3_{0}, 4_{0}, 3_{5}, 3_{6}\rangle &
\langle2_{1}, 3_{2}, 5_{5}, 5_{4}\rangle &
\langle0_{1}, 3_{3}, 1_{4}, 5_{6}\rangle &
\langle0_{0}, 2_{0}, 1_{4}, 3_{6}\rangle &
\end{array}$$ ]{}
$n=33$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{1}, 4_{6}, 1_{7}\rangle &
\langle\infty, 4_{0}, 4_{4}, 4_{5}\rangle &
\langle2_{1}, 2_{0}, 3_{4}, a_{1}\rangle &
\langle3_{3}, 3_{2}, 5_{5}, a_{1}\rangle &
\langle0_{3}, 5_{2}, 3_{5}, a_{2}\rangle &
\langle2_{1}, 1_{0}, 4_{4}, a_{2}\rangle &
\langle5_{2}, 1_{3}, 0_{5}, a_{3}\rangle &
\langle1_{1}, 5_{0}, 1_{4}, a_{3}\rangle \\
\langle2_{1}, 5_{0}, 0_{5}, a_{4}\rangle &
\langle5_{2}, 2_{3}, 5_{4}, a_{4}\rangle &
\langle0_{1}, 2_{0}, 4_{6}, a_{5}\rangle &
\langle0_{2}, 4_{3}, 4_{7}, a_{5}\rangle &
\langle5_{2}, 4_{3}, 2_{5}, a_{6}\rangle &
\langle4_{1}, 5_{0}, 3_{4}, a_{6}\rangle &
\langle1_{0}, 4_{2}, 4_{5}, a_{7}\rangle &
\langle0_{3}, 1_{1}, 3_{7}, a_{7}\rangle \\
\langle4_{0}, 4_{2}, 3_{5}, a_{8}\rangle &
\langle5_{1}, 1_{3}, 0_{7}, a_{8}\rangle &
\langle1_{1}, 5_{3}, 0_{5}, a_{9}\rangle &
\langle5_{0}, 4_{2}, 0_{7}, a_{9}\rangle &
\langle0_{3}, 1_{3}, 5_{4}, 1_{4}\rangle &
\langle2_{0}, 0_{2}, 5_{7}, 1_{7}\rangle &
\langle0_{1}, 4_{2}, 5_{6}, 4_{7}\rangle &
\langle0_{2}, 4_{2}, 3_{6}, 3_{4}\rangle \\
\langle3_{0}, 4_{2}, 4_{6}, 0_{6}\rangle &
\langle2_{1}, 5_{3}, 1_{7}, 5_{5}\rangle &
\langle0_{3}, 2_{3}, 5_{6}, 4_{6}\rangle &
\langle1_{0}, 5_{0}, 3_{7}, 3_{5}\rangle &
\langle2_{0}, 5_{3}, 1_{4}, 0_{6}\rangle &
\langle2_{0}, 1_{3}, 1_{6}, 2_{7}\rangle &
\langle3_{1}, 3_{2}, 0_{7}, 1_{4}\rangle &
\langle1_{1}, 0_{1}, 2_{6}, 1_{5}\rangle \\
\langle0_{1}, 2_{2}, 3_{4}, 0_{6}\rangle &
\end{array}$$ ]{}
$n=35$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{3}, 3_{6}, 4_{7}\rangle &
\langle\infty, 1_{0}, 5_{5}, 2_{4}\rangle &
\langle1_{3}, 0_{2}, 1_{7}, a_{1}\rangle &
\langle3_{1}, 5_{0}, 5_{5}, a_{1}\rangle &
\langle1_{1}, 0_{0}, 4_{4}, a_{2}\rangle &
\langle5_{3}, 2_{2}, 2_{5}, a_{2}\rangle &
\langle1_{2}, 3_{3}, 5_{4}, a_{3}\rangle &
\langle4_{0}, 0_{1}, 1_{6}, a_{3}\rangle \\
\langle2_{2}, 1_{1}, 5_{6}, a_{4}\rangle &
\langle5_{3}, 2_{0}, 2_{7}, a_{4}\rangle &
\langle3_{0}, 0_{1}, 5_{5}, a_{5}\rangle &
\langle1_{3}, 3_{2}, 5_{6}, a_{5}\rangle &
\langle0_{2}, 5_{3}, 1_{5}, a_{6}\rangle &
\langle4_{0}, 3_{1}, 0_{6}, a_{6}\rangle &
\langle0_{0}, 0_{2}, 4_{6}, a_{7}\rangle &
\langle2_{1}, 4_{3}, 2_{5}, a_{7}\rangle \\
\langle3_{3}, 3_{1}, 2_{4}, a_{8}\rangle &
\langle3_{2}, 4_{0}, 0_{7}, a_{8}\rangle &
\langle2_{3}, 5_{1}, 1_{7}, a_{9}\rangle &
\langle3_{2}, 2_{0}, 1_{5}, a_{9}\rangle &
\langle5_{2}, 1_{0}, 4_{7}, a_{10}\rangle &
\langle2_{3}, 3_{1}, 3_{6}, a_{10}\rangle &
\langle2_{2}, 2_{3}, 1_{6}, a_{11}\rangle &
\langle5_{0}, 5_{1}, 0_{7}, a_{11}\rangle \\
\langle4_{2}, 2_{0}, 5_{4}, 1_{4}\rangle &
\langle2_{2}, 5_{1}, 1_{4}, 4_{7}\rangle &
\langle0_{0}, 2_{0}, 1_{6}, 2_{4}\rangle &
\langle3_{2}, 4_{1}, 1_{7}, 2_{5}\rangle &
\langle3_{2}, 1_{1}, 3_{6}, 5_{4}\rangle &
\langle1_{3}, 0_{0}, 5_{7}, 1_{5}\rangle &
\langle5_{3}, 4_{3}, 1_{6}, 5_{4}\rangle &
\langle4_{2}, 4_{1}, 4_{4}, 4_{7}\rangle \\
\langle3_{0}, 5_{3}, 1_{7}, 0_{5}\rangle &
\langle5_{1}, 3_{2}, 4_{6}, 0_{5}\rangle &
\langle0_{1}, 4_{3}, 1_{4}, 3_{5}\rangle &
\end{array}$$ ]{}
$n=37$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{2}, 0_{9}, 1_{8}\rangle &
\langle\infty, 3_{1}, 4_{7}, 3_{6}\rangle &
\langle\infty, 4_{0}, 4_{5}, 2_{4}\rangle &
\langle0_{1}, 0_{0}, 1_{4}, a_{1}\rangle &
\langle0_{3}, 4_{2}, 3_{6}, a_{1}\rangle &
\langle3_{2}, 5_{2}, 4_{5}, 0_{4}\rangle &
\langle1_{2}, 4_{1}, 0_{4}, 0_{9}\rangle &
\langle0_{0}, 1_{0}, 0_{4}, 3_{9}\rangle \\
\langle1_{3}, 0_{1}, 1_{9}, 3_{5}\rangle &
\langle0_{1}, 5_{0}, 5_{7}, 5_{8}\rangle &
\langle0_{3}, 4_{0}, 2_{9}, 1_{8}\rangle &
\langle5_{2}, 5_{3}, 5_{8}, 3_{4}\rangle &
\langle2_{3}, 3_{1}, 0_{7}, 1_{7}\rangle &
\langle1_{1}, 3_{0}, 5_{6}, 1_{5}\rangle &
\langle0_{3}, 4_{3}, 2_{8}, 3_{4}\rangle &
\langle0_{2}, 5_{3}, 1_{6}, 2_{9}\rangle \\
\langle0_{0}, 3_{1}, 0_{6}, 5_{6}\rangle &
\langle2_{1}, 3_{1}, 0_{8}, 0_{4}\rangle &
\langle5_{1}, 3_{3}, 0_{8}, 5_{4}\rangle &
\langle0_{1}, 5_{2}, 0_{9}, 5_{4}\rangle &
\langle4_{2}, 3_{2}, 0_{8}, 1_{6}\rangle &
\langle5_{1}, 0_{2}, 1_{7}, 5_{8}\rangle &
\langle1_{1}, 4_{3}, 2_{6}, 3_{8}\rangle &
\langle4_{3}, 0_{2}, 4_{7}, 0_{7}\rangle \\
\langle1_{1}, 5_{0}, 0_{9}, 5_{9}\rangle &
\langle5_{3}, 4_{3}, 3_{9}, 5_{4}\rangle &
\langle4_{2}, 1_{0}, 2_{8}, 3_{7}\rangle &
\langle2_{3}, 5_{0}, 2_{5}, 2_{6}\rangle &
\langle2_{3}, 1_{0}, 5_{7}, 0_{5}\rangle &
\langle0_{3}, 5_{2}, 5_{6}, 1_{6}\rangle &
\langle3_{0}, 1_{0}, 5_{8}, 4_{7}\rangle &
\langle3_{0}, 5_{2}, 2_{9}, 2_{7}\rangle \\
\langle5_{0}, 5_{2}, 1_{4}, 1_{5}\rangle &
\langle0_{1}, 4_{2}, 0_{7}, 4_{5}\rangle &
\langle4_{0}, 3_{1}, 2_{6}, 5_{5}\rangle &
\langle4_{3}, 1_{2}, 5_{5}, 5_{9}\rangle &
\langle0_{1}, 2_{3}, 1_{5}, 5_{5}\rangle &
\end{array}$$ ]{}
$n=39$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{0}, 3_{9}, 0_{8}\rangle &
\langle\infty, 1_{1}, 3_{7}, 3_{6}\rangle &
\langle\infty, 3_{3}, 0_{4}, 4_{5}\rangle &
\langle0_{2}, 0_{3}, 0_{4}, a_{1}\rangle &
\langle4_{0}, 1_{1}, 5_{7}, a_{1}\rangle &
\langle4_{2}, 0_{3}, 5_{6}, a_{2}\rangle &
\langle4_{0}, 0_{1}, 1_{4}, a_{2}\rangle &
\langle1_{0}, 5_{1}, 3_{4}, a_{3}\rangle \\
\langle5_{2}, 3_{3}, 2_{7}, a_{3}\rangle &
\langle5_{3}, 3_{1}, 4_{8}, 0_{7}\rangle &
\langle5_{1}, 0_{1}, 2_{4}, 2_{8}\rangle &
\langle4_{3}, 3_{1}, 4_{9}, 0_{9}\rangle &
\langle0_{0}, 3_{2}, 0_{5}, 4_{5}\rangle &
\langle0_{1}, 2_{2}, 2_{5}, 4_{9}\rangle &
\langle4_{3}, 4_{0}, 5_{6}, 0_{6}\rangle &
\langle1_{1}, 4_{2}, 0_{8}, 0_{6}\rangle \\
\langle4_{1}, 5_{2}, 2_{6}, 3_{4}\rangle &
\langle1_{0}, 0_{3}, 4_{5}, 0_{6}\rangle &
\langle1_{0}, 2_{3}, 1_{9}, 1_{4}\rangle &
\langle5_{1}, 4_{3}, 0_{7}, 3_{5}\rangle &
\langle3_{2}, 0_{3}, 0_{8}, 4_{7}\rangle &
\langle0_{3}, 1_{3}, 2_{8}, 4_{9}\rangle &
\langle4_{2}, 4_{0}, 3_{4}, 5_{4}\rangle &
\langle3_{3}, 1_{0}, 0_{7}, 1_{6}\rangle \\
\langle0_{1}, 4_{1}, 0_{9}, 1_{5}\rangle &
\langle5_{0}, 4_{0}, 2_{6}, 2_{9}\rangle &
\langle5_{3}, 2_{1}, 2_{6}, 2_{5}\rangle &
\langle4_{2}, 5_{1}, 2_{6}, 5_{8}\rangle &
\langle4_{2}, 5_{2}, 4_{6}, 2_{9}\rangle &
\langle0_{0}, 5_{2}, 1_{5}, 3_{7}\rangle &
\langle5_{3}, 0_{2}, 0_{9}, 5_{5}\rangle &
\langle1_{1}, 1_{2}, 1_{7}, 0_{7}\rangle \\
\langle5_{0}, 1_{2}, 0_{9}, 3_{7}\rangle &
\langle1_{0}, 5_{0}, 1_{7}, 1_{8}\rangle &
\langle3_{0}, 3_{1}, 2_{5}, 1_{8}\rangle &
\langle2_{2}, 4_{1}, 4_{4}, 3_{9}\rangle &
\langle5_{0}, 3_{2}, 1_{5}, 2_{8}\rangle &
\langle3_{0}, 0_{3}, 1_{4}, 4_{8}\rangle &
\langle0_{2}, 1_{3}, 3_{4}, 4_{8}\rangle &
\end{array}$$ ]{}
$n=41$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{0}, 5_{5}, 1_{4}\rangle &
\langle\infty, 2_{3}, 5_{9}, 0_{8}\rangle &
\langle\infty, 1_{1}, 0_{6}, 3_{7}\rangle &
\langle4_{1}, 5_{0}, 4_{4}, a_{1}\rangle &
\langle5_{2}, 5_{3}, 2_{5}, a_{1}\rangle &
\langle5_{2}, 0_{3}, 2_{6}, a_{2}\rangle &
\langle4_{1}, 4_{0}, 5_{4}, a_{2}\rangle &
\langle1_{2}, 3_{3}, 2_{5}, a_{3}\rangle \\
\langle3_{1}, 1_{0}, 1_{7}, a_{3}\rangle &
\langle1_{2}, 4_{3}, 1_{4}, a_{4}\rangle &
\langle2_{1}, 5_{0}, 3_{5}, a_{4}\rangle &
\langle2_{1}, 4_{0}, 1_{5}, a_{5}\rangle &
\langle4_{2}, 2_{3}, 0_{4}, a_{5}\rangle &
\langle2_{1}, 3_{1}, 0_{8}, 5_{5}\rangle &
\langle1_{0}, 2_{1}, 4_{4}, 5_{9}\rangle &
\langle2_{1}, 1_{3}, 4_{8}, 2_{7}\rangle \\
\langle2_{2}, 2_{1}, 1_{4}, 0_{9}\rangle &
\langle0_{2}, 1_{2}, 0_{7}, 0_{6}\rangle &
\langle0_{2}, 5_{1}, 2_{7}, 5_{8}\rangle &
\langle0_{1}, 0_{3}, 5_{7}, 5_{9}\rangle &
\langle4_{2}, 3_{3}, 5_{7}, 4_{9}\rangle &
\langle2_{2}, 0_{0}, 1_{5}, 5_{8}\rangle &
\langle1_{1}, 3_{1}, 4_{6}, 1_{5}\rangle &
\langle5_{0}, 5_{3}, 5_{8}, 2_{6}\rangle \\
\langle4_{2}, 2_{2}, 2_{8}, 0_{5}\rangle &
\langle5_{0}, 4_{2}, 1_{7}, 2_{7}\rangle &
\langle2_{3}, 4_{1}, 5_{7}, 0_{9}\rangle &
\langle4_{0}, 2_{3}, 0_{6}, 2_{6}\rangle &
\langle3_{3}, 5_{3}, 5_{4}, 5_{5}\rangle &
\langle4_{0}, 4_{2}, 5_{9}, 4_{5}\rangle &
\langle4_{0}, 5_{0}, 2_{8}, 3_{7}\rangle &
\langle0_{2}, 5_{0}, 4_{6}, 1_{4}\rangle \\
\langle1_{3}, 0_{1}, 2_{6}, 1_{9}\rangle &
\langle0_{1}, 3_{2}, 0_{9}, 4_{6}\rangle &
\langle1_{3}, 4_{0}, 5_{7}, 3_{9}\rangle &
\langle1_{3}, 0_{0}, 2_{8}, 5_{5}\rangle &
\langle1_{3}, 4_{1}, 2_{4}, 3_{8}\rangle &
\langle4_{2}, 5_{1}, 2_{4}, 0_{8}\rangle &
\langle5_{0}, 1_{3}, 0_{6}, 0_{8}\rangle &
\langle1_{2}, 3_{0}, 3_{6}, 3_{9}\rangle \\
\langle0_{0}, 3_{2}, 0_{4}, 2_{9}\rangle &
\end{array}$$ ]{}
$n=43$: [$$\begin{array}{llllllllll}
\langle\infty, 4_{2}, 3_{4}, 2_{8}\rangle &
\langle\infty, 1_{1}, 4_{9}, 3_{6}\rangle &
\langle\infty, 5_{3}, 0_{5}, 3_{7}\rangle &
\langle2_{2}, 3_{3}, 4_{6}, a_{1}\rangle &
\langle2_{0}, 3_{1}, 5_{5}, a_{1}\rangle &
\langle2_{0}, 5_{3}, 4_{4}, a_{2}\rangle &
\langle4_{1}, 1_{2}, 2_{5}, a_{2}\rangle &
\langle0_{3}, 5_{0}, 5_{7}, a_{3}\rangle \\
\langle4_{1}, 3_{2}, 1_{5}, a_{3}\rangle &
\langle5_{3}, 5_{2}, 0_{7}, a_{4}\rangle &
\langle2_{0}, 4_{1}, 2_{6}, a_{4}\rangle &
\langle3_{0}, 4_{2}, 4_{4}, a_{5}\rangle &
\langle3_{1}, 3_{3}, 0_{6}, a_{5}\rangle &
\langle4_{0}, 1_{1}, 0_{6}, a_{6}\rangle &
\langle4_{3}, 2_{2}, 0_{7}, a_{6}\rangle &
\langle0_{3}, 1_{0}, 4_{4}, a_{7}\rangle \\
\langle2_{1}, 2_{2}, 1_{7}, a_{7}\rangle &
\langle1_{3}, 3_{1}, 4_{4}, 2_{9}\rangle &
\langle3_{1}, 0_{3}, 0_{8}, 2_{4}\rangle &
\langle5_{2}, 0_{0}, 0_{4}, 5_{6}\rangle &
\langle3_{2}, 3_{0}, 0_{9}, 0_{8}\rangle &
\langle2_{1}, 3_{0}, 3_{8}, 2_{4}\rangle &
\langle3_{1}, 2_{3}, 2_{5}, 5_{8}\rangle &
\langle0_{1}, 5_{1}, 1_{7}, 1_{9}\rangle \\
\langle5_{3}, 0_{2}, 2_{5}, 1_{6}\rangle &
\langle1_{0}, 0_{0}, 4_{6}, 2_{7}\rangle &
\langle3_{2}, 1_{2}, 5_{4}, 2_{9}\rangle &
\langle0_{2}, 3_{3}, 3_{7}, 3_{4}\rangle &
\langle2_{0}, 4_{3}, 3_{6}, 0_{8}\rangle &
\langle2_{2}, 4_{0}, 5_{5}, 4_{9}\rangle &
\langle3_{2}, 4_{2}, 1_{6}, 5_{8}\rangle &
\langle2_{2}, 0_{3}, 2_{9}, 1_{8}\rangle \\
\langle0_{1}, 2_{3}, 1_{5}, 3_{4}\rangle &
\langle2_{2}, 5_{0}, 4_{7}, 0_{9}\rangle &
\langle0_{1}, 2_{1}, 0_{8}, 4_{4}\rangle &
\langle2_{0}, 0_{1}, 4_{9}, 0_{7}\rangle &
\langle0_{0}, 2_{2}, 2_{5}, 2_{8}\rangle &
\langle5_{3}, 4_{1}, 2_{7}, 5_{6}\rangle &
\langle1_{3}, 0_{3}, 4_{9}, 5_{8}\rangle &
\langle3_{1}, 3_{0}, 0_{7}, 2_{8}\rangle \\
\langle4_{0}, 2_{3}, 2_{9}, 4_{5}\rangle &
\langle0_{0}, 0_{3}, 4_{5}, 5_{9}\rangle &
\langle0_{1}, 1_{2}, 0_{5}, 0_{6}\rangle &
\end{array}$$ ]{}
$n=45$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{3}, 3_{7}, 2_{6}\rangle &
\langle\infty, 4_{0}, 5_{4}, 3_{5}\rangle &
\langle\infty, 0_{2}, 0_{9}, 3_{8}\rangle &
\langle0_{3}, 0_{2}, 2_{7}, a_{1}\rangle &
\langle5_{0}, 2_{1}, 1_{6}, a_{1}\rangle &
\langle4_{1}, 3_{0}, 0_{7}, a_{2}\rangle &
\langle3_{2}, 4_{3}, 4_{6}, a_{2}\rangle &
\langle2_{0}, 1_{1}, 1_{6}, a_{3}\rangle \\
\langle3_{3}, 0_{2}, 5_{4}, a_{3}\rangle &
\langle0_{2}, 5_{3}, 2_{5}, a_{4}\rangle &
\langle0_{1}, 2_{0}, 4_{4}, a_{4}\rangle &
\langle0_{3}, 4_{2}, 5_{7}, a_{5}\rangle &
\langle4_{0}, 0_{1}, 2_{5}, a_{5}\rangle &
\langle4_{0}, 2_{3}, 1_{6}, a_{6}\rangle &
\langle5_{2}, 4_{1}, 5_{4}, a_{6}\rangle &
\langle0_{2}, 3_{0}, 1_{4}, a_{7}\rangle \\
\langle3_{1}, 2_{3}, 1_{5}, a_{7}\rangle &
\langle5_{3}, 1_{1}, 1_{5}, a_{8}\rangle &
\langle0_{0}, 5_{2}, 5_{7}, a_{8}\rangle &
\langle0_{3}, 3_{1}, 4_{5}, a_{9}\rangle &
\langle3_{2}, 1_{0}, 0_{4}, a_{9}\rangle &
\langle2_{0}, 4_{3}, 4_{9}, 5_{5}\rangle &
\langle4_{1}, 5_{3}, 1_{8}, 0_{4}\rangle &
\langle2_{0}, 5_{3}, 0_{9}, 3_{7}\rangle \\
\langle2_{2}, 0_{1}, 5_{5}, 4_{8}\rangle &
\langle0_{2}, 4_{3}, 4_{5}, 4_{7}\rangle &
\langle1_{1}, 1_{2}, 0_{7}, 5_{9}\rangle &
\langle3_{1}, 2_{1}, 5_{6}, 3_{9}\rangle &
\langle3_{0}, 5_{0}, 1_{7}, 2_{9}\rangle &
\langle3_{0}, 3_{3}, 3_{4}, 0_{8}\rangle &
\langle0_{1}, 0_{0}, 1_{8}, 3_{4}\rangle &
\langle4_{0}, 5_{2}, 2_{6}, 4_{8}\rangle \\
\langle0_{2}, 5_{2}, 1_{9}, 4_{6}\rangle &
\langle4_{1}, 3_{2}, 5_{6}, 3_{8}\rangle &
\langle0_{1}, 4_{2}, 4_{6}, 3_{9}\rangle &
\langle1_{0}, 0_{3}, 5_{8}, 1_{7}\rangle &
\langle4_{3}, 4_{1}, 3_{4}, 0_{9}\rangle &
\langle4_{3}, 2_{3}, 2_{8}, 5_{6}\rangle &
\langle0_{0}, 1_{0}, 1_{6}, 1_{5}\rangle &
\langle2_{0}, 3_{3}, 4_{8}, 2_{9}\rangle \\
\langle5_{2}, 3_{2}, 1_{4}, 4_{5}\rangle &
\langle2_{3}, 0_{1}, 0_{4}, 5_{9}\rangle &
\langle2_{0}, 0_{2}, 1_{8}, 3_{9}\rangle &
\langle2_{1}, 4_{1}, 5_{7}, 4_{8}\rangle &
\langle0_{1}, 3_{2}, 3_{5}, 0_{7}\rangle &
\end{array}$$ ]{}
$n=47$: [$$\begin{array}{llllllllll}
\langle\infty, 4_{3}, 5_{5}, 4_{4}\rangle &
\langle\infty, 3_{1}, 4_{7}, 4_{6}\rangle &
\langle\infty, 4_{0}, 3_{9}, 0_{8}\rangle &
\langle4_{0}, 4_{1}, 2_{7}, a_{1}\rangle &
\langle0_{2}, 3_{3}, 0_{5}, a_{1}\rangle &
\langle2_{3}, 1_{2}, 4_{6}, a_{2}\rangle &
\langle2_{0}, 3_{1}, 3_{4}, a_{2}\rangle &
\langle2_{3}, 0_{2}, 5_{7}, a_{3}\rangle \\
\langle4_{0}, 3_{1}, 5_{6}, a_{3}\rangle &
\langle4_{1}, 0_{2}, 2_{4}, a_{4}\rangle &
\langle5_{3}, 2_{0}, 5_{7}, a_{4}\rangle &
\langle2_{0}, 4_{1}, 5_{5}, a_{5}\rangle &
\langle5_{2}, 4_{3}, 0_{7}, a_{5}\rangle &
\langle4_{2}, 3_{0}, 2_{4}, a_{6}\rangle &
\langle5_{1}, 3_{3}, 5_{5}, a_{6}\rangle &
\langle0_{2}, 4_{0}, 1_{6}, a_{7}\rangle \\
\langle3_{3}, 0_{1}, 2_{4}, a_{7}\rangle &
\langle2_{0}, 2_{3}, 0_{4}, a_{8}\rangle &
\langle1_{1}, 0_{2}, 0_{7}, a_{8}\rangle &
\langle5_{0}, 4_{3}, 4_{5}, a_{9}\rangle &
\langle5_{1}, 3_{2}, 1_{7}, a_{9}\rangle &
\langle4_{3}, 3_{1}, 0_{4}, a_{10}\rangle &
\langle4_{2}, 4_{0}, 2_{5}, a_{10}\rangle &
\langle4_{0}, 3_{2}, 0_{7}, a_{11}\rangle \\
\langle1_{3}, 1_{1}, 2_{4}, a_{11}\rangle &
\langle5_{1}, 0_{1}, 5_{8}, 3_{6}\rangle &
\langle0_{0}, 4_{2}, 0_{9}, 5_{8}\rangle &
\langle5_{1}, 5_{2}, 4_{4}, 1_{8}\rangle &
\langle0_{2}, 4_{3}, 5_{8}, 2_{6}\rangle &
\langle4_{0}, 1_{2}, 5_{9}, 1_{4}\rangle &
\langle4_{1}, 0_{1}, 1_{9}, 3_{5}\rangle &
\langle1_{3}, 2_{1}, 2_{6}, 2_{9}\rangle \\
\langle1_{2}, 0_{2}, 0_{6}, 2_{5}\rangle &
\langle1_{0}, 4_{1}, 2_{5}, 5_{8}\rangle &
\langle1_{1}, 4_{2}, 5_{9}, 3_{9}\rangle &
\langle3_{0}, 1_{3}, 0_{9}, 5_{9}\rangle &
\langle5_{1}, 0_{2}, 4_{6}, 3_{8}\rangle &
\langle4_{0}, 0_{3}, 2_{9}, 0_{6}\rangle &
\langle2_{3}, 0_{1}, 5_{9}, 0_{7}\rangle &
\langle1_{0}, 5_{1}, 1_{5}, 2_{7}\rangle \\
\langle2_{2}, 0_{2}, 5_{5}, 0_{8}\rangle &
\langle1_{2}, 1_{3}, 4_{4}, 1_{9}\rangle &
\langle3_{0}, 4_{0}, 2_{6}, 3_{7}\rangle &
\langle4_{3}, 3_{0}, 3_{6}, 3_{8}\rangle &
\langle1_{0}, 3_{0}, 3_{4}, 4_{8}\rangle &
\langle5_{3}, 3_{3}, 4_{7}, 3_{8}\rangle &
\langle0_{3}, 1_{3}, 5_{5}, 3_{8}\rangle &
\end{array}$$ ]{}
$n=49$: [$$\begin{array}{llllllllll}
\langle\infty, 5_{2}, 1_{6}, 5_{7}\rangle &
\langle\infty, 3_{1}, 4_{10}, 5_{5}\rangle &
\langle\infty, 0_{0}, 3_{11}, 1_{8}\rangle &
\langle\infty, 4_{3}, 2_{9}, 2_{4}\rangle &
\langle5_{0}, 4_{1}, 5_{5}, a_{1}\rangle &
\langle1_{2}, 2_{3}, 2_{4}, a_{1}\rangle &
\langle3_{1}, 1_{3}, 2_{11}, 5_{8}\rangle &
\langle2_{0}, 3_{3}, 5_{10}, 4_{6}\rangle \\
\langle4_{0}, 1_{1}, 5_{9}, 0_{7}\rangle &
\langle0_{0}, 2_{3}, 1_{4}, 5_{10}\rangle &
\langle1_{2}, 2_{2}, 1_{9}, 3_{5}\rangle &
\langle1_{0}, 2_{1}, 2_{6}, 4_{9}\rangle &
\langle0_{3}, 2_{3}, 1_{10}, 5_{9}\rangle &
\langle2_{1}, 5_{3}, 0_{7}, 1_{4}\rangle &
\langle4_{3}, 3_{1}, 2_{6}, 0_{8}\rangle &
\langle3_{2}, 4_{1}, 4_{9}, 1_{6}\rangle \\
\langle0_{1}, 2_{2}, 1_{4}, 2_{11}\rangle &
\langle1_{1}, 4_{2}, 1_{7}, 5_{8}\rangle &
\langle3_{0}, 2_{3}, 1_{6}, 5_{5}\rangle &
\langle3_{1}, 3_{3}, 4_{8}, 2_{8}\rangle &
\langle0_{1}, 5_{1}, 2_{7}, 5_{10}\rangle &
\langle4_{0}, 5_{0}, 3_{4}, 4_{9}\rangle &
\langle1_{3}, 4_{0}, 3_{6}, 4_{8}\rangle &
\langle0_{1}, 5_{3}, 5_{9}, 5_{5}\rangle \\
\langle3_{0}, 0_{2}, 2_{8}, 2_{7}\rangle &
\langle3_{0}, 1_{1}, 5_{10}, 1_{8}\rangle &
\langle5_{2}, 3_{0}, 4_{5}, 5_{8}\rangle &
\langle5_{0}, 3_{3}, 5_{7}, 3_{11}\rangle &
\langle0_{0}, 0_{2}, 1_{7}, 3_{8}\rangle &
\langle2_{0}, 2_{1}, 3_{11}, 0_{5}\rangle &
\langle1_{2}, 0_{0}, 4_{10}, 2_{11}\rangle &
\langle0_{2}, 5_{3}, 2_{4}, 5_{8}\rangle \\
\langle1_{3}, 0_{3}, 5_{7}, 2_{9}\rangle &
\langle2_{2}, 4_{3}, 1_{11}, 5_{5}\rangle &
\langle1_{2}, 1_{3}, 5_{5}, 5_{11}\rangle &
\langle0_{0}, 2_{0}, 5_{5}, 4_{9}\rangle &
\langle0_{0}, 0_{3}, 5_{11}, 0_{10}\rangle &
\langle0_{1}, 2_{3}, 4_{11}, 2_{6}\rangle &
\langle2_{2}, 0_{3}, 2_{5}, 3_{6}\rangle &
\langle2_{2}, 4_{0}, 4_{11}, 1_{6}\rangle \\
\langle0_{0}, 5_{2}, 4_{7}, 3_{7}\rangle &
\langle1_{0}, 3_{1}, 3_{4}, 1_{6}\rangle &
\langle0_{1}, 2_{1}, 3_{9}, 4_{4}\rangle &
\langle5_{1}, 3_{2}, 2_{10}, 1_{10}\rangle &
\langle3_{1}, 3_{2}, 0_{4}, 0_{11}\rangle &
\langle0_{2}, 2_{2}, 2_{10}, 4_{9}\rangle &
\langle0_{2}, 3_{3}, 4_{4}, 1_{10}\rangle &
\langle0_0, 3_0, 0_4, 3_4\rangle^s \\
\langle0_1, 3_1, 0_5, 3_5\rangle^s &
\langle0_2, 3_2, 0_6, 3_6\rangle^s &
\langle0_3, 3_3, 0_7, 3_7\rangle^s &
\end{array}$$ ]{} Note that each of the codewords marked $s$ only generates three codewords.
T$(2,29;2,n;6) =7n$ for each odd $n$ and $29 \leq n \leq 57$.
Let $X_1=({{\mathbb{Z}}}_7\times \{0,1,2,3\})\cup \{\infty\}$. For $29 \leq n \leq 41$, let $X_2= ({{\mathbb{Z}}}_7\times \{4,5,6,7\})\cup (\{a\}\times \{1,\ldots,n-28\})$; for $43 \leq n \leq 55$, let $X_2= ({{\mathbb{Z}}}_7\times \{4,5,\ldots,9\})\cup (\{a\}\times \{1,\ldots,n-42\})$; for $n = 57$, let $X_2= ({{\mathbb{Z}}}_7\times \{4,5,\ldots,11\})\cup (\{a\}\times \{1\})$. Denote $X=X_1\cup X_2$. The desired codes of size $7n$ are constructed on ${{\mathbb{Z}}}_2^X$ and the base codewords are listed as follows.
$n=29$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{0}, 2_{5}, 5_{4}\rangle &
\langle\infty, 0_{1}, 0_{7}, 0_{6}\rangle &
\langle6_{2}, 5_{3}, 3_{5}, a_{1}\rangle &
\langle0_{1}, 6_{0}, 1_{4}, a_{1}\rangle &
\langle2_{2}, 5_{3}, 2_{7}, 1_{6}\rangle &
\langle2_{2}, 2_{3}, 2_{5}, 2_{6}\rangle &
\langle1_{3}, 2_{1}, 0_{5}, 0_{4}\rangle &
\langle1_{0}, 0_{3}, 6_{7}, 5_{4}\rangle \\
\langle0_{0}, 6_{1}, 1_{6}, 4_{6}\rangle &
\langle1_{2}, 2_{2}, 1_{4}, 3_{5}\rangle &
\langle3_{2}, 1_{3}, 6_{7}, 2_{5}\rangle &
\langle1_{0}, 5_{1}, 2_{7}, 1_{7}\rangle &
\langle0_{2}, 4_{2}, 2_{4}, 5_{6}\rangle &
\langle0_{1}, 4_{1}, 6_{7}, 3_{6}\rangle &
\langle0_{1}, 5_{1}, 0_{5}, 6_{5}\rangle &
\langle1_{0}, 1_{2}, 5_{7}, 3_{7}\rangle \\
\langle2_{1}, 2_{2}, 6_{6}, 6_{4}\rangle &
\langle1_{0}, 0_{0}, 2_{5}, 6_{4}\rangle &
\langle3_{0}, 1_{1}, 2_{7}, 6_{7}\rangle &
\langle2_{2}, 3_{3}, 5_{5}, 0_{5}\rangle &
\langle1_{0}, 1_{3}, 4_{5}, 3_{6}\rangle &
\langle1_{3}, 5_{3}, 2_{6}, 1_{7}\rangle &
\langle1_{2}, 3_{2}, 2_{7}, 4_{4}\rangle &
\langle1_{1}, 0_{1}, 3_{4}, 0_{4}\rangle \\
\langle2_{0}, 4_{1}, 5_{6}, 0_{5}\rangle &
\langle0_{0}, 2_{0}, 0_{6}, 6_{5}\rangle &
\langle2_{2}, 6_{3}, 5_{6}, 0_{7}\rangle &
\langle1_{3}, 6_{3}, 2_{4}, 3_{4}\rangle &
\langle0_{0}, 1_{3}, 1_{4}, 6_{6}\rangle &
\end{array}$$ ]{}
$n=31$: [$$\begin{array}{llllllllll}
\langle\infty, 5_{0}, 4_{5}, 5_{4}\rangle &
\langle\infty, 3_{1}, 0_{7}, 2_{6}\rangle &
\langle1_{3}, 1_{2}, 1_{7}, a_{1}\rangle &
\langle4_{1}, 2_{0}, 3_{4}, a_{1}\rangle &
\langle2_{0}, 3_{1}, 5_{4}, a_{2}\rangle &
\langle5_{3}, 4_{2}, 6_{6}, a_{2}\rangle &
\langle6_{3}, 4_{2}, 3_{7}, a_{3}\rangle &
\langle6_{1}, 2_{0}, 4_{4}, a_{3}\rangle \\
\langle5_{0}, 1_{1}, 6_{7}, 2_{7}\rangle &
\langle6_{0}, 4_{1}, 6_{7}, 2_{6}\rangle &
\langle5_{2}, 2_{3}, 0_{5}, 0_{7}\rangle &
\langle2_{2}, 2_{0}, 0_{7}, 5_{7}\rangle &
\langle4_{2}, 2_{3}, 5_{7}, 0_{4}\rangle &
\langle4_{0}, 1_{0}, 1_{6}, 2_{6}\rangle &
\langle4_{1}, 5_{0}, 6_{5}, 2_{5}\rangle &
\langle6_{1}, 3_{3}, 1_{6}, 3_{5}\rangle \\
\langle6_{0}, 4_{3}, 5_{4}, 1_{6}\rangle &
\langle1_{2}, 3_{2}, 2_{4}, 3_{4}\rangle &
\langle3_{1}, 6_{2}, 3_{7}, 3_{4}\rangle &
\langle4_{2}, 0_{3}, 2_{4}, 4_{5}\rangle &
\langle2_{3}, 1_{3}, 1_{4}, 5_{4}\rangle &
\langle2_{2}, 6_{2}, 3_{5}, 5_{6}\rangle &
\langle6_{0}, 0_{3}, 6_{5}, 1_{7}\rangle &
\langle3_{3}, 0_{3}, 2_{7}, 6_{6}\rangle \\
\langle3_{1}, 5_{1}, 6_{5}, 6_{6}\rangle &
\langle3_{3}, 5_{3}, 6_{5}, 5_{6}\rangle &
\langle3_{0}, 4_{0}, 1_{4}, 6_{5}\rangle &
\langle5_{0}, 0_{2}, 3_{5}, 4_{6}\rangle &
\langle1_{2}, 2_{1}, 2_{6}, 6_{6}\rangle &
\langle2_{1}, 5_{1}, 6_{4}, 1_{7}\rangle &
\langle0_{1}, 1_{2}, 0_{5}, 6_{5}\rangle &
\end{array}$$ ]{}
$n=33$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{0}, 3_{4}, 3_{5}\rangle &
\langle\infty, 1_{1}, 6_{6}, 4_{7}\rangle &
\langle5_{2}, 4_{3}, 4_{5}, a_{1}\rangle &
\langle3_{0}, 3_{1}, 4_{4}, a_{1}\rangle &
\langle6_{2}, 0_{3}, 0_{7}, a_{2}\rangle &
\langle6_{0}, 0_{1}, 2_{4}, a_{2}\rangle &
\langle2_{3}, 0_{2}, 5_{6}, a_{3}\rangle &
\langle5_{0}, 3_{1}, 3_{7}, a_{3}\rangle \\
\langle4_{2}, 0_{3}, 3_{7}, a_{4}\rangle &
\langle4_{0}, 3_{1}, 6_{4}, a_{4}\rangle &
\langle5_{0}, 1_{1}, 5_{7}, a_{5}\rangle &
\langle4_{2}, 1_{3}, 4_{4}, a_{5}\rangle &
\langle1_{1}, 2_{3}, 6_{7}, 0_{5}\rangle &
\langle1_{1}, 0_{1}, 2_{5}, 5_{4}\rangle &
\langle2_{1}, 0_{1}, 5_{5}, 1_{6}\rangle &
\langle0_{3}, 6_{3}, 1_{4}, 4_{4}\rangle \\
\langle1_{0}, 1_{3}, 0_{5}, 5_{5}\rangle &
\langle1_{1}, 3_{2}, 5_{5}, 4_{6}\rangle &
\langle1_{0}, 0_{3}, 2_{7}, 2_{5}\rangle &
\langle4_{0}, 1_{2}, 2_{5}, 0_{6}\rangle &
\langle1_{1}, 4_{3}, 1_{6}, 5_{6}\rangle &
\langle2_{1}, 2_{2}, 1_{4}, 4_{6}\rangle &
\langle3_{0}, 6_{2}, 2_{7}, 6_{7}\rangle &
\langle3_{2}, 2_{0}, 1_{4}, 0_{6}\rangle \\
\langle1_{0}, 2_{0}, 6_{4}, 4_{5}\rangle &
\langle0_{2}, 4_{2}, 0_{6}, 0_{5}\rangle &
\langle2_{0}, 2_{2}, 4_{7}, 6_{7}\rangle &
\langle1_{2}, 2_{2}, 5_{4}, 3_{4}\rangle &
\langle2_{2}, 6_{1}, 6_{5}, 0_{7}\rangle &
\langle0_{0}, 3_{0}, 4_{6}, 2_{6}\rangle &
\langle0_{3}, 4_{3}, 5_{7}, 6_{6}\rangle &
\langle1_{3}, 6_{3}, 2_{5}, 6_{6}\rangle \\
\langle0_{1}, 0_{3}, 0_{4}, 6_{7}\rangle &
\end{array}$$ ]{}
$n=35$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{2}, 0_{5}, 1_{4}\rangle &
\langle\infty, 1_{1}, 1_{6}, 1_{7}\rangle &
\langle6_{3}, 2_{2}, 6_{7}, a_{1}\rangle &
\langle6_{0}, 6_{1}, 0_{4}, a_{1}\rangle &
\langle5_{2}, 4_{3}, 6_{5}, a_{2}\rangle &
\langle5_{0}, 6_{1}, 1_{4}, a_{2}\rangle &
\langle5_{1}, 1_{3}, 5_{4}, a_{3}\rangle &
\langle0_{0}, 0_{2}, 2_{6}, a_{3}\rangle \\
\langle3_{3}, 3_{2}, 2_{4}, a_{4}\rangle &
\langle6_{0}, 2_{1}, 4_{5}, a_{4}\rangle &
\langle5_{2}, 1_{3}, 4_{7}, a_{5}\rangle &
\langle3_{0}, 5_{1}, 4_{5}, a_{5}\rangle &
\langle5_{0}, 3_{1}, 2_{4}, a_{6}\rangle &
\langle5_{3}, 4_{2}, 5_{6}, a_{6}\rangle &
\langle1_{0}, 0_{1}, 3_{7}, a_{7}\rangle &
\langle6_{2}, 4_{3}, 6_{6}, a_{7}\rangle \\
\langle1_{2}, 3_{3}, 6_{6}, 4_{5}\rangle &
\langle2_{1}, 5_{1}, 3_{7}, 3_{5}\rangle &
\langle3_{3}, 4_{1}, 0_{5}, 0_{6}\rangle &
\langle0_{3}, 2_{0}, 1_{7}, 6_{5}\rangle &
\langle3_{0}, 6_{3}, 3_{7}, 1_{7}\rangle &
\langle3_{2}, 5_{2}, 1_{7}, 6_{4}\rangle &
\langle0_{1}, 0_{3}, 6_{7}, 3_{4}\rangle &
\langle5_{3}, 1_{1}, 6_{6}, 3_{7}\rangle \\
\langle4_{3}, 5_{0}, 4_{5}, 2_{6}\rangle &
\langle4_{1}, 3_{1}, 1_{4}, 5_{6}\rangle &
\langle0_{2}, 5_{0}, 0_{5}, 3_{6}\rangle &
\langle2_{2}, 6_{0}, 6_{4}, 6_{5}\rangle &
\langle2_{3}, 5_{0}, 1_{6}, 3_{4}\rangle &
\langle4_{1}, 6_{2}, 1_{7}, 4_{5}\rangle &
\langle2_{3}, 4_{3}, 0_{5}, 4_{4}\rangle &
\langle2_{0}, 3_{0}, 3_{6}, 6_{7}\rangle \\
\langle5_{1}, 5_{2}, 2_{6}, 4_{6}\rangle &
\langle3_{2}, 2_{0}, 5_{5}, 3_{7}\rangle &
\langle0_{0}, 4_{2}, 2_{4}, 6_{4}\rangle &
\end{array}$$ ]{}
$n=37$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{1}, 2_{7}, 2_{6}\rangle &
\langle\infty, 1_{0}, 1_{5}, 1_{4}\rangle &
\langle5_{2}, 5_{3}, 2_{5}, a_{1}\rangle &
\langle5_{0}, 5_{1}, 6_{4}, a_{1}\rangle &
\langle6_{0}, 0_{1}, 2_{4}, a_{2}\rangle &
\langle4_{2}, 5_{3}, 5_{5}, a_{2}\rangle &
\langle1_{1}, 6_{0}, 1_{4}, a_{3}\rangle &
\langle5_{2}, 0_{3}, 1_{5}, a_{3}\rangle \\
\langle0_{1}, 4_{0}, 3_{4}, a_{4}\rangle &
\langle4_{2}, 0_{3}, 3_{5}, a_{4}\rangle &
\langle2_{2}, 6_{3}, 4_{6}, a_{5}\rangle &
\langle2_{0}, 6_{1}, 3_{5}, a_{5}\rangle &
\langle6_{2}, 4_{3}, 2_{7}, a_{6}\rangle &
\langle2_{1}, 4_{0}, 1_{4}, a_{6}\rangle &
\langle4_{0}, 3_{1}, 6_{5}, a_{7}\rangle &
\langle5_{2}, 4_{3}, 2_{4}, a_{7}\rangle \\
\langle2_{0}, 4_{2}, 0_{4}, a_{8}\rangle &
\langle2_{1}, 2_{3}, 4_{5}, a_{8}\rangle &
\langle1_{1}, 4_{3}, 4_{7}, a_{9}\rangle &
\langle3_{0}, 3_{2}, 3_{6}, a_{9}\rangle &
\langle0_{1}, 4_{2}, 2_{6}, 1_{7}\rangle &
\langle1_{0}, 0_{0}, 4_{5}, 2_{6}\rangle &
\langle2_{0}, 3_{2}, 4_{7}, 1_{5}\rangle &
\langle0_{2}, 6_{2}, 0_{4}, 5_{4}\rangle \\
\langle2_{1}, 4_{2}, 0_{6}, 6_{4}\rangle &
\langle1_{3}, 0_{3}, 6_{5}, 0_{4}\rangle &
\langle1_{0}, 5_{0}, 5_{7}, 6_{7}\rangle &
\langle0_{2}, 3_{2}, 4_{6}, 2_{7}\rangle &
\langle3_{1}, 4_{3}, 0_{7}, 1_{4}\rangle &
\langle1_{1}, 3_{3}, 5_{6}, 2_{6}\rangle &
\langle0_{1}, 2_{1}, 0_{5}, 1_{5}\rangle &
\langle0_{1}, 4_{1}, 6_{7}, 3_{6}\rangle \\
\langle0_{3}, 2_{3}, 3_{6}, 3_{4}\rangle &
\langle0_{0}, 2_{0}, 6_{6}, 5_{6}\rangle &
\langle0_{3}, 4_{3}, 4_{6}, 1_{7}\rangle &
\langle2_{0}, 6_{3}, 1_{7}, 5_{7}\rangle &
\langle0_{2}, 2_{2}, 2_{5}, 0_{7}\rangle &
\end{array}$$ ]{}
$n=39$: [$$\begin{array}{llllllllll}
\langle\infty, 6_{1}, 6_{7}, 6_{6}\rangle &
\langle\infty, 1_{0}, 1_{5}, 2_{4}\rangle &
\langle4_{1}, 4_{0}, 5_{5}, a_{1}\rangle &
\langle4_{2}, 4_{3}, 1_{6}, a_{1}\rangle &
\langle5_{0}, 6_{1}, 1_{4}, a_{2}\rangle &
\langle3_{2}, 4_{3}, 0_{5}, a_{2}\rangle &
\langle5_{2}, 4_{3}, 6_{6}, a_{3}\rangle &
\langle1_{0}, 0_{1}, 3_{5}, a_{3}\rangle \\
\langle2_{2}, 5_{3}, 0_{7}, a_{4}\rangle &
\langle4_{1}, 1_{0}, 0_{4}, a_{4}\rangle &
\langle2_{3}, 5_{2}, 5_{4}, a_{5}\rangle &
\langle4_{1}, 0_{0}, 5_{6}, a_{5}\rangle &
\langle2_{3}, 4_{2}, 4_{5}, a_{6}\rangle &
\langle1_{0}, 6_{1}, 5_{4}, a_{6}\rangle &
\langle4_{1}, 4_{2}, 5_{4}, a_{7}\rangle &
\langle0_{0}, 2_{3}, 3_{5}, a_{7}\rangle \\
\langle2_{3}, 2_{1}, 6_{4}, a_{8}\rangle &
\langle6_{0}, 3_{2}, 1_{6}, a_{8}\rangle &
\langle6_{0}, 0_{2}, 6_{6}, a_{9}\rangle &
\langle6_{1}, 4_{3}, 4_{4}, a_{9}\rangle &
\langle0_{2}, 2_{0}, 1_{5}, a_{10}\rangle &
\langle6_{1}, 0_{3}, 5_{7}, a_{10}\rangle &
\langle4_{3}, 1_{1}, 3_{5}, a_{11}\rangle &
\langle2_{2}, 2_{0}, 0_{4}, a_{11}\rangle \\
\langle4_{0}, 2_{0}, 3_{6}, 2_{7}\rangle &
\langle2_{0}, 2_{3}, 5_{7}, 0_{5}\rangle &
\langle2_{3}, 4_{3}, 3_{7}, 2_{6}\rangle &
\langle2_{3}, 6_{3}, 6_{5}, 6_{7}\rangle &
\langle2_{3}, 1_{0}, 1_{4}, 5_{6}\rangle &
\langle3_{0}, 0_{0}, 4_{7}, 2_{7}\rangle &
\langle1_{1}, 0_{1}, 5_{5}, 0_{5}\rangle &
\langle2_{1}, 0_{1}, 4_{6}, 5_{6}\rangle \\
\langle2_{2}, 1_{1}, 1_{4}, 3_{7}\rangle &
\langle1_{1}, 6_{2}, 5_{7}, 6_{7}\rangle &
\langle4_{1}, 3_{2}, 0_{7}, 3_{6}\rangle &
\langle1_{2}, 4_{2}, 6_{5}, 0_{5}\rangle &
\langle0_{0}, 4_{3}, 2_{4}, 3_{6}\rangle &
\langle1_{2}, 2_{2}, 4_{4}, 4_{7}\rangle &
\langle0_{2}, 2_{3}, 4_{4}, 3_{6}\rangle &
\end{array}$$ ]{}
$n=41$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{1}, 1_{5}, 1_{4}\rangle &
\langle\infty, 4_{2}, 1_{7}, 4_{6}\rangle &
\langle2_{1}, 2_{0}, 6_{4}, a_{1}\rangle &
\langle2_{2}, 2_{3}, 5_{6}, a_{1}\rangle &
\langle1_{2}, 2_{3}, 6_{6}, a_{2}\rangle &
\langle2_{1}, 1_{0}, 4_{4}, a_{2}\rangle &
\langle0_{3}, 5_{2}, 4_{5}, a_{3}\rangle &
\langle2_{0}, 4_{1}, 4_{4}, a_{3}\rangle \\
\langle0_{3}, 1_{2}, 1_{4}, a_{4}\rangle &
\langle1_{0}, 4_{1}, 1_{6}, a_{4}\rangle &
\langle4_{2}, 0_{3}, 0_{7}, a_{5}\rangle &
\langle1_{1}, 4_{0}, 4_{5}, a_{5}\rangle &
\langle3_{3}, 6_{2}, 1_{5}, a_{6}\rangle &
\langle0_{1}, 2_{0}, 3_{4}, a_{6}\rangle &
\langle6_{0}, 5_{1}, 6_{4}, a_{7}\rangle &
\langle1_{3}, 3_{2}, 3_{7}, a_{7}\rangle \\
\langle0_{1}, 5_{2}, 5_{5}, a_{8}\rangle &
\langle1_{3}, 3_{0}, 2_{6}, a_{8}\rangle &
\langle4_{3}, 2_{1}, 0_{4}, a_{9}\rangle &
\langle1_{2}, 4_{0}, 2_{5}, a_{9}\rangle &
\langle2_{3}, 3_{1}, 4_{6}, a_{10}\rangle &
\langle2_{2}, 2_{0}, 0_{4}, a_{10}\rangle &
\langle0_{3}, 2_{1}, 1_{7}, a_{11}\rangle &
\langle5_{0}, 4_{2}, 1_{6}, a_{11}\rangle \\
\langle5_{3}, 5_{1}, 0_{5}, a_{12}\rangle &
\langle4_{2}, 2_{0}, 3_{6}, a_{12}\rangle &
\langle6_{2}, 1_{0}, 0_{4}, a_{13}\rangle &
\langle3_{3}, 2_{1}, 0_{7}, a_{13}\rangle &
\langle3_{0}, 6_{0}, 5_{7}, 3_{7}\rangle &
\langle3_{0}, 5_{0}, 2_{5}, 1_{7}\rangle &
\langle2_{3}, 4_{3}, 1_{4}, 3_{5}\rangle &
\langle6_{1}, 3_{1}, 3_{5}, 6_{6}\rangle \\
\langle6_{1}, 3_{3}, 6_{7}, 1_{6}\rangle &
\langle6_{3}, 3_{3}, 1_{4}, 6_{5}\rangle &
\langle0_{0}, 1_{0}, 5_{6}, 3_{5}\rangle &
\langle3_{2}, 5_{2}, 2_{4}, 4_{7}\rangle &
\langle3_{3}, 2_{3}, 1_{7}, 2_{6}\rangle &
\langle2_{1}, 0_{1}, 4_{7}, 3_{7}\rangle &
\langle5_{2}, 2_{0}, 3_{5}, 3_{7}\rangle &
\langle3_{2}, 6_{1}, 5_{6}, 4_{6}\rangle \\
\langle0_{2}, 1_{2}, 3_{4}, 4_{5}\rangle &
\end{array}$$ ]{}
$n=43$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{3}, 5_{8}, 5_{9}\rangle &
\langle\infty, 0_{2}, 3_{6}, 2_{4}\rangle &
\langle\infty, 4_{0}, 2_{5}, 5_{7}\rangle &
\langle5_{3}, 4_{2}, 4_{5}, a_{1}\rangle &
\langle5_{0}, 5_{1}, 4_{6}, a_{1}\rangle &
\langle3_{1}, 5_{1}, 0_{6}, 5_{5}\rangle &
\langle2_{0}, 6_{0}, 4_{9}, 5_{9}\rangle &
\langle0_{0}, 4_{3}, 2_{6}, 4_{8}\rangle \\
\langle4_{2}, 4_{0}, 5_{6}, 4_{6}\rangle &
\langle5_{0}, 3_{2}, 3_{4}, 4_{7}\rangle &
\langle2_{2}, 5_{1}, 2_{8}, 4_{7}\rangle &
\langle6_{1}, 6_{3}, 4_{9}, 3_{7}\rangle &
\langle2_{3}, 0_{0}, 3_{4}, 2_{7}\rangle &
\langle4_{2}, 1_{1}, 5_{9}, 2_{4}\rangle &
\langle1_{3}, 5_{3}, 5_{9}, 2_{6}\rangle &
\langle0_{3}, 4_{0}, 3_{4}, 4_{4}\rangle \\
\langle3_{2}, 5_{2}, 4_{5}, 1_{8}\rangle &
\langle3_{3}, 0_{2}, 4_{7}, 4_{9}\rangle &
\langle2_{3}, 6_{1}, 2_{6}, 4_{6}\rangle &
\langle3_{1}, 0_{1}, 1_{7}, 6_{9}\rangle &
\langle3_{2}, 4_{2}, 6_{9}, 1_{6}\rangle &
\langle0_{0}, 1_{1}, 4_{7}, 1_{8}\rangle &
\langle5_{0}, 4_{2}, 3_{7}, 5_{8}\rangle &
\langle6_{1}, 0_{3}, 6_{6}, 0_{5}\rangle \\
\langle6_{0}, 0_{2}, 3_{4}, 0_{9}\rangle &
\langle0_{1}, 4_{3}, 2_{7}, 2_{8}\rangle &
\langle6_{0}, 5_{3}, 6_{5}, 0_{4}\rangle &
\langle2_{0}, 0_{1}, 2_{9}, 6_{5}\rangle &
\langle3_{0}, 5_{1}, 2_{5}, 5_{4}\rangle &
\langle0_{2}, 4_{3}, 2_{5}, 4_{4}\rangle &
\langle1_{1}, 2_{1}, 0_{4}, 5_{4}\rangle &
\langle2_{2}, 2_{3}, 4_{8}, 1_{4}\rangle \\
\langle2_{0}, 0_{0}, 5_{8}, 3_{5}\rangle &
\langle3_{1}, 4_{2}, 3_{9}, 5_{4}\rangle &
\langle0_{0}, 0_{3}, 3_{6}, 6_{8}\rangle &
\langle2_{2}, 2_{1}, 0_{5}, 5_{5}\rangle &
\langle5_{0}, 2_{1}, 0_{8}, 3_{6}\rangle &
\langle3_{3}, 2_{0}, 2_{7}, 5_{7}\rangle &
\langle6_{1}, 5_{3}, 2_{8}, 0_{9}\rangle &
\langle1_{3}, 3_{0}, 0_{9}, 5_{5}\rangle \\
\langle6_{2}, 2_{2}, 1_{6}, 2_{7}\rangle &
\langle6_{2}, 4_{1}, 3_{8}, 5_{8}\rangle &
\langle0_{2}, 2_{3}, 4_{5}, 5_{7}\rangle &
\end{array}$$ ]{}
$n=45$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{2}, 2_{9}, 2_{8}\rangle &
\langle\infty, 2_{0}, 2_{4}, 2_{5}\rangle &
\langle\infty, 2_{1}, 4_{6}, 3_{7}\rangle &
\langle5_{0}, 5_{1}, 6_{4}, a_{1}\rangle &
\langle1_{2}, 1_{3}, 1_{5}, a_{1}\rangle &
\langle3_{1}, 2_{0}, 5_{4}, a_{2}\rangle &
\langle4_{2}, 5_{3}, 2_{6}, a_{2}\rangle &
\langle4_{0}, 6_{1}, 6_{4}, a_{3}\rangle \\
\langle3_{2}, 5_{3}, 4_{5}, a_{3}\rangle &
\langle0_{0}, 5_{0}, 2_{9}, 3_{5}\rangle &
\langle0_{1}, 4_{1}, 5_{9}, 2_{5}\rangle &
\langle2_{1}, 5_{2}, 2_{9}, 2_{5}\rangle &
\langle3_{1}, 4_{3}, 0_{7}, 0_{9}\rangle &
\langle1_{0}, 2_{2}, 2_{6}, 5_{8}\rangle &
\langle1_{2}, 4_{3}, 5_{4}, 6_{7}\rangle &
\langle0_{1}, 3_{3}, 4_{5}, 6_{8}\rangle \\
\langle1_{0}, 4_{1}, 3_{6}, 0_{4}\rangle &
\langle2_{1}, 6_{3}, 6_{6}, 6_{8}\rangle &
\langle1_{0}, 4_{2}, 3_{5}, 3_{7}\rangle &
\langle0_{3}, 5_{3}, 6_{7}, 1_{6}\rangle &
\langle1_{0}, 0_{3}, 2_{9}, 5_{7}\rangle &
\langle3_{0}, 5_{2}, 2_{7}, 3_{8}\rangle &
\langle2_{0}, 0_{1}, 5_{6}, 2_{9}\rangle &
\langle1_{3}, 4_{3}, 2_{8}, 3_{6}\rangle \\
\langle2_{1}, 2_{3}, 2_{7}, 1_{9}\rangle &
\langle3_{0}, 3_{2}, 1_{9}, 6_{7}\rangle &
\langle0_{1}, 5_{2}, 5_{7}, 1_{6}\rangle &
\langle2_{1}, 1_{1}, 6_{4}, 4_{8}\rangle &
\langle1_{2}, 6_{2}, 4_{5}, 0_{8}\rangle &
\langle3_{1}, 0_{2}, 4_{8}, 2_{7}\rangle &
\langle1_{0}, 0_{1}, 2_{7}, 0_{6}\rangle &
\langle0_{2}, 6_{2}, 1_{9}, 1_{4}\rangle \\
\langle0_{0}, 1_{3}, 6_{9}, 4_{5}\rangle &
\langle0_{3}, 6_{3}, 2_{4}, 5_{4}\rangle &
\langle0_{2}, 4_{2}, 1_{6}, 6_{6}\rangle &
\langle2_{0}, 2_{3}, 4_{8}, 0_{6}\rangle &
\langle1_{1}, 1_{2}, 4_{9}, 0_{4}\rangle &
\langle3_{0}, 0_{1}, 3_{6}, 3_{7}\rangle &
\langle1_{2}, 6_{3}, 3_{8}, 6_{4}\rangle &
\langle0_{0}, 6_{0}, 4_{4}, 5_{8}\rangle \\
\langle4_{2}, 3_{3}, 3_{9}, 0_{4}\rangle &
\langle0_{1}, 5_{1}, 6_{5}, 5_{8}\rangle &
\langle3_{2}, 0_{3}, 5_{5}, 4_{7}\rangle &
\langle2_{0}, 4_{3}, 3_{8}, 5_{9}\rangle &
\langle0_{0}, 4_{3}, 1_{5}, 6_{5}\rangle &
\end{array}$$ ]{}
$n=47$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{2}, 3_{8}, 6_{9}\rangle &
\langle\infty, 3_{1}, 5_{7}, 1_{6}\rangle &
\langle\infty, 3_{0}, 3_{5}, 0_{4}\rangle &
\langle6_{1}, 2_{0}, 6_{5}, a_{1}\rangle &
\langle1_{2}, 4_{3}, 3_{7}, a_{1}\rangle &
\langle5_{3}, 0_{2}, 2_{5}, a_{2}\rangle &
\langle5_{1}, 4_{0}, 0_{4}, a_{2}\rangle &
\langle2_{3}, 0_{2}, 1_{5}, a_{3}\rangle \\
\langle5_{1}, 3_{0}, 6_{7}, a_{3}\rangle &
\langle2_{2}, 3_{3}, 5_{4}, a_{4}\rangle &
\langle6_{1}, 3_{0}, 0_{6}, a_{4}\rangle &
\langle0_{1}, 2_{0}, 3_{5}, a_{5}\rangle &
\langle0_{3}, 0_{2}, 1_{6}, a_{5}\rangle &
\langle2_{1}, 0_{1}, 2_{8}, 6_{5}\rangle &
\langle1_{2}, 0_{0}, 6_{7}, 0_{9}\rangle &
\langle4_{1}, 4_{3}, 2_{7}, 1_{9}\rangle \\
\langle6_{0}, 2_{0}, 0_{6}, 1_{9}\rangle &
\langle4_{1}, 2_{2}, 2_{4}, 3_{4}\rangle &
\langle4_{0}, 3_{3}, 5_{7}, 0_{8}\rangle &
\langle0_{2}, 1_{2}, 1_{7}, 1_{9}\rangle &
\langle3_{3}, 1_{3}, 6_{9}, 4_{7}\rangle &
\langle1_{2}, 3_{0}, 0_{7}, 4_{8}\rangle &
\langle4_{0}, 5_{0}, 3_{4}, 4_{8}\rangle &
\langle3_{1}, 3_{2}, 0_{4}, 6_{7}\rangle \\
\langle4_{2}, 1_{3}, 6_{6}, 2_{5}\rangle &
\langle6_{1}, 3_{1}, 1_{9}, 4_{8}\rangle &
\langle0_{3}, 4_{3}, 6_{6}, 6_{8}\rangle &
\langle2_{2}, 5_{0}, 0_{4}, 6_{9}\rangle &
\langle1_{2}, 2_{0}, 1_{5}, 0_{8}\rangle &
\langle4_{1}, 5_{2}, 3_{9}, 1_{6}\rangle &
\langle2_{3}, 4_{0}, 6_{6}, 2_{7}\rangle &
\langle1_{1}, 5_{3}, 4_{9}, 5_{8}\rangle \\
\langle1_{3}, 2_{1}, 5_{4}, 4_{5}\rangle &
\langle4_{3}, 6_{1}, 4_{5}, 6_{9}\rangle &
\langle5_{1}, 1_{3}, 5_{7}, 4_{8}\rangle &
\langle1_{1}, 2_{0}, 1_{6}, 4_{8}\rangle &
\langle1_{3}, 4_{0}, 1_{9}, 4_{4}\rangle &
\langle6_{2}, 2_{2}, 0_{8}, 5_{5}\rangle &
\langle2_{0}, 4_{2}, 6_{8}, 2_{6}\rangle &
\langle4_{3}, 5_{2}, 4_{4}, 2_{8}\rangle \\
\langle1_{1}, 3_{3}, 3_{6}, 1_{4}\rangle &
\langle3_{3}, 1_{0}, 2_{4}, 4_{9}\rangle &
\langle5_{1}, 2_{2}, 1_{6}, 6_{5}\rangle &
\langle3_{2}, 4_{1}, 5_{9}, 5_{4}\rangle &
\langle1_{3}, 1_{0}, 3_{5}, 4_{6}\rangle &
\langle2_{2}, 0_{1}, 6_{6}, 6_{7}\rangle &
\langle0_{0}, 2_{0}, 5_{5}, 2_{7}\rangle &
\end{array}$$ ]{}
$n=49$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{1}, 0_{4}, 1_{7}\rangle &
\langle\infty, 3_{0}, 2_{5}, 5_{8}\rangle &
\langle\infty, 3_{3}, 2_{6}, 4_{9}\rangle &
\langle0_{0}, 4_{2}, 0_{4}, a_{1}\rangle &
\langle4_{1}, 5_{3}, 5_{5}, a_{1}\rangle &
\langle1_{3}, 4_{2}, 0_{7}, a_{2}\rangle &
\langle3_{0}, 5_{1}, 4_{6}, a_{2}\rangle &
\langle4_{2}, 6_{0}, 1_{7}, a_{3}\rangle \\
\langle3_{1}, 0_{3}, 5_{4}, a_{3}\rangle &
\langle4_{1}, 3_{0}, 3_{5}, a_{4}\rangle &
\langle0_{3}, 0_{2}, 4_{6}, a_{4}\rangle &
\langle4_{0}, 2_{3}, 5_{4}, a_{5}\rangle &
\langle1_{1}, 6_{2}, 6_{7}, a_{5}\rangle &
\langle5_{3}, 5_{0}, 5_{7}, a_{6}\rangle &
\langle0_{1}, 1_{2}, 3_{5}, a_{6}\rangle &
\langle1_{3}, 6_{2}, 3_{4}, a_{7}\rangle \\
\langle0_{1}, 2_{0}, 5_{5}, a_{7}\rangle &
\langle5_{2}, 3_{3}, 5_{5}, 5_{9}\rangle &
\langle5_{0}, 4_{2}, 2_{7}, 2_{9}\rangle &
\langle6_{0}, 0_{2}, 5_{6}, 2_{8}\rangle &
\langle5_{0}, 6_{3}, 1_{6}, 4_{7}\rangle &
\langle1_{2}, 4_{3}, 0_{6}, 4_{6}\rangle &
\langle2_{1}, 2_{3}, 4_{8}, 5_{9}\rangle &
\langle6_{1}, 2_{1}, 1_{7}, 6_{8}\rangle \\
\langle0_{1}, 3_{2}, 4_{7}, 0_{5}\rangle &
\langle3_{0}, 6_{2}, 3_{8}, 1_{6}\rangle &
\langle0_{3}, 1_{0}, 6_{5}, 6_{8}\rangle &
\langle5_{1}, 3_{3}, 0_{5}, 6_{7}\rangle &
\langle1_{1}, 2_{1}, 6_{4}, 0_{9}\rangle &
\langle2_{2}, 3_{3}, 3_{9}, 0_{4}\rangle &
\langle4_{0}, 0_{3}, 1_{5}, 6_{4}\rangle &
\langle5_{2}, 5_{0}, 1_{9}, 6_{5}\rangle \\
\langle2_{2}, 5_{2}, 4_{9}, 1_{5}\rangle &
\langle1_{0}, 6_{0}, 0_{9}, 0_{8}\rangle &
\langle2_{2}, 0_{2}, 2_{4}, 5_{8}\rangle &
\langle1_{3}, 3_{3}, 5_{7}, 6_{8}\rangle &
\langle5_{1}, 1_{0}, 3_{6}, 6_{4}\rangle &
\langle2_{1}, 5_{3}, 2_{9}, 3_{6}\rangle &
\langle1_{1}, 0_{2}, 1_{4}, 5_{5}\rangle &
\langle5_{1}, 4_{3}, 5_{6}, 1_{8}\rangle \\
\langle4_{0}, 0_{0}, 3_{4}, 5_{7}\rangle &
\langle2_{3}, 3_{2}, 2_{4}, 3_{8}\rangle &
\langle3_{1}, 3_{2}, 0_{9}, 2_{8}\rangle &
\langle6_{2}, 4_{1}, 6_{6}, 0_{6}\rangle &
\langle2_{3}, 0_{1}, 3_{4}, 1_{9}\rangle &
\langle5_{1}, 2_{2}, 3_{8}, 1_{7}\rangle &
\langle5_{0}, 5_{1}, 2_{6}, 0_{9}\rangle &
\langle6_{3}, 4_{0}, 4_{9}, 0_{7}\rangle \\
\langle0_{0}, 4_{3}, 2_{5}, 4_{8}\rangle &
\end{array}$$ ]{}
$n=51$: [$$\begin{array}{llllllllll}
\langle\infty, 5_{2}, 6_{5}, 0_{8}\rangle &
\langle\infty, 2_{0}, 6_{6}, 6_{4}\rangle &
\langle\infty, 2_{3}, 4_{7}, 1_{9}\rangle &
\langle5_{0}, 5_{1}, 4_{7}, a_{1}\rangle &
\langle2_{3}, 0_{2}, 4_{5}, a_{1}\rangle &
\langle2_{0}, 5_{3}, 2_{4}, a_{2}\rangle &
\langle1_{2}, 6_{1}, 6_{6}, a_{2}\rangle &
\langle2_{0}, 3_{2}, 0_{7}, a_{3}\rangle \\
\langle4_{1}, 4_{3}, 0_{6}, a_{3}\rangle &
\langle3_{1}, 6_{0}, 4_{4}, a_{4}\rangle &
\langle3_{2}, 6_{3}, 6_{7}, a_{4}\rangle &
\langle0_{2}, 6_{1}, 1_{6}, a_{5}\rangle &
\langle2_{0}, 2_{3}, 4_{4}, a_{5}\rangle &
\langle5_{0}, 4_{2}, 0_{6}, a_{6}\rangle &
\langle3_{1}, 6_{3}, 4_{5}, a_{6}\rangle &
\langle2_{3}, 0_{0}, 6_{5}, a_{7}\rangle \\
\langle0_{1}, 4_{2}, 3_{4}, a_{7}\rangle &
\langle5_{1}, 0_{0}, 0_{7}, a_{8}\rangle &
\langle6_{3}, 2_{2}, 0_{5}, a_{8}\rangle &
\langle2_{3}, 1_{1}, 5_{7}, a_{9}\rangle &
\langle4_{0}, 1_{2}, 5_{6}, a_{9}\rangle &
\langle2_{0}, 4_{1}, 0_{9}, 4_{7}\rangle &
\langle2_{2}, 2_{3}, 1_{7}, 3_{9}\rangle &
\langle1_{3}, 2_{3}, 6_{7}, 4_{9}\rangle \\
\langle1_{2}, 1_{1}, 6_{9}, 4_{5}\rangle &
\langle4_{2}, 3_{2}, 2_{8}, 3_{5}\rangle &
\langle0_{0}, 1_{3}, 1_{9}, 3_{6}\rangle &
\langle3_{2}, 0_{2}, 5_{4}, 1_{7}\rangle &
\langle3_{3}, 6_{1}, 3_{6}, 3_{5}\rangle &
\langle0_{2}, 0_{0}, 6_{6}, 3_{8}\rangle &
\langle0_{0}, 4_{0}, 6_{8}, 2_{5}\rangle &
\langle3_{2}, 1_{0}, 4_{4}, 0_{4}\rangle \\
\langle1_{2}, 6_{3}, 3_{9}, 4_{4}\rangle &
\langle5_{3}, 0_{3}, 5_{8}, 1_{8}\rangle &
\langle6_{2}, 1_{2}, 5_{9}, 1_{6}\rangle &
\langle1_{0}, 3_{0}, 1_{6}, 3_{9}\rangle &
\langle1_{3}, 6_{1}, 4_{5}, 5_{6}\rangle &
\langle1_{0}, 2_{0}, 2_{5}, 5_{9}\rangle &
\langle0_{1}, 6_{3}, 4_{9}, 0_{4}\rangle &
\langle4_{2}, 5_{3}, 4_{8}, 6_{7}\rangle \\
\langle1_{0}, 0_{1}, 2_{4}, 0_{9}\rangle &
\langle0_{3}, 3_{3}, 3_{4}, 1_{6}\rangle &
\langle5_{0}, 6_{1}, 3_{8}, 1_{5}\rangle &
\langle3_{0}, 1_{3}, 0_{5}, 3_{8}\rangle &
\langle1_{1}, 4_{1}, 2_{6}, 4_{8}\rangle &
\langle3_{3}, 5_{1}, 2_{4}, 0_{8}\rangle &
\langle4_{0}, 0_{1}, 5_{8}, 1_{7}\rangle &
\langle1_{0}, 4_{2}, 4_{7}, 5_{8}\rangle \\
\langle3_{1}, 1_{1}, 2_{8}, 6_{7}\rangle &
\langle5_{1}, 3_{2}, 5_{5}, 6_{9}\rangle &
\langle0_{1}, 6_{2}, 6_{4}, 6_{9}\rangle &
\end{array}$$ ]{}
$n=53$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{1}, 1_{7}, 4_{6}\rangle &
\langle\infty, 0_{3}, 1_{9}, 0_{8}\rangle &
\langle\infty, 0_{0}, 0_{4}, 3_{5}\rangle &
\langle3_{2}, 3_{3}, 3_{5}, a_{1}\rangle &
\langle2_{1}, 4_{0}, 0_{6}, a_{1}\rangle &
\langle6_{2}, 0_{3}, 3_{5}, a_{2}\rangle &
\langle5_{1}, 4_{0}, 0_{4}, a_{2}\rangle &
\langle6_{2}, 1_{3}, 2_{6}, a_{3}\rangle \\
\langle4_{1}, 5_{0}, 5_{5}, a_{3}\rangle &
\langle4_{1}, 1_{0}, 5_{4}, a_{4}\rangle &
\langle6_{3}, 1_{2}, 1_{6}, a_{4}\rangle &
\langle1_{3}, 5_{2}, 5_{4}, a_{5}\rangle &
\langle0_{0}, 4_{1}, 1_{6}, a_{5}\rangle &
\langle3_{0}, 2_{2}, 0_{7}, a_{6}\rangle &
\langle2_{1}, 0_{3}, 1_{5}, a_{6}\rangle &
\langle5_{0}, 5_{1}, 0_{5}, a_{7}\rangle \\
\langle0_{3}, 1_{2}, 2_{7}, a_{7}\rangle &
\langle1_{2}, 1_{0}, 6_{4}, a_{8}\rangle &
\langle4_{3}, 1_{1}, 2_{7}, a_{8}\rangle &
\langle5_{3}, 6_{1}, 1_{6}, a_{9}\rangle &
\langle3_{0}, 1_{2}, 4_{7}, a_{9}\rangle &
\langle2_{2}, 1_{0}, 6_{7}, a_{10}\rangle &
\langle5_{3}, 3_{1}, 3_{4}, a_{10}\rangle &
\langle1_{0}, 4_{2}, 3_{6}, a_{11}\rangle \\
\langle6_{1}, 6_{3}, 5_{7}, a_{11}\rangle &
\langle0_{0}, 6_{3}, 3_{9}, 0_{8}\rangle &
\langle2_{3}, 5_{1}, 0_{9}, 3_{4}\rangle &
\langle0_{0}, 2_{2}, 6_{4}, 0_{6}\rangle &
\langle2_{0}, 6_{0}, 1_{7}, 0_{9}\rangle &
\langle1_{1}, 6_{2}, 1_{9}, 5_{4}\rangle &
\langle3_{2}, 1_{1}, 1_{5}, 5_{7}\rangle &
\langle1_{0}, 1_{3}, 0_{8}, 6_{6}\rangle \\
\langle6_{3}, 3_{3}, 3_{7}, 3_{6}\rangle &
\langle0_{1}, 1_{1}, 0_{6}, 4_{9}\rangle &
\langle4_{2}, 1_{1}, 2_{8}, 3_{8}\rangle &
\langle2_{3}, 0_{3}, 6_{5}, 2_{9}\rangle &
\langle2_{3}, 3_{3}, 2_{4}, 6_{8}\rangle &
\langle3_{2}, 6_{0}, 2_{7}, 4_{5}\rangle &
\langle0_{3}, 2_{0}, 6_{6}, 6_{9}\rangle &
\langle1_{2}, 5_{2}, 1_{9}, 2_{6}\rangle \\
\langle4_{2}, 6_{2}, 5_{9}, 0_{4}\rangle &
\langle5_{1}, 2_{2}, 3_{8}, 4_{4}\rangle &
\langle0_{2}, 1_{2}, 3_{5}, 4_{8}\rangle &
\langle4_{3}, 0_{2}, 2_{8}, 0_{7}\rangle &
\langle0_{3}, 4_{0}, 3_{9}, 5_{5}\rangle &
\langle1_{0}, 6_{0}, 1_{9}, 4_{8}\rangle &
\langle4_{2}, 5_{1}, 6_{6}, 3_{5}\rangle &
\langle0_{0}, 1_{0}, 2_{4}, 2_{8}\rangle \\
\langle1_{1}, 6_{0}, 5_{5}, 6_{7}\rangle &
\langle2_{1}, 0_{1}, 1_{9}, 6_{8}\rangle &
\langle1_{1}, 1_{2}, 1_{8}, 6_{9}\rangle &
\langle6_{3}, 4_{0}, 1_{5}, 1_{8}\rangle &
\langle0_{1}, 1_{3}, 3_{4}, 2_{7}\rangle &
\end{array}$$ ]{}
$n=55$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{3}, 0_{9}, 2_{6}\rangle &
\langle\infty, 0_{2}, 1_{5}, 4_{7}\rangle &
\langle\infty, 3_{0}, 3_{8}, 0_{4}\rangle &
\langle4_{1}, 6_{3}, 2_{7}, a_{1}\rangle &
\langle3_{0}, 0_{2}, 6_{6}, a_{1}\rangle &
\langle1_{0}, 2_{2}, 1_{4}, a_{2}\rangle &
\langle4_{1}, 5_{3}, 2_{5}, a_{2}\rangle &
\langle5_{1}, 1_{2}, 3_{4}, a_{3}\rangle \\
\langle4_{3}, 2_{0}, 3_{6}, a_{3}\rangle &
\langle4_{2}, 0_{3}, 5_{6}, a_{4}\rangle &
\langle0_{0}, 2_{1}, 3_{7}, a_{4}\rangle &
\langle1_{3}, 1_{2}, 1_{4}, a_{5}\rangle &
\langle0_{0}, 3_{1}, 4_{5}, a_{5}\rangle &
\langle2_{3}, 1_{0}, 4_{5}, a_{6}\rangle &
\langle1_{2}, 2_{1}, 5_{6}, a_{6}\rangle &
\langle3_{1}, 3_{2}, 5_{7}, a_{7}\rangle \\
\langle0_{0}, 3_{3}, 6_{4}, a_{7}\rangle &
\langle3_{3}, 0_{1}, 0_{7}, a_{8}\rangle &
\langle3_{0}, 2_{2}, 2_{5}, a_{8}\rangle &
\langle2_{3}, 4_{2}, 2_{7}, a_{9}\rangle &
\langle3_{1}, 5_{0}, 6_{4}, a_{9}\rangle &
\langle2_{1}, 0_{2}, 2_{5}, a_{10}\rangle &
\langle1_{0}, 1_{3}, 3_{7}, a_{10}\rangle &
\langle6_{2}, 4_{0}, 2_{6}, a_{11}\rangle \\
\langle1_{3}, 1_{1}, 0_{4}, a_{11}\rangle &
\langle2_{1}, 6_{2}, 6_{6}, a_{12}\rangle &
\langle5_{0}, 4_{3}, 3_{5}, a_{12}\rangle &
\langle1_{3}, 4_{0}, 3_{6}, a_{13}\rangle &
\langle0_{1}, 2_{2}, 3_{7}, a_{13}\rangle &
\langle2_{0}, 1_{1}, 3_{8}, 0_{7}\rangle &
\langle1_{0}, 6_{3}, 0_{7}, 4_{8}\rangle &
\langle4_{2}, 1_{3}, 0_{7}, 2_{4}\rangle \\
\langle0_{2}, 6_{1}, 5_{5}, 5_{9}\rangle &
\langle5_{2}, 1_{2}, 2_{8}, 0_{8}\rangle &
\langle6_{2}, 5_{3}, 2_{9}, 5_{5}\rangle &
\langle2_{2}, 4_{3}, 2_{7}, 6_{9}\rangle &
\langle2_{3}, 6_{3}, 3_{8}, 0_{9}\rangle &
\langle2_{1}, 0_{3}, 0_{6}, 3_{8}\rangle &
\langle1_{3}, 2_{1}, 4_{6}, 1_{8}\rangle &
\langle6_{1}, 5_{0}, 4_{8}, 5_{6}\rangle \\
\langle1_{0}, 1_{2}, 2_{9}, 3_{6}\rangle &
\langle3_{0}, 6_{0}, 3_{7}, 1_{8}\rangle &
\langle0_{0}, 6_{0}, 5_{9}, 3_{9}\rangle &
\langle1_{1}, 3_{1}, 3_{4}, 6_{9}\rangle &
\langle1_{2}, 0_{2}, 0_{9}, 4_{5}\rangle &
\langle0_{1}, 6_{1}, 0_{6}, 0_{9}\rangle &
\langle1_{1}, 1_{0}, 3_{9}, 5_{8}\rangle &
\langle0_{0}, 2_{0}, 2_{5}, 5_{4}\rangle \\
\langle5_{3}, 4_{2}, 0_{8}, 2_{6}\rangle &
\langle2_{1}, 6_{1}, 3_{4}, 2_{8}\rangle &
\langle1_{1}, 5_{3}, 3_{5}, 5_{9}\rangle &
\langle3_{0}, 1_{2}, 5_{4}, 3_{9}\rangle &
\langle2_{1}, 5_{0}, 6_{5}, 6_{7}\rangle &
\langle0_{2}, 2_{2}, 3_{4}, 0_{8}\rangle &
\langle0_{3}, 2_{3}, 4_{4}, 3_{5}\rangle &
\end{array}$$ ]{}
$n=57$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{0}, 6_{11}, 2_{5}\rangle &
\langle\infty, 0_{2}, 2_{7}, 1_{4}\rangle &
\langle\infty, 5_{1}, 2_{10}, 0_{9}\rangle &
\langle\infty, 4_{3}, 5_{6}, 2_{8}\rangle &
\langle1_{3}, 6_{2}, 3_{4}, a_{1}\rangle &
\langle0_{1}, 2_{0}, 6_{7}, a_{1}\rangle &
\langle0_{2}, 4_{2}, 6_{6}, 3_{11}\rangle &
\langle0_{1}, 6_{2}, 1_{5}, 5_{9}\rangle \\
\langle3_{0}, 4_{0}, 2_{10}, 5_{9}\rangle &
\langle0_{1}, 3_{3}, 3_{5}, 6_{9}\rangle &
\langle1_{2}, 1_{3}, 2_{9}, 1_{7}\rangle &
\langle0_{1}, 2_{2}, 0_{5}, 2_{5}\rangle &
\langle2_{2}, 4_{2}, 5_{7}, 0_{10}\rangle &
\langle2_{3}, 1_{2}, 2_{10}, 4_{9}\rangle &
\langle0_{3}, 1_{3}, 6_{5}, 3_{5}\rangle &
\langle1_{3}, 4_{2}, 3_{8}, 1_{9}\rangle \\
\langle1_{0}, 0_{1}, 4_{4}, 2_{10}\rangle &
\langle4_{0}, 6_{1}, 1_{4}, 3_{9}\rangle &
\langle4_{2}, 5_{0}, 1_{8}, 3_{5}\rangle &
\langle6_{0}, 1_{0}, 6_{9}, 4_{6}\rangle &
\langle1_{0}, 3_{3}, 3_{8}, 0_{6}\rangle &
\langle6_{0}, 2_{3}, 0_{4}, 1_{10}\rangle &
\langle1_{2}, 6_{3}, 0_{10}, 5_{11}\rangle &
\langle0_{3}, 3_{1}, 0_{11}, 4_{4}\rangle \\
\langle1_{1}, 2_{2}, 1_{7}, 4_{9}\rangle &
\langle2_{0}, 3_{1}, 0_{8}, 1_{5}\rangle &
\langle5_{3}, 0_{1}, 6_{8}, 3_{7}\rangle &
\langle4_{2}, 0_{3}, 5_{6}, 4_{9}\rangle &
\langle4_{2}, 3_{2}, 0_{5}, 5_{11}\rangle &
\langle2_{1}, 2_{3}, 6_{7}, 6_{5}\rangle &
\langle3_{1}, 0_{2}, 3_{4}, 5_{11}\rangle &
\langle4_{0}, 0_{1}, 6_{5}, 1_{6}\rangle \\
\langle6_{0}, 6_{3}, 3_{10}, 5_{8}\rangle &
\langle3_{3}, 2_{0}, 5_{10}, 5_{11}\rangle &
\langle2_{3}, 5_{3}, 4_{6}, 1_{7}\rangle &
\langle0_{1}, 6_{1}, 6_{10}, 1_{7}\rangle &
\langle0_{3}, 2_{3}, 3_{11}, 3_{4}\rangle &
\langle1_{2}, 3_{1}, 3_{8}, 6_{4}\rangle &
\langle4_{1}, 6_{1}, 3_{6}, 2_{10}\rangle &
\langle3_{0}, 0_{2}, 0_{8}, 5_{6}\rangle \\
\langle2_{2}, 2_{0}, 2_{10}, 2_{4}\rangle &
\langle0_{1}, 2_{3}, 2_{6}, 5_{6}\rangle &
\langle0_{0}, 2_{2}, 0_{8}, 3_{5}\rangle &
\langle2_{0}, 3_{2}, 3_{6}, 3_{11}\rangle &
\langle3_{0}, 0_{0}, 4_{5}, 2_{4}\rangle &
\langle1_{0}, 5_{3}, 0_{7}, 3_{11}\rangle &
\langle2_{1}, 1_{3}, 5_{8}, 0_{4}\rangle &
\langle3_{0}, 6_{2}, 4_{7}, 1_{4}\rangle \\
\langle6_{2}, 5_{3}, 1_{10}, 3_{10}\rangle &
\langle3_{1}, 3_{2}, 4_{8}, 2_{4}\rangle &
\langle3_{0}, 2_{3}, 3_{7}, 0_{9}\rangle &
\langle0_{1}, 3_{1}, 6_{11}, 5_{8}\rangle &
\langle0_{1}, 3_{2}, 0_{6}, 1_{9}\rangle &
\langle5_{1}, 6_{3}, 5_{9}, 3_{11}\rangle &
\langle4_{0}, 1_{1}, 4_{6}, 2_{11}\rangle &
\langle0_{0}, 0_{1}, 5_{7}, 0_{11}\rangle \\
\langle0_{0}, 5_{2}, 2_{7}, 1_{8}\rangle &
\end{array}$$ ]{}
T$(2,33;2,n;6) =8n$ for each odd $n$ and $33 \leq n \leq 65$.
Let $X_1=({{\mathbb{Z}}}_8\times \{0,1,2,3\})\cup \{\infty\}$. For $33 \leq n \leq 47$, let $X_2= ({{\mathbb{Z}}}_8\times \{4,5,6,7\})\cup (\{a\}\times \{1,\ldots,n-32\})$; for $49 \leq n \leq 63$, let $X_2= ({{\mathbb{Z}}}_8\times \{4,5,\ldots,9\})\cup (\{a\}\times \{1,\ldots,n-48\})$; for $n = 65$, let $X_2= ({{\mathbb{Z}}}_8\times \{4,5,\ldots,11\})\cup (\{a\}\times \{1\})$. Denote $X=X_1\cup X_2$. The desired codes of size $8n$ are constructed on ${{\mathbb{Z}}}_2^X$ and the base codewords are listed as follows.
$n=33$: [$$\begin{array}{llllllllll}
\langle\infty, 7_{1}, 2_{6}, 4_{7}\rangle &
\langle\infty, 0_{0}, 5_{5}, 0_{4}\rangle &
\langle1_{0}, 6_{1}, 3_{5}, a_{1}\rangle &
\langle4_{2}, 6_{3}, 3_{4}, a_{1}\rangle &
\langle3_{3}, 1_{0}, 2_{4}, 1_{7}\rangle &
\langle4_{2}, 0_{0}, 7_{4}, 0_{5}\rangle &
\langle3_{3}, 6_{3}, 4_{6}, 3_{7}\rangle &
\langle7_{1}, 6_{2}, 6_{6}, 5_{6}\rangle \\
\langle1_{2}, 7_{0}, 4_{7}, 2_{4}\rangle &
\langle0_{0}, 6_{0}, 4_{6}, 4_{7}\rangle &
\langle1_{3}, 5_{1}, 4_{5}, 5_{7}\rangle &
\langle4_{2}, 5_{2}, 6_{7}, 2_{5}\rangle &
\langle5_{3}, 3_{3}, 2_{5}, 6_{4}\rangle &
\langle3_{1}, 2_{1}, 3_{5}, 4_{4}\rangle &
\langle7_{0}, 2_{2}, 6_{6}, 1_{7}\rangle &
\langle7_{1}, 1_{1}, 5_{7}, 7_{4}\rangle \\
\langle3_{3}, 0_{2}, 7_{5}, 2_{6}\rangle &
\langle2_{0}, 3_{1}, 4_{6}, 5_{7}\rangle &
\langle4_{2}, 2_{3}, 2_{6}, 4_{5}\rangle &
\langle6_{1}, 1_{1}, 6_{6}, 4_{5}\rangle &
\langle6_{2}, 6_{1}, 2_{4}, 0_{5}\rangle &
\langle4_{0}, 7_{0}, 3_{5}, 5_{5}\rangle &
\langle4_{3}, 2_{1}, 6_{6}, 3_{7}\rangle &
\langle2_{2}, 7_{1}, 6_{7}, 3_{5}\rangle \\
\langle7_{0}, 6_{1}, 5_{4}, 0_{6}\rangle &
\langle7_{0}, 2_{3}, 7_{6}, 2_{5}\rangle &
\langle3_{0}, 3_{2}, 0_{4}, 0_{6}\rangle &
\langle0_{3}, 3_{2}, 3_{7}, 6_{5}\rangle &
\langle3_{3}, 7_{2}, 4_{7}, 7_{4}\rangle &
\langle0_{3}, 1_{0}, 2_{7}, 4_{6}\rangle &
\langle7_{2}, 7_{3}, 2_{6}, 1_{4}\rangle &
\langle0_{0}, 4_{3}, 2_{4}, 4_{4}\rangle \\
\langle0_{1}, 5_{2}, 3_{4}, 3_{7}\rangle &
\end{array}$$ ]{}
$n=35$: [$$\begin{array}{llllllllll}
\langle\infty, 4_{1}, 4_{6}, 4_{7}\rangle &
\langle\infty, 7_{0}, 7_{4}, 7_{5}\rangle &
\langle4_{3}, 2_{2}, 7_{5}, a_{1}\rangle &
\langle7_{1}, 1_{0}, 0_{4}, a_{1}\rangle &
\langle6_{2}, 4_{3}, 1_{5}, a_{2}\rangle &
\langle4_{0}, 5_{1}, 7_{4}, a_{2}\rangle &
\langle5_{2}, 5_{3}, 0_{7}, a_{3}\rangle &
\langle2_{1}, 5_{0}, 2_{4}, a_{3}\rangle \\
\langle3_{0}, 7_{3}, 2_{6}, 7_{6}\rangle &
\langle6_{2}, 5_{3}, 4_{7}, 5_{5}\rangle &
\langle4_{1}, 3_{2}, 7_{6}, 5_{6}\rangle &
\langle5_{1}, 7_{3}, 5_{5}, 1_{7}\rangle &
\langle1_{3}, 0_{3}, 6_{7}, 0_{4}\rangle &
\langle4_{1}, 7_{3}, 3_{7}, 1_{6}\rangle &
\langle7_{0}, 7_{3}, 4_{6}, 5_{6}\rangle &
\langle4_{2}, 0_{3}, 2_{5}, 1_{6}\rangle \\
\langle3_{3}, 6_{3}, 7_{4}, 2_{6}\rangle &
\langle5_{0}, 0_{2}, 5_{7}, 7_{6}\rangle &
\langle5_{1}, 6_{3}, 6_{7}, 3_{4}\rangle &
\langle1_{2}, 0_{2}, 1_{5}, 1_{6}\rangle &
\langle2_{1}, 1_{3}, 5_{5}, 7_{4}\rangle &
\langle1_{0}, 7_{0}, 3_{5}, 0_{5}\rangle &
\langle2_{2}, 0_{2}, 1_{4}, 2_{4}\rangle &
\langle2_{1}, 3_{1}, 0_{5}, 6_{4}\rangle \\
\langle7_{0}, 1_{1}, 3_{7}, 6_{7}\rangle &
\langle1_{2}, 4_{2}, 7_{6}, 0_{7}\rangle &
\langle3_{2}, 6_{3}, 5_{5}, 1_{4}\rangle &
\langle4_{0}, 1_{2}, 1_{7}, 2_{7}\rangle &
\langle4_{3}, 7_{1}, 6_{4}, 5_{7}\rangle &
\langle4_{0}, 7_{1}, 1_{5}, 5_{6}\rangle &
\langle3_{0}, 4_{2}, 7_{4}, 1_{4}\rangle &
\langle5_{1}, 0_{1}, 1_{5}, 7_{6}\rangle \\
\langle4_{0}, 5_{0}, 7_{7}, 6_{4}\rangle &
\langle1_{0}, 0_{1}, 7_{5}, 4_{6}\rangle &
\langle0_{0}, 7_{2}, 3_{5}, 1_{7}\rangle &
\end{array}$$ ]{}
$n=37$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{3}, 2_{6}, 7_{7}\rangle &
\langle\infty, 0_{0}, 0_{4}, 5_{5}\rangle &
\langle2_{0}, 7_{1}, 3_{6}, a_{1}\rangle &
\langle3_{2}, 1_{3}, 5_{5}, a_{1}\rangle &
\langle6_{2}, 1_{3}, 2_{5}, a_{2}\rangle &
\langle1_{1}, 3_{0}, 2_{6}, a_{2}\rangle &
\langle7_{1}, 7_{0}, 4_{6}, a_{3}\rangle &
\langle6_{3}, 2_{2}, 5_{7}, a_{3}\rangle \\
\langle1_{2}, 2_{3}, 0_{5}, a_{4}\rangle &
\langle0_{0}, 2_{1}, 5_{4}, a_{4}\rangle &
\langle7_{3}, 5_{2}, 3_{4}, a_{5}\rangle &
\langle2_{1}, 6_{0}, 7_{5}, a_{5}\rangle &
\langle0_{1}, 5_{1}, 0_{6}, 7_{6}\rangle &
\langle2_{3}, 7_{1}, 4_{7}, 7_{4}\rangle &
\langle7_{2}, 3_{0}, 2_{4}, 2_{5}\rangle &
\langle7_{3}, 2_{3}, 7_{5}, 5_{4}\rangle \\
\langle7_{0}, 1_{0}, 5_{5}, 1_{7}\rangle &
\langle6_{3}, 3_{0}, 5_{4}, 7_{4}\rangle &
\langle3_{2}, 5_{2}, 7_{4}, 0_{6}\rangle &
\langle5_{3}, 3_{3}, 6_{6}, 3_{6}\rangle &
\langle6_{3}, 0_{0}, 2_{6}, 6_{7}\rangle &
\langle2_{2}, 1_{0}, 4_{6}, 4_{7}\rangle &
\langle7_{1}, 6_{0}, 1_{4}, 7_{7}\rangle &
\langle4_{1}, 6_{2}, 3_{5}, 3_{7}\rangle \\
\langle4_{2}, 6_{1}, 2_{5}, 4_{5}\rangle &
\langle0_{1}, 1_{3}, 0_{5}, 6_{6}\rangle &
\langle0_{0}, 3_{2}, 7_{7}, 4_{7}\rangle &
\langle7_{3}, 1_{1}, 4_{7}, 5_{7}\rangle &
\langle3_{2}, 6_{0}, 4_{4}, 2_{6}\rangle &
\langle5_{3}, 6_{0}, 0_{5}, 7_{4}\rangle &
\langle2_{2}, 3_{0}, 3_{5}, 0_{7}\rangle &
\langle5_{1}, 6_{2}, 6_{4}, 6_{7}\rangle \\
\langle0_{3}, 1_{1}, 3_{7}, 2_{5}\rangle &
\langle1_{2}, 7_{0}, 5_{6}, 7_{6}\rangle &
\langle6_{3}, 7_{2}, 6_{4}, 0_{6}\rangle &
\langle5_{1}, 4_{2}, 1_{4}, 3_{7}\rangle &
\langle0_{1}, 1_{1}, 7_{4}, 3_{5}\rangle &
\end{array}$$ ]{}
$n=39$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{1}, 1_{7}, 1_{6}\rangle &
\langle\infty, 0_{0}, 0_{5}, 0_{4}\rangle &
\langle2_{0}, 2_{1}, 3_{4}, a_{1}\rangle &
\langle2_{2}, 2_{3}, 6_{6}, a_{1}\rangle &
\langle2_{0}, 3_{1}, 5_{4}, a_{2}\rangle &
\langle3_{2}, 4_{3}, 5_{6}, a_{2}\rangle &
\langle0_{2}, 2_{3}, 7_{6}, a_{3}\rangle &
\langle3_{0}, 5_{1}, 5_{4}, a_{3}\rangle \\
\langle3_{0}, 6_{1}, 1_{4}, a_{4}\rangle &
\langle1_{2}, 4_{3}, 4_{5}, a_{4}\rangle &
\langle1_{0}, 5_{1}, 2_{5}, a_{5}\rangle &
\langle4_{2}, 0_{3}, 4_{4}, a_{5}\rangle &
\langle2_{2}, 7_{3}, 3_{5}, a_{6}\rangle &
\langle3_{0}, 0_{1}, 7_{7}, a_{6}\rangle &
\langle0_{2}, 6_{3}, 1_{4}, a_{7}\rangle &
\langle4_{0}, 2_{1}, 6_{5}, a_{7}\rangle \\
\langle1_{0}, 3_{2}, 7_{7}, 2_{7}\rangle &
\langle1_{0}, 1_{2}, 3_{7}, 4_{7}\rangle &
\langle0_{0}, 1_{0}, 6_{5}, 4_{5}\rangle &
\langle0_{3}, 3_{3}, 6_{7}, 1_{5}\rangle &
\langle0_{0}, 5_{0}, 1_{6}, 7_{6}\rangle &
\langle1_{1}, 7_{2}, 4_{6}, 5_{7}\rangle &
\langle2_{0}, 0_{3}, 2_{7}, 7_{6}\rangle &
\langle0_{2}, 7_{3}, 7_{4}, 0_{7}\rangle \\
\langle1_{0}, 1_{3}, 0_{7}, 6_{7}\rangle &
\langle0_{3}, 7_{3}, 1_{4}, 6_{4}\rangle &
\langle0_{1}, 4_{2}, 7_{6}, 5_{7}\rangle &
\langle0_{1}, 0_{2}, 7_{5}, 6_{4}\rangle &
\langle0_{1}, 5_{2}, 2_{7}, 6_{6}\rangle &
\langle0_{1}, 5_{1}, 3_{5}, 2_{6}\rangle &
\langle2_{0}, 1_{1}, 5_{6}, 1_{5}\rangle &
\langle0_{2}, 2_{2}, 6_{5}, 0_{6}\rangle \\
\langle0_{1}, 2_{3}, 6_{7}, 7_{4}\rangle &
\langle1_{0}, 3_{0}, 0_{4}, 1_{6}\rangle &
\langle0_{3}, 2_{3}, 5_{5}, 0_{6}\rangle &
\langle0_{1}, 7_{3}, 1_{5}, 1_{6}\rangle &
\langle0_{1}, 3_{3}, 2_{5}, 3_{7}\rangle &
\langle0_{2}, 3_{2}, 5_{4}, 0_{5}\rangle &
\langle0_{1}, 1_{2}, 4_{4}, 5_{4}\rangle &
\end{array}$$ ]{}
$n=41$: [$$\begin{array}{llllllllll}
\langle\infty, 6_{1}, 6_{6}, 7_{7}\rangle &
\langle\infty, 4_{0}, 7_{5}, 4_{4}\rangle &
\langle7_{1}, 5_{0}, 2_{7}, a_{1}\rangle &
\langle6_{3}, 6_{2}, 2_{6}, a_{1}\rangle &
\langle3_{1}, 2_{0}, 5_{4}, a_{2}\rangle &
\langle5_{3}, 4_{2}, 6_{6}, a_{2}\rangle &
\langle0_{2}, 2_{3}, 6_{5}, a_{3}\rangle &
\langle0_{0}, 0_{1}, 1_{4}, a_{3}\rangle \\
\langle2_{1}, 7_{0}, 5_{4}, a_{4}\rangle &
\langle6_{3}, 3_{2}, 3_{7}, a_{4}\rangle &
\langle7_{3}, 3_{2}, 5_{4}, a_{5}\rangle &
\langle2_{1}, 6_{0}, 7_{5}, a_{5}\rangle &
\langle4_{1}, 7_{0}, 0_{7}, a_{6}\rangle &
\langle2_{2}, 7_{3}, 5_{5}, a_{6}\rangle &
\langle4_{3}, 6_{2}, 6_{6}, a_{7}\rangle &
\langle5_{1}, 7_{0}, 1_{5}, a_{7}\rangle \\
\langle0_{0}, 7_{1}, 5_{4}, a_{8}\rangle &
\langle5_{3}, 6_{2}, 6_{5}, a_{8}\rangle &
\langle6_{2}, 4_{0}, 2_{7}, a_{9}\rangle &
\langle3_{3}, 0_{1}, 1_{6}, a_{9}\rangle &
\langle6_{0}, 4_{3}, 4_{5}, 6_{7}\rangle &
\langle4_{3}, 2_{3}, 7_{6}, 3_{7}\rangle &
\langle4_{1}, 1_{3}, 4_{7}, 6_{5}\rangle &
\langle2_{0}, 5_{0}, 6_{6}, 4_{6}\rangle \\
\langle1_{1}, 4_{2}, 7_{7}, 5_{6}\rangle &
\langle5_{1}, 7_{2}, 4_{7}, 4_{5}\rangle &
\langle2_{1}, 6_{2}, 4_{6}, 1_{6}\rangle &
\langle7_{0}, 3_{2}, 3_{4}, 2_{6}\rangle &
\langle6_{1}, 4_{3}, 6_{4}, 5_{4}\rangle &
\langle1_{0}, 2_{0}, 7_{6}, 6_{5}\rangle &
\langle6_{0}, 3_{3}, 5_{5}, 1_{7}\rangle &
\langle6_{1}, 6_{3}, 1_{5}, 2_{4}\rangle \\
\langle5_{3}, 6_{3}, 5_{4}, 5_{6}\rangle &
\langle2_{2}, 0_{2}, 1_{5}, 4_{5}\rangle &
\langle1_{2}, 3_{0}, 7_{7}, 2_{7}\rangle &
\langle6_{1}, 6_{2}, 3_{4}, 0_{7}\rangle &
\langle3_{1}, 0_{1}, 6_{6}, 1_{5}\rangle &
\langle6_{1}, 7_{3}, 3_{7}, 6_{5}\rangle &
\langle3_{3}, 1_{0}, 3_{7}, 0_{4}\rangle &
\langle1_{2}, 6_{0}, 0_{4}, 6_{6}\rangle \\
\langle0_{2}, 3_{2}, 4_{4}, 6_{4}\rangle &
\end{array}$$ ]{}
$n=43$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{2}, 0_{5}, 5_{4}\rangle &
\langle\infty, 0_{1}, 0_{7}, 7_{6}\rangle &
\langle7_{1}, 6_{0}, 2_{5}, a_{1}\rangle &
\langle4_{3}, 1_{2}, 0_{4}, a_{1}\rangle &
\langle4_{2}, 5_{3}, 4_{7}, a_{2}\rangle &
\langle6_{0}, 6_{1}, 2_{4}, a_{2}\rangle &
\langle6_{2}, 0_{3}, 7_{6}, a_{3}\rangle &
\langle7_{1}, 3_{0}, 4_{5}, a_{3}\rangle \\
\langle4_{2}, 4_{3}, 4_{6}, a_{4}\rangle &
\langle6_{1}, 7_{0}, 6_{4}, a_{4}\rangle &
\langle5_{1}, 3_{0}, 2_{7}, a_{5}\rangle &
\langle6_{2}, 2_{3}, 2_{5}, a_{5}\rangle &
\langle4_{2}, 2_{3}, 6_{5}, a_{6}\rangle &
\langle5_{0}, 2_{1}, 0_{4}, a_{6}\rangle &
\langle7_{2}, 4_{3}, 1_{6}, a_{7}\rangle &
\langle3_{1}, 5_{0}, 7_{5}, a_{7}\rangle \\
\langle0_{0}, 0_{3}, 6_{4}, a_{8}\rangle &
\langle7_{1}, 3_{2}, 1_{6}, a_{8}\rangle &
\langle5_{2}, 5_{0}, 2_{4}, a_{9}\rangle &
\langle3_{3}, 2_{1}, 1_{7}, a_{9}\rangle &
\langle5_{2}, 4_{0}, 6_{4}, a_{10}\rangle &
\langle0_{1}, 6_{3}, 6_{7}, a_{10}\rangle &
\langle5_{3}, 1_{1}, 0_{5}, a_{11}\rangle &
\langle1_{0}, 6_{2}, 3_{6}, a_{11}\rangle \\
\langle2_{0}, 0_{0}, 0_{5}, 7_{5}\rangle &
\langle3_{3}, 1_{3}, 0_{5}, 4_{4}\rangle &
\langle3_{3}, 5_{0}, 1_{6}, 6_{7}\rangle &
\langle4_{1}, 4_{3}, 5_{6}, 0_{6}\rangle &
\langle4_{2}, 1_{0}, 0_{6}, 7_{6}\rangle &
\langle4_{2}, 2_{0}, 3_{6}, 6_{7}\rangle &
\langle1_{2}, 7_{2}, 1_{4}, 4_{7}\rangle &
\langle0_{1}, 3_{3}, 5_{4}, 3_{4}\rangle \\
\langle3_{1}, 0_{2}, 7_{7}, 5_{5}\rangle &
\langle0_{3}, 3_{1}, 3_{6}, 5_{7}\rangle &
\langle3_{0}, 6_{0}, 3_{7}, 3_{6}\rangle &
\langle0_{1}, 2_{3}, 7_{4}, 0_{5}\rangle &
\langle4_{0}, 6_{3}, 7_{5}, 7_{7}\rangle &
\langle4_{2}, 6_{0}, 7_{4}, 0_{7}\rangle &
\langle1_{2}, 1_{1}, 7_{5}, 2_{5}\rangle &
\langle0_{3}, 7_{0}, 7_{4}, 2_{6}\rangle \\
\langle5_{1}, 4_{1}, 2_{6}, 6_{4}\rangle &
\langle0_{3}, 1_{1}, 4_{7}, 2_{7}\rangle &
\langle0_{2}, 3_{2}, 3_{5}, 1_{7}\rangle &
\end{array}$$ ]{}
$n=45$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{0}, 3_{5}, 4_{4}\rangle &
\langle\infty, 6_{1}, 1_{7}, 5_{6}\rangle &
\langle6_{1}, 5_{2}, 2_{6}, a_{1}\rangle &
\langle6_{0}, 3_{3}, 6_{4}, a_{1}\rangle &
\langle2_{3}, 0_{2}, 1_{5}, a_{2}\rangle &
\langle0_{1}, 2_{0}, 5_{7}, a_{2}\rangle &
\langle7_{0}, 7_{1}, 7_{7}, a_{3}\rangle &
\langle4_{3}, 0_{2}, 7_{6}, a_{3}\rangle \\
\langle2_{0}, 5_{1}, 0_{4}, a_{4}\rangle &
\langle5_{3}, 4_{2}, 3_{7}, a_{4}\rangle &
\langle3_{3}, 0_{2}, 4_{5}, a_{5}\rangle &
\langle0_{1}, 3_{0}, 4_{4}, a_{5}\rangle &
\langle7_{3}, 0_{2}, 4_{7}, a_{6}\rangle &
\langle7_{1}, 0_{0}, 0_{6}, a_{6}\rangle &
\langle3_{3}, 3_{2}, 4_{4}, a_{7}\rangle &
\langle2_{0}, 3_{1}, 7_{5}, a_{7}\rangle \\
\langle3_{2}, 1_{3}, 7_{6}, a_{8}\rangle &
\langle2_{0}, 6_{1}, 5_{4}, a_{8}\rangle &
\langle5_{3}, 5_{1}, 0_{5}, a_{9}\rangle &
\langle7_{2}, 6_{0}, 1_{6}, a_{9}\rangle &
\langle0_{1}, 1_{3}, 2_{7}, a_{10}\rangle &
\langle0_{2}, 4_{0}, 0_{5}, a_{10}\rangle &
\langle4_{3}, 0_{1}, 5_{6}, a_{11}\rangle &
\langle0_{0}, 2_{2}, 2_{4}, a_{11}\rangle \\
\langle6_{1}, 0_{3}, 4_{4}, a_{12}\rangle &
\langle6_{2}, 7_{0}, 3_{7}, a_{12}\rangle &
\langle5_{1}, 3_{3}, 5_{6}, a_{13}\rangle &
\langle3_{2}, 6_{0}, 3_{7}, a_{13}\rangle &
\langle2_{3}, 7_{1}, 0_{5}, 4_{4}\rangle &
\langle3_{3}, 1_{3}, 5_{7}, 3_{5}\rangle &
\langle3_{0}, 5_{1}, 2_{5}, 1_{7}\rangle &
\langle0_{2}, 3_{1}, 5_{5}, 3_{5}\rangle \\
\langle0_{0}, 2_{3}, 6_{5}, 2_{6}\rangle &
\langle7_{0}, 0_{0}, 4_{6}, 1_{7}\rangle &
\langle4_{1}, 0_{2}, 6_{4}, 5_{4}\rangle &
\langle0_{0}, 0_{3}, 5_{4}, 7_{6}\rangle &
\langle4_{3}, 3_{0}, 2_{4}, 1_{6}\rangle &
\langle6_{2}, 0_{0}, 0_{5}, 1_{6}\rangle &
\langle1_{1}, 1_{2}, 7_{6}, 2_{7}\rangle &
\langle4_{1}, 6_{2}, 7_{6}, 6_{6}\rangle \\
\langle4_{3}, 0_{0}, 7_{7}, 1_{5}\rangle &
\langle7_{3}, 6_{3}, 6_{7}, 6_{4}\rangle &
\langle6_{1}, 5_{1}, 4_{7}, 4_{5}\rangle &
\langle3_{2}, 6_{2}, 1_{7}, 5_{4}\rangle &
\langle0_{2}, 1_{2}, 4_{4}, 7_{5}\rangle &
\end{array}$$ ]{}
$n=47$: [$$\begin{array}{llllllllll}
\langle\infty, 5_{1}, 3_{6}, 3_{7}\rangle &
\langle\infty, 5_{3}, 5_{5}, 1_{4}\rangle &
\langle7_{0}, 7_{1}, 4_{4}, a_{1}\rangle &
\langle3_{3}, 5_{2}, 7_{5}, a_{1}\rangle &
\langle6_{3}, 6_{0}, 4_{5}, a_{2}\rangle &
\langle3_{2}, 6_{1}, 3_{7}, a_{2}\rangle &
\langle4_{3}, 3_{2}, 2_{7}, a_{3}\rangle &
\langle0_{0}, 3_{1}, 1_{4}, a_{3}\rangle \\
\langle6_{3}, 1_{2}, 6_{7}, a_{4}\rangle &
\langle7_{1}, 5_{0}, 3_{4}, a_{4}\rangle &
\langle2_{3}, 6_{2}, 1_{6}, a_{5}\rangle &
\langle7_{1}, 1_{0}, 4_{5}, a_{5}\rangle &
\langle0_{2}, 3_{3}, 6_{7}, a_{6}\rangle &
\langle7_{0}, 6_{1}, 7_{6}, a_{6}\rangle &
\langle3_{3}, 1_{2}, 5_{4}, a_{7}\rangle &
\langle4_{0}, 1_{1}, 2_{7}, a_{7}\rangle \\
\langle1_{0}, 2_{1}, 1_{7}, a_{8}\rangle &
\langle3_{2}, 2_{3}, 3_{5}, a_{8}\rangle &
\langle4_{2}, 1_{0}, 4_{6}, a_{9}\rangle &
\langle6_{1}, 6_{3}, 5_{5}, a_{9}\rangle &
\langle6_{2}, 5_{0}, 7_{4}, a_{10}\rangle &
\langle5_{3}, 3_{1}, 7_{7}, a_{10}\rangle &
\langle4_{2}, 0_{0}, 3_{4}, a_{11}\rangle &
\langle6_{1}, 5_{3}, 0_{5}, a_{11}\rangle \\
\langle6_{0}, 3_{2}, 5_{4}, a_{12}\rangle &
\langle4_{1}, 7_{3}, 1_{6}, a_{12}\rangle &
\langle7_{0}, 6_{2}, 0_{7}, a_{13}\rangle &
\langle7_{3}, 2_{1}, 4_{6}, a_{13}\rangle &
\langle3_{1}, 4_{3}, 4_{4}, a_{14}\rangle &
\langle1_{0}, 1_{2}, 7_{6}, a_{14}\rangle &
\langle6_{0}, 4_{2}, 3_{5}, a_{15}\rangle &
\langle1_{3}, 5_{1}, 4_{6}, a_{15}\rangle \\
\langle5_{0}, 0_{0}, 1_{6}, 7_{6}\rangle &
\langle0_{0}, 7_{3}, 0_{4}, 3_{7}\rangle &
\langle6_{2}, 1_{2}, 5_{6}, 4_{4}\rangle &
\langle4_{3}, 2_{3}, 3_{7}, 1_{4}\rangle &
\langle7_{0}, 0_{0}, 4_{7}, 0_{5}\rangle &
\langle6_{2}, 5_{2}, 7_{6}, 2_{5}\rangle &
\langle0_{3}, 3_{3}, 1_{6}, 6_{4}\rangle &
\langle1_{0}, 3_{3}, 5_{5}, 0_{7}\rangle \\
\langle0_{1}, 5_{1}, 1_{5}, 0_{6}\rangle &
\langle6_{1}, 0_{1}, 0_{7}, 6_{5}\rangle &
\langle7_{2}, 1_{2}, 2_{7}, 2_{5}\rangle &
\langle7_{2}, 5_{1}, 4_{4}, 7_{4}\rangle &
\langle5_{0}, 2_{3}, 2_{6}, 7_{5}\rangle &
\langle3_{2}, 4_{1}, 0_{6}, 7_{7}\rangle &
\langle0_{0}, 4_{1}, 4_{4}, 7_{5}\rangle &
\end{array}$$ ]{}
$n=49$: [$$\begin{array}{llllllllll}
\langle\infty, 7_{0}, 3_{5}, 1_{4}\rangle &
\langle\infty, 3_{1}, 4_{7}, 0_{6}\rangle &
\langle\infty, 6_{2}, 1_{8}, 6_{9}\rangle &
\langle6_{3}, 4_{2}, 7_{7}, a_{1}\rangle &
\langle3_{0}, 6_{1}, 2_{4}, a_{1}\rangle &
\langle0_{0}, 0_{2}, 5_{5}, 2_{5}\rangle &
\langle0_{2}, 7_{3}, 7_{7}, 5_{4}\rangle &
\langle1_{3}, 1_{0}, 5_{8}, 5_{7}\rangle \\
\langle5_{0}, 1_{3}, 1_{6}, 6_{5}\rangle &
\langle4_{3}, 0_{1}, 4_{5}, 0_{9}\rangle &
\langle2_{2}, 2_{3}, 6_{6}, 5_{9}\rangle &
\langle6_{0}, 1_{3}, 6_{6}, 4_{4}\rangle &
\langle0_{0}, 5_{2}, 2_{6}, 3_{6}\rangle &
\langle1_{0}, 2_{0}, 7_{9}, 5_{9}\rangle &
\langle2_{0}, 1_{3}, 0_{7}, 3_{9}\rangle &
\langle2_{0}, 6_{2}, 2_{7}, 5_{5}\rangle \\
\langle5_{3}, 4_{0}, 6_{8}, 7_{7}\rangle &
\langle5_{1}, 2_{1}, 1_{6}, 3_{5}\rangle &
\langle2_{3}, 3_{3}, 2_{8}, 2_{4}\rangle &
\langle6_{3}, 5_{1}, 3_{8}, 1_{7}\rangle &
\langle3_{1}, 1_{3}, 5_{5}, 3_{5}\rangle &
\langle0_{2}, 7_{2}, 1_{9}, 5_{7}\rangle &
\langle6_{0}, 0_{1}, 3_{6}, 4_{8}\rangle &
\langle3_{0}, 1_{0}, 0_{8}, 1_{5}\rangle \\
\langle7_{1}, 1_{0}, 4_{8}, 2_{8}\rangle &
\langle4_{1}, 2_{2}, 5_{4}, 3_{8}\rangle &
\langle5_{3}, 2_{2}, 3_{7}, 3_{6}\rangle &
\langle0_{0}, 7_{2}, 7_{6}, 1_{6}\rangle &
\langle4_{1}, 3_{3}, 1_{9}, 0_{9}\rangle &
\langle2_{0}, 2_{1}, 0_{6}, 1_{9}\rangle &
\langle3_{1}, 5_{3}, 3_{8}, 2_{7}\rangle &
\langle3_{1}, 0_{3}, 5_{4}, 1_{9}\rangle \\
\langle4_{0}, 1_{0}, 1_{4}, 6_{7}\rangle &
\langle7_{1}, 6_{1}, 5_{4}, 4_{7}\rangle &
\langle5_{0}, 0_{2}, 0_{4}, 5_{9}\rangle &
\langle0_{0}, 4_{1}, 7_{7}, 7_{5}\rangle &
\langle3_{1}, 7_{2}, 3_{4}, 6_{4}\rangle &
\langle4_{2}, 1_{3}, 5_{4}, 0_{9}\rangle &
\langle0_{2}, 1_{3}, 2_{4}, 3_{6}\rangle &
\langle4_{3}, 6_{3}, 7_{6}, 5_{5}\rangle \\
\langle6_{0}, 4_{2}, 6_{8}, 2_{4}\rangle &
\langle3_{1}, 4_{2}, 2_{5}, 4_{8}\rangle &
\langle5_{0}, 4_{1}, 6_{7}, 7_{9}\rangle &
\langle1_{2}, 3_{2}, 3_{7}, 4_{5}\rangle &
\langle1_{2}, 5_{3}, 7_{8}, 0_{8}\rangle &
\langle7_{2}, 5_{3}, 5_{9}, 3_{5}\rangle &
\langle4_{1}, 1_{2}, 1_{5}, 6_{8}\rangle &
\langle0_{1}, 3_{3}, 5_{4}, 2_{6}\rangle \\
\langle0_{1}, 2_{2}, 1_{6}, 1_{9}\rangle &
\end{array}$$ ]{}
$n=51$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{2}, 1_{9}, 1_{8}\rangle &
\langle\infty, 4_{1}, 4_{6}, 2_{7}\rangle &
\langle\infty, 4_{0}, 3_{4}, 3_{5}\rangle &
\langle3_{2}, 3_{3}, 5_{7}, a_{1}\rangle &
\langle6_{0}, 6_{1}, 3_{6}, a_{1}\rangle &
\langle5_{3}, 4_{2}, 6_{5}, a_{2}\rangle &
\langle7_{0}, 0_{1}, 2_{4}, a_{2}\rangle &
\langle7_{0}, 2_{1}, 0_{4}, a_{3}\rangle \\
\langle0_{3}, 6_{2}, 7_{5}, a_{3}\rangle &
\langle1_{0}, 1_{3}, 4_{6}, 7_{4}\rangle &
\langle6_{2}, 1_{3}, 4_{4}, 3_{5}\rangle &
\langle3_{0}, 2_{1}, 1_{6}, 4_{5}\rangle &
\langle3_{2}, 0_{3}, 3_{7}, 7_{8}\rangle &
\langle1_{1}, 3_{2}, 6_{8}, 0_{4}\rangle &
\langle6_{0}, 5_{2}, 2_{7}, 0_{6}\rangle &
\langle7_{0}, 3_{3}, 0_{6}, 7_{5}\rangle \\
\langle1_{1}, 0_{1}, 1_{5}, 7_{5}\rangle &
\langle2_{0}, 3_{3}, 3_{8}, 1_{6}\rangle &
\langle2_{1}, 3_{3}, 1_{8}, 3_{6}\rangle &
\langle5_{1}, 3_{3}, 7_{6}, 2_{9}\rangle &
\langle4_{1}, 7_{2}, 0_{6}, 5_{8}\rangle &
\langle5_{0}, 7_{3}, 2_{5}, 0_{9}\rangle &
\langle1_{2}, 7_{2}, 3_{4}, 2_{4}\rangle &
\langle2_{0}, 1_{0}, 0_{8}, 6_{4}\rangle \\
\langle3_{1}, 3_{3}, 0_{7}, 6_{8}\rangle &
\langle6_{0}, 0_{1}, 0_{4}, 3_{7}\rangle &
\langle1_{0}, 6_{0}, 6_{9}, 4_{5}\rangle &
\langle2_{0}, 0_{0}, 6_{9}, 4_{8}\rangle &
\langle4_{1}, 7_{3}, 7_{5}, 5_{7}\rangle &
\langle3_{1}, 4_{2}, 0_{5}, 6_{6}\rangle &
\langle3_{2}, 0_{2}, 2_{8}, 7_{6}\rangle &
\langle4_{2}, 2_{3}, 3_{7}, 5_{9}\rangle \\
\langle7_{1}, 4_{2}, 3_{5}, 1_{8}\rangle &
\langle4_{1}, 7_{1}, 5_{9}, 0_{4}\rangle &
\langle3_{0}, 1_{3}, 5_{9}, 1_{7}\rangle &
\langle4_{2}, 0_{3}, 0_{9}, 1_{6}\rangle &
\langle4_{0}, 6_{2}, 6_{5}, 5_{9}\rangle &
\langle2_{2}, 3_{2}, 2_{4}, 0_{9}\rangle &
\langle3_{0}, 3_{2}, 3_{6}, 6_{7}\rangle &
\langle4_{1}, 0_{2}, 3_{9}, 4_{7}\rangle \\
\langle6_{0}, 3_{1}, 3_{8}, 6_{4}\rangle &
\langle2_{3}, 5_{3}, 2_{4}, 4_{6}\rangle &
\langle5_{1}, 3_{2}, 4_{7}, 5_{9}\rangle &
\langle1_{0}, 7_{1}, 3_{7}, 1_{7}\rangle &
\langle5_{2}, 4_{3}, 6_{8}, 3_{7}\rangle &
\langle1_{3}, 7_{3}, 0_{4}, 3_{4}\rangle &
\langle4_{1}, 2_{1}, 6_{9}, 0_{8}\rangle &
\langle4_{0}, 1_{3}, 5_{7}, 3_{9}\rangle \\
\langle1_{0}, 0_{3}, 1_{8}, 4_{8}\rangle &
\langle0_{0}, 6_{2}, 4_{5}, 4_{6}\rangle &
\langle0_{3}, 1_{3}, 6_{5}, 6_{9}\rangle &
\end{array}$$ ]{}
$n=53$: [$$\begin{array}{llllllllll}
\langle\infty, 5_{0}, 3_{5}, 4_{4}\rangle &
\langle\infty, 2_{3}, 3_{8}, 2_{9}\rangle &
\langle\infty, 5_{1}, 7_{6}, 3_{7}\rangle &
\langle7_{1}, 7_{0}, 0_{4}, a_{1}\rangle &
\langle3_{3}, 4_{2}, 1_{6}, a_{1}\rangle &
\langle1_{0}, 2_{1}, 4_{4}, a_{2}\rangle &
\langle2_{3}, 6_{2}, 2_{5}, a_{2}\rangle &
\langle2_{3}, 5_{2}, 2_{7}, a_{3}\rangle \\
\langle0_{0}, 2_{1}, 4_{5}, a_{3}\rangle &
\langle7_{1}, 4_{0}, 2_{4}, a_{4}\rangle &
\langle3_{3}, 1_{2}, 5_{7}, a_{4}\rangle &
\langle2_{1}, 6_{0}, 7_{5}, a_{5}\rangle &
\langle0_{2}, 3_{3}, 5_{4}, a_{5}\rangle &
\langle7_{0}, 3_{3}, 4_{7}, 6_{9}\rangle &
\langle0_{2}, 1_{1}, 6_{4}, 6_{7}\rangle &
\langle2_{2}, 5_{0}, 7_{5}, 2_{9}\rangle \\
\langle7_{0}, 7_{3}, 2_{5}, 3_{6}\rangle &
\langle1_{1}, 3_{1}, 5_{8}, 7_{5}\rangle &
\langle0_{3}, 0_{2}, 5_{8}, 6_{9}\rangle &
\langle0_{2}, 6_{3}, 2_{4}, 0_{6}\rangle &
\langle7_{0}, 0_{3}, 3_{7}, 1_{9}\rangle &
\langle2_{0}, 3_{0}, 3_{8}, 7_{4}\rangle &
\langle0_{2}, 1_{0}, 0_{7}, 6_{8}\rangle &
\langle3_{1}, 6_{2}, 3_{9}, 7_{7}\rangle \\
\langle4_{0}, 5_{2}, 4_{9}, 7_{7}\rangle &
\langle3_{0}, 0_{0}, 6_{9}, 0_{6}\rangle &
\langle1_{2}, 6_{2}, 5_{6}, 2_{8}\rangle &
\langle1_{2}, 5_{0}, 4_{8}, 5_{4}\rangle &
\langle0_{1}, 3_{3}, 5_{9}, 0_{5}\rangle &
\langle5_{0}, 7_{3}, 5_{7}, 6_{9}\rangle &
\langle4_{1}, 0_{3}, 0_{4}, 5_{9}\rangle &
\langle2_{2}, 4_{2}, 6_{9}, 5_{6}\rangle \\
\langle4_{1}, 1_{1}, 7_{8}, 3_{7}\rangle &
\langle4_{1}, 6_{3}, 7_{6}, 2_{9}\rangle &
\langle4_{2}, 7_{1}, 5_{4}, 7_{7}\rangle &
\langle7_{0}, 2_{2}, 2_{8}, 0_{6}\rangle &
\langle1_{1}, 2_{2}, 3_{9}, 2_{5}\rangle &
\langle3_{2}, 4_{2}, 5_{5}, 2_{5}\rangle &
\langle2_{1}, 7_{3}, 1_{8}, 5_{5}\rangle &
\langle7_{0}, 4_{1}, 3_{9}, 1_{6}\rangle \\
\langle7_{1}, 3_{2}, 6_{5}, 5_{6}\rangle &
\langle6_{1}, 6_{2}, 1_{9}, 5_{4}\rangle &
\langle1_{0}, 0_{3}, 7_{7}, 1_{5}\rangle &
\langle0_{1}, 1_{0}, 4_{6}, 3_{7}\rangle &
\langle4_{1}, 6_{2}, 5_{7}, 5_{8}\rangle &
\langle6_{0}, 4_{1}, 4_{6}, 5_{6}\rangle &
\langle1_{3}, 3_{3}, 4_{4}, 6_{6}\rangle &
\langle1_{0}, 7_{3}, 7_{8}, 6_{5}\rangle \\
\langle5_{1}, 4_{3}, 2_{8}, 4_{6}\rangle &
\langle1_{3}, 2_{3}, 7_{4}, 5_{8}\rangle &
\langle5_{0}, 2_{3}, 6_{7}, 4_{5}\rangle &
\langle0_{0}, 2_{2}, 2_{4}, 4_{8}\rangle &
\langle0_{1}, 1_{3}, 0_{4}, 0_{8}\rangle &
\end{array}$$ ]{}
$n=55$: [$$\begin{array}{llllllllll}
\langle\infty, 4_{0}, 5_{5}, 1_{4}\rangle &
\langle\infty, 3_{3}, 0_{9}, 3_{8}\rangle &
\langle\infty, 0_{1}, 3_{6}, 7_{7}\rangle &
\langle6_{3}, 6_{2}, 0_{5}, a_{1}\rangle &
\langle1_{1}, 1_{0}, 3_{4}, a_{1}\rangle &
\langle0_{1}, 5_{0}, 5_{7}, a_{2}\rangle &
\langle3_{3}, 2_{2}, 2_{5}, a_{2}\rangle &
\langle5_{1}, 0_{0}, 3_{6}, a_{3}\rangle \\
\langle5_{2}, 7_{3}, 3_{5}, a_{3}\rangle &
\langle7_{3}, 2_{2}, 1_{7}, a_{4}\rangle &
\langle0_{1}, 4_{0}, 3_{4}, a_{4}\rangle &
\langle6_{2}, 2_{3}, 0_{4}, a_{5}\rangle &
\langle7_{0}, 0_{1}, 7_{5}, a_{5}\rangle &
\langle4_{1}, 0_{3}, 4_{7}, a_{6}\rangle &
\langle3_{2}, 5_{0}, 2_{5}, a_{6}\rangle &
\langle0_{3}, 2_{0}, 1_{6}, a_{7}\rangle \\
\langle3_{1}, 1_{2}, 7_{4}, a_{7}\rangle &
\langle6_{0}, 5_{3}, 5_{5}, 2_{5}\rangle &
\langle2_{3}, 2_{0}, 6_{4}, 5_{7}\rangle &
\langle1_{3}, 4_{3}, 7_{5}, 1_{4}\rangle &
\langle1_{0}, 0_{2}, 4_{9}, 3_{5}\rangle &
\langle4_{0}, 2_{1}, 0_{8}, 0_{7}\rangle &
\langle1_{0}, 3_{3}, 2_{6}, 2_{4}\rangle &
\langle7_{1}, 6_{1}, 3_{8}, 0_{8}\rangle \\
\langle5_{2}, 0_{0}, 0_{6}, 3_{8}\rangle &
\langle3_{1}, 6_{3}, 1_{9}, 1_{4}\rangle &
\langle3_{0}, 7_{2}, 5_{7}, 2_{9}\rangle &
\langle1_{0}, 3_{2}, 3_{8}, 3_{6}\rangle &
\langle0_{3}, 5_{0}, 7_{9}, 1_{9}\rangle &
\langle2_{0}, 5_{0}, 3_{7}, 5_{4}\rangle &
\langle1_{0}, 2_{3}, 6_{8}, 0_{8}\rangle &
\langle2_{0}, 7_{3}, 7_{9}, 7_{7}\rangle \\
\langle6_{1}, 7_{3}, 7_{6}, 6_{8}\rangle &
\langle7_{1}, 0_{2}, 7_{4}, 2_{8}\rangle &
\langle0_{1}, 6_{1}, 4_{5}, 1_{7}\rangle &
\langle0_{2}, 3_{3}, 5_{4}, 4_{5}\rangle &
\langle0_{2}, 6_{1}, 2_{7}, 1_{9}\rangle &
\langle0_{0}, 2_{1}, 1_{9}, 3_{5}\rangle &
\langle1_{0}, 2_{0}, 6_{6}, 2_{8}\rangle &
\langle7_{0}, 6_{1}, 7_{9}, 5_{6}\rangle \\
\langle1_{1}, 3_{3}, 1_{9}, 0_{8}\rangle &
\langle2_{3}, 3_{2}, 3_{7}, 3_{4}\rangle &
\langle1_{1}, 6_{2}, 2_{4}, 6_{9}\rangle &
\langle6_{1}, 1_{1}, 1_{5}, 3_{5}\rangle &
\langle3_{1}, 0_{3}, 5_{7}, 3_{6}\rangle &
\langle7_{1}, 7_{2}, 1_{9}, 4_{6}\rangle &
\langle7_{2}, 4_{0}, 3_{7}, 2_{8}\rangle &
\langle0_{1}, 4_{2}, 5_{4}, 7_{4}\rangle \\
\langle6_{0}, 7_{2}, 4_{5}, 4_{9}\rangle &
\langle0_{2}, 2_{2}, 3_{7}, 4_{6}\rangle &
\langle4_{2}, 3_{2}, 2_{9}, 0_{8}\rangle &
\langle0_{2}, 6_{3}, 7_{8}, 5_{7}\rangle &
\langle4_{1}, 7_{2}, 0_{6}, 6_{6}\rangle &
\langle0_{3}, 7_{3}, 2_{8}, 5_{6}\rangle &
\langle0_{3}, 2_{3}, 4_{6}, 4_{9}\rangle &
\end{array}$$ ]{}
$n=57$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{1}, 4_{7}, 4_{6}\rangle &
\langle\infty, 2_{0}, 4_{5}, 2_{8}\rangle &
\langle\infty, 4_{3}, 4_{4}, 2_{9}\rangle &
\langle4_{1}, 6_{0}, 3_{4}, a_{1}\rangle &
\langle7_{3}, 0_{2}, 3_{6}, a_{1}\rangle &
\langle5_{0}, 7_{3}, 7_{7}, a_{2}\rangle &
\langle2_{2}, 5_{1}, 1_{4}, a_{2}\rangle &
\langle3_{2}, 5_{3}, 4_{6}, a_{3}\rangle \\
\langle2_{1}, 1_{0}, 2_{4}, a_{3}\rangle &
\langle0_{1}, 0_{3}, 6_{5}, a_{4}\rangle &
\langle2_{2}, 4_{0}, 6_{4}, a_{4}\rangle &
\langle0_{3}, 4_{2}, 1_{5}, a_{5}\rangle &
\langle6_{0}, 6_{1}, 4_{6}, a_{5}\rangle &
\langle6_{2}, 1_{3}, 3_{6}, a_{6}\rangle &
\langle0_{1}, 5_{0}, 5_{4}, a_{6}\rangle &
\langle5_{1}, 5_{2}, 4_{6}, a_{7}\rangle \\
\langle1_{3}, 5_{0}, 6_{7}, a_{7}\rangle &
\langle1_{1}, 4_{3}, 4_{6}, a_{8}\rangle &
\langle2_{2}, 6_{0}, 5_{4}, a_{8}\rangle &
\langle1_{2}, 6_{0}, 2_{4}, a_{9}\rangle &
\langle1_{1}, 3_{3}, 3_{5}, a_{9}\rangle &
\langle2_{1}, 1_{2}, 7_{5}, 3_{9}\rangle &
\langle7_{2}, 5_{3}, 0_{8}, 3_{6}\rangle &
\langle2_{1}, 3_{3}, 5_{4}, 2_{8}\rangle \\
\langle2_{2}, 5_{2}, 1_{8}, 7_{8}\rangle &
\langle7_{2}, 5_{0}, 6_{5}, 1_{5}\rangle &
\langle1_{0}, 2_{0}, 7_{8}, 6_{7}\rangle &
\langle0_{0}, 2_{1}, 3_{5}, 4_{8}\rangle &
\langle2_{0}, 1_{2}, 0_{9}, 1_{5}\rangle &
\langle5_{2}, 0_{0}, 6_{5}, 7_{6}\rangle &
\langle7_{0}, 2_{3}, 3_{9}, 3_{6}\rangle &
\langle1_{3}, 4_{3}, 7_{4}, 6_{9}\rangle \\
\langle1_{3}, 5_{1}, 4_{5}, 3_{8}\rangle &
\langle6_{3}, 0_{1}, 0_{5}, 5_{9}\rangle &
\langle2_{3}, 3_{1}, 0_{8}, 1_{7}\rangle &
\langle0_{1}, 5_{3}, 3_{7}, 1_{8}\rangle &
\langle5_{1}, 6_{0}, 1_{9}, 1_{7}\rangle &
\langle0_{1}, 5_{1}, 6_{4}, 2_{6}\rangle &
\langle0_{3}, 2_{3}, 1_{4}, 3_{7}\rangle &
\langle4_{0}, 1_{0}, 3_{8}, 1_{5}\rangle \\
\langle3_{0}, 1_{0}, 0_{9}, 4_{6}\rangle &
\langle1_{2}, 3_{2}, 1_{6}, 2_{7}\rangle &
\langle3_{0}, 4_{2}, 4_{8}, 5_{9}\rangle &
\langle3_{1}, 6_{2}, 1_{9}, 0_{7}\rangle &
\langle6_{3}, 7_{3}, 7_{8}, 3_{5}\rangle &
\langle1_{1}, 0_{1}, 0_{7}, 4_{8}\rangle &
\langle2_{2}, 2_{3}, 6_{7}, 2_{9}\rangle &
\langle5_{0}, 1_{1}, 3_{7}, 5_{6}\rangle \\
\langle2_{3}, 5_{0}, 6_{9}, 4_{7}\rangle &
\langle6_{2}, 5_{2}, 3_{7}, 3_{4}\rangle &
\langle6_{2}, 5_{1}, 4_{8}, 1_{5}\rangle &
\langle6_{2}, 4_{1}, 6_{4}, 3_{9}\rangle &
\langle0_{2}, 4_{1}, 6_{9}, 4_{9}\rangle &
\langle3_{0}, 0_{1}, 3_{9}, 0_{6}\rangle &
\langle0_{0}, 0_{2}, 0_{7}, 3_{8}\rangle &
\langle6_{3}, 7_{0}, 1_{6}, 2_{4}\rangle \\
\langle0_{2}, 5_{3}, 2_{4}, 4_{5}\rangle &
\end{array}$$ ]{}
$n=59$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{0}, 7_{7}, 5_{6}\rangle &
\langle\infty, 2_{1}, 4_{9}, 6_{4}\rangle &
\langle\infty, 3_{3}, 6_{5}, 1_{8}\rangle &
\langle3_{2}, 3_{3}, 4_{6}, a_{1}\rangle &
\langle4_{0}, 3_{1}, 4_{4}, a_{1}\rangle &
\langle3_{2}, 3_{0}, 2_{4}, a_{2}\rangle &
\langle6_{1}, 1_{3}, 6_{6}, a_{2}\rangle &
\langle6_{3}, 1_{1}, 1_{7}, a_{3}\rangle \\
\langle2_{2}, 7_{0}, 7_{6}, a_{3}\rangle &
\langle2_{3}, 3_{2}, 6_{7}, a_{4}\rangle &
\langle2_{0}, 7_{1}, 6_{6}, a_{4}\rangle &
\langle4_{1}, 2_{3}, 7_{7}, a_{5}\rangle &
\langle0_{2}, 7_{0}, 2_{4}, a_{5}\rangle &
\langle6_{3}, 2_{0}, 5_{6}, a_{6}\rangle &
\langle7_{1}, 0_{2}, 2_{5}, a_{6}\rangle &
\langle1_{1}, 7_{2}, 2_{5}, a_{7}\rangle \\
\langle3_{3}, 5_{0}, 6_{6}, a_{7}\rangle &
\langle1_{3}, 7_{1}, 5_{4}, a_{8}\rangle &
\langle6_{0}, 4_{2}, 6_{7}, a_{8}\rangle &
\langle4_{2}, 5_{0}, 1_{4}, a_{9}\rangle &
\langle2_{1}, 6_{3}, 4_{5}, a_{9}\rangle &
\langle3_{0}, 2_{3}, 2_{7}, a_{10}\rangle &
\langle2_{1}, 1_{2}, 1_{4}, a_{10}\rangle &
\langle6_{3}, 3_{2}, 0_{7}, a_{11}\rangle \\
\langle2_{1}, 7_{0}, 4_{6}, a_{11}\rangle &
\langle0_{0}, 7_{0}, 3_{8}, 5_{5}\rangle &
\langle3_{0}, 4_{3}, 4_{5}, 3_{8}\rangle &
\langle1_{1}, 4_{2}, 5_{5}, 1_{8}\rangle &
\langle4_{1}, 3_{0}, 5_{9}, 2_{5}\rangle &
\langle6_{0}, 0_{1}, 7_{7}, 7_{8}\rangle &
\langle1_{0}, 7_{0}, 1_{5}, 7_{9}\rangle &
\langle1_{0}, 6_{2}, 2_{4}, 4_{9}\rangle \\
\langle5_{2}, 7_{2}, 5_{8}, 4_{9}\rangle &
\langle0_{2}, 5_{3}, 6_{7}, 4_{7}\rangle &
\langle1_{0}, 5_{1}, 7_{7}, 4_{5}\rangle &
\langle6_{2}, 5_{2}, 0_{9}, 5_{6}\rangle &
\langle1_{1}, 2_{3}, 4_{9}, 4_{8}\rangle &
\langle2_{3}, 5_{3}, 6_{9}, 4_{5}\rangle &
\langle4_{0}, 2_{1}, 0_{9}, 6_{7}\rangle &
\langle1_{2}, 7_{0}, 2_{7}, 4_{8}\rangle \\
\langle4_{3}, 4_{0}, 2_{4}, 3_{9}\rangle &
\langle4_{1}, 6_{1}, 1_{4}, 2_{8}\rangle &
\langle0_{3}, 3_{0}, 2_{6}, 1_{8}\rangle &
\langle0_{1}, 0_{3}, 5_{5}, 4_{6}\rangle &
\langle2_{2}, 0_{1}, 1_{8}, 5_{6}\rangle &
\langle2_{2}, 4_{3}, 0_{5}, 3_{4}\rangle &
\langle2_{2}, 6_{1}, 6_{5}, 0_{4}\rangle &
\langle1_{2}, 5_{3}, 3_{6}, 5_{8}\rangle \\
\langle1_{2}, 7_{3}, 2_{8}, 7_{6}\rangle &
\langle1_{3}, 2_{3}, 2_{4}, 6_{8}\rangle &
\langle4_{2}, 5_{3}, 3_{7}, 7_{4}\rangle &
\langle5_{0}, 0_{3}, 6_{9}, 1_{5}\rangle &
\langle0_{2}, 5_{2}, 1_{9}, 5_{5}\rangle &
\langle1_{3}, 3_{3}, 6_{9}, 6_{4}\rangle &
\langle1_{0}, 6_{0}, 0_{8}, 3_{4}\rangle &
\langle7_{1}, 4_{1}, 3_{9}, 5_{7}\rangle \\
\langle7_{1}, 7_{2}, 1_{8}, 7_{9}\rangle &
\langle0_{0}, 0_{1}, 5_{9}, 6_{6}\rangle &
\langle0_{1}, 5_{2}, 1_{6}, 5_{7}\rangle &
\end{array}$$ ]{}
$n=61$: [$$\begin{array}{llllllllll}
\langle\infty, 7_{0}, 3_{4}, 0_{5}\rangle &
\langle\infty, 0_{2}, 0_{6}, 6_{7}\rangle &
\langle\infty, 5_{3}, 5_{9}, 1_{8}\rangle &
\langle1_{3}, 1_{0}, 7_{4}, a_{1}\rangle &
\langle0_{1}, 3_{2}, 5_{5}, a_{1}\rangle &
\langle0_{1}, 2_{3}, 1_{4}, a_{2}\rangle &
\langle4_{2}, 0_{0}, 1_{6}, a_{2}\rangle &
\langle3_{1}, 2_{2}, 7_{4}, a_{3}\rangle \\
\langle1_{0}, 7_{3}, 7_{5}, a_{3}\rangle &
\langle2_{1}, 2_{2}, 1_{7}, a_{4}\rangle &
\langle3_{0}, 6_{3}, 3_{4}, a_{4}\rangle &
\langle7_{1}, 3_{3}, 7_{7}, a_{5}\rangle &
\langle1_{0}, 7_{2}, 3_{6}, a_{5}\rangle &
\langle3_{3}, 3_{1}, 2_{5}, a_{6}\rangle &
\langle5_{0}, 2_{2}, 2_{7}, a_{6}\rangle &
\langle5_{1}, 3_{2}, 0_{7}, a_{7}\rangle \\
\langle3_{3}, 7_{0}, 7_{5}, a_{7}\rangle &
\langle3_{3}, 1_{2}, 5_{7}, a_{8}\rangle &
\langle2_{1}, 1_{0}, 1_{6}, a_{8}\rangle &
\langle7_{0}, 4_{3}, 1_{7}, a_{9}\rangle &
\langle6_{2}, 4_{1}, 7_{4}, a_{9}\rangle &
\langle5_{1}, 5_{0}, 7_{4}, a_{10}\rangle &
\langle1_{2}, 1_{3}, 2_{7}, a_{10}\rangle &
\langle3_{3}, 4_{1}, 1_{6}, a_{11}\rangle \\
\langle4_{2}, 1_{0}, 5_{5}, a_{11}\rangle &
\langle6_{0}, 5_{1}, 1_{5}, a_{12}\rangle &
\langle3_{3}, 4_{2}, 5_{6}, a_{12}\rangle &
\langle3_{0}, 2_{3}, 2_{7}, a_{13}\rangle &
\langle6_{2}, 1_{1}, 6_{4}, a_{13}\rangle &
\langle3_{3}, 0_{1}, 1_{8}, 6_{8}\rangle &
\langle1_{2}, 3_{2}, 1_{5}, 2_{8}\rangle &
\langle7_{0}, 3_{1}, 0_{7}, 2_{9}\rangle \\
\langle4_{3}, 7_{1}, 7_{4}, 3_{7}\rangle &
\langle2_{0}, 0_{1}, 1_{6}, 7_{4}\rangle &
\langle3_{2}, 7_{3}, 0_{8}, 5_{7}\rangle &
\langle4_{1}, 5_{2}, 6_{9}, 3_{8}\rangle &
\langle0_{3}, 6_{3}, 2_{4}, 1_{5}\rangle &
\langle2_{0}, 3_{2}, 6_{6}, 1_{4}\rangle &
\langle5_{2}, 2_{2}, 1_{5}, 0_{9}\rangle &
\langle5_{1}, 6_{1}, 2_{8}, 0_{5}\rangle \\
\langle4_{2}, 5_{3}, 6_{9}, 3_{9}\rangle &
\langle2_{1}, 0_{3}, 5_{8}, 3_{7}\rangle &
\langle6_{1}, 3_{1}, 4_{5}, 6_{9}\rangle &
\langle4_{0}, 6_{0}, 1_{8}, 2_{7}\rangle &
\langle6_{0}, 3_{0}, 5_{5}, 5_{9}\rangle &
\langle3_{1}, 0_{0}, 6_{6}, 4_{9}\rangle &
\langle0_{2}, 1_{2}, 5_{9}, 4_{4}\rangle &
\langle1_{1}, 5_{2}, 7_{4}, 7_{6}\rangle \\
\langle5_{0}, 7_{1}, 5_{9}, 5_{7}\rangle &
\langle0_{3}, 5_{3}, 7_{9}, 0_{6}\rangle &
\langle0_{1}, 1_{3}, 0_{6}, 5_{9}\rangle &
\langle6_{0}, 3_{1}, 7_{9}, 3_{5}\rangle &
\langle2_{1}, 0_{1}, 2_{8}, 4_{6}\rangle &
\langle5_{0}, 7_{3}, 0_{4}, 2_{9}\rangle &
\langle3_{0}, 4_{0}, 4_{8}, 2_{8}\rangle &
\langle6_{0}, 7_{3}, 4_{9}, 7_{4}\rangle \\
\langle0_{2}, 6_{3}, 7_{6}, 0_{8}\rangle &
\langle5_{0}, 4_{2}, 7_{8}, 2_{6}\rangle &
\langle4_{0}, 4_{2}, 7_{7}, 0_{8}\rangle &
\langle3_{3}, 0_{2}, 5_{5}, 2_{8}\rangle &
\langle0_{3}, 1_{3}, 6_{5}, 5_{6}\rangle &
\end{array}$$ ]{}
$n=63$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{0}, 1_{5}, 0_{4}\rangle &
\langle\infty, 1_{2}, 3_{8}, 2_{6}\rangle &
\langle\infty, 4_{3}, 0_{7}, 0_{9}\rangle &
\langle6_{2}, 1_{0}, 5_{6}, a_{1}\rangle &
\langle7_{1}, 4_{3}, 6_{4}, a_{1}\rangle &
\langle6_{0}, 7_{1}, 6_{5}, a_{2}\rangle &
\langle1_{2}, 2_{3}, 3_{6}, a_{2}\rangle &
\langle0_{0}, 2_{3}, 7_{5}, a_{3}\rangle \\
\langle0_{1}, 0_{2}, 6_{7}, a_{3}\rangle &
\langle1_{2}, 6_{0}, 5_{6}, a_{4}\rangle &
\langle3_{1}, 3_{3}, 7_{4}, a_{4}\rangle &
\langle1_{1}, 6_{0}, 4_{6}, a_{5}\rangle &
\langle4_{2}, 1_{3}, 4_{5}, a_{5}\rangle &
\langle6_{0}, 6_{1}, 3_{4}, a_{6}\rangle &
\langle1_{2}, 7_{3}, 5_{7}, a_{6}\rangle &
\langle6_{1}, 0_{3}, 0_{7}, a_{7}\rangle \\
\langle3_{2}, 7_{0}, 3_{4}, a_{7}\rangle &
\langle3_{0}, 0_{3}, 1_{4}, a_{8}\rangle &
\langle6_{1}, 7_{2}, 2_{7}, a_{8}\rangle &
\langle3_{1}, 1_{2}, 2_{7}, a_{9}\rangle &
\langle5_{0}, 6_{3}, 4_{4}, a_{9}\rangle &
\langle7_{2}, 2_{1}, 7_{6}, a_{10}\rangle &
\langle5_{0}, 3_{3}, 4_{7}, a_{10}\rangle &
\langle4_{2}, 0_{3}, 7_{4}, a_{11}\rangle \\
\langle2_{1}, 4_{0}, 4_{6}, a_{11}\rangle &
\langle1_{0}, 7_{2}, 5_{5}, a_{12}\rangle &
\langle5_{1}, 0_{3}, 3_{4}, a_{12}\rangle &
\langle4_{1}, 7_{0}, 7_{7}, a_{13}\rangle &
\langle5_{3}, 3_{2}, 1_{6}, a_{13}\rangle &
\langle1_{1}, 7_{3}, 2_{6}, a_{14}\rangle &
\langle2_{0}, 1_{2}, 3_{7}, a_{14}\rangle &
\langle7_{1}, 5_{0}, 3_{5}, a_{15}\rangle \\
\langle7_{3}, 0_{2}, 5_{6}, a_{15}\rangle &
\langle2_{1}, 6_{3}, 0_{8}, 5_{5}\rangle &
\langle4_{3}, 2_{3}, 6_{5}, 5_{8}\rangle &
\langle1_{0}, 1_{2}, 4_{5}, 7_{9}\rangle &
\langle3_{0}, 7_{1}, 0_{7}, 7_{8}\rangle &
\langle0_{2}, 0_{3}, 5_{9}, 5_{4}\rangle &
\langle0_{1}, 7_{2}, 4_{8}, 1_{4}\rangle &
\langle1_{0}, 4_{3}, 3_{7}, 4_{4}\rangle \\
\langle6_{2}, 7_{2}, 6_{7}, 7_{8}\rangle &
\langle0_{3}, 7_{3}, 0_{5}, 5_{8}\rangle &
\langle2_{0}, 1_{1}, 1_{9}, 7_{5}\rangle &
\langle1_{0}, 3_{2}, 6_{6}, 2_{8}\rangle &
\langle4_{1}, 5_{3}, 0_{9}, 5_{8}\rangle &
\langle7_{2}, 2_{2}, 0_{4}, 1_{9}\rangle &
\langle6_{1}, 2_{2}, 3_{9}, 1_{4}\rangle &
\langle0_{1}, 2_{2}, 5_{8}, 1_{5}\rangle \\
\langle0_{0}, 1_{2}, 2_{5}, 1_{9}\rangle &
\langle1_{2}, 7_{2}, 3_{5}, 5_{8}\rangle &
\langle6_{0}, 4_{0}, 0_{9}, 2_{7}\rangle &
\langle4_{0}, 3_{0}, 5_{4}, 2_{8}\rangle &
\langle3_{0}, 0_{0}, 5_{8}, 3_{8}\rangle &
\langle1_{0}, 5_{3}, 4_{6}, 4_{9}\rangle &
\langle4_{2}, 7_{3}, 1_{7}, 0_{9}\rangle &
\langle6_{1}, 3_{1}, 3_{7}, 5_{8}\rangle \\
\langle1_{0}, 1_{3}, 1_{9}, 4_{7}\rangle &
\langle2_{1}, 1_{3}, 7_{5}, 3_{9}\rangle &
\langle3_{0}, 2_{3}, 0_{9}, 4_{6}\rangle &
\langle0_{3}, 5_{3}, 4_{8}, 5_{6}\rangle &
\langle1_{2}, 6_{1}, 4_{9}, 6_{5}\rangle &
\langle0_{1}, 7_{1}, 7_{6}, 2_{9}\rangle &
\langle0_{1}, 2_{1}, 2_{4}, 6_{6}\rangle &
\end{array}$$ ]{}
$n=65$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{3}, 6_{10}, 1_{6}\rangle &
\langle\infty, 5_{0}, 6_{8}, 2_{7}\rangle &
\langle\infty, 3_{1}, 0_{5}, 4_{11}\rangle &
\langle\infty, 3_{2}, 0_{9}, 3_{4}\rangle &
\langle4_{2}, 7_{1}, 1_{5}, a_{1}\rangle &
\langle3_{0}, 4_{3}, 2_{6}, a_{1}\rangle &
\langle2_{0}, 1_{1}, 2_{10}, 4_{6}\rangle &
\langle2_{1}, 5_{1}, 1_{9}, 5_{4}\rangle \\
\langle0_{2}, 1_{2}, 1_{5}, 2_{11}\rangle &
\langle7_{3}, 7_{2}, 5_{9}, 3_{5}\rangle &
\langle1_{1}, 3_{2}, 4_{8}, 7_{4}\rangle &
\langle3_{1}, 6_{3}, 2_{10}, 4_{5}\rangle &
\langle2_{3}, 3_{3}, 2_{8}, 4_{4}\rangle &
\langle3_{0}, 1_{1}, 4_{7}, 4_{10}\rangle &
\langle5_{1}, 3_{3}, 2_{6}, 1_{8}\rangle &
\langle0_{0}, 7_{2}, 6_{10}, 4_{7}\rangle \\
\langle2_{2}, 4_{2}, 1_{6}, 2_{10}\rangle &
\langle5_{0}, 3_{0}, 5_{11}, 6_{5}\rangle &
\langle4_{2}, 7_{0}, 4_{11}, 2_{11}\rangle &
\langle0_{1}, 6_{1}, 0_{6}, 4_{11}\rangle &
\langle1_{1}, 6_{0}, 4_{11}, 5_{7}\rangle &
\langle0_{1}, 1_{3}, 5_{11}, 2_{11}\rangle &
\langle3_{1}, 5_{3}, 0_{7}, 7_{10}\rangle &
\langle4_{1}, 7_{2}, 1_{9}, 2_{5}\rangle \\
\langle3_{0}, 0_{0}, 0_{5}, 2_{10}\rangle &
\langle1_{0}, 3_{2}, 1_{8}, 0_{8}\rangle &
\langle5_{2}, 3_{3}, 2_{4}, 5_{8}\rangle &
\langle4_{0}, 1_{1}, 7_{6}, 1_{8}\rangle &
\langle3_{0}, 4_{0}, 5_{9}, 6_{8}\rangle &
\langle4_{2}, 7_{2}, 5_{4}, 0_{10}\rangle &
\langle3_{3}, 1_{2}, 4_{7}, 6_{11}\rangle &
\langle2_{2}, 1_{3}, 1_{11}, 3_{9}\rangle \\
\langle4_{0}, 0_{2}, 2_{6}, 1_{6}\rangle &
\langle0_{0}, 4_{1}, 2_{7}, 5_{9}\rangle &
\langle0_{1}, 4_{2}, 6_{10}, 0_{7}\rangle &
\langle2_{1}, 2_{3}, 3_{8}, 2_{10}\rangle &
\langle2_{2}, 3_{1}, 2_{7}, 1_{9}\rangle &
\langle4_{1}, 2_{0}, 3_{4}, 5_{4}\rangle &
\langle2_{0}, 3_{2}, 5_{7}, 2_{7}\rangle &
\langle0_{2}, 4_{3}, 3_{10}, 7_{5}\rangle \\
\langle3_{1}, 4_{1}, 3_{11}, 2_{8}\rangle &
\langle0_{1}, 7_{3}, 7_{5}, 2_{10}\rangle &
\langle0_{1}, 6_{2}, 1_{6}, 2_{9}\rangle &
\langle0_{3}, 1_{0}, 5_{6}, 5_{8}\rangle &
\langle6_{1}, 7_{2}, 3_{8}, 2_{4}\rangle &
\langle2_{1}, 6_{3}, 5_{5}, 2_{9}\rangle &
\langle6_{0}, 1_{3}, 2_{10}, 4_{9}\rangle &
\langle2_{0}, 0_{3}, 6_{11}, 1_{5}\rangle \\
\langle0_{0}, 6_{2}, 3_{10}, 4_{5}\rangle &
\langle3_{1}, 3_{2}, 4_{7}, 5_{8}\rangle &
\langle3_{3}, 1_{0}, 0_{9}, 7_{4}\rangle &
\langle0_{3}, 3_{3}, 3_{4}, 2_{7}\rangle &
\langle1_{0}, 4_{2}, 7_{8}, 4_{9}\rangle &
\langle1_{0}, 2_{1}, 1_{6}, 5_{9}\rangle &
\langle5_{2}, 6_{3}, 4_{4}, 0_{11}\rangle &
\langle3_{0}, 7_{3}, 1_{5}, 4_{11}\rangle \\
\langle0_{2}, 5_{3}, 4_{11}, 2_{5}\rangle &
\langle0_{0}, 0_{2}, 6_{7}, 2_{4}\rangle &
\langle3_{0}, 3_{1}, 0_{4}, 0_{10}\rangle &
\langle1_{3}, 4_{1}, 6_{4}, 6_{7}\rangle &
\langle1_{3}, 3_{3}, 2_{9}, 5_{6}\rangle &
\langle0_{0}, 0_{3}, 1_{6}, 0_{9}\rangle &
\langle0_{2}, 3_{3}, 6_{6}, 7_{8}\rangle &
\langle0_0, 4_0, 0_4, 4_4\rangle^s \\
\langle0_1, 4_1, 0_5, 4_5\rangle^s &
\langle0_2, 4_2, 0_6, 4_6\rangle^s &
\langle0_3, 4_3, 0_7, 4_7\rangle^s &
\end{array}$$ ]{} Note that each of the codewords marked $s$ only generates four codewords.
T$(2,37;2,n;6) =9n$ for each odd $n$ and $37 \leq n \leq 73$.
Let $X_1=({{\mathbb{Z}}}_9\times \{0,1,2,3\})\cup \{\infty\}$. For $37 \leq n \leq 53$, let $X_2= ({{\mathbb{Z}}}_9\times \{4,5,6,7\})\cup (\{a\}\times \{1,\ldots,n-36\})$; for $55 \leq n \leq 71$, let $X_2= ({{\mathbb{Z}}}_9\times \{4,5,\ldots,9\})\cup (\{a\}\times \{1,\ldots,n-54\})$; for $n = 73$, let $X_2= ({{\mathbb{Z}}}_9\times \{4,5,\ldots,11\})\cup (\{a\}\times \{1\})$. Denote $X=X_1\cup X_2$. The desired codes of size $9n$ are constructed on ${{\mathbb{Z}}}_2^X$ and the base codewords are listed as follows.
$n=37$: [$$\begin{array}{llllllllll}
\langle\infty, 4_{0}, 4_{4}, 4_{5}\rangle &
\langle\infty, 0_{1}, 0_{7}, 0_{6}\rangle &
\langle4_{2}, 4_{3}, 4_{5}, a_{1}\rangle &
\langle5_{0}, 8_{1}, 6_{4}, a_{1}\rangle &
\langle1_{0}, 7_{1}, 5_{6}, 6_{7}\rangle &
\langle1_{2}, 0_{3}, 4_{7}, 6_{4}\rangle &
\langle0_{3}, 3_{3}, 2_{7}, 5_{5}\rangle &
\langle1_{0}, 8_{1}, 3_{7}, 0_{7}\rangle \\
\langle0_{1}, 8_{1}, 8_{4}, 2_{4}\rangle &
\langle0_{2}, 7_{2}, 0_{7}, 1_{4}\rangle &
\langle2_{0}, 4_{1}, 0_{5}, 5_{6}\rangle &
\langle2_{3}, 6_{3}, 3_{5}, 3_{6}\rangle &
\langle3_{2}, 4_{3}, 8_{5}, 0_{4}\rangle &
\langle2_{2}, 6_{3}, 3_{7}, 7_{7}\rangle &
\langle2_{1}, 2_{2}, 4_{5}, 6_{4}\rangle &
\langle1_{1}, 3_{1}, 1_{5}, 7_{5}\rangle \\
\langle1_{2}, 0_{2}, 0_{4}, 4_{5}\rangle &
\langle1_{0}, 3_{0}, 0_{4}, 2_{5}\rangle &
\langle2_{0}, 7_{1}, 1_{6}, 5_{7}\rangle &
\langle0_{3}, 1_{3}, 0_{6}, 8_{4}\rangle &
\langle1_{0}, 2_{1}, 5_{5}, 8_{4}\rangle &
\langle3_{2}, 6_{3}, 2_{6}, 0_{5}\rangle &
\langle0_{0}, 5_{0}, 5_{6}, 6_{6}\rangle &
\langle2_{0}, 2_{3}, 0_{6}, 4_{6}\rangle \\
\langle0_{2}, 3_{2}, 1_{5}, 0_{6}\rangle &
\langle1_{1}, 4_{2}, 3_{7}, 7_{6}\rangle &
\langle1_{2}, 5_{2}, 3_{6}, 3_{4}\rangle &
\langle2_{2}, 4_{3}, 0_{7}, 7_{6}\rangle &
\langle2_{0}, 5_{0}, 7_{5}, 8_{5}\rangle &
\langle2_{0}, 2_{2}, 6_{7}, 8_{7}\rangle &
\langle0_{1}, 3_{3}, 1_{5}, 5_{4}\rangle &
\langle0_{1}, 7_{2}, 2_{6}, 8_{6}\rangle \\
\langle0_{0}, 8_{0}, 4_{4}, 2_{4}\rangle &
\langle0_{0}, 4_{1}, 1_{7}, 0_{7}\rangle &
\langle0_{3}, 7_{3}, 1_{4}, 7_{7}\rangle &
\langle4_{1}, 4_{3}, 7_{7}, 3_{5}\rangle &
\langle0_{1}, 1_{3}, 1_{4}, 5_{6}\rangle &
\end{array}$$ ]{}
$n=39$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{0}, 0_{4}, 0_{5}\rangle &
\langle\infty, 0_{1}, 0_{6}, 0_{7}\rangle &
\langle1_{2}, 1_{3}, 1_{5}, a_{1}\rangle &
\langle3_{0}, 3_{1}, 4_{4}, a_{1}\rangle &
\langle3_{0}, 4_{1}, 6_{4}, a_{2}\rangle &
\langle2_{2}, 3_{3}, 3_{7}, a_{2}\rangle &
\langle1_{2}, 6_{3}, 2_{5}, a_{3}\rangle &
\langle3_{0}, 5_{1}, 5_{4}, a_{3}\rangle \\
\langle0_{0}, 1_{0}, 7_{4}, 5_{4}\rangle &
\langle0_{0}, 5_{3}, 4_{7}, 7_{7}\rangle &
\langle0_{3}, 4_{3}, 7_{7}, 3_{5}\rangle &
\langle0_{3}, 8_{3}, 8_{4}, 6_{5}\rangle &
\langle1_{0}, 7_{1}, 2_{7}, 1_{7}\rangle &
\langle1_{2}, 7_{3}, 4_{7}, 7_{6}\rangle &
\langle0_{1}, 3_{1}, 8_{6}, 6_{6}\rangle &
\langle0_{0}, 6_{0}, 6_{6}, 3_{5}\rangle \\
\langle0_{3}, 3_{3}, 6_{6}, 1_{4}\rangle &
\langle0_{1}, 7_{1}, 7_{5}, 6_{5}\rangle &
\langle0_{3}, 2_{3}, 7_{6}, 5_{4}\rangle &
\langle0_{2}, 4_{3}, 0_{7}, 5_{6}\rangle &
\langle1_{2}, 4_{3}, 6_{4}, 8_{7}\rangle &
\langle0_{0}, 4_{3}, 8_{4}, 5_{5}\rangle &
\langle0_{2}, 2_{2}, 2_{6}, 7_{5}\rangle &
\langle0_{2}, 5_{2}, 7_{4}, 8_{6}\rangle \\
\langle1_{2}, 8_{3}, 0_{7}, 5_{4}\rangle &
\langle1_{1}, 1_{2}, 0_{4}, 5_{5}\rangle &
\langle1_{0}, 3_{3}, 5_{6}, 2_{6}\rangle &
\langle0_{0}, 5_{0}, 7_{6}, 8_{6}\rangle &
\langle0_{2}, 3_{2}, 4_{6}, 6_{4}\rangle &
\langle1_{0}, 8_{1}, 6_{7}, 4_{7}\rangle &
\langle1_{0}, 1_{2}, 3_{7}, 7_{7}\rangle &
\langle0_{0}, 7_{0}, 2_{5}, 8_{5}\rangle \\
\langle0_{1}, 1_{1}, 7_{4}, 4_{4}\rangle &
\langle0_{2}, 8_{2}, 0_{4}, 2_{5}\rangle &
\langle0_{0}, 4_{1}, 5_{6}, 7_{5}\rangle &
\langle0_{1}, 5_{1}, 7_{6}, 6_{7}\rangle &
\langle0_{1}, 4_{2}, 1_{5}, 8_{7}\rangle &
\langle1_{1}, 7_{2}, 6_{5}, 3_{7}\rangle &
\langle0_{1}, 0_{3}, 2_{5}, 4_{6}\rangle &
\end{array}$$ ]{}
$n=41$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{0}, 0_{4}, 0_{5}\rangle &
\langle\infty, 0_{1}, 0_{6}, 0_{7}\rangle &
\langle0_{2}, 0_{3}, 5_{5}, a_{1}\rangle &
\langle1_{0}, 1_{1}, 2_{4}, a_{1}\rangle &
\langle0_{0}, 1_{1}, 3_{4}, a_{2}\rangle &
\langle0_{2}, 1_{3}, 2_{5}, a_{2}\rangle &
\langle0_{2}, 2_{3}, 1_{5}, a_{3}\rangle &
\langle0_{0}, 7_{1}, 1_{7}, a_{3}\rangle \\
\langle0_{0}, 3_{1}, 6_{4}, a_{4}\rangle &
\langle0_{2}, 5_{3}, 1_{7}, a_{4}\rangle &
\langle0_{2}, 7_{3}, 1_{6}, a_{5}\rangle &
\langle4_{0}, 8_{1}, 3_{4}, a_{5}\rangle &
\langle0_{1}, 2_{1}, 8_{6}, 3_{5}\rangle &
\langle0_{0}, 8_{1}, 3_{7}, 7_{4}\rangle &
\langle0_{0}, 3_{0}, 0_{6}, 1_{6}\rangle &
\langle0_{2}, 3_{3}, 3_{7}, 0_{5}\rangle \\
\langle0_{1}, 1_{2}, 8_{7}, 3_{6}\rangle &
\langle0_{1}, 8_{2}, 7_{7}, 1_{7}\rangle &
\langle0_{2}, 4_{2}, 3_{6}, 3_{5}\rangle &
\langle1_{0}, 1_{2}, 6_{6}, 5_{7}\rangle &
\langle0_{3}, 4_{3}, 1_{6}, 2_{7}\rangle &
\langle1_{0}, 2_{2}, 7_{7}, 8_{7}\rangle &
\langle0_{1}, 1_{1}, 6_{4}, 0_{5}\rangle &
\langle1_{1}, 1_{3}, 5_{6}, 7_{7}\rangle \\
\langle0_{1}, 0_{2}, 6_{5}, 7_{4}\rangle &
\langle0_{3}, 2_{3}, 0_{6}, 6_{4}\rangle &
\langle0_{1}, 7_{3}, 2_{5}, 2_{7}\rangle &
\langle0_{1}, 3_{1}, 5_{6}, 7_{5}\rangle &
\langle0_{0}, 6_{1}, 2_{7}, 4_{6}\rangle &
\langle0_{2}, 1_{2}, 0_{4}, 2_{4}\rangle &
\langle0_{0}, 4_{0}, 0_{7}, 3_{6}\rangle &
\langle0_{2}, 3_{2}, 0_{6}, 7_{6}\rangle \\
\langle0_{2}, 7_{2}, 3_{4}, 4_{4}\rangle &
\langle0_{2}, 6_{3}, 4_{5}, 0_{7}\rangle &
\langle0_{3}, 1_{3}, 0_{4}, 3_{4}\rangle &
\langle0_{3}, 3_{3}, 2_{6}, 1_{4}\rangle &
\langle0_{0}, 2_{0}, 6_{5}, 5_{5}\rangle &
\langle0_{0}, 6_{3}, 8_{5}, 2_{6}\rangle &
\langle0_{0}, 7_{3}, 7_{5}, 8_{7}\rangle &
\langle0_{0}, 5_{1}, 5_{4}, 1_{5}\rangle \\
\langle0_{0}, 8_{3}, 4_{4}, 2_{5}\rangle &
\end{array}$$ ]{}
$n=43$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{1}, 1_{7}, 8_{6}\rangle &
\langle\infty, 2_{0}, 5_{4}, 3_{5}\rangle &
\langle3_{0}, 3_{1}, 4_{4}, a_{1}\rangle &
\langle1_{3}, 1_{2}, 1_{5}, a_{1}\rangle &
\langle0_{3}, 8_{2}, 2_{6}, a_{2}\rangle &
\langle6_{0}, 7_{1}, 4_{7}, a_{2}\rangle &
\langle4_{2}, 6_{3}, 5_{5}, a_{3}\rangle &
\langle0_{1}, 7_{0}, 4_{4}, a_{3}\rangle \\
\langle5_{2}, 3_{3}, 8_{7}, a_{4}\rangle &
\langle6_{0}, 0_{1}, 5_{4}, a_{4}\rangle &
\langle8_{0}, 4_{1}, 8_{5}, a_{5}\rangle &
\langle2_{2}, 6_{3}, 1_{7}, a_{5}\rangle &
\langle4_{2}, 0_{3}, 7_{5}, a_{6}\rangle &
\langle5_{0}, 3_{1}, 4_{6}, a_{6}\rangle &
\langle8_{0}, 5_{1}, 4_{7}, a_{7}\rangle &
\langle4_{2}, 1_{3}, 4_{4}, a_{7}\rangle \\
\langle1_{1}, 4_{3}, 8_{6}, 4_{6}\rangle &
\langle1_{3}, 5_{3}, 3_{4}, 2_{4}\rangle &
\langle1_{1}, 2_{1}, 1_{5}, 3_{7}\rangle &
\langle1_{1}, 5_{2}, 5_{7}, 1_{7}\rangle &
\langle1_{3}, 8_{3}, 2_{6}, 3_{5}\rangle &
\langle1_{0}, 3_{0}, 0_{5}, 8_{4}\rangle &
\langle2_{2}, 7_{0}, 0_{4}, 0_{7}\rangle &
\langle3_{1}, 5_{3}, 0_{4}, 8_{7}\rangle \\
\langle1_{3}, 3_{1}, 7_{6}, 6_{4}\rangle &
\langle2_{0}, 5_{0}, 3_{6}, 2_{6}\rangle &
\langle0_{0}, 2_{2}, 8_{7}, 6_{7}\rangle &
\langle1_{2}, 7_{2}, 8_{6}, 2_{4}\rangle &
\langle1_{0}, 0_{0}, 3_{5}, 5_{5}\rangle &
\langle4_{2}, 1_{1}, 3_{4}, 0_{4}\rangle &
\langle1_{3}, 2_{3}, 1_{4}, 8_{7}\rangle &
\langle1_{2}, 0_{2}, 3_{4}, 6_{5}\rangle \\
\langle1_{1}, 1_{2}, 3_{6}, 8_{5}\rangle &
\langle1_{1}, 2_{3}, 7_{5}, 1_{6}\rangle &
\langle0_{0}, 7_{3}, 3_{6}, 0_{7}\rangle &
\langle2_{1}, 3_{2}, 5_{5}, 5_{7}\rangle &
\langle3_{0}, 6_{2}, 3_{4}, 5_{6}\rangle &
\langle1_{2}, 6_{2}, 1_{6}, 5_{5}\rangle &
\langle1_{0}, 5_{3}, 5_{7}, 4_{7}\rangle &
\langle0_{1}, 8_{3}, 5_{5}, 2_{5}\rangle \\
\langle3_{0}, 0_{3}, 7_{6}, 1_{5}\rangle &
\langle5_{0}, 5_{2}, 1_{6}, 6_{7}\rangle &
\langle0_{1}, 2_{1}, 0_{4}, 8_{6}\rangle &
\end{array}$$ ]{}
$n=45$: [$$\begin{array}{llllllllll}
\langle\infty, 6_{0}, 4_{5}, 5_{4}\rangle &
\langle\infty, 6_{1}, 6_{7}, 5_{6}\rangle &
\langle4_{0}, 5_{1}, 0_{4}, a_{1}\rangle &
\langle1_{3}, 1_{2}, 7_{7}, a_{1}\rangle &
\langle1_{2}, 2_{3}, 6_{7}, a_{2}\rangle &
\langle1_{1}, 8_{0}, 8_{5}, a_{2}\rangle &
\langle8_{0}, 8_{1}, 0_{4}, a_{3}\rangle &
\langle7_{3}, 5_{2}, 7_{5}, a_{3}\rangle \\
\langle0_{0}, 3_{1}, 5_{7}, a_{4}\rangle &
\langle3_{2}, 6_{3}, 8_{5}, a_{4}\rangle &
\langle4_{2}, 8_{3}, 6_{7}, a_{5}\rangle &
\langle4_{0}, 2_{1}, 4_{4}, a_{5}\rangle &
\langle1_{2}, 6_{3}, 7_{6}, a_{6}\rangle &
\langle3_{0}, 7_{1}, 6_{4}, a_{6}\rangle &
\langle6_{3}, 0_{2}, 3_{4}, a_{7}\rangle &
\langle3_{0}, 0_{1}, 4_{5}, a_{7}\rangle \\
\langle0_{0}, 3_{2}, 3_{5}, a_{8}\rangle &
\langle0_{1}, 6_{3}, 4_{6}, a_{8}\rangle &
\langle3_{2}, 2_{3}, 4_{4}, a_{9}\rangle &
\langle1_{1}, 2_{0}, 6_{7}, a_{9}\rangle &
\langle7_{1}, 6_{2}, 1_{7}, 6_{7}\rangle &
\langle4_{2}, 7_{2}, 8_{6}, 5_{5}\rangle &
\langle5_{0}, 0_{0}, 2_{7}, 3_{7}\rangle &
\langle1_{2}, 3_{2}, 4_{7}, 7_{4}\rangle \\
\langle7_{3}, 6_{1}, 8_{5}, 2_{5}\rangle &
\langle8_{1}, 4_{2}, 4_{6}, 3_{7}\rangle &
\langle7_{1}, 2_{2}, 8_{5}, 4_{4}\rangle &
\langle8_{0}, 6_{2}, 6_{4}, 0_{6}\rangle &
\langle3_{1}, 8_{1}, 6_{6}, 0_{6}\rangle &
\langle7_{0}, 7_{2}, 6_{6}, 6_{5}\rangle &
\langle8_{1}, 2_{1}, 2_{4}, 8_{5}\rangle &
\langle8_{1}, 1_{3}, 6_{7}, 0_{7}\rangle \\
\langle4_{1}, 0_{3}, 4_{6}, 1_{7}\rangle &
\langle2_{3}, 8_{0}, 1_{5}, 5_{4}\rangle &
\langle1_{2}, 0_{0}, 3_{6}, 8_{7}\rangle &
\langle0_{2}, 0_{1}, 7_{4}, 5_{4}\rangle &
\langle3_{3}, 6_{3}, 2_{4}, 1_{4}\rangle &
\langle6_{0}, 7_{3}, 4_{6}, 7_{7}\rangle &
\langle1_{2}, 6_{0}, 6_{6}, 8_{6}\rangle &
\langle5_{3}, 7_{3}, 1_{6}, 3_{5}\rangle \\
\langle6_{3}, 5_{3}, 6_{4}, 5_{6}\rangle &
\langle2_{0}, 4_{0}, 8_{6}, 6_{4}\rangle &
\langle6_{0}, 8_{2}, 2_{5}, 3_{5}\rangle &
\langle6_{1}, 6_{3}, 8_{6}, 0_{5}\rangle &
\langle0_{0}, 7_{3}, 4_{5}, 0_{7}\rangle &
\end{array}$$ ]{}
$n=47$: [$$\begin{array}{llllllllll}
\langle\infty, 5_{0}, 7_{5}, 5_{4}\rangle &
\langle\infty, 4_{1}, 7_{7}, 3_{6}\rangle &
\langle4_{0}, 4_{1}, 5_{4}, a_{1}\rangle &
\langle8_{3}, 4_{2}, 2_{6}, a_{1}\rangle &
\langle5_{2}, 6_{3}, 7_{5}, a_{2}\rangle &
\langle5_{1}, 4_{0}, 1_{6}, a_{2}\rangle &
\langle1_{3}, 8_{2}, 6_{5}, a_{3}\rangle &
\langle8_{0}, 1_{1}, 4_{6}, a_{3}\rangle \\
\langle5_{1}, 2_{0}, 8_{4}, a_{4}\rangle &
\langle3_{3}, 0_{2}, 5_{5}, a_{4}\rangle &
\langle0_{2}, 0_{3}, 0_{5}, a_{5}\rangle &
\langle4_{0}, 8_{1}, 7_{4}, a_{5}\rangle &
\langle7_{1}, 2_{0}, 5_{6}, a_{6}\rangle &
\langle6_{3}, 1_{2}, 4_{5}, a_{6}\rangle &
\langle3_{3}, 6_{2}, 0_{7}, a_{7}\rangle &
\langle2_{0}, 8_{1}, 3_{5}, a_{7}\rangle \\
\langle0_{2}, 7_{3}, 2_{7}, a_{8}\rangle &
\langle2_{1}, 4_{0}, 0_{4}, a_{8}\rangle &
\langle1_{3}, 2_{2}, 6_{7}, a_{9}\rangle &
\langle8_{1}, 0_{0}, 8_{5}, a_{9}\rangle &
\langle4_{2}, 6_{0}, 1_{7}, a_{10}\rangle &
\langle4_{1}, 2_{3}, 5_{5}, a_{10}\rangle &
\langle3_{2}, 6_{0}, 4_{5}, a_{11}\rangle &
\langle0_{3}, 7_{1}, 4_{4}, a_{11}\rangle \\
\langle5_{2}, 2_{1}, 6_{7}, 4_{5}\rangle &
\langle3_{0}, 6_{0}, 3_{7}, 1_{4}\rangle &
\langle2_{3}, 8_{3}, 0_{7}, 0_{4}\rangle &
\langle3_{0}, 5_{0}, 8_{5}, 0_{5}\rangle &
\langle4_{0}, 0_{0}, 3_{7}, 2_{7}\rangle &
\langle4_{1}, 7_{3}, 2_{5}, 8_{6}\rangle &
\langle2_{2}, 3_{2}, 6_{4}, 4_{4}\rangle &
\langle4_{1}, 2_{1}, 7_{5}, 4_{4}\rangle \\
\langle1_{1}, 2_{1}, 3_{7}, 3_{6}\rangle &
\langle3_{3}, 1_{3}, 7_{6}, 3_{6}\rangle &
\langle3_{3}, 8_{3}, 5_{4}, 2_{7}\rangle &
\langle6_{1}, 7_{3}, 1_{4}, 3_{6}\rangle &
\langle4_{2}, 2_{2}, 6_{6}, 0_{4}\rangle &
\langle5_{1}, 0_{2}, 5_{6}, 4_{5}\rangle &
\langle4_{3}, 3_{3}, 3_{4}, 2_{6}\rangle &
\langle1_{1}, 6_{3}, 8_{7}, 6_{7}\rangle \\
\langle4_{0}, 8_{2}, 5_{6}, 2_{6}\rangle &
\langle5_{1}, 5_{3}, 2_{5}, 1_{4}\rangle &
\langle5_{0}, 4_{2}, 5_{6}, 4_{6}\rangle &
\langle1_{2}, 5_{0}, 1_{7}, 0_{6}\rangle &
\langle6_{0}, 0_{2}, 7_{7}, 6_{5}\rangle &
\langle4_{2}, 2_{0}, 1_{4}, 4_{4}\rangle &
\langle0_{1}, 1_{2}, 0_{7}, 6_{7}\rangle &
\end{array}$$ ]{}
$n=49$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{0}, 1_{5}, 1_{4}\rangle &
\langle\infty, 3_{1}, 7_{6}, 0_{7}\rangle &
\langle4_{2}, 4_{3}, 8_{5}, a_{1}\rangle &
\langle4_{1}, 4_{0}, 5_{4}, a_{1}\rangle &
\langle4_{3}, 3_{2}, 5_{5}, a_{2}\rangle &
\langle3_{1}, 2_{0}, 5_{4}, a_{2}\rangle &
\langle0_{1}, 7_{0}, 0_{4}, a_{3}\rangle &
\langle6_{2}, 8_{3}, 7_{5}, a_{3}\rangle \\
\langle1_{3}, 7_{2}, 3_{5}, a_{4}\rangle &
\langle1_{1}, 7_{0}, 0_{6}, a_{4}\rangle &
\langle3_{3}, 8_{2}, 6_{5}, a_{5}\rangle &
\langle8_{0}, 3_{1}, 5_{7}, a_{5}\rangle &
\langle6_{0}, 2_{1}, 1_{4}, a_{6}\rangle &
\langle1_{3}, 5_{2}, 6_{7}, a_{6}\rangle &
\langle4_{3}, 7_{2}, 7_{4}, a_{7}\rangle &
\langle8_{1}, 2_{0}, 3_{5}, a_{7}\rangle \\
\langle3_{3}, 5_{2}, 1_{7}, a_{8}\rangle &
\langle1_{1}, 3_{0}, 8_{4}, a_{8}\rangle &
\langle8_{1}, 0_{0}, 2_{5}, a_{9}\rangle &
\langle2_{3}, 3_{2}, 4_{4}, a_{9}\rangle &
\langle5_{1}, 5_{3}, 3_{5}, a_{10}\rangle &
\langle1_{2}, 2_{0}, 7_{7}, a_{10}\rangle &
\langle0_{3}, 4_{1}, 3_{7}, a_{11}\rangle &
\langle1_{2}, 1_{0}, 7_{6}, a_{11}\rangle \\
\langle4_{0}, 7_{2}, 2_{7}, a_{12}\rangle &
\langle6_{1}, 5_{3}, 2_{4}, a_{12}\rangle &
\langle6_{0}, 8_{2}, 0_{6}, a_{13}\rangle &
\langle6_{1}, 0_{3}, 2_{7}, a_{13}\rangle &
\langle2_{2}, 5_{2}, 2_{5}, 5_{6}\rangle &
\langle4_{1}, 5_{3}, 5_{5}, 2_{6}\rangle &
\langle1_{0}, 7_{0}, 8_{6}, 6_{6}\rangle &
\langle4_{1}, 6_{2}, 4_{6}, 8_{7}\rangle \\
\langle2_{1}, 3_{2}, 8_{6}, 5_{4}\rangle &
\langle1_{0}, 8_{0}, 5_{5}, 4_{5}\rangle &
\langle1_{3}, 2_{3}, 7_{5}, 1_{4}\rangle &
\langle4_{0}, 5_{2}, 5_{7}, 4_{6}\rangle &
\langle0_{2}, 8_{2}, 4_{4}, 7_{4}\rangle &
\langle2_{3}, 7_{1}, 1_{7}, 1_{6}\rangle &
\langle1_{0}, 8_{3}, 0_{7}, 3_{7}\rangle &
\langle5_{1}, 3_{2}, 0_{4}, 7_{6}\rangle \\
\langle2_{3}, 5_{3}, 7_{6}, 6_{4}\rangle &
\langle3_{1}, 1_{1}, 1_{7}, 0_{5}\rangle &
\langle1_{1}, 6_{1}, 2_{6}, 6_{5}\rangle &
\langle3_{3}, 5_{3}, 6_{6}, 3_{6}\rangle &
\langle3_{0}, 4_{3}, 0_{4}, 2_{4}\rangle &
\langle6_{0}, 6_{3}, 1_{6}, 6_{7}\rangle &
\langle8_{1}, 2_{2}, 0_{7}, 5_{4}\rangle &
\langle8_{0}, 0_{0}, 8_{5}, 3_{7}\rangle \\
\langle0_{2}, 4_{2}, 3_{5}, 3_{7}\rangle &
\end{array}$$ ]{}
$n=51$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{1}, 1_{6}, 1_{7}\rangle &
\langle\infty, 1_{0}, 3_{4}, 0_{5}\rangle &
\langle3_{2}, 3_{3}, 3_{5}, a_{1}\rangle &
\langle1_{0}, 1_{1}, 2_{4}, a_{1}\rangle &
\langle0_{0}, 1_{1}, 3_{4}, a_{2}\rangle &
\langle1_{2}, 2_{3}, 3_{5}, a_{2}\rangle &
\langle1_{2}, 3_{3}, 2_{5}, a_{3}\rangle &
\langle2_{0}, 4_{1}, 7_{7}, a_{3}\rangle \\
\langle2_{2}, 5_{3}, 5_{6}, a_{4}\rangle &
\langle2_{0}, 5_{1}, 8_{4}, a_{4}\rangle &
\langle3_{0}, 7_{1}, 2_{4}, a_{5}\rangle &
\langle2_{2}, 6_{3}, 0_{5}, a_{5}\rangle &
\langle2_{2}, 7_{3}, 5_{5}, a_{6}\rangle &
\langle5_{0}, 1_{1}, 0_{4}, a_{6}\rangle &
\langle4_{0}, 1_{1}, 5_{5}, a_{7}\rangle &
\langle1_{2}, 7_{3}, 0_{7}, a_{7}\rangle \\
\langle3_{2}, 1_{3}, 7_{5}, a_{8}\rangle &
\langle3_{0}, 1_{1}, 8_{4}, a_{8}\rangle &
\langle1_{2}, 0_{3}, 2_{4}, a_{9}\rangle &
\langle1_{0}, 0_{1}, 3_{5}, a_{9}\rangle &
\langle2_{1}, 2_{3}, 8_{7}, a_{10}\rangle &
\langle2_{0}, 2_{2}, 0_{4}, a_{10}\rangle &
\langle0_{1}, 8_{3}, 5_{4}, a_{11}\rangle &
\langle1_{0}, 2_{2}, 8_{5}, a_{11}\rangle \\
\langle2_{1}, 6_{3}, 2_{4}, a_{12}\rangle &
\langle1_{0}, 3_{2}, 1_{6}, a_{12}\rangle &
\langle1_{1}, 4_{3}, 6_{5}, a_{13}\rangle &
\langle1_{0}, 4_{2}, 3_{6}, a_{13}\rangle &
\langle2_{1}, 4_{3}, 1_{6}, a_{14}\rangle &
\langle1_{0}, 5_{2}, 4_{5}, a_{14}\rangle &
\langle0_{1}, 5_{3}, 6_{4}, a_{15}\rangle &
\langle2_{0}, 7_{2}, 3_{6}, a_{15}\rangle \\
\langle0_{0}, 1_{3}, 8_{6}, 6_{5}\rangle &
\langle0_{3}, 1_{3}, 0_{4}, 5_{6}\rangle &
\langle0_{3}, 6_{3}, 1_{7}, 8_{6}\rangle &
\langle0_{2}, 2_{2}, 2_{4}, 5_{4}\rangle &
\langle0_{0}, 7_{2}, 4_{7}, 8_{7}\rangle &
\langle0_{0}, 8_{2}, 5_{6}, 3_{7}\rangle &
\langle0_{1}, 6_{2}, 2_{5}, 6_{6}\rangle &
\langle0_{3}, 2_{3}, 0_{7}, 3_{6}\rangle \\
\langle0_{2}, 4_{2}, 8_{4}, 2_{7}\rangle &
\langle0_{3}, 4_{3}, 3_{7}, 7_{4}\rangle &
\langle2_{0}, 8_{2}, 0_{6}, 4_{7}\rangle &
\langle1_{1}, 6_{2}, 8_{6}, 0_{7}\rangle &
\langle1_{1}, 2_{2}, 6_{6}, 2_{7}\rangle &
\langle0_{1}, 1_{1}, 0_{5}, 7_{5}\rangle &
\langle0_{0}, 3_{0}, 1_{7}, 0_{7}\rangle &
\langle0_{0}, 1_{0}, 4_{6}, 5_{5}\rangle \\
\langle0_{1}, 3_{1}, 7_{7}, 5_{7}\rangle &
\langle0_{1}, 7_{1}, 2_{6}, 1_{6}\rangle &
\langle0_{0}, 5_{3}, 0_{4}, 0_{5}\rangle &
\end{array}$$ ]{}
$n=53$: [$$\begin{array}{llllllllll}
\langle\infty, 3_{1}, 3_{6}, 3_{7}\rangle &
\langle\infty, 4_{0}, 4_{4}, 4_{5}\rangle &
\langle0_{1}, 0_{0}, 1_{4}, a_{1}\rangle &
\langle3_{2}, 3_{3}, 4_{5}, a_{1}\rangle &
\langle5_{2}, 6_{3}, 1_{6}, a_{2}\rangle &
\langle0_{1}, 8_{0}, 2_{4}, a_{2}\rangle &
\langle4_{1}, 2_{0}, 4_{4}, a_{3}\rangle &
\langle5_{2}, 7_{3}, 7_{5}, a_{3}\rangle \\
\langle8_{0}, 2_{1}, 5_{4}, a_{4}\rangle &
\langle7_{2}, 1_{3}, 6_{5}, a_{4}\rangle &
\langle4_{0}, 8_{1}, 3_{4}, a_{5}\rangle &
\langle5_{2}, 0_{3}, 2_{7}, a_{5}\rangle &
\langle1_{1}, 5_{0}, 0_{4}, a_{6}\rangle &
\langle5_{2}, 1_{3}, 8_{5}, a_{6}\rangle &
\langle2_{1}, 5_{0}, 6_{5}, a_{7}\rangle &
\langle4_{2}, 1_{3}, 4_{4}, a_{7}\rangle \\
\langle6_{2}, 4_{3}, 6_{6}, a_{8}\rangle &
\langle6_{1}, 8_{0}, 4_{4}, a_{8}\rangle &
\langle8_{1}, 0_{0}, 2_{5}, a_{9}\rangle &
\langle7_{2}, 6_{3}, 1_{4}, a_{9}\rangle &
\langle8_{1}, 8_{3}, 5_{5}, a_{10}\rangle &
\langle4_{2}, 4_{0}, 2_{4}, a_{10}\rangle &
\langle8_{1}, 0_{3}, 6_{7}, a_{11}\rangle &
\langle6_{0}, 5_{2}, 0_{5}, a_{11}\rangle \\
\langle2_{2}, 8_{0}, 7_{7}, a_{12}\rangle &
\langle6_{1}, 0_{3}, 2_{4}, a_{12}\rangle &
\langle6_{1}, 4_{3}, 7_{7}, a_{13}\rangle &
\langle1_{2}, 6_{0}, 8_{6}, a_{13}\rangle &
\langle7_{0}, 0_{2}, 8_{7}, a_{14}\rangle &
\langle1_{3}, 8_{1}, 0_{6}, a_{14}\rangle &
\langle5_{0}, 1_{2}, 2_{7}, a_{15}\rangle &
\langle6_{1}, 2_{3}, 3_{4}, a_{15}\rangle \\
\langle1_{2}, 4_{0}, 7_{6}, a_{16}\rangle &
\langle4_{1}, 1_{3}, 6_{7}, a_{16}\rangle &
\langle4_{0}, 2_{2}, 3_{6}, a_{17}\rangle &
\langle3_{1}, 7_{3}, 7_{7}, a_{17}\rangle &
\langle1_{2}, 4_{2}, 0_{4}, 5_{4}\rangle &
\langle0_{2}, 5_{2}, 3_{7}, 2_{4}\rangle &
\langle1_{0}, 2_{0}, 6_{5}, 6_{6}\rangle &
\langle1_{3}, 0_{3}, 0_{4}, 7_{4}\rangle \\
\langle0_{0}, 5_{0}, 7_{7}, 0_{7}\rangle &
\langle2_{0}, 7_{3}, 8_{6}, 3_{6}\rangle &
\langle1_{2}, 3_{2}, 5_{6}, 8_{5}\rangle &
\langle3_{1}, 6_{2}, 8_{7}, 0_{6}\rangle &
\langle4_{1}, 2_{1}, 6_{6}, 7_{6}\rangle &
\langle1_{1}, 2_{1}, 1_{5}, 0_{6}\rangle &
\langle6_{0}, 7_{2}, 6_{6}, 4_{5}\rangle &
\langle0_{0}, 7_{0}, 6_{5}, 3_{7}\rangle \\
\langle2_{1}, 6_{1}, 5_{7}, 7_{5}\rangle &
\langle1_{3}, 5_{3}, 0_{7}, 4_{5}\rangle &
\langle3_{1}, 5_{2}, 0_{7}, 5_{5}\rangle &
\langle0_{3}, 6_{3}, 7_{7}, 3_{6}\rangle &
\langle0_{3}, 2_{3}, 4_{5}, 0_{6}\rangle &
\end{array}$$ ]{}
$n=55$: [$$\begin{array}{llllllllll}
\langle\infty, 4_{2}, 1_{8}, 4_{9}\rangle &
\langle\infty, 3_{1}, 3_{7}, 6_{6}\rangle &
\langle\infty, 1_{0}, 1_{4}, 0_{5}\rangle &
\langle4_{1}, 7_{0}, 3_{7}, a_{1}\rangle &
\langle4_{3}, 4_{2}, 3_{6}, a_{1}\rangle &
\langle3_{1}, 8_{1}, 5_{6}, 0_{8}\rangle &
\langle6_{0}, 4_{1}, 8_{7}, 5_{7}\rangle &
\langle4_{0}, 5_{3}, 8_{9}, 7_{9}\rangle \\
\langle5_{0}, 7_{3}, 7_{5}, 0_{6}\rangle &
\langle2_{1}, 8_{1}, 7_{4}, 6_{8}\rangle &
\langle3_{2}, 0_{3}, 6_{8}, 6_{7}\rangle &
\langle5_{0}, 5_{3}, 5_{9}, 3_{6}\rangle &
\langle3_{0}, 5_{2}, 3_{8}, 1_{7}\rangle &
\langle0_{1}, 1_{0}, 4_{5}, 0_{9}\rangle &
\langle2_{2}, 7_{0}, 5_{9}, 5_{5}\rangle &
\langle4_{0}, 3_{2}, 1_{5}, 4_{5}\rangle \\
\langle0_{3}, 1_{2}, 5_{4}, 3_{8}\rangle &
\langle4_{0}, 8_{1}, 8_{8}, 4_{7}\rangle &
\langle2_{0}, 3_{0}, 3_{6}, 8_{6}\rangle &
\langle2_{3}, 3_{3}, 8_{6}, 2_{4}\rangle &
\langle7_{1}, 6_{2}, 4_{7}, 4_{9}\rangle &
\langle6_{1}, 1_{3}, 5_{8}, 4_{7}\rangle &
\langle3_{0}, 1_{0}, 0_{4}, 6_{8}\rangle &
\langle2_{3}, 8_{3}, 1_{5}, 3_{9}\rangle \\
\langle4_{0}, 6_{1}, 7_{4}, 2_{8}\rangle &
\langle5_{1}, 6_{2}, 1_{6}, 1_{5}\rangle &
\langle5_{2}, 2_{2}, 4_{5}, 8_{6}\rangle &
\langle1_{1}, 3_{1}, 2_{5}, 5_{9}\rangle &
\langle6_{0}, 5_{3}, 2_{4}, 7_{8}\rangle &
\langle5_{1}, 5_{3}, 4_{9}, 0_{6}\rangle &
\langle6_{0}, 3_{3}, 8_{9}, 1_{4}\rangle &
\langle2_{1}, 8_{2}, 6_{4}, 8_{5}\rangle \\
\langle3_{2}, 2_{2}, 3_{6}, 2_{4}\rangle &
\langle6_{2}, 2_{3}, 8_{9}, 2_{6}\rangle &
\langle3_{2}, 6_{3}, 7_{8}, 4_{7}\rangle &
\langle5_{2}, 6_{3}, 6_{8}, 4_{9}\rangle &
\langle1_{1}, 0_{1}, 3_{4}, 3_{5}\rangle &
\langle6_{0}, 4_{3}, 5_{6}, 0_{7}\rangle &
\langle1_{0}, 6_{3}, 5_{7}, 2_{5}\rangle &
\langle5_{1}, 6_{3}, 3_{5}, 8_{7}\rangle \\
\langle2_{3}, 0_{3}, 4_{4}, 3_{4}\rangle &
\langle6_{0}, 7_{1}, 4_{4}, 7_{4}\rangle &
\langle8_{0}, 8_{2}, 5_{7}, 4_{5}\rangle &
\langle5_{0}, 8_{1}, 6_{9}, 7_{6}\rangle &
\langle8_{0}, 2_{3}, 1_{8}, 3_{5}\rangle &
\langle3_{1}, 1_{3}, 4_{6}, 6_{8}\rangle &
\langle5_{2}, 3_{3}, 1_{8}, 4_{7}\rangle &
\langle8_{1}, 2_{2}, 8_{5}, 1_{8}\rangle \\
\langle6_{2}, 6_{1}, 2_{9}, 8_{7}\rangle &
\langle5_{1}, 7_{2}, 3_{4}, 8_{9}\rangle &
\langle4_{0}, 0_{1}, 1_{9}, 7_{6}\rangle &
\langle2_{3}, 7_{3}, 5_{5}, 2_{7}\rangle &
\langle2_{2}, 4_{2}, 8_{9}, 5_{4}\rangle &
\langle2_{2}, 7_{2}, 4_{4}, 0_{6}\rangle &
\langle0_{0}, 6_{2}, 1_{7}, 6_{8}\rangle &
\end{array}$$ ]{}
$n=57$: [$$\begin{array}{llllllllll}
\langle\infty, 2_{0}, 2_{5}, 2_{4}\rangle &
\langle\infty, 3_{1}, 5_{7}, 6_{6}\rangle &
\langle\infty, 2_{2}, 3_{9}, 0_{8}\rangle &
\langle5_{2}, 5_{3}, 3_{6}, a_{1}\rangle &
\langle6_{1}, 6_{0}, 7_{4}, a_{1}\rangle &
\langle0_{1}, 1_{0}, 4_{4}, a_{2}\rangle &
\langle7_{3}, 6_{2}, 0_{6}, a_{2}\rangle &
\langle8_{2}, 3_{3}, 2_{5}, a_{3}\rangle \\
\langle2_{1}, 6_{0}, 1_{6}, a_{3}\rangle &
\langle2_{0}, 8_{3}, 4_{5}, 4_{8}\rangle &
\langle5_{0}, 5_{3}, 5_{8}, 4_{6}\rangle &
\langle5_{0}, 7_{2}, 4_{8}, 8_{7}\rangle &
\langle3_{2}, 3_{0}, 8_{8}, 8_{6}\rangle &
\langle4_{3}, 6_{3}, 7_{4}, 7_{8}\rangle &
\langle2_{0}, 6_{1}, 5_{8}, 6_{4}\rangle &
\langle5_{0}, 3_{1}, 8_{9}, 2_{7}\rangle \\
\langle1_{2}, 2_{2}, 1_{5}, 7_{9}\rangle &
\langle3_{0}, 2_{2}, 4_{7}, 7_{7}\rangle &
\langle3_{1}, 6_{1}, 7_{7}, 6_{8}\rangle &
\langle4_{2}, 0_{0}, 1_{6}, 7_{9}\rangle &
\langle2_{0}, 3_{3}, 4_{7}, 8_{9}\rangle &
\langle0_{1}, 4_{1}, 5_{8}, 1_{6}\rangle &
\langle3_{0}, 5_{3}, 7_{5}, 8_{7}\rangle &
\langle4_{0}, 7_{3}, 7_{6}, 6_{4}\rangle \\
\langle5_{1}, 6_{3}, 7_{5}, 8_{9}\rangle &
\langle6_{2}, 5_{3}, 7_{4}, 1_{4}\rangle &
\langle1_{2}, 6_{2}, 1_{9}, 6_{4}\rangle &
\langle2_{0}, 6_{0}, 6_{9}, 4_{6}\rangle &
\langle1_{1}, 8_{1}, 8_{6}, 8_{9}\rangle &
\langle1_{3}, 7_{3}, 4_{6}, 5_{9}\rangle &
\langle5_{2}, 1_{3}, 5_{6}, 0_{7}\rangle &
\langle4_{1}, 3_{2}, 0_{5}, 5_{9}\rangle \\
\langle2_{0}, 7_{2}, 8_{8}, 1_{7}\rangle &
\langle6_{1}, 0_{2}, 6_{7}, 8_{9}\rangle &
\langle4_{0}, 3_{3}, 0_{5}, 2_{8}\rangle &
\langle1_{3}, 6_{3}, 7_{9}, 6_{7}\rangle &
\langle4_{1}, 3_{3}, 7_{4}, 1_{8}\rangle &
\langle2_{0}, 8_{0}, 5_{5}, 8_{6}\rangle &
\langle2_{0}, 4_{0}, 3_{5}, 3_{9}\rangle &
\langle1_{1}, 6_{3}, 6_{4}, 4_{5}\rangle \\
\langle0_{1}, 6_{2}, 5_{7}, 4_{9}\rangle &
\langle7_{1}, 5_{3}, 0_{8}, 8_{5}\rangle &
\langle4_{1}, 4_{2}, 3_{4}, 6_{6}\rangle &
\langle7_{1}, 1_{3}, 5_{5}, 4_{9}\rangle &
\langle2_{0}, 1_{0}, 0_{4}, 7_{4}\rangle &
\langle6_{1}, 2_{2}, 4_{8}, 1_{6}\rangle &
\langle4_{0}, 1_{1}, 4_{7}, 5_{8}\rangle &
\langle8_{2}, 6_{3}, 6_{5}, 3_{8}\rangle \\
\langle3_{1}, 4_{2}, 0_{5}, 1_{4}\rangle &
\langle7_{1}, 4_{0}, 6_{9}, 2_{5}\rangle &
\langle4_{2}, 0_{1}, 2_{4}, 6_{4}\rangle &
\langle2_{2}, 8_{2}, 3_{6}, 2_{8}\rangle &
\langle2_{3}, 5_{2}, 8_{4}, 4_{8}\rangle &
\langle3_{2}, 6_{3}, 3_{7}, 1_{7}\rangle &
\langle3_{1}, 1_{2}, 3_{5}, 2_{5}\rangle &
\langle3_{1}, 3_{3}, 8_{6}, 1_{7}\rangle \\
\langle0_{0}, 5_{3}, 7_{7}, 5_{9}\rangle &
\end{array}$$ ]{}
$n=59$: [$$\begin{array}{llllllllll}
\langle\infty, 7_{1}, 0_{7}, 2_{6}\rangle &
\langle\infty, 0_{3}, 6_{9}, 4_{8}\rangle &
\langle\infty, 7_{0}, 7_{5}, 1_{4}\rangle &
\langle5_{1}, 2_{0}, 1_{5}, a_{1}\rangle &
\langle8_{3}, 8_{2}, 1_{6}, a_{1}\rangle &
\langle0_{1}, 4_{0}, 6_{4}, a_{2}\rangle &
\langle1_{3}, 5_{2}, 6_{6}, a_{2}\rangle &
\langle4_{2}, 6_{3}, 4_{5}, a_{3}\rangle \\
\langle3_{0}, 4_{1}, 7_{6}, a_{3}\rangle &
\langle4_{3}, 5_{2}, 1_{5}, a_{4}\rangle &
\langle6_{1}, 4_{0}, 4_{6}, a_{4}\rangle &
\langle7_{0}, 6_{1}, 6_{4}, a_{5}\rangle &
\langle8_{3}, 4_{2}, 2_{7}, a_{5}\rangle &
\langle3_{1}, 5_{2}, 7_{5}, 7_{7}\rangle &
\langle0_{3}, 6_{3}, 8_{9}, 1_{7}\rangle &
\langle8_{3}, 7_{1}, 0_{4}, 8_{5}\rangle \\
\langle3_{3}, 7_{3}, 7_{6}, 6_{8}\rangle &
\langle5_{0}, 6_{3}, 7_{9}, 7_{6}\rangle &
\langle8_{0}, 6_{2}, 1_{8}, 7_{7}\rangle &
\langle4_{0}, 0_{0}, 4_{8}, 7_{6}\rangle &
\langle4_{0}, 7_{2}, 2_{5}, 3_{8}\rangle &
\langle0_{1}, 0_{3}, 0_{8}, 5_{7}\rangle &
\langle5_{0}, 4_{0}, 2_{8}, 1_{7}\rangle &
\langle8_{3}, 8_{0}, 0_{5}, 8_{9}\rangle \\
\langle3_{0}, 6_{3}, 8_{5}, 8_{8}\rangle &
\langle2_{0}, 7_{2}, 6_{9}, 6_{5}\rangle &
\langle7_{2}, 1_{1}, 2_{4}, 0_{8}\rangle &
\langle2_{3}, 1_{3}, 4_{4}, 8_{4}\rangle &
\langle1_{1}, 2_{1}, 4_{9}, 2_{6}\rangle &
\langle6_{1}, 3_{1}, 4_{7}, 1_{8}\rangle &
\langle5_{0}, 3_{3}, 1_{6}, 8_{8}\rangle &
\langle1_{1}, 0_{2}, 3_{6}, 8_{4}\rangle \\
\langle1_{2}, 7_{2}, 5_{6}, 8_{8}\rangle &
\langle2_{1}, 6_{1}, 1_{4}, 7_{9}\rangle &
\langle4_{0}, 3_{3}, 5_{7}, 1_{9}\rangle &
\langle5_{0}, 0_{3}, 6_{8}, 5_{4}\rangle &
\langle3_{1}, 1_{1}, 0_{6}, 1_{9}\rangle &
\langle7_{2}, 8_{2}, 5_{9}, 1_{4}\rangle &
\langle2_{2}, 5_{3}, 0_{4}, 2_{7}\rangle &
\langle6_{2}, 3_{3}, 1_{7}, 6_{6}\rangle \\
\langle1_{2}, 8_{3}, 7_{7}, 4_{9}\rangle &
\langle2_{1}, 3_{2}, 3_{8}, 4_{5}\rangle &
\langle4_{3}, 6_{1}, 0_{5}, 2_{8}\rangle &
\langle8_{2}, 4_{2}, 7_{8}, 5_{4}\rangle &
\langle1_{0}, 7_{1}, 4_{7}, 7_{5}\rangle &
\langle6_{0}, 0_{0}, 1_{4}, 4_{7}\rangle &
\langle6_{2}, 2_{0}, 3_{6}, 4_{5}\rangle &
\langle2_{3}, 0_{1}, 6_{5}, 3_{8}\rangle \\
\langle2_{1}, 6_{2}, 6_{9}, 2_{7}\rangle &
\langle5_{2}, 3_{0}, 1_{4}, 1_{9}\rangle &
\langle6_{3}, 7_{1}, 5_{5}, 3_{6}\rangle &
\langle0_{0}, 6_{2}, 8_{9}, 6_{4}\rangle &
\langle8_{0}, 0_{2}, 5_{6}, 4_{9}\rangle &
\langle0_{1}, 5_{3}, 5_{4}, 8_{5}\rangle &
\langle5_{1}, 5_{2}, 4_{7}, 2_{8}\rangle &
\langle4_{1}, 8_{3}, 3_{9}, 7_{4}\rangle \\
\langle7_{3}, 4_{1}, 1_{9}, 7_{7}\rangle &
\langle2_{2}, 2_{0}, 1_{6}, 5_{5}\rangle &
\langle0_{0}, 2_{0}, 2_{7}, 3_{9}\rangle &
\end{array}$$ ]{}
$n=61$: [$$\begin{array}{llllllllll}
\langle\infty, 8_{3}, 0_{9}, 4_{8}\rangle &
\langle\infty, 5_{0}, 2_{4}, 6_{5}\rangle &
\langle\infty, 6_{1}, 1_{7}, 2_{6}\rangle &
\langle4_{1}, 3_{0}, 4_{4}, a_{1}\rangle &
\langle6_{2}, 1_{3}, 5_{6}, a_{1}\rangle &
\langle6_{1}, 7_{0}, 8_{7}, a_{2}\rangle &
\langle4_{2}, 5_{3}, 4_{5}, a_{2}\rangle &
\langle0_{0}, 4_{2}, 7_{6}, a_{3}\rangle \\
\langle1_{1}, 6_{3}, 7_{4}, a_{3}\rangle &
\langle4_{2}, 6_{0}, 0_{4}, a_{4}\rangle &
\langle2_{1}, 3_{3}, 2_{6}, a_{4}\rangle &
\langle1_{0}, 8_{1}, 6_{4}, a_{5}\rangle &
\langle6_{3}, 1_{2}, 0_{5}, a_{5}\rangle &
\langle6_{2}, 3_{3}, 4_{5}, a_{6}\rangle &
\langle0_{0}, 5_{1}, 3_{6}, a_{6}\rangle &
\langle5_{1}, 4_{3}, 7_{6}, a_{7}\rangle \\
\langle6_{0}, 2_{2}, 8_{7}, a_{7}\rangle &
\langle3_{0}, 7_{0}, 0_{9}, 8_{8}\rangle &
\langle7_{1}, 7_{0}, 6_{4}, 2_{6}\rangle &
\langle1_{1}, 2_{2}, 3_{8}, 0_{7}\rangle &
\langle6_{2}, 4_{0}, 4_{4}, 1_{5}\rangle &
\langle0_{3}, 0_{2}, 0_{8}, 1_{6}\rangle &
\langle3_{1}, 7_{3}, 2_{9}, 0_{9}\rangle &
\langle3_{1}, 5_{1}, 8_{8}, 2_{6}\rangle \\
\langle6_{2}, 0_{3}, 7_{7}, 0_{7}\rangle &
\langle2_{2}, 6_{1}, 7_{6}, 0_{9}\rangle &
\langle7_{1}, 5_{2}, 1_{5}, 4_{8}\rangle &
\langle0_{1}, 3_{2}, 2_{4}, 3_{6}\rangle &
\langle8_{0}, 0_{2}, 3_{4}, 6_{4}\rangle &
\langle2_{1}, 7_{0}, 2_{9}, 3_{7}\rangle &
\langle5_{3}, 8_{3}, 1_{5}, 8_{9}\rangle &
\langle2_{0}, 8_{1}, 4_{4}, 6_{8}\rangle \\
\langle8_{2}, 6_{1}, 5_{5}, 2_{9}\rangle &
\langle6_{3}, 0_{1}, 4_{4}, 3_{4}\rangle &
\langle5_{0}, 4_{3}, 8_{8}, 8_{7}\rangle &
\langle4_{3}, 2_{1}, 5_{7}, 3_{4}\rangle &
\langle0_{0}, 5_{3}, 4_{7}, 8_{7}\rangle &
\langle6_{3}, 8_{2}, 0_{4}, 3_{7}\rangle &
\langle0_{2}, 4_{2}, 2_{8}, 1_{9}\rangle &
\langle0_{0}, 2_{3}, 0_{9}, 8_{5}\rangle \\
\langle3_{0}, 1_{0}, 0_{6}, 1_{7}\rangle &
\langle6_{3}, 1_{3}, 6_{4}, 0_{8}\rangle &
\langle7_{1}, 1_{3}, 7_{8}, 3_{7}\rangle &
\langle0_{1}, 5_{1}, 7_{5}, 1_{5}\rangle &
\langle5_{3}, 5_{1}, 5_{5}, 0_{5}\rangle &
\langle0_{1}, 1_{1}, 7_{7}, 2_{9}\rangle &
\langle4_{3}, 6_{3}, 8_{4}, 3_{9}\rangle &
\langle4_{3}, 0_{0}, 6_{8}, 2_{6}\rangle \\
\langle3_{2}, 1_{2}, 4_{5}, 3_{4}\rangle &
\langle7_{0}, 8_{0}, 7_{8}, 2_{5}\rangle &
\langle1_{0}, 7_{0}, 3_{5}, 8_{9}\rangle &
\langle0_{1}, 3_{1}, 4_{8}, 7_{9}\rangle &
\langle4_{3}, 1_{0}, 0_{9}, 1_{6}\rangle &
\langle1_{2}, 7_{2}, 0_{7}, 3_{9}\rangle &
\langle1_{0}, 2_{3}, 7_{7}, 3_{8}\rangle &
\langle1_{2}, 6_{1}, 3_{5}, 6_{7}\rangle \\
\langle1_{0}, 1_{3}, 8_{8}, 8_{5}\rangle &
\langle7_{2}, 8_{0}, 2_{9}, 4_{6}\rangle &
\langle2_{0}, 8_{2}, 7_{9}, 3_{6}\rangle &
\langle0_{2}, 8_{2}, 3_{8}, 5_{8}\rangle &
\langle0_{2}, 2_{3}, 2_{6}, 7_{6}\rangle &
\end{array}$$ ]{}
$n=63$: [$$\begin{array}{llllllllll}
\langle\infty, 8_{3}, 6_{7}, 8_{6}\rangle &
\langle\infty, 1_{0}, 5_{4}, 5_{5}\rangle &
\langle\infty, 7_{2}, 2_{9}, 5_{8}\rangle &
\langle2_{1}, 7_{3}, 3_{4}, a_{1}\rangle &
\langle5_{0}, 0_{2}, 0_{7}, a_{1}\rangle &
\langle5_{2}, 3_{3}, 6_{7}, a_{2}\rangle &
\langle3_{1}, 7_{0}, 3_{4}, a_{2}\rangle &
\langle2_{0}, 2_{1}, 4_{5}, a_{3}\rangle \\
\langle0_{3}, 7_{2}, 2_{6}, a_{3}\rangle &
\langle2_{2}, 8_{3}, 2_{6}, a_{4}\rangle &
\langle3_{0}, 7_{1}, 2_{4}, a_{4}\rangle &
\langle8_{0}, 8_{2}, 7_{6}, a_{5}\rangle &
\langle6_{3}, 6_{1}, 3_{4}, a_{5}\rangle &
\langle1_{0}, 0_{3}, 7_{6}, a_{6}\rangle &
\langle8_{1}, 6_{2}, 5_{7}, a_{6}\rangle &
\langle0_{3}, 5_{1}, 1_{7}, a_{7}\rangle \\
\langle2_{0}, 0_{2}, 2_{6}, a_{7}\rangle &
\langle5_{3}, 5_{2}, 8_{5}, a_{8}\rangle &
\langle0_{0}, 2_{1}, 0_{4}, a_{8}\rangle &
\langle2_{0}, 5_{1}, 8_{5}, a_{9}\rangle &
\langle6_{2}, 0_{3}, 2_{4}, a_{9}\rangle &
\langle1_{3}, 6_{3}, 2_{8}, 7_{5}\rangle &
\langle7_{1}, 8_{2}, 2_{7}, 3_{8}\rangle &
\langle2_{1}, 7_{1}, 1_{6}, 4_{6}\rangle \\
\langle8_{0}, 4_{2}, 2_{5}, 3_{8}\rangle &
\langle6_{0}, 0_{0}, 1_{6}, 5_{9}\rangle &
\langle5_{0}, 2_{3}, 7_{6}, 2_{4}\rangle &
\langle4_{1}, 4_{2}, 3_{4}, 6_{7}\rangle &
\langle7_{2}, 5_{1}, 6_{5}, 8_{4}\rangle &
\langle1_{0}, 2_{0}, 0_{8}, 3_{7}\rangle &
\langle1_{1}, 2_{3}, 2_{8}, 2_{7}\rangle &
\langle6_{0}, 4_{0}, 7_{9}, 2_{6}\rangle \\
\langle6_{3}, 0_{1}, 5_{4}, 4_{5}\rangle &
\langle5_{1}, 4_{2}, 3_{9}, 7_{4}\rangle &
\langle0_{1}, 4_{2}, 5_{5}, 0_{5}\rangle &
\langle1_{2}, 8_{0}, 7_{7}, 5_{7}\rangle &
\langle4_{2}, 5_{0}, 8_{4}, 0_{9}\rangle &
\langle5_{1}, 8_{0}, 1_{9}, 4_{5}\rangle &
\langle7_{0}, 4_{2}, 6_{5}, 8_{5}\rangle &
\langle4_{2}, 0_{2}, 7_{9}, 1_{6}\rangle \\
\langle2_{1}, 3_{1}, 0_{5}, 5_{8}\rangle &
\langle5_{1}, 8_{1}, 3_{8}, 6_{6}\rangle &
\langle1_{0}, 5_{3}, 4_{7}, 3_{4}\rangle &
\langle2_{1}, 1_{3}, 2_{6}, 1_{9}\rangle &
\langle3_{3}, 3_{0}, 4_{4}, 6_{8}\rangle &
\langle2_{1}, 8_{2}, 8_{9}, 6_{6}\rangle &
\langle0_{3}, 7_{1}, 1_{9}, 5_{7}\rangle &
\langle4_{3}, 1_{0}, 7_{9}, 8_{4}\rangle \\
\langle7_{0}, 8_{3}, 7_{5}, 7_{8}\rangle &
\langle7_{2}, 1_{2}, 4_{8}, 7_{5}\rangle &
\langle0_{3}, 1_{3}, 7_{8}, 8_{9}\rangle &
\langle1_{2}, 2_{3}, 6_{8}, 6_{6}\rangle &
\langle2_{1}, 7_{2}, 8_{8}, 5_{7}\rangle &
\langle6_{0}, 2_{0}, 7_{8}, 2_{7}\rangle &
\langle4_{1}, 2_{3}, 6_{9}, 4_{8}\rangle &
\langle0_{3}, 5_{2}, 8_{6}, 2_{9}\rangle \\
\langle5_{0}, 3_{1}, 8_{6}, 2_{8}\rangle &
\langle4_{2}, 2_{2}, 4_{8}, 4_{4}\rangle &
\langle4_{0}, 6_{3}, 2_{9}, 2_{5}\rangle &
\langle4_{3}, 1_{1}, 1_{7}, 1_{9}\rangle &
\langle0_{2}, 8_{2}, 1_{9}, 6_{4}\rangle &
\langle7_{0}, 6_{1}, 5_{7}, 7_{9}\rangle &
\langle0_{3}, 2_{3}, 2_{5}, 4_{7}\rangle &
\end{array}$$ ]{}
$n=65$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{3}, 8_{9}, 3_{6}\rangle &
\langle\infty, 4_{2}, 6_{4}, 8_{7}\rangle &
\langle\infty, 0_{1}, 3_{5}, 5_{8}\rangle &
\langle6_{3}, 6_{2}, 7_{7}, a_{1}\rangle &
\langle8_{0}, 0_{1}, 5_{4}, a_{1}\rangle &
\langle1_{2}, 6_{0}, 0_{7}, a_{2}\rangle &
\langle1_{1}, 2_{3}, 0_{4}, a_{2}\rangle &
\langle2_{1}, 7_{2}, 5_{6}, a_{3}\rangle \\
\langle5_{0}, 6_{3}, 8_{4}, a_{3}\rangle &
\langle7_{3}, 7_{1}, 0_{6}, a_{4}\rangle &
\langle6_{0}, 7_{2}, 8_{4}, a_{4}\rangle &
\langle6_{3}, 5_{2}, 7_{6}, a_{5}\rangle &
\langle4_{0}, 2_{1}, 8_{4}, a_{5}\rangle &
\langle2_{1}, 3_{0}, 3_{4}, a_{6}\rangle &
\langle7_{2}, 2_{3}, 7_{5}, a_{6}\rangle &
\langle4_{2}, 3_{3}, 7_{4}, a_{7}\rangle \\
\langle2_{1}, 7_{0}, 3_{6}, a_{7}\rangle &
\langle7_{2}, 5_{1}, 3_{7}, a_{8}\rangle &
\langle7_{0}, 3_{3}, 4_{5}, a_{8}\rangle &
\langle2_{3}, 0_{1}, 4_{5}, a_{9}\rangle &
\langle0_{0}, 5_{2}, 2_{6}, a_{9}\rangle &
\langle8_{2}, 0_{1}, 0_{5}, a_{10}\rangle &
\langle0_{3}, 5_{0}, 5_{7}, a_{10}\rangle &
\langle3_{0}, 2_{2}, 6_{6}, a_{11}\rangle \\
\langle1_{1}, 6_{3}, 1_{7}, a_{11}\rangle &
\langle2_{1}, 0_{2}, 2_{9}, 8_{5}\rangle &
\langle2_{0}, 0_{2}, 2_{5}, 7_{8}\rangle &
\langle3_{2}, 0_{2}, 3_{7}, 3_{6}\rangle &
\langle0_{1}, 8_{1}, 0_{8}, 8_{6}\rangle &
\langle4_{1}, 2_{1}, 6_{9}, 8_{7}\rangle &
\langle4_{2}, 0_{1}, 5_{9}, 1_{9}\rangle &
\langle0_{0}, 3_{2}, 4_{8}, 7_{4}\rangle \\
\langle4_{0}, 6_{1}, 1_{8}, 5_{7}\rangle &
\langle5_{0}, 7_{0}, 0_{9}, 0_{5}\rangle &
\langle6_{1}, 0_{3}, 1_{4}, 8_{7}\rangle &
\langle6_{0}, 8_{3}, 8_{7}, 0_{9}\rangle &
\langle1_{2}, 0_{1}, 6_{6}, 8_{9}\rangle &
\langle2_{3}, 1_{3}, 4_{7}, 5_{8}\rangle &
\langle4_{2}, 2_{3}, 7_{8}, 3_{8}\rangle &
\langle7_{3}, 2_{2}, 7_{4}, 2_{9}\rangle \\
\langle1_{0}, 8_{3}, 7_{6}, 2_{9}\rangle &
\langle4_{0}, 1_{1}, 2_{5}, 0_{5}\rangle &
\langle1_{1}, 7_{2}, 7_{8}, 4_{7}\rangle &
\langle1_{3}, 7_{0}, 5_{6}, 6_{9}\rangle &
\langle0_{2}, 7_{2}, 8_{6}, 5_{5}\rangle &
\langle6_{1}, 1_{3}, 7_{7}, 8_{5}\rangle &
\langle0_{0}, 3_{1}, 1_{6}, 0_{9}\rangle &
\langle2_{2}, 3_{2}, 7_{9}, 1_{4}\rangle \\
\langle8_{3}, 1_{3}, 8_{6}, 8_{9}\rangle &
\langle6_{0}, 8_{2}, 2_{9}, 5_{5}\rangle &
\langle7_{0}, 7_{3}, 6_{4}, 4_{9}\rangle &
\langle6_{1}, 6_{0}, 2_{7}, 4_{9}\rangle &
\langle7_{0}, 2_{0}, 3_{4}, 6_{6}\rangle &
\langle7_{0}, 4_{2}, 0_{8}, 8_{5}\rangle &
\langle3_{2}, 7_{2}, 5_{7}, 0_{8}\rangle &
\langle4_{0}, 5_{0}, 3_{7}, 5_{8}\rangle \\
\langle1_{3}, 6_{3}, 0_{5}, 8_{8}\rangle &
\langle0_{0}, 8_{3}, 7_{8}, 3_{5}\rangle &
\langle8_{0}, 4_{1}, 8_{6}, 7_{8}\rangle &
\langle2_{0}, 8_{3}, 6_{7}, 5_{8}\rangle &
\langle8_{3}, 2_{2}, 5_{5}, 1_{9}\rangle &
\langle1_{3}, 2_{1}, 7_{6}, 1_{8}\rangle &
\langle7_{1}, 3_{1}, 5_{8}, 5_{4}\rangle &
\langle4_{1}, 2_{3}, 2_{5}, 7_{4}\rangle \\
\langle0_{2}, 3_{3}, 0_{4}, 6_{4}\rangle &
\end{array}$$ ]{}
$n=67$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{2}, 6_{9}, 0_{8}\rangle &
\langle\infty, 7_{1}, 3_{6}, 4_{7}\rangle &
\langle\infty, 2_{3}, 1_{4}, 0_{5}\rangle &
\langle8_{3}, 4_{0}, 5_{6}, a_{1}\rangle &
\langle2_{2}, 6_{1}, 7_{7}, a_{1}\rangle &
\langle1_{3}, 7_{0}, 4_{6}, a_{2}\rangle &
\langle8_{2}, 1_{1}, 4_{4}, a_{2}\rangle &
\langle5_{2}, 6_{3}, 0_{4}, a_{3}\rangle \\
\langle3_{0}, 7_{1}, 7_{7}, a_{3}\rangle &
\langle1_{1}, 7_{2}, 5_{5}, a_{4}\rangle &
\langle8_{3}, 8_{0}, 4_{7}, a_{4}\rangle &
\langle8_{0}, 2_{1}, 5_{7}, a_{5}\rangle &
\langle7_{3}, 3_{2}, 6_{5}, a_{5}\rangle &
\langle7_{3}, 6_{1}, 8_{5}, a_{6}\rangle &
\langle5_{0}, 1_{2}, 4_{4}, a_{6}\rangle &
\langle3_{3}, 2_{0}, 5_{4}, a_{7}\rangle \\
\langle1_{2}, 2_{1}, 5_{5}, a_{7}\rangle &
\langle0_{0}, 8_{1}, 7_{4}, a_{8}\rangle &
\langle5_{2}, 8_{3}, 4_{5}, a_{8}\rangle &
\langle2_{3}, 5_{0}, 2_{4}, a_{9}\rangle &
\langle1_{1}, 1_{2}, 0_{7}, a_{9}\rangle &
\langle1_{3}, 4_{1}, 8_{6}, a_{10}\rangle &
\langle5_{2}, 8_{0}, 0_{7}, a_{10}\rangle &
\langle3_{0}, 4_{2}, 1_{7}, a_{11}\rangle \\
\langle7_{3}, 2_{1}, 7_{5}, a_{11}\rangle &
\langle6_{1}, 8_{3}, 1_{7}, a_{12}\rangle &
\langle7_{0}, 1_{2}, 7_{5}, a_{12}\rangle &
\langle3_{1}, 1_{3}, 8_{7}, a_{13}\rangle &
\langle8_{2}, 1_{0}, 8_{6}, a_{13}\rangle &
\langle3_{0}, 7_{0}, 7_{4}, 8_{5}\rangle &
\langle3_{3}, 7_{3}, 3_{6}, 7_{9}\rangle &
\langle4_{1}, 0_{1}, 3_{6}, 2_{8}\rangle \\
\langle1_{2}, 0_{1}, 6_{6}, 7_{4}\rangle &
\langle4_{2}, 0_{0}, 8_{6}, 2_{9}\rangle &
\langle4_{3}, 0_{1}, 5_{8}, 0_{4}\rangle &
\langle1_{2}, 2_{2}, 0_{6}, 3_{8}\rangle &
\langle3_{0}, 8_{1}, 2_{9}, 6_{5}\rangle &
\langle1_{3}, 1_{1}, 2_{9}, 2_{4}\rangle &
\langle2_{0}, 1_{2}, 5_{8}, 7_{6}\rangle &
\langle8_{0}, 5_{1}, 8_{8}, 5_{5}\rangle \\
\langle1_{2}, 1_{3}, 0_{9}, 3_{6}\rangle &
\langle6_{0}, 7_{1}, 0_{9}, 5_{7}\rangle &
\langle1_{0}, 1_{2}, 2_{4}, 1_{9}\rangle &
\langle1_{3}, 5_{2}, 7_{7}, 3_{8}\rangle &
\langle4_{1}, 3_{1}, 0_{4}, 4_{8}\rangle &
\langle5_{3}, 7_{0}, 0_{6}, 5_{8}\rangle &
\langle6_{2}, 2_{2}, 2_{5}, 0_{7}\rangle &
\langle0_{0}, 2_{1}, 4_{6}, 8_{8}\rangle \\
\langle2_{3}, 6_{0}, 8_{8}, 8_{5}\rangle &
\langle2_{0}, 3_{0}, 8_{8}, 1_{5}\rangle &
\langle2_{2}, 0_{3}, 3_{5}, 3_{9}\rangle &
\langle2_{3}, 4_{3}, 6_{5}, 7_{8}\rangle &
\langle3_{0}, 6_{0}, 4_{9}, 6_{6}\rangle &
\langle3_{3}, 6_{2}, 7_{7}, 2_{8}\rangle &
\langle3_{0}, 5_{3}, 6_{7}, 7_{9}\rangle &
\langle4_{3}, 7_{3}, 2_{8}, 2_{4}\rangle \\
\langle6_{2}, 4_{0}, 6_{7}, 0_{9}\rangle &
\langle6_{1}, 0_{1}, 7_{6}, 6_{9}\rangle &
\langle4_{0}, 2_{1}, 8_{5}, 4_{7}\rangle &
\langle6_{2}, 8_{3}, 0_{6}, 7_{6}\rangle &
\langle0_{1}, 2_{2}, 8_{8}, 4_{9}\rangle &
\langle3_{0}, 3_{1}, 7_{8}, 5_{4}\rangle &
\langle6_{1}, 4_{1}, 5_{5}, 2_{9}\rangle &
\langle6_{3}, 7_{3}, 6_{7}, 3_{9}\rangle \\
\langle5_{3}, 6_{0}, 3_{9}, 2_{4}\rangle &
\langle3_{2}, 5_{2}, 3_{4}, 0_{9}\rangle &
\langle0_{2}, 3_{2}, 2_{4}, 3_{8}\rangle &
\end{array}$$ ]{}
$n=69$: [$$\begin{array}{llllllllll}
\langle\infty, 0_{2}, 5_{8}, 3_{4}\rangle &
\langle\infty, 7_{0}, 1_{7}, 5_{5}\rangle &
\langle\infty, 2_{3}, 5_{9}, 5_{6}\rangle &
\langle5_{2}, 0_{1}, 5_{5}, a_{1}\rangle &
\langle1_{0}, 3_{3}, 8_{7}, a_{1}\rangle &
\langle5_{2}, 8_{0}, 6_{6}, a_{2}\rangle &
\langle4_{3}, 8_{1}, 0_{4}, a_{2}\rangle &
\langle6_{2}, 5_{1}, 2_{7}, a_{3}\rangle \\
\langle0_{3}, 3_{0}, 6_{4}, a_{3}\rangle &
\langle3_{2}, 0_{0}, 5_{5}, a_{4}\rangle &
\langle4_{3}, 0_{1}, 1_{6}, a_{4}\rangle &
\langle8_{2}, 6_{0}, 5_{7}, a_{5}\rangle &
\langle5_{3}, 8_{1}, 4_{6}, a_{5}\rangle &
\langle3_{0}, 3_{1}, 6_{6}, a_{6}\rangle &
\langle8_{3}, 4_{2}, 5_{7}, a_{6}\rangle &
\langle0_{3}, 0_{2}, 0_{4}, a_{7}\rangle \\
\langle0_{1}, 4_{0}, 8_{7}, a_{7}\rangle &
\langle0_{1}, 6_{0}, 7_{5}, a_{8}\rangle &
\langle8_{2}, 7_{3}, 1_{4}, a_{8}\rangle &
\langle5_{0}, 3_{3}, 8_{5}, a_{9}\rangle &
\langle1_{1}, 8_{2}, 8_{6}, a_{9}\rangle &
\langle5_{3}, 5_{1}, 5_{5}, a_{10}\rangle &
\langle5_{2}, 6_{0}, 1_{4}, a_{10}\rangle &
\langle8_{1}, 2_{3}, 3_{6}, a_{11}\rangle \\
\langle0_{0}, 4_{2}, 1_{4}, a_{11}\rangle &
\langle1_{1}, 3_{2}, 2_{5}, a_{12}\rangle &
\langle4_{0}, 3_{3}, 5_{6}, a_{12}\rangle &
\langle8_{1}, 2_{0}, 8_{6}, a_{13}\rangle &
\langle3_{2}, 4_{3}, 6_{5}, a_{13}\rangle &
\langle3_{0}, 8_{2}, 2_{6}, a_{14}\rangle &
\langle0_{3}, 2_{1}, 4_{4}, a_{14}\rangle &
\langle1_{3}, 0_{0}, 5_{7}, a_{15}\rangle \\
\langle6_{1}, 1_{2}, 2_{4}, a_{15}\rangle &
\langle2_{2}, 5_{3}, 6_{7}, 4_{8}\rangle &
\langle7_{1}, 0_{3}, 0_{6}, 7_{7}\rangle &
\langle0_{2}, 1_{1}, 2_{7}, 3_{9}\rangle &
\langle4_{3}, 8_{0}, 6_{4}, 2_{8}\rangle &
\langle3_{2}, 8_{2}, 0_{5}, 0_{9}\rangle &
\langle5_{0}, 3_{1}, 2_{5}, 8_{9}\rangle &
\langle0_{0}, 4_{1}, 1_{9}, 8_{4}\rangle \\
\langle7_{0}, 1_{3}, 0_{7}, 7_{9}\rangle &
\langle6_{1}, 5_{3}, 7_{8}, 0_{8}\rangle &
\langle1_{1}, 6_{1}, 8_{7}, 4_{9}\rangle &
\langle0_{1}, 0_{2}, 6_{6}, 8_{4}\rangle &
\langle3_{0}, 4_{1}, 7_{5}, 1_{8}\rangle &
\langle3_{0}, 1_{2}, 8_{6}, 7_{8}\rangle &
\langle5_{0}, 0_{3}, 7_{4}, 6_{8}\rangle &
\langle5_{0}, 4_{0}, 4_{8}, 5_{7}\rangle \\
\langle6_{0}, 8_{0}, 8_{5}, 8_{6}\rangle &
\langle1_{1}, 3_{1}, 8_{8}, 7_{5}\rangle &
\langle0_{0}, 0_{2}, 5_{9}, 4_{6}\rangle &
\langle0_{2}, 6_{3}, 4_{9}, 7_{8}\rangle &
\langle0_{3}, 1_{3}, 4_{5}, 5_{6}\rangle &
\langle8_{0}, 3_{0}, 5_{9}, 5_{8}\rangle &
\langle6_{3}, 1_{3}, 6_{9}, 5_{9}\rangle &
\langle6_{0}, 5_{1}, 2_{4}, 5_{9}\rangle \\
\langle5_{1}, 2_{2}, 1_{7}, 4_{9}\rangle &
\langle8_{3}, 3_{2}, 1_{9}, 1_{7}\rangle &
\langle5_{3}, 7_{2}, 1_{8}, 6_{9}\rangle &
\langle2_{1}, 5_{1}, 5_{4}, 6_{9}\rangle &
\langle1_{2}, 0_{2}, 5_{5}, 1_{8}\rangle &
\langle3_{1}, 2_{1}, 2_{8}, 6_{7}\rangle &
\langle4_{0}, 7_{0}, 4_{4}, 2_{9}\rangle &
\langle2_{3}, 4_{3}, 3_{4}, 1_{5}\rangle \\
\langle3_{1}, 4_{3}, 2_{6}, 7_{8}\rangle &
\langle3_{0}, 4_{2}, 8_{8}, 2_{5}\rangle &
\langle0_{3}, 6_{3}, 7_{5}, 0_{7}\rangle &
\langle0_{2}, 3_{2}, 8_{6}, 3_{7}\rangle &
\langle0_{1}, 3_{2}, 7_{4}, 2_{8}\rangle &
\end{array}$$ ]{}
$n=71$: [$$\begin{array}{llllllllll}
\langle\infty, 7_{1}, 0_{9}, 1_{8}\rangle &
\langle\infty, 1_{3}, 6_{5}, 6_{4}\rangle &
\langle\infty, 2_{0}, 4_{7}, 2_{6}\rangle &
\langle4_{3}, 3_{1}, 7_{5}, a_{1}\rangle &
\langle6_{0}, 6_{2}, 0_{6}, a_{1}\rangle &
\langle3_{0}, 6_{1}, 8_{6}, a_{2}\rangle &
\langle2_{2}, 4_{3}, 4_{7}, a_{2}\rangle &
\langle0_{0}, 6_{1}, 7_{6}, a_{3}\rangle \\
\langle1_{3}, 1_{2}, 1_{5}, a_{3}\rangle &
\langle6_{2}, 4_{3}, 3_{6}, a_{4}\rangle &
\langle5_{1}, 0_{0}, 3_{4}, a_{4}\rangle &
\langle7_{1}, 8_{2}, 3_{6}, a_{5}\rangle &
\langle2_{3}, 7_{0}, 4_{5}, a_{5}\rangle &
\langle7_{0}, 1_{3}, 6_{6}, a_{6}\rangle &
\langle4_{2}, 5_{1}, 8_{4}, a_{6}\rangle &
\langle4_{0}, 6_{2}, 8_{5}, a_{7}\rangle \\
\langle2_{1}, 4_{3}, 8_{4}, a_{7}\rangle &
\langle4_{3}, 1_{2}, 2_{5}, a_{8}\rangle &
\langle1_{0}, 5_{1}, 3_{6}, a_{8}\rangle &
\langle2_{1}, 3_{0}, 3_{7}, a_{9}\rangle &
\langle3_{3}, 6_{2}, 5_{6}, a_{9}\rangle &
\langle5_{3}, 1_{1}, 2_{5}, a_{10}\rangle &
\langle6_{2}, 5_{0}, 0_{4}, a_{10}\rangle &
\langle5_{2}, 0_{0}, 3_{5}, a_{11}\rangle \\
\langle3_{3}, 3_{1}, 0_{6}, a_{11}\rangle &
\langle4_{1}, 7_{3}, 1_{7}, a_{12}\rangle &
\langle1_{0}, 7_{2}, 6_{4}, a_{12}\rangle &
\langle4_{1}, 1_{2}, 8_{6}, a_{13}\rangle &
\langle5_{3}, 7_{0}, 6_{7}, a_{13}\rangle &
\langle0_{0}, 0_{1}, 0_{5}, a_{14}\rangle &
\langle3_{3}, 2_{2}, 0_{7}, a_{14}\rangle &
\langle5_{0}, 4_{3}, 3_{4}, a_{15}\rangle \\
\langle0_{1}, 7_{2}, 7_{7}, a_{15}\rangle &
\langle7_{1}, 3_{3}, 1_{6}, a_{16}\rangle &
\langle5_{2}, 7_{0}, 0_{5}, a_{16}\rangle &
\langle6_{3}, 0_{1}, 5_{7}, a_{17}\rangle &
\langle1_{0}, 5_{2}, 3_{4}, a_{17}\rangle &
\langle0_{1}, 5_{1}, 8_{7}, 4_{8}\rangle &
\langle1_{0}, 2_{3}, 7_{9}, 2_{9}\rangle &
\langle0_{0}, 1_{1}, 8_{5}, 5_{7}\rangle \\
\langle1_{2}, 5_{2}, 4_{7}, 6_{8}\rangle &
\langle1_{0}, 7_{3}, 8_{9}, 6_{5}\rangle &
\langle5_{1}, 0_{2}, 5_{7}, 0_{4}\rangle &
\langle1_{2}, 5_{1}, 6_{9}, 7_{7}\rangle &
\langle4_{2}, 0_{3}, 8_{8}, 4_{6}\rangle &
\langle0_{1}, 7_{3}, 5_{9}, 1_{4}\rangle &
\langle1_{0}, 5_{0}, 6_{8}, 8_{7}\rangle &
\langle1_{1}, 3_{2}, 5_{9}, 2_{8}\rangle \\
\langle3_{0}, 1_{1}, 8_{9}, 0_{6}\rangle &
\langle6_{1}, 0_{1}, 0_{9}, 8_{4}\rangle &
\langle6_{3}, 0_{3}, 0_{6}, 1_{5}\rangle &
\langle2_{0}, 5_{0}, 6_{7}, 3_{5}\rangle &
\langle6_{1}, 5_{3}, 2_{4}, 6_{6}\rangle &
\langle0_{0}, 2_{0}, 1_{4}, 6_{8}\rangle &
\langle4_{1}, 3_{1}, 6_{5}, 0_{8}\rangle &
\langle0_{3}, 2_{3}, 4_{9}, 2_{8}\rangle \\
\langle3_{2}, 7_{3}, 4_{9}, 1_{9}\rangle &
\langle2_{0}, 3_{0}, 2_{9}, 1_{8}\rangle &
\langle5_{1}, 3_{0}, 3_{8}, 5_{8}\rangle &
\langle0_{2}, 1_{2}, 2_{4}, 6_{4}\rangle &
\langle8_{3}, 4_{3}, 6_{4}, 6_{7}\rangle &
\langle7_{0}, 6_{2}, 8_{6}, 1_{9}\rangle &
\langle8_{1}, 6_{1}, 5_{5}, 5_{9}\rangle &
\langle0_{0}, 0_{3}, 3_{8}, 0_{4}\rangle \\
\langle5_{2}, 5_{1}, 7_{8}, 1_{5}\rangle &
\langle1_{0}, 6_{3}, 7_{4}, 5_{9}\rangle &
\langle2_{2}, 4_{2}, 2_{8}, 1_{5}\rangle &
\langle2_{3}, 3_{2}, 6_{8}, 0_{8}\rangle &
\langle1_{2}, 7_{2}, 2_{7}, 7_{9}\rangle &
\langle1_{0}, 4_{2}, 3_{9}, 5_{6}\rangle &
\langle0_{3}, 1_{3}, 5_{7}, 6_{8}\rangle &
\end{array}$$ ]{}
$n=73$: [$$\begin{array}{llllllllll}
\langle\infty, 1_{3}, 5_{5}, 6_{6}\rangle &
\langle\infty, 4_{1}, 1_{10}, 5_{9}\rangle &
\langle\infty, 3_{0}, 6_{7}, 0_{11}\rangle &
\langle\infty, 0_{2}, 1_{8}, 1_{4}\rangle &
\langle3_{1}, 5_{0}, 8_{6}, a_{1}\rangle &
\langle4_{2}, 7_{3}, 8_{7}, a_{1}\rangle &
\langle2_{1}, 8_{0}, 2_{5}, 1_{9}\rangle &
\langle4_{2}, 2_{3}, 8_{9}, 2_{10}\rangle \\
\langle0_{1}, 5_{3}, 7_{4}, 8_{5}\rangle &
\langle1_{2}, 2_{0}, 3_{4}, 0_{6}\rangle &
\langle3_{1}, 3_{3}, 3_{8}, 5_{10}\rangle &
\langle1_{3}, 7_{0}, 0_{8}, 6_{7}\rangle &
\langle1_{0}, 0_{3}, 1_{11}, 8_{5}\rangle &
\langle1_{0}, 3_{3}, 0_{6}, 7_{7}\rangle &
\langle4_{2}, 0_{2}, 4_{8}, 0_{4}\rangle &
\langle6_{0}, 3_{1}, 5_{9}, 6_{8}\rangle \\
\langle1_{3}, 8_{3}, 3_{8}, 3_{6}\rangle &
\langle1_{1}, 2_{3}, 5_{8}, 5_{9}\rangle &
\langle7_{0}, 3_{1}, 8_{10}, 7_{7}\rangle &
\langle1_{0}, 8_{2}, 5_{9}, 7_{8}\rangle &
\langle2_{1}, 0_{2}, 1_{7}, 8_{9}\rangle &
\langle3_{1}, 2_{3}, 7_{10}, 6_{4}\rangle &
\langle3_{2}, 3_{3}, 4_{10}, 1_{7}\rangle &
\langle0_{0}, 5_{2}, 3_{8}, 1_{5}\rangle \\
\langle0_{0}, 1_{1}, 4_{10}, 7_{4}\rangle &
\langle3_{2}, 0_{2}, 3_{9}, 4_{6}\rangle &
\langle5_{0}, 5_{2}, 4_{4}, 1_{9}\rangle &
\langle1_{1}, 1_{0}, 8_{8}, 7_{6}\rangle &
\langle1_{3}, 0_{3}, 1_{4}, 8_{6}\rangle &
\langle5_{2}, 2_{3}, 6_{9}, 1_{7}\rangle &
\langle2_{0}, 8_{0}, 4_{10}, 4_{6}\rangle &
\langle2_{2}, 7_{3}, 8_{8}, 4_{8}\rangle \\
\langle1_{2}, 4_{0}, 1_{10}, 3_{10}\rangle &
\langle2_{2}, 7_{0}, 3_{5}, 1_{11}\rangle &
\langle5_{2}, 8_{1}, 3_{5}, 4_{7}\rangle &
\langle2_{2}, 1_{2}, 5_{5}, 8_{11}\rangle &
\langle2_{3}, 4_{1}, 7_{11}, 0_{4}\rangle &
\langle2_{0}, 3_{3}, 1_{8}, 8_{4}\rangle &
\langle3_{2}, 1_{1}, 1_{9}, 0_{6}\rangle &
\langle1_{1}, 6_{2}, 2_{8}, 8_{6}\rangle \\
\langle4_{0}, 7_{2}, 5_{6}, 2_{11}\rangle &
\langle3_{1}, 2_{2}, 3_{11}, 5_{11}\rangle &
\langle4_{3}, 2_{1}, 7_{11}, 4_{6}\rangle &
\langle1_{3}, 5_{3}, 6_{5}, 3_{11}\rangle &
\langle6_{0}, 3_{3}, 1_{5}, 2_{11}\rangle &
\langle5_{3}, 2_{1}, 0_{11}, 4_{4}\rangle &
\langle4_{2}, 6_{3}, 7_{6}, 6_{11}\rangle &
\langle3_{1}, 4_{2}, 4_{6}, 0_{11}\rangle \\
\langle1_{1}, 8_{1}, 4_{5}, 0_{4}\rangle &
\langle7_{2}, 7_{1}, 4_{5}, 7_{7}\rangle &
\langle3_{2}, 2_{0}, 6_{7}, 6_{8}\rangle &
\langle2_{0}, 0_{0}, 4_{4}, 4_{11}\rangle &
\langle1_{0}, 6_{0}, 1_{6}, 4_{10}\rangle &
\langle0_{0}, 2_{2}, 0_{4}, 5_{4}\rangle &
\langle2_{2}, 6_{3}, 8_{7}, 5_{10}\rangle &
\langle1_{1}, 4_{2}, 6_{9}, 1_{4}\rangle \\
\langle1_{0}, 1_{3}, 3_{5}, 6_{8}\rangle &
\langle1_{0}, 3_{1}, 1_{10}, 2_{11}\rangle &
\langle2_{1}, 6_{1}, 4_{5}, 0_{9}\rangle &
\langle0_{1}, 4_{3}, 1_{5}, 1_{11}\rangle &
\langle0_{0}, 4_{3}, 6_{9}, 7_{7}\rangle &
\langle4_{2}, 2_{2}, 8_{10}, 4_{5}\rangle &
\langle3_{0}, 1_{3}, 6_{9}, 1_{9}\rangle &
\langle4_{2}, 0_{1}, 0_{6}, 4_{11}\rangle \\
\langle4_{0}, 3_{1}, 5_{7}, 7_{4}\rangle &
\langle4_{1}, 1_{3}, 5_{10}, 4_{10}\rangle &
\langle4_{0}, 5_{0}, 4_{5}, 5_{9}\rangle &
\langle2_{3}, 5_{3}, 3_{9}, 2_{7}\rangle &
\langle5_{1}, 2_{1}, 4_{8}, 8_{7}\rangle &
\langle1_{2}, 0_{3}, 0_{5}, 6_{10}\rangle &
\langle0_{2}, 1_{3}, 8_{10}, 4_{4}\rangle &
\langle1_{0}, 5_{1}, 6_{7}, 3_{7}\rangle \\
\langle0_{1}, 1_{1}, 4_{6}, 6_{8}\rangle &
\end{array}$$ ]{}
SFSs of type $(4,2)^a(2,4)^b$ with $a+b\in\{5,6,7,8,9\}$
--------------------------------------------------------
To save space, we only list the non-empty cells of the SFSs. We use $(a,b;i,j)$ to denote a cell which is indexed by $(i,j)$ and contains a pair $\{a,b\}$. We use $I_n$ to denote the set $\{0,1,2,\ldots,n-1\}$.
There exists an SFS of type $(4,2)^a(2,2)^{5-a}$ for each $a\in\{0,1,2,3,4,5\}$.
Let $V=I_{10}$ and $S=I_{10+2a}$. $V$ can be partitioned as $V=\cup_{i=0}^4\{2i,2i+1\}$ and $S$ can be partitioned as $S=(\cup_{i=0}^{a-1}\{4i,4i+1,4i+2,4i+3\})\cup (\cup_{i=a}^{4}\{2i,2i+1\})$. The required SFSs are presented as follows.
$a=0$: [$$\begin{array}{llllllllll}
(1, 6; 9, 3) &
(1, 8; 7, 2) &
(5, 3; 9, 6) &
(5, 9; 3, 0) &
(0, 8; 3, 5) &
(2, 4; 0, 7) &
(6, 9; 4, 2) &
(9, 3; 1, 7) &
(2, 7; 9, 5) &
(6, 3; 0, 5) \\
(0, 4; 2, 9) &
(5, 0; 8, 7) &
(2, 6; 8, 1) &
(0, 2; 4, 6) &
(7, 8; 4, 0) &
(9, 1; 6, 5) &
(7, 4; 3, 8) &
(1, 3; 4, 8) &
(4, 8; 1, 6) &
(5, 7; 1, 2) \\
\end{array}$$ ]{}
$a=1$: [$$\begin{array}{lllllllll}
(7, 0; 7, 11) &
(6, 3; 10, 1) &
(0, 4; 8, 10) &
(6, 9; 6, 3) &
(4, 2; 1, 11) &
(0, 8; 5, 6) &
(6, 1; 7, 4) &
(8, 2; 8, 3) &
(3, 8; 7, 0)\\
(8, 5; 4, 1) &
(4, 8; 2, 9) &
(6, 4; 5, 0) &
(7, 9; 1, 5) &
(4, 7; 4, 3) &
(5, 3; 9, 3) &
(0, 9; 4, 9) &
(9, 5; 8, 0) &
(5, 1; 10, 5) \\
(1, 3; 11, 8) &
(9, 2; 7, 2) &
(7, 2; 10, 0) &
(1, 2; 6, 9) &
(3, 7; 2, 6) &
(5, 6; 2, 11) &
\end{array}$$ ]{}
$a=2$: [$$\begin{array}{lllllllll}
(2, 5; 2, 10) &
(6, 9; 4, 2) &
(8, 7; 6, 2) &
(4, 2; 0, 13) &
(6, 3; 9, 3) &
(8, 2; 11, 9) &
(5, 6; 6, 0) &
(7, 1; 8, 4) &
(0, 7; 7, 9) \\
(2, 6; 1, 12) &
(0, 9; 6, 11) &
(8, 6; 8, 7) &
(9, 7; 5, 0) &
(9, 1; 9, 10) &
(0, 4; 5, 10) &
(0, 5; 4, 13) &
(9, 2; 8, 3) &
(4, 8; 4, 3) \\
(4, 1; 6, 12) &
(3, 7; 13, 1) &
(5, 7; 12, 3) &
(8, 3; 10, 0) &
(3, 4; 2, 11) &
(3, 0; 8, 12) &
(4, 9; 1, 7) &
(1, 5; 7, 11) &
(1, 6; 5, 13) \\
(5, 8; 1, 5) &
\end{array}$$ ]{}
$a=3$: [$$\begin{array}{lllllllll}
(4, 0; 7, 15) &
(9, 3; 3, 10) &
(3, 7; 15, 11) &
(7, 8; 7, 3) &
(9, 0; 8, 6) &
(5, 2; 15, 3) &
(7, 5; 14, 6) &
(8, 3; 12, 9) &
(4, 3; 14, 0) \\
(1, 7; 8, 4) &
(8, 2; 0, 8) &
(6, 3; 8, 1) &
(6, 4; 4, 3) &
(6, 1; 9, 15) &
(5, 6; 7, 0) &
(5, 9; 4, 1) &
(4, 1; 6, 12) &
(7, 2; 1, 10) \\
(2, 9; 2, 12) &
(4, 7; 5, 2) &
(0, 8; 4, 11) &
(7, 9; 0, 9) &
(8, 6; 2, 6) &
(6, 0; 14, 10) &
(5, 3; 13, 2) &
(9, 6; 5, 11) &
(2, 1; 11, 14) \\
(0, 5; 12, 5) &
(8, 4; 1, 13) &
(0, 2; 9, 13) &
(1, 8; 5, 10) &
(1, 9; 7, 13) &
\end{array}$$ ]{}
$a=4$: [$$\begin{array}{lllllllll}
(4, 0; 13, 6) &
(6, 1; 7, 11) &
(7, 5; 6, 3) &
(9, 2; 13, 10) &
(8, 2; 11, 3) &
(2, 1; 14, 17) &
(4, 7; 16, 7) &
(9, 1; 9, 6) &
(8, 6; 6, 1) \\
(3, 1; 13, 8) &
(6, 9; 5, 0) &
(8, 0; 7, 14) &
(3, 6; 10, 2) &
(0, 9; 15, 8) &
(4, 1; 5, 15) &
(1, 8; 10, 12) &
(2, 4; 0, 12) &
(5, 3; 0, 14) \\
(9, 4; 1, 14) &
(5, 2; 1, 15) &
(7, 0; 5, 10) &
(8, 7; 0, 8) &
(5, 1; 4, 16) &
(6, 2; 8, 16) &
(3, 0; 16, 11) &
(9, 7; 11, 4) &
(0, 5; 12, 17) \\
(6, 0; 9, 4) &
(8, 5; 5, 13) &
(9, 5; 2, 7) &
(6, 4; 17, 3) &
(9, 3; 12, 3) &
(4, 8; 2, 4) &
(7, 3; 1, 17) &
(2, 7; 2, 9) &
(3, 8; 9, 15) \\
\end{array}$$ ]{}
$a=5$: [$$\begin{array}{lllllllll}
(0, 7; 17, 7) &
(4, 3; 1, 17) &
(3, 5; 3, 12) &
(5, 8; 0, 14) &
(8, 7; 4, 2) &
(5, 7; 5, 16) &
(0, 3; 9, 16) &
(2, 6; 17, 11) &
(8, 4; 3, 7) \\
(0, 4; 19, 13) &
(3, 8; 10, 15) &
(1, 6; 6, 18) &
(1, 2; 15, 16) &
(1, 4; 14, 5) &
(6, 0; 4, 8) &
(5, 0; 15, 18) &
(6, 4; 2, 16) &
(6, 5; 1, 7) \\
(6, 9; 10, 3) &
(9, 2; 0, 8) &
(1, 3; 11, 13) &
(9, 3; 14, 2) &
(8, 6; 5, 9) &
(7, 1; 10, 19) &
(2, 5; 19, 2) &
(0, 9; 12, 5) &
(2, 8; 13, 1) \\
(2, 7; 9, 3) &
(0, 8; 6, 11) &
(5, 9; 13, 6) &
(9, 1; 9, 7) &
(9, 4; 15, 4) &
(2, 0; 14, 10) &
(4, 2; 18, 12) &
(7, 9; 11, 1) &
(1, 8; 12, 8) \\
(5, 1; 4, 17) &
(3, 7; 8, 18) &
(3, 6; 0, 19) &
(4, 7; 0, 6) &
\end{array}$$ ]{}
There exists an SFS of type $(4,2)^a(2,2)^{6-a}$ for each $a\in\{0,1,\ldots,6\}$.
Let $V=I_{12}$ and $S=I_{12+2a}$. $V$ can be partitioned as $V=\cup_{i=0}^5\{2i,2i+1\}$ and $S$ can be partitioned as $S=(\cup_{i=0}^{a-1}\{4i,4i+1,4i+2,4i+3\})\cup (\cup_{i=a}^{5}\{2i,2i+1\})$. The required SFSs are presented as follows.
$a=0$: [$$\begin{array}{llllllllll}
(4, 7; 10, 1) &
(5, 0; 11, 7) &
(3, 1; 5, 11) &
(8, 4; 11, 6) &
(9, 5; 3, 10) &
(6, 3; 10, 0) &
(10, 8; 0, 5) &
(11, 9; 6, 5) &
(3, 0; 6, 9) \\
(10, 7; 3, 4) &
(2, 1; 7, 10) &
(11, 5; 9, 1) &
(7, 2; 5, 9) &
(0, 8; 10, 4) &
(5, 7; 0, 8) &
(1, 5; 2, 6) &
(0, 6; 5, 3) &
(6, 8; 1, 2) \\
(2, 9; 4, 0) &
(3, 11; 8, 4) &
(2, 10; 6, 1) &
(3, 9; 1, 7) &
(9, 7; 2, 11) &
(1, 6; 4, 9) &
(4, 1; 8, 3) &
(6, 2; 11, 8) &
(8, 11; 3, 7) \\
(4, 10; 9, 7) &
(0, 10; 2, 8) &
(4, 11; 0, 2) &
\end{array}$$ ]{}
$a=1$: [$$\begin{array}{llllllllll}
(1, 10; 5, 6) &
(1, 7; 11, 4) &
(0, 6; 11, 7) &
(1, 3; 13, 10) &
(5, 6; 0, 5) &
(11, 3; 9, 7) &
(2, 10; 1, 7) &
(5, 10; 2, 10) &
(7, 4; 13, 1) \\
(2, 0; 9, 13) &
(4, 3; 12, 0) &
(0, 8; 12, 4) &
(11, 9; 2, 5) &
(4, 2; 11, 2) &
(7, 9; 12, 3) &
(9, 10; 0, 9) &
(10, 3; 11, 3) &
(4, 0; 5, 10) \\
(9, 1; 7, 8) &
(8, 7; 7, 5) &
(8, 4; 9, 3) &
(9, 5; 13, 4) &
(6, 8; 2, 13) &
(2, 5; 8, 3) &
(5, 11; 11, 1) &
(6, 2; 12, 10) &
(7, 3; 2, 6) \\
(11, 6; 4, 3) &
(9, 6; 6, 1) &
(8, 3; 8, 1) &
(4, 10; 8, 4) &
(7, 11; 0, 10) &
(0, 11; 6, 8) &
(1, 5; 9, 12) &
(2, 8; 0, 6) &
\end{array}$$ ]{}
$a=2$: [$$\begin{array}{llllllllll}
(4, 8; 14, 1) &
(9, 3; 1, 9) &
(7, 4; 5, 12) &
(2, 9; 11, 3) &
(8, 2; 2, 8) &
(11, 8; 6, 0) &
(3, 10; 11, 2) &
(0, 7; 13, 6) &
(1, 8; 9, 5) \\
(11, 1; 4, 8) &
(6, 0; 9, 4) &
(4, 11; 7, 11) &
(0, 8; 11, 15) &
(6, 11; 1, 5) &
(3, 11; 12, 3) &
(3, 4; 0, 10) &
(1, 5; 6, 11) &
(2, 11; 10, 13) \\
(2, 7; 1, 15) &
(0, 5; 7, 10) &
(8, 6; 3, 7) &
(9, 5; 5, 2) &
(6, 5; 0, 14) &
(3, 5; 15, 13) &
(7, 9; 7, 0) &
(2, 0; 14, 12) &
(8, 10; 4, 10) \\
(10, 5; 12, 1) &
(6, 9; 8, 6) &
(2, 10; 0, 9) &
(3, 7; 8, 14) &
(1, 6; 12, 15) &
(4, 9; 15, 4) &
(5, 7; 4, 3) &
(1, 9; 10, 14) &
(7, 11; 9, 2) \\
(4, 10; 6, 3) &
(0, 10; 5, 8) &
(1, 10; 7, 13) &
(4, 6; 2, 13) &
\end{array}$$ ]{}
$a=3$: [$$\begin{array}{llllllllll}
(6, 10; 10, 4) &
(4, 10; 1, 5) &
(6, 2; 9, 0) &
(9, 6; 16, 1) &
(9, 2; 8, 17) &
(11, 5; 0, 4) &
(8, 11; 11, 12) &
(7, 2; 11, 1) &
(3, 11; 1, 8) \\
(9, 0; 12, 9) &
(10, 0; 11, 14) &
(8, 2; 10, 2) &
(1, 3; 16, 12) &
(0, 3; 10, 13) &
(8, 5; 1, 7) &
(11, 2; 13, 3) &
(8, 0; 16, 4) &
(6, 3; 15, 2) \\
(1, 10; 15, 8) &
(5, 2; 16, 14) &
(11, 6; 7, 14) &
(8, 1; 6, 9) &
(7, 3; 17, 14) &
(10, 9; 13, 7) &
(5, 10; 12, 2) &
(0, 6; 8, 6) &
(11, 9; 10, 5) \\
(6, 1; 11, 5) &
(4, 2; 15, 12) &
(8, 6; 17, 3) &
(9, 3; 11, 0) &
(4, 11; 2, 6) &
(7, 1; 7, 10) &
(4, 0; 7, 17) &
(7, 10; 0, 6) &
(4, 1; 4, 14) \\
(4, 8; 0, 13) &
(10, 3; 3, 9) &
(11, 7; 15, 9) &
(8, 7; 8, 5) &
(5, 1; 13, 17) &
(0, 5; 15, 5) &
(4, 7; 3, 16) &
(5, 9; 3, 6) &
(7, 9; 2, 4) \\
\end{array}$$ ]{}
$a=4$: [$$\begin{array}{llllllllll}
(8, 11; 2, 12) &
(3, 8; 10, 19) &
(5, 1; 14, 6) &
(4, 11; 14, 5) &
(11, 2; 1, 11) &
(4, 10; 12, 1) &
(10, 7; 7, 11) &
(9, 3; 11, 15) &
(9, 7; 1, 10) \\
(0, 8; 18, 11) &
(5, 7; 2, 19) &
(9, 5; 12, 5) &
(2, 7; 3, 8) &
(1, 10; 17, 13) &
(3, 6; 0, 18) &
(0, 2; 16, 15) &
(6, 2; 9, 2) &
(11, 9; 3, 6) \\
(4, 7; 4, 18) &
(5, 2; 18, 17) &
(0, 10; 5, 10) &
(10, 5; 0, 16) &
(8, 2; 0, 14) &
(11, 3; 16, 8) &
(6, 0; 19, 7) &
(6, 8; 1, 5) &
(7, 11; 0, 9) \\
(1, 6; 11, 4) &
(11, 6; 17, 10) &
(4, 3; 2, 17) &
(1, 7; 16, 5) &
(4, 6; 16, 3) &
(1, 8; 7, 8) &
(9, 0; 14, 8) &
(8, 5; 3, 4) &
(9, 4; 7, 0) \\
(2, 1; 12, 10) &
(11, 5; 15, 7) &
(9, 2; 13, 19) &
(10, 8; 15, 9) &
(8, 4; 6, 13) &
(10, 6; 8, 6) &
(4, 1; 15, 19) &
(3, 10; 14, 3) &
(1, 9; 9, 18) \\
(9, 10; 2, 4) &
(7, 0; 6, 17) &
(0, 11; 4, 13) &
(0, 3; 9, 12) &
(3, 5; 1, 13) &
\end{array}$$ ]{}
$a=5$: [$$\begin{array}{llllllllll}
(5, 9; 1, 13) &
(10, 0; 16, 8) &
(9, 0; 5, 21) &
(4, 10; 1, 7) &
(5, 3; 20, 3) &
(1, 3; 16, 15) &
(6, 2; 21, 1) &
(9, 4; 3, 6) &
(7, 10; 10, 6) \\
(2, 9; 8, 0) &
(0, 4; 17, 15) &
(11, 8; 10, 7) &
(1, 8; 13, 6) &
(8, 5; 15, 0) &
(7, 5; 2, 19) &
(10, 6; 18, 2) &
(5, 11; 17, 6) &
(4, 11; 2, 13) \\
(6, 3; 17, 10) &
(1, 5; 18, 21) &
(2, 8; 20, 2) &
(4, 1; 14, 20) &
(9, 10; 11, 15) &
(1, 2; 9, 19) &
(2, 0; 14, 10) &
(6, 0; 6, 20) &
(2, 11; 15, 18) \\
(11, 0; 19, 11) &
(4, 2; 16, 12) &
(7, 8; 1, 5) &
(9, 11; 4, 14) &
(5, 6; 4, 16) &
(7, 11; 3, 16) &
(1, 9; 10, 12) &
(1, 10; 17, 4) &
(10, 5; 5, 14) \\
(4, 6; 5, 19) &
(7, 0; 4, 9) &
(3, 8; 14, 11) &
(3, 10; 0, 19) &
(8, 4; 21, 4) &
(9, 3; 9, 2) &
(8, 6; 8, 3) &
(0, 5; 7, 12) &
(7, 2; 11, 17) \\
(6, 11; 0, 9) &
(10, 2; 13, 3) &
(1, 6; 7, 11) &
(9, 7; 20, 7) &
(11, 1; 8, 5) &
(10, 8; 12, 9) &
(3, 7; 8, 21) &
(0, 3; 13, 18) &
(3, 11; 1, 12) \\
(4, 7; 0, 18) &
\end{array}$$ ]{}
$a=6$: [$$\begin{array}{llllllllll}
(8, 5; 3, 7) &
(10, 2; 17, 0) &
(9, 10; 3, 8) &
(5, 0; 4, 18) &
(0, 8; 12, 11) &
(1, 11; 4, 11) &
(6, 3; 0, 18) &
(7, 9; 1, 9) &
(8, 10; 9, 5) \\
(6, 9; 2, 21) &
(4, 7; 18, 5) &
(4, 8; 21, 0) &
(7, 8; 22, 10) &
(1, 6; 8, 17) &
(4, 6; 20, 3) &
(7, 2; 19, 8) &
(7, 11; 0, 16) &
(2, 11; 18, 3) \\
(8, 6; 23, 4) &
(3, 7; 3, 11) &
(10, 6; 7, 11) &
(1, 4; 23, 14) &
(6, 5; 6, 1) &
(7, 10; 4, 2) &
(1, 3; 19, 9) &
(8, 11; 6, 8) &
(11, 3; 17, 14) \\
(5, 3; 21, 13) &
(4, 2; 13, 2) &
(2, 9; 11, 23) &
(4, 10; 6, 12) &
(1, 7; 21, 7) &
(0, 4; 7, 17) &
(3, 0; 23, 8) &
(8, 1; 13, 20) &
(9, 4; 15, 4) \\
(2, 1; 12, 22) &
(10, 1; 18, 15) &
(0, 10; 16, 13) &
(5, 7; 23, 17) &
(9, 0; 5, 14) &
(6, 2; 9, 16) &
(1, 5; 16, 5) &
(8, 2; 1, 14) &
(5, 11; 12, 2) \\
(7, 0; 6, 20) &
(5, 9; 0, 22) &
(0, 11; 9, 15) &
(11, 4; 1, 19) &
(4, 3; 22, 16) &
(5, 10; 19, 14) &
(9, 3; 12, 20) &
(5, 2; 20, 15) &
(11, 9; 7, 13) \\
(9, 1; 6, 10) &
(10, 3; 10, 1) &
(8, 3; 15, 2) &
(0, 6; 19, 22) &
(0, 2; 10, 21) &
(6, 11; 5, 10) &
\end{array}$$ ]{}
There exists an SFS of type $(4,2)^a(2,2)^{7-a}$ for each $a\in\{0,1,\ldots,7\}$.
Let $V=I_{14}$ and $S=I_{14+2a}$. $V$ can be partitioned as $V=\cup_{i=0}^6\{2i,2i+1\}$ and $S$ can be partitioned as $S=(\cup_{i=0}^{a-1}\{4i,4i+1,4i+2,4i+3\})\cup (\cup_{i=a}^{6}\{2i,2i+1\})$. The required SFSs are presented as follows.
$a=0$: [$$\begin{array}{llllllllll}
(3, 7; 12, 0) &
(11, 1; 4, 6) &
(7, 11; 8, 2) &
(13, 2; 9, 0) &
(9, 5; 11, 3) &
(9, 10; 2, 6) &
(3, 6; 11, 9) &
(5, 10; 7, 12) &
(6, 5; 10, 0) \\
(8, 3; 6, 10) &
(0, 12; 3, 6) &
(2, 7; 11, 1) &
(12, 8; 0, 11) &
(9, 1; 10, 12) &
(2, 0; 8, 10) &
(0, 3; 7, 13) &
(13, 7; 4, 10) &
(4, 12; 7, 10) \\
(3, 9; 1, 4) &
(8, 1; 2, 7) &
(2, 4; 13, 6) &
(13, 0; 2, 11) &
(6, 1; 13, 3) &
(4, 1; 11, 8) &
(4, 10; 3, 0) &
(2, 9; 5, 7) &
(12, 5; 2, 1) \\
(13, 5; 8, 6) &
(11, 0; 12, 5) &
(1, 12; 9, 5) &
(4, 6; 2, 12) &
(12, 6; 4, 8) &
(11, 13; 3, 7) &
(3, 10; 8, 5) &
(4, 11; 1, 9) &
(9, 11; 0, 13) \\
(8, 7; 3, 5) &
(6, 13; 1, 5) &
(5, 7; 9, 13) &
(10, 0; 4, 9) &
(2, 8; 4, 12) &
(8, 10; 1, 13) &
\end{array}$$ ]{}
$a=1$: [$$\begin{array}{llllllllll}
(11, 5; 2, 11) &
(2, 12; 1, 12) &
(11, 6; 6, 1) &
(1, 11; 14, 7) &
(12, 9; 0, 4) &
(10, 5; 14, 10) &
(1, 10; 15, 8) &
(1, 5; 13, 4) &
(11, 13; 4, 8) \\
(9, 1; 12, 5) &
(0, 6; 12, 11) &
(8, 4; 13, 3) &
(12, 7; 7, 13) &
(5, 2; 3, 15) &
(2, 9; 6, 14) &
(13, 2; 9, 13) &
(6, 4; 15, 4) &
(11, 8; 5, 15) \\
(8, 12; 6, 2) &
(13, 7; 12, 2) &
(5, 8; 9, 12) &
(2, 11; 0, 10) &
(11, 9; 9, 3) &
(0, 12; 10, 5) &
(12, 3; 3, 8) &
(4, 3; 12, 0) &
(9, 5; 8, 1) \\
(13, 10; 0, 6) &
(13, 4; 11, 5) &
(0, 9; 13, 15) &
(8, 10; 1, 4) &
(13, 6; 10, 3) &
(6, 9; 7, 2) &
(8, 0; 8, 7) &
(3, 6; 14, 13) &
(0, 4; 14, 9) \\
(10, 3; 9, 2) &
(13, 3; 1, 7) &
(4, 2; 2, 8) &
(3, 1; 10, 6) &
(7, 3; 15, 11) &
(7, 4; 10, 1) &
(7, 0; 6, 4) &
(12, 1; 9, 11) &
(2, 10; 7, 11) \\
(7, 8; 0, 14) &
(5, 6; 0, 5) &
(7, 10; 3, 5) &
\end{array}$$ ]{}
$a=2$: [$$\begin{array}{llllllllll}
(11, 4; 3, 7) &
(7, 12; 12, 3) &
(6, 9; 4, 3) &
(12, 6; 2, 6) &
(8, 10; 6, 1) &
(2, 1; 10, 8) &
(7, 1; 7, 13) &
(2, 7; 17, 15) &
(5, 13; 5, 2) \\
(8, 5; 7, 17) &
(0, 12; 4, 11) &
(10, 12; 8, 7) &
(12, 5; 0, 15) &
(0, 6; 7, 14) &
(10, 6; 12, 16) &
(8, 7; 0, 5) &
(8, 2; 11, 3) &
(0, 11; 16, 10) \\
(5, 9; 1, 10) &
(9, 13; 7, 15) &
(4, 10; 10, 5) &
(5, 2; 14, 12) &
(2, 10; 9, 2) &
(4, 1; 15, 16) &
(13, 10; 4, 0) &
(2, 6; 13, 1) &
(11, 3; 2, 17) \\
(10, 9; 11, 17) &
(2, 9; 0, 16) &
(6, 11; 0, 8) &
(4, 3; 11, 0) &
(8, 13; 10, 14) &
(9, 7; 2, 8) &
(11, 13; 13, 11) &
(11, 12; 1, 5) &
(0, 4; 13, 17) \\
(8, 4; 4, 2) &
(5, 7; 16, 4) &
(1, 11; 12, 4) &
(8, 6; 15, 9) &
(7, 3; 1, 14) &
(3, 12; 10, 13) &
(11, 7; 6, 9) &
(3, 0; 15, 12) &
(13, 4; 12, 1) \\
(0, 9; 5, 9) &
(6, 1; 5, 17) &
(1, 5; 6, 11) &
(10, 5; 13, 3) &
(1, 12; 9, 14) &
(4, 9; 6, 14) &
(13, 3; 3, 9) &
(0, 13; 6, 8) &
(3, 8; 8, 16) \\
\end{array}$$ ]{}
$a=3$: [$$\begin{array}{llllllllll}
(6, 3; 18, 15) &
(4, 3; 16, 12) &
(11, 0; 11, 5) &
(4, 12; 13, 1) &
(11, 2; 2, 18) &
(10, 6; 0, 6) &
(3, 8; 0, 11) &
(5, 8; 5, 18) &
(4, 11; 15, 6) \\
(7, 8; 8, 17) &
(12, 8; 2, 16) &
(8, 1; 6, 19) &
(8, 13; 13, 3) &
(12, 11; 4, 8) &
(5, 11; 12, 0) &
(2, 0; 13, 19) &
(4, 0; 18, 14) &
(1, 10; 4, 18) \\
(13, 9; 4, 0) &
(7, 2; 3, 9) &
(6, 4; 3, 5) &
(3, 7; 14, 2) &
(12, 0; 6, 9) &
(9, 3; 1, 8) &
(10, 12; 11, 12) &
(10, 3; 13, 10) &
(11, 9; 3, 10) \\
(12, 7; 5, 0) &
(7, 9; 18, 16) &
(1, 6; 10, 17) &
(10, 5; 3, 19) &
(0, 9; 12, 17) &
(0, 13; 7, 15) &
(1, 7; 11, 7) &
(1, 5; 15, 16) &
(4, 9; 2, 7) \\
(13, 7; 10, 6) &
(5, 9; 6, 13) &
(3, 12; 3, 17) &
(0, 8; 10, 4) &
(10, 8; 7, 9) &
(6, 9; 19, 11) &
(7, 10; 1, 15) &
(10, 2; 8, 14) &
(7, 4; 4, 19) \\
(3, 11; 9, 19) &
(13, 1; 8, 12) &
(6, 11; 1, 7) &
(4, 2; 17, 0) &
(13, 6; 9, 14) &
(10, 13; 5, 2) &
(0, 6; 8, 16) &
(12, 2; 15, 10) &
(1, 9; 9, 5) \\
(6, 5; 2, 4) &
(5, 12; 14, 7) &
(1, 11; 14, 13) &
(13, 2; 11, 16) &
(2, 8; 1, 12) &
(5, 13; 1, 17) &
\end{array}$$ ]{} $a=4$: [$$\begin{array}{llllllllll}
(2, 5; 1, 18) &
(12, 6; 1, 17) &
(4, 9; 19, 2) &
(10, 3; 10, 0) &
(6, 5; 2, 20) &
(10, 0; 12, 8) &
(1, 7; 11, 20) &
(9, 0; 9, 20) &
(6, 4; 18, 21) \\
(2, 1; 9, 14) &
(8, 10; 6, 15) &
(10, 12; 11, 5) &
(0, 5; 15, 19) &
(3, 13; 15, 18) &
(10, 9; 1, 14) &
(2, 9; 21, 11) &
(1, 6; 10, 6) &
(13, 4; 1, 7) \\
(13, 2; 17, 8) &
(4, 0; 14, 4) &
(5, 7; 17, 7) &
(5, 3; 21, 16) &
(10, 1; 17, 21) &
(8, 1; 4, 13) &
(8, 6; 0, 9) &
(10, 7; 4, 16) &
(4, 12; 6, 16) \\
(0, 8; 7, 18) &
(13, 0; 6, 13) &
(7, 8; 2, 8) &
(11, 6; 4, 8) &
(12, 1; 18, 8) &
(9, 5; 13, 5) &
(0, 11; 11, 17) &
(4, 11; 3, 20) &
(1, 9; 12, 7) \\
(4, 2; 0, 12) &
(9, 3; 8, 3) &
(13, 9; 4, 10) &
(4, 1; 15, 5) &
(3, 8; 20, 1) &
(4, 3; 13, 17) &
(9, 11; 0, 15) &
(3, 11; 2, 14) &
(8, 13; 12, 11) \\
(5, 13; 0, 14) &
(12, 5; 3, 4) &
(2, 12; 15, 2) &
(7, 13; 3, 5) &
(3, 6; 19, 11) &
(8, 11; 21, 5) &
(11, 5; 6, 12) &
(12, 8; 10, 14) &
(2, 10; 20, 13) \\
(3, 12; 9, 12) &
(10, 6; 7, 3) &
(8, 2; 3, 19) &
(6, 0; 5, 16) &
(12, 11; 7, 13) &
(1, 13; 16, 19) &
(7, 9; 6, 18) &
(7, 12; 19, 0) &
(13, 10; 2, 9) \\
(11, 2; 10, 16) &
(0, 7; 10, 21) &
(7, 11; 1, 9) &
\end{array}$$ ]{} $a=5$: [$$\begin{array}{llllllllll}
(2, 6; 18, 21) &
(6, 1; 7, 9) &
(10, 13; 9, 17) &
(12, 8; 5, 13) &
(13, 8; 14, 8) &
(3, 4; 22, 17) &
(2, 9; 13, 10) &
(6, 4; 5, 1) &
(2, 8; 23, 2) \\
(7, 11; 3, 5) &
(5, 11; 19, 1) &
(4, 0; 19, 6) &
(5, 0; 13, 18) &
(3, 12; 21, 16) &
(12, 0; 15, 8) &
(13, 0; 5, 16) &
(13, 2; 19, 3) &
(1, 13; 6, 12) \\
(10, 9; 5, 14) &
(10, 0; 7, 22) &
(0, 11; 9, 14) &
(7, 0; 10, 20) &
(13, 3; 1, 10) &
(12, 7; 18, 7) &
(7, 1; 4, 17) &
(6, 12; 19, 10) &
(13, 5; 7, 15) \\
(2, 11; 8, 17) &
(0, 6; 23, 17) &
(5, 12; 17, 20) &
(12, 4; 0, 4) &
(2, 4; 20, 14) &
(8, 7; 9, 6) &
(7, 5; 2, 16) &
(8, 5; 12, 22) &
(3, 1; 20, 19) \\
(9, 5; 21, 6) &
(9, 13; 20, 4) &
(5, 10; 3, 4) &
(3, 6; 8, 3) &
(5, 3; 14, 0) &
(10, 8; 1, 11) &
(12, 11; 6, 2) &
(12, 9; 3, 12) &
(4, 1; 13, 21) \\
(5, 1; 23, 5) &
(1, 10; 18, 10) &
(8, 0; 21, 4) &
(7, 13; 0, 21) &
(2, 7; 11, 22) &
(1, 9; 8, 22) &
(0, 3; 12, 11) &
(10, 7; 19, 8) &
(4, 10; 23, 15) \\
(1, 12; 14, 11) &
(11, 8; 10, 15) &
(6, 9; 11, 2) &
(10, 2; 0, 12) &
(13, 11; 13, 11) &
(8, 6; 0, 20) &
(9, 3; 9, 15) &
(7, 9; 23, 1) &
(4, 8; 7, 3) \\
(11, 3; 23, 18) &
(4, 13; 2, 18) &
(9, 11; 7, 0) &
(10, 6; 6, 16) &
(1, 2; 15, 16) &
(2, 12; 1, 9) &
(3, 10; 2, 13) &
(4, 11; 12, 16) &
(6, 11; 4, 22) \\
\end{array}$$ ]{}
$a=6$: [$$\begin{array}{llllllllll}
(2, 12; 0, 23) &
(1, 3; 12, 11) &
(10, 9; 14, 2) &
(1, 5; 14, 21) &
(7, 3; 18, 8) &
(2, 11; 13, 2) &
(0, 10; 6, 10) &
(4, 8; 21, 13) &
(10, 12; 15, 11) \\
(13, 11; 17, 15) &
(12, 7; 9, 1) &
(7, 4; 23, 25) &
(12, 9; 12, 5) &
(4, 12; 18, 22) &
(9, 5; 22, 25) &
(1, 6; 25, 17) &
(2, 0; 22, 15) &
(7, 13; 21, 2) \\
(13, 10; 5, 18) &
(6, 10; 4, 9) &
(6, 3; 24, 10) &
(1, 11; 18, 24) &
(3, 0; 25, 9) &
(4, 1; 15, 16) &
(13, 0; 16, 12) &
(8, 0; 14, 4) &
(1, 13; 6, 23) \\
(5, 3; 20, 0) &
(12, 1; 8, 7) &
(4, 9; 0, 6) &
(9, 3; 23, 15) &
(8, 1; 9, 5) &
(13, 6; 11, 22) &
(6, 0; 21, 7) &
(0, 9; 11, 24) &
(5, 2; 12, 17) \\
(8, 11; 25, 11) &
(2, 13; 3, 9) &
(12, 8; 20, 3) &
(6, 11; 16, 0) &
(5, 12; 2, 4) &
(12, 3; 21, 16) &
(9, 2; 21, 1) &
(10, 3; 1, 13) &
(7, 8; 10, 7) \\
(7, 11; 3, 6) &
(7, 10; 17, 0) &
(0, 7; 20, 19) &
(11, 4; 12, 4) &
(9, 11; 7, 9) &
(1, 2; 19, 10) &
(3, 8; 22, 2) &
(10, 5; 3, 16) &
(13, 4; 20, 1) \\
(4, 2; 24, 14) &
(5, 13; 13, 7) &
(8, 5; 6, 15) &
(9, 13; 10, 4) &
(6, 4; 5, 2) &
(6, 12; 19, 6) &
(8, 6; 1, 23) &
(9, 1; 13, 20) &
(0, 5; 23, 18) \\
(8, 10; 24, 12) &
(11, 0; 5, 8) &
(13, 3; 14, 19) &
(7, 5; 24, 5) &
(11, 12; 10, 14) &
(7, 1; 22, 4) &
(11, 5; 1, 19) &
(12, 0; 13, 17) &
(3, 4; 17, 3) \\
(13, 8; 8, 0) &
(10, 4; 7, 19) &
(2, 7; 11, 16) &
(2, 6; 20, 18) &
(2, 10; 8, 25) &
(6, 9; 3, 8) &
\end{array}$$ ]{}
$a=7$: [$$\begin{array}{llllllllll}
(9, 6; 0, 9) &
(5, 12; 4, 2) &
(3, 12; 8, 14) &
(6, 1; 20, 7) &
(0, 11; 26, 18) &
(2, 6; 3, 11) &
(8, 13; 1, 12) &
(4, 11; 13, 17) &
(3, 8; 22, 24) \\
(5, 1; 26, 12) &
(7, 4; 27, 18) &
(4, 1; 19, 23) &
(7, 12; 23, 1) &
(4, 8; 15, 26) &
(10, 0; 9, 6) &
(9, 11; 12, 2) &
(1, 12; 17, 21) &
(12, 2; 22, 19) \\
(6, 0; 17, 25) &
(13, 6; 22, 2) &
(3, 13; 0, 18) &
(0, 13; 8, 20) &
(7, 3; 16, 2) &
(5, 6; 19, 1) &
(3, 0; 10, 13) &
(13, 10; 19, 4) &
(12, 9; 7, 10) \\
(6, 4; 4, 24) &
(8, 7; 8, 6) &
(9, 4; 6, 25) &
(6, 11; 27, 8) &
(2, 1; 18, 15) &
(1, 11; 16, 4) &
(4, 3; 20, 1) &
(5, 0; 27, 15) &
(7, 9; 26, 4) \\
(5, 3; 21, 25) &
(1, 7; 11, 22) &
(8, 6; 5, 21) &
(7, 5; 20, 17) &
(5, 8; 3, 23) &
(13, 2; 17, 14) &
(0, 7; 19, 24) &
(6, 10; 16, 10) &
(13, 7; 21, 9) \\
(5, 13; 7, 16) &
(11, 3; 19, 11) &
(5, 9; 22, 5) &
(9, 2; 27, 20) &
(0, 2; 16, 23) &
(6, 3; 23, 26) &
(6, 12; 6, 18) &
(8, 12; 13, 20) &
(2, 5; 13, 0) \\
(1, 13; 6, 13) &
(12, 10; 11, 15) &
(10, 3; 17, 12) &
(2, 7; 10, 25) &
(11, 5; 14, 6) &
(12, 11; 3, 9) &
(11, 8; 25, 7) &
(2, 10; 26, 8) &
(1, 10; 5, 25) \\
(0, 8; 11, 4) &
(8, 2; 9, 2) &
(2, 11; 24, 1) &
(3, 9; 15, 3) &
(8, 10; 0, 27) &
(9, 0; 14, 21) &
(0, 4; 7, 22) &
(8, 1; 14, 10) &
(10, 7; 3, 7) \\
(12, 0; 12, 5) &
(9, 13; 11, 23) &
(13, 11; 15, 10) &
(2, 4; 21, 12) &
(4, 13; 5, 3) &
(1, 3; 27, 9) &
(12, 4; 16, 0) &
(4, 10; 2, 14) &
(7, 11; 0, 5) \\
(10, 5; 18, 24) &
(1, 9; 8, 24) &
(9, 10; 1, 13) &
\end{array}$$ ]{}
There exists an SFS of type $(4,2)^a(2,2)^{8-a}$ for each $a\in\{0,1,\ldots,8\}$.
Let $V=I_{16}$ and $S=I_{16+2a}$. $V$ can be partitioned as $V=\cup_{i=0}^7\{2i,2i+1\}$ and $S$ can be partitioned as $S=(\cup_{i=0}^{a-1}\{4i,4i+1,4i+2,4i+3\})\cup (\cup_{i=a}^{7}\{2i,2i+1\})$. The required SFSs are presented as follows.
$a=0$: [$$\begin{array}{llllllllll}
(1, 9; 3, 14) &
(14, 7; 9, 1) &
(13, 11; 5, 14) &
(8, 7; 13, 14) &
(13, 5; 8, 7) &
(1, 3; 5, 15) &
(1, 12; 4, 2) &
(7, 11; 15, 12) &
(7, 15; 10, 8) \\
(6, 8; 12, 4) &
(0, 5; 13, 3) &
(14, 6; 5, 8) &
(8, 2; 5, 7) &
(10, 15; 0, 7) &
(6, 12; 10, 0) &
(11, 3; 13, 1) &
(5, 8; 11, 15) &
(6, 11; 2, 9) \\
(3, 15; 4, 6) &
(8, 13; 1, 6) &
(9, 10; 5, 2) &
(8, 0; 10, 2) &
(15, 12; 5, 1) &
(14, 9; 13, 0) &
(9, 11; 4, 7) &
(3, 13; 0, 9) &
(13, 9; 15, 10) \\
(1, 14; 12, 7) &
(3, 4; 10, 12) &
(13, 7; 4, 3) &
(4, 8; 3, 0) &
(12, 5; 14, 6) &
(0, 4; 7, 14) &
(9, 0; 6, 12) &
(5, 10; 1, 12) &
(12, 3; 7, 11) \\
(4, 1; 6, 8) &
(15, 4; 13, 2) &
(12, 11; 8, 3) &
(15, 2; 12, 9) &
(7, 5; 0, 2) &
(1, 5; 9, 10) &
(13, 14; 2, 11) &
(9, 4; 11, 1) &
(4, 12; 15, 9) \\
(10, 14; 3, 6) &
(10, 6; 15, 13) &
(14, 2; 10, 4) &
(2, 6; 1, 14) &
(0, 10; 4, 9) &
(10, 3; 8, 14) &
(6, 15; 11, 3) &
(2, 11; 0, 6) &
(0, 2; 8, 15) \\
(0, 7; 5, 11) &
(1, 2; 11, 13) &
\end{array}$$ ]{}
$a=1$: [$$\begin{array}{llllllllll}
(4, 14; 1, 5) &
(14, 0; 6, 11) &
(13, 2; 2, 6) &
(6, 9; 0, 15) &
(1, 13; 5, 13) &
(7, 5; 17, 0) &
(12, 4; 4, 2) &
(5, 1; 14, 10) &
(14, 7; 13, 10) \\
(8, 6; 2, 13) &
(4, 3; 12, 14) &
(8, 3; 16, 6) &
(7, 13; 3, 11) &
(6, 4; 3, 10) &
(6, 11; 5, 11) &
(2, 8; 7, 0) &
(8, 1; 8, 4) &
(1, 3; 9, 7) \\
(12, 15; 13, 0) &
(6, 12; 12, 7) &
(12, 11; 6, 3) &
(10, 14; 15, 4) &
(9, 4; 13, 8) &
(3, 5; 11, 1) &
(0, 2; 10, 9) &
(1, 4; 17, 11) &
(13, 6; 1, 16) \\
(13, 10; 8, 7) &
(5, 9; 5, 2) &
(2, 12; 11, 8) &
(15, 6; 4, 6) &
(3, 13; 10, 0) &
(10, 12; 10, 5) &
(6, 10; 17, 14) &
(11, 14; 9, 0) &
(1, 9; 6, 12) \\
(15, 5; 8, 3) &
(0, 8; 5, 14) &
(8, 12; 17, 1) &
(10, 8; 3, 9) &
(2, 11; 15, 17) &
(11, 0; 16, 8) &
(13, 0; 17, 12) &
(5, 13; 4, 9) &
(3, 14; 8, 2) \\
(11, 7; 14, 2) &
(10, 4; 0, 16) &
(9, 14; 7, 14) &
(9, 11; 1, 4) &
(4, 15; 9, 15) &
(10, 7; 6, 1) &
(10, 15; 2, 11) &
(9, 3; 17, 3) &
(12, 9; 9, 16) \\
(0, 7; 7, 4) &
(15, 11; 10, 7) &
(2, 15; 1, 14) &
(3, 0; 13, 15) &
(14, 2; 12, 3) &
(5, 2; 13, 16) &
(7, 15; 5, 12) &
(1, 7; 15, 16) &
(5, 8; 12, 15) \\
\end{array}$$ ]{}
$a=2$: [$$\begin{array}{llllllllll}
(14, 1; 8, 7) &
(1, 8; 11, 14) &
(12, 9; 8, 14) &
(3, 15; 3, 12) &
(0, 10; 10, 5) &
(4, 13; 10, 19) &
(3, 9; 19, 11) &
(14, 13; 3, 6) &
(6, 13; 7, 2) \\
(4, 10; 16, 1) &
(11, 1; 9, 16) &
(4, 8; 4, 3) &
(13, 11; 1, 5) &
(0, 4; 13, 6) &
(0, 8; 19, 15) &
(1, 4; 15, 17) &
(0, 5; 7, 11) &
(11, 6; 3, 13) \\
(13, 9; 9, 15) &
(7, 10; 3, 9) &
(7, 4; 14, 0) &
(13, 3; 8, 0) &
(2, 12; 18, 3) &
(15, 10; 4, 2) &
(7, 11; 19, 8) &
(14, 3; 17, 13) &
(9, 6; 1, 4) \\
(11, 14; 4, 10) &
(15, 12; 15, 10) &
(11, 15; 6, 17) &
(11, 2; 0, 11) &
(9, 15; 0, 5) &
(13, 5; 14, 12) &
(14, 10; 11, 12) &
(2, 15; 14, 9) &
(12, 3; 9, 1) \\
(6, 10; 17, 19) &
(14, 5; 15, 1) &
(4, 11; 7, 12) &
(7, 2; 1, 13) &
(8, 14; 0, 9) &
(14, 6; 16, 14) &
(4, 6; 5, 18) &
(12, 1; 5, 13) &
(7, 3; 15, 16) \\
(6, 0; 9, 12) &
(9, 2; 2, 17) &
(9, 7; 18, 7) &
(2, 8; 16, 10) &
(15, 13; 11, 13) &
(0, 15; 8, 16) &
(10, 12; 0, 7) &
(10, 5; 13, 18) &
(14, 7; 5, 2) \\
(11, 8; 2, 18) &
(5, 6; 6, 0) &
(7, 0; 4, 17) &
(5, 9; 16, 3) &
(1, 9; 10, 6) &
(8, 10; 6, 8) &
(6, 2; 8, 15) &
(7, 12; 6, 12) &
(1, 13; 4, 18) \\
(15, 8; 1, 7) &
(1, 2; 19, 12) &
(0, 3; 18, 14) &
(8, 5; 17, 5) &
(12, 5; 4, 19) &
(3, 5; 2, 10) &
(4, 12; 2, 11) &
\end{array}$$ ]{}
$a=3$: [$$\begin{array}{llllllllll}
(15, 13; 12, 9) &
(11, 14; 7, 11) &
(5, 10; 13, 6) &
(7, 14; 0, 4) &
(9, 0; 9, 19) &
(10, 1; 8, 21) &
(7, 11; 18, 9) &
(12, 3; 1, 9) &
(12, 9; 20, 0) \\
(9, 10; 1, 12) &
(15, 0; 15, 17) &
(2, 8; 3, 9) &
(4, 6; 14, 16) &
(5, 8; 12, 2) &
(3, 7; 2, 10) &
(1, 9; 4, 13) &
(6, 15; 19, 11) &
(15, 12; 13, 16) \\
(7, 5; 1, 21) &
(8, 12; 5, 21) &
(8, 13; 13, 17) &
(13, 11; 1, 5) &
(3, 10; 11, 20) &
(5, 13; 20, 4) &
(12, 2; 11, 2) &
(0, 12; 10, 4) &
(10, 0; 5, 18) \\
(6, 0; 7, 21) &
(6, 13; 6, 0) &
(6, 1; 9, 5) &
(13, 2; 21, 16) &
(15, 11; 2, 4) &
(2, 4; 13, 18) &
(9, 13; 3, 11) &
(6, 9; 8, 17) &
(8, 10; 7, 19) \\
(1, 7; 15, 11) &
(15, 8; 8, 0) &
(5, 14; 5, 17) &
(2, 11; 12, 14) &
(0, 3; 16, 12) &
(14, 6; 18, 2) &
(12, 7; 3, 8) &
(1, 14; 6, 12) &
(10, 13; 15, 2) \\
(8, 0; 11, 6) &
(3, 5; 15, 0) &
(5, 1; 19, 16) &
(1, 13; 7, 10) &
(3, 14; 3, 13) &
(15, 5; 18, 3) &
(4, 10; 4, 3) &
(3, 1; 14, 17) &
(9, 7; 7, 16) \\
(4, 14; 19, 15) &
(8, 14; 16, 10) &
(0, 13; 8, 14) &
(7, 15; 5, 14) &
(11, 3; 8, 19) &
(11, 4; 21, 0) &
(4, 15; 1, 7) &
(7, 4; 6, 20) &
(4, 9; 2, 5) \\
(4, 12; 17, 12) &
(9, 3; 18, 21) &
(8, 6; 1, 4) &
(12, 5; 14, 7) &
(10, 2; 10, 0) &
(14, 2; 8, 1) &
(15, 9; 10, 6) &
(7, 2; 19, 17) &
(8, 1; 20, 18) \\
(11, 12; 6, 15) &
(14, 10; 14, 9) &
(11, 0; 13, 20) &
(2, 6; 15, 20) &
(6, 11; 3, 10) &
\end{array}$$ ]{}
$a=4$: [$$\begin{array}{llllllllll}
(0, 8; 14, 7) &
(7, 1; 6, 10) &
(2, 14; 17, 0) &
(12, 11; 14, 22) &
(2, 4; 12, 21) &
(6, 9; 3, 23) &
(13, 9; 8, 1) &
(5, 3; 20, 3) &
(15, 6; 20, 6) \\
(8, 2; 2, 15) &
(11, 5; 2, 23) &
(3, 7; 8, 19) &
(2, 9; 9, 18) &
(7, 2; 3, 22) &
(9, 4; 5, 0) &
(8, 1; 22, 9) &
(6, 1; 4, 19) &
(13, 6; 7, 18) \\
(15, 8; 3, 10) &
(10, 9; 11, 15) &
(8, 13; 23, 11) &
(12, 3; 15, 1) &
(15, 12; 2, 9) &
(11, 4; 20, 15) &
(5, 10; 1, 16) &
(1, 12; 18, 23) &
(8, 11; 0, 8) \\
(4, 14; 16, 2) &
(12, 5; 12, 6) &
(4, 7; 18, 1) &
(5, 9; 19, 22) &
(2, 15; 1, 19) &
(1, 15; 15, 16) &
(10, 3; 0, 22) &
(15, 13; 14, 0) &
(12, 6; 10, 0) \\
(3, 14; 18, 11) &
(11, 6; 21, 9) &
(13, 10; 2, 13) &
(4, 8; 6, 19) &
(4, 13; 4, 22) &
(13, 1; 12, 5) &
(0, 12; 8, 4) &
(9, 11; 7, 13) &
(12, 10; 5, 17) \\
(0, 11; 10, 16) &
(8, 3; 13, 21) &
(5, 15; 17, 13) &
(0, 5; 5, 15) &
(15, 10; 7, 8) &
(4, 3; 23, 14) &
(4, 1; 17, 7) &
(7, 5; 0, 7) &
(10, 2; 10, 14) \\
(0, 10; 21, 6) &
(6, 0; 11, 22) &
(11, 14; 1, 12) &
(8, 10; 20, 12) &
(8, 6; 5, 1) &
(14, 1; 13, 8) &
(0, 13; 19, 17) &
(0, 14; 9, 20) &
(13, 11; 6, 3) \\
(9, 7; 20, 2) &
(15, 0; 12, 18) &
(4, 12; 3, 13) &
(2, 1; 11, 20) &
(1, 5; 21, 14) &
(13, 14; 10, 15) &
(11, 7; 4, 17) &
(14, 12; 7, 19) &
(6, 2; 8, 16) \\
(3, 9; 10, 12) &
(10, 14; 4, 3) &
(6, 3; 17, 2) &
(15, 11; 11, 5) &
(0, 2; 23, 13) &
(14, 7; 21, 5) &
(5, 8; 4, 18) &
(9, 15; 21, 4) &
(9, 14; 6, 14) \\
(3, 13; 9, 16) &
(7, 10; 9, 23) &
(7, 12; 11, 16) &
\end{array}$$ ]{}
$a=5$: [$$\begin{array}{llllllllll}
(14, 9; 22, 15) &
(1, 3; 10, 19) &
(4, 2; 16, 1) &
(7, 13; 5, 9) &
(7, 0; 16, 7) &
(14, 10; 12, 2) &
(1, 11; 5, 15) &
(12, 10; 1, 5) &
(5, 1; 7, 23) \\
(10, 8; 7, 14) &
(8, 13; 11, 20) &
(5, 8; 3, 6) &
(14, 8; 9, 0) &
(3, 11; 25, 3) &
(10, 9; 8, 3) &
(15, 12; 0, 10) &
(4, 0; 23, 19) &
(13, 10; 13, 25) \\
(5, 3; 22, 16) &
(4, 12; 7, 25) &
(2, 7; 8, 19) &
(10, 7; 6, 23) &
(1, 10; 9, 16) &
(3, 6; 18, 11) &
(14, 2; 11, 14) &
(12, 9; 12, 11) &
(3, 9; 13, 23) \\
(8, 11; 22, 2) &
(0, 3; 9, 14) &
(12, 5; 15, 2) &
(2, 10; 10, 18) &
(6, 2; 20, 9) &
(3, 13; 0, 17) &
(4, 1; 12, 6) &
(10, 3; 24, 15) &
(9, 7; 4, 25) \\
(5, 15; 4, 19) &
(14, 11; 23, 18) &
(14, 4; 13, 5) &
(1, 13; 8, 18) &
(5, 6; 21, 25) &
(9, 2; 2, 21) &
(1, 15; 11, 13) &
(1, 14; 17, 21) &
(6, 15; 22, 5) \\
(4, 13; 3, 15) &
(15, 13; 21, 7) &
(8, 1; 4, 24) &
(13, 0; 10, 24) &
(8, 0; 25, 5) &
(7, 14; 10, 20) &
(0, 6; 8, 4) &
(9, 15; 9, 6) &
(8, 15; 1, 15) \\
(4, 3; 20, 2) &
(6, 10; 0, 19) &
(11, 6; 1, 6) &
(10, 4; 4, 17) &
(4, 15; 18, 14) &
(12, 14; 16, 4) &
(5, 7; 1, 24) &
(5, 2; 0, 12) &
(12, 6; 17, 24) \\
(1, 2; 25, 22) &
(0, 15; 20, 12) &
(15, 7; 2, 17) &
(11, 13; 4, 12) &
(12, 1; 14, 20) &
(3, 14; 1, 8) &
(15, 2; 23, 3) &
(11, 15; 8, 16) &
(4, 9; 0, 24) \\
(14, 6; 3, 7) &
(7, 11; 11, 0) &
(2, 0; 17, 15) &
(6, 13; 2, 16) &
(11, 2; 13, 24) &
(11, 12; 19, 9) &
(9, 11; 10, 7) &
(12, 7; 3, 18) &
(5, 0; 13, 18) \\
(5, 9; 5, 20) &
(5, 11; 17, 14) &
(7, 4; 22, 21) &
(13, 9; 14, 1) &
(6, 8; 23, 10) &
(14, 13; 19, 6) &
(12, 0; 6, 21) &
(0, 10; 11, 22) &
(3, 8; 12, 21) \\
(8, 12; 8, 13) &
\end{array}$$ ]{}
$a=6$: [$$\begin{array}{llllllllll}
(10, 2; 10, 13) &
(14, 12; 18, 9) &
(9, 14; 10, 6) &
(3, 11; 10, 1) &
(14, 5; 16, 21) &
(12, 8; 8, 13) &
(4, 2; 15, 0) &
(10, 7; 3, 16) &
(3, 1; 17, 21) \\
(9, 6; 11, 21) &
(14, 7; 0, 20) &
(7, 9; 7, 8) &
(2, 14; 14, 23) &
(8, 13; 11, 5) &
(15, 0; 8, 22) &
(5, 2; 24, 3) &
(2, 9; 22, 25) &
(13, 14; 15, 2) \\
(12, 15; 14, 21) &
(3, 14; 8, 19) &
(13, 4; 19, 21) &
(11, 15; 9, 2) &
(5, 7; 25, 27) &
(13, 10; 4, 8) &
(1, 5; 20, 13) &
(4, 14; 25, 4) &
(4, 12; 7, 20) \\
(5, 0; 6, 15) &
(12, 3; 16, 0) &
(2, 15; 18, 11) &
(15, 8; 3, 6) &
(13, 11; 6, 13) &
(3, 13; 27, 12) &
(7, 2; 9, 21) &
(1, 11; 26, 25) &
(5, 10; 5, 0) \\
(6, 0; 25, 18) &
(7, 1; 11, 23) &
(1, 14; 12, 24) &
(6, 10; 6, 24) &
(13, 1; 16, 14) &
(11, 9; 12, 0) &
(9, 4; 5, 26) &
(15, 9; 24, 4) &
(9, 10; 27, 2) \\
(2, 8; 20, 2) &
(6, 11; 16, 8) &
(8, 7; 10, 26) &
(0, 8; 21, 27) &
(12, 6; 27, 3) &
(0, 7; 4, 19) &
(0, 11; 7, 14) &
(1, 15; 10, 15) &
(14, 8; 22, 7) \\
(13, 15; 0, 7) &
(6, 1; 9, 5) &
(5, 11; 18, 4) &
(4, 0; 23, 24) &
(14, 11; 5, 3) &
(3, 7; 2, 24) &
(12, 1; 22, 4) &
(12, 7; 6, 1) &
(9, 3; 13, 9) \\
(2, 13; 1, 17) &
(3, 4; 3, 14) &
(12, 0; 5, 10) &
(8, 11; 15, 24) &
(6, 13; 23, 10) &
(8, 4; 12, 1) &
(4, 6; 22, 2) &
(0, 2; 12, 16) &
(13, 9; 3, 20) \\
(12, 10; 26, 17) &
(10, 14; 11, 1) &
(14, 0; 13, 17) &
(6, 5; 7, 17) &
(15, 7; 5, 17) &
(3, 0; 11, 20) &
(11, 4; 27, 17) &
(0, 13; 9, 26) &
(12, 11; 19, 11) \\
(9, 5; 14, 1) &
(5, 12; 2, 12) &
(6, 15; 20, 1) &
(13, 7; 22, 18) &
(10, 3; 15, 18) &
(8, 10; 14, 9) &
(4, 1; 6, 18) &
(3, 8; 25, 23) &
(6, 2; 19, 26) \\
(8, 6; 0, 4) &
(15, 4; 13, 16) &
(9, 12; 15, 23) &
(2, 1; 27, 8) &
(3, 5; 26, 22) &
(10, 15; 25, 12) &
(1, 10; 7, 19) &
(5, 15; 19, 23) &
\end{array}$$ ]{}
$a=7$: [$$\begin{array}{llllllllll}
(3, 12; 16, 8) &
(15, 8; 15, 3) &
(11, 5; 13, 2) &
(7, 13; 6, 9) &
(5, 8; 0, 6) &
(9, 7; 29, 2) &
(4, 1; 7, 28) &
(10, 9; 11, 4) &
(4, 14; 27, 3) \\
(1, 15; 18, 6) &
(8, 13; 11, 23) &
(4, 12; 6, 15) &
(2, 14; 2, 18) &
(15, 7; 23, 8) &
(3, 11; 15, 28) &
(2, 5; 14, 29) &
(4, 6; 19, 24) &
(10, 14; 24, 14) \\
(1, 10; 17, 26) &
(11, 8; 12, 9) &
(0, 8; 14, 26) &
(0, 9; 8, 21) &
(4, 10; 5, 13) &
(10, 6; 8, 0) &
(0, 6; 11, 22) &
(9, 2; 26, 28) &
(12, 8; 2, 4) \\
(0, 15; 16, 10) &
(3, 7; 17, 25) &
(10, 8; 7, 1) &
(5, 0; 28, 24) &
(3, 13; 29, 1) &
(15, 4; 26, 4) &
(14, 1; 21, 11) &
(9, 15; 7, 24) &
(12, 10; 28, 18) \\
(2, 7; 24, 11) &
(4, 3; 0, 21) &
(4, 7; 16, 1) &
(14, 8; 25, 8) &
(0, 7; 27, 7) &
(9, 3; 10, 13) &
(2, 0; 15, 20) &
(9, 1; 9, 15) &
(7, 12; 3, 21) \\
(1, 8; 29, 13) &
(13, 1; 12, 8) &
(13, 0; 13, 19) &
(11, 7; 26, 0) &
(15, 13; 22, 14) &
(0, 11; 5, 25) &
(9, 6; 6, 23) &
(3, 10; 9, 2) &
(1, 7; 4, 19) \\
(4, 2; 12, 22) &
(15, 2; 0, 13) &
(11, 14; 6, 19) &
(10, 13; 15, 10) &
(8, 3; 24, 22) &
(4, 11; 17, 29) &
(5, 1; 20, 5) &
(5, 10; 25, 3) &
(7, 14; 10, 5) \\
(14, 0; 4, 9) &
(2, 6; 17, 9) &
(2, 11; 8, 3) &
(15, 10; 19, 12) &
(5, 7; 22, 18) &
(12, 14; 23, 13) &
(12, 9; 20, 0) &
(11, 1; 16, 24) &
(10, 0; 6, 29) \\
(12, 11; 14, 11) &
(6, 8; 27, 5) &
(9, 13; 5, 3) &
(14, 6; 20, 16) &
(1, 3; 27, 14) &
(9, 4; 25, 14) &
(5, 6; 1, 26) &
(3, 6; 18, 3) &
(3, 15; 11, 20) \\
(1, 12; 22, 10) &
(15, 6; 25, 2) &
(15, 5; 21, 17) &
(13, 4; 2, 20) &
(13, 11; 7, 18) &
(12, 2; 19, 1) &
(7, 8; 20, 28) &
(6, 12; 7, 29) &
(3, 5; 23, 19) \\
(2, 8; 10, 21) &
(12, 0; 12, 17) &
(13, 14; 0, 17) &
(6, 13; 21, 28) &
(4, 0; 18, 23) &
(2, 1; 23, 25) &
(14, 9; 22, 1) &
(13, 5; 4, 16) &
(12, 15; 9, 5) \\
(10, 2; 27, 16) &
(11, 15; 27, 1) &
(11, 6; 10, 4) &
(5, 14; 15, 7) &
(3, 14; 12, 26) &
(5, 9; 12, 27) &
\end{array}$$ ]{}
$a=8$: [$$\begin{array}{llllllllll}
(15, 7; 25, 22) &
(2, 7; 26, 30) &
(2, 0; 21, 25) &
(7, 10; 31, 9) &
(3, 13; 8, 19) &
(5, 7; 20, 3) &
(6, 5; 2, 7) &
(12, 5; 21, 16) &
(6, 10; 25, 19) \\
(1, 4; 4, 25) &
(13, 7; 17, 0) &
(5, 2; 1, 14) &
(7, 12; 4, 29) &
(4, 11; 19, 28) &
(14, 4; 6, 22) &
(5, 10; 30, 17) &
(10, 13; 18, 1) &
(1, 8; 9, 6) \\
(5, 14; 4, 26) &
(4, 6; 31, 18) &
(7, 9; 6, 1) &
(14, 0; 9, 12) &
(6, 3; 1, 9) &
(0, 8; 13, 22) &
(6, 2; 27, 10) &
(11, 13; 9, 3) &
(12, 3; 13, 23) \\
(15, 3; 12, 18) &
(8, 12; 30, 0) &
(15, 1; 21, 27) &
(14, 10; 8, 5) &
(4, 2; 15, 29) &
(12, 11; 2, 18) &
(9, 12; 9, 5) &
(11, 15; 24, 5) &
(15, 4; 0, 26) \\
(9, 3; 22, 24) &
(5, 8; 25, 28) &
(8, 2; 2, 11) &
(14, 12; 7, 17) &
(3, 11; 31, 25) &
(2, 11; 13, 17) &
(0, 6; 29, 24) &
(3, 0; 30, 10) &
(8, 15; 1, 7) \\
(15, 0; 17, 4) &
(9, 11; 0, 11) &
(10, 8; 10, 29) &
(10, 2; 24, 0) &
(9, 6; 23, 26) &
(11, 7; 7, 10) &
(12, 10; 6, 14) &
(15, 13; 23, 2) &
(0, 7; 18, 28) \\
(15, 9; 14, 10) &
(12, 1; 10, 12) &
(8, 3; 26, 20) &
(13, 9; 21, 29) &
(2, 12; 22, 3) &
(3, 10; 3, 28) &
(6, 12; 11, 28) &
(8, 4; 14, 23) &
(10, 1; 15, 26) \\
(13, 14; 10, 15) &
(13, 8; 31, 12) &
(13, 4; 13, 7) &
(9, 4; 30, 3) &
(1, 11; 8, 30) &
(9, 5; 27, 31) &
(0, 12; 31, 8) &
(6, 14; 0, 20) &
(0, 9; 20, 15) \\
(4, 3; 17, 27) &
(0, 5; 19, 23) &
(6, 15; 3, 8) &
(6, 13; 30, 4) &
(11, 8; 4, 15) &
(14, 8; 24, 3) &
(9, 10; 13, 4) &
(10, 15; 16, 11) &
(2, 13; 28, 16) \\
(4, 0; 5, 16) &
(2, 15; 20, 9) &
(1, 2; 31, 19) &
(7, 8; 8, 27) &
(5, 13; 22, 5) &
(7, 1; 5, 23) &
(6, 11; 16, 6) &
(7, 14; 19, 11) &
(15, 12; 19, 15) \\
(11, 14; 27, 1) &
(10, 0; 7, 27) &
(14, 2; 23, 18) &
(11, 5; 29, 12) &
(7, 3; 16, 2) &
(1, 6; 22, 17) &
(5, 3; 0, 15) &
(3, 1; 29, 11) &
(3, 14; 21, 14) \\
(4, 10; 12, 2) &
(4, 12; 20, 1) &
(4, 7; 21, 24) &
(0, 13; 11, 6) &
(9, 14; 25, 2) &
(9, 2; 12, 8) &
(11, 0; 14, 26) &
(8, 6; 21, 5) &
(13, 1; 20, 14) \\
(1, 5; 18, 24) &
(1, 9; 7, 28) &
(1, 14; 13, 16) &
(5, 15; 6, 13) &
\end{array}$$ ]{}
There exists an SFS of type $(4,2)^a(2,2)^{9-a}$ for each $a\in\{0,1,\ldots,9\}$.
Let $V=I_{18}$ and $S=I_{18+2a}$. $V$ can be partitioned as $V=\cup_{i=0}^8\{2i,2i+1\}$ and $S$ can be partitioned as $S=(\cup_{i=0}^{a-1}\{4i,4i+1,4i+2,4i+3\})\cup (\cup_{i=a}^{8}\{2i,2i+1\})$. The required SFSs are presented as follows.
$a=0$: [$$\begin{array}{llllllllll}
(7, 4; 13, 1) &
(9, 6; 1, 14) &
(10, 12; 3, 9) &
(9, 17; 0, 3) &
(9, 0; 4, 15) &
(10, 17; 5, 7) &
(0, 3; 5, 10) &
(0, 17; 11, 13) &
(13, 5; 11, 9) \\
(12, 15; 1, 7) &
(15, 5; 3, 12) &
(14, 16; 3, 4) &
(17, 15; 9, 2) &
(5, 0; 17, 14) &
(12, 3; 16, 14) &
(11, 4; 3, 14) &
(16, 1; 12, 14) &
(3, 1; 7, 17) \\
(10, 7; 2, 14) &
(9, 14; 12, 5) &
(5, 10; 0, 15) &
(1, 12; 8, 4) &
(14, 2; 6, 1) &
(15, 4; 16, 8) &
(8, 6; 16, 12) &
(2, 12; 15, 5) &
(8, 13; 4, 14) \\
(7, 13; 8, 10) &
(16, 15; 0, 5) &
(14, 10; 17, 8) &
(15, 2; 4, 11) &
(13, 2; 0, 7) &
(8, 0; 3, 6) &
(1, 4; 11, 15) &
(13, 6; 3, 17) &
(4, 3; 0, 6) \\
(9, 12; 11, 17) &
(11, 8; 0, 2) &
(13, 10; 16, 6) &
(0, 4; 9, 7) &
(10, 6; 4, 13) &
(14, 1; 9, 13) &
(14, 0; 16, 2) &
(2, 5; 16, 13) &
(10, 3; 1, 12) \\
(16, 8; 11, 1) &
(2, 4; 17, 12) &
(7, 1; 16, 3) &
(5, 12; 6, 2) &
(8, 7; 17, 5) &
(7, 14; 11, 0) &
(13, 1; 2, 5) &
(11, 17; 6, 4) &
(17, 5; 1, 10) \\
(16, 6; 15, 2) &
(8, 14; 7, 10) &
(6, 12; 0, 10) &
(17, 2; 8, 14) &
(7, 17; 12, 15) &
(4, 9; 2, 10) &
(15, 11; 13, 17) &
(8, 3; 13, 15) &
(11, 13; 1, 15) \\
(6, 11; 5, 9) &
(6, 3; 8, 11) &
(16, 2; 9, 10) &
(3, 7; 4, 9) &
(9, 16; 13, 6) &
(15, 1; 10, 6) &
(5, 16; 7, 8) &
(0, 11; 8, 12) &
(9, 11; 7, 16) \\
\end{array}$$ ]{}
$a=1$: [$$\begin{array}{llllllllll}
(4, 11; 5, 1) &
(7, 4; 10, 15) &
(14, 4; 9, 18) &
(9, 3; 13, 0) &
(2, 1; 17, 14) &
(11, 8; 0, 7) &
(4, 0; 19, 14) &
(6, 12; 16, 5) &
(8, 10; 14, 5) \\
(11, 16; 6, 15) &
(15, 17; 4, 9) &
(6, 13; 17, 7) &
(7, 17; 16, 7) &
(11, 1; 10, 4) &
(12, 11; 17, 3) &
(4, 2; 16, 0) &
(16, 3; 17, 2) &
(5, 15; 8, 0) \\
(2, 14; 3, 6) &
(14, 11; 11, 2) &
(15, 7; 1, 12) &
(0, 8; 6, 16) &
(1, 7; 18, 13) &
(1, 5; 9, 5) &
(14, 3; 19, 10) &
(0, 14; 12, 4) &
(13, 11; 9, 19) \\
(17, 3; 1, 15) &
(12, 8; 9, 2) &
(14, 5; 1, 14) &
(11, 3; 18, 8) &
(1, 15; 19, 6) &
(16, 10; 1, 9) &
(13, 8; 1, 8) &
(9, 12; 1, 19) &
(5, 0; 15, 17) \\
(14, 16; 8, 13) &
(6, 16; 10, 0) &
(4, 17; 17, 11) &
(3, 6; 12, 6) &
(0, 15; 5, 13) &
(10, 9; 6, 17) &
(9, 11; 16, 14) &
(16, 13; 5, 11) &
(16, 7; 14, 3) \\
(6, 4; 4, 2) &
(3, 10; 3, 16) &
(6, 10; 15, 19) &
(5, 8; 12, 19) &
(0, 13; 10, 18) &
(7, 13; 6, 2) &
(6, 15; 3, 11) &
(13, 9; 3, 12) &
(2, 12; 13, 10) \\
(12, 17; 0, 6) &
(8, 15; 18, 15) &
(0, 2; 7, 8) &
(10, 15; 10, 2) &
(15, 3; 14, 7) &
(17, 9; 5, 8) &
(14, 1; 7, 15) &
(4, 8; 13, 3) &
(14, 7; 5, 0) \\
(7, 2; 19, 11) &
(9, 16; 4, 7) &
(9, 2; 15, 9) &
(2, 6; 18, 1) &
(8, 7; 4, 17) &
(12, 10; 18, 7) &
(3, 0; 9, 11) &
(9, 5; 2, 18) &
(6, 17; 13, 14) \\
(17, 5; 3, 10) &
(2, 17; 12, 2) &
(5, 13; 13, 16) &
(10, 13; 0, 4) &
(1, 16; 16, 12) &
(12, 4; 8, 12) &
(1, 10; 8, 11) &
(5, 12; 4, 11) &
\end{array}$$ ]{}
$a=2$: [$$\begin{array}{llllllllll}
(12, 4; 21, 15) &
(8, 1; 8, 17) &
(10, 3; 1, 13) &
(1, 7; 7, 21) &
(4, 2; 16, 18) &
(12, 16; 7, 2) &
(9, 15; 8, 4) &
(12, 7; 14, 1) &
(12, 10; 6, 3) \\
(12, 1; 10, 4) &
(14, 8; 11, 16) &
(2, 7; 3, 19) &
(17, 0; 9, 7) &
(0, 7; 5, 8) &
(16, 13; 11, 3) &
(15, 17; 3, 14) &
(14, 12; 5, 9) &
(1, 14; 6, 20) \\
(1, 13; 5, 14) &
(6, 15; 15, 12) &
(10, 4; 19, 7) &
(0, 13; 18, 12) &
(5, 6; 3, 18) &
(4, 8; 20, 3) &
(11, 6; 20, 5) &
(11, 16; 4, 19) &
(5, 12; 19, 13) \\
(4, 3; 14, 12) &
(16, 9; 17, 14) &
(14, 3; 21, 3) &
(6, 3; 17, 0) &
(8, 0; 4, 14) &
(11, 9; 7, 3) &
(2, 8; 9, 2) &
(17, 9; 5, 2) &
(0, 10; 17, 11) \\
(17, 6; 16, 4) &
(3, 12; 8, 18) &
(8, 5; 5, 21) &
(8, 15; 7, 10) &
(16, 1; 16, 13) &
(17, 14; 15, 17) &
(10, 5; 20, 4) &
(15, 4; 1, 5) &
(9, 7; 9, 18) \\
(11, 15; 16, 9) &
(2, 13; 20, 0) &
(5, 2; 12, 11) &
(17, 12; 12, 0) &
(16, 7; 6, 12) &
(16, 14; 8, 0) &
(0, 2; 13, 15) &
(2, 15; 17, 21) &
(6, 4; 2, 6) \\
(17, 2; 10, 1) &
(12, 9; 20, 11) &
(17, 8; 6, 19) &
(7, 15; 13, 20) &
(11, 14; 2, 12) &
(0, 3; 16, 20) &
(5, 11; 10, 17) &
(13, 11; 6, 21) &
(9, 10; 21, 0) \\
(4, 7; 4, 17) &
(10, 1; 12, 9) &
(0, 6; 19, 21) &
(13, 6; 7, 13) &
(9, 5; 1, 16) &
(11, 17; 13, 8) &
(0, 9; 10, 6) &
(7, 10; 2, 16) &
(9, 1; 19, 15) \\
(10, 13; 8, 10) &
(16, 10; 5, 18) &
(3, 16; 10, 15) &
(5, 14; 14, 7) &
(1, 17; 11, 18) &
(2, 6; 14, 8) &
(3, 13; 19, 9) &
(15, 5; 6, 0) &
(4, 14; 13, 10) \\
(15, 3; 11, 2) &
(6, 16; 9, 1) &
(5, 13; 2, 15) &
(13, 14; 1, 4) &
(8, 11; 1, 18) &
(4, 11; 0, 11) &
(7, 8; 0, 15) &
\end{array}$$ ]{}
$a=3$: [$$\begin{array}{llllllllll}
(8, 12; 2, 9) &
(0, 14; 15, 8) &
(17, 10; 8, 2) &
(12, 1; 16, 15) &
(3, 16; 17, 9) &
(11, 1; 10, 12) &
(8, 1; 7, 19) &
(7, 9; 1, 10) &
(17, 11; 15, 11) \\
(3, 5; 22, 1) &
(10, 13; 13, 11) &
(11, 9; 4, 2) &
(15, 3; 0, 15) &
(14, 9; 22, 7) &
(0, 8; 18, 11) &
(17, 8; 17, 3) &
(11, 6; 22, 14) &
(13, 16; 7, 15) \\
(16, 15; 11, 19) &
(13, 14; 2, 10) &
(3, 10; 12, 20) &
(4, 15; 2, 14) &
(12, 2; 0, 22) &
(2, 10; 9, 15) &
(15, 0; 6, 13) &
(12, 16; 1, 6) &
(16, 7; 2, 21) \\
(13, 0; 17, 22) &
(3, 1; 13, 18) &
(3, 6; 2, 11) &
(15, 8; 8, 22) &
(5, 1; 20, 14) &
(4, 6; 20, 15) &
(0, 11; 9, 20) &
(17, 13; 20, 1) &
(5, 17; 19, 13) \\
(14, 6; 16, 9) &
(5, 2; 2, 12) &
(3, 8; 10, 16) &
(8, 11; 1, 5) &
(0, 17; 7, 12) &
(6, 13; 5, 0) &
(12, 3; 21, 8) &
(14, 12; 11, 4) &
(11, 7; 19, 8) \\
(0, 10; 5, 14) &
(9, 12; 20, 13) &
(2, 16; 20, 8) &
(13, 1; 4, 8) &
(14, 16; 14, 12) &
(16, 4; 13, 3) &
(6, 9; 6, 8) &
(13, 3; 14, 23) &
(7, 17; 0, 4) \\
(6, 15; 10, 17) &
(7, 13; 9, 3) &
(0, 7; 16, 23) &
(4, 17; 16, 21) &
(7, 5; 5, 15) &
(17, 12; 10, 14) &
(15, 10; 7, 23) &
(7, 2; 18, 14) &
(4, 8; 23, 4) \\
(14, 4; 18, 1) &
(5, 15; 3, 4) &
(1, 17; 9, 6) &
(6, 0; 21, 4) &
(6, 10; 19, 1) &
(5, 12; 17, 23) &
(15, 17; 18, 5) &
(12, 11; 3, 7) &
(5, 6; 7, 18) \\
(9, 16; 5, 16) &
(14, 11; 6, 23) &
(2, 11; 21, 13) &
(2, 15; 16, 1) &
(14, 8; 13, 0) &
(10, 5; 21, 0) &
(6, 2; 3, 23) &
(12, 4; 12, 5) &
(0, 2; 19, 10) \\
(9, 1; 21, 23) &
(9, 10; 3, 18) &
(7, 1; 11, 22) &
(15, 9; 9, 12) &
(11, 16; 0, 18) &
(14, 3; 3, 19) &
(4, 9; 19, 0) &
(10, 16; 10, 4) &
(1, 14; 5, 17) \\
(8, 13; 12, 21) &
(5, 13; 16, 6) &
(2, 9; 11, 17) &
(4, 7; 17, 7) &
(4, 10; 6, 22) &
(7, 8; 6, 20) &
\end{array}$$ ]{}
$a=4$: [$$\begin{array}{llllllllll}
(13, 6; 11, 23) &
(5, 10; 4, 23) &
(17, 13; 22, 17) &
(15, 12; 9, 18) &
(9, 14; 10, 25) &
(17, 0; 21, 5) &
(13, 1; 13, 24) &
(8, 13; 9, 25) &
(4, 13; 2, 4) \\
(11, 16; 6, 14) &
(4, 15; 12, 6) &
(5, 7; 5, 25) &
(7, 4; 24, 0) &
(6, 10; 16, 10) &
(1, 15; 8, 4) &
(16, 0; 22, 4) &
(3, 16; 11, 17) &
(8, 0; 8, 13) \\
(14, 3; 20, 1) &
(9, 10; 11, 5) &
(14, 1; 5, 19) &
(7, 15; 7, 19) &
(17, 2; 20, 8) &
(15, 10; 1, 25) &
(17, 8; 23, 7) &
(10, 12; 6, 13) &
(4, 17; 1, 15) \\
(9, 7; 6, 2) &
(2, 11; 25, 3) &
(6, 9; 9, 4) &
(1, 5; 6, 18) &
(17, 9; 12, 18) &
(6, 16; 18, 7) &
(15, 5; 16, 21) &
(17, 3; 9, 2) &
(8, 7; 20, 10) \\
(3, 10; 14, 22) &
(14, 16; 2, 21) &
(6, 4; 22, 5) &
(14, 13; 8, 14) &
(9, 3; 15, 21) &
(2, 16; 23, 0) &
(14, 7; 17, 18) &
(9, 0; 24, 20) &
(6, 1; 17, 25) \\
(0, 4; 16, 18) &
(11, 12; 2, 11) &
(11, 6; 1, 24) &
(14, 11; 13, 4) &
(13, 11; 5, 16) &
(2, 14; 16, 11) &
(11, 7; 22, 21) &
(4, 9; 3, 14) &
(0, 15; 11, 14) \\
(12, 4; 17, 19) &
(16, 12; 8, 5) &
(14, 8; 6, 15) &
(15, 2; 24, 17) &
(6, 8; 21, 3) &
(9, 5; 22, 1) &
(15, 17; 3, 13) &
(11, 5; 17, 12) &
(16, 13; 12, 1) \\
(7, 10; 3, 8) &
(12, 3; 10, 24) &
(12, 0; 23, 25) &
(10, 8; 2, 24) &
(5, 2; 2, 13) &
(10, 0; 17, 7) &
(1, 2; 14, 10) &
(10, 1; 20, 15) &
(6, 3; 8, 0) \\
(11, 17; 0, 10) &
(8, 2; 18, 22) &
(5, 14; 7, 24) &
(12, 17; 14, 16) &
(14, 10; 0, 9) &
(5, 16; 3, 20) &
(1, 16; 9, 16) &
(7, 3; 23, 16) &
(6, 15; 20, 2) \\
(17, 7; 11, 4) &
(15, 16; 15, 10) &
(12, 8; 1, 4) &
(10, 2; 21, 12) &
(2, 7; 1, 9) &
(3, 13; 3, 18) &
(8, 15; 5, 0) &
(9, 13; 7, 0) &
(5, 12; 0, 15) \\
(2, 13; 15, 19) &
(8, 5; 19, 14) &
(9, 16; 19, 13) &
(4, 3; 25, 13) &
(11, 0; 15, 9) &
(0, 13; 6, 10) &
(4, 1; 23, 21) &
(9, 11; 8, 23) &
(17, 6; 6, 19) \\
(0, 3; 19, 12) &
(4, 11; 7, 20) &
(12, 1; 7, 22) &
(1, 8; 11, 12) &
(12, 14; 3, 12) &
\end{array}$$ ]{}
$a=5$: [$$\begin{array}{llllllllll}
(16, 11; 22, 14) &
(0, 2; 15, 16) &
(6, 1; 10, 27) &
(17, 10; 2, 25) &
(2, 4; 27, 24) &
(16, 12; 19, 20) &
(16, 10; 9, 15) &
(5, 14; 0, 23) &
(10, 8; 13, 10) \\
(12, 10; 0, 16) &
(7, 11; 26, 11) &
(14, 10; 1, 5) &
(7, 2; 2, 17) &
(13, 3; 21, 24) &
(10, 2; 14, 18) &
(6, 12; 9, 5) &
(15, 1; 14, 9) &
(15, 10; 12, 23) \\
(0, 10; 17, 26) &
(12, 4; 21, 14) &
(6, 3; 25, 3) &
(8, 6; 1, 11) &
(6, 9; 24, 26) &
(9, 5; 20, 22) &
(12, 5; 15, 3) &
(9, 7; 10, 6) &
(17, 6; 18, 7) \\
(9, 12; 25, 12) &
(11, 14; 4, 13) &
(1, 8; 6, 12) &
(13, 2; 12, 3) &
(15, 11; 27, 16) &
(4, 7; 7, 0) &
(9, 16; 21, 8) &
(15, 3; 20, 8) &
(3, 17; 14, 11) \\
(13, 9; 0, 27) &
(8, 16; 24, 5) &
(4, 0; 25, 5) &
(14, 4; 18, 3) &
(14, 9; 14, 7) &
(12, 7; 18, 8) &
(15, 9; 11, 2) &
(3, 4; 19, 12) &
(2, 1; 26, 20) \\
(11, 3; 17, 1) &
(14, 3; 16, 9) &
(13, 4; 26, 13) &
(0, 14; 19, 6) &
(0, 8; 7, 27) &
(6, 5; 16, 21) &
(4, 11; 6, 15) &
(16, 0; 11, 23) &
(1, 13; 5, 19) \\
(2, 17; 23, 9) &
(0, 9; 13, 9) &
(10, 6; 22, 6) &
(15, 16; 3, 4) &
(11, 1; 23, 7) &
(11, 12; 2, 24) &
(0, 3; 22, 18) &
(2, 14; 22, 11) &
(7, 17; 20, 24) \\
(15, 12; 7, 26) &
(8, 11; 3, 9) &
(14, 6; 2, 8) &
(0, 6; 20, 4) &
(12, 2; 1, 10) &
(10, 7; 19, 3) &
(2, 11; 8, 0) &
(15, 5; 6, 18) &
(0, 17; 12, 8) \\
(13, 10; 11, 7) &
(17, 9; 5, 3) &
(4, 9; 4, 1) &
(15, 6; 19, 0) &
(9, 3; 23, 15) &
(16, 3; 0, 10) &
(17, 11; 10, 19) &
(2, 8; 25, 21) &
(7, 14; 21, 27) \\
(1, 4; 16, 22) &
(17, 13; 16, 4) &
(12, 1; 11, 4) &
(5, 1; 17, 25) &
(15, 7; 22, 5) &
(14, 16; 17, 12) &
(13, 16; 2, 6) &
(12, 3; 27, 13) &
(11, 5; 12, 5) \\
(3, 5; 2, 26) &
(5, 0; 24, 14) &
(13, 14; 10, 20) &
(8, 17; 0, 22) &
(17, 1; 21, 15) &
(8, 7; 23, 4) &
(5, 2; 19, 13) &
(16, 1; 13, 18) &
(13, 15; 15, 17) \\
(17, 15; 13, 1) &
(12, 17; 17, 6) &
(14, 8; 26, 15) &
(4, 8; 20, 2) &
(15, 0; 10, 21) &
(1, 10; 24, 8) &
(5, 10; 4, 27) &
(13, 8; 14, 8) &
(6, 4; 17, 23) \\
(11, 13; 18, 25) &
(7, 16; 25, 16) &
(5, 16; 1, 7) &
(7, 13; 1, 9) &
\end{array}$$ ]{}
$a=6$: [$$\begin{array}{llllllllll}
(6, 1; 24, 7) &
(4, 15; 29, 12) &
(0, 10; 27, 13) &
(14, 17; 22, 25) &
(12, 6; 3, 19) &
(16, 4; 6, 21) &
(15, 0; 14, 8) &
(3, 14; 28, 8) &
(3, 10; 12, 3) \\
(16, 10; 24, 16) &
(5, 2; 19, 26) &
(10, 12; 5, 26) &
(0, 6; 5, 10) &
(9, 6; 4, 8) &
(15, 11; 4, 10) &
(7, 11; 1, 8) &
(1, 17; 15, 26) &
(10, 5; 15, 2) \\
(7, 16; 4, 26) &
(1, 10; 11, 28) &
(13, 15; 16, 6) &
(16, 14; 19, 15) &
(6, 4; 1, 28) &
(13, 6; 11, 21) &
(8, 1; 6, 8) &
(0, 16; 11, 22) &
(17, 2; 8, 24) \\
(2, 7; 25, 20) &
(3, 4; 19, 23) &
(10, 9; 29, 6) &
(1, 5; 4, 16) &
(14, 11; 3, 5) &
(5, 9; 22, 1) &
(2, 13; 1, 29) &
(6, 14; 18, 23) &
(9, 1; 5, 20) \\
(12, 9; 9, 15) &
(7, 10; 7, 18) &
(4, 14; 24, 13) &
(12, 2; 21, 13) &
(12, 16; 2, 8) &
(8, 12; 4, 27) &
(14, 0; 16, 7) &
(5, 15; 18, 28) &
(5, 14; 12, 20) \\
(13, 11; 2, 19) &
(13, 16; 5, 0) &
(6, 8; 9, 0) &
(13, 0; 15, 4) &
(3, 5; 24, 0) &
(1, 13; 12, 22) &
(12, 0; 28, 23) &
(14, 10; 4, 9) &
(1, 14; 29, 14) \\
(13, 10; 17, 8) &
(2, 9; 27, 23) &
(1, 4; 27, 25) &
(12, 15; 7, 22) &
(12, 11; 18, 29) &
(11, 16; 14, 25) &
(4, 17; 4, 3) &
(3, 9; 13, 26) &
(7, 4; 5, 22) \\
(15, 10; 0, 25) &
(13, 7; 23, 9) &
(2, 11; 28, 0) &
(15, 1; 19, 21) &
(12, 5; 14, 6) &
(12, 17; 12, 11) &
(0, 3; 9, 21) &
(5, 11; 17, 13) &
(7, 17; 27, 0) \\
(8, 13; 10, 28) &
(16, 2; 12, 10) &
(15, 6; 2, 20) &
(3, 12; 20, 10) &
(16, 15; 3, 9) &
(8, 14; 21, 2) &
(9, 17; 2, 7) &
(9, 15; 24, 11) &
(11, 8; 24, 12) \\
(13, 4; 18, 26) &
(8, 16; 7, 20) &
(6, 2; 22, 16) &
(8, 15; 5, 13) &
(3, 7; 16, 2) &
(12, 14; 17, 1) &
(4, 0; 20, 17) &
(4, 11; 15, 7) &
(15, 2; 15, 17) \\
(5, 7; 29, 3) &
(7, 9; 21, 28) &
(14, 9; 10, 0) &
(17, 15; 1, 23) &
(1, 2; 9, 18) &
(7, 1; 10, 17) &
(0, 7; 24, 19) &
(10, 17; 10, 19) &
(0, 9; 25, 12) \\
(16, 1; 13, 23) &
(11, 3; 27, 11) &
(16, 6; 17, 27) &
(13, 5; 27, 7) &
(17, 5; 21, 5) &
(17, 13; 20, 13) &
(2, 4; 14, 2) &
(14, 7; 11, 6) &
(11, 17; 9, 16) \\
(2, 8; 3, 11) &
(4, 12; 0, 16) &
(9, 13; 3, 14) &
(0, 8; 26, 29) &
(6, 3; 29, 25) &
(3, 17; 14, 17) &
(0, 17; 6, 18) &
(6, 11; 6, 26) &
(3, 8; 15, 22) \\
(3, 16; 1, 18) &
(5, 8; 23, 25) &
(8, 10; 1, 14) &
\end{array}$$ ]{}
$a=7$: [$$\begin{array}{llllllllll}
(10, 17; 13, 28) &
(5, 7; 5, 17) &
(2, 8; 29, 10) &
(15, 6; 9, 27) &
(1, 3; 24, 20) &
(7, 9; 9, 31) &
(6, 10; 25, 18) &
(10, 2; 30, 17) &
(14, 2; 25, 16) \\
(6, 11; 1, 31) &
(13, 2; 9, 15) &
(11, 16; 12, 4) &
(17, 1; 29, 26) &
(1, 9; 25, 30) &
(9, 11; 3, 28) &
(12, 17; 17, 7) &
(8, 15; 1, 24) &
(12, 1; 6, 16) \\
(8, 4; 13, 30) &
(7, 12; 3, 29) &
(3, 9; 21, 12) &
(2, 7; 0, 26) &
(16, 6; 21, 3) &
(11, 13; 5, 8) &
(2, 1; 11, 14) &
(15, 5; 16, 23) &
(6, 8; 28, 5) \\
(12, 4; 5, 0) &
(2, 17; 1, 23) &
(9, 2; 24, 13) &
(6, 2; 2, 8) &
(14, 13; 2, 12) &
(9, 14; 0, 11) &
(14, 6; 30, 7) &
(17, 14; 24, 19) &
(7, 3; 30, 27) \\
(3, 4; 1, 15) &
(13, 15; 4, 20) &
(15, 2; 22, 3) &
(10, 13; 19, 29) &
(10, 3; 3, 8) &
(12, 3; 22, 28) &
(9, 6; 26, 22) &
(14, 4; 31, 23) &
(14, 7; 22, 6) \\
(3, 8; 2, 23) &
(4, 6; 6, 20) &
(4, 0; 22, 19) &
(15, 17; 10, 2) &
(15, 1; 5, 18) &
(8, 14; 4, 27) &
(3, 16; 29, 17) &
(9, 16; 10, 6) &
(1, 6; 23, 17) \\
(1, 4; 7, 27) &
(3, 5; 13, 25) &
(11, 12; 30, 15) &
(10, 1; 12, 10) &
(17, 9; 5, 27) &
(8, 10; 9, 6) &
(7, 15; 11, 21) &
(16, 4; 24, 18) &
(12, 6; 10, 19) \\
(15, 10; 14, 0) &
(0, 5; 21, 27) &
(5, 2; 31, 12) &
(15, 16; 7, 13) &
(8, 16; 0, 25) &
(14, 3; 14, 10) &
(11, 2; 19, 27) &
(16, 14; 5, 20) &
(5, 13; 30, 1) \\
(13, 17; 0, 22) &
(0, 13; 17, 6) &
(0, 15; 30, 12) &
(7, 4; 4, 25) &
(7, 1; 8, 19) &
(11, 7; 2, 24) &
(3, 15; 19, 31) &
(8, 5; 14, 22) &
(16, 10; 27, 11) \\
(17, 5; 6, 15) &
(9, 10; 1, 7) &
(15, 11; 25, 6) &
(12, 0; 11, 13) &
(4, 10; 2, 16) &
(6, 17; 4, 11) &
(6, 3; 0, 16) &
(3, 13; 11, 18) &
(0, 9; 4, 29) \\
(14, 12; 1, 8) &
(0, 11; 16, 10) &
(11, 8; 26, 11) &
(4, 13; 28, 21) &
(5, 11; 7, 0) &
(9, 5; 2, 20) &
(12, 9; 14, 23) &
(11, 17; 9, 18) &
(0, 17; 25, 14) \\
(0, 3; 26, 9) &
(4, 17; 3, 12) &
(16, 5; 26, 19) &
(16, 12; 2, 9) &
(11, 4; 14, 29) &
(13, 16; 14, 16) &
(10, 12; 4, 31) &
(14, 1; 9, 21) &
(12, 2; 18, 21) \\
(5, 1; 4, 28) &
(4, 15; 26, 17) &
(8, 0; 31, 15) &
(17, 7; 16, 20) &
(8, 12; 12, 20) &
(14, 10; 15, 26) &
(16, 1; 15, 22) &
(1, 13; 31, 13) &
(7, 0; 7, 18) \\
(6, 5; 29, 24) &
(7, 13; 23, 10) &
(13, 8; 3, 7) &
(8, 17; 8, 21) &
(14, 11; 13, 17) &
(9, 15; 8, 15) &
(0, 10; 24, 5) &
(14, 5; 3, 18) &
(16, 7; 1, 28) \\
(0, 2; 20, 28) &
(0, 16; 8, 23) &
\end{array}$$ ]{}
$a=8$: [$$\begin{array}{llllllllll}
(7, 9; 28, 6) &
(0, 16; 10, 21) &
(2, 7; 31, 10) &
(17, 1; 21, 6) &
(11, 0; 4, 18) &
(1, 2; 19, 23) &
(13, 0; 19, 6) &
(4, 16; 24, 1) &
(3, 13; 16, 30) \\
(16, 1; 27, 22) &
(10, 6; 10, 30) &
(4, 0; 17, 30) &
(4, 3; 13, 23) &
(17, 3; 10, 18) &
(3, 16; 3, 20) &
(6, 3; 25, 28) &
(17, 13; 4, 8) &
(1, 14; 10, 7) \\
(8, 3; 22, 24) &
(5, 7; 30, 4) &
(1, 7; 16, 26) &
(13, 8; 29, 21) &
(14, 8; 20, 11) &
(15, 8; 14, 0) &
(7, 10; 1, 8) &
(9, 10; 5, 26) &
(12, 14; 30, 6) \\
(15, 11; 26, 10) &
(8, 12; 10, 15) &
(15, 10; 13, 16) &
(8, 7; 23, 3) &
(16, 2; 8, 16) &
(5, 17; 7, 27) &
(10, 16; 15, 6) &
(14, 17; 22, 17) &
(4, 8; 12, 6) \\
(2, 13; 9, 20) &
(9, 1; 13, 20) &
(3, 9; 0, 15) &
(5, 13; 13, 1) &
(13, 16; 23, 14) &
(12, 10; 28, 9) &
(12, 2; 22, 13) &
(2, 14; 27, 3) &
(4, 7; 5, 22) \\
(5, 2; 2, 29) &
(9, 15; 21, 27) &
(1, 4; 31, 29) &
(17, 7; 24, 2) &
(11, 7; 29, 9) &
(1, 11; 11, 17) &
(2, 17; 12, 0) &
(17, 8; 1, 5) &
(4, 12; 16, 0) \\
(15, 12; 4, 3) &
(0, 9; 14, 7) &
(8, 6; 27, 31) &
(12, 3; 11, 29) &
(0, 15; 15, 24) &
(8, 10; 7, 2) &
(3, 14; 26, 1) &
(13, 15; 5, 31) &
(9, 11; 3, 30) \\
(6, 9; 2, 9) &
(17, 6; 29, 16) &
(10, 4; 18, 14) &
(12, 5; 23, 18) &
(10, 14; 12, 4) &
(16, 5; 0, 28) &
(6, 12; 8, 19) &
(2, 11; 1, 14) &
(5, 10; 31, 3) \\
(3, 15; 2, 12) &
(10, 3; 19, 27) &
(9, 17; 25, 11) &
(13, 9; 22, 10) &
(13, 14; 2, 15) &
(0, 7; 27, 20) &
(1, 13; 18, 12) &
(16, 14; 5, 13) &
(13, 10; 0, 11) \\
(6, 1; 24, 5) &
(5, 9; 12, 24) &
(8, 16; 26, 9) &
(2, 6; 26, 21) &
(8, 2; 25, 30) &
(7, 12; 21, 7) &
(0, 3; 31, 8) &
(0, 6; 11, 23) &
(6, 16; 7, 18) \\
(0, 17; 13, 28) &
(11, 14; 31, 24) &
(14, 9; 8, 23) &
(14, 5; 19, 14) &
(8, 11; 13, 8) &
(9, 16; 29, 4) &
(5, 1; 15, 25) &
(17, 15; 23, 9) &
(14, 4; 21, 25) \\
(12, 17; 14, 20) &
(12, 0; 5, 12) &
(4, 6; 4, 20) &
(1, 8; 28, 4) &
(14, 7; 0, 18) &
(0, 10; 25, 29) &
(6, 13; 3, 17) &
(9, 12; 1, 31) &
(11, 6; 6, 0) \\
(16, 12; 17, 2) &
(17, 11; 19, 15) &
(11, 5; 5, 16) &
(5, 3; 17, 21) &
(1, 15; 30, 8) &
(15, 6; 1, 22) &
(3, 1; 9, 14) &
(7, 16; 11, 19) &
(15, 5; 6, 20) \\
(0, 5; 26, 22) &
(0, 14; 16, 9) &
(11, 16; 12, 25) &
(2, 10; 17, 24) &
(7, 15; 25, 17) &
(15, 4; 7, 19) &
(17, 4; 26, 3) &
(2, 15; 11, 18) &
(11, 4; 27, 2) \\
(2, 4; 15, 28) &
(11, 13; 7, 28) &
\end{array}$$ ]{}
$a=9$: [$$\begin{array}{llllllllll}
(17, 2; 9, 0) &
(3, 0; 10, 12) &
(6, 10; 2, 4) &
(15, 6; 34, 3) &
(4, 9; 34, 20) &
(8, 12; 13, 2) &
(3, 6; 33, 23) &
(7, 0; 30, 6) &
(16, 10; 3, 29) \\
(3, 5; 26, 19) &
(8, 1; 26, 33) &
(11, 17; 11, 30) &
(11, 4; 29, 33) &
(16, 7; 19, 28) &
(1, 2; 35, 12) &
(12, 15; 19, 32) &
(16, 3; 24, 16) &
(0, 10; 14, 26) \\
(1, 7; 31, 25) &
(15, 8; 0, 10) &
(9, 6; 26, 30) &
(0, 2; 34, 11) &
(11, 0; 25, 16) &
(5, 12; 18, 5) &
(15, 5; 24, 6) &
(15, 4; 25, 15) &
(12, 1; 9, 23) \\
(13, 1; 30, 21) &
(13, 15; 16, 12) &
(2, 9; 29, 10) &
(4, 3; 30, 2) &
(16, 5; 30, 14) &
(17, 3; 20, 25) &
(4, 6; 35, 16) &
(1, 6; 24, 29) &
(6, 5; 22, 32) \\
(12, 2; 20, 1) &
(8, 16; 12, 5) &
(11, 12; 4, 8) &
(14, 3; 11, 21) &
(6, 11; 31, 0) &
(13, 16; 31, 11) &
(2, 5; 31, 27) &
(4, 13; 0, 18) &
(17, 0; 5, 17) \\
(13, 17; 19, 2) &
(0, 4; 24, 28) &
(5, 11; 3, 12) &
(0, 5; 4, 29) &
(5, 9; 1, 23) &
(0, 16; 23, 18) &
(15, 2; 21, 26) &
(4, 1; 5, 32) &
(9, 17; 12, 24) \\
(11, 14; 1, 15) &
(13, 10; 35, 17) &
(6, 0; 19, 27) &
(9, 12; 33, 28) &
(9, 10; 27, 11) &
(11, 15; 17, 27) &
(10, 7; 5, 24) &
(6, 12; 7, 17) &
(8, 0; 15, 20) \\
(3, 1; 28, 17) &
(1, 17; 16, 27) &
(15, 10; 13, 8) &
(1, 10; 34, 18) &
(17, 10; 6, 10) &
(17, 15; 23, 4) &
(7, 15; 20, 18) &
(17, 12; 3, 31) &
(13, 9; 4, 14) \\
(3, 12; 0, 29) &
(15, 3; 35, 1) &
(14, 12; 14, 34) &
(9, 14; 35, 8) &
(9, 3; 3, 13) &
(13, 8; 7, 29) &
(2, 4; 14, 3) &
(14, 5; 16, 2) &
(11, 9; 32, 6) \\
(8, 5; 35, 25) &
(7, 5; 17, 34) &
(11, 1; 10, 13) &
(1, 15; 14, 11) &
(4, 14; 17, 12) &
(8, 10; 1, 32) &
(13, 11; 5, 28) &
(7, 3; 32, 9) &
(7, 17; 29, 22) \\
(9, 1; 15, 22) &
(8, 3; 8, 27) &
(5, 17; 15, 21) &
(6, 14; 5, 10) &
(16, 9; 25, 0) &
(2, 13; 15, 23) &
(4, 12; 22, 6) &
(2, 16; 13, 17) &
(3, 13; 34, 22) \\
(6, 17; 18, 8) &
(10, 3; 15, 31) &
(7, 14; 0, 33) &
(7, 8; 3, 4) &
(16, 14; 27, 20) &
(6, 2; 25, 28) &
(10, 12; 12, 30) &
(16, 15; 9, 22) &
(10, 2; 16, 33) \\
(16, 11; 2, 7) &
(4, 17; 13, 1) &
(12, 0; 21, 35) &
(14, 13; 3, 6) &
(4, 8; 31, 21) &
(17, 8; 28, 14) &
(1, 14; 4, 19) &
(2, 7; 8, 2) &
(12, 7; 16, 11) \\
(6, 13; 9, 20) &
(11, 3; 14, 18) &
(12, 16; 10, 15) &
(2, 8; 30, 22) &
(10, 14; 25, 9) &
(16, 1; 8, 6) &
(5, 10; 28, 0) &
(5, 1; 20, 7) &
(6, 8; 6, 11) \\
(7, 4; 23, 27) &
(14, 0; 13, 22) &
(4, 10; 19, 7) &
(0, 9; 9, 31) &
(4, 16; 26, 4) &
(8, 14; 23, 24) &
(2, 11; 24, 19) &
(7, 11; 26, 35) &
(9, 15; 2, 5) \\
(14, 2; 32, 18) &
(14, 17; 7, 26) &
(9, 7; 7, 21) &
(13, 7; 10, 1) &
(11, 8; 9, 34) &
(16, 6; 21, 1) &
(13, 0; 32, 8) &
(0, 15; 7, 33) &
(5, 13; 13, 33) \\
\end{array}$$ ]{}
HSAS$(s,v;3,3)$ and HSAS$(s,v;5,3)$ with $v\in\{11,15,19\}$
-----------------------------------------------------------
There exists an HSAS$(s,11;3,3)$ for each $s\in \{11,13,15,17,19\}$.
Let $V=I_{11}$ and $S=I_s$. Let $W=\{8,9,10\}$ and $T=\{s-3,s-2,s-1\}$. The desired HSASs filled with pairs of points from $V$ and indexed by $S$ are presented as follows.
$s=11$: [$$\begin{array}{lllllllllll}
(3, 5; 2, 10) &
(6, 2; 4, 3) &
(4, 8; 1, 2) &
(8, 7; 6, 5) &
(2, 9; 1, 0) &
(6, 3; 0, 8) &
(4, 7; 8, 3) &
(3, 7; 9, 1) &
(5, 10; 3, 7) &
(2, 1; 6, 9) \\
(8, 3; 4, 7) &
(5, 6; 5, 9) &
(0, 6; 6, 10) &
(3, 9; 5, 3) &
(1, 8; 3, 0) &
(4, 10; 6, 0) &
(0, 2; 8, 2) &
(1, 10; 2, 4) &
(4, 2; 5, 10) &
(0, 4; 7, 9) \\
(1, 7; 7, 10) &
(10, 0; 1, 5) &
(6, 9; 7, 2) &
(5, 1; 8, 1) &
(0, 7; 0, 4) &
(5, 9; 4, 6) &
\end{array}$$ ]{}
$s=13$: [$$\begin{array}{lllllllllll}
(10, 5; 2, 3) &
(9, 0; 9, 0) &
(10, 4; 5, 7) &
(2, 4; 4, 12) &
(0, 6; 7, 3) &
(0, 7; 12, 2) &
(7, 6; 9, 5) &
(3, 10; 8, 4) &
(8, 1; 1, 4) &
(2, 1; 2, 11) \\
(5, 1; 9, 7) &
(9, 3; 1, 7) &
(1, 6; 12, 8) &
(5, 0; 4, 10) &
(10, 1; 6, 0) &
(4, 0; 6, 11) &
(7, 3; 0, 11) &
(9, 2; 5, 3) &
(8, 3; 9, 3) &
(8, 6; 6, 2) \\
(3, 5; 12, 6) &
(6, 5; 11, 1) &
(4, 7; 1, 3) &
(2, 5; 8, 0) &
(1, 3; 10, 5) &
(9, 4; 8, 2) &
(4, 6; 0, 10) &
(10, 2; 9, 1) &
(9, 7; 6, 4) &
(8, 0; 5, 8) \\
(2, 7; 7, 10) &
\end{array}$$ ]{}
$s=15$: [$$\begin{array}{lllllllllll}
(7, 4; 8, 7) &
(5, 6; 4, 9) &
(3, 4; 3, 5) &
(4, 5; 0, 14) &
(7, 0; 14, 10) &
(4, 6; 12, 11) &
(0, 3; 11, 1) &
(2, 8; 6, 1) &
(1, 8; 11, 5) &
(9, 7; 6, 2) \\
(0, 9; 8, 9) &
(8, 7; 0, 9) &
(8, 3; 7, 10) &
(2, 7; 5, 12) &
(9, 2; 7, 11) &
(3, 9; 4, 0) &
(0, 6; 6, 0) &
(4, 10; 9, 1) &
(10, 6; 7, 5) &
(3, 6; 14, 8) \\
(5, 8; 3, 8) &
(1, 3; 12, 6) &
(9, 6; 10, 3) &
(1, 10; 8, 4) &
(10, 2; 2, 0) &
(5, 3; 13, 2) &
(4, 8; 4, 2) &
(7, 10; 3, 11) &
(5, 9; 1, 5) &
(2, 1; 9, 14) \\
(10, 5; 6, 10) &
(1, 4; 13, 10) &
(6, 7; 13, 1) &
(0, 1; 3, 2) &
(0, 2; 4, 13) &
(0, 5; 7, 12) &
\end{array}$$ ]{}
$s=17$: [$$\begin{array}{lllllllllll}
(2, 10; 1, 7) &
(2, 7; 16, 9) &
(2, 6; 14, 11) &
(5, 3; 3, 16) &
(2, 5; 10, 0) &
(1, 3; 4, 5) &
(0, 5; 12, 2) &
(10, 1; 13, 2) &
(4, 2; 4, 6) &
(8, 6; 10, 4) \\
(10, 0; 8, 5) &
(5, 4; 14, 5) &
(7, 6; 12, 0) &
(8, 3; 2, 11) &
(8, 2; 5, 3) &
(10, 7; 11, 10) &
(4, 0; 11, 0) &
(9, 1; 11, 12) &
(10, 6; 3, 6) &
(4, 6; 16, 2) \\
(7, 3; 15, 6) &
(5, 10; 4, 9) &
(4, 8; 12, 7) &
(7, 9; 5, 2) &
(4, 9; 8, 10) &
(6, 1; 8, 1) &
(8, 1; 0, 6) &
(6, 9; 13, 9) &
(9, 5; 1, 6) &
(0, 8; 1, 9) \\
(0, 2; 15, 13) &
(1, 0; 7, 16) &
(4, 1; 9, 15) &
(8, 7; 13, 8) &
(4, 3; 13, 1) &
(3, 9; 0, 7) &
(0, 3; 10, 14) &
(5, 6; 15, 7) &
(3, 2; 12, 8) &
(7, 1; 3, 14) \\
(0, 9; 3, 4) &
\end{array}$$ ]{}
$s=19$: [$$\begin{array}{lllllllllll}
(6, 5; 13, 2) &
(2, 9; 6, 1) &
(6, 3; 1, 16) &
(3, 0; 7, 5) &
(5, 9; 12, 3) &
(4, 6; 12, 18) &
(0, 9; 0, 15) &
(6, 7; 17, 4) &
(6, 9; 7, 10) &
(7, 2; 12, 14) \\
(1, 0; 17, 1) &
(8, 0; 13, 12) &
(4, 9; 11, 5) &
(0, 6; 3, 14) &
(10, 6; 0, 6) &
(4, 2; 0, 13) &
(4, 5; 16, 15) &
(3, 2; 18, 8) &
(5, 8; 0, 11) &
(9, 1; 8, 13) \\
(1, 4; 4, 6) &
(4, 0; 10, 2) &
(5, 2; 17, 5) &
(0, 7; 11, 18) &
(8, 3; 10, 6) &
(1, 5; 18, 10) &
(8, 1; 14, 2) &
(3, 4; 17, 14) &
(4, 10; 9, 7) &
(9, 7; 9, 2) \\
(1, 10; 12, 5) &
(2, 8; 15, 4) &
(6, 1; 15, 11) &
(3, 10; 2, 15) &
(5, 3; 9, 4) &
(2, 10; 11, 10) &
(2, 1; 3, 7) &
(5, 10; 1, 14) &
(1, 7; 16, 0) &
(10, 7; 13, 3) \\
(0, 2; 16, 9) &
(8, 6; 9, 5) &
(10, 0; 4, 8) &
(8, 4; 3, 8) &
(7, 8; 1, 7) &
(5, 7; 6, 8) &
\end{array}$$ ]{}
There exists an HSAS$(s,15;3,3)$ for each $s\in \{15,17,\ldots,27\}$.
Let $V=I_{15}$ and $S=I_s$. Let $W=\{12,13,14\}$ and $T=\{s-3,s-2,s-1\}$. The desired HSASs filled with pairs of points from $V$ and indexed by $S$ are presented as follows.
$s=15$: [$$\begin{array}{lllllllllll}
(12, 9; 1, 0) &
(7, 8; 5, 13) &
(10, 12; 6, 9) &
(6, 14; 3, 9) &
(5, 1; 10, 1) &
(4, 13; 6, 10) &
(11, 2; 14, 1) &
(11, 9; 13, 6) &
(9, 6; 11, 5) &
(0, 5; 0, 9) \\
(0, 8; 8, 14) &
(2, 0; 13, 3) &
(10, 1; 0, 13) &
(11, 4; 11, 2) &
(2, 1; 6, 12) &
(10, 4; 14, 3) &
(12, 2; 7, 2) &
(3, 5; 12, 11) &
(4, 5; 8, 13) &
(7, 5; 7, 4) \\
(8, 6; 0, 6) &
(2, 14; 0, 10) &
(1, 8; 3, 2) &
(0, 10; 11, 7) &
(11, 12; 4, 8) &
(12, 8; 10, 11) &
(7, 1; 9, 14) &
(2, 3; 9, 8) &
(10, 7; 2, 12) &
(12, 5; 5, 3) \\
(14, 1; 4, 11) &
(11, 6; 7, 12) &
(9, 13; 3, 4) &
(2, 4; 4, 5) &
(8, 10; 1, 4) &
(13, 7; 11, 0) &
(0, 14; 6, 5) &
(4, 8; 12, 9) &
(11, 13; 5, 9) &
(9, 3; 14, 7) \\
(14, 9; 8, 2) &
(6, 3; 13, 4) &
(0, 13; 2, 1) &
(13, 1; 8, 7) &
(3, 11; 0, 3) &
(0, 9; 10, 12) &
(3, 7; 1, 6) &
(6, 5; 2, 14) &
(7, 6; 8, 10) &
(3, 10; 5, 10) \\
(4, 14; 1, 7) &
\end{array}$$ ]{}
$s=17$: [$$\begin{array}{lllllllllll}
(12, 10; 1, 0) &
(3, 5; 14, 1) &
(11, 13; 10, 0) &
(7, 0; 1, 5) &
(0, 6; 7, 6) &
(13, 8; 1, 4) &
(9, 5; 9, 15) &
(2, 10; 5, 14) &
(5, 4; 12, 16) &
(8, 14; 10, 13) \\
(3, 2; 2, 12) &
(11, 0; 8, 16) &
(10, 7; 11, 16) &
(1, 11; 15, 7) &
(4, 3; 11, 15) &
(5, 1; 0, 3) &
(4, 13; 5, 9) &
(2, 8; 16, 7) &
(5, 12; 5, 10) &
(9, 0; 4, 12) \\
(11, 9; 1, 11) &
(2, 6; 0, 15) &
(3, 13; 7, 8) &
(5, 7; 4, 6) &
(9, 2; 10, 8) &
(11, 8; 14, 6) &
(10, 5; 8, 2) &
(10, 8; 3, 12) &
(1, 3; 13, 9) &
(6, 1; 2, 16) \\
(7, 14; 12, 8) &
(2, 11; 4, 9) &
(8, 1; 8, 11) &
(3, 8; 0, 5) &
(12, 11; 12, 13) &
(13, 1; 12, 6) &
(12, 4; 2, 6) &
(13, 9; 2, 13) &
(10, 6; 9, 10) &
(2, 14; 1, 6) \\
(5, 14; 11, 7) &
(12, 6; 4, 11) &
(6, 7; 13, 14) &
(13, 0; 3, 11) &
(9, 4; 0, 14) &
(9, 3; 16, 6) &
(7, 8; 15, 2) &
(1, 14; 5, 4) &
(7, 3; 10, 3) &
(12, 7; 7, 9) \\
(1, 0; 10, 14) &
(10, 4; 4, 7) &
(4, 6; 8, 1) &
(6, 9; 5, 3) &
(4, 2; 3, 13) &
(0, 14; 9, 0) &
(0, 10; 13, 15) &
(11, 14; 2, 3) &
\end{array}$$ ]{}
$s=19$: [$$\begin{array}{lllllllllll}
(7, 12; 7, 6) &
(2, 12; 8, 2) &
(0, 9; 4, 2) &
(10, 2; 12, 15) &
(3, 5; 10, 6) &
(6, 13; 10, 4) &
(5, 12; 1, 9) &
(7, 13; 8, 12) &
(7, 14; 15, 4) \\
(6, 0; 16, 0) &
(10, 12; 14, 4) &
(11, 7; 0, 2) &
(5, 8; 16, 8) &
(13, 10; 13, 2) &
(7, 0; 9, 14) &
(14, 9; 0, 12) &
(4, 13; 0, 1) &
(10, 3; 1, 5) \\
(8, 14; 10, 14) &
(9, 10; 8, 9) &
(11, 6; 18, 7) &
(3, 4; 3, 18) &
(3, 12; 13, 11) &
(11, 3; 9, 12) &
(3, 13; 14, 7) &
(4, 8; 2, 11) &
(1, 0; 11, 6) \\
(0, 11; 15, 1) &
(10, 6; 17, 11) &
(4, 1; 17, 14) &
(5, 1; 7, 4) &
(6, 5; 2, 12) &
(2, 8; 4, 6) &
(10, 7; 18, 10) &
(11, 9; 17, 10) &
(4, 12; 10, 12) \\
(13, 5; 5, 11) &
(8, 7; 13, 1) &
(1, 3; 16, 15) &
(9, 6; 14, 15) &
(0, 10; 3, 7) &
(14, 11; 3, 13) &
(8, 6; 5, 9) &
(1, 6; 8, 3) &
(9, 7; 11, 16) \\
(14, 6; 1, 6) &
(9, 1; 18, 13) &
(8, 3; 0, 17) &
(0, 14; 8, 5) &
(4, 2; 16, 9) &
(5, 4; 13, 15) &
(1, 2; 10, 1) &
(5, 7; 17, 3) &
(12, 1; 5, 0) \\
(8, 12; 15, 3) &
(0, 8; 18, 12) &
(1, 14; 9, 2) &
(4, 11; 8, 4) &
(10, 11; 16, 6) &
(9, 13; 3, 6) &
(5, 2; 0, 18) &
(2, 0; 17, 13) &
(4, 9; 5, 7) \\
(2, 11; 14, 5) &
(2, 14; 7, 11) &
\end{array}$$ ]{}
$s=21$: [$$\begin{array}{lllllllllll}
(8, 14; 13, 6) &
(12, 6; 15, 1) &
(0, 7; 17, 20) &
(6, 11; 8, 13) &
(7, 11; 4, 18) &
(7, 1; 14, 10) &
(0, 13; 8, 0) &
(8, 7; 19, 2) &
(5, 2; 18, 15) \\
(6, 8; 18, 10) &
(8, 0; 15, 4) &
(12, 5; 12, 6) &
(13, 10; 10, 16) &
(4, 1; 4, 20) &
(3, 11; 5, 15) &
(6, 3; 9, 19) &
(1, 13; 15, 6) &
(7, 2; 16, 11) \\
(4, 9; 8, 17) &
(9, 8; 9, 16) &
(4, 11; 19, 3) &
(3, 14; 8, 12) &
(6, 2; 6, 4) &
(3, 10; 3, 14) &
(5, 8; 8, 3) &
(4, 6; 2, 5) &
(4, 5; 1, 0) \\
(8, 11; 12, 17) &
(11, 12; 14, 16) &
(10, 4; 6, 18) &
(0, 4; 12, 13) &
(0, 10; 1, 19) &
(1, 6; 16, 12) &
(13, 11; 1, 7) &
(9, 5; 4, 5) &
(9, 11; 11, 6) \\
(2, 14; 17, 3) &
(1, 10; 8, 7) &
(3, 13; 17, 4) &
(12, 0; 5, 11) &
(11, 10; 9, 20) &
(7, 14; 7, 5) &
(2, 11; 0, 10) &
(8, 13; 14, 5) &
(4, 2; 9, 14) \\
(0, 3; 2, 6) &
(4, 14; 16, 15) &
(9, 6; 20, 7) &
(2, 10; 12, 5) &
(0, 1; 9, 18) &
(9, 2; 19, 13) &
(5, 14; 10, 9) &
(7, 12; 3, 0) &
(10, 14; 11, 4) \\
(14, 1; 0, 2) &
(13, 5; 2, 13) &
(2, 8; 20, 1) &
(0, 5; 14, 7) &
(5, 1; 19, 11) &
(12, 2; 2, 8) &
(3, 9; 0, 18) &
(14, 9; 14, 1) &
(6, 13; 3, 11) \\
(8, 3; 7, 11) &
(3, 5; 20, 16) &
(9, 10; 2, 15) &
(7, 13; 9, 12) &
(9, 0; 10, 3) &
(6, 10; 17, 0) &
(12, 4; 7, 10) &
(1, 12; 13, 17) &
(3, 7; 1, 13) \\
\end{array}$$ ]{}
$s=23$: [$$\begin{array}{lllllllllll}
(9, 4; 18, 4) &
(13, 6; 8, 12) &
(4, 1; 15, 12) &
(7, 6; 15, 13) &
(10, 7; 17, 11) &
(10, 2; 15, 20) &
(14, 9; 19, 0) &
(0, 2; 21, 17) &
(13, 9; 2, 9) \\
(9, 1; 6, 8) &
(11, 12; 5, 8) &
(7, 13; 0, 14) &
(12, 7; 6, 12) &
(8, 0; 8, 20) &
(6, 3; 6, 21) &
(14, 0; 5, 16) &
(7, 5; 4, 1) &
(1, 7; 20, 7) \\
(3, 0; 2, 19) &
(8, 14; 9, 17) &
(14, 5; 11, 6) &
(8, 5; 2, 18) &
(10, 4; 5, 22) &
(13, 11; 13, 16) &
(14, 2; 12, 18) &
(1, 12; 0, 3) &
(3, 8; 13, 7) \\
(3, 14; 4, 8) &
(7, 9; 3, 21) &
(12, 9; 17, 13) &
(4, 3; 3, 20) &
(4, 11; 21, 19) &
(2, 1; 14, 13) &
(8, 13; 11, 3) &
(5, 13; 5, 17) &
(6, 5; 20, 19) \\
(6, 1; 16, 17) &
(0, 13; 18, 15) &
(5, 3; 10, 14) &
(2, 7; 19, 16) &
(7, 8; 10, 5) &
(6, 8; 0, 1) &
(10, 1; 19, 18) &
(10, 11; 14, 2) &
(5, 12; 16, 15) \\
(4, 8; 16, 6) &
(11, 5; 12, 9) &
(11, 1; 1, 22) &
(0, 9; 12, 14) &
(14, 4; 14, 1) &
(9, 6; 11, 5) &
(0, 10; 6, 1) &
(12, 6; 18, 10) &
(12, 4; 7, 2) \\
(6, 2; 3, 2) &
(1, 14; 10, 2) &
(3, 10; 12, 16) &
(4, 7; 8, 9) &
(2, 9; 7, 1) &
(9, 11; 20, 10) &
(5, 2; 8, 22) &
(6, 0; 9, 22) &
(9, 8; 22, 15) \\
(4, 2; 11, 0) &
(1, 3; 5, 9) &
(5, 0; 3, 7) &
(6, 10; 7, 4) &
(0, 4; 10, 13) &
(3, 11; 17, 0) &
(11, 14; 15, 7) &
(10, 5; 0, 21) &
(8, 12; 19, 14) \\
(3, 7; 18, 22) &
(12, 3; 11, 1) &
(1, 8; 21, 4) &
(10, 14; 3, 13) &
(11, 0; 4, 11) &
(2, 13; 6, 10) &
(2, 12; 4, 9) &
\end{array}$$ ]{}
$s=25$: [$$\begin{array}{lllllllllll}
(3, 7; 3, 17) &
(9, 8; 11, 1) &
(5, 10; 8, 1) &
(9, 1; 16, 22) &
(8, 11; 21, 24) &
(4, 3; 5, 24) &
(4, 5; 17, 23) &
(8, 14; 5, 8) &
(12, 0; 1, 20) \\
(14, 10; 6, 10) &
(0, 13; 13, 4) &
(14, 0; 21, 19) &
(11, 4; 11, 10) &
(4, 10; 19, 20) &
(1, 7; 1, 24) &
(8, 10; 2, 7) &
(12, 10; 13, 0) &
(13, 4; 2, 8) \\
(0, 8; 22, 15) &
(5, 8; 16, 14) &
(11, 13; 9, 1) &
(12, 5; 6, 12) &
(2, 14; 3, 13) &
(2, 13; 16, 0) &
(6, 13; 10, 7) &
(6, 11; 13, 23) &
(7, 11; 7, 22) \\
(8, 12; 19, 3) &
(1, 8; 9, 13) &
(3, 10; 22, 9) &
(2, 4; 22, 1) &
(9, 0; 17, 8) &
(7, 10; 14, 15) &
(9, 3; 7, 18) &
(6, 0; 24, 11) &
(14, 6; 2, 18) \\
(7, 0; 2, 10) &
(2, 8; 20, 4) &
(1, 13; 15, 11) &
(7, 14; 0, 4) &
(1, 14; 17, 20) &
(12, 11; 5, 17) &
(5, 3; 15, 4) &
(6, 5; 22, 21) &
(11, 14; 14, 12) \\
(11, 0; 16, 18) &
(12, 7; 11, 9) &
(9, 2; 12, 21) &
(6, 10; 16, 12) &
(9, 4; 14, 0) &
(14, 4; 15, 9) &
(10, 13; 17, 21) &
(3, 14; 1, 16) &
(7, 13; 12, 19) \\
(3, 13; 14, 20) &
(2, 6; 15, 8) &
(4, 0; 12, 3) &
(2, 5; 5, 10) &
(6, 8; 17, 0) &
(1, 4; 7, 21) &
(2, 1; 14, 2) &
(6, 7; 5, 20) &
(7, 4; 13, 16) \\
(2, 10; 18, 24) &
(4, 8; 18, 6) &
(0, 2; 9, 7) &
(7, 2; 6, 23) &
(1, 11; 3, 4) &
(11, 3; 6, 8) &
(12, 6; 14, 4) &
(9, 12; 10, 15) &
(1, 6; 19, 6) \\
(9, 13; 6, 5) &
(10, 9; 23, 4) &
(5, 13; 18, 3) &
(1, 0; 23, 5) &
(8, 3; 12, 23) &
(5, 9; 24, 13) &
(9, 6; 9, 3) &
(12, 1; 18, 8) &
(5, 14; 11, 7) \\
(2, 3; 19, 11) &
(3, 12; 2, 21) &
(11, 9; 2, 19) &
(1, 3; 0, 10) &
(5, 11; 0, 20) &
\end{array}$$ ]{}
$s=27$: [$$\begin{array}{lllllllllll}
(3, 12; 0, 23) &
(1, 7; 0, 22) &
(6, 13; 6, 17) &
(7, 2; 17, 23) &
(11, 8; 0, 11) &
(1, 11; 24, 10) &
(14, 2; 3, 11) &
(3, 14; 12, 15) &
(6, 8; 12, 14) \\
(14, 8; 9, 20) &
(8, 5; 18, 26) &
(13, 1; 2, 11) &
(6, 7; 24, 1) &
(1, 14; 13, 18) &
(6, 0; 8, 22) &
(7, 9; 16, 5) &
(12, 5; 11, 12) &
(5, 2; 21, 15) \\
(5, 3; 2, 8) &
(8, 1; 16, 3) &
(13, 10; 23, 5) &
(0, 8; 23, 4) &
(3, 1; 7, 26) &
(5, 1; 9, 1) &
(5, 7; 6, 13) &
(14, 11; 16, 1) &
(2, 13; 0, 10) \\
(3, 7; 4, 11) &
(10, 5; 10, 16) &
(7, 14; 8, 14) &
(11, 6; 15, 23) &
(12, 6; 18, 16) &
(7, 10; 26, 15) &
(6, 9; 26, 11) &
(8, 13; 21, 1) &
(1, 4; 5, 14) \\
(5, 13; 4, 22) &
(14, 9; 10, 2) &
(0, 9; 15, 24) &
(9, 12; 6, 14) &
(10, 1; 12, 19) &
(12, 4; 13, 10) &
(10, 9; 17, 1) &
(3, 2; 14, 1) &
(5, 6; 25, 3) \\
(9, 1; 23, 8) &
(9, 8; 22, 19) &
(13, 3; 20, 19) &
(11, 4; 18, 2) &
(4, 14; 22, 23) &
(8, 7; 7, 25) &
(8, 10; 2, 13) &
(0, 3; 13, 25) &
(8, 12; 15, 8) \\
(11, 7; 19, 21) &
(1, 2; 20, 25) &
(10, 3; 21, 24) &
(12, 11; 3, 20) &
(9, 4; 21, 4) &
(0, 5; 0, 14) &
(4, 5; 24, 17) &
(4, 13; 7, 15) &
(11, 0; 5, 7) \\
(0, 1; 21, 6) &
(0, 2; 2, 16) &
(2, 12; 19, 5) &
(10, 0; 9, 3) &
(8, 2; 6, 24) &
(1, 12; 4, 17) &
(10, 2; 18, 22) &
(13, 0; 18, 12) &
(0, 7; 10, 20) \\
(10, 6; 4, 20) &
(6, 2; 7, 13) &
(4, 3; 16, 6) &
(4, 0; 26, 1) &
(3, 8; 5, 10) &
(9, 5; 20, 7) &
(7, 12; 9, 2) &
(10, 11; 8, 6) &
(6, 14; 5, 21) \\
(4, 6; 19, 0) &
(13, 11; 9, 14) &
(9, 3; 18, 9) &
(10, 4; 11, 25) &
(14, 0; 17, 19) &
(3, 11; 22, 17) &
(2, 11; 4, 26) &
(9, 13; 3, 13) &
(9, 11; 25, 12) \\
(4, 2; 8, 9) &
(4, 7; 3, 12) &
(10, 14; 0, 7) &
\end{array}$$ ]{}
There exists an HSAS$(s,19;3,3)$ for each $s\in \{19,21,\ldots,35\}$.
Let $V=I_{19}$ and $S=I_s$. Let $W=\{16,17,18\}$ and $T=\{s-3,s-2,s-1\}$. The desired HSASs filled with pairs of points from $V$ and indexed by $S$ are presented as follows.
$s=19$: [$$\begin{array}{lllllllllll}
(2, 13; 3, 17) &
(3, 7; 0, 16) &
(14, 8; 10, 14) &
(13, 16; 11, 9) &
(9, 1; 17, 7) &
(14, 7; 12, 17) &
(0, 15; 0, 2) &
(15, 6; 4, 1) &
(9, 6; 14, 18) \\
(13, 0; 18, 5) &
(4, 12; 5, 2) &
(14, 16; 15, 3) &
(8, 3; 9, 13) &
(5, 1; 2, 8) &
(16, 10; 13, 10) &
(15, 12; 11, 3) &
(9, 11; 16, 3) &
(15, 11; 8, 17) \\
(17, 7; 2, 10) &
(17, 13; 8, 6) &
(12, 5; 7, 18) &
(8, 7; 3, 18) &
(7, 10; 11, 7) &
(13, 14; 0, 1) &
(8, 6; 5, 16) &
(4, 1; 3, 4) &
(11, 12; 0, 9) \\
(5, 13; 10, 16) &
(18, 1; 12, 15) &
(18, 5; 3, 0) &
(0, 2; 4, 14) &
(2, 6; 2, 13) &
(6, 18; 10, 11) &
(11, 1; 5, 10) &
(3, 5; 14, 11) &
(14, 18; 13, 6) \\
(13, 10; 14, 15) &
(6, 4; 17, 0) &
(14, 2; 9, 8) &
(10, 8; 0, 8) &
(4, 11; 7, 6) &
(17, 4; 1, 9) &
(12, 2; 1, 10) &
(14, 12; 16, 4) &
(0, 4; 13, 8) \\
(10, 2; 6, 16) &
(10, 3; 3, 5) &
(7, 5; 9, 4) &
(9, 8; 1, 11) &
(18, 8; 7, 4) &
(14, 3; 18, 2) &
(7, 1; 13, 1) &
(0, 9; 6, 9) &
(1, 0; 16, 11) \\
(17, 12; 14, 13) &
(15, 1; 9, 18) &
(5, 9; 13, 12) &
(10, 18; 2, 9) &
(14, 17; 5, 11) &
(3, 12; 17, 6) &
(8, 15; 12, 6) &
(15, 4; 14, 16) &
(5, 10; 17, 1) \\
(9, 18; 5, 8) &
(13, 15; 13, 7) &
(3, 0; 10, 12) &
(17, 3; 15, 4) &
(12, 16; 8, 12) &
(4, 2; 12, 18) &
(1, 16; 6, 0) &
(17, 6; 12, 3) &
(11, 18; 1, 14) \\
(16, 0; 7, 1) &
(9, 4; 15, 10) &
(10, 11; 18, 4) &
(8, 0; 17, 15) &
(5, 15; 5, 15) &
(16, 7; 5, 14) &
(2, 11; 15, 11) &
(16, 9; 2, 4) &
(17, 2; 0, 7) \\
(13, 11; 2, 12) &
(7, 6; 15, 6) &
(3, 6; 7, 8) &
\end{array}$$ ]{}
$s=21$: [$$\begin{array}{lllllllllll}
(3, 16; 12, 5) &
(16, 8; 17, 0) &
(7, 17; 9, 0) &
(2, 3; 15, 17) &
(11, 3; 4, 19) &
(13, 0; 8, 20) &
(13, 6; 17, 12) &
(12, 8; 6, 19) &
(14, 12; 20, 17) \\
(13, 11; 9, 5) &
(16, 14; 8, 9) &
(8, 7; 16, 4) &
(18, 2; 0, 16) &
(15, 5; 16, 19) &
(9, 4; 1, 3) &
(15, 16; 7, 4) &
(3, 17; 7, 8) &
(10, 5; 18, 3) \\
(2, 1; 7, 20) &
(1, 15; 3, 15) &
(4, 16; 16, 14) &
(6, 2; 3, 6) &
(10, 14; 0, 15) &
(5, 11; 11, 20) &
(18, 13; 15, 4) &
(4, 10; 13, 20) &
(16, 2; 1, 2) \\
(6, 10; 16, 7) &
(4, 0; 7, 18) &
(15, 11; 2, 6) &
(6, 3; 14, 10) &
(16, 7; 3, 10) &
(15, 6; 20, 1) &
(11, 4; 10, 15) &
(14, 8; 7, 3) &
(11, 17; 16, 1) \\
(1, 16; 11, 13) &
(5, 2; 10, 8) &
(7, 4; 17, 11) &
(2, 4; 19, 12) &
(17, 9; 6, 12) &
(14, 6; 5, 19) &
(3, 8; 20, 2) &
(18, 9; 14, 2) &
(10, 11; 8, 17) \\
(15, 2; 11, 9) &
(8, 18; 11, 8) &
(7, 14; 14, 6) &
(3, 0; 16, 3) &
(3, 9; 11, 0) &
(4, 15; 5, 8) &
(13, 17; 3, 11) &
(12, 4; 2, 9) &
(18, 10; 5, 10) \\
(12, 6; 4, 11) &
(17, 8; 10, 13) &
(2, 12; 13, 18) &
(3, 13; 6, 13) &
(7, 1; 19, 8) &
(5, 18; 17, 6) &
(1, 0; 0, 14) &
(12, 5; 5, 7) &
(13, 12; 1, 0) \\
(12, 9; 8, 15) &
(18, 1; 9, 12) &
(7, 9; 20, 5) &
(11, 12; 3, 14) &
(11, 8; 18, 12) &
(0, 17; 17, 2) &
(12, 1; 16, 10) &
(1, 10; 1, 6) &
(6, 1; 2, 18) \\
(0, 16; 6, 15) &
(5, 17; 14, 15) &
(0, 15; 12, 13) &
(7, 10; 12, 2) &
(9, 13; 7, 19) &
(0, 8; 5, 1) &
(0, 10; 9, 19) &
(5, 4; 4, 0) &
(1, 17; 4, 5) \\
(2, 10; 4, 14) &
(18, 7; 7, 13) &
(9, 15; 10, 17) &
(6, 11; 13, 0) &
(13, 15; 18, 14) &
(6, 8; 15, 9) &
(3, 7; 1, 18) &
(9, 14; 4, 18) &
(14, 0; 10, 11) \\
(14, 13; 2, 16) &
(5, 9; 9, 13) &
(5, 14; 1, 12) &
\end{array}$$ ]{}
$s=23$: [$$\begin{array}{lllllllllll}
(6, 7; 15, 7) &
(2, 5; 12, 20) &
(13, 1; 14, 4) &
(2, 18; 14, 15) &
(14, 9; 12, 21) &
(8, 10; 15, 5) &
(1, 17; 7, 12) &
(4, 13; 11, 10) &
(4, 10; 9, 19) \\
(16, 9; 4, 17) &
(7, 9; 11, 2) &
(3, 4; 4, 20) &
(6, 18; 8, 13) &
(0, 16; 12, 9) &
(17, 2; 6, 16) &
(0, 3; 19, 2) &
(15, 0; 5, 16) &
(5, 15; 0, 3) \\
(5, 17; 14, 17) &
(5, 3; 21, 16) &
(1, 12; 13, 22) &
(13, 14; 15, 22) &
(10, 15; 10, 8) &
(11, 15; 21, 7) &
(6, 1; 10, 0) &
(14, 4; 16, 18) &
(3, 2; 8, 17) \\
(15, 1; 6, 1) &
(12, 0; 11, 18) &
(12, 17; 9, 10) &
(8, 12; 21, 2) &
(9, 17; 5, 18) &
(14, 10; 7, 20) &
(10, 11; 1, 17) &
(18, 13; 9, 2) &
(16, 12; 0, 15) \\
(9, 8; 20, 16) &
(6, 8; 9, 11) &
(3, 14; 9, 5) &
(10, 0; 22, 14) &
(3, 13; 12, 18) &
(1, 4; 21, 5) &
(16, 5; 2, 1) &
(0, 9; 13, 15) &
(5, 11; 9, 15) \\
(2, 0; 4, 21) &
(14, 11; 13, 4) &
(7, 4; 22, 3) &
(7, 1; 8, 19) &
(15, 17; 19, 13) &
(2, 15; 11, 22) &
(2, 6; 2, 5) &
(8, 13; 19, 1) &
(15, 4; 15, 2) \\
(14, 16; 6, 3) &
(0, 1; 17, 20) &
(6, 13; 17, 21) &
(6, 9; 19, 3) &
(11, 16; 10, 14) &
(10, 3; 13, 11) &
(12, 9; 6, 7) &
(15, 7; 17, 9) &
(16, 2; 18, 19) \\
(7, 13; 6, 20) &
(10, 17; 0, 2) &
(18, 0; 10, 3) &
(13, 11; 3, 16) &
(17, 7; 1, 4) &
(5, 6; 6, 4) &
(15, 6; 20, 14) &
(14, 7; 14, 0) &
(9, 5; 10, 22) \\
(2, 9; 0, 9) &
(16, 1; 11, 16) &
(13, 16; 5, 8) &
(12, 10; 3, 4) &
(0, 4; 8, 6) &
(13, 0; 7, 0) &
(7, 8; 10, 12) &
(14, 2; 1, 10) &
(5, 8; 18, 8) \\
(4, 16; 13, 7) &
(5, 7; 5, 13) &
(17, 14; 8, 11) &
(12, 14; 19, 17) &
(7, 10; 18, 21) &
(8, 3; 22, 7) &
(15, 18; 4, 18) &
(17, 3; 15, 3) &
(11, 6; 22, 12) \\
(5, 18; 19, 7) &
(9, 3; 1, 14) &
(1, 11; 18, 2) &
(3, 11; 6, 0) &
(4, 12; 12, 14) &
(11, 18; 11, 5) &
(2, 8; 3, 13) &
(12, 11; 8, 20) &
(12, 6; 16, 1) \\
(18, 10; 16, 12) &
(18, 8; 6, 17) &
(4, 18; 0, 1) &
\end{array}$$ ]{}
$s=25$: [$$\begin{array}{lllllllllll}
(8, 16; 17, 18) &
(0, 3; 0, 22) &
(10, 11; 4, 14) &
(12, 13; 23, 20) &
(5, 6; 2, 23) &
(9, 10; 13, 16) &
(1, 17; 5, 18) &
(15, 4; 17, 12) &
(5, 0; 15, 20) \\
(7, 8; 12, 0) &
(16, 3; 15, 13) &
(18, 1; 13, 12) &
(18, 13; 15, 17) &
(9, 0; 12, 1) &
(6, 7; 4, 16) &
(14, 11; 23, 13) &
(6, 16; 8, 5) &
(17, 11; 8, 21) \\
(12, 2; 13, 11) &
(5, 18; 10, 18) &
(12, 14; 4, 17) &
(15, 6; 22, 13) &
(14, 10; 15, 1) &
(11, 1; 17, 0) &
(10, 0; 5, 9) &
(7, 4; 5, 20) &
(2, 10; 2, 22) \\
(11, 6; 20, 7) &
(17, 15; 0, 15) &
(1, 0; 3, 21) &
(5, 9; 4, 3) &
(12, 1; 2, 19) &
(15, 13; 3, 18) &
(4, 2; 15, 21) &
(8, 17; 10, 16) &
(7, 14; 21, 22) \\
(14, 9; 5, 10) &
(14, 0; 18, 8) &
(7, 1; 23, 14) &
(11, 12; 15, 5) &
(13, 4; 8, 13) &
(3, 7; 8, 2) &
(3, 5; 5, 17) &
(0, 13; 4, 24) &
(3, 13; 16, 1) \\
(5, 2; 8, 0) &
(11, 3; 9, 19) &
(17, 13; 2, 6) &
(15, 1; 20, 1) &
(6, 2; 24, 18) &
(18, 12; 6, 0) &
(9, 16; 20, 0) &
(16, 15; 21, 16) &
(17, 2; 7, 17) \\
(15, 9; 19, 23) &
(16, 1; 4, 7) &
(18, 3; 3, 11) &
(8, 0; 19, 13) &
(17, 5; 1, 13) &
(12, 10; 24, 8) &
(15, 18; 5, 14) &
(9, 8; 2, 24) &
(5, 4; 9, 14) \\
(18, 8; 21, 20) &
(16, 7; 6, 19) &
(14, 8; 7, 9) &
(2, 14; 3, 20) &
(7, 11; 24, 1) &
(11, 0; 10, 2) &
(9, 3; 21, 6) &
(2, 11; 6, 12) &
(8, 15; 8, 4) \\
(16, 13; 9, 12) &
(11, 5; 16, 11) &
(14, 4; 19, 24) &
(4, 18; 7, 1) &
(12, 6; 21, 12) &
(15, 12; 10, 9) &
(13, 5; 19, 22) &
(6, 4; 11, 0) &
(10, 8; 23, 11) \\
(0, 6; 6, 17) &
(1, 4; 10, 22) &
(2, 3; 23, 10) &
(14, 15; 11, 6) &
(0, 4; 16, 23) &
(17, 0; 11, 14) &
(3, 14; 12, 14) &
(13, 7; 10, 11) &
(18, 14; 2, 16) \\
(12, 8; 3, 22) &
(10, 4; 6, 18) &
(9, 13; 14, 7) &
(9, 11; 18, 22) &
(6, 10; 10, 19) &
(10, 13; 21, 0) &
(17, 3; 4, 20) &
(9, 1; 11, 15) &
(17, 6; 9, 3) \\
(7, 10; 3, 17) &
(8, 6; 14, 15) &
(9, 18; 9, 8) &
(16, 4; 3, 2) &
(15, 3; 7, 24) &
(12, 7; 18, 7) &
(1, 2; 9, 16) &
(5, 10; 7, 12) &
(18, 2; 4, 19) \\
(5, 1; 24, 6) &
(8, 2; 1, 5) &
(12, 16; 1, 14) &
\end{array}$$ ]{}
$s=27$: [$$\begin{array}{lllllllllll}
(8, 5; 17, 23) &
(16, 9; 11, 7) &
(6, 15; 0, 10) &
(1, 3; 11, 25) &
(1, 10; 1, 22) &
(16, 14; 13, 22) &
(2, 10; 5, 0) &
(4, 11; 18, 2) &
(4, 10; 9, 23) \\
(8, 6; 9, 2) &
(11, 0; 24, 15) &
(0, 8; 26, 18) &
(5, 16; 20, 12) &
(3, 13; 26, 12) &
(5, 11; 8, 7) &
(10, 17; 14, 19) &
(7, 15; 16, 18) &
(3, 8; 20, 21) \\
(15, 0; 12, 8) &
(6, 9; 12, 13) &
(13, 12; 4, 16) &
(8, 10; 10, 24) &
(2, 16; 17, 2) &
(18, 3; 0, 23) &
(10, 3; 6, 7) &
(15, 2; 6, 21) &
(14, 17; 10, 2) \\
(4, 16; 16, 10) &
(17, 7; 1, 12) &
(17, 13; 17, 18) &
(2, 6; 22, 26) &
(9, 13; 21, 1) &
(6, 5; 6, 1) &
(16, 10; 18, 15) &
(12, 0; 9, 1) &
(17, 2; 4, 23) \\
(18, 4; 14, 3) &
(7, 18; 13, 5) &
(11, 15; 25, 19) &
(18, 10; 4, 2) &
(5, 14; 16, 0) &
(13, 6; 5, 3) &
(17, 0; 21, 16) &
(3, 16; 14, 4) &
(17, 4; 8, 6) \\
(1, 8; 12, 19) &
(5, 7; 10, 11) &
(16, 7; 0, 9) &
(15, 3; 5, 9) &
(5, 9; 14, 5) &
(18, 0; 17, 19) &
(15, 8; 1, 14) &
(8, 13; 6, 13) &
(4, 9; 22, 25) \\
(15, 12; 2, 22) &
(5, 13; 19, 9) &
(12, 11; 14, 21) &
(4, 6; 4, 11) &
(0, 14; 14, 6) &
(13, 2; 15, 25) &
(12, 14; 20, 23) &
(1, 2; 16, 7) &
(9, 11; 3, 0) \\
(2, 4; 24, 19) &
(15, 10; 17, 13) &
(7, 12; 7, 24) &
(6, 18; 21, 7) &
(7, 13; 14, 2) &
(9, 2; 18, 10) &
(4, 7; 26, 20) &
(7, 14; 8, 19) &
(7, 3; 15, 3) \\
(2, 14; 12, 3) &
(11, 1; 20, 4) &
(18, 8; 11, 16) &
(16, 8; 3, 8) &
(14, 15; 4, 7) &
(13, 0; 7, 0) &
(6, 14; 25, 18) &
(11, 2; 11, 1) &
(11, 16; 6, 5) \\
(11, 18; 12, 10) &
(9, 3; 16, 2) &
(17, 1; 0, 13) &
(15, 9; 23, 24) &
(1, 13; 8, 23) &
(1, 12; 10, 17) &
(5, 10; 26, 3) &
(9, 8; 4, 15) &
(5, 17; 15, 22) \\
(17, 15; 11, 3) &
(12, 4; 5, 15) &
(18, 13; 20, 22) &
(1, 15; 26, 15) &
(5, 3; 24, 18) &
(12, 10; 12, 11) &
(6, 10; 20, 8) &
(17, 8; 7, 5) &
(13, 14; 24, 11) \\
(14, 11; 26, 9) &
(16, 6; 19, 23) &
(17, 9; 20, 9) &
(0, 3; 10, 22) &
(18, 12; 18, 6) &
(11, 7; 22, 23) &
(1, 0; 3, 2) &
(12, 3; 8, 13) &
(18, 2; 8, 9) \\
(14, 1; 21, 5) &
(12, 8; 0, 25) &
(1, 6; 14, 24) &
(0, 5; 25, 4) &
(2, 0; 20, 13) &
(11, 6; 17, 16) &
(9, 12; 19, 26) &
(4, 5; 13, 21) &
(9, 7; 17, 6) \\
(7, 10; 21, 25) &
(14, 18; 1, 15) &
(3, 4; 1, 17) &
\end{array}$$ ]{}
$s=29$: [$$\begin{array}{lllllllllll}
(14, 3; 12, 3) &
(2, 4; 23, 20) &
(17, 12; 18, 16) &
(12, 4; 28, 11) &
(1, 11; 14, 24) &
(3, 10; 8, 26) &
(14, 10; 21, 22) &
(18, 8; 3, 20) &
(5, 15; 26, 11) \\
(2, 6; 2, 4) &
(8, 1; 5, 17) &
(12, 11; 4, 8) &
(8, 5; 4, 14) &
(4, 9; 19, 26) &
(11, 4; 21, 18) &
(9, 15; 6, 18) &
(8, 11; 23, 27) &
(7, 14; 10, 6) \\
(8, 15; 21, 7) &
(9, 16; 10, 17) &
(16, 15; 9, 15) &
(1, 6; 10, 27) &
(15, 2; 12, 0) &
(5, 6; 19, 22) &
(7, 0; 16, 0) &
(15, 10; 23, 1) &
(0, 17; 21, 12) \\
(16, 14; 4, 18) &
(13, 7; 24, 26) &
(15, 1; 16, 28) &
(12, 6; 3, 17) &
(9, 13; 21, 14) &
(6, 17; 24, 0) &
(15, 7; 27, 5) &
(11, 2; 3, 10) &
(10, 11; 25, 20) \\
(7, 12; 2, 14) &
(16, 7; 19, 23) &
(18, 14; 17, 9) &
(4, 8; 22, 12) &
(7, 9; 13, 11) &
(6, 11; 16, 26) &
(12, 18; 21, 19) &
(3, 9; 27, 24) &
(7, 1; 21, 1) \\
(17, 11; 9, 22) &
(3, 4; 2, 7) &
(4, 18; 1, 4) &
(6, 3; 5, 21) &
(2, 8; 26, 13) &
(15, 18; 8, 2) &
(17, 4; 8, 6) &
(16, 10; 7, 6) &
(10, 17; 14, 5) \\
(14, 0; 27, 14) &
(18, 5; 7, 24) &
(0, 18; 11, 10) &
(0, 10; 24, 18) &
(0, 13; 20, 8) &
(13, 8; 19, 18) &
(15, 3; 14, 20) &
(7, 18; 18, 22) &
(6, 7; 12, 28) \\
(8, 9; 8, 16) &
(16, 12; 13, 20) &
(3, 11; 15, 1) &
(0, 15; 4, 3) &
(9, 12; 15, 0) &
(3, 5; 9, 18) &
(9, 5; 28, 25) &
(5, 12; 1, 10) &
(6, 18; 15, 25) \\
(1, 13; 11, 12) &
(3, 2; 17, 28) &
(10, 4; 16, 10) &
(9, 17; 23, 3) &
(14, 1; 26, 23) &
(5, 2; 27, 21) &
(13, 2; 7, 9) &
(13, 10; 28, 0) &
(3, 18; 6, 23) \\
(8, 3; 11, 0) &
(6, 0; 7, 1) &
(0, 16; 5, 25) &
(13, 17; 1, 17) &
(6, 16; 14, 11) &
(12, 2; 24, 25) &
(16, 8; 24, 1) &
(10, 8; 15, 2) &
(14, 2; 8, 1) \\
(0, 11; 17, 19) &
(13, 15; 10, 22) &
(8, 0; 28, 6) &
(13, 6; 13, 6) &
(1, 4; 3, 25) &
(14, 11; 28, 7) &
(17, 14; 2, 11) &
(13, 12; 5, 23) &
(16, 5; 12, 16) \\
(9, 1; 4, 7) &
(10, 12; 27, 12) &
(5, 7; 8, 3) &
(10, 1; 9, 13) &
(9, 2; 22, 5) &
(16, 13; 2, 3) &
(13, 3; 25, 16) &
(9, 11; 2, 12) &
(3, 17; 4, 10) \\
(11, 18; 0, 13) &
(2, 10; 11, 19) &
(17, 7; 7, 15) &
(18, 2; 16, 14) &
(17, 1; 20, 19) &
(17, 15; 13, 25) &
(12, 0; 26, 9) &
(7, 10; 4, 17) &
(5, 14; 15, 20) \\
(13, 4; 27, 15) &
(5, 11; 5, 6) &
(1, 2; 18, 15) &
(8, 7; 9, 25) &
(9, 6; 9, 20) &
(4, 5; 13, 17) &
(3, 0; 22, 13) &
(15, 14; 24, 19) &
(4, 14; 0, 5) \\
(5, 0; 2, 23) &
(12, 1; 6, 22) &
(1, 16; 0, 8) &
\end{array}$$ ]{}
$s=31$: [$$\begin{array}{lllllllllll}
(0, 6; 25, 1) &
(3, 9; 3, 15) &
(0, 9; 12, 5) &
(8, 18; 26, 22) &
(11, 8; 19, 1) &
(11, 7; 10, 30) &
(18, 12; 21, 27) &
(3, 12; 11, 17) &
(13, 3; 1, 12) \\
(4, 3; 21, 6) &
(10, 1; 26, 13) &
(10, 18; 11, 15) &
(14, 15; 21, 5) &
(11, 3; 20, 28) &
(18, 4; 2, 16) &
(17, 8; 27, 23) &
(5, 0; 21, 20) &
(12, 10; 9, 10) \\
(7, 5; 26, 6) &
(0, 10; 14, 18) &
(0, 12; 24, 7) &
(16, 15; 4, 16) &
(3, 5; 16, 27) &
(15, 10; 8, 1) &
(17, 13; 15, 0) &
(3, 10; 30, 0) &
(9, 8; 16, 13) \\
(18, 11; 5, 8) &
(9, 14; 25, 9) &
(9, 12; 29, 0) &
(6, 2; 12, 15) &
(16, 12; 3, 8) &
(2, 16; 7, 23) &
(13, 7; 29, 25) &
(6, 12; 14, 6) &
(15, 2; 29, 17) \\
(6, 5; 24, 28) &
(8, 7; 24, 5) &
(18, 13; 4, 23) &
(5, 16; 0, 25) &
(0, 15; 0, 6) &
(2, 4; 11, 27) &
(0, 1; 28, 8) &
(10, 16; 6, 24) &
(6, 16; 11, 26) \\
(16, 4; 5, 9) &
(6, 17; 18, 21) &
(3, 14; 8, 23) &
(11, 5; 4, 12) &
(11, 15; 22, 9) &
(14, 2; 2, 24) &
(5, 4; 13, 1) &
(8, 1; 21, 0) &
(17, 12; 1, 16) \\
(1, 4; 4, 29) &
(15, 9; 24, 11) &
(4, 8; 25, 18) &
(14, 13; 28, 22) &
(1, 11; 17, 7) &
(3, 6; 5, 13) &
(14, 6; 4, 27) &
(9, 5; 23, 18) &
(17, 10; 4, 20) \\
(14, 5; 15, 29) &
(18, 1; 12, 6) &
(17, 14; 11, 6) &
(1, 2; 22, 25) &
(13, 15; 27, 10) &
(2, 11; 16, 0) &
(16, 14; 13, 17) &
(2, 18; 10, 14) &
(3, 7; 18, 9) \\
(1, 17; 9, 14) &
(11, 17; 2, 25) &
(18, 15; 3, 18) &
(16, 11; 15, 18) &
(14, 0; 19, 16) &
(17, 0; 22, 10) &
(15, 1; 23, 15) &
(7, 2; 21, 8) &
(13, 10; 16, 17) \\
(2, 9; 28, 6) &
(3, 1; 24, 19) &
(9, 1; 20, 30) &
(17, 9; 26, 7) &
(8, 16; 20, 12) &
(0, 4; 17, 23) &
(9, 18; 17, 1) &
(6, 1; 16, 10) &
(16, 3; 22, 2) \\
(9, 16; 19, 10) &
(0, 11; 29, 27) &
(15, 12; 25, 12) &
(14, 8; 30, 3) &
(7, 14; 7, 20) &
(7, 1; 11, 3) &
(16, 13; 14, 21) &
(8, 15; 28, 2) &
(6, 13; 30, 7) \\
(10, 7; 2, 12) &
(7, 9; 14, 4) &
(5, 13; 9, 11) &
(4, 7; 19, 0) &
(12, 5; 19, 22) &
(0, 2; 30, 9) &
(6, 8; 17, 9) &
(12, 2; 20, 13) &
(6, 4; 3, 22) \\
(12, 13; 26, 5) &
(7, 12; 28, 23) &
(10, 2; 3, 5) &
(7, 0; 13, 15) &
(12, 4; 30, 15) &
(8, 13; 8, 6) &
(10, 9; 22, 27) &
(17, 4; 24, 8) &
(3, 2; 4, 26) \\
(13, 11; 24, 13) &
(10, 11; 23, 21) &
(9, 6; 8, 2) &
(14, 11; 26, 14) &
(18, 6; 20, 0) &
(2, 13; 19, 18) &
(8, 0; 11, 4) &
(3, 18; 25, 7) &
(17, 5; 3, 17) \\
(4, 14; 10, 12) &
(4, 10; 28, 7) &
(7, 16; 1, 27) &
(1, 5; 5, 2) &
(14, 1; 18, 1) &
(0, 13; 3, 2) &
(15, 5; 14, 30) &
(15, 17; 19, 13) &
(6, 10; 19, 29) \\
(3, 8; 14, 29) &
(4, 15; 20, 26) &
(5, 8; 7, 10) &
\end{array}$$ ]{}
$s=33$: [$$\begin{array}{lllllllllll}
(2, 4; 22, 0) &
(17, 11; 23, 7) &
(5, 15; 9, 28) &
(6, 14; 10, 23) &
(9, 15; 22, 8) &
(12, 5; 26, 29) &
(17, 2; 1, 13) &
(2, 13; 16, 31) &
(13, 6; 0, 28) \\
(4, 16; 27, 5) &
(10, 2; 14, 7) &
(9, 11; 26, 21) &
(16, 1; 22, 10) &
(11, 1; 0, 17) &
(14, 12; 18, 1) &
(17, 13; 3, 19) &
(2, 7; 25, 24) &
(7, 17; 11, 22) \\
(5, 1; 11, 25) &
(6, 1; 29, 13) &
(17, 1; 21, 14) &
(6, 12; 32, 3) &
(14, 16; 15, 8) &
(9, 3; 18, 24) &
(1, 14; 19, 5) &
(13, 15; 18, 21) &
(0, 3; 21, 3) \\
(12, 7; 0, 12) &
(5, 8; 32, 0) &
(11, 0; 16, 18) &
(16, 13; 20, 17) &
(1, 2; 28, 6) &
(9, 4; 1, 10) &
(2, 5; 20, 27) &
(1, 13; 32, 8) &
(11, 18; 27, 3) \\
(5, 4; 31, 15) &
(17, 6; 16, 5) &
(12, 1; 24, 4) &
(9, 16; 23, 0) &
(8, 13; 6, 27) &
(1, 4; 3, 23) &
(15, 4; 6, 30) &
(4, 11; 19, 2) &
(10, 6; 25, 21) \\
(16, 11; 13, 6) &
(0, 10; 0, 6) &
(3, 8; 13, 5) &
(6, 3; 31, 6) &
(10, 15; 1, 27) &
(18, 2; 18, 5) &
(3, 18; 12, 25) &
(18, 7; 15, 1) &
(12, 13; 5, 7) \\
(5, 10; 3, 30) &
(10, 11; 20, 10) &
(5, 16; 21, 4) &
(14, 13; 2, 14) &
(11, 3; 32, 1) &
(10, 14; 32, 12) &
(0, 16; 24, 1) &
(10, 9; 28, 5) &
(18, 8; 21, 10) \\
(7, 4; 28, 32) &
(6, 0; 11, 9) &
(15, 0; 32, 2) &
(8, 7; 19, 29) &
(3, 13; 10, 11) &
(10, 13; 24, 9) &
(14, 5; 24, 7) &
(6, 7; 2, 30) &
(7, 9; 9, 3) \\
(0, 18; 17, 23) &
(0, 2; 10, 19) &
(11, 14; 28, 30) &
(18, 4; 29, 24) &
(14, 8; 31, 3) &
(15, 6; 7, 12) &
(14, 18; 9, 26) &
(5, 13; 1, 22) &
(17, 0; 8, 28) \\
(7, 13; 4, 26) &
(15, 17; 20, 24) &
(5, 3; 23, 8) &
(9, 2; 11, 32) &
(6, 2; 17, 26) &
(9, 1; 12, 30) &
(6, 18; 4, 19) &
(4, 6; 8, 20) &
(15, 7; 10, 13) \\
(13, 4; 25, 13) &
(11, 8; 25, 15) &
(10, 3; 17, 4) &
(15, 2; 29, 15) &
(0, 5; 5, 12) &
(7, 11; 14, 8) &
(7, 1; 27, 31) &
(0, 9; 31, 29) &
(18, 5; 16, 2) \\
(16, 8; 28, 14) &
(12, 9; 13, 16) &
(17, 9; 25, 4) &
(18, 15; 11, 14) &
(5, 6; 18, 14) &
(9, 5; 6, 19) &
(14, 4; 11, 16) &
(7, 14; 21, 6) &
(0, 8; 7, 22) \\
(10, 1; 18, 2) &
(17, 5; 10, 17) &
(1, 3; 15, 9) &
(7, 10; 16, 23) &
(1, 0; 26, 20) &
(3, 12; 14, 30) &
(13, 11; 29, 12) &
(12, 4; 21, 17) &
(12, 8; 11, 20) \\
(17, 12; 15, 6) &
(10, 16; 11, 29) &
(18, 10; 8, 13) &
(15, 14; 25, 17) &
(0, 4; 4, 14) &
(9, 18; 7, 20) &
(17, 4; 18, 9) &
(16, 7; 7, 18) &
(3, 17; 2, 27) \\
(12, 10; 31, 19) &
(3, 4; 26, 7) &
(11, 6; 22, 24) &
(12, 18; 22, 28) &
(8, 15; 4, 23) &
(11, 15; 31, 5) &
(3, 15; 19, 0) &
(9, 8; 2, 17) &
(1, 8; 16, 1) \\
(16, 2; 12, 3) &
(16, 15; 26, 16) &
(2, 11; 9, 4) &
(9, 6; 27, 15) &
(14, 17; 0, 29) &
(14, 3; 20, 22) &
(2, 8; 30, 8) &
(13, 0; 15, 30) &
(0, 14; 13, 27) \\
(17, 8; 26, 12) &
(12, 16; 25, 9) &
(2, 12; 2, 23) &
\end{array}$$ ]{}
$s=35$: [$$\begin{array}{lllllllllll}
(9, 4; 22, 15) &
(0, 2; 14, 34) &
(13, 18; 25, 8) &
(11, 16; 4, 30) &
(6, 18; 3, 17) &
(13, 15; 16, 26) &
(2, 6; 27, 23) &
(3, 1; 19, 15) &
(3, 13; 22, 7) \\
(6, 1; 6, 31) &
(5, 17; 16, 5) &
(14, 1; 26, 32) &
(9, 6; 12, 34) &
(1, 17; 3, 22) &
(5, 3; 9, 31) &
(11, 3; 20, 1) &
(11, 15; 23, 12) &
(17, 12; 0, 20) \\
(4, 6; 16, 9) &
(16, 1; 9, 29) &
(12, 14; 15, 27) &
(6, 8; 30, 15) &
(3, 7; 6, 34) &
(11, 4; 7, 10) &
(15, 17; 27, 9) &
(10, 8; 5, 27) &
(7, 8; 8, 22) \\
(16, 15; 8, 15) &
(9, 12; 31, 16) &
(9, 13; 2, 14) &
(12, 18; 21, 12) &
(8, 2; 31, 32) &
(12, 6; 8, 10) &
(1, 13; 5, 34) &
(12, 5; 28, 34) &
(13, 5; 6, 13) \\
(5, 10; 17, 15) &
(12, 2; 4, 26) &
(10, 2; 22, 18) &
(6, 11; 28, 32) &
(4, 17; 4, 12) &
(16, 0; 16, 2) &
(6, 7; 1, 26) &
(7, 2; 10, 2) &
(18, 9; 28, 0) \\
(13, 11; 18, 17) &
(10, 15; 34, 2) &
(16, 3; 26, 27) &
(2, 1; 12, 17) &
(9, 11; 27, 8) &
(6, 14; 25, 18) &
(9, 17; 21, 17) &
(18, 11; 9, 26) &
(10, 17; 1, 30) \\
(8, 0; 6, 12) &
(14, 4; 34, 21) &
(7, 15; 24, 32) &
(14, 9; 33, 19) &
(17, 14; 28, 8) &
(12, 11; 11, 13) &
(10, 4; 13, 0) &
(4, 2; 25, 33) &
(2, 3; 21, 0) \\
(18, 0; 18, 15) &
(10, 13; 12, 33) &
(16, 2; 20, 24) &
(14, 11; 2, 31) &
(3, 12; 25, 17) &
(1, 15; 11, 25) &
(1, 12; 2, 30) &
(12, 4; 19, 6) &
(9, 7; 25, 7) \\
(3, 9; 3, 29) &
(2, 13; 1, 28) &
(18, 15; 6, 30) &
(15, 4; 31, 5) &
(16, 13; 19, 0) &
(12, 8; 33, 18) &
(4, 18; 1, 24) &
(3, 17; 10, 18) &
(6, 10; 21, 24) \\
(5, 2; 30, 8) &
(10, 14; 16, 23) &
(9, 8; 9, 20) &
(2, 17; 7, 19) &
(2, 18; 11, 16) &
(15, 8; 1, 19) &
(11, 2; 5, 15) &
(3, 8; 23, 28) &
(5, 6; 0, 11) \\
(15, 14; 10, 0) &
(0, 3; 30, 33) &
(5, 11; 19, 3) &
(6, 17; 2, 13) &
(14, 16; 22, 11) &
(1, 18; 23, 20) &
(0, 4; 3, 26) &
(9, 1; 13, 1) &
(12, 15; 29, 7) \\
(3, 10; 8, 32) &
(0, 5; 10, 1) &
(11, 0; 21, 22) &
(0, 10; 19, 4) &
(13, 12; 32, 3) &
(13, 14; 9, 30) &
(17, 13; 31, 15) &
(16, 10; 25, 28) &
(11, 8; 34, 0) \\
(7, 18; 19, 5) &
(17, 8; 14, 11) &
(5, 18; 22, 27) &
(8, 14; 4, 17) &
(18, 8; 2, 29) &
(14, 7; 3, 20) &
(12, 16; 1, 14) &
(5, 1; 33, 21) &
(15, 2; 13, 3) \\
(11, 17; 6, 24) &
(7, 10; 11, 9) &
(0, 9; 11, 32) &
(13, 7; 21, 23) &
(3, 14; 12, 24) &
(0, 12; 9, 23) &
(7, 1; 4, 28) &
(14, 2; 6, 29) &
(9, 5; 24, 23) \\
(5, 4; 2, 32) &
(8, 16; 21, 3) &
(1, 0; 24, 8) &
(18, 10; 10, 31) &
(15, 9; 4, 18) &
(17, 0; 29, 25) &
(10, 9; 6, 26) &
(15, 6; 22, 33) &
(0, 14; 13, 5) \\
(7, 0; 27, 0) &
(9, 16; 5, 10) &
(13, 4; 20, 11) &
(3, 18; 4, 13) &
(14, 18; 7, 14) &
(4, 16; 23, 17) &
(5, 15; 14, 20) &
(7, 4; 18, 30) &
(10, 11; 29, 14) \\
(6, 0; 7, 20) &
(0, 15; 28, 17) &
(13, 6; 29, 4) &
(11, 7; 33, 16) &
(6, 3; 14, 5) &
(5, 16; 7, 18) &
(13, 8; 24, 10) &
(1, 4; 27, 14) &
(8, 1; 16, 7) \\
(16, 7; 31, 13) &
(5, 7; 12, 29) &
(5, 8; 25, 26) &
\end{array}$$ ]{}
There exists an HSAS$(s,v;5,3)$ for each $(s,v)\in \{(21,11),(29,15),(37,19)\}$.
Let $V=I_v$ and $S=I_s$. Let $W=\{v-3,v-2,v-1\}$ and $T=\{s-5,s-4,s-3,s-2,s-1\}$. The desired HSASs filled with pairs of points from $V$ and indexed by $S$ are presented as follows.
$(s,v)=(21,11)$: [$$\begin{array}{lllllllllll}
(1, 10; 4, 12) &
(3, 6; 20, 6) &
(4, 8; 12, 3) &
(2, 0; 1, 18) &
(7, 5; 5, 17) &
(8, 7; 13, 8) &
(3, 1; 19, 0) &
(6, 1; 11, 5) &
(9, 7; 11, 1) &
(0, 6; 17, 0) \\
(2, 6; 3, 9) &
(2, 3; 17, 15) &
(10, 7; 6, 9) &
(7, 1; 15, 18) &
(0, 10; 13, 2) &
(1, 5; 7, 9) &
(6, 10; 1, 10) &
(4, 7; 0, 20) &
(10, 4; 14, 11) &
(5, 0; 8, 12) \\
(5, 8; 11, 10) &
(0, 3; 16, 11) &
(4, 2; 10, 19) &
(2, 9; 4, 7) &
(4, 5; 6, 16) &
(7, 6; 16, 7) &
(0, 7; 3, 19) &
(3, 5; 18, 14) &
(4, 9; 2, 9) &
(4, 0; 7, 5) \\
(2, 8; 6, 0) &
(3, 10; 7, 3) &
(2, 1; 13, 16) &
(9, 3; 5, 13) &
(4, 6; 13, 18) &
(3, 4; 4, 8) &
(8, 0; 15, 4) &
(1, 4; 17, 1) &
(8, 6; 2, 14) &
(10, 2; 5, 8) \\
(0, 9; 6, 10) &
(1, 0; 14, 20) &
(5, 2; 2, 20) &
(7, 3; 2, 10) &
(5, 6; 19, 4) &
(3, 8; 1, 9) &
(5, 10; 15, 0) &
(7, 2; 14, 12) &
(9, 6; 12, 15) &
(1, 9; 3, 8) \\
\end{array}$$ ]{}
$(s,v)=(29,15)$: [$$\begin{array}{lllllllllll}
(3, 0; 16, 25) &
(8, 4; 28, 23) &
(4, 9; 24, 8) &
(13, 8; 0, 15) &
(12, 5; 8, 2) &
(11, 5; 4, 28) &
(10, 5; 14, 21) &
(2, 7; 22, 11) &
(4, 12; 3, 15) \\
(6, 7; 5, 19) &
(12, 10; 18, 6) &
(3, 1; 6, 24) &
(3, 11; 27, 5) &
(6, 1; 28, 2) &
(9, 13; 16, 5) &
(14, 7; 18, 4) &
(11, 0; 24, 22) &
(12, 6; 22, 7) \\
(12, 7; 16, 14) &
(10, 4; 13, 2) &
(10, 6; 23, 17) &
(10, 9; 28, 3) &
(10, 0; 27, 19) &
(14, 10; 22, 8) &
(3, 6; 21, 13) &
(3, 12; 20, 0) &
(7, 0; 28, 0) \\
(13, 6; 1, 8) &
(5, 8; 22, 6) &
(4, 1; 1, 11) &
(7, 11; 17, 3) &
(12, 1; 21, 4) &
(2, 14; 3, 21) &
(13, 4; 14, 4) &
(11, 9; 26, 14) &
(13, 2; 23, 2) \\
(14, 0; 15, 2) &
(14, 9; 19, 1) &
(10, 7; 20, 26) &
(8, 1; 10, 26) &
(2, 0; 10, 6) &
(1, 7; 9, 15) &
(6, 0; 11, 26) &
(1, 14; 17, 5) &
(14, 8; 20, 12) \\
(2, 12; 9, 12) &
(13, 3; 10, 18) &
(11, 1; 0, 16) &
(0, 1; 20, 3) &
(10, 3; 4, 12) &
(3, 8; 11, 3) &
(13, 7; 6, 21) &
(6, 14; 9, 0) &
(5, 6; 24, 12) \\
(0, 8; 7, 8) &
(1, 9; 27, 7) &
(4, 7; 25, 7) &
(9, 2; 25, 20) &
(2, 1; 14, 8) &
(12, 11; 10, 1) &
(9, 12; 13, 11) &
(10, 11; 25, 9) &
(6, 8; 14, 25) \\
(3, 7; 8, 23) &
(11, 2; 19, 15) &
(2, 4; 26, 0) &
(4, 14; 16, 6) &
(3, 2; 1, 28) &
(6, 9; 15, 4) &
(5, 7; 1, 27) &
(8, 2; 4, 27) &
(9, 7; 12, 10) \\
(5, 1; 25, 19) &
(2, 6; 18, 16) &
(14, 3; 7, 14) &
(9, 8; 9, 18) &
(14, 11; 23, 13) &
(5, 3; 15, 26) &
(13, 1; 22, 13) &
(4, 3; 22, 19) &
(13, 5; 3, 9) \\
(0, 4; 21, 9) &
(7, 8; 24, 13) &
(0, 12; 5, 23) &
(13, 11; 11, 7) &
(5, 0; 18, 13) &
(0, 13; 12, 17) &
(14, 5; 11, 10) &
(10, 2; 24, 5) &
(12, 8; 19, 17) \\
(6, 11; 20, 6) &
(4, 6; 10, 27) &
(11, 4; 12, 18) &
(5, 9; 0, 23) &
(10, 8; 16, 1) &
(8, 11; 21, 2) &
(3, 9; 2, 17) &
(5, 4; 20, 5) &
(2, 5; 7, 17) \\
\end{array}$$ ]{}
$(s,v)=(37,19)$: [$$\begin{array}{lllllllllll}
(5, 11; 24, 2) &
(0, 7; 32, 21) &
(15, 17; 1, 26) &
(0, 17; 24, 4) &
(5, 16; 12, 30) &
(5, 1; 26, 13) &
(11, 16; 1, 13) &
(5, 2; 1, 22) \\
(11, 4; 12, 0) &
(11, 7; 36, 6) &
(13, 2; 8, 29) &
(0, 6; 23, 25) &
(6, 17; 20, 15) &
(18, 9; 0, 13) &
(8, 2; 16, 0) &
(8, 6; 21, 31) \\
(15, 8; 24, 23) &
(9, 15; 29, 15) &
(15, 16; 11, 7) &
(1, 4; 4, 25) &
(11, 3; 26, 22) &
(13, 1; 11, 34) &
(1, 15; 9, 18) &
(0, 3; 33, 0) \\
(7, 8; 15, 2) &
(14, 12; 6, 27) &
(10, 18; 15, 31) &
(12, 2; 36, 3) &
(6, 7; 35, 22) &
(9, 12; 17, 22) &
(6, 13; 10, 32) &
(17, 3; 13, 2) \\
(8, 1; 36, 19) &
(0, 5; 17, 6) &
(14, 8; 35, 7) &
(5, 3; 15, 21) &
(4, 3; 7, 36) &
(13, 0; 22, 36) &
(12, 11; 28, 25) &
(0, 4; 27, 8) \\
(13, 16; 17, 4) &
(0, 12; 26, 11) &
(16, 10; 16, 22) &
(11, 2; 14, 9) &
(7, 3; 12, 1) &
(18, 8; 17, 5) &
(7, 18; 8, 19) &
(9, 7; 28, 20) \\
(15, 13; 19, 27) &
(0, 10; 7, 19) &
(6, 12; 24, 33) &
(12, 13; 12, 9) &
(18, 0; 16, 1) &
(6, 16; 29, 6) &
(9, 10; 11, 27) &
(17, 10; 9, 17) \\
(6, 15; 4, 28) &
(17, 1; 22, 3) &
(2, 10; 4, 23) &
(16, 1; 24, 14) &
(4, 15; 2, 30) &
(8, 11; 27, 32) &
(6, 5; 7, 27) &
(14, 5; 19, 9) \\
(5, 12; 31, 23) &
(11, 10; 34, 29) &
(14, 0; 13, 29) &
(13, 7; 25, 30) &
(17, 4; 14, 5) &
(17, 2; 21, 27) &
(14, 10; 8, 18) &
(1, 12; 15, 32) \\
(7, 14; 4, 14) &
(15, 0; 14, 3) &
(13, 5; 16, 14) &
(7, 1; 29, 0) &
(6, 14; 26, 0) &
(12, 15; 35, 0) &
(16, 3; 18, 27) &
(5, 8; 18, 4) \\
(3, 9; 16, 5) &
(2, 18; 11, 28) &
(1, 2; 6, 30) &
(0, 16; 5, 15) &
(1, 18; 27, 20) &
(14, 18; 25, 22) &
(11, 13; 7, 15) &
(15, 5; 34, 10) \\
(6, 4; 17, 1) &
(3, 1; 28, 23) &
(1, 0; 10, 31) &
(8, 4; 11, 22) &
(0, 8; 12, 34) &
(9, 5; 3, 32) &
(14, 11; 30, 17) &
(3, 13; 6, 35) \\
(11, 9; 4, 33) &
(18, 4; 9, 29) &
(9, 14; 34, 31) &
(3, 6; 34, 30) &
(4, 12; 34, 13) &
(9, 4; 19, 24) &
(16, 2; 31, 19) &
(15, 11; 16, 20) \\
(1, 6; 16, 8) &
(14, 15; 36, 5) &
(17, 14; 12, 23) &
(15, 18; 21, 6) &
(14, 3; 32, 11) &
(5, 10; 0, 36) &
(10, 4; 6, 32) &
(9, 16; 25, 2) \\
(9, 6; 14, 36) &
(13, 17; 28, 0) &
(12, 16; 21, 8) &
(14, 1; 2, 33) &
(4, 5; 28, 35) &
(13, 9; 21, 23) &
(17, 9; 6, 18) &
(15, 10; 33, 25) \\
(10, 12; 20, 30) &
(13, 10; 5, 24) &
(7, 2; 34, 18) &
(3, 2; 17, 25) &
(5, 17; 25, 29) &
(16, 4; 10, 20) &
(9, 1; 12, 7) &
(18, 12; 7, 18) \\
(2, 0; 20, 2) &
(3, 12; 29, 4) &
(8, 9; 26, 8) &
(15, 7; 13, 17) &
(13, 14; 20, 1) &
(6, 11; 3, 19) &
(16, 14; 28, 3) &
(12, 7; 10, 5) \\
(1, 10; 1, 35) &
(4, 14; 21, 16) &
(3, 8; 20, 9) &
(0, 9; 30, 9) &
(6, 10; 2, 12) &
(4, 13; 18, 31) &
(6, 2; 5, 13) &
(12, 8; 1, 14) \\
(11, 1; 21, 5) &
(4, 2; 26, 33) &
(11, 0; 35, 18) &
(8, 17; 10, 30) &
(18, 11; 23, 10) &
(17, 7; 7, 31) &
(15, 3; 8, 31) &
(2, 14; 15, 24) \\
(7, 16; 23, 9) &
(7, 5; 11, 33) &
(10, 3; 14, 10) &
(11, 17; 8, 11) &
(7, 10; 3, 26) &
(2, 9; 35, 10) &
(2, 15; 32, 12) &
(17, 12; 19, 16) \\
(18, 3; 3, 24) &
(13, 18; 2, 26) &
(10, 8; 13, 28) &
(8, 13; 3, 33) &
\end{array}$$ ]{}
[^1]: The research of H. Wei was supported by the Post-Doctoral Science Foundation of China under Grant No. 2015M571067, and Beijing Postdoctoral Research Foundation.
The research of G. Ge was supported by the National Natural Science Foundation of China under Grant Nos. 61171198, 11431003 and 61571310, and the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions.
[^2]: X. Wang is with the School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China (e-mail: 11235062@zju.edu.cn).
[^3]: H. Wei is with the School of Mathematical Sciences, Capital Normal University, Beijing 100048, China (e-mail: ven0505@163.com).
[^4]: C. Shangguan is with the School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China (e-mail: 11235061@zju.edu.cn).
[^5]: G. Ge is with the School of Mathematical Sciences, Capital Normal University, Beijing 100048, China (e-mail: gnge@zju.edu.cn). He is also with Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing, 100048, China.
|
---
abstract: 'In terms of the parton hadron string dynamics (PHSD) approach — including the retarded electromagnetic field — we investigate the role of fluctuations of the correlation function in the azimuthal angle $\psi$ of charged hadrons that is expected to be a sensitive signal of local strong parity violation. For the early time we consider fluctuations in the position of charged spectators resulting in electromagnetic field fluctuations as well as in the position of participant baryons defining the event plane. For partonic and hadronic phases in intermediate stages of the interaction we study the possible formation of excited matter in electric charge dipole and quadrupole form as generated by fluctuations. The role of the transverse momentum and local charge conservation laws in the observed azimuthal asymmetry is investigated, too. All these above-mentioned effects are incorporated in our analysis based on event-by-event PHSD calculations. Furthermore, the azimuthal angular correlations from Au+Au collisions observed in the recent STAR measurements within the Relativistic Heavy Ion Collider (RHIC) Beam Energy Scan (BES) program are studied. It is shown that the STAR correlation data at the collision energies of $\sqrt{s_{NN}}=$ 7.7 and 11.5 GeV can be reasonably reproduced within the PHSD. At higher energies the model fails to describe the $\psi$ correlation data resulting in an overestimation of the partonic scalar field involved. We conclude that an additional transverse anisotropy fluctuating source is needed which with a comparable strength acts on both in- and out-of-plane components.'
author:
- 'V. D. Toneev'
- 'V. P. Konchakovski'
- 'V. Voronyuk'
- 'E. L. Bratkovskaya'
- 'W. Cassing'
title: 'Event-by-event background in estimates of the chiral magnetic effect'
---
Introduction
============
A fundamental property of the non-Abelian gauge theory is the existence of nontrivial topological configurations in the QCD vacuum. Spontaneous transitions between topologically different states occur with a change of the topological quantum number characterizing these states and induce anomalous processes like local violation of the ${\cal P}$ and ${\cal CP}$ symmetry. The interplay of topological configurations with (chiral) quarks shows the local imbalance of chirality. Such a chiral asymmetry when coupled to a strong magnetic field induces a current of electric charge along the direction of the magnetic field which leads to a separation of oppositely charged particles with respect to the reaction plane [@KMcLW07; @Kh09; @FKW08].
This strong magnetic field can convert topological charge fluctuations in the QCD vacuum into a global electric charge separation with respect to the reaction plane. Thus, as argued in Refs. [@KZ07; @KMcLW07; @FKW08; @KW09], the topological effects in QCD might be observed in heavy-ion collisions directly in the presence of very intense external electromagnetic fields due to the “chiral magnetic effect” (CME) as a manifestation of spontaneous violation of the ${\cal CP}$ symmetry. Indeed, it was shown that electromagnetic fields of the required strength can be created in relativistic heavy-ion collisions [@KMcLW07; @SIT09; @EM_HSD] by the charged spectators in peripheral collisions.
The first experimental evidence for the CME, identified via the charge asymmetry, was obtained by the STAR Collaboration at the Relativistic Heavy Ion Collider (RHIC) at $\sqrt{s_{NN}}=$ 200 and 62 GeV [@Vol09; @STAR-CME; @STAR-CME2] and confirmed qualitatively by the PHENIX Collaboration [@PHENIX]. Recently, these measurements were extended, from one side, below the nominal RHIC energy down to $\sqrt{s_{NN}}=$ 7.7 GeV within the RHIC Beam Energy Scan (BES) program [@BES11] and, from the other side, preliminary results for the maximal available energy $\sqrt{s_{NN}}=$ 2.76 TeV were announced from the Large Hadron Collider (LHC) [@Ch11; @LHC_CS12]. Though at first sight, some features of these data appear to be consistent with an expectation from the local parity violation phenomenon, the interpretation of the observed effect is still under intense discussion [@Wa09; @BKL09; @Pr10-1; @Pr10; @Pr09; @AMM10; @LKB10; @BKL10; @Lo11; @Wa12].
The fluctuation nature of the CME will give rise to a vanishing expectation value of a ${\cal P}$-odd observable and due to that, as proposed by Voloshin [@Vol04], the azimuthal angle two-particle correlator related to charge asymmetry with respect to the reaction plane is measured in experiments [@Vol09; @STAR-CME; @STAR-CME2; @BES11; @Ch11; @LHC_CS12]. Accompanying these experiments hadronic estimates of the dynamical background in these experimental papers including only statistical (hadronic) fluctuations do not involve the electromagnetic field at all. The electromagnetic field — created in heavy-ion collisions — was calculated in different dynamical approaches in Refs. [@KMcLW07; @SIT09; @EM_HSD; @BES-HSD; @TV10; @OL11]. In two of them [@EM_HSD; @BES-HSD] calculations were carried out in comparison with the CME observable. However, in all these studies only the mean electromagnetic field was presented, being averaged over the whole ensemble of colliding nuclei.
As noted in Ref. [@BS11] event-by-event fluctuations of the electromagnetic field in off-central heavy-ion collisions can reach rather high values comparable with the average values. The presence of large fluctuations was then confirmed in a more elaborated model in Ref. [@DH12]. In this study — based on the parton-hadron-string dynamics (PHSD) kinetic approach — we analyze event-by-event fluctuations in the electromagnetic fields as well as in transverse momentum, multiplicity and conserved quantities and the influence of these fluctuations on physics observables relevant to measurements of the CME.
The paper is organized as follows. After a short recapitulation of the PHSD approach in Sec. \[Sec:PHSD\] we sequentially (Sec. \[Sec:Sources\]) consider the manifestation of the initial geometry fluctuations in spectator protons and participant nucleons as well as in the charged quasiparticle geometry at some later stage. These effects are relevant for fluctuations in the electromagnetic fields, the event plane orientation and the possible formation of a fluctuating electric charge dipole/quadrupole transient subsystem, respectively. Conservation of the transverse momentum and local charge is analyzed as an alternative explanation of the observed azimuthal asymmetry. In Sec. \[Sec:Observable\] we discuss the role and importance of these effects in the azimuthal angle correlations and their dependence on collision energy. Our conclusions are summarized in Sec. \[Sec:Summary\].
Reminder of the PHSD approach {#Sec:PHSD}
=============================
Here we analyze the dynamics of partons, hadrons and strings in relativistic nucleus-nucleus collisions within the parton hadron string dynamics approach [@PHSD]. In this transport approach the partonic dynamics is based on Kadanoff-Baym equations for Green functions with self-energies from the dynamical quasiparticle model (DQPM) [@Cassing06; @Cassing07] which describes QCD properties in terms of ‘resummed’ single-particle Green functions. In Ref. [@BCKL11], the actual three DQPM parameters for the temperature-dependent effective coupling were fitted to the recent lattice QCD results of Ref. [@aoki10]. The latter leads to a critical temperature $T_c \approx$ 160 MeV which corresponds to a critical energy density of $\epsilon_c \approx$ 0.5 GeV/fm$^3$. In PHSD the parton spectral functions $\rho_j$ ($j=q, {\bar q}, g$) are no longer $\delta-$functions in the invariant mass squared as in conventional cascade or transport models but depend on the parton mass and width parameters which were fixed by fitting the lattice QCD results from Ref. [@aoki10]. We recall that the DQPM allows one to extract a potential energy density $V_p$ from the space-like part of the energy-momentum tensor as a function of the scalar parton density $\rho_s$. Derivatives of $V_p$ with respect to $\rho_s$ then define a scalar mean-field potential $U_s(\rho_s)$ which enters into the equation of motion for the dynamic partonic quasiparticles. Thus, one should avoid large local fluctuations in the potential $V_p$ which indeed is solved in the parallel ensemble method by averaging the mean-field over many events. In the present study we modify the default PHSD approach by evaluating the electromagnetic fields for each event without averaging the charge currents over many (parallel) events.
The transition from partonic to hadronic degrees of freedom (d.o.f.) (and vice versa) is described by covariant transition rates for the fusion of quark-antiquark pairs or three quarks (antiquarks), respectively, obeying flavor current-conservation, color neutrality as well as energy-momentum conservation [@PHSD; @BCKL11]. Since close to the phase transition the dynamical quarks and antiquarks become very massive, the formed resonant ‘prehadronic’ color-dipole states ($q\bar{q}$ or $qqq$) are of high invariant mass, too, and sequentially decay to the ground-state meson and baryon octets increasing the total entropy.
On the hadronic side PHSD includes explicitly the baryon octet and decouplet, the $0^-$- and $1^-$-meson nonets as well as selected higher resonances as in the hadron string dynamics (HSD) approach [@Ehehalt; @HSD]. Note that PHSD and HSD merge at low energy density, in particular below the critical energy density $\epsilon_c\approx$ 0.5 GeV/fm$^{3}$.
The PHSD approach has been applied to nucleus-nucleus collisions from $\sqrt{s_{NN}}\sim$ 5 to 200 GeV in Refs. [@PHSD; @BCKL11] in order to explore the space-time regions of ‘partonic matter’. It was found that even central collisions at the top-SPS energy of $\sqrt{s_{NN}}=$ 17.3 GeV show a large fraction of nonpartonic, [*i.e.*]{}, hadronic or string-like matter, which can be viewed as a hadronic corona. This finding implies that neither hadronic nor only partonic ‘models’ can be employed to extract physical conclusions in comparing model results with data. All these previous findings provide promising perspectives to use PHSD in the whole range from about $\sqrt{s_{NN}}=$ 5 to 200 GeV for a systematic study of azimuthal asymmetries of hadrons produced in relativistic nucleus-nucleus collisions. This expectation has been realized, in particular, in the successful description of various flow harmonics in the transient energy range [@KBCTV11; @KBCTV12].
The collision geometry for a peripheral collision is displayed in Fig. \[tr-pl\] in the transverse $(x-y)$ plane. The reaction plane is defined as the $(z-x)$ plane. The overlapping strongly interacting region (participants) has an “almond”-like shape. The nuclear region outside this “almond” corresponds to spectator matter which is the dominant source of the electromagnetic field at the very beginning of the nuclear collision. Note that in the PHSD approach the particles are subdivided into target and projectile spectators and participants not geometrically but dynamically: spectators are nucleons which suffered yet no hard collision.
As in Refs. [@SIT09; @EM_HSD; @BS11; @DH12] the electric and magnetic fields at the relative position $\mathbf{R}_n={\bf r}-{\bf r}_n$ are calculated according to the retarded $(t_n=t-|{\bf r}-{\bf r}_n|$) Liénard-Wiechert equations for a charge moving with velocity $\mathbf{v}$: \[LWeq1\] e(, t) &=& \_n Z\_n (1-v\^2) ,\
\[LWeq2\] e(, t) &=& \_n Z\_n (1-v\^2) , where the summation runs over all charged quasiparticles in the system, both spectators and participants, $Z_n$ is the charge of the particle and $\alpha=e^2/4\pi=1/137$ is the electromagnetic constant. By including explicitly the participants — created during the heavy-ion reaction and being propagated in time also under the influence of the retarded electromagnetic fields — we also consider the back reaction of the particles on the retarded fields. Equations (\[LWeq1\]), (\[LWeq2\]) have singularities for $R_n=0$ and in the calculations we regularize them by the condition $R_n>$ 0.3 fm.
However, if the produced matter, after the short early-stage evolution, is in the QGP phase, the electric conductivity is not negligible. Strictly speaking our estimates of the magnetic and electric fields in Eqs. (\[LWeq1\]), (\[LWeq2\]) are strictly valid only at the early stage of the collision. At later stages we have neglected the collective electromagnetic response of the matter produced in the collision by assuming that the produced matter is ideally electrically insulating. Here, the magnetic response from the created medium is expected to become increasingly important [@LL84] and in principle may substantially influence the time evolution of the electromagnetic fields in the QGP. In particular, a non-trivial electromagnetic response — as studied within generalized Maxwell equations including the permeability and permittivity of the QGP — can lead to a slowdown of the decrease of the magnetic field at later times of 2-4 fm/c [@DH12; @Tu10]. It is of interest to recall that for a peripheral Au+Au collision at $\sqrt{s_{NN}}=$ 200 GeV our kinetic model with the retarded electromagnetic field predicts a flattening of the strong time dependence for the magnetic field at $t\approx$1 fm/c (see Fig. 4 in Ref. [@EM_HSD]). It is also noteworthy that the magnetic field strength at this time is by three orders of magnitude lower than the maximal field strength. Therefore it is not likely that there will be a noticeable influence of the effect discussed above on observables for later times. Furthermore, we mention that according to Faraday’s law a strongly decreasing magnetic field induces an electric field circulating around the direction of the magnetic field. In turn this electric field generates an electric current that produces a magnetic field pointing in the positive z direction according to the Lenz rule [@Tu11]. All these nontrivial responses of charged matter to intense electromagnetic fields are of great interest and more elaborated studies are required; however, this is beyond the aim of this present paper.
Sources of background fluctuations {#Sec:Sources}
==================================
Fluctuations in the proton spectator positions {#glasma}
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Let us consider geometrical fluctuations in the electromagnetic field taking into account fluctuations in the position of [*spectator*]{} protons. The retarded electric and magnetic field evaluated according to Eqs. (\[LWeq1\]),(\[LWeq2\]) are presented in Fig. \[Pfield\] for off-central Au+Au collisions at the collision energy of $\sqrt{s_{NN}}=$ 200 GeV. The PHSD results (including contributions of all quasiparticles) are given for the time of the maximal overlap of the compressed colliding nuclei which corresponds to $t\simeq$ 0.05 fm/c. As noted above the main contribution is coming from spectator protons. In peripheral collisions the average magnetic component orthogonal to the reaction plane $<B_y>$ is dominant. The dimensionless field magnitude $e<B_y>/ {m_\pi}^2\simeq$ 5 and its dispersion are in a reasonable agreement (discrepancy is less than 10%) with recent calculations results within the partonic HIJING model [@DH12]. The difference in the calculated electromagnetic field between HIJING and our PHSD approach is due to different regularization procedures used for Eqs. (\[LWeq1\]), (\[LWeq2\]). In [@DH12] all field contributions resulting in a numerical overflow were taken away while we used a constraint on the closest distance $R_n>0.3$ fm. The agreement between these two models demonstrates a very week sensitivity of the results for reasonable values $R_n$. Our present results are consistent also with earlier calculations within the hadronic dynamics of the ultrarelativistic quantum-molecular dynamics (UrQMD) [@SIT09] and the hadron string dynamics (HSD) [@EM_HSD] models.
If one looks at the field variance \[Fig. \[Pfield\](a)\] the full width of the $E_y, E_x, B_x$ distributions is about $\sigma \sim 2
/m_\pi^2$ for all transverse field components being consistent with Ref. [@DH12]. Here, additional results are plotted also for the restricted case when the electromagnetic field is averaged over all events in the parallel ensemble as explained in the previous section. This procedure has been used before in Ref. [@EM_HSD]. As seen from Fig. \[Pfield\](a) this leads to a suppression of the variance for all field distributions by a factor of about 3.
In Fig. \[Pfield\](b) we mimic results of the schematic model in Ref. [@BS11] considering a nuclear colliding system at the time of the maximal overlap as an infinitely thin disk. This was simulated numerically by an artificial shift of the position of the longitudinal components of all protons at this moment to the plane $z=0$. As is seen in Fig. \[Pfield\](b) all field distributions indeed increase in width by a factor of about two. A direct comparison of our results to those of Ref. [@BS11] gives a factor of three or even more. This finding completely coincides with the results of Ref. [@DH12] as to both the value of the width and its origin.
![(Color online) Time dependence of the momentum increment of forward moving ($p_z>0$) partons due to the electromagnetic field created in Au+Au ($\sqrt{s_{NN}}=$ 200 GeV) collisions with the impact parameter $b=$ 10 fm.[]{data-label="Dp-150"}](inc-comp.eps){width="37.00000%"}
{height="6.0truecm"} {height="6.0truecm"} {height="6.0truecm"} {height="6.0truecm"}
The estimated strength of the electromagnetic fields provides no information about their action on the quasiparticle transport. Let us look at the early time dynamics in more detail and introduce a momentum increment $\Delta {\bf p}$ as a sum of the mean particular increases of the quasiparticle momentum $d{\bf p}$ due to the action of the electric and magnetic forces, \[force\] [**F**]{}\_[em]{}=e[**E**]{}+(e/c) [**v**]{}, during the short time interval at the expense of the given source, \[dt\] (t)=\_[t\_i]{}\^t d[**p**]{}(t\_i) . Equation (\[dt\]) is considered on an event-by-event basis and for each event the mean momentum increase during a time-step $\langle
d{\bf p}(t_i) \rangle$ is calculated over all particles involved. In Fig. \[Dp-150\] the average momentum change of forward moving quarks $(p_z>0)$ is shown for three components of the electromagnetic force at $\sqrt{s_{NN}}=$ 200 GeV. Note the different scales for the solid lines in Fig. \[Dp-150\] that give the net momentum change at this energy. It is a remarkable fact that the transverse electric and magnetic components compensate each other almost completely.
Two remarks are in order: First, due to the linearity of the electromagnetic force (\[force\]) with respect to the electric and magnetic field, one should not expect a difference in quark transport calculations with and without taking into account electromagnetic field fluctuations. This was demonstrated for quasiparticles earlier in terms of the HSD model [@BES-HSD]. Second, if transverse fluctuations are characterized by the average strength of the fields, $\langle|E_{x,y}|\rangle$ and $\langle|B_{x,y}|\rangle$, certain equalities between components like $\langle|E_{x}|\rangle\approx
\langle|E_{y}|\rangle\approx \langle|B_{x}|\rangle$ — as numerically obtained in Ref. [@BS11] and confirmed in Ref. [@DH12] — imply that similar equalities should hold for the fluctuations. Indeed, similar relations follow from our PHSD calculations, see Fig. \[Pfield\](a) where the increment functions for appropriate field components practically coincide. We emphasize again that the PHSD transverse field components are not only of comparable strength but their action on the quarks \[see Eq. (\[force\])\] approximately compensate each other. One should note that this is the compensation effect rather than the short lifetime of the electromagnetic interaction which leads to a very weak sensitivity of observables as has been demonstrated recently in terms of the hadronic HSD transport model in Ref. [@BES-HSD]. For a quasiparticle moving along the trajectory $x=x(t)$, this compensation in a simplified 1D case can be illustrated by a short calculation as e E=-e \~-e \~-eBv , \[comp\] [*i.e.*]{}, the action of the electric and magnetic transverse components is roughly equal and directed oppositely.
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The important advantage of the PHSD approach relative to hadron-string models is the inclusion of partonic degrees of freedom. In particular, the involved partonic fields (of scalar and vector type) showed up to be essential to describe the elliptic flow excitation function from lower SPS to top RHIC energies [@KBCTV11; @KBCTV12] and to be a key quantity in analyzing the CME.
The evolution of momentum increments for partonic forces is presented in Fig. \[Dp-cromo-comp\] for off-central Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV. It is seen that (marked by the subscript $``c"$) the transverse ‘electric’ $E_c$ and ‘magnetic’ field $B_c$ of the partonic field components \[Fig. \[Dp-cromo-comp\](a),(b)\] almost compensate each other. The $z$ component is practically vanishing and for $t\gsim 8$ fm/c all quark increments stay roughly constant, [ *i.e.*]{}, the quark phase ends here. The final action of the partonic forces is defined by the sum of the forces (d) which is dominated by the scalar one.
Apart from the average forces (momentum increments) the fluctuations of the forces are of further interest. As seen from Fig. \[dp1\] the distribution in the quark momentum deviation $\delta {\bf p}={\bf
p}-\langle {\bf p}\rangle$ in case of scalar forces is well collimated with respect to the average trajectory $\langle {\bf
p}\rangle$ presented in Fig. \[Dp-cromo-comp\] but its width increases by about a factor of three when proceeding from $t=$ 0.05 to 3.0 fm/c \[Fig. \[dp1\](a)-(c)\]. This spread is slightly larger in the $x$ component since the derivatives of the scalar mean-field are higher in the $x$ than in the $y$ direction. The influence of the electromagnetic force on quarks and charged pions is visible more clearly \[Fig. \[dp1\](d)-(i)\] in the early time corresponding to the maximal overlap of the colliding nuclei ($t=$ 0.05 fm/c) when the created electromagnetic field is maximal. Here, the $\langle \delta
p_x\rangle$ component is shifted for quarks (d)-(f) and even more for mesons (g)-(i). This shift decreases in time and disappears for $t=$ 3 fm/c; at this time the deviation distributions for all three components of the electromagnetic force are close to a $\delta$ function.
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Some general considerations on parity violation in heavy-ion reactions are in order here: Since the magnetic field is odd under time reversal (or equivalently, under the combined charge conjugation and parity ${\cal CP}$ transformation), the time reversal symmetry of a quantum system is broken in the presence of an external magnetic field. A magnetic field ${\bf B}$ can also combine with an electric field ${\bf
E}$ to form the Lorentz invariant $({\bf E}\cdot {\bf B})$ which changes the sign under a parity transformation. In the normal QCD vacuum with its spontaneously broken chiral symmetry the leading interaction involves the invariant $({\bf E}\cdot {\bf B})$ which enters [*e.g.*]{}, into the matrix element that mediates the two-photon decay of the neutral pseudoscalar mesons. In the deconfined chirally symmetric phase of QCD, the leading interaction term is proportional to $\alpha\alpha_s({\bf E}\cdot {\bf B})({\bf E^a}\cdot
{\bf B^a})$, where ${\bf E^a}$ and $ {\bf B^a}$ denote the chromoelectric and chromomagnetic fields, respectively, and $\alpha_s$ is the strong QCD coupling. Both interactions are closely related to the electromagnetic axial anomaly, which in turn relates the divergence of the isovector axial current to the pseudoscalar invariant of the electromagnetic field (see Ref. [@MS10]). The evolution of the electromagnetic invariant ${\bf E}\cdot {\bf B}$ is shown in Fig. \[EtoB\]. The case of Au+Au ($\sqrt{s_{NN}}=$ 200 GeV) collisions at impact parameter $b=$ 10 fm is considered. As seen from Fig. \[EtoB\] the electromagnetic invariant $({\bf E}\cdot {\bf B})$ is non-zero only in the initial time $t\lsim$ 0.5 fm/c where the $({\bf E}\cdot {\bf B})$ distribution is quite irregular and its nonzero values correlate well with the location of the nuclear overlap region. For later times this electromagnetic invariant vanishes in line with the electric field space-time distributions [@EM_HSD]. Note that the quantities plotted in Fig. \[EtoB\] are dimensionless and the scaling factor $m_\pi^4 \ [\text{GeV}^4]$ is quite small.
One should note that in addition to the strong electromagnetic fields [@KMcLW07; @SIT09] present in noncentral collisions, very strong color electric ${\bf E}^a$ and color magnetic ${\bf H}^a$ fields are produced in the very beginning of these collisions as shown in the non-Abelian field theory [@Fu12]. These fields can be characterized by a gluon saturation momentum $Q_s$ and the time $\sim
1/Q_s$. Both fields are parallel to each other and directed along the $z$ axis. This leads to a nonzero topological charge $Q \sim ({\bf
E}^a\cdot {\bf B}^a)\ne$0. Since gauge fields with $Q\ne$ 0 generate chirality, they also can induce electromagnetic currents along a magnetic field [@FKW10] resulting in the CME. Though a large amount of topological charge might be produced through the mechanism of sphaleron transitions, the primary mechanism for topological charge $Q$ generation at the early stage is by fluctuations of color electric and magnetic fields. The decay of these fields is essentially governed by the non-Abelian dynamics of the glasma [@Fu12; @LMcL06] which ultimately produces the QGP (close to equilibrium). Unfortunately, this possible mechanism for the CME is beyond the potential of the PHSD model used. We thus may speculate about but not prove this mechanism.
Fluctuations in the position of participant nucleons
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As noted above (see Fig. \[tr-pl\]), the interaction region after averaging over many events has an almond-like shape; the averaged spatial initial asymmetry of the participant matter is symmetric with respect to the reaction plane. Actual collision profiles, however, are not smooth and the symmetry axis in an individual event is tilted due to fluctuations (cf. Fig. \[C2\]). The geometry fluctuations in the location of the [*participant*]{} nucleons lead to fluctuations of the participant plane (PP) from one event to another, rendering larger coordinate space eccentricities which due to pressure gradients are translated into elliptic flow for the final state particles. Thus, the system of the elliptical almond-like shape expands predominantly along the minor axis.
![(Color online) Projection of a single peripheral Au+Au (200 GeV) collision on the transverse plane. Spectator and participant nucleons are plotted by empty and filled circles, respectively. The reaction plane (RP) projection corresponds to $x$ axis. Transverse axes of the participant plane (PP) are marked by stars ($x^\star,y^\star$).[]{data-label="C2"}](coll.eps){width="\linewidth"}
Depending on the location of the participant nucleons in the colliding nuclei at the time of the collision, the actual shape of the overlap area may vary. As is seen from Fig. \[C2\], due to fluctuations the overlap area in a single event can have, for example, a rotated triangular rather than an almond shape. Note that an almond shape is regained by averaging over many events for the same impact parameter. However, in experiment the collective flows are measured with respect to a third plane, the so-called [*event*]{} plane defined by observable charged participants in momentum space through the harmonic/multipole analysis. More precisely, the flow coefficients $v_n$ are defined as the $n$th Fourier harmonic of the particle momentum distribution with respect to the particular momentum event plane $\Psi_n$, \[vn\] v\_n= , where $\psi = \arctan(p_y/p_x)$ is the azimuthal angle of the particle momentum ${\bf p}$ in the c.m. frame and angular brackets denote a statistical average over many events.
![(Color online) Distribution in the event plane angle for different harmonics $\Psi_n$ calculated with retarded magnetic and electric fields. Grey histograms show the results for respective calculations without fields.[]{data-label="Psi"}](EventPlane.eps){width="45.00000%"}
One should note that all azimuthal correlations are not only due to the collective flow. The early-time two-particle spatial correlations probe both the event geometry (fluctuating in individual events) and genuine local pair correlations referred to as ‘nonflow’ correlations. The Fourier decomposition (\[vn\]) is not enough to disentangle these two contributions. A possible solution of the connection between flow fluctuations and initial state correlations is given by the cumulant expansion method [@BDO01] using two- and four-particle correlation measurements of the harmonic flow coefficients. However, this method is beyond the scope of the present study.
The distributions in the event plane angle for different harmonics are shown in Fig. \[Psi\] for the freeze-out case. All distributions are symmetric with respect to the point $\Psi_n=0$ which corresponds to the true reaction plane. As is seen, the event plane angle $\Psi_n$ determined from the $n$th harmonic is in the range $0\leq \Psi_n <
2\pi/n$ and fluctuations of several lowest order harmonics have comparable magnitudes. Inside this region $\Psi_1$ has two maxima at $\Psi_n=0$ and $\pi$ corresponding to forward-backward emission. The even components $\Psi_2,\Psi_4$ have a rather prominent maximum for $\Psi_n =0$ indicating the local nature of fluctuations, but the odd harmonics $\Psi_3, \Psi_5$ are practically flat. This may be easily understood since the odd moments of the spatial anisotropy purely originate from fluctuations while the even ones are combined effects of fluctuations and geometry. As a consequence, if one defines the spatial anisotropy parameters with respect to the pre-determined reaction plane, the event-averaged ones vanish for all odd moments but not for the even moments.
The histograms in Fig. \[Psi\] are calculated from a sample of $3\times 10^4$ events taking into account magnetic and electric field fluctuations. Similar calculations without fields are shown in the same figure by the grey histograms which are hard to distinguish from the previous ones. In other words, there is no additional “tilting” effect by electromagnetic fields as expected in Refs. [@BS11; @DH12]. This is due to the compensation of the transverse electromagnetic components as explained above.
Two-particle angular correlations
---------------------------------
An experimental signal of the local spontaneous parity violation is a charged particle separation with respect to the reaction plane [@Vol05]. It is characterized by the two-body correlator in the azimuthal angles, $$\begin{aligned}
\label{cos}\gamma_{ij}&\equiv&\langle \cos (\psi_i+\psi_j-2\Psi_{RP})
\rangle \\ \nonumber &=&\langle \cos (\psi_i-\Psi_{RP})\ \cos (\psi_j -\Psi_{RP})\rangle
\\ \nonumber
&-& \langle\sin (\psi_i-\Psi_{RP})\ \sin (\psi_j-\Psi_{RP})\rangle\end{aligned}$$ where $\Psi_{RP}$ is the azimuthal angle of the reaction plane defined by the beam axis and the line joining the centers of the colliding nuclei and subscripts of $\gamma_{ij}$ represent the signs of electric charges being positive or negative. The averaging in Eq. (\[cos\]) is carried out over the whole event ensemble and $\cos$ and $\sin$ terms in Eq. (\[cos\]) correspond to out-of-plane and in-plane projections of $\gamma_{ij}$.
As was proposed in Refs. [@Pr10-1; @Pr09] and more elaborated in Ref. [@BKL10], a possible source of azimuthal correlations among participants is the conservation of the transverse momentum which might give rise to a contribution comparable with the measured CME. Transverse momentum conservation (TMC) introduces back-to-back correlations for particle pairs because they tend to balance each other in transverse momentum space. A large multiplicity of particles will dilute the effect of these two-particle correlation. Furthermore, this correlation should be stronger in plane than out of plane due to the presence of the elliptic flow. Nevertheless, the TMC provides a background for the CME that should be properly quantified.
From quite general considerations — making use of the central limit theorem and describing particles thermodynamically — one can derive the following simple expression for the two-particle correlator [@BKL10]: \[TMC\] \_[ij]{} = (\_i+\_j)=- , where $N=N_++N_0+N_-$ is the total number of all produced particles (in full phase space). At $\sqrt{s_{NN}}=$ 200 GeV it can be approximated as $N \approx (3/2)\ N_{ch} \approx 21
\ N_{part}$ [@BKL10] where $N_{part}$ is calculated dynamically in our model as well as the momentum-dependent factors for full phase space and the ratio to the measured accepted phase space. It is of interest to note that the proportionality of the CME to the elliptic flow $v_2$ seen in Eq. (\[TMC\]) follows also from more elaborated considerations. In particular, the chiral magnetic effect in the hydrodynamic approach and in terms of a holographic gravity dual model (see Ref. [@GKK12]) predicts a linear dependence of the CME on $v_2$ with more sophisticated coefficients which depend on the axial anomaly coefficient and the axial chemical potential as well as on dynamics of fluids through the particle density, baryon chemical potential and pressure.
![(Color online) PHSD centrality dependence of the elliptic flow (a) and angular correlators $\gamma_{ss}$ and $\delta_{ss}$ of charged particles from Au+Au at $\sqrt{s_{NN}}=$ 200 (b) and 7 (c) GeV from the transverse momentum conservation according to Eqs. (\[ss\]) and (\[delta\]). The experimental data points for $ v_2$ and $\gamma_{ss},\ \delta_{ss}$ are from Refs. [@Back:2004mh] and [@BES11], respectively.[]{data-label="TMC-cos200"}](approx200flow.eps "fig:"){width="40.00000%"} ![(Color online) PHSD centrality dependence of the elliptic flow (a) and angular correlators $\gamma_{ss}$ and $\delta_{ss}$ of charged particles from Au+Au at $\sqrt{s_{NN}}=$ 200 (b) and 7 (c) GeV from the transverse momentum conservation according to Eqs. (\[ss\]) and (\[delta\]). The experimental data points for $ v_2$ and $\gamma_{ss},\ \delta_{ss}$ are from Refs. [@Back:2004mh] and [@BES11], respectively.[]{data-label="TMC-cos200"}](approx200cme.eps "fig:"){width="40.00000%"} ![(Color online) PHSD centrality dependence of the elliptic flow (a) and angular correlators $\gamma_{ss}$ and $\delta_{ss}$ of charged particles from Au+Au at $\sqrt{s_{NN}}=$ 200 (b) and 7 (c) GeV from the transverse momentum conservation according to Eqs. (\[ss\]) and (\[delta\]). The experimental data points for $ v_2$ and $\gamma_{ss},\ \delta_{ss}$ are from Refs. [@Back:2004mh] and [@BES11], respectively.[]{data-label="TMC-cos200"}](approx7cme.eps "fig:"){width="40.00000%"}
Experimentally, the same-sign correlator $\gamma_{ss}$ is defined as the average of $\gamma_{++}$ and $\gamma_{--}$ by assuming that the momentum balance is shared equally among the charges \[ss\] \_[ss]{}=(\_[++]{}+\_[–]{})=- , where the subscripts “full" and “acc" imply that average values should be calculated in the “full" phase space or in the proper “acceptance region", respectively. In practice, only a subset of particles is measured. In this case some of the momentum balance stems from unmeasured particles and one might expect $\gamma_{ss} \ll
v_2/N$ [@Pr09]. In the STAR experiment [@Aggarwal:2010ya] tracks were measured for the central two units of rapidity. However, the initial colliding beams approached with $\pm 5.5$ units of rapidity and more than 50% of the charged particles tracks have rapidities outside the STAR acceptance. These particles can serve as a source of momentum, which may quench the momentum conservation condition thus reducing the magnitude of $\gamma_{ss}$. However, the transverse momentum of a given track is more likely to be balanced by neighboring particles, which have similar rapidities. This is particularly true when considering the components of the momenta responsible for elliptic flow. We conclude that this effect should be more essential for lower collision energy.
The direct comparison of the momentum conservation effect (\[ss\]) on the CME observable is presented in Fig. \[TMC-cos200\] for the top RHIC energy. The total (rather than transverse) momentum conservation is inherent in the PHSD model. In the actual calculations the experimental acceptance $p_t>$ 200 MeV/c is taken into account; as seen from the upper part of Fig. \[TMC-cos200\] the centrality dependence of the elliptic flow $\langle v_2\rangle$ for charged particles is rather well reproduced by PHSD. However, the experimental same-sign correlator $\gamma_{ss}$ is underestimated substantially. We note that the experimental acceptance essentially influences the momentum-dependent ratio $\langle p_t\rangle^2_{acc}/\langle
p_t^2\rangle_{full}$. In reality the difference in $\gamma_{ss}$ should be even larger as discussed above. This point is in agreement with the full HSD calculation of the hadronic background within the CME studies in Ref. [@BES-HSD].
A similar analysis for the lower energy $\sqrt{s_{NN}}=$ 7.7 GeV is presented in Fig. \[TMC-cos200\](c). Unfortunately, measured data for the centrality dependence of the elliptic flow are not available at this energy but the PHSD calculated average $\langle v_2 \rangle$ for minimum bias collisions is only slightly below the experiment [@KBCTV12] due to neglecting a baryon mean-field potential (see also the end of Sec. \[ch-fl\]). The calculated correlation $\gamma_{ss}$ strongly differs from the measured values having even the opposite sign. One should note that in this case the same and opposite sign components are almost equal to each other ([ *i.e.*]{}, there is no charge separation effect). This observation is also nicely reproduced within the HSD model at this energy (cf. Ref. [@BES-HSD]).
It is of further interest to consider the average cosine of the transverse angle difference which is independent of the reaction plane \[delta\] \_[ij]{} (\_i-\_j)=- , where the last equality is obtained from the transverse momentum conservation [@BKL10]. As follows from the comparison between Eqs. (\[TMC\]) and (\[delta\]), the correlator $\delta_{ij}$ differs from $\gamma_{ij}$ only by the elliptic flow coefficient $v_2$ and is expected to be more sensitive to the TMC. As one can see from Fig. \[TMC-cos200\] this estimate of $\delta_{ss}$ is too large and hardly consistent with appropriate experimental data from Fig. \[CMEm\].
Thus, the considered angular correlation $\gamma_{ss}$ is generated by a combination of momentum conservation, which causes particles to be preferably generated in the opposite direction, and elliptic flow which gives more particles in the $\pm x$ direction than in the $\pm
y$ direction. However, this source is by far not able to explain the observed pion asymmetry in the angular correlation. In addition, the considered TMC is blind to the particle charge and cannot disentangle same-sign and opposite-sign pair correlations.
Electric charge fluctuations in the transient stage {#ch-fl}
---------------------------------------------------
The almond-like fireball created in the early collision phase then expands in an anisotropic way, however, the spatial anisotropy is reduced with increasing time. In this transient stage the electromagnetic field is strongly reduced since the spectator matter is flying away from the formed fireball. The pressure gradients act predominantly in the reaction plane resulting in elliptic flow $v_2$.
Strong interactions in this phase might produce significant fluctuations in energy density (temperature), transverse momentum, multiplicity and conserved quantities such as the net charge. In the plasma phase in a magnetic field an electric quadrupole can be formed due to chiral anomaly and as a signal of that the elliptic flow difference between $\pi^+$ and $\pi^-$ mesons is predicted $v_2(\pi^+)< v_2(\pi^-)$ [@BKLY11]. Certainly, to check that the influence of hadronic transport on observables should be taken into account.
Furthermore, the CME [@KMcLW07] predicts that in the presence of a strong electromagnetic field at the early stage of the collision the sphaleron transitions in a hot and dense QCD matter induce a separation of charges along the direction of the magnetic field which is perpendicular to the reaction $(x-z)$ plane. This charge separation results in the formation of an electric dipole in momentum space which breaks parity. Being interested essentially in the quark phase, we investigate in this subsection to what extent such an electric dipole can be generated by background statistical and electromagnetic field fluctuations.
![(Color online) Probability distribution in the magnitude $Q_{c1}$ of the generated electric dipole at freeze-out for all partons. The panels (a) and (b) correspond to calculations without and with the electromagnetic field, respectively. The system is Pb + Pb at $\sqrt{s_{NN}}=$ 200 GeV at impact parameter $b=$ 10 fm.[]{data-label="Q-c1"}](Q-mf0.eps "fig:"){width="45.00000%"} ![(Color online) Probability distribution in the magnitude $Q_{c1}$ of the generated electric dipole at freeze-out for all partons. The panels (a) and (b) correspond to calculations without and with the electromagnetic field, respectively. The system is Pb + Pb at $\sqrt{s_{NN}}=$ 200 GeV at impact parameter $b=$ 10 fm.[]{data-label="Q-c1"}](Q-mf5.eps "fig:"){width="45.00000%"}
Let us quantify the dipole defining the plane $\hat {Q}_{c1}$ of the quark distribution in the transverse momentum space. The magnitude $Q_{c1}$ and azimuthal angle $\Psi_{c1}$ of this vector can be determined in a given event as follows: Q\_[c1]{} \_[c1]{}=\_i q\_i \_i ,\
Q\_[c1]{} \_[c1]{}=\_i q\_i \_i , \[Q1\] where the summation runs over all charged particles in the event with the electric charge $q_i$ and azimuthal angle $\psi_i$ of each particle. Note that Eq. (\[Q1\]) describes the dipole shape of charged particles (quarks or hadrons) without any reference to the charge separation.
As seen from Fig. \[Q-c1\], the average magnitude of the electric dipole $\bar{Q}_{c1}$ at the moment of the maximum nuclear overlap ($t=$ 0.05 fm/c) is about 4 charge units with dispersion $\sigma_{Q1}\approx 2$. At this moment the system is in the quark phase having on average an almond-like shape and therefore is expanding preferentially along the $x$ axis. Note that according to Eq. (\[Q1\]) quark net electric charges with $|q_i|<1$ are considered. Thus, the number of quarks involved in the dipole is large. In the next step ($t=0.1$ fm/c) of the expansion stage (dot-dashed line in Fig. \[Q-c1\]) the $Q_{c1}$ distribution is getting broader with a noticeable increase of $\bar{Q}_{c1}$. At $t=$ 10 fm/c the quark-gluon phase transforms predominantly into the hadronic phase through the dynamical coalescence mechanism and the $Q_{c1}$ distribution becomes narrower again. The influence of the electromagnetic field on this evolution is very weak \[compare the top (a) and bottom (b) panels in Fig. \[Q-c1\]\].
![(Color online) Electric dipole evolution of charged partons in $Q_{c1}-t$ presentation for Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV with taking into account the electromagnetic field. []{data-label="Q-c1-t"}](Q-3d-mf5.eps){width="45.00000%"}
The whole evolution of the electric dipole is seen more clearly in the 3D representation in Fig. \[Q-c1-t\]. Indeed, the $Q_{c1}$ distribution has a pronounced peak formed shortly after the collision, then the magnitude of the $Q_{c1}$ charge distribution is minimal during about 3 fm/c to testify that a large-in-charge subsystem is formed. After that the $Q_{c1}$ distribution is getting narrower because during the expansion the $Q_{c1}$ value slightly decreases due to parton hadronization; the maximum of the probability distribution increases and then stabilizes after $t\approx$ 10 fm/c.
![(Color online) Distribution in the magnitude of the charge dipole (a) and quadrupole (b) calculated for charged particles at times $t=$ 0.05, 1, 10, and 40 fm/c. The system is Au + Au at $\sqrt{s_{NN}}=$ 200 GeV and impact parameter $b=$ 10 fm.[]{data-label="Q1-Q2"}](dipQ1.eps "fig:"){width="45.00000%"} ![(Color online) Distribution in the magnitude of the charge dipole (a) and quadrupole (b) calculated for charged particles at times $t=$ 0.05, 1, 10, and 40 fm/c. The system is Au + Au at $\sqrt{s_{NN}}=$ 200 GeV and impact parameter $b=$ 10 fm.[]{data-label="Q1-Q2"}](dipQ2.eps "fig:"){width="45.00000%"}
Similarly to the dipole, one can define a charged quadrupole formed in heavy-ion collisions as follows: Q\_[c2]{} 2\_[c2]{}=\_i q\_i 2\_i ,\
Q\_[c2]{}2\_[c2]{}=\_i q\_i 2\_i . \[Q2\]
![(Color online) The same as in Fig. \[Q1-Q2\] but for the distribution in the event plane angle of the charge dipole (a) and quadrupole (b) at times $t=$ 0.05, 1, 10, and 40 fm/c.[]{data-label="Psi1-2"}](dipPsi1.eps "fig:"){width="45.00000%"} ![(Color online) The same as in Fig. \[Q1-Q2\] but for the distribution in the event plane angle of the charge dipole (a) and quadrupole (b) at times $t=$ 0.05, 1, 10, and 40 fm/c.[]{data-label="Psi1-2"}](dipPsi2.eps "fig:"){width="45.00000%"}
Characteristics of the time evolution of the electric dipole and quadrupole (formed in semicentral Au+Au at $\sqrt{s_{NN}}=$ 200 GeV collisions) for all (parton and hadrons) charged quasiparticles in the midrapidity range are presented in the next two figures. The $Q_{c1}$ minimum observed in the pure partonic case (see Figs. \[Q-c1\] and \[Q-c1-t\]) survives also in this case. The magnitude of the $n=2$ quadrupole harmonic (presented in Fig. \[Q1-Q2\]) is close to that for the dipole $n=1$, [*i.e.*]{}, $Q_{c2}\approx Q_{c1}$ and their maximal values extend to values of 30–40. As is seen from Fig. \[Psi1-2\], the distributions in the reaction plane angle for the electric quadrupole $\Psi_{c2}$ are rather flat during the whole evolution while the electric dipole angle $\Psi_{c1}$ distribution is flat only in the partonic phase (see $t=$ 0.05 fm/c in Fig. \[Psi1-2\]) but in the hadronic phase the distribution resembles that for the directed flow ([*c.f.*]{} Fig. \[Psi\]). The main axis of $\Psi_{c1}$ and $\Psi_{c2}$ can randomly be parallel or antiparallel to the minor axis of the almond. Like in the dipole case (cf. Fig. \[Q-c1\]) the electromagnetic field has no sizable influence on the characteristics of the electric quadrupole. When the collision energy $\sqrt{s_{NN}}$ decreases the behavior of the dipole and quadrupole distributions in the magnitudes $Q_{c1}$ and $Q_{c2}$ and the angle practically do not change besides some structure in the $\Psi_{c1}$ distribution. As seen from Fig. \[Qc-11\] at $\sqrt{s_{NN}}=$ 11.5 GeV back-to-back correlations — as specific for the directed flow — are manifested. This is mainly due to the proton contribution which becomes noticeable at low collision energy.
![(Color online) Angular distribution of charged particles for the charge quadrupole (solid line) and dipole (dashed) subsystems calculated for Au+Au ($\sqrt{s_{NN}}=$ 11.5 GeV) collisions. []{data-label="Qc-11"}](dipPsi11GeV.eps){width="48.00000%"}
Thus, the statistical fluctuations of “normal” matter in the presence of the retarded electromagnetic field do not result in a sizable formation of a deformed subsystem of dipole- or quadrupole-shape during the evolution of the heavy-ion collision.
We point out, however, that such subsystems might be formed in nontrivial topological systems due to the chiral anomaly effect. In particular, it happens when a quark experiences both a strong magnetic field and a topologically nontrivial gluonic field such as an instanton [@BDK12]. The inherent asymmetry — when both instanton and magnetic field are present — can lead to the development of an electric dipole moment. Physically, it can be understood as the result of two competing effects: the spin projection produced by a magnetic field and the chirality projection produced by an instanton field. Such a consideration is beyond the scope of our present microscopic study.
It is of interest that the axial anomaly in a strong external magnetic field induces not only the CME but also the separation of the chiral charge. The coupling of the density waves of electric and chiral charge results in the ‘chiral magnetic wave’ and can induce a static quadrupole moment of the electric charge density [@BKLY11]. This chiral magnetic wave results in the degeneracy between the elliptic flows of positive and negative pions leading to $v_2(\pi^-)>
v_2(\pi^+)$, which was estimated theoretically on the level of $\sim
30\%$ for midcentral Au+Au collisions at $\sqrt{s_{NN}}=$ 11 GeV [@BKLY11]. Our PHSD calculations give about 6% which is quite comparable with the recently measured value of 10% [@Moh11] and essentially smaller than the prediction of Ref. [@BKLY11]. Noteworthy that the $v_2$ degeneracy in the PHSD version used is only due to different elastic and inelastic cross sections for $\pi^+$ and $\pi^-$ mesons but without taking into consideration the (small) mean-field pion-nucleus potential. The elliptic flow analysis of the difference between particles and antiparticles (including kaons and baryons alongside with pions) shows that this difference is coming mainly from the hadronic mean-field potential [@XCKL12]. Recently these $v_2$ data have been also successfully explained in terms of a hybrid model, which combines the fluid dynamics of a fireball evolution with a transport treatment of the initial and final hadronic states [@SKB12]. Therefore, there is not much room for the contribution from a transient charged quadrupole due to the chiral magnetic wave.
Charge balance functions
------------------------
In the formation of the charged dipole and quadrupole there is no information about a possible charge separation which could result in an electric driving force. In principle such information can be provided by the balance function which is based on the idea that charge is locally conserved when particles are produced pair-wise. In the subsequent expansion of the system and rescattering of the charge carriers, which in principle can be hadronic or partonic, the balancing partners are then spread out within some finite distance to each other. The original correlation in space-time transforms into a correlation in momentum space in the final hadronic emission profile. Therefore, the motion of the balancing partners suffers from the collective expansion of the system and diffusion due to the collisions with other particles. The study of charge-balance correlations hence gives insight into the production and diffusion of charge. In particular, it is expected that the balance function is sensitive to the delayed transition of the quark-gluon phase to a hadronic phase [@BDP00].
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Thus, whenever a positive charge is created, a negative charge arises from the same point in space-time and both particles then tend to be focused in the same rapidity and azimuthal angle by collective flow. This results in a correlation between positive and negative charges, [*i.e.*]{}, for every positively charged particle emitted at an angle $\psi_+$, there tends to be a negatively charged particle emitted with $\psi^-\approx\psi^+$ and similar rapidity. Charge balance functions [@BDP00] represent a measure of such correlations, and have already been investigated as a function of relative rapidity for identified particles and for relative pseudorapidity $\eta$ for nonidentified particles [@Aggarwal:2010ya; @STAR-BF03]. Generally, the balance function $B(p_a|p_b)$ is a six-dimensional function of the particle momenta. In the context of studies of the separation of balancing charges, the discussion is reduced to the difference $({\bf p}_1 -
{\bf p}_2)$. In particular we will focus on the charge balance function in relative pseudorapidity $\delta\eta$, [*i.e.*]{}, $B(p_a|p_b) \to B(\delta \eta,\eta_w)$ and similarly for the azimuthal angle $\psi$ [@Bo05].
Charge balance functions are constructed in such a way that like-sign subtractions statistically isolate the charge balancing partners, \[bal.f\] B(,\_w)&=& ( .\
&+&. ), where the conditional probability $N_{+-}(\delta\eta,\eta_w )$ counts pairs with opposite charge which satisfy the criteria that their relative pseudorapidity $\delta\eta=\eta_+-\eta_-$ in a given pseudorapidity window is $\eta_w$, $(\delta\eta \in \eta_w)$, whereas $N_+ (N_-)$ is the number of positive (negative) particles in the same interval. Similarly for $N_{++}$, $N_{--}$, and $N_{-+}$. The factor 1/2 ensures the normalization of $B(\delta \eta, \eta_w)$. All terms in Eq. (\[bal.f\]) are calculated within PHSD using pairs from a given event and the resulting distributions are summed over all events.
Both balance-function and charge-fluctuation observables are generated from one-body and two-body observables which necessitates that they may be expressed in terms of spectra and two-particle correlation functions. The charge fluctuation is a global measure of the charge correlation and the balance function is a differential measure of the charge correlation; it therefore carries more information. Writing $N_{\pm}= \langle N_{\pm}\rangle_{\eta_w}+\delta N_\pm$, where $\langle \dots \rangle_{\eta_w}$ denotes the average in the phase-space region $\eta_w$, it is easy to show [@JP02] that \[dif\] 1 - \_0\^[\_w]{} d B(|\_w ) , where $Q_{ch} = N_+ - N_-$ and $N_{ch} = N_+ + N_-$.
From this example, one can readily understand how balance functions identify balancing charges. For any positive charge, there exists only one negatively charged particle whose negative charge originates from the point at which the positive charge was created. By subtracting from the numerator the same object created with positive-positive pairs, one is effectively subtracting the uncorrelated contribution from the distribution.
It is expected that charge balance functions are sensitive to the separation of balancing charges in momentum space and give insight into the dynamics of hadronization [@BDP00]. Indeed, such a pair is composed of a positive and negative particle (or particle and antiparticle) whose charge originates from the same point in space-time. According to [@BDP00], if a quark-gluon plasma results in a large production of new charges (quark-antiquark pairs) late in the reaction, a tight correlation between the balancing charge-anticharge pairs would provide evidence for the creation of this novel state of matter.
The time evolution of the $\delta \eta$- and $\delta\phi$-dependent charge balance function is demonstrated in Fig. \[BF-t\] for central \[(a),(b)\] and peripheral \[(c),(d)\] Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV. The times $t=$ 0.05 and 1 fm/c correspond to the developed quark phase which ends at about $t=$ 10 fm/c (cf. Fig. \[Dp-cromo-comp\]) while at $t=$ 40 fm/c the system is in a purely hadronic phase. We recall that hadronization in the PHSD model is realized via a crossover transition and quasiparticle rescattering is included in both the partonic and hadronic phase.
As follows from Fig. \[BF-t\], there is a very small difference in the time evolution of charge balance function $B(\delta \eta)$ at both centralities. As to $B(\delta \phi)$ a rather clear enhancement is seen for central collisions at $\delta\eta\sim\delta\phi\sim 0$ while this dependence is essentially weaker for peripheral collisions. This observation is in qualitative agreement with experiment but enhancement effect is too small. Thus the expectation of a high sensitivity of the balance function to the hadron phase transition and particle diffusion seems somewhat too optimistic.
The direct comparison with experiment of the charge balance function is presented in Fig. \[BF\] for Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV calculated within PHSD for central and peripheral ($b=$ 10 fm) collisions. Here the charge conservation law is locally fulfilled in each quasiparticle collision. Hadronic resonance decays are taken into account in PHSD but corrections due to final state interactions for small $\delta\phi$ are computationally very involved (and uncertain).
![(Color online) The balance function for pseudorapidity (a) and azimuthal angle (b) of charged pions with $|\eta_{+/-}|<1$ and $p_t>$ 0.2 GeV/c from central and peripheral Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV. The dotted lines for $\delta \phi$ correspond to calculations including the electromagnetic field effects. The experimental data points are from Ref. [@Aggarwal:2010ya].[]{data-label="BF"}](bal_eta.eps "fig:"){width="45.00000%"} ![(Color online) The balance function for pseudorapidity (a) and azimuthal angle (b) of charged pions with $|\eta_{+/-}|<1$ and $p_t>$ 0.2 GeV/c from central and peripheral Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV. The dotted lines for $\delta \phi$ correspond to calculations including the electromagnetic field effects. The experimental data points are from Ref. [@Aggarwal:2010ya].[]{data-label="BF"}](bal_phi.eps "fig:"){width="45.00000%"}
As is seen from Fig. \[BF\] there is a maximum for $\delta \eta \sim
\delta \phi \sim 0$ but no large difference is observed for different centralities (apart from the $\delta \phi$ distribution). This finding is in agreement with the kinetic results of UrQMD and HIJING model calculations [@Aggarwal:2010ya] of the width of the balance function in terms of $\delta \eta$ which also show no narrowing of the peak for central collisions as observed in experiments. The inclusion of the electromagnetic field \[the dotted line in Fig. \[BF\](b) for $b=$ 10 fm\] practically shows no influence and does not result in any additional focusing in central collisions.
We find that PHSD does not describe quantitatively the experimental balance functions. There are some claims that the blast-wave model can resolve this discrepancy [@Pr10-1; @Pr10; @Aggarwal:2010ya]. The blast-wave model is in fact a parametrization of the kinetic freeze-out configuration motivated by a hydrodynamical model for the system described in local thermal equilibrium. The system is then completely characterized by the collective velocity profile, freeze-out temperature, and the freeze-out surface which is usually associated with some volume. Generally, the blast-wave model parameters may be varied in a large parameter space to fit experimental data. These single-particle freeze-out properties can, [*e.g.*]{}, be parametrized as suggested in [@BW-STAR05] to study, for example, the evolution of flow. However, the change in the kinetic freeze-out temperature and the increase of collective flow alone fail to explain the observed focusing of the balance function for more central collisions [@Pr10].
With regard to charge-balance correlations, the blast-wave model needs additionally to incorporate local charge conservation. This can be achieved in the following way: Instead of generating a single particle at a time, an ensemble of particles with exactly conserved charges is generated in such a way to remain unchanged the single-particle distributions. For the relative distribution of the pairs within an ensemble, a Gaussian distribution is assumed with dispersions $\sigma^2_\eta$ and $\sigma^2_\phi$ for rapidity and transverse angle, respectively. Treating these dispersions as free parameters at every centrality it is possible to tune the narrowing effect for central events [@Pr10]. It is of interest that at the exactly central collision $\sigma^2_\eta=\sigma^2_\phi=0$ and they strongly grow with impact parameter reaching $\sigma^2_\eta\approx 0.6$ and $\sigma^2_\phi/\pi\approx 0.4$ for centralities about 70% [@Pr10]. However, the additional assumption that balancing charges at the freeze-out are strongly correlated in momentum space is in conflict with the basic model assumption on thermodynamic equilibrium of the system.
In Ref. [@Pr10] the charge balance function was applied for the analysis of the CME. The charge separation between opposite-charge and same-charge two-pion correlators $\gamma_P$ was defined as \[pr\] \_P (2 \_[+-]{}- \_[++]{}- \_[–]{}) = \_[+-]{}- \_[ss]{} , where the angle brackets in Eq. (\[cos\]) include the balance function $B(p_+|p_-)$ as a weight factor for the balancing charges. The quantity $\gamma_P$ can be estimated from available experimental data [@STAR-CME]. Since the PHSD is not successful in reproducing the charge balance function, there is not much sense to apply it for the charge separation $\gamma_P$. As follows from the comparison between the STAR data and the blast-wave model (including correlations and rescaling) results in reproducing the experimental normalization; the charge balance correlations for the relativistic charge separation are of the same size as the experimental signal and exhibit a similar qualitative behavior with respect to the centrality dependence [@Pr10]. The authors of Ref. [@Pr10] claim that their results are solid on the level of 10–20%. The calculation of uncertainties originates predominantly from the particular parametrization of both the blast-wave model itself and, in particular, of the centrality dependence of the charge separation $B(p_a|p_b)$ in the azimuthal angle. However, the considered breakup physics differs significantly from more realistic scenarios as has been shown recently in Ref. [@Kumar:2012xa]; the freeze-out temperature and baryon chemical potential — defining the chemical composition of the system — noticeably depend on centrality. Furthermore, there is some inconsistency in using a reaction-plane independent fit of the balance function for the CME signal where azimuthal angles are measured with respect to the reaction plane. It is also unclear how the extracted parameters change with the collision energy, however, the first preliminary STAR data on the collision-energy dependence of the balance function have been published recently [@Wa11]. Thus, a further careful study of this issue is needed.
The CME observable {#Sec:Observable}
==================
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The experimental signal of the possible CME is the azimuthal angle correlator calculated according to Eq. (\[cos\]). The experimental acceptance $|\eta|<1$ and 0.20 $<p_t<2$ GeV has been also incorporated in the theoretical PHSD calculations. Note that the theoretical reaction plane is fixed exactly by the initial conditions rather than by a correlation with a third charged particle as in the experiment [@BES11]. Thus, within PHSD we calculate the observable (\[cos\]) as a function of the impact parameter $b$ or the centrality of the nuclear collisions which should be considered as a background of the CME signal. A comparison of the measured angular correlator with result of calculations is presented in Fig. \[C2Hpr\]. We mention that the calculation of this correlation is a very CP time consuming process and the proper statistical error bars are shown in Fig. \[C2Hpr\].
At the lowest measured energy $\sqrt{s_{NN}}=$ 7.7 GeV the results for oppositely and same-charged pions are very close to each other and show some enhancement in very peripheral collisions. The centrality distributions of $\gamma_{ij} $ are reasonably reproduced by the PHSD and HSD calculations presented in the same picture. Note that the scalar quark potential is not zero at this low energy but absent in the HSD model. The striking result is that the case of $\sqrt{s_{NN}}=$ 7.7 GeV drastically differs from $\sqrt{s_{NN}}=$ 200 GeV \[cf. panels (a) and (d) in Fig. \[C2Hpr\]\]. The picture quantitatively changes only slightly when one proceeds to $\sqrt{s_{NN}}=$ 11.5 GeV \[see the panel (b) in Fig. \[C2Hpr\]\] though the value of $\gamma_{ij}$ at the maximum (centrality 70%) decreases a little bit in the calculations. Experimental points at larger centrality are not available but are of great interest. In addition, one may indicate a weak charge separation effect in the experimental data because statistical error bars are very small (less than the symbol size). Unfortunately, the calculated error bars are rather large to specify the charge separation effect. The influence of the electromagnetic field here is negligible. The calculated and measured correlation functions for oppositely and same charged pions are shown in Fig. \[C2Hpr\] for the available three BES energies. The case for the top RHIC energy $\sqrt{s_{NN}}=$ 200 GeV is also presented for comparison.
If one looks now at the results for $\sqrt{s_{NN}}=$ 39 GeV, the measured same- and oppositely charged pion lines are clearly separated, being positive for the same-charged and negative for the oppositely charged pions to be strongly suppressed. The PHSD model is not able to describe this picture and overestimates the data with increasing energy. These growing large values of $\gamma_{ij}$ are due to the scalar parton potential which increases with the collision energy. The HSD version predicts a very small effect in qualitative agreement with our earlier analysis [@BES-HSD]. Though both models provide the charge separation essentially smaller than the measured one, the PHSD has a satisfying feature: the same-charge points are above the oppositely charged ones to be in agreement with experiment. The same situation is observed in the case of $\sqrt{s_{NN}}=$ 200 GeV; a small difference between them is seen in very peripheral collisions: the oppositely charged correlation jumps to zero at centrality $\sim$ 70% for $\sqrt{s_{NN}}=$ 39 GeV while corresponding data at 200 GeV are not available.
![(Color online) Angular correlations of opposite- and same-charge pions for the cosine of the difference in the azimuthal angles for Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV as a function of centrality. The experimental data points are from [@STAR-CME].[]{data-label="CMEm"}](cme200m.eps){width="42.00000%"}
Though the results at $\sqrt{s_{NN}}=$ 7.7 and 11.5 GeV roughly can be considered as a background of the CME, at higher energies it is impossible to identify the true effect of the local parity violation as the difference between measured and PHSD results. The PHSD model [@PHSD] includes directly the dynamics of quark-gluon degrees of freedom which are becoming more important with increasing energy. We recall that the growing importance of the repulsive partonic mean field — illustrated earlier by the rise of the elliptic flow explained convincingly in the PHSD model [@KBCTV11; @KBCTV12] — results here in an overestimation of the CME background.
![(Color online) Angular correlations of $\cos$ (out-of-plane) and $\sin$ (in-plane) projections for Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV as a function of centrality. The experimental data points are from [@STAR-CME].[]{data-label="CMEproj"}](termsS.eps "fig:"){width="45.00000%"} ![(Color online) Angular correlations of $\cos$ (out-of-plane) and $\sin$ (in-plane) projections for Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV as a function of centrality. The experimental data points are from [@STAR-CME].[]{data-label="CMEproj"}](termsO.eps "fig:"){width="45.00000%"}
In Fig. \[CMEm\] the results for the average cosine of the difference in the azimuthal angles $\delta_{ij}$ are presented. The measured centrality dependence for the same charge pions is flat and practically consistent with zero while that for oppositely charged particles is a monotonic increasing function with impact parameter. However, the PHSD calculations clearly overestimate the experimental points [@STAR-CME]. We note in passing that the PHSD results for Au+Au collisions at the energy $\sqrt{s_{NN}}=$ 200 GeV turn out to be astonishingly close to the appropriate experimental data at $\sqrt{s_{NN}}=$ 2.76 TeV [@Ch11; @LHC_CS12]. This fact indicates that the strength of the repulsive scalar quark potential in PHSD might be presently overestimated.
In accordance with Eq. (\[cos\]), one can separate the in-plane and out-of-plane components using experimental results for $\gamma_{ij}$ and $\delta_{ij}$. Such separation together with PHSD calculation results is presented in Fig. \[CMEproj\] for the same charge and opposite charge pions. As was first noted in Ref. [@BKL09] and is seen in Fig. \[CMEproj\](a), for the same-charge pairs the sinus term is essentially zero whereas the cosine term is finite. This tells us that the observed correlations are actually in-plane rather than out-of-plane. This is contrary to the expectation from the chiral magnetic effect, which results in same-charge correlations out of plane. In addition, since the cosine term is negative, the in-plane correlations are stronger for back-to-back pairs than for small angle pairs. The PHSD does not reproduce these features. We see also that for opposite-charge pairs the in-plane and out-of-plane correlations are virtually identical. As was stated in [@BKL09], this is difficult to comprehend since there is a sizable elliptic flow in these collisions. Nevertheless, the PHSD model predicts very close in-plane and out-of-plane distributions for opposite-charge pairs due to scalar parton potential and at the same time nicely reproduces the various harmonics of charged particles [@KBCTV11; @KBCTV12]. This feature is not reproduced in the HSD.
We close this section with some more general remarks. As follows from the results presented in Figs. \[C2Hpr\],\[CMEm\],\[CMEproj\] an additional sizable source of asymmetry is needed for both in-plane and out-of-plane components rather than only an out-of-plane component as expected from the CME. As discussed in the Introduction, the vacuum nontrivial topological structure (as a genuine source of the CME) leads to the picture of a topological $\theta$ vacuum of non-Abelian gauge theories. The $\theta$ term in the QCD Lagrangian explicitly breaks ${\cal P}$ and ${\cal CP}$ symmetries of QCD. However, stringent limits on the value of $\theta < 3\times 10^{-10}$ deduced from the experimental bounds on the electric dipole moment of the neutron [@Ba06] practically indicate the absence of [*global*]{} ${\cal P}$ and ${\cal CP}$ violation in QCD. Reference to the [ *local*]{} ${\cal P}$- and ${\cal CP}$-odd effects due to the topological fluctuations characterized by an effective $\theta\equiv
\theta({\bf x},t)$ varying in space and time [@Kharzeev:2004ey] does not provide much hope. In addition, partons near the phase transition are not chiral (as typically assumed) but massive degrees of freedom in the PHSD in agreement with lattice QCD calculations. The finite mass of the partons washes out the chirality effect.
Summary and outlook {#Sec:Summary}
===================
In this study we have investigated several effects that might contribute to the observed chiral magnetic effect (CME) in relativistic nucleus-nucleus collisions on the basis of event-by-event calculations within the PHSD transport approach. The individual results can be summarized as follows:
- [Our study shows that fluctuations in the position of quasiparticles can manifest themselves in different interaction stages and in different ways. Since the electromagnetic field generated by [*spectators*]{} is dominant at the early stage, the fluctuation in their position results in a noticeable fluctuation in the strength of the electromagnetic field. However, the fluctuation spread is not so large as expected in the estimate from Ref. [@BS11] and its influence on observables is negligible; in particular, the event plane angle is not tilted due to these electromagnetic field fluctuations!]{}
- [Early time fluctuations in the position of [*participant*]{} baryons were discussed in the past as a source of the impact parameter fluctuation. Its influence survives till the freeze-out resulting in a considerable difference between the theoretical reaction plane and the measured event plane. This effect leads to an increase in the magnitude of the elliptic flow and generates nonvanishing odd flow harmonics.]{}
- [We have found out that within the PHSD model the retarded strong electromagnetic field — created during nucleus-nucleus collisions — turns out to be not so important as has been expected before. Similarly to the HSD results in Ref. [@EM_HSD], the electromagnetic field has almost no influence on observables. The reason is not a shortness of the interaction time, when the electromagnetic field is maximal, but the compensation of the mutual action of transverse electric and magnetic components. This compensation effect might be important, for example, if an additional induced electric field (as a source of the CME) is available in the system since this field will not be entangled due to other electromagnetic sources.]{}
- [Another important point emerging from the compensation effect of electric and magnetic forces is worth mentioning: A significance of an external magnetic field in astrophysics is largely accepted. There are many studies where various effects of external magnetic fields are discussed in the application to astrophysics ([*e.g.*]{}, see the Introduction in Ref. [@EM_HSD] and references in [@GMS11]. It is correct in this particular problems, however, in many cases it is concluded by a statement like “the same effect should be observed in high-energy heavy-ion collisions” which does not hold true due the compensation effect as demonstrated in the present work.]{}
- [In the intermediate stage of the heavy-ion collision the statistical fluctuations of charged quasiparticles in momentum space can generate charge dipoles or even charge quadrupoles. However, the magnitudes $Q_{c1}$ and $Q_{c2}$ are small; their orientation is distributed almost uniformly and the direction of the main axis is changed from event to event. The influence of the electromagnetic field here is negligible again.]{}
- [The transverse momentum conservation — proposed as an alternative mechanism for an explanation of the observed azimuthal asymmetry — shows a correlation of the CME and the elliptic flow. However, the effect estimated at $\sqrt{s_{NN}}=$ 200 GeV is too small and insensitive to the charge separation.]{}
- [A possible charge separation of balancing charges has been addressed by the charge balance function. We note that the PHSD model fails to describe the focusing effect of the balance function for central Au+Au collisions. Certainly, further investigations of this problem are needed, both in theory and experiment especially at lower energies.]{}
The PHSD approach naturally takes into account the main alternative mechanisms of the CME: the momentum conservation and local charge conservation as well as clusters (mini jets, strings, prehadrons, resonances). At the moderate energies $\sqrt{s_{NN}}=$ 7.7 and 11.5 GeV the PHSD model results are close to the experiment since partonic degrees-of-freedom are subleading. However, at higher collision energy the PHSD model fails to reproduce the observed azimuthal asymmetry. In contrast with our earlier analysis within the HSD model [@BES-HSD], the PHSD overestimates the measured centrality dependence of azimuthal distributions due to an increasing action of the repulsive scalar parton potential which generates the collective flow harmonics in accordance with experiment. This finding suggests that a new source of azimuthal anisotropy fluctuation is needed beyond the ‘standard’ interactions incorporated in PHSD. The new source does not dominate in out-of-plane direction as could be expected for the CME but both in-plane and out-of-plane components contribute with a comparable strength. In this respect the interpretation of the CME STAR measurements is still puzzling.
The present PHSD model is already quite elaborated, however, as our analysis has shown, color degrees of freedom or intimate peculiarities of non-Abelian Yang-Mills theory should additionally be taken into consideration. In particular, this concerns the very early stage of the nuclear interaction. In this initial state the highly compressed strongly interacting matter is dense and though the QCD coupling constant is small, gluonic states have high occupation numbers, [ *i.e.*]{}, the partons begin to overlap in phase space which leads to some saturated state. Strong color forces might create strong chromoelectric and chromomagnetic fields producing a new state, a [ *glasma*]{} [@Fu12; @LMcL06; @La11] as mentioned above in context of the discussion in Sec. \[glasma\], or forming new objects like “string ropes” described in the framework of Yang-Mills theory [@MCS02]. We are planning to include these effects into the PHSD model in near future.
Another class of strong fields relevant to the chirality and confinement of dynamical quarks is the long range (or soft) vacuum gluon field configurations. Long range vacuum gluon fields can be seen as an origin of nonzero gluon condensate and topological susceptibility of QCD vacuum [@Shifman:1978bx; @Minkowski:1981ma]. Soft fields arise in the consideration of the global minima of the QCD effective action[@Leutwyler:1980ev] and are known to play an important role in hadron phenomenology at zero temperature [@Kalloniatis:2003sa]. The nonzero gluon condensate survives at high temperature as demonstrated by QCD lattice calculations [@D'Elia:2002ck]. Interplay of strong electromagnetic and vacuum long-range gluon fields can lead to the qualitatively new effects in high energy heavy ion collisions [@Galilo:2011nh]. However these effects are beyond the scope of this paper.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We are thankful to Che Ming Ko, Sergei Molodtsov, Sergei Nedelko, Oleg Teryaev, Sergei Voloshin, and Harmen Warringa for illuminating discussions. This work has been supported by the LOEWE Center HIC for FAIR, a Heisenberg-Landau grant, RFFI grant no. 11-02-01538-a, and by DFG.
[99]{} D.E. Kharzeev, L.D. McLerran and H.J. Warringa, Nucl. Phys. [**A803**]{}, 227 (2008). D. E. Kharzeev, Annals Phys. [**325**]{}, 205 (2010). K. Fukushima, D.E. Kharzeev and H.J. Warringa, Phys. Rev. [**D78**]{}, 074033 (2008). D. Kharzeev and A. Zhitnitsky, Nucl. Phys. [**A797**]{}, 67 (2007). D.E. Kharzeev and H.J. Warringa, Phys. Rev. [**D80**]{}, 034028 (2009). V. Skokov, A. Illarionov and V. Toneev, Int. J. Mod. Phys. [A **24**]{}, 5925 (2009). V. Voronyuk, V.D. Toneev, W. Cassing, E.L. Bratkovskaya, V.P. Konchakovski and S.A. Voloshin, Phys. Rev. [**C 83**]{}, 054911 (2011). I. Selyuzhenkov (for the STAR Collaboration), Rom. Rep. Phys. [**58**]{}, 049 (2006); S. Voloshin (for the STAR Collaboration), Nucl. Phys. [**A**830]{}, 377c (2009). B.I. Abelev, [*et al.*]{} (for the STAR Collaboration), Phys. Rev. [**C81**]{}, 054908 (2010). B. I. Abelev, [*et al.*]{} (for the STAR Collaboration), Phys. Rev. Lett. **103**, 251601 (2009). A. Ajitanand, S. Esumi and R. Lacey \[PHENIX Collaboration\], Proceedings of the RBRC Workshops, Vol. 96, 2010: “P- and CP-odd effects in hot and dense matter”; http://quark.phy.bnl.gov/$\sim$kharzeev/cpodd/ . D. Gangadharan (for the STAR Collaboration), J. Phys. G: Nucl. Part. Phys. [**38**]{}, 124166 (2011); Ilya Selyuzhenkov (for the ALICE Collaboration), PoS WPCF [**2011**]{}, 044 (2011). P. Christakoglou, J. Phys. [**G38**]{} 124165 (2011). ATLAS Collaboration, Phys.Rev. [**C86**]{} 014907 (2012). F. Wang, Phys. Rev. [**C81**]{}, 064902 (2010). A. Bzdak, V. Koch and J. Liao, Phys. Rev. **C81**, 031901 (2010). S. Pratt, arXiv:1002.1758. S. Schlichting and S. Pratt, arXiv:1005.5341; Phys. Rev. [**C83**]{}, 014913 (2011). S. Pratt, S. Schlichting and S. Gavin, Phys. Rev. [**C84**]{}, 024909 (2011). M. Asakawa, A. Majumder and B. Müller, Phys. Rev. [**C81**]{}, 064912 (2010). J. Liao, V. Koch and A. Bzdak, Phys. Rev. [**C82**]{}, 054902 (2010). A. Bzdak, V. Koch and J. Liao, Phys. Rev. [**C83**]{}, 014905 (2011). R. Longacre, arXiv:1112.2139. Quan Wang, arXiv:1205.4638. S.A. Voloshin, Phys. Rev. [**C 70**]{}, 057901 (2004). V. D. Toneev, V. Voronyuk, E. L. Bratkovskaya, W. Cassing, V. P. Konchakovski, S. A. Voloshin, Phys. Rev. [**C 85**]{}, 034910 (2012). V. Toneev and V. Voronyuk, Phys. Part. Nucl. Lett. [**8**]{}, 938 (2011); Phys. Part. Nucl. Lett. [**8**]{}, 938 (2011); Phys. Atom. Nucl. [**75**]{}, 607 (2012); Acta Phys. Pol. 5, 1001 (2012). L. Ou and B. A. Li, Phys. Rev. [**C 84**]{}, 064605 (2011). A. Bzdak and V. Skokov, Phys. Lett. [**B 710**]{}, 171 (2012). W.-T. Deng and X.-G. Huang, Phys. Rev. [**C 85**]{}, 044907 (2012). W. Cassing and E.L. Bratkovskaya, Phys. Rev. [**C78**]{}, 034919 (2008); W. Cassing and E.L. Bratkovskaya, Nucl. Phys. [**A831**]{}, 215 (2009). W. Cassing, Nucl. Phys. [**A791**]{}, 365 (2007). W. Cassing, Nucl. Phys. [**A795**]{}, 70 (2007). E. L. Bratkovskaya [*et al.*]{}, Nucl. Phys. [**A856**]{}, 162 (2011). Y. Aoki [*et al.*]{}, JHEP [**0906**]{}, 088 (2009). W. Ehehalt and W. Cassing, Nucl. Phys. [**A602**]{}, 449 (1996). W. Cassing and E.L. Bratkovskaya, Phys. Rep. [**308**]{}, 65 (1999). V.P. Konchakovski, E.L. Bratkovskaya, W. Cassing, V.D. Toneev and V. Voronyuk, Phys. Rev. [**C 85**]{}, 011902 (2012). V. P. Konchakovski, E. L. Bratkovskaya, W. Cassing, V. D. Toneev, S. A. Voloshin, and V. Voronyuk, Phys. Rev. [**C 85**]{}, 044922 (2012). L. D. Landau and E. M. Lifshitz, [*Electrodynamics of Continuous Media*]{}, Addison-Welsey, Reading, MA (1984), Sect. 58. K. Tuchin, Phys. Rev. [**C 82**]{}, 034904 (2010) \[Erratum-ibid. [**C 83**]{}, 039903 (2011)\]. K. Tuchin, J. Phys. G [**39**]{}, 025010 (2012). B. Müller and A. Schäfer, Phys. Rev. [**C 82**]{}, 057902 (2010). K. Fukushima, Acta Phys. Polon. [**B42**]{}, 2697 (2011). K. Fukushima, D.E. Kharzeev and H.J. Warringa, Phys. Rev. Lett. [**104**]{}, 212001 (2010). T. Lappi and L. McLerran, Nucl. Phys. [**A772**]{}, 200 (2006). N. Borghini, P. M. Dinh and J.-Y. Ollitrault, Phys. Rev. [**C63**]{}, 054906 (2001); Phys. Rev. [**C64**]{}, 054901 (2001). S. A. Voloshin, Phys. Lett. [**B632**]{}, 490 (2006); Nucl. Phys. [**A639**]{}, 287 (2005). I. Gahramanov, T. Kalaydzhyan and I. Kirsch, Phys. Rev. [**D 85**]{}, 126013 (2012). B. B. Back [*et al.*]{} (for the PHOBOS Collaboration), Phys. Rev. C [**72**]{}, 051901 (2005). M. M. Aggarwel [*et al.*]{} (for the STAR Collaboration), Phys. Rev. [**C 82**]{}, 024905 (2010). Y. Burnier, D.E. Kharzeev, J. Liao, and H.-U. Yee, Phys. Rev. Lett. [**107**]{}, 052303 (2011). G. Basar, G. V. Dunne, D. E. Kharzeev, Phys. Rev. [**D85**]{}, 045026 (2012). B. Mohanty (for the STAR Collaboration), J. Phys. [**G38**]{}, 124023 (2011). J. Xu, L.-W. Chen, Ch. M. Ko, and Zi-Wei Lin, Phys. Rev. [**C 85**]{}, 041901(R) (2012). J. Steinheimer, V. Koch and M. Bleicher, Phys. Rev. C [**86**]{}, 044903 (2012). S. A. Bass, P. Danielewicz and S. Pratt, Phys. Rev. Lett. [**85**]{}, 2689 (2000). J. Adams [*et al.*]{} (for the STAR Collaboration), Phys. Rev. Lett. [**90**]{}, 172301 (2003). P. Bozek, Phys. Lett. [**B609**]{}, 247 (2005). S. Jeon and S. Pratt, Phys. Rev. [**C 65**]{}, 044902 (2002). Adams [*at al.*]{} (for the STAR Collaboration), . Phys. Rev. [**C72**]{}, 14904 (2005). L. Kumar (for the STAR Collaboration), Central Eur. J. Phys. [**10**]{}, 1274 (2012). H. Wang (for the STAR Collaboration), J. Phys. Conf. Ser. [**316**]{} 012021 (2011). C.A. Baker, D.D. Doyle, P. Geltenbort [*et al.*]{}, Phys. Rev. Lett. [**97**]{}, 131801 (2006). D. Kharzeev, R.D. Pisarski and M.H.G. Tytgat, Phys. Rev. Lett. [**81**]{}, 512 (1998). E.V. Gorbar, V.A. Miransky and I.A. Shovkovy, Prog. Part. Nucl. Phys. [**67**]{}, 547 (2012). T. Lappi, Int. J. Mod. Phys. [**E20**]{}, 1 (2011). V.K. Magas, L.P. Csernai and D.D. Strottman, Phys. Rev. [**C64**]{}, (2001) 014901; Nucl. Phys. [**A712**]{} (2002) 167. M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. [**B147**]{}, 385 (1979). P. Minkowski, Phys. Lett. [**B76**]{}, 439 (1978); Nucl. Phys. [**B177**]{}, 203 (1981). H. Leutwyler, Phys. Lett. [**B96**]{}, 154 (1980). A. C. Kalloniatis and S. N. Nedelko, Phys. Rev. [**D69**]{}, 074029 (2004) [*ibid.*]{} [**D70**]{}, 119903 (2004); J. V. Burdanov, G. V. Efimov, S. N. Nedelko and S. A. Solunin, [*ibid.*]{} [**D54**]{}, 4483 (1996). M. D’Elia, A. Di Giacomo and E. Meggiolaro, Phys. Rev. [**D67**]{}, 114504 (2003). B. V. Galilo and S. N. Nedelko, Phys. Rev. [**D84**]{}, 094017 (2011).
|
---
abstract: |
The Border algorithm and the iPred algorithm find the Hasse diagrams of FCA lattices. We show that they can be generalized to arbitrary lattices. In the case of iPred, this requires the identification of a join-semilattice homomorphism into a distributive lattice.
, Hasse diagrams, border algorithms
author:
- José L Balcázar
- Cristina Tîrnăucă
bibliography:
- 'bibfile.bib'
title: |
Border Algorithms for Computing\
Hasse Diagrams of Arbitrary Lattices[^1]
---
Introduction
============
Lattices are mathematical structures with many applications in computer science; among these, we are interested in fields like data mining, machine learning, or knowledge discovery in databases. One classical use of lattice theory is in formal concept analysis (FCA) [@GanWil99], where the concept lattice with its diagram graph allows for the visualization and summarization of data in a more concise representation. In the Data Mining community, the same mathematical notions (often under additional “frequency” constraints that bound from below the size of the support set) are studied under the banner of Closed-Set Mining (see e.g. [@ZakHsi05]).
In these applications, data consists of *transactions*, also called *objects*, each of which, besides having received a unique identifier, consists of a set of *items* or *attributes* taken from a previously agreed finite set. A concept is a pair formed by a set of transactions —the *extent* set or *support set* of the concept— and a set of attributes —the *intent* set of the concept— defined as the set of all those attributes that are shared by all the transactions present in the extent. Some data analysis processes are based on the family of all intents (the “closures” stemming from the dataset); but others require to determine also their order relation, which is a finite lattice, in the form of a line graph (the *Hasse diagram*).
Existing algorithms can be divided into three main types: the ones that only generate the set of concepts, the ones that first generate the set of concepts and then construct the Hasse diagram, and the ones that construct the diagram while computing the lattice elements (see [@ZakHsi05], and also [@GoMiAl95; @KuzObi01] and the references therein). The goal is to obtain the concept lattice in linear time in the number of concepts because this number is, most of the times, already exponential in the number of attributes, making the task of getting polynomial algorithms in the size of the input rather impossible.
One widespread use of concepts or closures is the generation of implications or of partial implications (also called association rules). Several data mining algorithms aim at processing large datasets in time linear in the size of the closure space, and explore closed sets individually; these solutions tend to drown the user under a deluge of partial implications. More sophisticated works attempt at providing selected “bases” of partial implications; the early proposal in [@Luxe91] requires to compute immediate predecessors, that is, the Hasse diagram. Alternative proposals such as the Essential Rules of [@AggYu01b] or the equivalent Representative Rules of [@Krys98] (of which a detailed discussion with new characterizations and an alternative basis proposal appears in [@Balc10c]) require to process predecessors of closed sets obeying tightly certain support inequalities; these algorithms also benefit from the Hasse diagram, as the slow alternatives are blind repeated traversal of the closed sets in time quadratic in the size of the closure space, or storage of all predecessors of each closed set, which soon becomes large enough to impose a considerable penalty on the running times.
The problem of constructing the Hasse diagram of an arbitrary finite lattice is less studied. One algorithm that has a better worst case complexity than various previous works is described in [@NouRay99]. From our “arbitrary lattices” perspective, its main drawback is that it requires the availability of a *basis* from which each element of the lattice can be derived. In the absence of such a subset, one may still use this algorithm (at a greater computational cost) to output the Dedekind-MacNeille completion [@DavPri91] of the given lattice, which in our case is isomorphic to the lattice itself. The algorithm is also easily adaptable to concept lattices, where indeed a basis is available immediately from the dataset transactions.
We consider of interest to have available further, faster algorithms for arbitrary finite lattices; we have two reasons for this aim. First, many (although not all) algorithms constructing Hasse diagrams traverse concepts in layers defined by the size of the intents; our explorations about association rules sometimes require to follow different orderings, so that a more abstract approach is helpful; second, we keep in mind the application area corresponding to certain variants of implications and database dependencies that are characterized by lattices of equivalence relations, so that we are interested in laying a strong foundation that gives us a clear picture of the applicability requirements for each algorithm constructing Hasse diagrams in lattices other than powerset sublattices.
Of course, we expect that FCA-oriented algorithms could be a good source of inspiration for the design of algorithms applicable in the general case. An example that such an extension can be done is the algorithm in [@VaMiLe00] (see Section \[s:Border\] for more details), whose highest-level description matches the general case of arbitrary lattices; nevertheless, the actual implementation described in [@VaMiLe00] works strictly for formal concept lattices, so that further implementations and complexity analyses are not readily available for arbitrary finite lattices.
The contribution of the present paper supports the same idea: we show how two existing algorithms that build the Hasse diagrams of a concept lattice can be adapted to work for arbitrary lattices. Both algorithms have in common the notion of *border*, which we (re-)define and formalize in Section \[s:Border\], after presenting some preliminary notions about lattice theory in Section \[s:Prel\]; our approach has the specific interest that the notion of border is given just in terms of the ordering relation, and not in terms of a set of elements already processed as in previous references ([@BSVG09; @MarEkl08; @VaMiLe00]); yet, the notions are equivalent. We state and prove properties of borders and describe the *Generalized Border Algorithm*; whereas the algorithm reads, in high level, exactly as in previous references, its validation is new, as previous ones depended on the lattice being an FCA lattice. In Section \[s:iPA\] we introduce the *Generalized iPred Algorithm*, exporting the iPred algorithm of FCA lattices [@BSVG09] to arbitrary lattices, after arguing its correctness. This task is far from trivial and is our major contribution, since the existing rendering and validation of the iPred algorithm relies again extensively on the fact that it is being applied to an FCA lattice, and even performs operations on difference sets that may not belong to the closure space. Concluding remarks and future work ideas are presented in Section \[s:Conc\].
Preliminaries {#s:Prel}
=============
We develop all our work in terms of lattices and semilattices; see [@DavPri91] as main source. All our structures are *finite*. A *lattice* is a partially ordered set in which every nonempty subset has a meet (greatest lower bound) and a join (lowest upper bound). If only one of these two operations is guaranteed to be available a priori, we speak of a *join-semilattice* or a *meet-semilattice* as convenient. Top and bottom elements are denoted $\top$ and $\bot$, respectively. Lower case letters, possibly with primes, and taken usually from the end of the latin alphabet denote lattice elements: $x$, $y'$. Note that Galois connections are not explicitly present in this paper, so that the “prime” notation does not refer to the operations of Galois connections.
Finite semilattices can be extended into lattices by addition of at most one further element [@DavPri91]; for instance, if $(\LL,\leq,\lor)$ is a join-semilattice with bottom element $\bot$, one can define a meet operation as follows: $\bigwedge X = \bigvee \{y \st \hbox{$\forall x \in X,$} y \leq x\}$; the element $\bot$ ensures that this set is nonempty. Thus, if the join-semilattice lacks a bottom element, it suffices to add an “artificial” one to obtain a lattice. A dual process is obviously possible in meet-semilattices.
Given two join-semilattices $(S,\lor)$ and $(T, \lor)$, a *homomorphism* is a function $f: S \rightarrow T$ such that $f(x \lor y) = f(x) \lor f(y)$. Hence $f$ is just a homomorphism of the two semigroups associated with the two semilattices. If $S$ and $T$ both include a bottom element $\bot$, then $f$ should also be a monoid homomorphism, i.e. we additionally require that $f(\bot) = \bot$. Homomorphisms of meet-semilattices and of lattices are defined similarly. It is easy to check that $x\leq y \Rightarrow f(x)\leq f(y)$ for any homomorphism $f$; the converse implication, thus the equivalence $x\leq y \Leftrightarrow f(x)\leq f(y)$, is also true for injective $f$ but not guaranteed in general.
We must point out here a simple but crucial fact that plays a role in our later developments: given a homomorphism $f$ between two join-semilattices $S$ and $T$, if we extend both into lattices as just indicated, then $f$ is *not* necessarily a lattice homomorphism; for instance, there could be elements of $T$ that do not belong to the image set of $f$, and they may become meets of subsets of $T$ in a way that prevents them to be the image of the corresponding meet of $S$. For one specific example, see Figure \[fg:lattices\]: consider the two join-semilattices defined by the solid lines, where the numbering defines an injective homomorphism from the join-semilattice in (a) to the join-semilattice in (b). Both lack a bottom element. Upon adding it, as indicated by the broken lines, in lattice (a) the meets of 1 and 2 and of 1 and 3 coincide, but the meets of their corresponding images in (b) do not; for this reason, the homomorphism cannot be extended to the whole lattices.
(2,0) – (0,2) – (-2,0) (0,2) – (0,0); (2,0) – (0,-2) – (-2,0) (0,-2) – (0,0); (0,2) node \[fill=white\] [$\top$]{}; (-2,0) node \[fill=white\] [1]{}; (0,0) node \[fill=white\] [2]{}; (2,0) node \[fill=white\] [3]{}; (0,-2) node \[fill=white\] [$\bot$]{}; (0,-3) node [(a)]{}; (6,2) – (4,0); (6,2) – (6,0); (6,2) node \[fill=white\] [$\top$]{} – (8,0); (4,0) node \[fill=white\] [1]{} – (5,-1); (6,0) node \[fill=white\] [2]{} – (5,-1); (5,-1) node \[fill=white\] [4]{} – (6,-2); (8,0) node \[fill=white\] [3]{} – (6,-2) node \[fill=white\] [$\bot$]{}; (6,-3) node [(b)]{};
However, the following does hold:
\[l:joinmorph\] Consider two join-semilattices $S$ and $T$, and let $f :S \to T$ be a homomorphism. After extending both semilattices into lattices, $f(\bigwedge Y) \leq \bigwedge f(Y)$ for all $Y\subseteq S$.
This is immediate to see by considering that $\bigwedge Y \leq y$ for all $y\in Y$, hence $f(\bigwedge Y) \leq f(y)$ for all such $y$, and the claimed inequality follows.
We employ $x<y$ as the usual shorthand: $x\leq y$ and $x\neq y$. We denote as $x\prec y$ the fact that $x$ is an immediate predecessor of $y$ in $\LL$, that is, $x < y$ and, for all $z$, $x < z \leq y$ implies $z=y$ (equivalently, $x\leq z < y$ implies $x=z$).
We focus on algorithms that have access to an underlying finite lattice $\LL$ of size $|\LL|=n$, with ordering denoted $\leq$; abusing language slightly, we denote by $\LL$ as well its carrier set. The *width* $w(\LL)$ of the lattice $\LL$ is the maximum size of an antichain (a subset of $\LL$ formed by pairwise incomparable elements). The lattice is assumed to be available for our algorithms in the form of an abstract data type offering an iterator that traverses all the elements of the carrier set, together with the operations of testing for the ordering (given $x$, $y\in\LL$, find out whether $x\leq y$) and computing the meet $x\land y$ and join $x\lor y$ of $x$, $y\in\LL$; also the constants $\top\in\LL$ and $\bot\in\LL$ are assumed available.
The algorithms we consider are to perform the task of constructing explicitly the Hasse diagram (also known as the reflexive and transitive reduction) of the given lattice: $H(\LL)=\{(x,y)\st x\prec y\}$. By projecting the Hasse diagram along the first or the second component we find our crucial ingredients: the well-known upper and lower covers.
\[def:covers\] The upper cover of $x\in\LL$ is $\uc(x) = \{ y \st x\prec y \}$. The lower cover of $y\in\LL$ is $\lc(y) = \{ x \st x\prec y \}$.
The following immediate fact is stated separately just for purposes of easy later reference:
\[p:covers\] If $x<y$ then there is $z\in\uc(x)$ such that $x\prec z\leq y$; and there is $z'\in\lc(y)$ such that $x\leq z'\prec y$.
We will use as well yet another easy technicality:
\[l:twoprecs\] If $x_1 \prec y$ and $x_2 \prec y$, with $x_1\neq x_2$ then $x_1 \lor x_2 = y$.
Since $y \geq x_1$ and $y \geq x_2$ we have $y \geq x_1 \lor x_2$. Then, $x_1\neq x_2$ implies that they are mutually incomparable, since otherwise the smallest is not an immediate predecessor of $y$; this implies that $y \geq x_1 \lor x_2 > x_1$, whence $y = x_1 \lor x_2$ as $x_1\prec y$.
The Border Algorithm in Lattices {#s:Border}
================================
The algorithms we are considering here have in common the fact that they traverse the lattice and explicitly maintain a subset of the elements seen so far: those that still might be used to identify new Hasse edges. This subset is known as the “border” and, as it evolves during the traversal, actually each element $x\in\LL$ “gets its own border” associated as the algorithm reaches it. The border associated to an element may be potentially used to construct new edges touching it (although these edges may not touch the border elements themselves): more precisely, operations on the border for $x$ will result in $\uc(x)$, hence in the Hasse edges of the form $(x,z)$.
In previous references the border is defined in terms of the elements already processed, and its properties are mixed with those of the algorithm that uses it. Instead, we study axiomatically the properties of the notion of “border” on itself, always as a function of the element for which the border will be considered as a source of Hasse edges, in a manner that is independent of the fact that one is traversing the lattice. This allows us to clarify which abstract properties are necessary for border-based algorithms, so that we can generalize them to arbitrary lattices, traversed in flexible ways. Our key definition is, therefore:
\[def:border\] Given $x\in\LL$ and $B\subseteq\LL$, $B$ is a *border for* $x$ if the following properties hold:
1. $\forall y\in B \, (y \not\leq x)$;
2. $\forall z \, (x\prec z \Rightarrow \exists y \in B \, (y\leq z))$.
That is, $x$ is never above an element of a border, but each upper cover of $x$ is; this last condition is equivalent to: all elements strictly above $x$ are greater than or equal to some element of the border. Since $x\leq (x\lor y)$ always holds and $x = (x\lor y)$ if and only if $y\leq x$, we get:
\[l:easy\] Let $B$ be a border for $x$. Then $\forall y \in B \, (x < x\lor y)$.
All our borders will fulfill an extra “antichain” condition; the only use to be made of this fact is to bound the size of every border by the width of the lattice.
\[def:properborder\] A border $B$ is *proper* if every two different elements of $B$ are mutually incomparable.
The key property of borders, that shows how to extract Hasse edges from them, is the following:
\[th:useborder\] Let $B$ be a border for $x_0$. For all $x_1$ with $x_0 < x_1$, the following are equivalent:
1. $x_1\in\uc(x_0)$ (that is, $x_0\prec x_1$);
2. there is $y\in B$ such that $x_1 = (x_0 \lor y)$ and, for all $z\in B$, if $(x_0\lor z)\leq(x_0\lor y)$ then $(x_0\lor z) = (x_0\lor y)$.
Given $x_0\prec x_1$, we can apply the second condition in the definition of border for $x_0$: $\exists y \in B \, (y\leq x_1)$. Using Lemma \[l:easy\], $x_0 < (x_0 \lor y) \leq x_1$, implying $(x_0 \lor y) = x_1$ since $x_0\prec x_1$. Additionally, assuming $(x_0\lor z)\leq(x_0\lor y)$ for some $z\in B$ leads likewise to $x_0 < (x_0\lor z) \leq (x_0\lor y) = x_1$ and the same property applies to obtain $(x_0\lor z) = (x_0\lor y) = x_1$.
Conversely, again Lemma \[l:easy\] gives $x_0 < (x_0 \lor y) = x_1$. By Proposition \[p:covers\], there is $z_0\in\uc(x_0)$ with $x_0 \prec z_0 \leq (x_0 \lor y) = x_1$. We apply the second condition of borders to $x_0\prec z_0$ to obtain $z_1 \in B$ with $z_1\leq z_0$, whence $(x_0 \lor z_1) \leq z_0 \leq (x_0 \lor y) = x_1$, allowing us to apply the hypothesis of this direction: $(x_0 \lor z_1) \leq (x_0 \lor y)$ with $z_1 \in B$ implies $(x_0 \lor z_1) = (x_0 \lor y)$ and, therefore, $(x_0 \lor z_1) = z_0 = (x_0 \lor y) = x_1$. That is, $x_1 = z_0 \in\uc(x_0)$.
Therefore, given an arbitrary element $x_0$ of the lattice, any candidate for being an element of its upper cover has to be obtainable as a join between $x_0$ and a border element ($x_1 = x_0 \lor y$ for some $y \in B$). Moreover, among these candidates, only those that are minimals represent immediate successors: they come from those $y$ where $(x_0\lor z)\leq(x_0\lor y)$ implies $(x_0\lor z) = (x_0\lor y)$, for all $z \in B$.
Advancing Borders {#ss:Border}
-----------------
There is a naturally intuitive operation on borders; if we have a border $B$ for $x$, and we use it to compute the upper cover of $x$, then we do not need $B$ as such anymore; to update it, seeing that we no longer need to forbid the membership of $x$, it is natural to consider adding $x$ to the border. If we had a proper border, and we wished to preserve the antichain property, the elements to be removed would be exactly the upper cover just computed, as these are, as we argue below, the only elements comparable to $x$ that could be in a proper border. (All elements other than $x$ are mutually incomparable, as the border was proper to start with.)
Given $x\in\LL$ and a border $B$ for $x$, the standard step for $B$ and $x$ is $B\cup\{x\}-\uc(x)$.
Note that this is [*not*]{} to say that $\uc(x)\subseteq B$; elements of $\uc(x)$ may or may not appear in $B$. We will apply the standard step always when $B$ is a border for $x$, but let us point out that the definition would be also valid without this constraint, as it consists of just some set-theoretic operations.
Let $B$ be a proper border for $x$. Then the standard step for $B$ and $x$ is also an antichain.
Elements of the standard step different from $x$ and from all elements of $\uc(x)$ were already in the previous proper border and are, therefore, mutually incomparable. None of them is below $x$, by the first border property. If $y > x$ for some $y\in B$, then $y\geq z \succ x$ for some $z\in B$, and the antichain property of $B$ tells us that $y=z$ so that it gets removed with $\uc(x)$.
However, we are left with the problem that we have now a candidate border but we lack the lattice element for which it is intended to be a border. In [@MarEkl08] and [@BSVG09], the algorithm moves on to an intent set of the same cardinality as $x$, whenever possible, and to as small as possible a larger intent set if all intents of the same cardinality are exhausted. In [@VaMiLe00] it is shown that, for their variant of the Border algorithm, it suffices to follow a (reversed) linear embedding of the lattice. Here we follow this more flexible approach, which is easier now that we have stated the necessary properties of borders with no reference to the order of traversal: there is no need of considering intent sets and their cardinalities.
Both lattices and their Hasse diagrams can be seen as directed acyclic graphs, by orienting the inequalities in either direction; here we choose to visualize edges $(x,y)$ as corresponding to $x\leq y$. A linear embedding corresponds to the well-known operation of topological sort of directed acyclic graphs, which we will employ for lattices in a “reversed” way:
A reverse topological sort of $\LL$ is a total ordering $x_1,\ldots,x_n$ of $\LL$ such that $x_i\leq x_j$ always implies $j\leq i$.
All our development could be performed with a standard topological sort, not reversed, that is, a linear embedding of the lattice’s partial order. However, as it is customary in FCA to guide the visualization through the comparison of extents, the algorithms we build on were developed with a sort of “built-in reversal” that we inherit through reversing the topological sort (see the similar discussion in Section 2.1 of [@BSVG09]). A reversed topological sort must start with $\top$, hence the initialization is easy:
\[p:initial\] $B=\emptyset$ is a border for $\top\in\LL$.
Both conditions in the definition of border become vacuously true: the first one as $B=\emptyset$ and the second one as the top element has no upper covers.
\[th:advancing\] Let $x_1,\ldots,x_n$ be a reverse topological sort of $\LL$. Starting with $B_1 = \emptyset$, define inductively $B_{k+1}$ as the standard step for $B_k$ and $x_k$. Then, for each $k$, $B_k$ is a border for $x_k$.
For clarity, we factor off the proof of the following inductive technical fact, where we use the same notation as in the previous statement.
$B_k\subseteq\{ x_1,\ldots,x_{k-1} \}$ and, for all $x_j$ with $j<k$, there is $y\in B_k$ with $y\leq x_j$.
For $k=1$, the statements are vacuously true. Assume it true for $k$, and consider $B_{k+1} = B_k\cup\{x_k\}-\uc(x_k)$, the standard step for $B_k$ and $x_k$. The first statement is clearly true. For the second, $x_k$ is itself in $B_{k+1}$ and, for the rest, inductively, there is $y\in B_k$ with $y\leq x_j$. We consider two cases; if $y\notin\uc(x_k)$, then the same $y$ remains in $B_{k+1}$; otherwise, $x_k\prec y\leq x_j$, and $x_k$ is the corresponding new $y$ in $B_{k+1}$.
Again by induction on $k$; we see that the basis is Proposition \[p:initial\]. Assuming that $B_k$ is a border for $x_k$, we consider $B_{k+1} = B_k\cup\{x_k\}-\uc(x_k)$. Applying the lemma, $B_{k+1}\subseteq\{ x_1,\ldots,x_k \}$, which ensures immediately that $\forall y\in B_{k+1} \, (y \not\leq x_{k+1})$ by the property of the reverse topological sort, and the first condition of borders follows. For the second, pick any $z\in\uc(x_{k+1})$; by the condition of reverse topological sort, $z$, being a strictly larger element than $x_{k+1}$, must appear earlier than it, so that $z=x_j$ with $j<k+1$. Then, again the lemma tells us immediately that there is $y\in B_{k+1}$ with $y\leq x_j=z$, as we need to complete the proof.
The Generalized Border Algorithm {#ss:GBA}
--------------------------------
The algorithm we end up validating through our theorems has almost the same high-level description as the rendering in [@BSVG09]; the most conspicuous differences are: first, that a reverse topological sort is used to initialize the traversal of the lattice; and, second, that the “reversed lattice” model in [@BSVG09] has the consequence that their set-theoretic intersection in computing candidates becomes a lattice join in our generalization. Another minor difference is that Proposition \[p:initial\] spares us the separate handling of the first element of the lattice.
RevTopSort($\LL$) $B$ = $\emptyset$ $H$ = $\emptyset$
Theorem \[th:advancing\] and Proposition \[p:initial\] tell us that the following invariant is maintained: $B$ is a border for $x$. Then, the Hasse edges are computed and added to $H$ according to Theorem \[th:useborder\], in two steps: first, we prepare the list of joins $x \lor y$ and, then, we keep only the minimal elements in it. In essence, this process is the same as described (in somewhat different renderings) in [@BSVG09], [@MarEkl08] or [@VaMiLe00]; however, while the definition of border given in [@VaMiLe00] (and recalled in [@BSVG09]) leads, eventually, to the same notion employed in this paper, further development of a general algorithm that works outside the formal concept analysis framework is dropped off from [@VaMiLe00] on efficiency considerations. Moreover, the border algorithm described in [@MarEkl08] works exclusively on the set of intents and assumes the elements are sorted sizewise. The validations of the algorithms in these references rely very much, at some points, on the fact that the lattice is a sublattice of a powerset and contains formal concepts, explicitly operating set-theoretically on their intents. Theorem \[th:useborder\] captures the essence of the notion of border and lifts the algorithm to arbitrary lattices.
One additional difference comes from the fact that the cost of computing the meet and join operations plays a role in the complexity analysis, but is not available in the general case. If we assume that meet and join operations take constant time, then the total running time of the algorithm (except for the sort initialization, which takes $\mathcal{O}(|\LL|\log|\LL|)$) is bounded by $\mathcal{O}(|\LL|w(\LL)^2)$. By comparison with [@VaMiLe00], one can see that one factor of the formula given in [@VaMiLe00] gets dropped under the constant time assumption for computing meet and join. However, this assumption may be unreasonable in certain applications; the same reference indicates that their FCA target case requires a considerable amount of graph search for the same operations. Nevertheless, in absence of further information about the specific lattice at hand, it is not possible to provide a finer analysis.
We must point out that, in our implementation, we have employed a heapsort-based version that keeps providing us the next element to handle by means of an iterator, instead of completing the sorting step for the initialization.
Distributivity and the iPred Algorithm {#s:iPA}
======================================
In [@BSVG09], an extra sophistication is introduced that, as demonstrated both formally in the complexity analysis of the algorithm and also practically, leads to a faster algorithm; namely, if some further information is maintained along, once the candidates are available there is a constant-time test to pick those that are in the cover, by employing the duality $y\in\uc(x)\Leftrightarrow x\in\lc(y)\Leftrightarrow x\prec y$. Constant time also suffices to maintain the additional information. This gives the iPred algorithm. However, it seems that the unavoidable price is to work on formal concepts, as the extra information is heavily set-theoretic (namely, a union of set differences of previously found cover sets for the candidate under study).
Again we show that a fully abstract, lattice-theoretic interpretation exists, and we show that the essential property that allows for the algorithm to work is distributivity: be it due to a distributive $\LL$, or, as in fact happens in iPred, due to the embedding of the lattice into a distributive lattice, in the same way as concept lattices (possibly nondistributive) can be embedded in the distributive powerset lattice.
We start treating the simplest case, of very limited usefulness in itself but good as stepping stone towards the next theorem. The property where distributivity can be applied later, if available, is as follows:
\[p:nondist\] Consider two comparable elements, $x < z$, from $\LL$; let $Y\subseteq\lc(z)$ be the set of lower covers of $z$ that show up in the reverse topological sort before $x$ (it could be empty). Then, $x\in\lc(z)$ if and only if $\bigwedge_{y\in Y} (x \lor y) \geq z$.
Applying Proposition \[p:covers\], we know that there is some $y\in\lc(z)$ such that $x\leq y\prec z$. Any such $y$, if different from $x$, must appear before $x$ in the reverse topological sort.
Suppose first that no lower covers of $z$ appear before $x$, that is, $Y=\emptyset$. Then, no such $y$ different from $x$ can exist; we have that both $x = y \prec z$ and $\bigwedge_{y\in Y} (x \lor y) = \top \geq z$ trivially hold.
In case $Y$ is nonempty, assume first $x \prec z$; we can apply Lemma \[l:twoprecs\]: $x \lor y=z$ for every $y\in Y$, hence $\bigwedge_{y\in Y} (x \lor y) = z$. To argue the converse, assume $x\notin\lc(z)$ and let $x\leq y'\prec z$ as before, where we know further that $x\neq y'$: then $y'\in Y$, so that $\bigwedge_{y\in Y} (x \lor y) \leq (x \lor y') = y' < z$.
This means that the test for minimality of Algorithm \[a:GBA\] can be replaced by checking the indicated inequality; but it is unclear that we really save time, as a number of joins have to be performed (between the current element $x$ and all the elements in the lower cover of the candidate $z$ that appeared before $x$ in the reverse topological sort) and the meet of their results computed. However, clearly, in distributive lattices the test can be rephrased in the following, more convenient form:
Assume $\LL$ distributive. In the same conditions as in the previous proposition, $x$ is in the lower cover of $z$ if and only if $x \lor (\bigwedge_{y\in Y} y) \geq z$.
This last version of the test is algorithmically useful: as we keep identifying elements $Y = \{y_1,\ldots,y_m\}$ of $\lc(z)$, we can maintain the value of $y = \bigwedge_{i\in \{1,\ldots,m\}} y_i$; then, we can test a candidate $z$ by computing $x\lor y$ and comparing this value to $z$. Afterwards, we update $y$ to $y\land x$ if $x = y_{m+1}$ is indeed in the cover. This may save the loop that tests for minimality at a small price.
However, unfortunately, if the lattice is not distributive, this faster test may fail: given $Y \subseteq \lc(z)$, the cover elements found so far along the reverse topological sort, it is always true that $x$ is in the lower cover of $z$ if $x \lor (\bigwedge_{y \in Y} y) \geq z$, because $z \leq x \lor (\bigwedge_{y \in Y} y) \leq \bigwedge_{y\in Y} (x \lor y)$ and, then, one of the directions of Proposition \[p:nondist\] applies; but the converse does not hold in general. Again an example is furnished by Figure \[fg:lattices\](a), one of the basic, standard examples of a small nondistributive lattice; assume that the traversal follows the natural ordering of the labels, and consider what happens after seeing that 1 and 2 are indeed lower covers of $z=\top$. Upon considering $x=3$, we have $Y = \{1,2\}$, so that $x \lor (\bigwedge Y) = x \lor \bot = x < z$, yet $x$ is a lower cover of $z$ and, in fact, $\bigwedge_{y\in Y} (x \lor y) = (3\lor 1)\land(3\lor 2) = \top$. Hence, the distributivity condition is necessary for the correctness of the faster test.
The Generalized iPred Algorithm {#ss:GiPA}
-------------------------------
The aim of this subsection is to show the main contribution of this paper: we can spare the loop that tests candidates for minimality in an indirect way, whenever a distributive lattice is available where we can embed $\LL$. However, we must be careful in how the embedding is performed: the right tool is an injective homomorphism of join-semilattices. Recall that, often, this will *not* be a lattice morphism. Such an example is the identity morphism having as domain the carrier set of a concept lattice $\LL$ over the set of attributes $X$, and as range, $\mathcal{P}(X)$ (see Section \[s:Conc\] for more details on this particular case).
\[thm:iPred\] Let $(\LL',\leq,\lor)$ be a distributive join-semilattice and $f : \LL \rightarrow \LL'$ an injective homomorphism. Consider two comparable elements, $x < z$, from $\LL$; let $Y\subseteq\lc(z)$ be the set of lower covers of $z$ that show up in the reverse topological sort before $x$.Then, $x\prec z$ if and only if $f(x) \lor (\bigwedge_{y\in Y} f(y)) \geq f(z)$.
If $Y = \emptyset$ we have $x \prec z$ as in Proposition \[p:nondist\]; for this case, $\bigwedge_{y\in Y} f(y) = \top$ (of $\LL'$) and $f(x) \lor (\bigwedge_{y\in Y} f(y))
= f(x) \lor \top = \top \geq f(z)$.
For the case where $Y\neq\emptyset$, assume first $x\prec z$ and apply Proposition \[p:nondist\]: we have that $\bigwedge_{y\in Y} (x \lor y) \geq z$ whence $f(\bigwedge_{y\in Y} (x \lor y)) \geq f(z)$. By Lemma \[l:joinmorph\], we obtain $f(z) \leq f(\bigwedge_{y\in Y} (x \lor y)) \leq
\bigwedge_{y\in Y} f(x \lor y) =
\bigwedge_{y\in Y} (f(x) \lor f(y)) =
f(x) \lor \bigwedge_{y\in Y} f(y)$, where we have applied that $f$ commutes with join and that $\LL'$ is distributive.
For the converse, arguing along the same lines as in Proposition \[p:nondist\], assume $x\notin\lc(z)$ and let $x\leq y'\prec z$ with $x\neq y'$ so that $y'\in Y$: necessarily $\bigwedge_{y\in Y} f(y) \leq f(y')$, so that $f(x) \lor (\bigwedge_{y\in Y} f(y)) \leq f(x) \lor f(y')
= f(x\lor y') = f(y') < f(z)$, where the last step makes use of injectiveness.
The generalized iPred algorithm is based on this theorem, which proves it correct. In it, the homomorphism $f$ is assumed available, and table LC keeps, for each $z$, the meet of the $f(x)$’s for all the lower covers $x$ of $z$ seen so far.
RevTopSort($\LL$) $B$ = $\emptyset$ $H$ = $\emptyset$
In the Appendix below, we provide some example runs for further clarification. Regarding the time complexity, again we lack information about the cost of meets, joins, and comparisons in both lattices, and also about the cost of computing the homomorphism. Assuming constant time for these operations, the running time of the generalized iPred algorithm is $\mathcal{O}(|\LL|w(\LL))$ (plus sorting): the main loop (line 4-15) is repeated $|\LL|$ times, and then for each of the at most $w(\LL)$ candidates, the algorithm checks if a certain condition is met (in constant time) and updates the diagram and the border in the positive case.
If meets and joins do not take constant time, there is little to say at this level of generality; however, for the particular case of the original iPred, which only works for lattices of formal concepts, see [@BSVG09]: in the running time analysis there, one extra factor appears since the meet operation (corresponding to a set union plus a closure operation) is not guaranteed to work in constant time.
Conclusions and Future Work {#s:Conc}
===========================
We have provided a formal framework for the task of computing Hasse diagrams of arbitrary lattices through the notion of “border associated with a lattice element”. Although the concept of *border* itself is not new, our approach provides a different, more “axiomatic” point of view that facilitates considerably the application of this notion to algorithms that construct Hasse diagrams outside the formal concept analysis world.
While Algorithm \[a:GBA\] is a clear, straightforward generalization of the Border algorithm of [@VaMiLe00; @BSVG09] (although the correctness proof is far less straightforward), we consider that we should explain further in what sense the iPred algorithm comes out as a particular case of Algorithm \[a:GiPred\]. In fact, the iPred algorithm uses set-theoretic operations and, therefore, is operating with sets that do not belong to the closure space: effectively, it has moved out of the concept lattice into the (distributive) powerset lattice. Starting from a concept lattice $(\LL,\leq, \lor,\wedge)$ on a set $X$ of attributes, we can define:
- $x \leq y \Leftrightarrow x \supseteq y$
- $x \lor y := x \cap y$
- $x \land y := \bigvee \{z \in \LL \st z \leq x, z \leq y\} = \bigcap \{z \in \LL \st z \supseteq x, z \supseteq y\}$
- $\top:=\emptyset, \bot:=X$
Thus, $\LL$ is a join-subsemilattice of the (reversed) powerset on $X$, and we can define $f : \LL \rightarrow \mathcal{P}(X)$ as the identity function: it is injective, and it is a join-homomorphism since $\LL$, being a concept lattice, is closed under set-theoretic intersection. Therefore, Theorem \[thm:iPred\] can be translated to: $x \in \lc(z)$ if and only if $x \cap (\bigcup_{y \in Y} y) \subseteq z$, where $Y$ is the set of lower covers of $z$ already found; this is fully equivalent to the condition behind algorithm iPred of [@BSVG09] (see Proposition 1 on page 169 in [@BSVG09]). Additionally, iPred works on one specific topological sort, where all intents of the same cardinality appear together; our generalization shows that this is not necessary: any linear embedding suffices.
A further application we have in mind refers to various forms of implication known as multivalued dependency clauses [@SDPF81; @SDPF87]; in [@Baix07; @Baix08; @BaiBal06], these clauses are shown to be related to partition lattices in a similar way as implications are related to concept lattices through the Guigues-Duquenne basis ([@GanWil99; @GuiDuq86]); further, certain database dependencies (the degenerate multivalued dependencies of [@SDPF81; @SDPF87]) are related to these clauses in the same way as functional dependencies correspond to implications. Data Mining algorithms that extract multivalued dependencies do exist [@SavFla00] but we believe that alternative ones can be designed using Hasse diagrams of the corresponding partition lattices or related structures like split set lattices [@Baix07]. The task is not immediate, as functional and degenerate multivalued dependencies are of the so-called “equality-generating” sort but full-fledged multivalued dependencies are of the so-called “tuple-generating” sort, and their connection to lattices is more sophisticated (see [@Baix07]); but we still hope that further work along this lattice-theoretic approach to Hasse diagrams would allow us to create a novel application to multivalued dependency mining.
Appendix
========
We exemplify here some runs of iPred, for the sake of clarity. First we see how it operates on the lattice in Figure \[fg:lattices\](a), denoted $\LL$ here, using as $f$ the injective homomorphism into the distributive lattice of Figure \[fg:lattices\](b) provided by the labels. The run is reported in Table \[tb:run\], where we can see that we identify the respective upper covers of each of the lattice elements in turn. The linear order is assumed to be $( \top, 1, 2, 3, \bot )$. Only the last loop has more than one candidate, in fact three. The snapshots of the values of $B$, $H$, and $\LC$ reported in each row (except the initialization) are taken at the end of the corresponding loop, so that each reported value of $B$ is a border for the next row. In the Hasse edges $H$, thin lines represent edges that are yet to be found, and thick lines represent the edges found so far. Recall that the values of $\LC$ are actually elements of the distributive lattice of Figure \[fg:lattices\](b), and not from $\LL$.
$\LL$ $B$ $H$ cand $\LC[\top]$ $\LC[1]$ $\LC[2]$ $\LC[3]$ $\LC[\bot]$
-------- ------------- ----- ------------- ------------- ---------- ---------- ---------- -------------
init $\0$
$\top$ $\{\top\}$ $\0$ $\top$
1 $\{1\}$ $\{\top\}$ 1 $\top$
2 $\{1,2\}$ $\{\top\}$ 4 $\top$ $\top$
3 $\{1,2,3\}$ $\{\top\}$ $\bot$ $\top$ $\top$ $\top$
$\bot$ $\0$ $\{1,2,3\}$ $\bot$ $\bot$ $\bot$ $\bot$ $\top$
: Example run of the iPred algorithm using the lattices in Figure \[fg:lattices\][]{data-label="tb:run"}
All along the run we can see that $\LC[z]$ indeed maintains the meet of the set of predecessors found so far for $f(z)$ in the distributive embedding lattice; of course, this meet is $\top$ whenever the set is empty.
Let us compare with the run on the distributive lattice in Figure \[fg:distlattice\], where the homomorphism $f$ is now the identity. Observe that the only different Hasse edge is the one above 3 which now goes to 2 instead of going to $\top$. Again the linear sort follows the order of the labels.
(0,0) – (-1,1) – (0,2) – (1,1) – (0,0) – (1,-1) – (2,0) – (1,1); (0,2) node \[fill=white\] [$\top$]{} (-1,1) node \[fill=white\] [1]{} (1,1) node \[fill=white\] [2]{} (2,0) node \[fill=white\] [3]{} (0,0) node \[fill=white\] [4]{} (1,-1) node \[fill=white\] [$\bot$]{};
Due to the similarity among the Hasse diagrams, the run of generalized iPred on this lattice starts exactly like the one already given, up to the point where node 3 is being processed. At that point, 2 is candidate and will indeed create an edge, but 1 leads to candidate $1\lor3=\top$ for which the test fails, as $\LC[\top] = 4$ at that point, and $3\lor 4 = 2 < \top$. Hence, this candidate has no effect. After this, the visits to 4 and $\bot$ complete the Hasse diagram with their corresponding upper covers.
$\LL$ $B$ $H$ cand $\LC[\top]$ $\LC[1]$ $\LC[2]$ $\LC[3]$ $\LC[4]$ $\LC[\bot]$
-------- ------------ ----- -------------- ------------- ---------- ---------- ---------- ---------- -------------
init $\0$
$\top$ $\{\top\}$ $\0$ $\top$
1 $\{1\}$ $\{\top\}$ 1 $\top$
2 $\{1,2\}$ $\{\top\}$ 4 $\top$ $\top$
3 $\{1,3\}$ $\{\top,2\}$ 4 $\top$ 3 $\top$
4 $\{3,4\}$ $\{1,2\}$ 4 4 $\bot$ $\top$ $\top$
$\bot$ $\0$ $\{3,4\}$ 4 $\bot$ $\bot$ $\bot$ $\bot$ $\top$
: Example run of the iPred algorithm on the lattice in Figure \[fg:distlattice\][]{data-label="tb:secondrun"}
[^1]: This work has been partially supported by project FORMALISM (TIN2007-66523) of Programa Nacional de Investigación, Ministerio de Ciencia e Innovación (MICINN), Spain, by the Juan de la Cierva contract JCI-2009-04626 of the same ministry, and by the Pascal-2 Network of the European Union.
|
---
abstract: 'In this paper we show how to find a closed form solution for third order difference operators in terms of solutions of second order operators. This work is an extension of previous results on finding closed form solutions of recurrence equations and a counterpart to existing results on differential equations. As motivation and application for this work, we discuss the problem of proving positivity of sequences given merely in terms of their defining recurrence relation. The main advantage of the present approach to earlier methods attacking the same problem is that our algorithm provides human-readable and verifiable, i.e., certified proofs.'
author:
- |
Yongjae Cha[^1]\
Johannes Kepler University\
4040 Linz (Austria)\
`ycha@risc.jku.at`
bibliography:
- 'myrefs.bib'
title: Closed form solutions of linear difference equations in terms of symmetric products
---
Introduction
============
This paper presents an extension of the algorithm [*solver*]{} [@YC11; @CH09; @CHG10] that returns closed form solutions for second order linear difference equations to third order linear difference equations. The solutions that we are looking for are in terms of (finite) sums of squares. This is motivated by applying the algorithm for proving inequalities on special functions, i.e., on expressions that may be defined in terms of linear difference equations with polynomial coefficients. Conjectures about positivity of special functions inequalities arise in many applications in mathematics and science. Proving them usually requires profound knowledge on relations between these special functions. It is well known that there exist many algorithms for proving and finding special function identities [@Zeil90a; @ChyzakDM; @AeqB; @KoutschHF]. For automated proving of special functions inequalities only few approaches exist. Gerhold and Kauers [@GKIneq; @MKSumCracker] introduced a method that is based on Cylindrical Algebraic Decomposition (CAD). This method has been proven to work well on many non-trivial examples [@MKTuran; @VPSI], but even though correctness is easy to be seen, termination cannot be guaranteed, hence it is not an algorithm in the strict sense. A first attempt to clarify the latter issue has been made in [@MKVP10]. One of the features of proofs of special functions identities is that they usually come with a certificate, i.e., some easy to check identity that verifies the proof. The CAD-based approach can not hope to have a similar certificate in the near future. The method presented here is a first step toward human readable proofs of special functions inequalities, although admittedly a representation in terms of sums of squares with positive coefficients is not expected to exist for any given input. Besides this application, the results presented are of independent interest as they provide difference case counterparts to results obtained for the differential case [@MS85; @vH07].
First we review the available results in the differential case. Let $k$ be a differential field and $L_d \in k[\partial],\partial=d/dx$ be a linear homogeneous third order differential operator. Singer [@MS85] characterizes when solutions of $L_d$ can be written in terms of solutions of a second order operator in $\bar{k}[\partial]$. Van Hoeij [@vH07] handles the similar problem when the coefficients of the second order operator are restricted to $k$ and shows that it will be either of the following cases.
Case 1
: $L_d$ is the symmetric square of a second order operator $K_d \in
k[\partial]$
Case 2
: $L_d$ is gauge equivalent to a symmetric square of a second order operator $K_d \in k[\partial]$
The definitions of symmetric products and gauge equivalence are recalled in sections \[sec:sp\] and \[sec:ge\] below. The algorithm given in [@vH07] returns a second order differential operator, $K_d \in k[d/dx]$, and a gauge transformation in $k[\partial]$ that sends solutions of the symmetric squares of $K_d$ to solutions of $L_d$ for Case 2.
In the differential case, the symmetric square of $L_d$ has order 5 if and only if we are in Case 1. In this case, there is a simple formula that gives $K_d$. Case 2 is equivalent to the symmetric square of $L_d$ having order 6 and a first order right-hand side factor in $k[\partial]$ as well as a certain conic of $L_d$([@MS85 Equation 4.2.1]) having a non-zero solution in $k$. Since for $k={{\mathbb{C}}}(x)$ this conic is solvable over ${{\mathbb{C}}}(x)$, the last condition becomes trivial in this case. The algorithm given in [@vH07] in the first step checks the order of the symmetric square of $L_d$ to distinguish the cases. The difference case behaves differently; here we denote by $D={{\mathbb{C}}}(x)[\tau]$ the ring of linear difference operators, where $\tau$ denotes the shift operator. Example \[example:symp\] shows that the cases can not be distinguished according to the order of the symmetric squares when the coefficients are in ${{\mathbb{C}}}(x)$. To set up a counterpart theorem for difference equations, this example shows that we need one more transformation than that in the differential case. Furthermore in Case 1, the algorithm for finding the second order operator is more complicated than in the differential case. Summarizing, the ideas used in the differential case can not be carried over immediately to the difference case. Furthermore our aim is to have a closed form solution of the given input. Hence, if a factorization is found that is not solvable, this fails to satisfy our goal. Thus we build on the ideas of the algorithm [*solver*]{} [@YC11; @CH09; @CHG10]. Here we say that a function is in closed form if it is a linear combination of elementary functions, special functions or hypergeometric functions over ${{\mathbb{C}}}(x)$. For instance the modified Bessel function of the first kind is a closed form solution of the second order operator $L_b:=z\tau^2-(2x+2)\tau+x+z$.
The algorithm [*solver*]{} returns closed form solutions for second order linear difference operators. The main idea of [*solver*]{} is to map the given operator $L_1$ to an operator $L_2$ of which a solution is known. This transformation is a bijective map, called GT-transformation, that sends solutions of $L_1$ to solutions of $L_2$. If a closed form solution to one of the operators is known, then by means of this transformation the solution of the second operator can be constructed. For this purpose a table with second order operators including parameters together with characteristic data (local data) has been constructed. This local data can be computed for the given operator, the corresponding equivalent operator is found by table look-up. Then by comparing parameters of the local data the GT-transformation can be constructed. The characteristic data is described in Section \[sec:ld\]. To cover the extension described here the table has been extended so that we can give closed form solutions of certain third order linear difference operators.
Preliminary
===========
In this section we introduce notations used in this paper and recall some known facts [@CH09; @CHG10; @AeqB; @PS97] about difference operators. Additionally Cases 1 and 2 above are carried over to the difference case for algebraic extensions in Theorem \[thm:ord3dec\] below.
Ring of difference oprators
---------------------------
Let $D:={{\mathbb{C}}}(x)[\tau]$ be the ring of linear difference operators with coefficients in ${{\mathbb{C}}}(x)$, where $\tau$ is the shift operator acting on $x$ by $ \tau(x)=x+1$. Then $D$ is a noncommutative ring where $$\tau
\cdot \tau^{i-1}=\tau^i \ \text{for} \ i \in {{\mathbb{N}}}, \ \tau \cdot f= \tau(f)\tau \ \text{for}
\ f \in {{\mathbb{C}}}(x).$$ For $L=a_d(x)\tau^d+\cdots+a_1(x)\tau+a_0(x) \in D$ with $a_d \neq 0$, we say that $L$ has order $d$ and write ${\mathrm{ord}}(L)=d$. If furthermore $a_0\ne0$ then $L$ is said to be a [*normal*]{} operator. In this paper we will assume all operators to be normal.
The adjoint operator of $L$ is defined by $L^* = \sum_{i=0}^d a_{d-i}(x+i)\tau^i$. Suppose $L=M\cdot N$ for some $M, N \in D$. Then $L^*=(M\cdot N)^*=(\tau^{d_1}\cdot
N^*\cdot \tau^{-d_1})\cdot M^*$, where $d_1={\mathrm{ord}}(M)$ and thus right-hand side factors of $L$ correspond to left-hand side factors of $L^*$. We say that a third order operator $L$ is irreducible in $D$ if both $L$ and $L^*$ have no first order right-hand side factor in $D$.
A second order operator $K=b_2\tau^2+b_1\tau+b_0 \in D$ is called a [ *full*]{} operator if $b_2b_1b_0 \neq 0 $. Thus, if $K$ is a normal but not full operator, then $b_1=0$.
Ring of sequences
-----------------
Let ${{\mathbb{C}}}^{{\mathbb{N}}}:=\{ f \mid f:{{\mathbb{N}}}\rightarrow {{\mathbb{C}}}\}$. Then an element $v \in {{\mathbb{C}}}^{{\mathbb{N}}}$ corresponds to a sequence $v:=(v(1), v(2), v(3), \ldots)$. ${{\mathbb{C}}}$ is embedded in ${{\mathbb{C}}}^{{\mathbb{N}}}$ as a subring via constant sequences. Suppose $v_1, v_2 \in {{\mathbb{C}}}^{{\mathbb{N}}}$, then we define $$\begin{split}
v_1+v_2&:=(v_1(1)+v_2(1), v_1(2)+v_2(2), \ldots ) \\
v_1v_2&:=(v_1(1)v_2(1), v_1(2)v_2(2), \ldots ).
\end{split}$$ With the above termwise addition and multiplication, ${{\mathbb{C}}}^{{\mathbb{N}}}$ forms a ${{\mathbb{C}}}$-algebra. We define the action of $\tau$ on ${{\mathbb{C}}}^{{\mathbb{N}}}$ by $\tau(v):=(v(2), v(3), v(4), \ldots)$.
Let $\mathbf{S} := {{\mathbb{C}}}^{{{\mathbb{N}}}}/_\sim$ where $s_1 \sim s_2$ if there exists $N \in {{\mathbb{N}}}$ such that, for all $i > N$, $s_1(i) = s_2(i)$. Then it is easy to verify that $s$ is a unit in ${\mathbf{S}}$, i.e. $s$ is invertible in ${\mathbf{S}}$, if and only if $s \in {\mathbf{S}}$ has only finitely many zeros. If $f \in {{\mathbb{C}}}(x)$, then the image of $f$ in ${\mathbf{S}}$ and the action of $\tau$ on ${\mathbf{S}}$ are well defined. This way we can embed ${{\mathbb{C}}}(x)$ to ${\mathbf{S}}$ and call $s \in {\mathbf{S}}$ rational if there exist $g(x) \in {{\mathbb{C}}}(x)$ and $N \in {{\mathbb{N}}}$ such that $g(i)=s(i)$ for all $i
\geq N$. ${\mathbf{S}}[\tau]$ forms a ring of difference operators and $D$ is embedded in ${\mathbf{S}}[\tau]$.
We say $L(v)=0$ for $v \in {\mathbf{S}}, L=a_d(x)\tau^d+\cdots+a_0(x) \in {\mathbf{S}}[\tau]$ if there is $n_0 \in {{\mathbb{N}}}$ such that $$a_d(i)v(i+d)+a_{d-1}(i)v(i+d-1)+\cdots+a_0(i)v(i)=0 \quad \text{for \ all} \ i \geq n_0.$$
$h \in {\mathbf{S}}$ is called hypergeometric if $r=\tau(h)/h \in {\mathbf{S}}\setminus \{0\}$ is rational and $r$ is called the certificate of $h.$
If $h \in {\mathbf{S}}$ is hypergeometric then $(\tau-r)(h)=0$ where $r$ is the certificate of $h$. We define $V(L):=\{ u\in \mathbf{S} \mid L(u)=0 \}$.
[@AeqB Theorem 8.2.1] $\dim_{{\mathbb{C}}}(V(L))={\mathrm{ord}}(L)$ for a normal difference operator $L \in D$.
Thus for a normal operator $L \in D$, $V(L)$ forms a ${{\mathbb{C}}}$-vector space with a basis $\{ v_i \in {\mathbf{S}}\mid 1 \leq i \leq {\mathrm{ord}}(L) \}$.
Term equivalence {#sec:sp}
----------------
\[def:sp\] The symmetric product, $M \circledS N$, of operators $M$ and $N \in D$ is an order-minimal and monic operator such that $\mu \nu \in V(M \circledS N)$ for all $\mu
\in V(M)$ and $\nu \in V(N)$.
There is a simple formula if one of the operators has order $1$. Let $L
=a_d(x)\tau^d+\cdots+\dots+a_1(x)\tau+a_0(x) \in D$ and $r(x) \in {{\mathbb{C}}}(x)$. Then $$\label{sym}
\begin{split}
& L \circledS (\tau-r(x))=\displaystyle \sum_{i=0}^d b_i \tau^i, \ \text{where} \ b_d(x)=a_d(x) \ \text{and}\\
& b_i(x)=a_i(x)\displaystyle\prod_{j=i}^{d-1} \tau^j(r(x)) \ \text{for} \ i=0,\ldots,d-1.
\end{split}$$ Thus, $(\tau-a(x)) \circledS (\tau-b(x)) =\tau-a(x)b(x)$ for any $a(x), b(x) \in {{\mathbb{C}}}(x)$.
Suppose $L \in D$ and $s \in {\mathbf{S}}$. Then the above formula gives an operator $\tilde{L}=L {\circledS}(\tau-s) \in {\mathbf{S}}[\tau]$ such that $V(\tilde{L})=\{ hu \mid L(u)=0 \}$ where $h \in {\mathbf{S}}$ is a solution of $\tau-s$. If $L {\circledS}(\tau-s) \in D$ then it is easy to see that $s$ is rational.
$L_1, L_2 \in D$ are said to be term equivalent if there exists $T=\tau-r \in D$ such that $V(L_2)=V( L_1 {\circledS}(\tau-r))$, denoted by $L_1 \sim_t L_2$. Such a T is called the term transformation from $L_1$ to $L_2$.
If $L_1$ and $L_2$ are term equivalent and $\tau-r $ is the term transformation then $V(L_2)=\{ h v \mid h \in V(\tau-r), v \in V(L_1) \}$. Suppose $L_1$ and a term transformation $T$ are given, then $L_2$ can be obtained by .
Gauge equivalence {#sec:ge}
-----------------
Let $L_1,L_2\in D$ be two given operators, where a closed form solution $u$ of $L_1$ is known. If furthermore an operator $G\in D$ can be determined sucht that $G(u)$ is solution of $L_2$, then a closed form solution of $L_2$ can be written as a linear combination of shifts of $u$ over ${{\mathbb{C}}}(x)$. Such a transformation $G$ is called a gauge transformation and $L_1$ and $L_2$ are said to be gauge equivalent if such a transformation exists.
\[def:gt\] Let $L_1, L_2 \in D$ have the same order. $G \in D$ is called a [*gauge transformation*]{} from $L_1$ to $L_2$ iff $G: V(L_1) \rightarrow V(L_2)$ is a bijection.
Note that $G$ is not required to be a normal operator.
Suppose we are given a gauge transformation $G$ where ${\mathrm{ord}}(G) \geq {\mathrm{ord}}(L_1)$. Then there exist $Q, \hat{G} \in D$ with ${\mathrm{ord}}(\hat{G}) < {\mathrm{ord}}(L_1)$ such that $G=QL_1+\hat{G}$. The remainder $\hat{G}$ is also a gauge transformation that acts in the same way as $G$ on $V(L_1)$. Hence, w.l.o.g., we may assume that ${\mathrm{ord}}(G)<{\mathrm{ord}}(L_1)$.
Let ${\mathrm{GCRD}}(L, M)$ denote the greatest common right divisor of $L, M \in D$. Since $G$ is a bijection, any non zero solution $u$ of $L_1$ does not map to zero by $G$. Thus, $L_1$ and $G$ have no nontrivial common right hand factor, i.e. ${\mathrm{GCRD}}(L_1, G)=1$. Using the extended Euclidean algorithm $\tilde{G},\tilde{L_1}\in D$ can be determined such that $\tilde{G}G+\tilde{L_1}L_1=1$. Then $\tilde{G}G$ is the identity on $V(L_1)$ and $\tilde{G}$ is an inverse of $G$ that sends $V(L_2) \rightarrow V(L_1)$ bijectively.
\[def:ge\] Two operators $L_1$ and $L_2$ with the same order are called gauge equivalent if there exists a gauge transformation $G:V(L_1)\rightarrow V(L_2)$ and we use the notation $L_1
\sim_g L_2$.
Suppose $L_1 \sim_g L_2$ where the gauge transformation from $L_1$ to $L_2$ is a single term operator, $c(x)\tau^n$ for $n < {\mathrm{ord}}(L_1)$. Then $\tau^n\cdot L_1 \cdot\tau^{-n}$ is term equivalent to $L_2$ where the term transformation from $\tau^n\cdot L_1 \cdot \tau^{-n}$ to $L_2$ is $\tau-\frac{c(x+1)}{c(x)}$.
### How to compute the gauge transformation {#hom}
Suppose $L_1$ and $L_2$ are gauge equivalent and $G$ is a gauge transformation from $L_1$ to $L_2$. Then there is an operator $H \in D, {\mathrm{ord}}(H) < {\mathrm{ord}}(L_2)$ such that $H \cdot
L_1=L_2 \cdot G$.
The algorithm that was used to find the gauge transformation in [@CH09; @CHG10; @GH10] works as follows:
1. For given operators $L_1$ and $L_2$, we set up the ansatz $G:=\sum_{i=0}^{{\mathrm{ord}}(L_1)-1} c_i(x)\tau^i$, where the $c_i(x)$ are undetermined coefficients.
2. \[step3\] Right divide $L_2 \cdot G$ by $L_1$ and set the remainder to zero. This way we obtain a system $A$ of difference equations for the unknown coefficients $c_i(x)$.
3. Compute the rational solutions of the system $A$ to determine the values for the $c_i(x)$.
This algorithm was efficient for second order operators, but for operators of order three and higher, computing a solution of the system $A$, we get at Step \[step3\], is very costly. Hence in the current implementation we use the new algorithm HOM to compute the gauge transformations that give the set of homomorphisms ${\mathrm{Hom}}_D(V(L_1),V(L_2))$ in $D$ sending $V(L_1)$ to $V(L_2)$ for any $L_1,L_2\in D$. This means in particular that we can drop the condition on the orders, ${\mathrm{ord}}(L_1) = {\mathrm{ord}}(L_2)$.
In short, the algorithm HOM works as follows: For $L=\sum_{i=0}^d a_i(x)\tau^i \in D$, $a_d(x)=1$, we define the $\vee$-adjoint operator $L^\vee:=\sum_{i=0}^d
a_{d-i}(x+i-1)\tau^i$. Then there is a one to one correspondence between ${\mathrm{Hom}}(L_1, L_2)$ and rational (invariant under the difference Galois group) elements of $V(L_1^\vee)
\otimes V(L_2)$. We define a space $\mathcal{M}(L_1^\vee, L_2)$ that is isomorphic to $V(L_1^\vee) \otimes V(L_2)$. Then rational elements of $\mathcal{M}(L_1^\vee, L_2)$ correspond bijectively to elements of ${\mathrm{Hom}}(L_1, L_2)$. Thus, we compute rational elements of $\mathcal{M}(L_1^\vee, L_2)$. This is done by working directly with $L_1^\vee$ and $L_2$, and we avoid computing large operators such as the symmetric product of $L_1^\vee$ and $L_2$ (whose solution space is a homomorphic image of $\mathcal{M}(L_1^\vee, L_2)$.)
Note that if $L_1$ and $L_2$ are of the same order, then HOM returns exactly the gauge transformations. The algorithm HOM is available at <http://www.risc.jku.at/people/ycha/Hom.txt> and more details can be found in [@TensorRatSol]. This is joint work of Yongjae Cha and Mark van Hoeij.
GT-equivalence
--------------
Suppose there is a gauge transformation $G$ and a term transformation $T=\tau-r(x)$ such that the composition of $G$ and $T$, $G \circ T$, maps $V(L_1)$ to $V(L_2)$, i.e. $G: V(L_1 {\circledS}(\tau-r(x)) \rightarrow V(L_2)$. Then $L_1$ and $L_2$ are called GT-equivalent, denoted by $L_1 \sim_{gt}
L_2$, and the composition of $G$ and $T$ is refered to as the GT-transformation from $L_1$ to $L_2$.
Suppose there is a map $\overline{GT}$ which is a multiple composition of gauge transformations and term transformations. Then [@Le10 Theorem 3.3.] shows that we can find a gauge transformation $G$ and a term transformation $T$ such that $\overline{GT}(V(L_1))=G\circ T( V(L_1))$.
### How to compute the GT-Transformation
\[SNFdef\] Let $C$ be a subfield of ${{\mathbb{C}}}$ and $r(x) = cp_1(x)^{e_1} \dotsm p_j(x)^{e_j} \in C(x)$, for some $e_i \in {{\mathbb{Z}}}$, monic irreducible in $p_i(x) \in C[x]$, and let $s_i \in C$ equal the sum of the roots of $p_i(x)$.
$r(x)$ is said to be in [*shift normal form*]{} if $-\deg(p_i(x)) < {\rm Re}(s_i)
\leqslant 0$, for $i=1, \dotsc, j$. We denote by ${\mathrm{SNF}}(r(x))$ the shift normalized form of $r(x)$, which is obtained by replacing each $p_i(x)$ by $p_i(x+k_i)$ for some $k_i
\in {{\mathbb{Z}}}$ such that $p_i(x+k_i)$ is in shift normal form.
${\mathrm{SNF}}(r(x))$ is unique up to the choice of $C$. In the algorithm given in Section \[sec:algo\] we assume $C={{\mathbb{Q}}}$. For $L = a_d(x)\tau^d+ \cdots + a_0(x) \in D $, we denote by $\det(L)$ the determinant of the companion matrix of $L$, which is $(-1)^da_0(x)/a_d(x)$.
[@YC11 Theorem 2.3.9]\[tpsuff\] Suppose $L_1 \sim_{gt} L_2$ for $L_1, L_2 \in C(x)[\tau]$ where $C$ is a subfield of ${{\mathbb{C}}}$. Then there exists a gauge transformation $G \in C(x)[\tau]$ from $L_1 \otimes
(\tau-r(x))$ to $L_2$ for some $r(x) \in C(x)$ where $$r(x)^d =
{\mathrm{SNF}}(\det(L_2)/\det(L_1)), \quad {\mathrm{ord}}(L_1)=d.$$
The original statement of the above theorem uses $C={{\mathbb{C}}}$, but the same proof works for any subfield $C$ of ${{\mathbb{C}}}$.
Suppose we know that $L_1 \sim_{gt} L_2$ for $L_1, L_2 \in {{\mathbb{Q}}}(x)[\tau]$ and we want to find the GT-transformation. By the above theorem there exists $r(x) \in {{\mathbb{Q}}}(x)$ such that ${\mathrm{SNF}}(\det(L_2)/\det(L_1))=r(x)^d$ where $d={\mathrm{ord}}(L_1)$. When $d$ is even $L_1 {\circledS}(\tau-r(x))$ or $L_1 {\circledS}(\tau+r(x))$ can be gauge equivalent to $L_2$. Thus the algorithm Hom will return a non-empty set for either of the two. Furthermore, when $d$ is odd, $L_1 {\circledS}(\tau-r(x))$ is gauge equivalent to $L_2$.
Symmetric powers of operators
-----------------------------
Given an operator $L\in D$ that annihilates a function $u$, then in order to obtain an operator $M\in D$ that annihilates $u^2$ we need the symmetric square of $L$. In this section we state some facts about these operators.
By $L^{\circledS m}$ we denote the $m^\text{th}$ symmetric power of $L$, i.e.,we define $L^{\circledS 1}=L$ and $L^{\circledS m} = L{\circledS}L^{\circledS(m-1)}$. $K$ is called a symmetric square root of $L$ if $L=K^{\circledS 2}$.
Suppose $K_d$ is a differential operator of order 2 then it is known that the order of $L_d^{\circledS m}$ is $m+1$ [@MS85 Lemma 3.2, (b)]. However the following lemma shows that this is not true for difference operators.
[@GH10 Lemma 3]\[lemma:sym-power2\] Let $K= a_2(x)\tau^2+a_1(x)\tau+a_0(x) \in D$. Then
1. if $a_1(x) \neq 0$ then $$K^{\circledS 2}=b_3(x)\tau^3+b_2(x)\tau^2+b_1(x)\tau+b_0(x),$$ where $$\begin{aligned}
\null\kern-1.5em b_3(x)&=a_1(x)a_2(x+1)^2a_2(x)\\
\null\kern-1.5em b_2(x)&=a_1(x+1)a_2(x)(a_0(x+1)a_2(x)-a_1(x+1)a_1(x))\\
\null\kern-1.5em b_1(x)&=a_0(x+1)a_1(x)(a_1(x+1)a_1(x)-a_0(x+1)a_2(x))\\
\null\kern-1.5em b_0(x)&=-a_1(x+1)a_0(x+1)a_0(x)^2.
\end{aligned}$$
2. if $a_1(x) =0$ then $K^{\circledS 2}=a_2(x)^2\tau^2-a_0(x)^2$.
The formulas above give order-minimal operators for both cases.
If a full operator $K=a_2(x)\tau^2+a_1(x)\tau+a_0(x)$ is a symmetric square root of a third order operator $L$, then also $\overline{K} =
a_2(x)\tau^2-a_1(x)\tau+a_0(x)$ is a symmetric square root of $L$. If $u$ is a solution of $K$, then $(-1)^x u$ is a solution of $\overline{K}$. We say $K$ and $\overline{K}$ are conjugates if $K \sim_t \overline{K}$ where the term transformation is $\tau+1$.
Solutions of an equation of type 2 are called Liouvillian solutions [@CH09; @HS99; @GH10]. Suppose $u_1$ is a solution of $K=a_2(x)\tau^2-a_0(x)$ then $\{ u_1, u_2 \}$, where $u_2=
(-1)^xu_1$, forms a basis of $V(K)$ and $u_1^2=u_2^2$. Also, it is easy to verify that for arbitrary orders $m$ it holds that $K^{\circledS m}=a_2(x)^m\tau^2+(-1)^{m+1}a_0(x)^m$ with a similar proof to the one of Lemma \[lemma:sym-power2\].
A second order operator $K$ is called a unity free operator if the solution space of $K$ does not admit a basis $\{ v_1, v_2 \}$ such that $v_1^n=v_2^n$ for some $n \in {{\mathbb{N}}}$.
Let $K=\left( {x}^{2}+x \right) {\tau}^{2}+ \left( 2\,x+{x}^{2} \right)
\tau+{x}^{2}+3\,x+2$. Then a basis of the solution space of $K$ is $\{xw_1^x, xw_2^x \}$ where $w_1$ and $w_2$ are solutions of $z^2+z+1$ in ${{\mathbb{C}}}$. Since $ (xw_1^x)^3=
(xw_2^x)^3=x^3$ for $x \in {{\mathbb{N}}}$, $K$ is not a unity free operator.
\[lm:unfr\] If $K \in D$ is an irreducible second order operator then it is a unity free operator.
We prove this by contraposition. Suppose $K \in D$ is not a unity free operator. Then we may assume $K$ is monic and $V(K)$ admits a basis $\{ v_1, v_2 \}$ such that $v_1^n=v_2^n$ for some $n \in {{\mathbb{Z}}}$. Let $n_0 \in {{\mathbb{Z}}}_{>0}$ be the smallest integer that satisfies $v_1^{n_0}=v_2^{n_0}$, then we may assume $v_1=(u_{n_0}^a)^xf,
v_2=(u_{n_0}^b)^xf$ for some $f \in {\mathbf{S}}$ where $u_{n_0}$ denotes $n_0$th root of unity and $n_0, a, b$ are pairwise relatively prime. Thus $K=(\tau^2-(u_{n_0}^a+u_{n_0}^b)\tau+u_{n_0}^au_{n_0}^b) {\circledS}(\tau-r)$ where $r=\tau(f)/f$. Since $K$ is an element in $D$ and by equation , $r$ is in ${{\mathbb{C}}}(x)$ and this implies $K$ is reducible in D.
\[lm:zd\] If $v \in {\mathbf{S}}, v \neq 0$ satisfies a full second order operator $K=b_2(x)\tau^2+b_1(x)\tau+b_0(x) \in D$ then $v$ is not a zero divisor in ${\mathbf{S}}$.
We will prove that $v$ has only finitely many zeros. Since $K(v)=0$ there is $n_0 \in {{\mathbb{N}}}$ such that $$\label{eq:zd}
b_2(x)v(x+2)+b_1(x)v(x+1)+b_0(x)v(x)=0$$ and $b_i(x)$ has no poles or roots for all $x \geq n_0,i=1,2$. Suppose $v(n_1)=v(n_1+1)=0$ for some $n_1 \geq n_0$. Then by , $v(x)=0$ for all $x \geq n_1$ and this contradicts that $v \neq 0$. Suppose $v(n_2)=0, v(n_2+1) \neq 0$ for some $n_2 \geq n_0$. Then again by , $v(x) \neq 0$ for all $x \geq n_2+1$. Thus $v(x) \neq 0$ for $x$ large enough and hence $v$ is a unit.
\[thm:nonvan\]
If $L=K^{{\circledS}m}$ for some irreducible full second order operator $K \in D$ then ${\mathrm{ord}}(L)=m+1$
Let $\{ v_1, v_2 \}$ be a basis of $V(K)$. We will show that then $\{v_1^iv_2^{m-i} \mid
i=0..m \}$ are linearly independent. Suppose there exist $c_i$ in ${{\mathbb{C}}}$, not all zero such that $c_m v_1^m+c_{m-1} v_1^{m-1}v_2+\cdots+c_0 v_2^m=0$.
By Lemma \[lm:zd\], $v_2$ is not a unit and since $K$ is irreducible operator, by Lemma \[lm:unfr\], $v_1^{n}/v_2^{n} \neq 1$ for any $n \in {{\mathbb{N}}}$. Let $z:=v_1/v_2 \in
{\mathbf{S}}$ and $f(y):=c_m y^m+c_{m-1}y^{m-1}+\cdots+c_0$ then $f(z)=0$, i.e. $f(z(x))=0$ for all $x \in {{\mathbb{N}}}$. Thus, $z(x) \in \{ c \in {{\mathbb{C}}}\mid f(c)=0 \}$ for all $x \in {{\mathbb{N}}}$ and $v_1=zv_2$. Suppose $z$ is not a constant sequence. Since $K$ is an irreducible full operator in $D$, it contradicts that $v_1$ is a solution of $K$. Suppose $z$ is a constant sequence. Then it contradicts that $v_1$ are $v_2$ linearly independent.
In the differential case it holds that if the symmetric square of a third order differential operator $L_d \in {{\mathbb{C}}}(x)[\partial]$ has order 5, then $L_d=K_d^{{\circledS}2}$ for some second order operator $K_d \in {{\mathbb{C}}}(x)[\partial]$. However, the following example shows that this does not hold in the difference case.
\[example:symp\] Let $E:=( x+1) {\tau}^{3}+ ( -28{x}^{3}-4{x}^{4}-36-84 x-73{x}^{2}) {\tau}^{2}+ (
-69x-77{x}^{3}-18-104{x }^{2}-4{x}^{5}-28{x}^{4} ) \tau+{x}^{4}+5{x}^{3}+8{x}^{2 }+4x \in
D$. Then ${\mathrm{ord}}(E^{{\circledS}2})=5$. A solution of $E$ is $x I_x(1)^2$ where $I_x(z)$ denotes the modified Bessel function of the first kind. Then the symmetric square roots of $E$ are $K_{1}=\tau^2+(2+2x)\sqrt{x+1}\tau-\sqrt{x(x+1)}$ and $K_{2}=\tau^2-(2+2x)\sqrt{x+1}\tau-\sqrt{x(x+1)}$, which are not in $D$. A solution of $K_{1}$ and $K_{2}$ are $\sqrt{x}I_x(1)$ and $-\sqrt{x}I_x(1)$, respectively.
Let $B:=z\tau^2-(2x+2)\tau+x+z$. Then a solution of $B$ is $I_x(z)$. Then $E=K_{1}^{{\circledS}2}
\sim_t B^{{\circledS}2}$, but $B$ and $K_{1}$ are not gauge equivalent in $D$, i.e, there is no operator in $D$ that sends $V(B)$ to $V(K_{1})$. Since $\sqrt{x}$ is not a solution of any shift operator in $D$, [@FlajoletGerholdSalvy05 Theorem 5.2]and [@Chen2012111 Lemma A.2], $K_{1}$ is not a symmetric product of $B$ and a difference operator in $D$.
In the differential case, suppose $L_d \sim_t K_d^{{\circledS}2}$, i.e, the solution of $L_d$ can be obtained by multiplying a hyperexponential term $h$ to the solutions of $K_d^{{\circledS}2}$. Let $\{ u_1, u_2\}$ be a basis of the solution space of $K_d$, then $L_d$ admits a basis of the solution space $\{ gu_1^2, gu_1u_2, gu_2^2\}$. However, if $g$ is hyperexponential then $\sqrt{g}$ is also hyperexponential. Thus, $L_d=\tilde{K}^{{\circledS}2}$ for $\tilde{K} \in {{\mathbb{C}}}(x)[\delta]$ such that $\tilde{K}=K {\circledS}(\partial-\frac12 \frac{g'}{g})$ where $g'=\frac{d}{dx} h$. However if $h$ is a hypergeometric term, $\sqrt{h}$ is not guaranteed to be a solution of an operator in $D$.
\[def:gauge-equ\] An irreducible operator $L$ is said to be [*s*olvable in terms of second order in $D$]{} if it is GT-equivalent to $K_1 \circledS K_2 \cdots \circledS K_d$ where the $K_i$’s are irreducible and full second order operators in $D$.
We need the following Lemma to prove Theorem \[thm:ord3dec\].
\[symbs\] Let $K_1, K_2 \in D$ be full second order operators. If ${\mathrm{ord}}(K_1 {\circledS}K_2)=3$ then we can choose a basis $\{ v_1, v_2 \}$ of $V(K_1)$, and a basis $\{
w_1, w_2 \}$ of $V(K_2)$, such that $v_1w_2=v_2 w_1$.
Let $\{ v_1, v_2 \}$ be a basis of $V(L_1)$ and $\{ w_1, w_2 \}$ be a basis of $V(L_2)$. Since ${\mathrm{ord}}(K_1 {\circledS}K_2)=3$, the ${{\mathbb{C}}}$-vector space generated by $\{ v_1w_1, v_1w_2,
v_2w_1, v_2w_2 \}$ has dimension 3. Then there exists $a_1, a_2, a_3 \in {{\mathbb{C}}}$, which are not all zero, such that $$v_1w_2=a_1v_1w_1+a_2v_2w_1+a_3v_2w_2.$$ Suppose $a_1=a_2=0$ and $a_3 \neq 0$ then it contradicts that $v_1$ and $v_2$ are linearly independent. Likewise, if $a_2= a_3=0$ and $a_1 \neq 0$ then it contradicts that $w_1$ and $w_2$ are linearly independent. If $a_1=a_3=0$ and $a_2 \neq 0$ then we have the desired form. So, the remaining cases are either only one of the coefficients $a_1, a_2, a_3$ is zero, or all $a_1, a_2, a_3$ are non-zero. Here, we will prove the case when $a_2$ is the only zero coefficient. Let $\{\tilde{w}_1, \tilde{w}_2 \}$ be another basis of $V(L_2)$ such that $$\begin{pmatrix}
\tilde{w}_1 \\
\tilde{w}_2
\end{pmatrix}
=
\begin{pmatrix}
0 & a_3 \\
-a_1 & 1
\end{pmatrix}
\begin{pmatrix}
w_1 \\
w_2
\end{pmatrix}$$ Then for $\{\tilde{w_1},\tilde{w_2}\}$ we have $v_1\tilde{w_2}=v_2\tilde{w_1}$ as claimed.
\[thm:ord3dec\] Let $L$ be an operator of order 3, irreducible and solvable in terms of second order in $D$. Then $L \sim_{gt} K^{\circledS 2}$ for some irreducible full second order operator $K \in D$ and furthermore
(a) \[c1\] $L \sim_t K^{{\circledS}2}$ then ${\mathrm{ord}}(L^{\circledS 2})=5$.
(b) \[c2\] if the gauge transformation of $L \sim_{gt} K^{{\circledS}2}$ is not a single term operator then ${\mathrm{ord}}(L^{\circledS 2})=6$.
Let $L$ be a third order, irreducible operator that is solvable in terms of second order in $D$. Then by definition (and the restriction of the order), there exist two irreducible full second order operators $K_1, K_2 \in D$ such that $L\sim_{gt}K_1{\circledS}K_2$. By Lemma \[symbs\], a suitable basis $\{v_1,v_2\}$ for $K_1$, and a suitable basis $\{w_1,w_2\}$ for $K_2$ can be chosen, such that $v_1w_2=v_2w_1$. Let $h=w_1/v_1=w_2/v_2$, then $h \in {\mathbf{S}}$ and $w_1=hv_1$, and $w_2=hv_2$. Since $\{ w_1, w_2\}$ is a basis for an operator in $D$, $h$ is hypergeometric and this implies that $K_1 \sim_t K_2$ with term transformation $\tau-r$, where $r$ is the certificate of the hypergeometric term $h$. Summarizing, by Lemma \[lemma:sym-power2\], $L \sim_{gt} K^{\circledS 2}$ for some full operator $K
\in D$.
\(a) Let $\{ v_1, v_2 \}$ be a basis of $V(K)$ and $\tau-r$ be the term transformation from $K^{{\circledS}2}$ to $L$ and $h$ be a solution of $\tau-r$. Then $\{ hv_1^2, hv_1v_2, hv_2^2 \}$ forms a basis of $L$ and thus ${\mathrm{ord}}(L^{\circledS 2})=5$ by Lemma \[thm:nonvan\].
\(b) Let $\{ v_1, v_2 \}$ be a basis of $V(K)$. Then $\{ G(hv_1^2), G(hv_1v_2), G(hv_2^2)
\}$ forms a basis of $V(L)$, where $G=c_2(x)\tau^2+c_1(x)\tau+c_0(x) \in D$ is a non single term operator and $h$ is a hypergeometric term. Then $G(hv_1^2)G(hv_2^2) \neq
G(hv_1v_2)^2$ and this implies ${\mathrm{ord}}(L^{\circledS 2})=6$.
Suppose $L_d$ is a differential operator of order 3 and $L_d = K_d^{{\circledS}2}$ for some second order differential operator $K_d$. Then it is well known that there exists a formula to construct this $K_d$, see [@MS85 Lemma 3.4]. The case where only gauge-equivalence holds, i.e., $L_d\sim_g K_d^{{\circledS}2}$, is more interesting. In [@vH07] third order operators are treated with a focus on determining both $K_d$ and a gauge transformation.
It is possible to implement a similar algorithm for difference equations which returns the second order operator $K$ to which the given $L$ can be reduced to and a gauge transformation. However, in the difference case, in order to give a closed form solution of $K$ other algorithms need to be applied or a table look-up. Also, even if we are in case , finding $K$ is not as simple as in the differential case, in particular if there is a parameter included in the input. Morever to distinguish the cases, the symmetric square of a third order operator needs to be computed which can become costly if many parameters are involved.
Local data {#sec:ld}
==========
The local data that we are using are the valuation growths at finite singularities in ${{\mathbb{C}}}/{{\mathbb{Z}}}$ and generalized exponents at the point of infinity. This data is invariant under GT transformations. In this section, we review the definition and an invariance property (Theorem \[gp\], Theorem \[genexp\]) of local data from [@YC11; @CH09; @CHG10; @HO99]. We omit proofs in this paper.
Finite singularities
--------------------
Valuation growth was first introduced in [@HO99] and an algorithm to compute it was given in the same paper. Let $L=a_d\tau^d+\cdots+a_0\tau^0 \in D$. After multiplying $L$ from the left by a suitable element of ${{\mathbb{C}}}(x)$, we may assume that the $a_i$ are in ${{\mathbb{C}}}[x]$ and gcd$(a_0,\ldots,a_d)=1$. Then $q \in {{\mathbb{C}}}$ is called a [ *problem point*]{} of $L$ if $q$ is a root of the polynomial $a_0(x)a_d(x-d)$ and $p \in
{{\mathbb{C}}}/ {{\mathbb{Z}}}$ is called a [*finite singularity*]{} of $L$ if $L$ has a problem point in $p$ (i.e. $p=q+{{\mathbb{Z}}}$ for some problem point $q$). Let $p \in {{\mathbb{C}}}/{{\mathbb{Z}}}$. For $a,b \in p \subset
{{\mathbb{C}}}$ we say $a > b$ iff $a-b$ is a positive integer.
Let $\varepsilon$ be a new indeterminant, i.e., transcendental over ${{\mathbb{C}}}$. We define $L_\varepsilon:=\sum_{i=0}^d a_i(x+\varepsilon)\tau^i$ which is obtained by substituting $x$ with $x+\varepsilon$ in $L$. The map $L \mapsto L_\varepsilon$ defines an embedding (as non-commutative rings) of ${{\mathbb{C}}}(x)[\tau]$ in ${{\mathbb{C}}}(x,\varepsilon)[\tau]$. Hence, if $L=MN$, then $L_\varepsilon=M_\varepsilon N_\varepsilon$.
Let $a \in \overline{C}(\epsilon)$ and $\overline{C}[[\epsilon]]$ be the ring of formal power series over $\overline{C}$ in $\epsilon$. The $\varepsilon$-valuation $v_\varepsilon(a)$ of $a$ at $\varepsilon=0$ is the element of ${{\mathbb{Z}}}\cup {\infty}$ defined as follows: if $a\neq0$ then $v_\varepsilon(a)$ is the largest integer $m \in
{{\mathbb{Z}}}$ such that $a/\varepsilon^m \in \overline{C}[[\varepsilon]]$, and $v_\varepsilon(0)=\infty$.
We define an ${\mathrm{ord}}(L)$ dimensional ${{\mathbb{C}}}(\varepsilon)$-vector space $$V_p(L_\varepsilon):=\{\tilde{u}:p \rightarrow {{\mathbb{C}}}(\varepsilon) \mid L_\varepsilon(\tilde{u})=0\}.$$ Let $q_l$ be the smallest root of $a_0(x)a_d(x-d)$ in $p$, so $q_l$ is the smallest problem point in $p$. Likewise we define $q_r$ to be the largest root of $a_0(x)a_d(x-d)$ in $p$. If $p$ is not a singularity, that is, if $a_0$ and $a_d$ have no roots in $p$, then choose two arbitrary elements in $p$ and define $q_l, q_r$ to be those two elements.
\[box\] For non-zero $\tilde{u} \in V_p(L_\varepsilon)$ and for $a,
b \in {{\mathbb{C}}}$ if $b=a+d-1$, where $d={\mathrm{ord}}(L_\varepsilon)$, we define the [*box-valuation*]{} $$v^a_b(\tilde{u})=\min\{v_\varepsilon(\tilde{u}(m))|m=a,a+1,\ldots,b\}.$$
\[vl\] With $q_l, q_r$ chosen as above, we have $$v_{q-1}^{q-d}(\tilde{u})=v_{q_l-1}^{q_l-d}(\tilde{u}) \ \ \text{\rm for all} \ q \in \{q_l-1,q_l-2,q_l-3,\ldots \},$$ $$v_{q+d}^{q+1}(\tilde{u})=v_{q_r+d}^{q_r+1}(\tilde{u}) \ \ \text{\rm for all} \ q \in \{q_r+1,q_r+2,q_r+3,\ldots \}$$.
We define $v_{\varepsilon,l}(\tilde{u})$ as $v_{q_l-1}^{q_l-d}(\tilde{u})$ which, by Lemma \[vl\], equals the box valuation of any box on the left of $q_l$. Likewise we define $v_{\varepsilon,r}(\tilde{u})$ as $v_{q_r+d}^{q_r+1}(\tilde{u})$.
\[def:valg\] Define the [*valuation growth*]{} of non-zero $\tilde{u} \in V_p(L_\varepsilon)$ as $$g_{p,\varepsilon}(\tilde{u})=v_{\varepsilon,r}(\tilde{u})-v_{\varepsilon,l}(\tilde{u}) \in {{\mathbb{Z}}}.$$ Define the [*set of valuation growths*]{} of $L$ at $p$ as $$\overline{g}_p(L)=\{g_{p,\varepsilon}(\tilde{u}) \mid \tilde{u}\in V_p(L_\varepsilon),\tilde{u}\neq0\}\subset {{\mathbb{Z}}}.$$
If $L$ is a first operator operator then $\overline{g}_p(L)$ has only one element.
\[apartsing\] Let $L$ be a difference operator and $p \in {{\mathbb{C}}}/{{\mathbb{Z}}}$ be a finite singularity of $L$. If $\overline{g}_p(L)$ has more than one element then $p$ is called an [*essential singularity*]{}.
The algorithm given in [@HO99] determines the set $$\{ \overline{g_p}(L) \mid p \text{ \ is an essential singularity of}\ L \}.$$
\[gp\][@CH09 Theorem 1] If $L_1$ and $L_2$ are gauge equivalent then $$\max(\overline{g_p}(L_1))=\max( \overline{g_p}(L_2) ) \quad \text{and} \quad \min(\overline{g_p}(L_1))=\min(\overline{g_p}(L_2))$$ for all $p \in {{\mathbb{C}}}/{{\mathbb{Z}}}$.
The following lemma is an immediate consequence of Definition \[def:valg\].
For each $p \in {{\mathbb{C}}}/{{\mathbb{Z}}}$, $$\max(\overline{g_p}(L^{{\circledS}2})=2\max(\overline{g_p}(L)) \quad \text{and} \quad
\min(\overline{g_p}(L^{{\circledS}2})=2\min(\overline{g_p}(L)).$$
The above theorem only gives invariance under gauge equivalence. To have invariance under GT-equivalence, we need to define one more set. Suppose $L_1 \sim_{gt} L_2$, then $L_1 {\circledS}(\tau-r(x)) \sim_g L_2$ for some $r(x) \in {{\mathbb{C}}}(x)$. Then $$\max(\overline{g_p}(L_2))=\max( \overline{g_p}(L_1) )+d \quad \text{and} \quad \min(\overline{g_p}(L_2))=\min( \overline{g_p}(L_1) )+d$$ where $\{ d\}=\overline{g_p}(\tau-r(x))$, $d \in {{\mathbb{Z}}}$. So $d_p(L)=\max(\overline{g_p}(L))-\min(\overline{g_p}(L))$ is invariant under GT-equivalence. Thus, for a difference operator $L \in $, we define a set of ordered pairs $${\mathrm{ValG}}:=\{ (p, d_p(L)) \in {{\mathbb{C}}}/{{\mathbb{Z}}}\ \times \ {{\mathbb{Z}}}_{\geq 0} \mid p \text{ \ is an essential singularity of \ } L \}.$$
Singularity at infinity
-----------------------
Let ${{\mathbb{K}}}:={{\mathbb{C}}}((t)), x=1/t$ be the field of formal Laurent series and ${{\mathbb{K}}}_r={{\mathbb{C}}}((t^{1/r}))$ for $r \in {{\mathbb{N}}}$. We define the valuation for $a \in {{\mathbb{K}}}$ as the smallest power of $a$ whose coefficient is non-zero and denote it by $v(a)$. This definition can be extended to $\hat{D}={{\mathbb{K}}}[\tau]={{\mathbb{K}}}[\Delta]$, where $\Delta:=\tau-1$ denotes the forward difference, by setting $$v(L)=\min \{ v(a_i)+i\mid L=a_0+\cdots+a_d \Delta^d\}$$ for any operator $L\in\hat{D}$.
\[ind\] Let $L \in {{\mathbb{K}}}[\tau]$. There exists a polynomial $P$ such that for every $n \in {{\mathbb{Z}}}$ we have $$\label{tn}
L(t^n) = P(n) t^{n + v(L)} + \cdots$$ where the dots refer to terms of valuation $> n + v(L)$.
${\mathrm{Ind}}_L(n)$, the [*indicial polynomial*]{} of $L$, is the polynomial $P(n)$ in Lemma \[ind\] .
Lemma 9.2 in [@CHG10] states that if $N \in {{\mathbb{Z}}}$ is a root of ${\mathrm{Ind}}_L(n)$ then there is $u \in
{{\mathbb{K}}}$ such that $L(u)=0$ and $v(u)=N$. However, there is no one-to-one correspondence between solutions of $L$ in ${{\mathbb{K}}}$ and integer roots of ${\mathrm{Ind}}_L(n)$. For this matter, we introduce the ring ${{\mathbb{K}}}[l]$, where $l$ is a solution of $\tau(l) - l =
t$, see [@LF99] for existence of $l$. We extend valuation on ${{\mathbb{K}}}$ to ${{\mathbb{K}}}[l]$ by: for $a=a_1t^d+\cdots \in {{\mathbb{K}}}[l]$, $a_i \in {{\mathbb{C}}}[l]$, $d \in {{\mathbb{Z}}}$, and $a_1 \neq 0$, we let $v(a)=d$. With this notion we obtain the following theorem which is equivalent to [@YC11 Theorem 3.2.10] and [@PS97 Lemma 6.1].
$p \in {{\mathbb{Z}}}$ is a solution of ${\mathrm{Ind}}_L(n)$ if and only if $L$ has a solution $u \in {{\mathbb{K}}}[l]$ with $v(u)=p$.
An immediate consequence of the above theorem is the following corollary.
\[indsym\] If $p_1$ and $p_2 \in {{\mathbb{Z}}}$ are the solutions of the indicial equations of $L_1$ and $L_2$, respectively, then $p_1+p_2$ is a solution of the indicial equation of $L_1 \circledS L_2$.
Define the action of $\tau$ on ${{\mathbb{K}}}_r$ as: $$\label{acttau}
\begin{split}
\tau(t^{\frac1r}) &= t^{\frac1r} (1+t)^{-\frac1r}\\
&=t^{\frac1r} (1 - \frac{1}{1!}\frac1r t + \frac{1}{2!}\frac1r(\frac1r+1) t^2 \\
& -\frac{1}{3!}\frac1r(\frac1r+1)(\frac1r+2) t^3 + \cdots ) \in {{\mathbb{K}}}_r.
\end{split}$$
Since we have defined the action of $\tau$ on ${{\mathbb{K}}}_r$, we can now apply the formula for the term symmetric product in to ${{\mathbb{K}}}_r[\tau]$. Let $E_r$ and $\tilde{G}_r$ be the following subset and subgroup, respectively, of ${{\mathbb{K}}}_r^*$: $$E_r=\biggr\{a \in K_r^* \mid a=ct^v(1+\displaystyle\sum_{i=1}^r a_i t^{i/r}), a_i \in {{\mathbb{C}}},
c \in {{\mathbb{C}}}^*, v \in \tfrac1r{{\mathbb{Z}}}\biggl\},$$ $$\tilde{G}_r=\biggr\{ a \in K_r^* \mid a=1+\displaystyle\sum_{i=r+1}^{\infty} a_i t^{i/r} , \ a_i \in {{\mathbb{C}}}\biggl\}.$$
Now $E_r$ is a set of representatives for ${{\mathbb{K}}}^*_r / \tilde{G}_r$. The composition of the natural maps ${{\mathbb{K}}}^*_r \rightarrow {{\mathbb{K}}}^*_r/\tilde{G}_r \rightarrow E_r$ defines a natural map $${\mathrm{Trunc}}:{{\mathbb{K}}}^*_r \rightarrow E_r .$$ Let $$G_r=\{ a \in {{\mathbb{K}}}_r^* \mid a=1+\frac{m}{r}t+\displaystyle\sum_{i=r+1}^{\infty} a_i t^{i/r} , \ a_i \in {{\mathbb{C}}}, \ m \in {{\mathbb{Z}}}\}.$$
\[def:req\] Let $r \in {{\mathbb{N}}}$ then for $a, b \in E_r$, we say $a$ is $r$-equivalent to $b$, $a \thicksim_r b$, when $a/b \in G_r$.
Note that $a \thicksim_r b$ if and only if $a_r \equiv b_r\!\mod \frac1r {{\mathbb{Z}}}$ with $a_r$ as in the definition of $E_r$, $a_i=b_i$ for $i < r$, and $c,v$ matching as well.
Let $g \in E_r$ for some $r \in {{\mathbb{N}}}$. We say that $g$ is a [*generalized exponent*]{} of $L$ with multiplicity $m$ if and only if zero is a root of ${\mathrm{Ind}}_{\tilde{L}}(n)$ with multiplicity m where $\tilde{L}= L \circledS (\tau-\frac1g)$. We denote by ${\mathrm{GenExp}}(L)$ the set of generalized exponents of $L$.
Suppose $L=\tau-r(x) \in D$ then ${\mathrm{GenExp}}(L)=\{ {\mathrm{Trunc}}(r(t)) \}$.
\[genexp\] If two operators $L_1$ and $L_2$ are gauge equivalent then for each $g_1 \in
{\mathrm{GenExp}}(L_1)$ there is a $g_2 \in {\mathrm{GenExp}}(L_2)$ such that $g_2$ is equivalent to $g_1$.
This theorem has been proven first in [@CHG10]. An alternative proof can be found in [@YC11].
\[symgen\] Suppose $L, L' \in D$ then $${\mathrm{GenExp}}(L \circledS L')= \{ {\mathrm{Trunc}}(gg') \mid g \in {\mathrm{GenExp}}(L), g' \in {\mathrm{GenExp}}(L') \}
$$
$L \circledS L' \circledS (\tau-\frac{1}{gg'})=L \circledS (\tau-\frac{1}{g}) \circledS
L' \circledS (\tau-\frac{1}{g'})$ and since 0 is a solution of $L \circledS
(\tau-\frac{1}{g})$ and $L \circledS (\tau-\frac{1}{g'})$, 0 is also a solution of the indicial equation of $L \circledS L' \circledS (\tau-\frac{1}{{\mathrm{Trunc}}(gg')})$ by Lemma \[indsym\]
Likewise for the valuation growth, we need to define one more set to have invariance for GT-equivalence. Suppose $L_1 {\circledS}(\tau-r(x)) \sim_g L_2$ for some $r(x) \in {{\mathbb{C}}}(x)$. Then $${\mathrm{GenExp}}(L_2)=\{ g_r g \mid g \in {\mathrm{GenExp}}(L_1), \ \{g_r\}={\mathrm{GenExp}}(\tau-r(x)) \}.$$
Thus we define the following set, $${\mathrm{Gquo}}(L):=\{ {\mathrm{Trunc}}(g_i/g_j) \mid g_i \neq g_j, g_i, g_j \in {\mathrm{GenExp}}(L)\}$$ and then ${\mathrm{Gquo}}(L_1)={\mathrm{Gquo}}(L_2)$ if $L_1 \sim_{gt} L_2$.
Table of base equations
=======================
In [@YC11; @CHG10], we have formed a table of base equations of order 2, call it TB, as follows;
- collect equations with known solution from [@abst; @AG76].
- for any closed form expression that shows up often in the literature, generate a base equation with existing algorithms [@ChyzakDM; @KoutschHF].
For the algorithm given in Section \[sec:algo\], we have computed symmetric squares of each base equation in TB yielding an entry in TB2 of a base equations of third orders. Moreover we have generated further base equations as follows:
Suppose $u(x)$ is a solution of an operator $L=\sum_{i=0}^d a_i(x)\tau^i$. Then $u(x/m)$ is a solution of the operator $$\label{eq:Tm}
L_{(m)}=\sum_{i=0}^d a_i(x/m)\tau^{mi}.$$ As input for our algorithm we accept only operators of order three and the above equation may be of higher order. One way of obtaining the base equation for $u(x/m)$ in this case is using $L_{(m)}$ when it is a multiple of an operator $M\in D$ for which $M(u(x/m))=0$. Since $L_{(m)}$ as constructed above is not guaranteed to be the minimial order operator we compute ${\mathrm{Hom}}(L_{(m)},L_{(m)})$. If the algorithm HOM returns the identity map this means that $L_{(m)}$ is in fact order-minimal. These cases are neglected and we use $L_{(m)}$ as a base equation only if HOM returns a non-trivial map.
For instance for the squared hypergeometric function in the table below, ${}_2F_1\Big[\genfrac{}{}{0pt}{}{-x/2+a, \ x/2+b}{c} ; z \Big]^2$, an annihilating operator $L_{(2)}$ can be obtained starting from an operator $L_{(1)}$ annihilating ${}_2F_1\Big[\genfrac{}{}{0pt}{}{-x+a, \ x+b}{c} ; z \Big]^2$ using . Then the order-minimality of $L_{(2)}$ is checked with the algorithm HOM. In this case HOM returns a non-identity map and hence we save $L_{(2)}$ in the table.
If $ct^vf \in {\mathrm{GenExp}}(L)$, then $z_c{\mathrm{Trunc}}(g_m^vf) \in {\mathrm{GenExp}}(L_{(m)})$, where $z_c$ is a root of $z^m=c$ and $g_m^v \in {\mathrm{GenExp}}(\tau^m-(\frac{x}{m})^v)$. Thus, we can detect whether an input operator may have a solution $u(x/m)$ if a base equation for $u(x)$ is in our table. However, it is more efficient to compute the base equation for small values of $m$.
Example of base equations {#sec:base}
-------------------------
Here we list a small part of the table which is needed in Section \[sec:algo\] and \[apps\]. In the following table they are listed under (a) a solution (b) the corresponding ${\mathrm{Gquo}}$, and (c) the ${\mathrm{ValG}}$. The full table can be found at <http://www.risc.jku.at/people/ycha/TB2.txt>.
1. \[ex2\]
1. ${}_2F_1\Big[\genfrac{}{}{0pt}{}{-x/2+a, \ x/2+b}{c} ; z \Big]^2$
2. $ \left\{ -1,- \left( 2\,z-1\pm2\,\sqrt {{z}^{2}-z} \right) ^{2}, \pm(2\,z-1\pm2\,\sqrt {{z}^{2}-z}) \right\}
$
3. $\{ (-2b, 2), (2a, 2), (2a-2c, 2), (2c-2b, 2) \}$
2. 1. $P_x(z)^2$ (Legendre polynomials squared)
2. $\left\{ -1+2\,{z}^{2}\pm2\,\sqrt {-{z}^
{2}+{z}^{4}}, \left( -1+2\,{z}^{2}\pm2\,\sqrt {-{z}^{2}+{z}^{4}} \right) ^{
-1}, \frac{ -1+2\,{z}^{2}\mp2\,\sqrt {-{z}^{2}+{z}^{4}}}{ -1
+2\,{z}^{2}\pm2\,\sqrt {-{z}^{2}+{z}^{4}} } \right\}
$
3. $\{(0, 4) \}$
3. 1. $H_x(z)^2$ (Hermite polynomials squared)
2. $\left\{ -1\pm\sqrt {-2\,{z}^{2}}T+{z}^{2}{T}^{2},1\pm2\,\sqrt {-2\,{z}^{2}}T-4\,{z}^{2}{T}^{2}\right\}
$
3. $\{(0, 2) \}$
Algorithm {#sec:algo}
=========
The basic structure of the algorithm is the same that was given in [@YC11]. Here we use an extended table of base equations and a more efficient algorithm for computing the gauge transformation, as mentioned in Section \[hom\].
Suppose $L$ is the input operator with local data $${\mathrm{Gquo}}(L)=\{ a, \overline{a}, b, \overline{b}, c, \overline{c} \} , \quad {\mathrm{ValG}}(L)=\{ (0,4) \}$$ for $a, b, c \in {{\mathbb{C}}}$. By comparing the corresponding data in TB2, we can find that local data of $L$ matches with the data of in Section \[sec:base\]. Let $L_{lgd}$ be the operator of which $P_x(z)^2$ is a solution. To compute the parameter $z$, we compare $a$ with each entry of ${\mathrm{Gquo}}(L_{lgd})$ and compute the set of candidates of possible values for $z$ which is, $$\left\{ \pm\frac12\,\sqrt {{\frac {2\,a\pm\sqrt {2\,{a}^{2}+{a}^{3}+a}}{a}}}, \
\pm\frac12\,{\frac {a+1}{\sqrt {a}}} \right\}.$$ Substituting $z$ by each of the values of the above set, a set of equations $cdd2$ is obtained. It remains to cheek for each of the equations in $cdd2$ whether there is a GT-transformation to $L$ and if so then we return the closed form solution by applying the GT-transformation to $P_x(z)^2$.\
[**Algorithm [*solver2*]{}**]{}\
[**Input**]{}: A third order normal operator $L_I \in {{\mathbb{Q}}}[x, \tau]$.\
[**Output**]{}: Either at least one closed form solution of $L$ in the form of $c_0(x)u(x)^2+c_1(x)u(x+1)^2+c_2(x)u(x+2)^2$ where $c_i(x)$ are hypergeometric terms and $u(x)^2$ is a solution in TB2 or otherwise the empty set.
1. $cdd1:=\{\}, GQ:={\mathrm{Gquo}}(L_I), VG:={\mathrm{ValG}}(L_I)$ .
2. Find the base equations in TB2 by comparing $GQ$ and $VG$ with the corresponding data in the table.
1. if there is no match then return ‘Not solvable within the Table’.
2. if there is a matching equation $L_c$, $cdd1:=cdd1 \cup \{L_c\}$.
3. \[four\] For each $L_c \in cdd1$, compute candidate values for the parameters using $GQ$ and $VG$.
4. Construct a set $cdd2$ by substituting parameters by the values determined in Step.\[four\]
5. For each $L_p \in cdd2$ check if there exists a GT-Transformation from $L_p$ to $L_I$.
1. if there is a GT-transformation then apply $GT$ to the known solution of $L_c$ and return the solution.
2. if there is no GT-transformation found return ‘Not solvable within the Table’.
Improvement
===========
A similar approach can be applied to higher order operators that are solvable in terms of order two. Suppose $L_4$ is a fourth order operator that is solvable in terms of order two in $D$. Then $L_4$ is equal or gauge equivalent to either $K_1^{{\circledS}3}$ or $K_1{\circledS}K_2$ for some second order operators $K_1,K_2\in D$ with nonvanishing coefficients. The candidates for $K_i$ can be detected analogously using Theorem \[symgen\].
Concerning the applications to proving positivity of special functions inequalities it has to be noted that representations in terms of finite linear combination of squares with non-negative coefficients need not exist on the full range of validity of a given inequality, as can be seen below. Further investigations of the applicability of this approach as well as an implementation of the above mentioned extension to higher order recurrences are ongoing work.
Applications {#apps}
============
Our main motivation to extend finding closed form solutions of difference equations in terms of symmetric products is to develop an algorithmic approach for proving special functions inequalities. Existing symbolic methods [@GKIneq; @MKSumCracker; @MKVP10] are based on using Cylindrical Algebraic Decomposition (CAD) which in several examples has proven to be an effective way for proving positivity of sequences that are given only in terms of their defining sequences. However, it is sometimes unsatisfiable to have a proof that only comes with “True” without any certificate. Some classical proofs of inequalities are using rewriting of the given expression as linear combination of easy to verify positive objects such as sums of squares. The present work tries to make this approach algorithmic. Certainly it will not provide answers for any special functions inequality, but it is a first step in a new direction of automatic inequality proving. Below we give two examples, one for each of the cases distinguished above, of classical problems that can be solved fully or at least partially using the presented algorithm. Note that all of these identities stated can be proven easily using existing algorithms for symbolic summation. The novelty is the automatic discovery of certain closed form expressions for sequences that are given only in terms of their defining recurrence relation. In this sense it is comparable to the above mentioned algorithms based on CAD.
Clausen’s formula {#Clausen}
-----------------
Proofs of special function inequalities often depend on a variety of classical techniques such as argument transformations, integral representations of hypergeometric series and many more. For instance in the proof of the Askey-Gasper inequality [@AG76],which played a key role in the proof of the Bieberbach conjecture by de Branges [@deBranges], Clausen’s formula $$\begin{aligned}
\label{cl}
&{}_3F_2\biggl[\genfrac{}{}{0pt}{}{-x, x+\alpha+1, \frac{\alpha+1}{2}}{\alpha+1, \frac{\alpha+2}{2}} ; z \biggl] \\
&\quad= {}_2F_1\biggl[\genfrac{}{}{0pt}{}{-\frac12x, \frac12x+\frac12(\alpha+1)}{\frac{\alpha}{2}+1} ; z \biggl]^2\nonumber\end{aligned}$$ entered at a central point. Zeilberger [@SBE93] has shown how this identity can be proven using symbolic summation. By means of the algorithm presented here, Clausen’s formula can be discovered entirely automatic.
The hypergeometric function in satisfies a third order recurrence that is given by the operator $L_3$ and is too large to be displayed here. It can however be found easily common symbolic summation algorithms [@Zeil90a; @ChyzakDM; @KoutschHF]. This difference operator is the input for our procedure and we start by determining the local data given by $$\begin{aligned}
&{\mathrm{Gquo}}(L_3)= \left\{ -\left( 2\,z-1\pm2\,\sqrt {{z}^{2}-z} \right) ^{2},-2\,z+1\pm2\,
\sqrt {{z}^{2}-z} \right\}
,\\
& {\mathrm{ValG}}(L_3)=\left\{ (0, 2),(-\alpha, 2) \right\}.
\end{aligned}$$ A table look-up shows that this local data is compatible with \[ex2\] in Section \[sec:base\]. Comparing local data and solving the system mod ${{\mathbb{Z}}}$ the following candidates for $a,b$ and $c$ can be found: $$a\in\{0,\tfrac12\},\quad b \in \{\tfrac12 \alpha, \tfrac12 \alpha+\tfrac12\},\quad c \in \{\tfrac12 \alpha+1, \tfrac12\alpha+\tfrac32 \}.$$ There is no term transformation for these operators and an application of HOM shows that we obtain a constant map if $a=0$, $b=\frac12
\alpha+\frac12$ and $c=\frac12 \alpha+1$.
Turán inequality for Hermite polynomials {#turan}
----------------------------------------
The positivity of Turán determinants has been proven for many different families of orthogonal polynomials. The first Turán inequality was formulated for Legendre polynomials [@TP50] and Szegö [@SG48] has given four different proofs of this inequality. Szwarc [@Szwarc] has provided a more general approach for proving Turán type inequalities based on the mere knowledge of the recurrence coefficients satisfied by the given sequence. Gerhold and Kauers [@MKTuran] have proven and improved this type of inequalities using their CAD-based method. The approach presented here does not give a full proof for Turán type inequalities, however it gives a representation of the given determinant in sums of squares derived from the third order annihilating operator of the determinant. In the case of Hermite polynomials this yields a representation that gives positivity in the limit for $n$ tending to infinity.
Turán’s inequality for Hermite polynomials $H_x(z)$ reads as follows: $$\Delta_x(z)=H_{x+1}(z)^2- H_{x}(z)H_{x+2}(z) \ge 0,\quad n\ge0,\ z\in\mathbb{R}.$$ Then an annihilating operator of $\Delta_x(z)$ is $L_h:=\tau^3+(2x+2-4z^2)\tau^2-4(x+2)(x-2z^2+4)\tau-8(1+x)(x+2)^2$ and the local data of this operator is $$\begin{aligned}
&{\mathrm{Gquo}}(L_h)= \left\{ -1\pm\sqrt {-2\,{z}^{2}}t^\frac12+ \left( {z}^{2}\pm1 \right) t,
1\pm2\,\sqrt {-2\,{z}^{2}}t^\frac12-4\,{z}^{2}t \right\}
,\\
& {\mathrm{ValG}}(L_h)=\left\{ (0, 2) \right\}.
\end{aligned}$$
$-1\pm\sqrt {-2\,{z}^{2}}t^\frac12+ \left( {z}^{2}\pm1 \right)t$ are elements in ${\mathrm{Gquo}}(L_h)$ and these are equivalent to $-1\pm\sqrt {-2\,{z}^{2}}t^\frac12+ {z}^{2}t$ under $\sim_2$, see Definition \[def:req\] for $\sim_2$. Thus the local data of $L_h$ correspond to those of the third entry of the table given in Section \[sec:base\].
Using the algorithm described above a gauge transformation can be found that applied to $H_x(z)^2$ yields the following equivalent formulation $$\Delta_x(z) = \tfrac12 H_{x+1}^2(z)+2(x+1-z^2) H_x(z)^2 + 2x^2 H_{x-1}^2(x).$$ This representation gives the positivity of Turán’s inequality for $$z\in[-\sqrt{x+1},\sqrt{x+1}\ ],\quad x\ge0.$$
[^1]: Supported by the Austrian Science Fund (FWF) under grant P22748-N18
|
---
abstract: 'We consider two quantities that measure complexity of binary strings: $\operatorname{\mathit{KM}}(x)$ is defined as the minus logarithm of continuous a priori probability on the binary tree, and $\operatorname{\mathit{K}}(x)$ denotes prefix complexity of a binary string $x$. In this paper we answer a question posed by Joseph Miller and prove that there exists an infinite binary sequence $\omega$ such that the sum of $2^{\operatorname{\mathit{KM}}(x)-\operatorname{\mathit{K}}(x)}$ over all prefixes $x$ of $\omega$ is infinite. Such a sequence can be chosen among characteristic sequences of computably enumerable sets.'
author:
- 'Mikhail Andreev, Akim Kumok'
title: 'The sum $2^{\operatorname{\mathit{KM}}(x)-\operatorname{\mathit{K}}(x)}$ over all prefixes $x$ of some binary sequence can be infinite'
---
Introduction
============
Algorithmic information theory tries to define the notion of *complexity* of a finite object and the related notion of its *a priori probability*. Both notions have different versions, and many of these versions can be used to define algorithmic randomness. To explain why the result of this paper could be interesting, let us start with a short survey of these notions and related results; for the detailed exposition of the related definitions and results see, e.g., [@kolmbook; @ShenNotes].
A notion of prefix complexity was introduced by Levin (see [@LevinThesis; @LevinPpi; @GacsSymmetry] and later by Chaitin [@ChaitinPrefix] (in different forms). Let $D$ be a computable function whose arguments and values are binary strings. This function is called *prefix-free* if its domain is prefix-free, i.e., does not contain both a string and its non-trivial prefix. Define $\operatorname{\mathit{K}}_D(x)$ the minimal length of $p$ such that $D(p)=x$. Among all functions $\operatorname{\mathit{K}}_D$ for all computable prefix-free $D$ there exists a mininal one (up to $O(1)$ additive term); one of them is fixed and called $\operatorname{\mathit{K}}(x)$, the prefix-free complexity of $x$. (Another version, which gives the same function $\operatorname{\mathit{K}}$ with $O(1)$-precision, uses *prefix-stable* functions $D$: this means that if $D(x)$ is defined, then $D(xz)$ is defined and equals $D(x)$ for all $z$).
The prefix complexity is closely related with the *discrete a priori probability* [@LevinThesis; @LevinPpi; @ChaitinPrefix]. Consider a non-negative total real function $m$ defined on binary strings. We call $m$ a *discrete semimeasure* if $\sum_x m(x)\le 1$. We say also that $m$ is *lower semicomputable* if $m(x)$ can be represented as a limit of a non-decreasing sequence $M(x,0),M(x,1),\ldots$ where $M$ is a non-negative total function of two arguments with rational values. Levin introduced this notion and showed that there exist a maximal (up to $O(1)$-factor) lower semicomputable semimeasure, and this semimeasure is equal to $2^{-\operatorname{\mathit{K}}(x)+O(1)}$. We fix some maximal lower semicomputable semimeasure, call it the *discrete a priori probability* (see below about the continuous a priori probability), and denote it in the sequel by $\mathbf{m}(x)$.
Discrete lower semicomputable semimeasures are exactly the output distributions of probabilistic machines without input that produce their output at once (say, write a binary string and then terminate). We can also consider probabilistic machines without input that produce their output bit by bit (and never terminate explicitly, though it may happen that they produce only finitely many output bits). The output distributions of such machines are described by lower semicomputable *continuous semimeasures* (=semimeasures on a binary tree), introduced in [@ZvonkinLevin]. By a continuous semimeasure we mean a non-negative total function $a$ that is defined on binary strings and has the following two properties:
- $a(\Lambda)=1$, where $\Lambda$ is an empty string;
- $a(x)\ge a(x0)+a(x1)$ for every string $x$.
There exists a maximal (up to $O(1)$-factor) lower semicomputable continuous semimeasure; it is called the *continuous a priori probability* and is denoted by $\operatorname{\mathbf{a}}(x)$ in the sequel. The quantity $-\log_2 \operatorname{\mathbf{a}}(x)$ is ofter called *a priori complexity* and sometimes denoted $\operatorname{\mathit{KM}}(x)$.
Now we have defined all the quantities involved in our main result, but to explain its informal meaning we should say more about algorithmic randomness. (These explanations are not needed to understand the statement and the proof of the main result, so the reader may jump to the next section.)
The notion of a random sequence was introduced by Martin-Löf in 1966 (see [@MartinLofDefinition]). Let $P$ be a computable measure on the Cantor space $\Omega=\{0,1\}^\infty$ of infinite binary sequences; this means that the values $P(x\Omega)$ of the cylinders (here $x\Omega$ is the set of all infinite extensions of a binary string $x$) can be effectively computed with arbitrary precision. An *effectively open* subset of $\Omega$ is a union of a (computably) enumerable set of cylinders. A *Martin-Löf test* (with respect to $P$) is an uniformly enumerable decreasing sequence of effectively open sets $$U_1\supset U_2 \supset U_3\supset\ldots$$ such that $P(U_i)\le 2^{-i}$. A sequence $\omega\in\Omega$ *passes* this test if it does not belong to $\bigcap_i U_i$. *Martin-Löf random sequences* are sequences that pass all tests (with respect to $P$).
In 1970s Levin and Gacs found an useful reformulation of this definition in terms of *randomness deficiency* function. Consider a lower semicomputable function $t$ on the Cantor space with non-negative real values (possible infinite). Lower semicomputability means that this the set $\{\omega|t(\omega)> r\}$ is effectively open for all positive rational $r$ uniformly in $r$. A *Levin–Gacs test* with respect to $P$ is such a function with finite integral $\int t(\omega)\,dP(\omega)$. For a given $P$ there is a maximal (up to $O(1)$-factor) Levin–Gacs test; Martin-Löf random sequnces are exactly the sequences for which this test is finite.
There is a formula[^1] that expresses a maximal test in terms of a priori probability: $$\operatorname{\mathbf{t}}(\omega)=\sum_{x\sqsubset \omega}\frac{\operatorname{\mathbf{m}}(x)}{P(x)};$$ here $x\sqsubset\omega$ means that binary string $x$ is a prefix of an infinite binary sequence $\omega$; note that $\operatorname{\mathbf{t}}$ in the left-hand side depends on $P$ (though this is not reflected in the notation). Moreover, the sum in this formula can be replaced by the supremum.
For the uniform Lebesgue measure on the Cantor space this result can be rewritten as follows: $$\operatorname{\mathbf{t}}(\omega)=\sum_n 2^{n-\operatorname{\mathit{K}}(\omega_1\ldots\omega_n)}=2^{\sup_n( n-\operatorname{\mathit{K}}(\omega_1\ldots\omega_n))}.$$ This equation implies both Schnorr–Levin criterion of randomness (see [@SchnorrCrit; @LevinCrit]; its version with prefix complexity saying $\omega$ is Martin-Löf random with respect to the uniform measure iff $n-\operatorname{\mathit{K}}(\omega_1\ldots\omega_n)$ is bounded, is mentioned in [@ChaitinPrefix]) and the Miller–Yu *ample excess* lemma ([@MillerYu], section 2) saying that the sum in the right hand side is finite for random $\omega$.
There were many attempts to generalize a notion of randomness to a broader class of distributions, not only computable measures. The notion of uniform test (a function of two arguments: a sequence and a measure) was introduced by Levin (see [@LevinCrit; @LevinUniform]); it was used to define *uniform randomness* with respect to arbitrary (not necessarily computable) measure $P$. Levin proved that there exists a *neutral measure* $N$ such that every sequence is uniformly random with respect to $N$ (and even has uniform randomness deficiency at most $1$), see [@longpaper] for the exposition of these results.
One could also try to extend the definition to continuous lower semicomputable *semi*measures (a broader class than computable measures where $a(x)=a(x0)+a(x1)$). Such a semimeasure is an output distribution of a probabilistic machine and one may ask which sequences are “plausible outcomes” for such a machine. In this case there is no universally accepted definition; one of the desirable properties of such a definition is that every sequence should be random with respect to continuous a priori probability $\operatorname{\mathbf{a}}(\cdot)$ (that corresponds to a probabilistic machine for which we do not have any a priori information).
One of the possibilities would be to use Gacs’ formula as a definition and say that a sequence $\omega$ is random with respect to a continuous semimeasure $A$ if the sum $\sum_{x\sqsubset\omega} \operatorname{\mathbf{m}}(x)/A(x)$ is finite, or if the supremum $\sup_{x\sqsubset\omega} \operatorname{\mathbf{m}}(x)/A(x)$ is finite. If $A$ is the continuous a priori probability, the supremum is always finite (and uniformly bounded: it is easy to see that $\operatorname{\mathbf{m}}(x)/\operatorname{\mathbf{a}}(x)\le O(1)$ for all $x$). Moreover, in 2010 Lempp, Miller, Ng and Turetsky (unpublished; we thank J. Miller who kindly provided a copy of this note) have shown that for every $\omega$ the ratio $\operatorname{\mathbf{m}}(x)/\operatorname{\mathbf{a}}(x)$ tends to zero for prefixes $x\sqsubset\omega$ (though it is $\Theta(1)$, say, for strings of the form $0^n1$).
In this paper we show (Theorem \[th:main\] in Section \[sec:main\]) that this result cannot be strengthened to show that the sum of $\operatorname{\mathbf{m}}(x)/\operatorname{\mathbf{a}}(x)$ along every sequence is bounded. So the first of the suggested definitions of randomness with respect to semimeasure (with the sum instead of supremum) differs from the second one: not all sequences are random with respect to $\operatorname{\mathbf{a}}$, according to this definition.
It would be interesting to understand better for which sequences the sum $$\sum_{x\sqsubset\omega}\operatorname{\mathbf{m}}(x)/\operatorname{\mathbf{a}}(x)$$ is finite. Are they related somehow to K-trivial sequences (where $\operatorname{\mathbf{m}}(x)$ is equal to $\operatorname{\mathbf{m}}(|x|)$ up to $O(1)$-factor)? We do not know the answer; we can show only (see Section \[sec:enumerable\]) that one can find a computably enumerable set whose characteristic sequence has this property.
Our result about the sum of $\operatorname{\mathbf{m}}(x)/\operatorname{\mathbf{a}}(x)$ is of computational nature: if we allow more computational power for $\operatorname{\mathbf{a}}(x)$, the sum becomes finite, as the following simple proposition shows.
\[prop:offline\] Let $\operatorname{\mathbf{a}}'=\operatorname{\mathbf{a}}^{\mathbf{0}'}$ be the relativized continuous a priori probability using $\mathbf{0}'$ as an oracle. Then the sum $\sum_{x\sqsubset\omega}{\frac{\operatorname{\mathbf{m}}(x)}{\operatorname{\mathbf{a}}'(x)}}$ is bounded for all $\omega$ by a constant (not depending on $\omega$).
It is enough to construct a $\mathbf{0}'$-computable *measure* $a'$ such that $\sum_{x\sqsubset\omega}{\frac{\operatorname{\mathbf{m}}(x)}{a'(x)}}\le 1$ for all $\omega$. (Then we can note that $\operatorname{\mathbf{a}}'(x)$ is an upper bound for $a'$.) One can describe such a measure explicitly. Let us add all the a priori probabilities of all strings $x$ that start with $0$ and with $1$: $$M_0=\sum_u \operatorname{\mathbf{m}}(0u); \quad M_1=\sum_u \operatorname{\mathbf{m}}(1u).$$ (Note that $M_0+M_1+\operatorname{\mathbf{m}}(\Lambda)\le 1$, where $\Lambda$ denotes the empty string, the root of the tree.) Now let us split $1$ into $a'(0)+a'(1)$ in the same proportion, i.e., let $$a'(0)=\frac{M_0}{M_0+M_1},\quad a'(1)=\frac{M_1}{M_0+M_1}.$$ Then we continue in the same way, splitting $a'(0)$ into $a'(00)$ and $a'(01)$ in the proportion $M_{00}:M_{01}$, and so on. Here $M_z$, defined for every string $z$, denotes the sum $\sum_u \operatorname{\mathbf{m}}(zu)$.
The numbers $M_z$ are lower semicomputable, so they are $\mathbf{0}'$-computable (and positive), and the measure $a'$ is well defined and $\mathbf{0}'$-computable. It remains to check that it is large enough, so the sum in question is bounded by $1$.
It is enough to prove this bound for finite sums (when only vertices below some level $N$ are considered), so we can argue by induction and assume that the similar statement is true for the left and right subtrees of the root, with appropriate scaling. [^2] The sum of $m(x)$ in the left subtree is bounded by (actually, is equal to) $M_0$, instead of $1$ in the entire tree; the sum in the right subtree in bounded by $M_1$. On the other hand, the values of $a'$ at the roots of these trees, i.e., $a'(0)$ and $a'(1)$, are also smaller. So the induction assumption says that for each path in the left subtree the sum of $\operatorname{\mathbf{m}}(x)/a'(x)$ is bounded by $M_0/a'(0)$, and for each path in the right subtree the sum is bounded by $M_1/a'(1)$. Therefore, it remains to show that $$\frac{M_0}{a'(0)}+\operatorname{\mathbf{m}}(\Lambda) \le 1, \qquad
\frac{M_1}{a'(1)}+\operatorname{\mathbf{m}}(\Lambda) \le 1.$$
Recall that we defined $a'(0)$ and $a'(1)$ in such a way that they are proportional to $M_0$ and $M_1$ respectively, and the sum $a'(0)+a'(1)=1$. So the both fractions in the last formula are equal to $M_0+M_1$, and it remains to note that $M_0+M_1+\operatorname{\mathbf{m}}(\Lambda)$ is the sum of $\operatorname{\mathbf{m}}(x)$ over all strings $x$ and is bounded by $1$.
**Remark**. Laurent Bienvenu noted that this (simple) computation can be replaced by references to some known facts and techniques. Namely, we know that there exists a neutral measure $N$ such that every binary sequence $\omega$ has uniform deficiency at most $1$ with respect to $N$. This deficiency can be rewritten as $\sum_{x\sqsubset\omega} \frac{\operatorname{\mathbf{m}}(x|N)}{N(x)}$ (see [@longpaper] for details). Using low-basis argument, we can choose a $\mathbf{0}'$-computable neutral measure $N$; then $\operatorname{\mathbf{a}}'$ is greater that this $N$. And (in any case) $\operatorname{\mathbf{m}}(x|N)$ is greater than $\operatorname{\mathbf{m}}(x)$, so we get a desired result.
Main result and the proof sketch: the game argument {#sec:main}
===================================================
\[th:main\] There exists an infinite binary sequence $\omega$ such that $$\sum_{x\sqsubset\omega} \frac{\operatorname{\mathbf{m}}(x)}{\operatorname{\mathbf{a}}(x)} =\infty.$$
This is the main result of the paper. The proof uses (now quite standard) game technique. In this section we describe some infinite game and show how the main result follows from the existence of a computable winning strategy for one of the players (called Mathematician, or **M**) in this game. Then, in Section \[sec:finite-use\] we reduce this game to a finite game (more precisely, to a class of finite games), and show that if all these games uniformly have a computable winning strategy for **M**, then the infinite game has a computable winning strategy. Finally, in Section \[sec:finite-win\] we construct (inductively) winning strategies for finite games. (This will be the most technical part of the proof: we even need to compute some integral!)
Let us describe an infinite game with full information between two players, the Mathematician (**M**) and the Adversary (**A**). This game is played on an infinite binary tree.
Mathematician assigns some non-negative rational weights to the tree vertices (=binary strings). Initially all the weights are zeros; at each move **M** can increase finitely many weights but cannot decrease any of them. The total weight used by **M** (the sum of her weights) should never exceed $1$. (We may assume that **M** loses the game immediately if her weights become too big.) The current **M**’s weight of some vertex $x$ will be denoted by $m(x)$, so the requirement says that $\sum_x m(x)\le 1$ at any moment of the game (otherwise **M** loses immediately).
Adversary also assigns increasing non-negative rational weights to the tree vertices. Initially all they are zeros, except for the root weight which is $1$. But the condition is different: for every vertex $x$ the inequality $a(x0)+a(x1)\le a(x)$ should be true. Informally, one can interpret $a(x)$ as a (pre)flow that comes to vertex $x$. The flow $1$ arrives to the root. From the root some parts $a(0)$ and $a(1)$ are shipped to the left and right sons of the root (while the remaining part $1-a(0)-a(1)$ is reserved for future use. At the next level, e.g., in the vertex $0$, the incoming flow $a(0)$ is split into $a(00)$, $a(01)$ and the (non-negative) reserve $a(0)-a(00)-a(01)$, and so on. As the time goes, the incoming flow (from the father) increases, and it can be used to increase the outgoing flow (to the sons) or kept as a reserve. Again, if **A** violates the restriction (the inequality $a(x0)+a(x1)\le a(x)$), she loses immediately.
One may assume that the players alternate, though it is not really important: the outcome of the (infinite) game is determined by the limit situation, and postponing some move never hurts (and even can simplify the player’s task, since more information about the opponent’s moves is then available). We say that **M** wins if there exist a branch in the tree, an infinite binary sequence $\omega$, such that $$\sum_{x\sqsubset \omega} \frac{m(x)}{a(x)}=\infty,$$ where $m(x)$ and $a(x)$ are limit values of the **M**’s and **A**’s weights respectively. One should agree also what happens if some values are zeros. It is not really important since each of the players can easily make her weights positive. However, it is convenient to assume that $m/0=\infty$ for $m\ne 0$ and $0/0=0$.
Now the game is fully defined. Since all the moves are finite objects, one can speak about computable strategies. The following lemma is the main step in the proof of Theorem \[th:main\].
\[lem:infinite\] **M** has a computable winning strategy in this game.
The proof of this lemma will be given in the next two sections. In the rest of this section we explain how the statement of the lemma implies Theorem \[th:main\]. This is a standard argument useg in all the game proofs. Consider an ignorant Adversary who does not even look on our (Mathematician’s) moves, and just enumerates from below (lower semicomputes) the values of the continuous a priori probability $\operatorname{\mathbf{a}}(x)$. (They are lower semicomputable; some additional care is needed to ensure that $a(x)\ge a(x0)+a(x1)$ is true not only for the limit values, but for approximations at every step, but this is done in a standard way, we can increase $a(x)$ going from the leaves to the root.)
The actions of **A** are computable. Let **M** uses her computable winning strategy against such an adversary. Then **M**’s behavior is computable, too. So the limit values of $m(x)$ form a lower semicomputable function, and the winning condition guarantees that $\sum_{s\sqsubset \omega} m(x)/\operatorname{\mathbf{a}}(x)$ is infinite for some sequence $\omega$. It remains to note that the discrete a priori probability $\operatorname{\mathbf{m}}(x)$ is an upper bound (up to $O(1)$-factor) for every lower semicomputable function $m(x)$.
Finite games are enough {#sec:finite-use}
=======================
To construct the winning strategy for **M** in the infinite game described in the previous section, we combine winning strategies for finite games of similar nature. A finite game is determined by two parameters $N$ and $k$; the value of $N$ is the height of the finite full binary tree on which the game is played, and $k$ is the value of the sum that **M** should achieve to win the game. Here $N$ is a positive integer, and $k\ge 1$ is a rational number.
Initially all vertices (=all strings of length at most $N$) have zero $a$- and $m$-weights, except for the root that has unit $a$-weight: $a(\Lambda)=1$. The players alternate; at every move each player may increase her weights (rational numbers), but both players should obey the restrictions: the sum of $m$-weights should not exceed $1$; for every $x$ that is not a leaf the inequality $a(x)\ge a(x0)+a(x1)$ should be true; the value of $a(\Lambda)$ remains equal to $1$. The position of a game is *winning* for **M** if there exists a leaf $w$ such that the sum $\sum_{x\sqsubset w} m(x)/a(x)$ is at least $k$. Otherwise the position is winning for **A**. Each player, making a move, should create a winning position (for her), otherwise she loses the game. (She may also lose the game by violating the restrictions for her moves.)
\[lem:finite\] For every positive rational $k$ there exists some $N$ and a winning strategy for **M** that guarantees that **M** wins after a bounded number of steps. (The bound depends on $k$, but not on **A**’s moves.) The value of $N$ and the strategy are computable given $k$.
[r]{}[0pt]{} 
The proof of this lemma will be given in the next section. In the rest of this section we show how we can use winning strategies for finite games to win the infinite game of the previous section (and therefore to finish the proof of our main result, Theorem \[th:main\]). Let us make first several simple remarks.
First, note that if **M** has a winning strategy for some $N$, she has also a winning strategy for all larger $N$ (just ignore the vertices that have height greater than $N$). So the words “there exists some $N$” can be replaced by “for every sufficiently large $N$”.
Second, one can scale the game, bounding the total **M**-weights by some quota $M$ (instead of $1$) and letting $a(\Lambda)$ be some $A$ (also instead of $1$). Then, if **M** was able to achieve the sum $k$ in the original game, she can use essentially the same strategy in the new game to achieve $kM/A$. For that she should imagine that the actual moves of **A** are divided by $A$, and multiply by $M$ the moves recommended by the strategy.
Since $k$ in Lemma \[lem:finite\] is arbitrary, **M** can achieve arbitrary large sum even if her weights are limited by arbitrary small constant $\mu>0$ (known in advance); the size $N$ of the tree then depends both on the sum we want to achieve, and on the allowed quota $\mu$. This simple remark allows **M** to run in parallel several strategies on some subtrees, allocating quotas $\mu_1,\mu_2,\mu_3,\ldots$ to them, where $\sum \mu_i \le 1$ is some converging series, e.g., $\mu_i=2^{-i}$. These strategies achieve sum $1$ in each subtree. It is indeed possible: the flow generated by the adversary can be considered separately on each subtree: if the total flow starting from the root is at most $1$, the flow in every vertex, including the root of a subtree, is also at most $1$. (Note the using $a(\Lambda)<1$ in the root instead of $1$ makes the task of adversary harder, so **M** can win in every subtree.) These subtrees are chosen as shown on Fig. \[subtrees\].
Knowing $\mu_1$, we choose the height of the first subtree; knowing the number of leaves in the first subtree and the corresponding $\mu_i$, we choose the appropriate heights for the second layer subtrees (one can choose the same height for all of them to make the picture nicer); then, knowing the number of leaves in all of then, we look at the corresponding $\mu_i$ and select the height for the third layer, etc. The games are played (and won) independently in each subtree. In each subtree there is a path with $\sum m(x)/a(x)\ge 1$, and we can combine these paths into an infinite path starting from the root.
How to win the finite game {#sec:finite-win}
==========================
In this section we provide the proof of Lemma \[lem:finite\], therefore finishing the proof of our main result, Theorem \[th:main\]. As we have seen, the winning strategy for Mathematician should rely on the on-line nature of the game: if **M** makes only one move and then stops, Adversary could win by splitting the flow proportional to the weights of the subtrees (see the proof of Proposition \[prop:offline\]).
![First move of **M** and the reaction of **A**.[]{data-label="game5"}](game-5.pdf)
To construct the winning strategy for **M** in the finite game, first let us start with a toy example and show how she can make the sum $\sum_{x\sqsubset w} m(x)/a(x)$ greater than $1$. For this, tree of height $2$ is enough (in fact only some part of it is needed).
**M** starts by putting weights $\frac{1}{4}$ to vertices $0$ and $00$ (Figure \[game5\]). Then **A** has to decide how much flow she wants to send to $0$ and $00$. There are several possibilities:
- The flow to $0$ is small: $a(0)<\frac{1}{2}$. In this case $a(00)$ is obviously also less than $\frac{1}{2}$, so $$\frac{m(\Lambda)}{a(\Lambda)}+\frac{m(0)}{a(0)}+\frac{m(00)}{a(00)}>0+\frac{1}{2}+\frac{1}{2}>1,$$ and this move does not create a winning position for **A**.
![A winning move of **M** in the second case.[]{data-label="game6"}](game-6.pdf)
- The flow to $0$ is big: $a(0)>\frac{1}{2}$. In this case **A** may get a winning position (for now). However, **M** still can win. Indeed, $a(1)\leq 1-a(0)$ is less than $\frac{1}{2}$ and remains less than $1/2$ forever. Then **M** puts weight $\frac{1}{2}$ to vertex $1$ (Figure \[game6\]), making the sum there greater than $1$, and **A** cannot do anything.
![A winning move of **M** in the third case.[]{data-label="game7"}](game-7.pdf)
- The intermediate case: $a(0)=\frac{1}{2}$. In this case $a(00)$ should be also $\frac{1}{2}$, otherwise the sum in $00$ will still exceed $1$ and **A** does not get a winning position. But if $a(00)=a(0)=1/2$, **M** can put weight $\frac{1}{2}$ to vertex $01$, and **A** cannot send more than $1/2$ to $01$ (since $1/2$ is already directed to $00$). Then, $$\frac{m(\Lambda)}{a(\Lambda)}+\frac{m(0)}{a(0)}+\frac{m(01)}{a(01)}\geq 0+\frac{1}{4}+\frac{1/2}{1/2}=\frac{5}{4}>1.$$
More careful analysis shows that using this idea **M** can get a winning strategy for $k=17/16$. But we need an arbitrary large $k$ anyway, so we do not go into details, and provide another construction.
The winning strategy for arbitrary $k$ will be recursive: we assume that **M** has winning strategy for some $k$ and then use this strategy to construct **M**’s winning strategy for some $k'=k+\varepsilon$, where $\varepsilon>0$. The increase $\varepsilon$ depends on $k$ and is rather small, but has a lower bound $f(k)$ which is a positive continuous function of $k$.
Iterating this construction, we get $k_i$-winning strategies where $k_1=1$ (for $k=1$ the winning strategy is trivial) and $$k_{i+1}\ge k_i + f(k_i).$$ We see now that $k_i\to\infty$ and $i\to\infty$; indeed, if $k_i\to K$ for some finite $K$, then $k_{i+1}\ge k_i+f(k_i)\to K+f(K)$, a contradiction.
[r]{}[0cm]{} 
To explain the idea of this construction, let us first comment on the toy example explained above (how to get sum greater than $1$). Making her first move, **M** keeps some reserve that can be later put into the vertex $1$. This possibility creates a constant threat for **A** that prevents her from directing too much flow to $0$. The same kind of threat will be used in the final construction; again vertex $1$ will be used as “threat vertex”. If **A** directs to much flow to the left (vertex $0$), **M** sees this and uses all the reserve to win in the right subtree.
However, now the strategy is more complicated. There are two main improvements. First, instead of placing some weight in a vertex as before, **M** uses scaled $k$-strategy in the subtree rooted at that vertex, so the weight is used more efficiently (with factor $k$). This is done both in the left subtree and in the threat vertex $1$. (The subtrees where $k$-strategy is played, are shown in grey in Figure \[game8\].) Second, in the left subtree (of sufficient height) **M** uses sequentially $n$ vertices $z_1,\ldots,z_n$ (and $n$ corresponding subtrees) for large enough $n$. (We will discuss later how $n$ is chosen.)
Let us describe the $(k+\varepsilon)$-strategy in more details.
First of all, **M** puts weight $\varepsilon$ into vertex $0$ (after that **M** will never add weight there, so vertex $0$ always has weight $\varepsilon$).[^3]
After that **M** still has weight $1-\varepsilon$ available. It is divided into $n$ equal parts, $(1-\varepsilon)/n$ each. These parts are used sequentially in subtrees with roots $z_1,\ldots,z_n$ (Figure \[game8\]). In these subtrees **M** uses scaled $k$-strategy; the coefficient is $(1-\varepsilon)/n$. In this way **M** forces **A** to direct a lot of flow to these $n$ subtrees or lose the $k$-game in one of this subtrees (and therefore lose $(k+\varepsilon)$-game in the entire tree, if the parameters are chosen correctly).
The threat vertex $1$ is used as follows: if at some point (after $i$ games for some $i$) the flow directed by **A** to $0$ is too large, **M** changes her strategy and use all the remaining weight, which is $(1-\varepsilon)(1-\frac{i}{n})$, for $k$-strategy in the $1$-subtree (and wins, if the parameters are chosen correctly).
Now we have to quantify the words “a lot of flow” and “too large flow” by choosing some thresholds. Assume that after $i$ games **A** directed some weight $d$ to $0$. Then she can use only $1-d$ for the game in the threat vertex. Using $k$-strategy with reserve $(1-\varepsilon)(1-\frac{i}{n})$, **M** can achieve sum (along some path in the right subtree) $$\frac{k(1-\varepsilon)(1-\frac{i}{n})}{1-d},$$ so the threshold $d_i$ is obtained from the equation $$\frac{k(1-\varepsilon)(1-\frac{i}{n})}{1-d_i}=k+\varepsilon.\eqno(*)$$ If the flow to the left vertex $0$ is at least $d_i$, **M** stops playing games in the left subtree and wins the entire game by switching to $k$-strategy in the right subtree and using all remaining weight there.
What happens if **A** does not exceed the thresholds $d_i$? Then the vertex $0$ adds $\varepsilon/d_i$ to the sum in the $i$-th game, and to win the entire game **M** needs to get the sum $(k+\varepsilon)-\varepsilon/d_i$ in the $i$-th game. This can be achieved using (scaled) $k$-strategy with weight $(1-\varepsilon)/n$ unless **A** directs $a_i$ to $z_i$-subtree, where $a_i$ is determined by the equation $$k+\varepsilon-\frac{\varepsilon}{d_i}=k\frac{(1-\varepsilon)/n}{a_i}
\eqno(**)$$ We need to prove, therefore, that for some $\varepsilon$ (depending on $k$) and for large enough $n$ the values $a_i$ determined by $(**)$, where $d_i$ is determined by $(*)$, satify the inequality $$\sum_{i=1}^{n} a_i>1.$$ Then **A** is unable to direct $a_i$ in $z_i$-subtree for all $i$ and loses the game.
This sum can be rewritten as follows: $$\sum_{i=1}^n a_i=
\frac{1}{n}\sum_{i=1}^n\frac{kd_i(1-\varepsilon)}{d_i(k+\varepsilon)-\varepsilon} .$$ Note that $d_i$ depends only on $k$, $\varepsilon$ and $u=\frac{i}{n}$, so this sum is the Riemann sum for the integral $$\int_0^1 k(1-\varepsilon)\frac{d(u)}{(k+\varepsilon)d(u)-\varepsilon}du$$ where $$d(u)=1-\frac{k}{k+\varepsilon}(1-\varepsilon)(1-u).$$ Note that we integrate a rational function of the form $(Au+B)/(Cu+D)$, so it is not a problem, and we get $$\begin{gathered}
\int_0^1 \frac{(1-\varepsilon)(ku- ku\varepsilon+k\varepsilon+\varepsilon)}{(k+\varepsilon)(u+\varepsilon-u\varepsilon)}du=\\=\frac{k(u+\varepsilon-\varepsilon u)+\varepsilon\log(u(\varepsilon-1)-\varepsilon)}{k+\varepsilon}\biggr|_0^1=\\=\frac{k(1-\varepsilon)+\varepsilon\cdot\log(1/\varepsilon)}{k+\varepsilon}.\end{gathered}$$ Note that for $\varepsilon\to 0$ this expression can be rewritten as $$1+\varepsilon\cdot(\log(1/\varepsilon)-k-1)/k+O(\varepsilon^2),$$ so for sufficiently small $\varepsilon>0$ this integral will be greater than $1+\varepsilon$, and we can choose $n$ large enough to make the Riemann sum greater than $1$. It is easy to get a positive lower bound for $\varepsilon$ (depending on $k$) and find the corresponding $n$ effectively.
This finishes the proof of Lemma \[lem:finite\] and therefore the proof of our main result, Theorem \[th:main\].
**Remark**. Let us repeat the crucial point of this argument: during the initial phase of the strategy, when **M**’s reserve is large, **A** cannot direct a lot of flow into $0$, so the weight $\varepsilon$ placed into this vertex is taken with a large coefficient. Without this threat **A** could place all the flow in the left subtree, and then the weight $\varepsilon$ would not help: on the contrary, the same weight could have been used $k$ times more efficiently in the subtrees, and we get no increase in $k$.
Improvement: how to find a c. e. set with infinite sum {#sec:enumerable}
======================================================
The construction can be adjusted to guarantee some additional properties of the sequence $\omega$ with $\sum_{x\sqsubset\omega}\operatorname{\mathbf{m}}(x)/\operatorname{\mathbf{a}}(x)=\infty$.
\[th:enumerable\] There exists a computably enumerable set $X$ such that for its characterstic sequence $\omega_X$ (where $\omega_i=1$ for $i\in X$ and $\omega_i=0$ for $i\notin X$) the sum $\sum_{x\sqsubset\omega_X}\operatorname{\mathbf{m}}(x)/\operatorname{\mathbf{a}}(x)$ is infinite.
We start by modification of the finite game of Lemma \[lem:finite\]. Let us agree that **M** should (for each her move) not only achieve a winning position, but also explicitly mark one of the nodes of the tree where the sum is at least $1$ (according to the definition of the winning position). If there are several nodes where the (current) sum reaches $1$, **M** can choose any of them. During the game, **M** can change the marked node, but monotonicity is required: the marked nodes should form an increasing sequence in a coordinate-wise ordering (for each node of a binary tree we consider a sequence of zeros and ones that leads to this node, and add infinitely many trailing zeros; in this way we get an infinite sequence, and when the node changes, the new sequence should be obtained from the previous one by some $0\to 1$ replacements).
This requirement could be satisfied by minor changes in construction of winning strategy for $(k+\varepsilon)$ game. Note that the winning strategy for $(k+\varepsilon)$ game calls the winning $k$-strategy for vertices $z_1,\ldots,z_n$, and maybe for the threat vertex. Using induction, we may assume that $k$-strategy satisfies the monotonicity requirement. This is not enough: **M** needs also to guarantee monotonicity while switching from the $k$-strategy in $z_i$-subtree to the $k$-strategy in $z_{i+1}$-subtree (or in the threat vertex). To achieve this, some precautions are needed. First of all, we choose $z_1,\ldots,z_n$ in such a way that $z_1\le z_2\le\ldots z_n$ coordinate-wise. Moreover, while playing the game above $z_1$, **M** makes some bits equal to $1$ (according to the winning $k$-strategy in the subtree). These bits cannot be reversed back, but this is not a problem: for example, one can add several $1$s at the end of $z_2$ (to cover all the bits changed while playing above $z_1$), and use a subtree rooted there, then do the similar trick for $z_3$, etc. (see Figure \[monotone2\]). Finally, the same can be done for the threat vertex.
![Special precautions needed to preserve the monotonicity during the induction step.[]{data-label="monotone2"}](game-9.pdf)
Infinite game of Lemma \[lem:infinite\] can also be adjusted. Here **M** should after each move maintain a *current branch*, an infinite path in the binary tree that contains only finitely many ones (so it is essentially a finite object and can be specified by **M** explicitly). The current branch may change during the game but in a monotone way: it should increase *coordinate-wise*. In other words, if the previous branch went right at some level, the next one should do it too (at the same level). This monotonicity requirement guarantees that there exists a limit branch, and **M** wins the (infinite) game if the sum is infinite along this branch.
![Subtrees where the games are started. When a marked leaf changes, all the subtrees above it are abandoned, and new subtree is chosen with all $1$s inbetween.[]{data-label="mon_subtrees"}](game-10.pdf)
We claim that **M** has a computable winning strategy in this game. Knowing this, we can easily construct an enumerable set required by Theorem \[th:enumerable\]. Again we use the computable winning strategy against a “blind” adversary that enumerates from below the values of the continuous a priori semimeasure. Then the behavior of the adversary is computable, the behavior of the computable winning strategy is also computable, and the limit branch will be a characteristic sequence of a (computably) enumerable set.
It remains to explain how one can combine winning strategies for finite games (modified) to get a winning strategy for the infinite game. We cannot run the strategies on subtrees in parallel as we did before, because the candidate branches provided by the strategies at the same level will not be related, and switching from one game to another will violate the ordering condition. Instead, we start first the game in the root subtree. The strategy makes some move, in particular, marks some leaf of this subtree (“current candidate”). Then we start the strategy on a subtree that is above this marked leaf. This strategy marks some leaf in this subtree, and we start a third game above it, etc. (See Figure \[mon\_subtrees\].)
At some point one of these strategies may change its marked leaf. Then all the games started above this (now discarded) leaf are useless, and we start a new game above the new marked leaf. To satisfy the monotonicity condition, we should start the new game high enough and put $1$s in all positions below the starting point of the new game. This will guarantee that all $1$s that were already in the current branch will remain there. (We assume that at every moment only finitely many games are started, and the current branch has only finitely many ones.)
One can see that in the limit we still have a branch with infinite sum. Indeed, in the root game the current marked leaf can only increase in the coordinate-wise ordering, and only finitely many changes of marked leaf are possible. Therefore, some leaf will remain marked forever. The game started above this leaf will never be discarded, but the leaves marked in this game may change (monotonically). This happens finitely many times, and after that the marked leaf remains the same, the game above it is never discarded, etc.
The monotonicity is guaranteed both for the elements inside the tree where the marked leaf changed (according to the monotonicity for finite games) and for outside elements (since we replace the bits in the discarded parts by $1$s only).
[99]{}
L. Bienvenu, P. Gács, M. Hoyrup, C. Rojas, A. Shen, Algorithmic tests and randomness with respect to a class of measures, *Proc. of the Steklov Institute of Mathematics*, v. 274 (2011), p. 41–102. See also: `arXiv:1103.1529v2`.
A. Shen, *Algoritmic information theory and Kolmogorov complexity*, lecture notes, `http://www.it.uu.se/research/publications/reports/2000-034`.
N.K. Vereshchagin, V.A. Uspensky, A. Shen, Kolmogorov complexity and algorithmic randomness, Moscow: MCCME Publishers, 2013. (In Russian)
Leonid Levin. *Some theorems about the algorithmic approach to probability theory and information theory*. Ph.D. thesis, Moscow State University, 1971. 53 pp. (In Russian)
L.A. Levin, Laws on information conservation (nongrowth) and aspects of the foundation of probability theory. *Problems of Information Transmission*, vol. 10 (1974), p. 206–210.
P. G' acs \[P. Gač\], On the symmetry of algorithmic information, *Soviet Math. Dokl.*. vol. 15 (1974), No. 5, p. 1477–1480.
G.J. Chaitin, A theory of program size formally identical to information theory, *Journal of the ACM*, vol. 22 (1975), no. 3, p. 329–340.
A.K. Zvonkin and L.A. Levin, The complexity of finite objects and the developments of the concepts of information and randomness by means of the theory of algorithms. *Russian Mathematical Surveys*, 1970, vol. 25, issue 6(156), p. 83–124.
Per Martin-Löf, The definition of random sequences, *Information and Control*, v. 9 (1966), p. 602–619.
Peter Gács, Exact expressions for some randomness tests, *Zeitschrift f. Math. Logik und Grundlagen d. Math.*, 1979, vol. 26, p. 385–394.
C. P. Schnorr, Process complexity and effective random tests, *Journal of Computer and System Sciences*, v. 7 (1973), p. 376–388. Preliminary version: *Proc. 4th ACM Symp. on Theory of Computing* (STOC), 1972, p. 168–176.
L.A. Levin, On the notion of a random sequence, *Soviet. Math. Dokl.*, v. 14 (1973), p. 1413–1416.
L.A. Levin, Uniform tests of randomness, *Soviet Math. Dokl.*, v. 17 (1976), p. 337.
Joseph S. Miller and Liang Yu, On initial segment complexity and degrees of randomness, *Transaction of the AMS*, v. 360 (2008), issue 6, p. 3193-3210.
[^1]: It goes back to P. Gács paper [@GacsExactExpression], but Gács used a different and rather cumbersome notation there. See [@longpaper] for the detailed exposition.
[^2]: The summation is stopped at the same level $N$, so the tree height is less by $1$ and we can apply the induction assumption. The base of induction is trivial: in the root the ratio $m/a$ is at most $1$ for evident reasons.
[^3]: The weight of vertex $0$ in the strategy is equal to the desired increase in $k$; there are no deep reasons for this choice, but it simplifies the computations.
|
---
abstract: 'An extremely simple and unified base for physics comes out by starting all over from a single postulate on the common nature of matter and stationary forms of radiation quanta. Basic relativistic, gravitational (G) and quantum mechanical properties of a standing wave particle model have been derived. This has been done from just dual properties of radiation’s and strictly homogeneous relationships for nonlocal cases in G fields. This way reduces the number of independent variables and puts into relief (and avoid) important inhomogeneity errors of some G theories. It unifies and accounts for basic principles and postulates physics. The results for gravity depend on linear radiation properties but not on arbitrary field relations. They agree with the conventional tests. However they have some fundamental differences with current G theories. The particle model, at a difference of the conventional theories, also fixes well-defined cosmological and astrophysical models that are different from the rather conventional ones. They have been described and tested with the astronomical observations. These tests have been resumed in a separated work.'
author:
- |
Rafael A. Vera[^1]\
Deptartamento de Fisica\
Universidad de Concepcion\
Casilla 4009. Concepcion. Chile
date: 'Aug. 29, 1995'
title: |
Unified Relativistic Physics\
from a Standing Wave Particle Model
---
gr-qc/9509014
Introduction
============
This is a review on a self consistent theory based on the simplest kind of [*particle model*]{} that in principle can account for the basic properties of uncharged matter an it’s gravitational (G) field.
The first steps of this work were published by first time in 1977 in [*Atenea*]{}, a yearly book on science and art of the University of Concepcion, Chile [@V77]. The base of this theory was also presented in [*The Einstein centennial symposium of fundamental physics*]{}, in 1979. This one was published in 1981, in the corresponding proceedings[@V81a]. In it, the emphasis was made on demonstrating that it is the body, but not the field, the one that puts on the energy during G work. This means that the currently presumed energy exchange between static G fields, which has never been demonstrated, is not strictly true. Then the present work is in disagreement with the arguments used by Einstein for his G field equation [@E55]. A more detailed work was also published in 1981 [@V81b].
The next works have been aimed to prove that both physics and astrophysics can be unified by starting all over from a single postulate on the nature of matter [@V86], [@V95a], [@V95b].
This theory has been conceived as independent as possible from current theories. Thus only some of the most elementary and unquestionable properties of light have been used. In this way, in principle it is possible to get new more self-consistent and unified viewpoints that cannot be contaminated with current but non well-proved assumptions.
According to the rather single postulate of this theory [*particles are stationary forms of the radiation’s*]{}. Thus the original particle model, called the [*light box model*]{}, is a kind of wave cavity with one or more quanta of radiation confined themselves as standing-waves (SWs)[^2]. For general purposes, it is not necessary to know the exact shape and symmetry of the particle model. It is interesting that [*torus*]{} shaped models have angular momentum’s that are obviously consistent with those of [*bosons*]{}.
Nonlocal relativity
-------------------
A better defined and more general relativistic language must be used for establishing strictly homogeneous nonlocal (NL) relations in G fields. This can be inferred from [*G time dilation*]{} (GTD) experiments and from the fact that in them [*the fly time of light is negligible compared with the measured time intervals*]{}. From them it is simple to conclude that
[ ]{} - [*GTD occurs in the light source but not during the time of fly*]{}.
[ ]{} - [*The atoms and the unit systems of observers at rest in different G field potentials are different relative to each other, respectively*]{}.
[ ]{} - [*Current comparisons of quantities measured in different G field potentials are inhomogeneous*]{}. They may be as meaningless as to compare prices in two countries without reducing them to a common money.
It is trivial that an observer cannot be at two places at the same time. Then, to measure a nonlocal object (in a different G field potential), he and his instruments must move up to the NL object. Thus any general change occurring to the bodies would also hold for his instrument and for the objects. Thus, strictly, [*it is not possible to make nonlocal measurements in G fields because bodies and instruments would change in identical proportions after identical changes of G field potential*]{}. Thus the ordinary nonlocal relations made up with local (measured) quantities have not well defined meanings because they do not take into account any kind of general change occurring to the bodies when they change of field potentials.
Then it may be concluded that [*to relate quantities measured in different G fields’ potentials, all they must be referred to some common standard that has not had the same velocity and G field potential changes of the objects*]{} [@V81b].
For these NL relations, [*a theoretical (T) observer*]{} (and his standard body) can be imagined to be located in some fixed and well-defined field potential or position. Only in this way the theoretical reference framework is well-defined and in principle invariable. Only in this way all kinds of phenomena can be described, theoretically, [*regardless on whether some quantities cannot be measured by such observer*]{}.
These [*NL quantities*]{} are, normally, functions on velocities ($\beta
=v/c$) and field potentials (or NL position’s $r$) of the object and of the observer. The fixed field potential of the common reference standard, or it’s NL position, can be stated by means of a sub index[^3].
Due to the common nature of uncharged particles and SWs, the relativity postulates get reduced to the single fact that [*the bodies and the instruments, including any SW used for measurements, must change in the same way and in the same proportion after the same velocity and field potential changes*]{}. Since the local (relative) values don’t change, then the local (measurable) physical laws also remain unchanged[^4].
This approach was used before by Vera[@V81b] both to detect and to avoid current errors coming from inhomogeneous relations of the form \[$x_A(A)-
x_B(B)$\]. The corresponding homogeneous differences, of the form \[$x_A(A)-x_A(B)$\], were obtained from conservative properties of [*light*]{}.
They can also be obtained from GTD experiments. From them, the NL periods emitted by an atom at rest in a field potential $B$, relative to a standard in a field potential, $A$, are given by: $$\label{1.1}
T_A(0,B)\simeq T_A(0,A)[1+\Delta \phi ]^{-1}$$ \[1.2\] Then the NL frequencies are given by: $$\label{1.21}
\nu _A(0,B)=T_A(0,B)^{-1}\simeq \nu _A(0,A)[1+\Delta \phi]$$ Since these are just the values observed in G redshift (GRS) experiments, after neglecting the cosmological kind of redshift, it is concluded that [*all the GRS has occurred in the source of light but nearly nothing during the light trip*]{}. Then definitively, the NL frequency of light, relative to a fixed observer, remains constant during its trip from B up to A[^5].
In a similar way it is erroneous to say that [*the (relativistic) mass of a body increases during a free G fall*]{} because this means a difference like $m_A(\beta ,A)-m_B(0,B)$, which is [*inhomogeneous*]{}. Only strictly homogeneous (NL) differences, like $m_A(\beta ,A)-m_A(0,B)$, can have physical meaning. In such case, only its first term is trivial, according to local relativity: $$\label{1.3}
m_A(\beta ,A)= \gamma m_A(0,A)\simeq m_A(0,A)[1+\Delta \phi ],$$ However $m_A(0,B)$ is unknown. This one, for example, can be derived from the fact that the energy lost by an atom after a photon emission is a constant fraction of its atomic mass regardless of the local G field potential. Then [*atomic masses and their photon frequencies must change in the same proportion after identical changes of time units or G field potential*]{}. $$\label{1.4}
m_A(0,B)\simeq m_A(0,A)[1+\Delta \phi ] \simeq m_A(\beta ,A)$$ By comparing Eq. \[1.3\] with Eq. \[1.4\] it is concluded [*the NL mass of a body, referred to a fixed standard, does not increase during the fall but remains constant*]{}.
Relations similar to the above ones, carried out for charges in electric (E) fields that [*do not show E time dilation*]{}, prove that the NL mass does increase during acceleration produced by E fields.
Then it may be concluded that [*static G fields, just on the opposite of E fields, do not exchange energy with the test bodies*]{}[^6]
Quantum mechanical properties
=============================
According to optical principles, the model wavelets must interfere constructively within it and its short range field. In this form the net wavelet amplitudes would fix the most probable quantum position. Thus the model [*dual*]{} properties would come from [*the dual properties of its radiation’s*]{}.
For a single quantum, it is most useful to define a [*NL frequency vector*]{} (in its propagation direction) and a [*NL wavelength vector*]{} (parallel to the first one). The scalar product of these [*dual vectors*]{} is the NL [*speed of the quantum*]{}. $$\label{2.1}
c_{r^{\prime }}(r)= {\vec \nu }_{r'}(r).{\vec \lambda }_{r'}(r)$$ For a single quantum particle model, the deductions are simplified by using a transversal model with two NL frequency vectors symmetrical relative to the movement representing waves traveling in opposite directions[^7].
For models moving with velocities $\beta =v/c$ relative to the observer, it is also useful to define [*NL quantum vectors*]{} as a fixed multiple ($h$) of their net NL frequency vectors[^8]. Thus from Doppler shift or plain vector geometry, the net model quantum vector turns out to be equal to: $$\label{2.2}
{\vec Q}(\beta ,r)=\sum_jh{\vec \nu }(j)= 2h \nu '(\beta ,r){\vec \beta}m(\beta,r){\vec \beta} = {\vec p}(\beta,r) c(r).$$ $$\label{2.3}
\nu '(\beta ,r)=\frac 12\sum_j\nu (j) \,\,\,\,\,\,;\,\,\,\,\,\
m(\beta ,r)=h\sum_j\nu (j)=2h\nu '(\beta ,r)$$ in which $m(\beta ,r)$ is [*the model NL mass relative to the observer*]{}. For simplicity, the sub indexes have been omitted and only two vectors representing, each one, half of the model energy. For a body made up of several particle models, it is assumed that any binding energy (field) between the models would have stationary forms that would keep the bodies with well-defined phases. Thus their quantum vectors can be summed up. Thus the body behaves as a single quantum with a net quantum vector equal to the sum of its components.
According to Eq. \[2.2\], the local momentum conservation’s corresponds to the limiting case of NL quantum vector conservation for $r=r'$. From Eq. \[2.3\], the [*NL mass-energy conservation*]{} can be stated by saying that [*the sum of the NL frequencies of all the quanta confined in a closed system, relative to a single and well-defined standard, remain constant*]{}. Since the NL time unit is invariable, this means that the net number of [*quantum cycles*]{} is also invariable, i.e., the quantum cycles occurring in a system remain unchanged.
It has been found [@V81b] that the final NL wavelengths of the model waves that result from interference of the Doppler shifted wavelets) are given by[^9]. $$\label{2.31}
\lambda ^{\prime }(\beta ,r)=\lambda (\beta ,r)/\beta (r)$$ Form Eq. \[2.2\], this turns out to be equal to $h/p(\beta ,r)$, which corresponds with the conventional (De Broglie) ones.
It seems evident that the well-defined NL frequency (energy) of the SWs is also consistent with the well-defined energy levels in atoms.
Gravity
=======
The model G field turns out to be due to the [*long range properties of radiation confined as SWs*]{}. These properties can be deduced from the experiments on light interference with single photons. They prove that the quanta propagate themselves according to interference of wavelets that have not been destroyed during previous interferences. Then the wavelets diverging from the model quanta would travel rather indefinitely in the space, interfering each other [*out of phase*]{}. Thus [*the model G field can only come from the gradient of the relative perturbation rate in the space produced by out of phase or random phase wavelets*]{}[^10].
From the fact that the net wavelet amplitude doesn’t increase by increasing the number of random phase sources, it is inferred, again, that [*G fields would not have real energy*]{}. In other terms, just to the contrary of current beliefs, the static G fields do not give up real energy to the falling bodies. The las ones are self-propelled a way similar to a car in a static road. They use their own energy to accelerate, i.e., the field do not provide the energy but only the momentum needed for releasing energy confined in bodies[^11]. This agrees with previous results[@V81b].
The (uncharged) model SW can accelerate by itself only if a gradient of the NL speed of light exists in the field. The relations between such gradient and other model NL gradients, and with its acceleration, have been derived in detail by [@V81b].
A short cut can be done by using the fact that the frequency and wavelength vectors of the model are always parallel relative to each other. Since their local ratios, for observers at rest in different G field potentials, is always the same, then both the NL frequencies and the NL wavelengths must change in the same proportions under the same field potential changes either of the object or (of some observer). From this fact and Eq. Eq. \[2.1\] and Eq. \[2.3\], $$\label{3.1}
\frac{\bigtriangledown \nu (0,r)}{\nu (0,r)}=\frac{\bigtriangledown
m(0,r)}{m(0,r)}=\frac{\bigtriangledown \lambda (0,r)}{\lambda
(0,r)}=\frac{\bigtriangledown c(r)}{2c(r)}=\bigtriangledown \phi (r)$$ The first two members correspond to the phenomena of [*GRS and to the mass-energy released after G work*]{}. The two next ones describe [*G contraction*]{} and [*G refraction*]{}, respectively. The last one defines a dimensionless point function $\phi (r)$ called [*NL field potential*]{}
The NL field potential
======================
The relative contribution of some quantum $j$ to the perturbation rate at some point $i$, compared with that of a universe of uniform density, may be called $dw(i)$. This one would be proportional both to the NL frequency and to the NL amplitude of the wavelets crossing such point.
On the other hand it is simple to prove that in order that gravity may exist, the NL frequencies of the wavelets must decrease while they propagate themselves through long distances. If this were not so, the space would become saturated with wavelets coming from all over the universe[^12].
Then the fraction of redshift per unit of NL distance, after assuming a uniform universe, should be constant. Thus $\delta \nu /\nu $, should be proportional to the NL distances ($r$). The net relative perturbation rate at some point $i$, compared with a universe of uniform density, turns out to be[^13]: $$\label{3.2}
w(i)=K\sum_j^\infty [h\nu (r^j)][\exp (r^{ij}/R)][r^{ij}]^{-1}=G\sum_k^\infty
[m(r^k)][\exp (r^{ik}/R)][r^{ik}]^{-1}$$ The last member can be divided into two main components. The first one is the rather constant contribution of the long range universe of rather uniform density, called $w(U)$, which is nearly unity. The second one is the mass in excess over the first distribution, which is practically equal to the variable contribution of the relatively local inhomogeneities. $$\label{3.21}
w(i) = 1 + w(L)$$ By comparing $w(i)$ with the earlier value of $\phi (i)$ derived from Poison equation [@V81b], $$\label{3.3}
\phi (i) = - w(i) = - \sum_{k=1}^lG\frac{m(j)}{r(ij)}\exp (r^{ij}/R)$$ Thus the minimum $w(r)$, for a space free of inhomogeneities, is just one. This would give a maximum NL field potential of $-1$. this value and the Hubble radius ($R$), after integration of Eq. \[3.3\], the value of the average density of the universe turns out to be roughly consistent with the values derived dynamically in astronomy [@V95b].
The two main kind of interactions
---------------------------------
The differences betweeen gravitational and short range interactions are most clear for a transversal model falling between potentials B and A followed by a stop at A.
[*During a free fall*]{}, as shown above, the NL mass of the model remain constant. This means that[*the average modulus of it’s NL frequency vectors remain constant*]{}[^14]. Then, if the model falls along some direction OX, [*the model vectors would rotate without changing their average moduli*]{}, after angles with OX given by $\sin \theta =s\beta$. This rotation generates a NL momentum along the OX direction, which is given by Eq. \[2.2\].
[*The local stop*]{}, just to the contrary of the above case, occurs within a space in which $\phi(r)$ is constant and, therefore. $c_{r'}(r)$ is also constant. According to NL momentum conservation, the model should give up its forward momentum to some other body else, or to some photons, after some kind of electromagnetic like interaction. This does not changes the transversal components of the model quantum vectors, along the OY direction.
Then, after the stop, the final model quantum vectors are just equal to their projections in the OY (transversal) orientation, i.e., equal to: $$\label{3.31}
\nu '(0,r) = \nu '(\beta ,r) cos \theta$$ Thus the final vectors are smaller than the orignial ones. The same holds for: $$\label{3.32}
m(0,r) = m(\beta , r)[1-\beta^2]^{1/2}$$ The mass difference, equal to $2\nu '(\beta , r) - 2\nu '(0, r)$ is equal to [*the fraction of the mass-energy is released or given away during the stop*]{}. Then changes of the NL momentum of the body are associated with the conventional changes mass-energy.
Then the G interaction occurring during the fall is fundamentally different from the short range interactions occurring during the stop.
It is simple to prove that the above relations do not depend on the model orientation.Then, in general, the quantitative relations obtained from plain vector geometry, NL mass-energy (or NL frequency) conservation and Eq. \[2.2\], can be written in the form: $$\label{3.4}
\cos\theta =\frac{\nu _A{'}(0,A)}{\nu _A{'}(\beta ,A)}=\frac{\nu _A{'}(0,A)}{\nu
_A{'}(0,B)} =\frac{m_A(0,A)}{m_A(\beta
,A)}=\frac{m_A(0,A)}{m_A(0,B)}={\gamma }^{-1}.$$ $$\label{3,41}
[m_A(\beta ,A)]^2=[m _A(0 ,A)]^2 + [p_A(\beta ,A) c(r)]^2$$ The consistency with local relativity is obvious.
Free orbits in static G fields
------------------------------
For central fields, according to NL mass-energy conservation, Eq. \[3.3\] and Eq. \[3.4\], $$\label{3.5}
m_{r'}(\beta ,r) = \gamma m_{r'}(0,r)=[1-{\beta}^2]^{-1/2}m_{r'}(0,r') e^{\phi
(r)-\phi (r')} = Constant$$ where $$\label{3.51}
m_{r'}(0,r') =m_r(0,r) =m$$ They are the constant local values of the masses.
The NL field potential is: $$\label{3.52}
-\phi (r) \simeq 1+GM/r$$ For the universe of uniform density, $$\phi (U) \simeq -1$$
Thus the model orbits in [*static central fields*]{} are fixed by Eq. \[3.5\] and NL angular momentum conservation. The last one has also been derived from optical principles and has the form [@V81b]: $$\label{3.6}
\vec {j}=\frac {\vec {L}}{m(\beta, r)}={\vec r}\times {\vec {v}(r)}[c(r)]^{-1}$$ The new relationships are obviously linear. In spite of this fundamental difference with General Relativity, they are entirely consistent with the conventional tests for G theories.
Cosmological tests
==================
On the other hand, the same as in the case of the G field equation, the new cosmological and astrophysical contexts would be fixed, definitively, by the particle model properties [@V81b], [@V86].
Such contexts do not depend on arbitrary assumptions and, therefore, are well defined. However they would have some fundamental differences with the rather conventional ones because the model is not disconnected from the universe but depends, entirely, on it.
Effectively, in these works it has been proved that it is not possible to find a free SW model that does not expand in the same proportion as the universe. Then [*the relative values and physical laws in the universe must remain unchanged after some uniform universe expansion*]{}. This turns out to be a trivial generalization of NL relativity for the case of universe expansion.
A black hole (BH), on the other hand, after recovering the energy lost by its matter after condensation, would vaporize itself into new hydrogen that would turn into new stellar-like subjects. The last ones, soon or later, would become condensed again as BHs and so on.
According to this, the universe must evolve, indefinitely, in rather closed cycles in which the radiation emitted by the condensation of matter is absorbed by the BHs resulting from such condensation.
This brings out a new cosmological context, which is a new kind of [*conservative steady state*]{} that is quite different from the rather conventional one.
Consequently, another important test of this theory comes out after comparing the new astrophysical context with the observed facts. For reasons of space, they have been treated in a separated work [@V95b]. However it is interesting to mention here that all the bodies and cosmic radiation backgrounds that should result from the evolution cycles of matter, between BHs and gas and vice versa, are clearly consistent with those observed, directly and indirectly, in astronomy. This includes, of course, the low temperature cosmic background.
Conclusions
===========
The SW particle model can be used a base for [*a new kind of physics based on just properties of light*]{}. This one makes possible to describe the phenomena in terms of a minimum number of parameters and by using the most elemental properties of light.
On the other hand NL relativity makes possible a more trivial, and complete description of the physical phenomena, according to strictly homogeneous relations, regardless on whether they can be measured or not.
By using both the model and NL relativity, it is possible:
[ ]{}- To remove and prevent ambiguities and errors coming from current but inhomogeneous relations between quantities measured in different G field potentials.
[ ]{}- To account for and to unify a wide range of physical phenomena occurring in systems ranging from single uncharged particles up to the universe, including some eventual expansion of the last one.
[ ]{} - To get new physical and cosmological contexts that are fixed by a single hypothesis on the nature of matter.
[ ]{} - To reduce the net number of fundamental hypotheses and arbitrary assumptions normally made not only in physics but also in astrophysics and cosmology.
The new global contexts in physics, astrophysics and cosmology turn out to be most simple, self-consistent and consistent with a wide variety of local and nonlocal phenomena in nature, mainly with:
[ ]{} - Fundamental physics
[ ]{} - The current tests for G theories
[ ]{} - The astronomical observations
On the other hand these contexts have some fundamental differences both with some current assumptions normally used in current G theories, like GR, and with the rather conventional cosmological models. Their differences are, mainly
[ ]{}- Matter properties and free space properties that are linearly related each other, i.e., a linear G field equation.
[ ]{}- More fundamental differences between E fields and G fields. For example,
[ ]{}- G fields do not give up energy to the bodies (Self-propelled bodies).
[ ]{}- G fields without a true energy density.
[ ]{}- New conservative properties of the BHs
[ ]{}- Universe expansion produce [*matter expansion*]{}, in same proportions.
[ ]{}- A Conservative and steady universe in which relative values remain constant, indefinitely [@V81b], [@V86], [@V95a], [@V95b].
According to the nature of the SW particle model, all of them, the uncharged systems of particles and the short range fields between them, can be described as stationary forms of the radiation’s.
In more detail, they can also be described by [*rather coherent wavelets interfering each other constructively*]{}. Away from them, in the free space, they would interfere [*out of phase*]{} or randomly, i. e, the free space and long range fields would not have a true energy density but a high perturbation rate (high density of rather random phase wavelets). This point, proven from different viewpoints, makes an important difference between this theory and GR (or quantum gravity).
The net NL field potential, $\phi (r)$, turns out to be fixed by the negative value of the NL perturbation rate of the free space, also called $w(r)$ or [*wavelet density*]{}. Thus [*the NL G field potential is also a measure of the percentage of relative capacity of the space for admitting wavelets up to saturation*]{}. This percentage is extremely small due to the average wavelet contribution of the universe. This parameter, in turn, would fix the values of the NL speed of light. In this way $w(r)$ also fixes the values of both, the NL frequency and NL wavelength of each stationary wave in matter. Then the gradients of $w(r)$ would account for all of the basic NL G phenomena in the space, mainly G refraction of light, and all of the phenomena induced by it on matter, mainly NL G redshift (or GTD) and NL G contraction of particles.
This theory not only stands out the importance of the [*wavelets*]{} but also provides new interesting hints on the nature and properties of them.
NL refraction turns out to be most important because wavelet, light and bodies would propagate themselves, preferentially, towards toward lower NL speed of light, i.e., towards higher densities of mass-energy. Thus, for example, [*critical reflections*]{}, due to gradients of $c(r)$, would tend to keep the energy (coherent wavelets) in condensed forms. This may also prevent the energy spread from photons and from stable particles, in a way similar to that in the new kind of BH [@V81b].
It is reasonable that coherent wavelets of the same phase and orientation can interact each other much more strongly than the random ones. In other words, interference of coherent wavelets may produce a higher decrease of $c(r)$. In this way, for example, the gradient of the coherent wavelet density that should exist in the boundary of a single photon would produce, just temporally, some gradient of $c(r)$ that may prevent the photon spread. This one, in turn, would be consistent with a global conservation of the total NL mass-energy in the universe.
Something similar is likely to occur in particle models[^15]. Due to the high gradients of the coherent wavelet density existing in the model and its boundary, its radiation could not escape from it. Thus the radiation in the model would always travel in closed stationary paths under angles below the critical reflection angle.
Thus [*short range fields*]{} can in principle be produced by the coherent wavelets escaping rather temporarily from the models, according to the phenomenon of fustrated reflections. Their amplitudes would decrease rather exponentially with the distance, according to the phenomenon of [*dielectric reflections*]{}. Since they would have some energy density, then the field associated field should be of higher order of magnitude than the G fields[^16].
In the region between two models, the mutual local changes of the NL refraction indexes would produce [*frustrated reflections*]{}, i.e., rather stationary radiation between the models with well-defined phases with the models. This would be consistent with the well defined binding energies and distances between particles.
According to this, the universe would be as a [*wavelet’s sea*]{} of random and coherent wavelets associated with all of them: free radiation’s, particles and more massive bodies. Matter would normally be at places in which the wavelets would remain, after longer times, coherent each other. Then it is reasonable that the wavelets associated to a free quantum would have some non null probability to interact with other wavelets and, throughout this, with other bodies associated to the last ones. Since a free quantum depends only on a single proper parameter, then any wavelet lost by a quantum most probably would produce a small [*redshift*]{} of its NL frequency.
In ordinary diffraction experiments, these frequency changes would be negligible. However they may become important in cosmological ranges of distances because they would produce an average redshift of light proportional to the NL distances. This would be another alternative for the existence of WRS, HRS and, therefore, of gravityAs shown above, G field gradients could not exist without some WRS (or HRS).
The good consistency with the observations would indicate that most of the physical phenomena in nature would be determined by space perturbations currently described in optics as [*wavelets*]{}. They would interfere each other constructively in some places and destructively in other ones[^17]. They would reconstruct quanta and particles, in different NL positions and NL times.
Something similar occurs in X-ray crystallography and holography. The detailed three-dimensional picture of the structure of matter is virtually reconstructed after interference of waves of well-defined amplitudes and frequencies. It is amazing how large is the number of similarities existing in nature.
Strictly, these wavelet wavelet interactions would also make small changes in the energy distribution in the system (universe), without changing its total energy. This means that the energy lost from HRS, in one way or another, would appear in other bodies like in BHs.
Of course the present theory, due to its high simplicity, may look very primitive. However there is a large and fascinating research field on this line and a lot of work to do[@V95c]. On the other hand, as in anything made up by human beings, this work may also contain some eventual errors. Thus any suggestion, constructive critic, are highly welcomed.
[99]{}
A. Einstein, [*The Meaning of Relativity*]{}. Princeton University Press (1955).
R. A. Vera. [*Una Nueva Concepcion del Universo*]{}. Atenea. Yearly book on science and art. Publ. by Universidad de Concepcion. Vol 435, pages 103-127 (1977)
R. A. Vera. [*What does the work in gravitational fields?*]{} Proceedings of the Einstein Centennial Symposium on Fundamental Physics. Eds. S. M. Moore, A. M. Rodriguez-Vargas, A. Rueda, G. Viollini. Universidad de los Andes, Bogota, and Universita di Roma, pages 597 - 626 (1981)
R. A. Vera. International. Journal. of Th. Phys, [**20**]{}, 19- 50 (1981)
R. A. Vera. [*Theoretical properties of gravitational fields derived from roperties of light.*]{} Proceedings of the 4th Marcel Grossmann Meeting on Gen.Rel. Eds R. Ruffini. Publ. Elsevier Science. Pages 1743- 1752 (1986)
R. A. Vera. [*Advances on a unified theory of Physics and Astrophysics based on a Standing Wave Particle Model*]{}. To appear in the proceedings of the 7th Marcel Grossman Meeting on General Relativity. World Scientific Bublishing Co. (1995).
R. A. [*The new kind of universe fixed by a standing-wave particle model*]{}. To appear in the Proceedings of the. 6th Canadian Conf. on Gen. Relativity and Relativistic Astrophys ( Fields Institute Communications) and in gr-qc xxx.lanl. gov. (1995)
Vera, R. A. [*From a photon up to the universe*]{}, a unified theory for physics and astrophysics based on properties of light. (Book in preparation)
[^1]: email: rvera@buho.dpi.udec.cl
[^2]: For the moment it is not necessary to know the mechanism for the confinement of the radiation because the most probable one can be inferred from the new results coming out from the present work, at the conclusion stage. The model does not include any external matter like mirrors. In experimental tests, the contribution of the mass of some external mirrors can be subtracted from the total mass.
[^3]: The constant quantities used in special relativity, like $m$, would be limiting cases of these NL functions, like $m_{r'}(\beta,r)$, for $\beta =0$ and $r'=r$
[^4]: This is a fact that can be verified in a Michelson-Morley gedanken experiment made up with particle models instead arms. In it, the model SWs and the light SW waves between the end mirrors must change in identical proportions after identical changes of velocity and field potential. The relative numbers of wavelengths remain unchanged.
[^5]: In a first step, for non cosmological purposes, the Hubble redshift (HRS) is neglected.
[^6]: Notice that this fundamental difference between these fields has been demonstrated from the fact that G fields do show GTD and that E fields don’t. The same result was obtained before from more exact theoretical methods [@V81a], [@V81b].
[^7]: These two waves traveling in opposite directions would correspond to halves of a single quantum. They are most probably related with fermions and antiferimions that would have opposite phases relative to each other.
[^8]: For simplicity $h=1$
[^9]: This is also trivial in a transversal model after drawing wavefronts after each wavelength.
[^10]: The real existence of these wavelets is evident in the phenomenon of [*frustrated reflection*]{}.
[^11]: This makes an important difference with short range fields that would depend on [*coherent wavelet interferences*]{}. The last ones do account for a real field energy and for the energy exchange between the field and the charges.
[^12]: In other terms the G field equation would diverge. This would mean that contribution of local bodies, compared with the universe one, would be null, i.e., gravity could not exist
[^13]: R is the typical distance for a WRS factor $1/e$. It corresponds with the Hubble Radius.
[^14]: G refraction occurring during the back and forth reflections would deviate the vectors from their original orientations. This would cause the model acceleration
[^15]: For a better understanding of this, it is better to think on SW models of [*torus*]{} shapes.
[^16]: Observe that they should have well-defined phase relations with the models SWs. This seems very interesting for the case of electric fields.
[^17]: This is also consistent with the ideas proposed by T. W. Andrews (Personal communications).
|
---
abstract: 'Heat kernels are used in this paper to express the analytic index of projectively invariant Dirac type operators on $\Gamma$ covering spaces of compact manifolds, as elements in the K-theory of certain unconditional completions of the twisted group algebra of $\Gamma$. This is combined with V. Lafforgue’s results in the untwisted case, to compute the range of the trace on the K-theory of these algebras, under the hypothesis that $\Gamma$ is in the class C’ (defined by V. Lafforgue).'
address: 'Department of Mathematics, University of Adelaide, Adelaide 5005, Australia'
author:
- Varghese Mathai
title: Heat kernels and the range of the trace on completions of twisted group algebras
---
Introduction {#introduction .unnumbered}
============
For $\Gamma$ a torsion-free discrete group, one formulation of a standing conjecture of Kaplansky and Kadison states that the range of the canonical trace on the $K$-theory of the reduced $C^*$-algebra of $\Gamma$, is contained in the integers. However, if we twist the convolution by a multiplier (i.e. a normalized $\Uu$-valued 2-cocycle on $\Gamma$) then this is no longer true, as shown by Pimsner-Voiculescu [@PV] and Rieffel [@Rieff], who computed the precise range of the canonical trace on the twisted group $C^*$-algebra of ${\bbZ^2}$ (which turns out to be the noncommutative torus). The author with Marcolli in [@MM] settled the case of surface groups by identifying the range of the canonical trace on the twisted group $C^*$-algebra. In the present paper, we study the range of the trace on the $K$-theory of [*good*]{} unconditional completions $\cA(\Gamma, \sigma)$ of the twisted group algebra $\bbC(\Gamma,\sigma)$ (see section \[RD\]) - an example of such a completion is the $\ell^1$ completion of $\bbC(\Gamma, \sigma)$. Our approach is to study a twisted analogue of the assembly map, viewed as a homomorphism $$\label{twistedassembly}
\mu_{\sigma}^{\cA} : K^\Gamma_* (\underline{E}\Gamma) \rightarrow
K_*(\cA(\Gamma, \sigma)),$$ whenever the Dixmier-Douady invariant $ \delta(\sigma)$ of the multiplier $\sigma$ on $\Gamma$ is trivial. The map $\mu_\sigma^\cA$ is a twisted version of the assembly map defined by Lafforgue, and the definition uses Lafforgue’s Banach $KK$-theory. We use the method of heat kernels to study an analytic version of the map , called the twisted analytic Baum-Connes map in the paper, and a standard index theorem in section \[equivalence\] establishes that both definitions are equivalent. Using fundamental results in [@Laff], we prove in Theorem \[main1\] that if $\Gamma$ is a discrete group in Lafforgue’s class $\cC'$, then the twisted assembly map $\mu_\sigma^\cA$ is an isomorphism. The class $\cC'$ will be described later, but we mention here that it contains all discrete subgroups of connected Lie groups, word hyperbolic groups and amenable groups. Together with a twisted version of an $L^2$-index theorem given in [@Ma2] it is then a straightforward corollary to obtain a formula for the range of the trace on the $K$-theory of $\cA(\Gamma, \sigma)$ in terms of classical characteristic classes on the the classifying space $\underline{B}\Gamma$ for proper actions, as explained in section \[sect:range\]. Although this formula is computationally challenging, it can be explicitly computed in low dimensional cases, e.g. in the case when $\Gamma$ is torsion-free and $B\Gamma$ is a smooth compact oriented manifold of dimension less than or equal to 4, which is done in section \[3&4\] of the paper. This generalizes earlier results of [@CHMM], [@MM] , [@PV], [@Rieff], [@BC].
If in addition $\Gamma$ has the Rapid Decay property (property RD), then we can choose the good unconditional completion $\cA(\Gamma,\sigma)$ to be the Sobolev completion $H^s(\Gamma, \sigma)$ of $\bbC(\Gamma, \sigma)$, for $s$ large and $H^s(\Gamma, \sigma)$ being a dense $*$-subalgebra of the reduced $C^*$-algebra $C^*_r(\Gamma, \sigma)$, stable under the holomorphic functional calculus, the twisted assembly map reduces to the usual twisted assembly map cf. [@Ma], $$\label{twistedassembly2}
\mu_{\sigma}: K^\Gamma_i (\underline{E}\Gamma) \rightarrow
K_i(C^*_r(\Gamma, \sigma)).$$ We prove in Theorem \[main1\] that if $\Gamma$ is a discrete group in the class $\cC'$ and $\Gamma$ has property RD, then the twisted assembly map $\mu_\sigma$ in (\[twistedassembly2\]) is an isomorphism. The groups that have property RD include all finitely generated groups of polynomial growth, all finitely generated free groups, all word hyperbolic groups, and certain property $(T)$ groups such as cocompact lattices in ${\bf SL}(3, \mathbb F)$ or the exceptional group ${\bf E}_6$, where $\mathbb F$ is a non-discrete locally compact field or the quaternions. All of these groups are also in the class $\cC'$. Using the earlier mentioned procedure, we also obtain a formula for the range of the trace on the $K$-theory of $C^*_r(\Gamma, \sigma)$ in terms of classical characteristic classes on $\underline{B}\Gamma$.
The last section of the paper is devoted to studying the degree one of the assembly map, and following the construction of Natsume [@Natsume] and Valette et. al. in [@Bettaieb-Matthey], we determine the explicit generators of $K_1(\cA(\Gamma, \sigma))$ and $K_1(C^*_r(\Gamma, \sigma))$, whenever $\Gamma$ is a torsion-free cocompact Fuchsian group.
The appendix, written by Indira Chatterji, establishes useful results on the twisted rapid decay property for $(\Gamma, \sigma)$ that are used in the text. One interesting result there is that if $\Gamma$ has property RD, then it has the twisted RD property for any multiplier $\sigma$ on $\Gamma$. This means in particular that we [*do not*]{} have to appeal to the Baum-Connes conjecture [*with coefficients*]{}, which is a technical improvement of results in [@Ma]. The author thanks Indira Chatterji for helpful discussions.
Basics
======
In this section $\sigma$ is a multiplier on $\Gamma$ a discrete group, that is a map $\sigma:\Gamma\times\Gamma\to\Uu$ satisfying the following identity for all $\gamma, \mu,\delta\in \Gamma$:
- $\sigma(\gamma,\mu)\sigma(\gamma\mu,\delta)=\sigma(\gamma,\mu\delta)\sigma(\mu,\delta)$.
- $\sigma(\gamma,1)=\sigma(1,\gamma)=1$.
Recall that the *Dixmier-Douady invariant* of a multiplier $\sigma$ is the cohomology class $\delta(\sigma)\in
H^3(\Gamma,\bbZ)$, the image of $[\sigma]$ obtained under the map $\delta$ arising in the long exact sequence in cohomology derived from the short exact sequence of coefficients $$0\to\bbZ\to\bbR\to\Uu\to 0.$$ We denote by $E\Ga$ the universal cover of $B\Ga$, the classifying space for $\Ga$. The following lemma will be used later.
\[twistedIdentities\] Let $\alpha\in Z^2(B\Ga,\bbR)$ and $X\subset E\Ga$ a cocompact $\Ga$-space. Then there is a map $\varphi:\Ga\to C_0(X)$ such that;
- $\varphi_{\gamma}(x)+\varphi_{\mu}(\gamma x)-\varphi_{\mu\gamma}(x)$ is independent of $x\in X$.
- There is $x_0\in X$ such that $\varphi_{\gamma}(x_0)=0$ for any $\gamma\in\Ga$.
- $\lambda(\gamma,\mu)= \varphi_{\gamma}(\mu x_0)$ is an $\bbR$-valued 2 cocycle that is cohomologous to $\alpha$.
Let $p:E\Ga\to B\Ga$ be the canonical projection and take a lift $\tilde{\alpha}=p^*(\alpha)\in Z^2(E\Ga,\bbR)$. Since $E\Ga$ is contractible, there is a $\Lambda\in C^1(E\Ga,\bbR)$ such that $\tilde{\alpha}=d\Lambda$. By definition of $\tilde{\alpha}$, we have $$0=\gamma^*\tilde{\alpha}-\tilde{\alpha}=d(\gamma^*\Lambda-\Lambda)\ \ \hbox{ for any }\gamma\in\Ga.$$ The element $\eta_{\gamma}=\gamma^*\Lambda-\Lambda$ hence belongs to $Z^1(E\Ga,\bbR)$, so that there exists $c_{\gamma}\in C^0(E\Ga,\bbR)$ with $\eta_{\gamma}=dc_{\gamma}$. Let us show that $\mu^*c_{\gamma}+c_{\mu}-c_{\gamma\mu}\in C^0(E\Ga,\bbR)$ is a constant. To do so, it is enough to see that $d(\mu^*c_{\gamma}+c_{\mu}-c_{\gamma\mu})=0$. We compute: $$\begin{aligned}
dc_{\gamma\mu}&=&\eta_{\gamma\mu}=(\gamma\mu)^*\Lambda-\Lambda=\mu^*\gamma^*\Lambda-\gamma^*\Lambda+\gamma^*\Lambda-\Lambda\\
&=&\mu^*\eta_{\gamma}+\eta_{\mu}=d(\mu^*c_{\gamma}+c_{\mu})\end{aligned}$$ Let $x_0\in X$, we now define $$\begin{aligned}
\varphi:\Ga &\to & C_0(X)\\
\gamma &\mapsto &\varphi_{\gamma},\end{aligned}$$ where $\varphi_\gamma(x) = c_{\gamma}(x)-c_{\gamma}(x_0)$. Then $\varphi$ satisfies (i) and (ii). In particular, from the $\bbR$-valued closed 2-form $\alpha$ on $B\Gamma$, we have produced an $\bbR$-valued group 2-cocycle $\lambda(\gamma,\mu)= \varphi_{\gamma}(\mu x_0)$ on $\Gamma$. Also, the group extension corresponding to $\lambda$ can be described as follows. Let $\Gamma^\lambda = \Gamma \times \bbR$ with product given by $(\gamma, r)(\gamma', r') = (\gamma\gamma', \lambda(\gamma, \gamma') + r + r')$. If $g_{ij}$ are transition functions for the principal bundle $p: E\Gamma \to B\Gamma$, define the lift $\hat g_{ij} = (g_{ij}, 0) \in \Gamma^\lambda$ Then $ t_{ijk} = \hat g_{ij} \hat g_{jk} \hat g_{ki} = \lambda( g_{ij}, g_{jk}) + \lambda(g_{ik}, g_{ki}) $ is the $\bbR$-valued Cech 2-cocycle on $B\Gamma$ that is associated to the $\bbR$-valued group 2-cocycle $\lambda$ on $\Gamma$.
If $\alpha|_{U_i} = d \theta_i$, $(\theta_i - \theta_j)|_{U_i\cap U_j} = d f_{ij}$, $ (f_{ij} + f_{jk}
+ f_{ki}) = t_{ijk} \in \mathbb R$, then the Cech cohomology 2-cocycle corresponding to the de Rham closed 2-form $\alpha$ is $t$, by the well known Cech-de Rham isomorphism.
This shows that $[\alpha] = [t] = [\lambda] \in H^2(B\Gamma, \bbR)$.
In the notation of Lemma \[twistedIdentities\] above, one verifies that $\sigma(\gamma, \mu) = \exp(-i\varphi_\gamma(\mu x_0))$ defines a multiplier on $\Gamma$. The map $\varphi$ is called a *phase* associated to $\sigma$.
Let $A$ be a $\Gamma$-$C^*$-algebra, we denote by $\C(\Gamma, A, \sigma)$ the $*$-algebra of finitely supported maps from $\Gamma$ to $A$ endowed with a $\sigma$-twisted convolution given as follows: for all $a, b \in A$ and $\gamma. \mu \in \Gamma$, $$aT_{\gamma}*_\sigma bT_{\mu}=a\alpha_{\gamma}(b)\sigma(\gamma,\mu)T_{\gamma\mu},$$ where $\alpha$ denotes the action of $\Gamma$ on $A$. Here we think of elements of $\C(\Gamma, A, \sigma)$ as finite sums $\sum a_\gamma T_\gamma$, where $a_\gamma \in A$, $T_\gamma$ is a formal letter satisfying $T_{\gamma}T_{\mu} = \sigma(\gamma,\mu)T_{\gamma\mu}$, $T_\gamma a T_\gamma^* = \alpha_\gamma (a)$ and $T_\gamma^* = \sigma(\gamma, \gamma^{-1}) T_{\gamma^{-1}}$.
Given a Banach norm $\|\ \|_{B}$ on $\C(\Gamma, A, \sigma)$, we denote by $B(\Gamma,A,\sigma)$ the completion of $\C(\Gamma, A, \sigma)$ with respect to the norm $\|\ \|_{B}$.
In case where $A=\C$ (with a trivial $\Ga$-action) we simply write $\bbC(\Gamma,\sigma)$. We often represent it as the $\bbC$-subalgebra of ${\mathcal
B}(\ldg)$ generated by $\{T_{\gamma}|{\gamma\in\Gamma}\}$, where for $\gamma\in\Gamma$ $$\begin{aligned}
T_{\gamma}:\ldg & \to & \ldg,\\
\delta_\mu & \mapsto &
\sigma(\gamma,\mu)\delta_{\gamma\mu},\end{aligned}$$ so that an element in $\bbC(\Gamma,\sigma)$ is a finite $\bbC$-linear combination of the operators $T_{\gamma}$, and the convolution reads (for $\gamma,\mu\in\Gamma$) $$T_{\gamma}*_{\sigma}T_{\mu}=\sigma(\gamma,\mu)T_{\gamma\mu}.$$ We shall consider several completions of $\bbC(\Gamma,\sigma)$ that we now explain. The *$\ell^1$-completion* (given by the norm $\|\sum_{\gamma\in\Gamma}a_{\gamma}T_{\gamma}\|_1=\sum_{\gamma\in\Gamma}
|a_{\gamma}|$) yields the *$\ell^1$-twisted Banach algebra* denoted by $\ell^1(\Gamma,\sigma)$, which is the completion of $\bbC(\Gamma,\sigma)$ with respect to this $\ell^1$-norm. It is a straightforward computation to show that it is indeed a Banach algebra, contained in ${\mathcal B}(\ldg)$. Next we shall consider the *operator norm*, given by $$\|f\|_{op}=\sup\{\|f(\xi)\|_2 : \|\xi\|_2=1\},$$ and the completion of $\bbC(\Gamma,\sigma)$ with respect to this norm is the *twisted reduced $C*$-algebra* $C^*_r(\Gamma,\sigma)$. Recall that a *length function on $\Gamma$* is a map $\ell:\Gamma\to{\bbR}_+$ satisfying:
- $\ell(1)=0$, where 1 denotes the neutral element in $\Gamma$,
- $\ell(\gamma)=\ell(\gamma^{-1})$ for any $\gamma\in\Gamma$,
- $\ell(\gamma\mu)\leq \ell(\gamma)+\ell(\mu)$ for any $\gamma,\mu\in\Gamma$.
For $\ell$ a length function on $\Gamma$ and $s$ a positive real number, the *$s$-weighted $\ell^2$-norm* is defined by $$\|\sum_{\gamma\in\Gamma}a_{\gamma}T_{\gamma}\|_{s}=\sqrt{\sum_{\gamma\in\Gamma}|a_{\gamma}|^2(1+\ell(\gamma))^{2s}}$$ and the *$s$-Sobolev space* is the completion of $\bbC(\Gamma,\sigma)$ with respect to this norm, denoted by $H^s_\ell (\Gamma,\sigma)$. If the length function is chosen to be the word length with respect to a finite set of generators for $\Gamma$, then we just write $H^s(\Gamma,\sigma)$, omitting $\ell$ in the notation. Finally, the *space of rapidly decreasing functions (with respect to the length $\ell$)* is given by $$H^{\infty}_\ell (\Gamma,\sigma)=\bigcap_{s\geq 0}H^s_\ell (\Gamma,\sigma).$$ $H^{\infty}_\ell (\Gamma,\sigma)$ is not an algebra in general, but it is one if $\Gamma$ has the Rapid Decay property (with respect to the length $\ell$), see Definition \[sRD\]. In fact, if $\Gamma$ has the Rapid Decay property (with respect to the length $\ell$), then $H^s_\ell (\Gamma,\sigma)$ is an algebra for $s$ large enough, cf. Corollary \[remarque\].
\[supnormRD\] If $\Gamma$ has polynomial volume growth (with respect to the length $\ell$), then the space of rapidly decreasing functions (with respect to the length $\ell$) has the following sup norm characterization: $$H^{\infty}_\ell (\Gamma,\sigma)=\displaystyle\left\{ f: \Gamma \to \bbC: \displaystyle\sup_{\gamma\in\Gamma} \left(|f({\gamma})|(1+\ell(\gamma))^{s}\right)<\infty \quad \forall s\in \bbN
\right\}$$
If $f \in H^{\infty}_\ell(\Gamma,\sigma)$, then for all $s\in \bbN$, one sees that the function $\gamma \mapsto |f({\gamma})|^2(1+\ell(\gamma))^{2s}$ is bounded on $\Gamma$, therefore $\gamma \mapsto |f({\gamma})|(1+\ell(\gamma))^{s}$ is also a bounded function on $\Gamma$.
Conversely, suppose that $ f: \Gamma \to \bbC$ is such that $$\sup_{\gamma\in\Gamma} \left(|f({\gamma})|(1+\ell(\gamma))^{r}\right) = C_r<\infty \quad \forall r\in \bbN.$$ Then we estimate, $$\sum_{\gamma\in \Gamma}|f({\gamma})|^2(1+\ell(\gamma))^{2s}\le C_r^2
\sum_{\gamma\in \Gamma} (1+\ell(\gamma))^{2(s-r)}.$$ Since $\Gamma$ has polynomial growth (with respect to the length $\ell$), we see that by choosing $r$ sufficiently large, the right hand side is finite, proving the lemma.
An important step is to compute the $K$-theory of $C^*_r(\Gamma,\sigma)$. The following lemma shows that we only need to know the cohomology class of the multiplier.
\[iso\]Let $\sigma,\sigma'\in H^2(\Gamma,\Uu)$ be two cohomologous $2$-cocycles. Then there exists an isomorphism $$\varphi:B(\Gamma,\sigma)\to B(\Gamma,\sigma'),$$ inducing the identity map on $K$-theory. Here $B(\Ga,\sigma)$ is any $*$-Banach completion of $\bbC(\Ga,\sigma)$.
That the cocycles $\sigma$ and $\sigma'$ are cohomologous means that there exists $f:\Gamma\to\Uu$ such that $\sigma'=\sigma df$, where for $\gamma_1,\gamma_2\in\Gamma$, $df(\gamma_1,\gamma_2)=f(\gamma_1\gamma_2)f(\gamma_1)^{-
1}f(\gamma_2)^{-1}$. We shall define the map $\varphi:B(\Gamma,\sigma)\to
B(\Gamma,\sigma')$ on the generators $\{T_{\gamma}\}_{\gamma\in\Gamma}$ by $\varphi(T_{\gamma})=f(\gamma)T'_{\gamma}$, extend it by linearity to $\bbC(\Gamma,\sigma)$ and by continuity to $B(\Gamma,\sigma)$. Indeed, it is a \*-homomorphism: $$\begin{aligned}
\varphi(T_{\gamma}T_{\mu})&=&\sigma(\gamma,\mu)\varphi(
T_{\gamma\mu})=\sigma(\gamma,\mu)f(\gamma\mu)T'_{\gamma\mu}\\
&=&\sigma'(\gamma,\mu)f(\gamma)f(\mu)T'_{\gamma\mu}=
f(\gamma)f(\mu)T'_{\gamma}T'_{\mu}=\varphi(T_{\gamma})\varphi(T_{\mu}),\end{aligned}$$ bijective (it is bijective on the generators), hence induces an isomorphism in $K$-theory, which is the identity since $[T'_{\gamma}]=[f(\gamma)T'_{\gamma}]$ in $K_1(B(\Gamma,\sigma'))$ for any $\gamma\in\Gamma$, the homotopy being given by a path in $\Uu$ between $f(\gamma)$ and $1$. Similarly at the level of $K_0$.
Good unconditional completions
==============================
\[good\] Following Lafforgue [@Laff], we say that a norm $\|\ \|_{\cA}$ on $\bbC(\Gamma,\sigma)$ is *unconditional* if for any two elements $A=\sum_{\gamma\in\Gamma}a_\gamma T_\gamma$ and $B=\sum_{\gamma\in\Gamma}b_\gamma T_\gamma$ in $\bbC(\Gamma,\sigma)$, $|a_\gamma|\leq |b_\gamma|$ implies $\|A\|_{\cA}\leq\|B\|_{\cA}$. Given an unconditional norm $\|\
\|_{\cA}$ on $\bbC(\Gamma,\sigma)$, we denote by $\cA(\Gamma,\sigma)$ the completion.
For technical reasons, since we use the heat kernel approach in this paper, we introduce the following special case. Assume that an unconditional completion $\cA(\Gamma,
\sigma)$ of $\C(\Gamma, \sigma)$ is such that $$\|T_g\|_{\cA} \le C_1 e^{C_2 \ell(g)^p}, \qquad \forall g\in\Gamma,$$ for some positive constants $C_1, C_2$ independent of $g\in \Gamma$ and for some $p$ such that $1\le p<2$ which is also independent of $g\in
\Gamma$. We shall call such an unconditional completion a [*good unconditional completion*]{} of $\C(\Gamma, \sigma)$.
Note that $\ell^1$ is trivially a good unconditional completion, and that it is straightforward to see that the Sobolev completions are good unconditional completions as well. The operator norm is not unconditional, which means that the reduced (twisted) group $C^*$-algebra is [*not*]{} an unconditional completion.
[[H]{}]{}
Let $\ell^2(\Gamma, \cH)$ denote the space of $\cH$-valued square summable functions on the group $\Gamma$, where $\cH$ is a separable Hilbert space with the trivial action of $\Gamma$. Let $\U_\cH(\Gamma, \sigma)$ denote the von Neumann algebra of all bounded linear operators on $\ell^2(\Gamma, \cH)$ that commute with the $(\Gamma,
\sigma)$-action. It is a standard observation that any element $A\in\U_\cH(\Gamma,\sigma)$ can be represented by a strongly convergent series, $$A=\sum_{\gamma\in\Gamma}T_\gamma\otimes A(\gamma),$$ where $A(\gamma)\in \B(\cH)$ is a bounded linear operator on $\cH$, defined by the formula $$A(\delta_e\otimes v)=\sum_{\gamma\in\Gamma}\delta_\gamma\otimes
A(\gamma)v, \quad v\in \cH.$$ One has the following useful sufficient condition, where $\otimes$ denotes the projective tensor product in the entire paper.
[\[D\]]{} Let $\cA(\Gamma, \sigma)$ be a good unconditional completion of $\C(\Gamma, \sigma)$ and ${\mathcal K}$ denote the algebra of compact operators on the Hilbert space $\cH$. If $A\in \U_\cH(\Gamma, \sigma)$, $A=\sum_{\gamma\in\Gamma}T_\gamma\otimes A(\gamma)$ is such that $A(\gamma) \in {\mathcal K}$ and also satisfies $
\|A(\gamma)\|< C_5 e^{-C_6 \ell(\gamma)^2}$ for some positive constants $C_5, C_6$, then $A \in
\cA(\Gamma, \sigma) \otimes{\mathcal K}$.
Observe that one has the estimate $$\label{e:d1}
\# \left\{\gamma\in\Gamma\; |\;
\ell(\gamma) \le n \right\} \le C_7 e^{C_8 n},$$ for some positive constants $C_7, C_8$, since the growth rate of the volume of balls in $\Gamma$ is at most exponential. We compute, $$\begin{aligned}
|| A ||_{\cA\otimes {\mathcal K}} & = & ||\sum_{\gamma\in\Gamma}T_\gamma\otimes
A(\gamma)||_{\cA\otimes {\mathcal K}} \le\sum_{\gamma\in\Gamma} ||T_\gamma||_\cA
||A(\gamma)|| \\[+7pt]
& \le & \sum_{\gamma\in\Gamma} C_1 e^{C_2 \ell(\gamma)^p}C_5 e^{-C_6
\ell(\gamma)^2} = \sum_{n\in\N} \sum_{\ell(\gamma) \le n} C_1 e^{C_2
\ell(\gamma)^p}C_5 e^{-C_6
\ell(\gamma)^2} \\[+7pt]
& \le & \sum_{n\in\N} C_7 e^{C_8 n} C_1 e^{C_2 n^p}C_5 e^{-C_6
n^2} < \infty.\end{aligned}$$ The last sum is convergent since $0\le p<2$ by the good unconditional hypothesis.
In [@PR], Packer and Raeburn, inspired by A. Wasserman’s thesis, established a stabilization (or untwisting) trick. We will present a good unconditional version of this, in the simple case of a discrete group $\Gamma$, that we need in this paper. Let $\sigma$ be a multiplier on $\Gamma$ and $\mathcal K$ be the algebra of compact operators on $\ell^2(\Gamma)$. Observe that for any $\Gamma$-$C^*$-algebra $A$, one has the following canonical isomorphism, $$\label{pre-PR}
\C(\Gamma, A, \sigma) \otimes {\mathcal K} \cong
\C(\Gamma, A\otimes {\mathcal K}),$$ where $\Gamma$ acts diagonally on the tensor product $A\otimes {\mathcal K}$, and is given by the given action of $\Gamma$ on $A$ and the adjoint action, $\gamma \mapsto {\rm Ad}(T_\gamma)$. That is, the twisted convolution on the left hand side of becomes an ordinary convolution on the right hand side of : for all $a, b \in A$, for all $U, V \in {\mathcal K}$ and for all $\gamma. \mu \in \Gamma$, $$a \otimes {\rm Ad}(T_{\gamma}) U * b \otimes {\rm Ad}(T_{\mu}) V =a\alpha_{\gamma}(b) \otimes {\rm Ad}(T_{\gamma\mu}) UV,$$ where $\alpha$ denotes the action of $\Gamma$ on $A$.
Recall that for any good unconditional completion $\cA(\Gamma, A, \sigma)$ of $\C(\Gamma, A, \sigma)$ there are positive constants $C_1, C_2$ independent of $g\in \Gamma$ such that $$\|T_g\|_{\cA} \le C_1 e^{C_2 \ell(g)^p}, \qquad \forall g\in\Gamma,$$ for some $p$ such that $1\le p<2$ which is also independent of $g\in
\Gamma$. But this implies that $$\|{\rm Ad}(T_g)\|_{\cA} \le C_1^2 e^{2C_2 \ell(g)^p}, \qquad \forall g\in\Gamma,$$ and conversely. Therefore for any good unconditional completion, one has the canonical isomorphism $$\label{unconditional-PR}
\cA(\Gamma, A, \sigma) \otimes {\mathcal K} \cong
\cA(\Gamma, A\otimes {\mathcal K}).$$ This isomorphism is clearly also true for general unconditional completions.
Heat kernels and the analytic twisted Baum-Connes map {#analytical}
=====================================================
Spin$^\bbC$ manifolds and twisted spin$^\bbC$ Dirac operators {#Dirac}
-------------------------------------------------------------
Let $M$ be a smooth $\Gamma$ manifold without boundary. A choice of $\Gamma$-invariant Riemannian metric $g$ on $M$ defines a bundle of Clifford algebras, with fibre at $z\in M$ the complexified Clifford algebra $$\begin{gathered}
\Cl_z(M)=\left(\bigoplus_{k=0}^\infty (T^*_z M \otimes \bbC)^k\right)/\langle \alpha
\otimes\beta +\beta \otimes\alpha -2(\alpha ,\beta )_g,\ \alpha ,\beta
\in T^*_z M\rangle.
\end{gathered}$$ If $\dim M=2n,$ this complexified algebra is isomorphic to the matrix algebra on $\bbC^{2^n}.$ In particular the Clifford bundle is an associated bundle to the metric coframe bundle, the principal ${\operatorname{SO}}(2n)$-bundle $\cF,$ where the action of ${\operatorname{SO}}(2n)$ on the Euclidean Clifford algebra $\Cl(2n)$ is through the spin group, which may be identified within the Clifford algebra as $$\Spin(2n)=\{v_1v_2\cdots v_{2k}\in\Cl(2n);v_i\in \bbR^{2n},\ |v_i|=1\}.$$ The non-trivial double covering of ${\operatorname{SO}}(2n)$ is realized through the mapping of $v$ to the reflection $R(v)\in\operatorname{O}(2n)$ in the plane orthogonal to $v$ $$p:\Spin(2n)\ni a=v_1\cdots v_{2k}\longmapsto R(v_1)\cdots R(v_{2k})=R\in{\operatorname{SO}}(2n).$$ The ${\operatorname{Spin}^{\bbC}}(2n)$ group, defined as $${\operatorname{Spin}^{\bbC}}(2n)=\{cv_1v_2\cdots v_{2k}\in\Cl(2n);v_i\in \bbR^{2n},\ |v_i|=1,
c\in\bbC,\ |c|=1\},$$ is a central extension of ${\operatorname{SO}}(2n),$ $$\UU(1)\longrightarrow {\operatorname{Spin}^{\bbC}}(2n)\longrightarrow {\operatorname{SO}}(2n),$$ where the quotient map is consistent with the covering of ${\operatorname{SO}}(2n)$ by $\Spin(2n),$ i.e. $${\operatorname{Spin}^{\bbC}}(2n) = \Spin(2n) \times_{\bbZ_2} \UU(1).$$
The manifold $M$ is said to have a $\Gamma$-equivariant ${\operatorname{Spin}^{\bbC}}$ structure, if there is an extension of the coframe bundle to a principal ${\operatorname{Spin}^{\bbC}}(2n)$-bundle $$\xymatrix{\UU(1)\ar[d] \ar[r]^{=}& \UU(1)\ar[d]\\
{\operatorname{Spin}^{\bbC}}(2n)\ar[r]\ar[d]&\cF_{L}\ar[d]\ar[r] & M\ar[d]^{||}\\\
{\operatorname{SO}}(2n)\ar[r]&\cF \ar[r] & M,}
\label{mms3.67}$$ where $\cF_L,$ the ${\operatorname{Spin}^{\bbC}}(2n)$ bundle over $M$, may also be viewed as a circle bundle over $\cF$, compatible with the $\Gamma$-action. The associated bundles of half spinors on $M$ are defined as $${\mathcal S}^\pm = \cF_L \times_{{\operatorname{Spin}^{\bbC}}(2n)} S^\pm,$$ where $S^\pm$ are the fundamental half spin representations of ${\operatorname{SO}}(2n)$. The $\Gamma$-invariant Levi-Civita connection determines a connection 1-form on $\cF$, and together with the choice of a $\Gamma$-invariant connection 1-form on the circle bundle $\cF_L$ over $\cF$, they determine a connection 1-form on the principal ${\operatorname{Spin}^{\bbC}}$ bundle $\cF_L$ over $M$, which is $\Gamma$-invariant. That is, one gets a connection $$\nabla^{\cS\otimes E} : C^\infty(M, {\mathcal S}^+\otimes E) \to C^\infty(M, T^*M \otimes {\mathcal S}^+ \otimes E),$$ defined as $\nabla^{\cS\otimes E} = \nabla^{\mathcal S} \otimes 1 + 1 \otimes \nabla^E$, where $\nabla^E$ is a $\Gamma$-invariant connection on the $\Gamma$-invariant vector bundle $E$ over $M$. Now the contraction given by Clifford multiplication defines a map $$C : C^\infty(M, T^*M \otimes {\mathcal S}^+\otimes E) \to C^\infty(M, {\mathcal S}^-\otimes E).$$ The $\Gamma$-equivariant Spin$^{\mathbb C}$ Dirac operator with coefficients in $E$ is defined as the composition $$\np^{{\mathbb C}+}_E = C \circ \nabla^{\cS\otimes E}.$$
In this section, we will define the analytic index map for an arbitrary torsion-free discrete group $\Gamma$ and for an arbitrary multiplier $\sigma$ on $\Gamma$ with trivial Dixmier-Douady invariant $\delta(\sigma)$. Now let $M$ be a manifold without boundary with a given smooth proper cocompact $\Gamma$ action and a $\Gamma$-equivariant Spin$^{\mathbb C}$ structure, $E\to
M$ a $\Gamma$-equivariant complex vector bundle on $M$, and $\phi :
M \to \uE\Gamma$ a $\Gamma$-equivariant continuous map. We will view the $\Gamma$-equivariant Spin$^{\mathbb C}$ Dirac operator with coefficients in $E$ as an operator on the $L^2$-spaces, $\np^{\mathbb C}_E: L^2(M, {\mathcal S}^+\otimes E) \to
L^2(M,{\mathcal S}^-\otimes E)$.
Let $c$ be an $\R$-valued $\Gamma$-equivariant Cech 2-cocycle on $\uE\Gamma$ and $\omega$ be a $\Gamma$-equivariant closed 2-form on $M$ such that the $\Gamma$-equivariant cohomology class of $\omega$ is equal to $\phi^*(c)$. Note that $\omega$ is *exact*, $\omega = d\eta$, since $\uE\Gamma$ is contractible. Define $\nabla=d+\,i\eta$. Then $\nabla$ is a Hermitian connection on the trivial line bundle $\cL$ over ${M}$, and the curvature of $\nabla$ is $\ (\nabla)^2=i\,{\omega}$. Then $\nabla$ defines a projective action of $\Gamma$ on $L^2$ spinors as follows:
For $u \in L^2({M}, {\mathcal{S}}\otimes {E}\otimes \cL )$, let $S_\gamma u = e^{i\varphi_\gamma}u$ (where $\varphi$ is the phase for $\sigma$ as explained in Lemma \[twistedIdentities\]), $\;U_\gamma u
={\gamma^{-1}}^*u$, and $T_\gamma=U_\gamma S_\gamma$ be the composition, for all $\gamma\in \Gamma$. Then $T$ defines a projective $(\Gamma,\sigma)$-action on $L^2({M}, {\mathcal{S}}\otimes {E} \otimes \cL)$, meaning that for any $\gamma,\gamma'\in\Gamma$ one has $$T_\gamma
T_{\gamma'}=\sigma(\gamma,\gamma')T_{\gamma\gamma'}.$$ Let $\np^{{\mathbb C}+}_{E\otimes\cL}: L^2(M, {\mathcal S}^+\otimes E\otimes \cL) \to
L^2(M,{\mathcal S}^-\otimes E\otimes \cL)$ denote the twisted $\Gamma$-equivariant Spin$^{\mathbb C}$ Dirac operator.
The twisted $\Gamma$-equivariant Spin$^{\mathbb C}$ Dirac operator on ${M}$, $$\np^{{\mathbb C}+}_{E\otimes\cL}: L^2(M, {\mathcal S}^+\otimes E\otimes \cL) \to
L^2(M,{\mathcal S}^-\otimes E\otimes \cL),$$ commutes with the projective $(\Gamma,\sigma)$-action.
To simplify notation, set $D_\eta = \np^{{\mathbb C}+}_{E\otimes\cL}$ and $D_0 = \np^{{\mathbb C}+}_{E}$ where we emphasize the dependence on $\eta$. Then $D_\eta = D_0 + ic(\eta)$, where $c(\eta)$ denotes Clifford multiplication by the one-form $\eta$. An easy computation establishes that $U_\gamma D_\eta = D_{{\gamma^{-1}}^*\eta} U_\gamma$ and that $S_\gamma D_{{\gamma^{-1}}^*\eta} = D_\eta S_\gamma \quad \text{for
all}\;
\gamma\in\Gamma$. Then $T_\gamma D_\eta =D_{\eta} T_\gamma$, where $T_\gamma = U_\gamma S_\gamma$ denotes the projective $(\Gamma,\sigma)$-action.
Heat kernels and the analytic index
-----------------------------------
Recall the following well-known smoothness properties and Gaussian off-diagonal estimates for the heat kernel, cf. [@Greiner; @Ko].
\[E\] The Schwartz kernels $k_\pm(t,x,y)$ of the heat operators $e^{-tD^\pm
D^\mp}$ are smooth for all $t>0$. Moreover, for any $t>0$ there are positive constants $C_1, C_2$ such that the following off-diagonal estimate holds $$|k_\pm (t,x,y)| \le C_1 e^{-C_2 d(x,y)^2},\quad x\in M,\quad y\in M,$$ where $d$ denotes the Riemannian distance function on $M$
For fixed $t>0$, we will use Lemma \[E\] to show the following.
\[off-diagonal\] Let $\cA(\Gamma, \sigma)$ be a good unconditional completion of $\C(\Gamma, \sigma)$. Then for fixed $t>0$, the heat operators $e^{-tD^-D^+}$ and $e^{-tD^+D^-} $ belong to $ \cA(\Gamma, \sigma)\otimes
{\mathcal K}_+$ and $\cA(\Gamma, \sigma)\otimes
{\mathcal K}_-$ respectively, where ${\mathcal K}_\pm$ denotes the algebra of compact operators on the Hilbert space $\cH_\pm= L^2(\F, \mathcal S^\pm \otimes E|_\F)$, and $\F$ denotes a connected fundamental domain of the action of $\Gamma$ on $M.$
We have $e^{-tD^\pm D^\mp} \in\U_{\cH_\mp}(\Gamma, \sigma)$, so that $$e^{-tD^\pm D^\mp}
=\sum_{\gamma\in\Gamma}T_\gamma\otimes h_{t}^\pm(\gamma),$$ where $h_{t}^\pm (\gamma)\in\B(\cH_\pm)$ has Schwartz kernel $k_\pm(t,
x, \gamma y)$ for $x, y \in \F$. By Lemma \[E\], we have $$||h_{t}^\pm (\gamma)|| \le ||k_\pm(t, x,
\gamma y)|_\F||_\infty
\le C_1 e^{-C_2 d(\gamma)^2},$$ where $d(\gamma) = \inf\{ d(x, \gamma y): x, y \in \F\}$.
It is well known that $$\label{e:d}
\ell(\gamma) \le C_4 (d(\gamma)+1),$$ for some positive constant $C_4$. From (\[e:d\]) and Lemma \[E\], we get $$\label{e:2}
\| h_{t}^\pm (\gamma) \|\le C_5 e^{-C_6
\ell(\gamma)^2},$$ for some positive constants $C_5, C_6$. We conclude using Lemma \[D\].
For fixed $t>0$, define the idempotent $$e_t(D^+)\in
M_2(\cA(\Gamma,\sigma)\otimes\tilde{\mathcal K})$$ as follows: $$e_t(D^+) = \begin{pmatrix}
e^{-tD^-D^+} & \displaystyle
e^{-{\frac{t}{2}}D^-D^+}\frac{(1-e^{-tD^-D^+})}
{D^-D^+} D^+ \\[+11pt]
e^{-{\frac{t}{2}}D^+D^-}{D^+} &
1- e^{-tD^+D^-}
\end{pmatrix},$$ where $\cA(\Gamma,\sigma)\otimes
\tilde{\mathcal K}$ denotes the unital algebra associated with $\cA(\Gamma,\sigma)\otimes
{\mathcal K}$. It is the analogue of the Wasserman idempotent, see e.g. Connes and Moscovici [@ConnesMosc]. Since $\cA(\Gamma,\sigma)$ is a Banach algebra, one has the invariance property of $K$-theory under stable isomorphism, $K_\bullet(\cA(\Gamma,\sigma))
\cong K_\bullet(\cA(\Gamma,\sigma)\otimes
{\mathcal K}).
$ Using this isomorphism, the $\cA$-[*twisted analytic index*]{} is defined as $$\label{bc}
a\!-\!\index_\sigma^\cA(D^+) =
[e_t(D)] - [E_0] \in K_0(\cA(\Gamma,\sigma)),$$ where $t>0$ and $E_0$ is the idempotent $$E_0 = \begin{pmatrix} 0 & 0 \\ 0 &
1
\end{pmatrix} \in M_2(\cA(\Gamma,\sigma)\otimes
\tilde{\mathcal K}).$$
Since the difference $e_t(D^+)- E_0$ is in $M_2(\cA(\Gamma,\sigma)\otimes
{\mathcal K}), $ we see that the right hand side of equation (\[bc\]) is in $K_0
(\cA(\Gamma,\sigma))$ as asserted.
Topological $K$-homology and the analytic twisted Baum-Connes map {#khomology}
-----------------------------------------------------------------
We shall now give a brief description of the Baum-Connes-Douglas [@BC], [@bd] version of the $K$-homology groups $K_j^\Gamma(\uE\Gamma)$ ($j=0,1$).
The basic objects are $\Gamma$-equivariant $K$-cycles. A *$\Gamma$-equivariant $K$-cycle* on $\uE\Gamma$ is a triple $(M,
E, \phi)$, where:
- $M$ is a manifold without boundary with a smooth proper cocompact $\Gamma$-action and a $\Gamma$-equivariant Spin$^{\mathbb C}$ structure.
- $E\to M$ is a $\Gamma$-equivariant complex vector bundle on $M$.
- $\phi : M \to \uE\Gamma$ is a $\Gamma$-equivariant continuous map.
Two $\Gamma$-equivariant $K$-cycles $(M, E, \phi)$ and $(M', E', \phi')$ are said to be [*isomorphic*]{} if there is a $\Gamma$-equivariant diffeomorphism $h: M \to M'$ preserving the $\Gamma$-equivariant Spin$^{\mathbb C}$ structures on $M, M'$ such that $h^*(E') \cong E$ and $h^*\phi' = \phi$. Let $\Pi^\Gamma(\uE\Gamma)$ denote the collection of all $\Gamma$-equivariant $K$-cycles on $\uE\Gamma$. The following operations on $\Gamma$-equivariant $K$-cycles will enable us to define an equivalence relation on $\Pi^\Gamma(\uE\Gamma)$.
[*Bordism*]{}: Two $\Gamma$-equivariant $K$-cycles $(M_i,
E_i,\phi_i) \in \Pi^\Gamma(\uE\Gamma)$ ($i=0,1,$) are said to be [ *bordant*]{} if there is a triple $(W, E, \phi)$, where $W$ is a manifold with boundary $\partial W$, with a smooth proper cocompact $\Gamma$-action and a $\Gamma$-equivariant Spin$^{\mathbb C}$ structure; $E\to W$ is a $\Gamma$-equivariant complex vector bundle on $W$ and $\phi:W\to X$ is a $\Gamma$-equivariant continuous map such that $(\partial W, E\big|_{\partial W}, \phi\big|_{\partial W})$ is isomorphic to the disjoint union $(M_0, E_0, \phi_0) \cup (-M_1, E_1,
\phi_1)$. Here $-M_1$ denotes $M_1$ with the reversed $\Gamma$-equivariant Spin$^{\mathbb C}$ structure.
[*Direct sum*]{}: Suppose that $(M,E, \phi) \in
\Pi^\Gamma(\uE\Gamma)$ and that $E=E_0\oplus E_1$. Then $(M, E,\phi)$ is isomorphic to $(M, E_0,
\phi)\cup (M, E_1,\phi)$.
[*Vector bundle modification*]{}: Let $(M, E, \phi) \in \Pi^\Gamma(\uE\Gamma)$ and $H$ be an even dimensional $\Gamma$-equivariant Spin$^{\mathbb C}$ vector bundle over M. Let $\widehat M = S(H\oplus 1)$ denote the sphere bundle of $H\oplus 1$. Then $\widehat M$ is canonically a $\Gamma$-equivariant Spin$^{\mathbb C}$ manifold. Let ${\mathcal S}$ denote the $\Gamma$-equivariant bundle of spinors on $H$. Since $H$ is even dimensional, ${\mathcal S}$ is ${\mathbb Z}_2$-graded, $${\mathcal S} = {\mathcal S}^+ \oplus {\mathcal S}^-,$$ into $\Gamma$-equivariant bundles of $1/2$-spinors on $M$. Define $\widehat E = \pi^*({\mathcal S}^{+*} \otimes E)$, where $\pi : \widehat M \to M$ is the projection. Finally, set $\widehat
\phi = \pi^*\phi$. Then $(\widehat M,\widehat E, \widehat\phi) \in \Pi^\Gamma(\uE\Gamma)$ is said to be obtained from $(M, E, \phi)$ and $H$ by [*vector bundle modification*]{}.
Let $\;\sim\;$ denote the equivalence relation on $\Pi^\Gamma(\uE\Gamma)$ generated by the operations of bordism, direct sum and vector bundle modification. Notice that $\;\sim\;$ preserves the parity of the dimension of the $K$-cycle. Let $$K_0^\Gamma(\uE\Gamma)=\Pi^\Gamma_{even}(\uE\Gamma)/\sim,$$ where $\Pi^\Gamma_{even}(\uE\Gamma)$ denotes the set of all even dimensional $\Gamma$-equivariant $K$-cycles in $\Pi^\Gamma(\uE\Gamma)$, and let $$K_1^\Gamma(\uE\Gamma)=\Pi^\Gamma_{odd}(\uE\Gamma)/\sim,$$ where $\Pi^\Gamma_{odd}(\uE\Gamma)$ denotes the set of all odd dimensional $\Gamma$-equivariant $K$-cycles in $\Pi^\Gamma(\uE\Gamma)$.
The analytic twisted Baum-Connes map is defined as $$\label{eq:index}
a\!-\!\mu_\sigma^{\cA} : K_0^\Gamma(\uE\Gamma) \to K_0(\cA(\Gamma,\sigma))$$ $$a\!-\!\mu_\sigma^{\cA}([M,E, \phi]) =a\!-\!\index_\sigma^{\cA} (D^+),$$ where $D^+$ is the twisted $\Gamma$-equivariant ${\operatorname{Spin}^{\bbC}}$ Dirac operator defined as in section \[Dirac\].
Twisted Baum-Connes conjecture in Lafforgue’s settings
======================================================
On Lafforgue’s Banach $KK$-theory
---------------------------------
We here recall Lafforgue’s definitions of Banach $KK$-theory in [@Laff], and its compatibility with Kasparov $KK$-theory. A Banach algebra $A$ is called a $\Gamma$-Banach algebra if $\Gamma$ acts on $A$ by isometric automorphisms. We shall briefly sketch how Lafforgue associates to a pair of $\Gamma$-Banach algebras $(A,B)$ an abelian group $$KK^{ban}_{\Gamma}(A,B).$$ An important concept in this setting is the notion of *$\Gamma-(A,B)$-Banach bimodule* $E$: to start with, $E$ is a *$B$-pair*, that is a pair of Banach spaces $E=(E^<,E^>)$ each of which is endowed with a $B$-action (left and right respectively), and with a $B$-valued and ${\bbC}$-linear bracket satisfying $$\left<bx,y\right>=b\left<x,y\right>,\
\left<x,yb\right>=\left<x,y\right>b,\
\|\left<x,y\right>\|_B\leq\|x\|\,\|y\|,$$ (where the norms of $x$ and $y$ are taken in $E^<$ and $E^>$ respectively). A $B$-pair $E$ is called an $(A,B)$-bimodule if it is endowed with a Banach algebra morphism from $A$ into ${\mathcal L}(E)$. If $A$ and $B$ are $\Gamma$-Banach algebras, then a $B$-pair $E$ endowed with an isometric $\Gamma$-action is called a $\Gamma$-$B$-pair, and a $\Gamma-(A,B)$-Banach bimodule if $E$ is both an $(A,B)$-bimodule and a $\Gamma$-$B$-pair such that the morphism $A\to{\mathcal L}(E)$ is $\Gamma$-equivariant. Denote by $E^{ban}(A,B)$ the isomorphism classes of pairs $\alpha=(E,T)$, where $E$ is a ${\bbZ}_2$-graded $\Gamma-(A,B)$-Banach bimodule and $T\in{\mathcal L}(E)$ an operator reversing the graduation and such that for any $a\in A$, $[a,T]$ and $a(Id_E-T^2)$ are compact operator on $E$. Two cycles $\alpha$ and $\beta$ in $E^{ban}(A,B)$ are said *homotopic* if they are the image of the evaluation in 0 and 1 respectively of a single element in $E^{ban}(A,B[0,1])$, where $B[0,1]$ denotes the Banach algebra of continuous maps from the interval $[0,1]$ into $B$. $KK^{ban}_{\Gamma}(A,B)$ is the quotient of $E^{ban}(A,B)$ by the equivalence relation induced by homotopy. This defines an abelian group, and Lafforgue’s Banach $KK^{ban}$ theory is compatible with Kasparov’s $KK$-theory in the sense that the forgetful morphism $\iota :E_{\Gamma}(A,B)\to E^{ban}_{\Gamma}(A,B)$ induces a well-defined morphism $\iota :KK_{\Gamma}(A,B)\to KK^{ban}_{\Gamma}(A,B)$ which is functorial in $A$ and $B$ in the case where those are $\Gamma$-$C^*$-algebras. It is well-known that $K_*^{\Gamma}(X)\simeq
KK_{\Gamma}(C_0(X),\bbC)$, where $X$ is any $\Gamma$-CW-complex.
Twisted Assembly map - the idempotent method {#idemptent-method}
--------------------------------------------
For any separable $\Gamma$-$C^*$-algebra $C$, there is a [*dilation*]{} homomorphism $$\tau_{C, \Gamma} : KK_\Gamma^{ban}(A,B) \to KK_\Gamma^{ban}(C\otimes A,C \otimes B),$$ where as in the entire paper, $\otimes$ denotes the projective tensor product, cf. [@Laff]. The following stability property is also proved in [@Laff]: $$\label{Kstability}
KK^{ban}(A,B) \cong KK^{ban}({\mathcal K}\otimes A,{\mathcal K}\otimes B),$$ where ${\mathcal K}$ denotes the Banach algebra of compact operators on a Hilbert space, such as $\ell^2(\Gamma)$.
\[descent\] For any two $\Gamma$-$C^*$-algebras $A$ and $B$ there is a twisted descent map $$\label{eq:descent}
j_{\Gamma,\cA,\sigma}:KK^{ban}_{\Gamma}(A,B)\to
KK^{ban}\left(\cA(\Gamma,A,\sigma),\cA(\Gamma,B,\sigma)\right),$$ which is compatible with the canonical homomorphism $j_{\Gamma,\sigma}$ of Proposition 2.1 in [@Ma].
The twisted descent map is defined as the composition of the following three homomorphisms. The first is the dilation homomorphism, $$\label{eq:descent1}
\tau_{{\mathcal K}, \Gamma} : KK_\Gamma^{ban}(A,B) \to KK_\Gamma^{ban}({\mathcal K}\otimes A,{\mathcal K}\otimes B),$$ where the action of $\Gamma$ on ${\mathcal K}$ is determined by $\sigma$ and is given as in the unconditional version of the Packer-Raeburn stabilization theorem, . The second is Lafforgue’s descent homomorphism [@Laff], $$\label{eq:descent2}
j_{\Gamma, \cA} : KK_\Gamma^{ban}({\mathcal K}\otimes A,{\mathcal K}\otimes B) \to
KK^{ban}(\cA(\Gamma, {\mathcal K}\otimes A), \cA(\Gamma, {\mathcal K}\otimes B)),$$ where $\Gamma$ acts diagonally on ${\mathcal K}\otimes A$ and on ${\mathcal K}\otimes B$. The third isomomorphism is obtained as a result of the unconditional version of the Packer-Raeburn stabilization theorem , together with stability of $KK^{ban}$ as above , $$\label{eq:descent3}
KK^{ban}(\cA(\Gamma, {\mathcal K}\otimes A), \cA(\Gamma, {\mathcal K}\otimes B)) \cong
KK^{ban}\left(\cA(\Gamma,A,\sigma),\cA(\Gamma,B,\sigma)\right).$$ The composition of the homomorphisms , , and yields the twisted descent map in equation .
To follow Lafforgue’s construction of the assembly map we shall now define a canonical element in $KK(\bbC,\cA(\Gamma,C_0(X),\sigma))\simeq
K_0(\cA(\Gamma,C_0(X),\sigma))$, where $X\subset E\Gamma$ is a free cocompact $\Gamma$-CW-complex.
\[thm:idempotent\] Take $h\in C_0(X)$ such that $\sum_{\gamma\in\Gamma}h(\gamma x)^2=1$ and let $\varphi$ be the phase associated to the cocycle $\sigma$. The element $$\label{eq:idempotent}
e(\gamma,x)=h(x)h(\gamma^{-1}x)e^{-i\varphi_{\gamma}(\gamma^{-1}x)}\in\cA(\Gamma,C_0(X),\sigma),$$ is an idempotent, which defines a class $[e] \in K_0(\cA(\Gamma,C_0(X),\sigma))$ that is independent of the choice of $h$.
That $e$ belongs to $\cA(\Gamma,C_0(X),\sigma)$ is clear since it is finitely supported. We now compute $$\begin{aligned}
(e*e)(\gamma,x)&=&\sum_{g\in\Gamma}h(x)
h(g^{-1}x)e^{-i\varphi_g(g^{-1}x)}h(g^{-1}x)
h(\gamma^{-1}x)e^{-i\varphi_{g^{-1}\gamma}(\gamma^{-1}x)}\sigma(g,g^{-1}\gamma)\\
&=&h(x)h(\gamma^{-1}x)\sum_{g\in\Gamma}h(g^{-1}x)^2e^{-i(\varphi_g(g^{-1}x)+\varphi_{g^{-1}\gamma}(\gamma^{-1}x))}\sigma(g,g^{-1}\gamma)\\
&=&h(x)h(\gamma^{-1}x)\sum_{g\in\Gamma}h(g^{-1}x)^2e^{i\varphi_{\gamma}(\gamma^{-1}x)}\\
&=&h(x)h(\gamma^{-1}x) e^{i\varphi_{\gamma}(\gamma^{-1}x)} = e(\gamma, x),\end{aligned}$$ where the last equality follows from the relations described under Lemma \[twistedIdentities\]. Since the set of all $h$ as in the lemma is convex, one sees that the class $[e] \in
K_0(\cA(\Gamma,C_0(X),\sigma))$ is independent of the choice of $h$.
We denote by $$p:KK^{ban}\left(\cA(\Gamma,C_0(X),\sigma),\cA(\Gamma,\sigma)\right)\to
K_0(\cA(\Gamma,\sigma)),$$ the map determined by the idempotent $e$, i.e. $p(\xi) = [e] \otimes_{\cA(\Gamma,C_0(X),\sigma)} \xi \in K_0(\cA(\Gamma,\sigma))$ for all $\xi \in KK^{ban}\left(\cA(\Gamma,C_0(X),\sigma),\cA(\Gamma,\sigma\right)$, cf. Lemma \[eq:idempotent\], as done by Lafforgue in [@Laff] page 42 in the untwisted case.
The [*twisted assembly map*]{}, $$\label{eq:iassembly}
t\!-\!\mu_{\sigma}^{\cA}:K_*^{\Gamma}(\uE\Gamma)\simeq KK_{\Gamma}(C_0(\uE\Gamma),\bbC)\to
K_*(\cA(\Gamma,\sigma)),$$ is then defined as the inductive limit over cocompact $\Gamma$-CW-complexes $X$ of the following maps: $$t\!-\!\mu_{\sigma}^{\cA, X}:K_*^{\Gamma}(X)\simeq KK_{\Gamma}(C_0(X),\bbC)\to
K_*(\cA(\Gamma,\sigma)),$$ where each map $t\!-\!\mu_{\sigma}^{\cA, X}$ is given as the composition $p\circ j_{\Gamma,\cA,\sigma}\circ\iota$, that is, [$$KK_{\Gamma}(C_0(X),\bbC)\stackrel{\iota}{\to} KK^{ban}_{\Gamma}(C_0(X),\bbC)
)\stackrel{ j_{\Gamma,\cA,\sigma}}{\to} KK^{ban}\left(\cA(\Gamma,C_0(X),\sigma),\cA(\Gamma,\sigma)\right)
\stackrel{p}{\to}
K_*(\cA(\Gamma,\sigma)).$$]{}
On the equivalence of the analytic twisted Baum-Connes and twisted assembly maps {#equivalence}
--------------------------------------------------------------------------------
We sketch the equivalence of the twisted assembly maps given by equations and .
As in section \[khomology\], let $(M, E, \phi)$ denote a $\Gamma$ equivariant $K$-cycle. Then the analytic twisted Baum-Connes map is defined as in in terms of the analytic index, $$a\!-\!\mu_\sigma^{\cA} : K_0^\Gamma(\uE\Gamma) \to K_0(\cA(\Gamma,\sigma))$$ $$a\!-\!\mu_\sigma^{\cA}([M,E, \phi]) = a\!-\!\index_\sigma^{\cA} (D^+),$$ where $D$ is the twisted ${\operatorname{Spin}^{\bbC}}$ Dirac operator defined as in section \[Dirac\].
On the other hand, section \[idemptent-method\] defines a twisted assembly map in terms of the class of an idempotent, $[e] \in K_0(\cA(\Gamma,C_0(X),\sigma))$, as $$t\!-\!\mu_\sigma^{\cA} : K_0^\Gamma(\uE\Gamma) \to K_0(\cA(\Gamma,\sigma))$$ $$t\!-\!\mu_\sigma^{\cA}([M,E, \phi]) = t\!-\!\index_\sigma^{\cA} (D^+),$$ where $ t\!-\!\index_\sigma^{\cA} (D^+) = [e] \otimes_{\cA(\Gamma,C_0(X),\sigma)}
j_{\Gamma,\cA,\sigma}([M, E, \phi]) \in K_0(\cA(\Gamma,\sigma))$.
A direct application of the scheme of section 4, [@Ka2], establishes the following index theorem, $$\label{a-index=t-index}
a\!-\!\index_\sigma^{\cA} (D^+) = t\!-\!\index_\sigma^{\cA} (D^+) \in K_0(\cA(\Gamma,\sigma)).$$
Therefore the analytic twisted Baum-Connes map $a\!-\!\mu_\sigma^{\cA}$ and the twisted assembly map $t\!-\!\mu_\sigma^{\cA}$ are equal, so we will henceforth denote either of these by $\mu_\sigma^{\cA}$.
Unconditional analog of the twisted Baum-Connes conjecture
----------------------------------------------------------
The following conjecture is natural in view of the above computations combined with Lafforgue’s work, and it amounts to a twisted Bost conjecture in case where we choose the unconditional completion to be $\ell^1$.
\[TwistedBost\] Let $\Gamma$ be a countable group and $\sigma$ a multiplier on $\Gamma$ with trivial Dixmier-Douady invariant. Then for any unconditional completion $\cA(\Gamma,\sigma)$ of $\C(\Gamma,\sigma),$ the twisted assembly map $$\mu_\sigma^{\cA} : K_j^\Gamma( \uE\Gamma) \rightarrow K_j
(\cA(\Gamma,
\sigma)),\qquad j=0,1,$$ is an isomorphism.
This conjecture is strongly related to a twisted version of the Baum-Connes conjecture (see [@BC]).
\[TwistedBC\] Let $\Gamma$ be a countable group and $\sigma$ a multiplier on $\Gamma$ with trivial Dixmier-Douady invariant. Then the twisted assembly map $$\mu_\sigma : K_j^\Gamma( \uE\Gamma) \rightarrow
K_j(C^*_r(\Gamma,\sigma)),\qquad j=0,1,$$ is an isomorphism.
To prove Conjecture \[TwistedBC\] in some cases, we first prove Conjecture \[TwistedBost\] and deduce Conjecture \[TwistedBC\] from Conjecture \[TwistedBost\] when the groups in addition have property RD, using Proposition \[KtheoryIso\]. To prove Conjecture \[TwistedBost\] in case where the group $\Gamma$ is in Lafforgue’s class ${\mathcal C}'$ we need to first recall some facts and definitions. Let $A$ be a proper $\Gamma$-C\*-algebra. Then a Dirac element $\alpha\in
KK^\Gamma_0(A,{\bbC})$ and a dual Dirac element $\beta\in KK^\Gamma_0({\bbC},A)$ satisfy the following conditions, $$\begin{array}{lcl}
\alpha \otimes_{\bbC} \beta & = & 1 \in KK^\Gamma_0(A, A)\\[+7pt]
\beta \otimes_{A} \alpha & = & \gamma \in KK^\Gamma_0(\bbC, \bbC),
\end{array}$$ where $\gamma$ is the idempotent as defined by Kasparov in [@Kasparov], Lafforgue in [@Laff] or Valette in [@Valette]. The Dirac element $\alpha$ gets its name as it is constructed using a ${\operatorname{Spin}^{\bbC}}$ Dirac operator.
We say that a group $\Gamma$ has the *Banach Dirac-Dual Dirac property* if the element $\gamma\in KK_\Gamma
(\bbC,\bbC)$ is trivial in $KK^{
ban}_\Gamma (\bbC,\bbC)$.
Recall that Lafforgue’s class ${\mathcal C}'$ defined in [@Laff] contains all countable discrete groups acting properly and by isometries either on a Hilbert space (those are said to have the Haagerup property, see [@les_welches], which include amenable groups, free groups and the property is closed under free and direct products), on a strongly bolic space (e.g. CAT(0) groups, hyperbolic groups due to Mineyev and Yu [@Mineyev_Yu]), or on some non-positively curved Riemannian manifolds (as linear groups).
Any group $\Gamma$ in the class ${\mathcal C}'$ has the Banach Dirac-Dual Dirac property.
\[main1\] Suppose that $\Gamma$ is a discrete group that has the [Banach Dirac-Dual Dirac property]{}, and that $\sigma$ is a multiplier on $\Gamma$ with trivial Dixmier-Douady invariant. Then Conjecture \[TwistedBost\] is true.
If in addition, $\Gamma$ has property RD, then Conjecture \[TwistedBC\] is true.
It follows from Lafforgue’s work that we can find a proper $\Gamma$-C\*-algebra $A$ and a Dirac element $\alpha\in
KK^\Gamma_i(A,{\bf C})$ and a dual Dirac element $\beta\in KK^\Gamma_i({\bf C},A)$ such that $$\gamma=\beta\otimes_A\alpha=1\ \ \ \hbox{in}\ KK_\Gamma^{ban}({\bf
C},{\bf C}).$$ Then consider the following commutative diagram: $$\xymatrix{
K^\Gamma_*(\underline{E}\Gamma)\ar[rr]^-{\otimes_{\bf
C}\beta}\ar[d]^{\mu_{\sigma}^{\cA}} & &
KK^\Gamma_*(\underline{E}\Gamma,A)\ar[d]^{\mu_*^{\Gamma,A}}_{\simeq}\ar[rr]^-{\otimes_A\alpha} & &
K^\Gamma_*(\underline{E}\Gamma)\ar[d]^{\mu_{\sigma}^{\cA}}\\
K_*(\cA(\Gamma,\sigma))\ar[rr]_-{\bigotimes j_\Gamma(\beta)} & &
K_*(\cA(\Gamma,A,\sigma))\ar[rr]_-{\bigotimes j_\Gamma(\alpha)} & &
K_*(\cA(\Gamma,\sigma)),
}$$ with the fact that, using Proposition \[descent\], composites on the top and the bottom lines are identity.
On the range of the trace, conjectures and applications
=======================================================
Characteristic classes
----------------------
We recall some basic facts about some well-known characteristic classes that will be used in this paper, cf. [@Hir].
Let $E\to M$ be a Hermitian vector bundle over the compact manifold $M$ that has dimension $n=2m$. The [*Chern classes*]{} of $E$, $c_j(E)$, are by definition [*integral* ]{} cohomology classes. The [*Chern character*]{} of $E$, ${{\operatorname{Ch}}}(E)$, is a rational cohomology class $${{\operatorname{Ch}}}(E)=\sum_{r=0}^{m} {{\operatorname{Ch}}}_r(E),$$ where ${{\operatorname{Ch}}}_r(E)$ denotes the component of ${{\operatorname{Ch}}}(E)$ of degree $2r$. Then ${{\operatorname{Ch}}}_0(E) = {\operatorname{rank}}(E)$, ${{\operatorname{Ch}}}_1(E) = c_1(E)$ and in general $${{\operatorname{Ch}}}_r(E) = \frac{1}{r!} P_r(E)
\in H^{2r}(M, \mathbb Q),$$ where $P_r(E) \in H^{2r}(M, \mathbb Z)$ is a polynomial in the Chern classes of degree less than or equal to $r$ with [*integral*]{} coefficients, that is determined inductively by the Newton formula $$P_r(E) - c_1(E) P_{r-1}(E)+\ldots +
(-1)^{r-1}c_{r-1}(E) P_1(E)
+(-1)^{r} r c_{r}(E) =0,$$ and by $P_0(E) = {\operatorname{rank}}(E)$. The next two terms are $P_1(E) = c_1(E)$, $P_2(E) = c_1(E)^2 - 2 c_2(E)$.
The Todd-genus characteristic class of the Hermitian vector bundle $E$ is a rational cohomology class in $H^{2\bullet}(M,\mathbb Q)$, $${{\operatorname{Todd}}}(E) = \sum_{r=0}^{m}{{\operatorname{Todd}}}_r(E),$$ where ${{\operatorname{Todd}}}_r(E) $ denotes the component of ${{\operatorname{Todd}}}(E)$ of degree $2r$. Then ${{\operatorname{Todd}}}_r(E) = B_r Q_r(E)$, where $Q_r(E)$ is a polynomial in the Chern classes of degree less than or equal to $r$, with [*integral*]{} coefficients, and $B_r \ne 0, B_r \in \mathbb Q$ are the Bernoulli numbers. In particular, ${{\operatorname{Todd}}}_0(E) = B_0 Q_0 = 1$.
[*For the rest of this section, we use the notation of Section \[Dirac\]*]{}.
An $L^2$ index theorem {#l2index}
----------------------
Let $\;\tau\;$ be the canonical trace on $\cA(\Gamma,\sigma)\;$ defined by evaluation at the identity element of $\Gamma$. It induces a linear map $$[\tau] : K_0 (\cA(\Gamma,
\sigma)) \to \mathbb R,$$ which is called the [*trace map*]{} in $K$-theory. Explicitly, first $\;\tau\;$ extends to matrices with entries in $\cA(\Gamma, \sigma)\;$ as (with Trace denoting matrix trace): $$\tau(f\otimes r) = {\mbox{Trace}}(r)
\tau(f).$$
Then the extension of $\;\tau\;$ to $K_0$ is given by $\;[\tau]([e]-[f]) = \tau(e) - \tau(f),\;$ where $e$ and $f$ are idempotent matrices with entries in $\cA(\Gamma,\sigma)$.
We will compute $[\tau]\circ \mu_\sigma^\cA(K_0^\Gamma(\uE\Gamma))$ as follows. $$\begin{array}{rcl}
[\tau]\circ \mu_\sigma^\cA ([M,E, \phi]) & = & [\tau]( [ e_t(D) ] -
[E_0])\\[+7pt]
& = & \tau(e^{-tD^-D^+} ) - \tau(e^{-tD^+D^-} ) \\[+7pt]
& = & c_0 \displaystyle \int_{M/\Gamma} {{\operatorname{Todd}}}(M)\wedge e^{\omega}
\wedge{{\operatorname{Ch}}}(E),
\end{array}$$ where $D^+ = \np^{{\mathbb C}+}_{E\otimes\cL}$ denotes the twisted $\Gamma$-equivariant Dirac operator and $D^-$ denotes its adjoint. Here the local index theorem is used to deduce the last line, cf. the Appendix in [@Ma2]. Here $c_0= {1}/{(2\pi)^{n/2}}$ is the universal constant determined by the Atyiah-Singer index theorem, see [@AtSi] , $n = \dim M$, ${{\operatorname{Todd}}}$ and ${{\operatorname{Ch}}}$ denote the Todd-genus and the Chern character respectively, $\omega$ is the curvature of the connection on the trivial line bundle $\cL$ that is described in Section \[Dirac\]. This theorem is also a consequence of section \[equivalence\].
Range of the canonical trace {#sect:range}
----------------------------
Here we will present some consequences of the twisted Bost conjecture above and the twisted $L^2$ index theorem described in subsection \[l2index\] above.
The following result is an easy modification of a result in [@Ma].
\[range\] Suppose that $(\Gamma, \sigma)$ satisfies Conjecture \[TwistedBost\] . Then the range of the canonical trace on $K_0(\cA(\Gamma, \sigma))$ is given by $$\left\{ c_0 \int_{M/\Gamma}{{\operatorname{Todd}}}(M)\wedge e^{\omega}
\wedge{{\operatorname{Ch}}}(E) ;\; \mbox{\rm for all}\;(M, E,
\phi)
\in \Pi_{\rm even}^\Gamma(\uE\Gamma)\right\}.$$
The set $$\left\{ c_0 \int_{M/\Gamma} {{\operatorname{Todd}}}(M)\wedge e^{\omega}
\wedge{{\operatorname{Ch}}}(E) ;
\;\mbox{\rm for all}\; (M, E, \phi)
\in \Pi_{\rm even}^\Gamma(\uE\Gamma)\right\},$$ is a countable discrete subgroup of $\R$, but it is not in general a subgroup of $\Z$.
When $\Gamma$ is the fundamental group of a compact Riemann surface of positive genus, it follows from [@Rieff] in the genus one case, and [@CHMM] in the general case, that the set $$\left\{ c_0 \int_{M/\Gamma} {{\operatorname{Todd}}}(M)\wedge
e^{\omega}
\wedge{{\operatorname{Ch}}}(E) ; \;\mbox{\rm for all}\; (M, E, \phi)
\in
\Pi_{\rm even}^\Gamma(\uE\Gamma)\right\},$$ reduces to the countable discrete group $\Z + \theta \Z$, where $\theta \in [0,1)$ corresponds to the multiplier $\sigma$ under the isomorphism $H^2(\Gamma; {\rm\bf U}(1))\cong \R/\Z$.
By hypothesis, the twisted assemby map $\mu_\sigma^\cA$ is an isomorphism. Therefore to compute the range of the trace map on $K_0(\cA(\Gamma, \sigma))$, it suffices to compute the range of the trace map on elements of the form $$\mu_\sigma^\cA([M, E, \phi]), \qquad
[M, E, \phi] \in K_0^\Gamma(\uE\Gamma).$$ Here $(M, E, \phi)\in \Pi_{\rm even}^\Gamma(\uE\Gamma)$. By the $L^2$ index theorem described in section \[l2index\] above, one has $$[\tau](\mu_\sigma^\cA([M, E, \phi])) = c_0 \int_{M/\Gamma} {{\operatorname{Todd}}}(M)\wedge e^{\omega}\wedge{{\operatorname{Ch}}}(E),$$ as desired.
Therefore we deduce the following.
Suppose that $(\Gamma, \sigma)$ satisfies Conjecture \[TwistedBC\], then the range of the trace map on $K_0(C^*_r(\Gamma, \sigma))$ is $$\left\{ c_0 \int_{M/\Gamma}{{\operatorname{Todd}}}(M)\wedge e^{\omega}
\wedge{{\operatorname{Ch}}}(E) ;\; \text{for all}\;(M, E,
\phi)
\in \Pi_{even}^\Gamma(\uE\Gamma)\right\}.$$
The 3 and 4 dimensional cases {#3&4}
-----------------------------
We explicitly determine the range of the trace in the special case when $\Ga$ is torsion-free and $B\Gamma$ is either a three or a four dimensional smooth compact manifold. In the three dimensional case we get the following.
\[3D\] Let $\Ga$ be a torsion-free group such that $(\Gamma, \sigma)$ satisfies Conjecture \[TwistedBost\], and such that $B\Gamma$ is a smooth, compact oriented three dimensional manifold. Then the range of the trace map is $$\label{eq:3D}
[{\operatorname{tr}}] (K_0 (\cA(\Gamma, \sigma)) ) = \Z + \sum_{i=1}^{b_1} \Z \theta_i,$$ where for $i=1,\dots, b_1$, $\theta_i= c_0 \left<\eta_i\cup\omega,[B\Ga]\right>$ and the $\eta_i$’s are generators for $H^1(B\Ga,\Z) \cap H^1(B\Gamma, \bbR) $, $b_1 = \dim H^1(B\Gamma, \bbR) $ and $\sigma=e^{\omega}$.
Since any smooth, compact oriented three dimensional manifold is a spin manifold, it satisfies Poincar' e duality, $$K_0(B\Gamma) \cong K^1(B\Gamma).$$ The range of the trace $[{\operatorname{tr}}] (K_0 (\cA(\Gamma, \sigma)) ) $ simplifies to $$\label{eq:3D1}
\left\{ c_0 \int_{B\Gamma} {{\operatorname{Todd}}}(B\Gamma)\wedge e^{\omega} \wedge{{\operatorname{Ch}}}^{odd}(E) ;\; \text{for all}\; E
\in K^1(B\Gamma) \right\}.$$ For dimension reasons, ${{\operatorname{Todd}}}(B\Gamma) = 1$, $e^{\omega} = 1 + \omega$ and ${{\operatorname{Ch}}}^{odd}(E) = c^{odd}_1(E) + {{\operatorname{Ch}}}^{odd}_3(E)$. Therefore equation reduces to $$\left\{ c_0 \int_{B\Gamma} c^{odd}_1(E) \wedge \omega + c_0 \int_{B\Gamma} {{\operatorname{Ch}}}^{odd}_3 (E) ;\; \text{for all}\; E
\in K^1(B\Gamma) \right\}.$$ By the Atiyah-Singer index theorem [@AtSi], one knows that $$c_0 \int_{B\Gamma} {{\operatorname{Ch}}}^{odd}_3 (E) \in \mathbb Z \qquad \text{for all}\; E
\in K^1(B\Gamma).$$ The proof is concluded from the fact that $c^{odd}_1(E) = c^{odd}_1(\det E) \in H^1(B\Ga,\Z) \cap H^1(B\Gamma, \bbR)$.
We now turn to the four dimensional case. Let $Q(a,b) = \langle a\cup b, [B\Gamma]\rangle$, for $a,b \in
H^2(\Gamma, \R)$, be the intersection form on $B\Gamma$. Define the linear functional $T_\omega : H^2(\Gamma, \Z) \to \R$ as $T_\omega
(a) = Q(\omega,a)$. Then the following is a consequence of Theorem \[range\] and the proof of Theorem 2.5 in [@MM].
Let $\Ga$ be a torsion-free group such that $(\Gamma, \sigma)$ satisfies Conjecture \[TwistedBost\], and such that $B\Gamma$ is a smooth, compact oriented four dimensional manifold. Then the range of the trace map is $$\label{4D}
[{\operatorname{tr}}] (K_0 (\cA(\Gamma, \sigma)) ) = \Z\theta + \Z + B,$$ where $2(2\pi)^2\theta = \langle[\omega\cup \omega], [B\Gamma]\rangle$, and $B= {\rm range}(T_\omega)$.
Here $\omega$ is as in subsection \[l2index\]. If $a_1, \dots, a_{r}$ are generators of $H^2(B\Gamma, \Z) \cap H^2(B\Gamma, \bbR) $, where $r = \dim H^2(B\Gamma, \bbR) $, then we can express equation (\[4D\]) as, $$[{\operatorname{tr}}] (K_0 (\cA(\Gamma, \sigma)) ) = \Z\theta + \Z + \sum_{j=1}^{r}
\Z\theta_j,$$ where $\theta_j = \langle \omega\cup a_j, [B\Gamma]\rangle$ for $j=1, \dots, r$.
The trace conjecture for unconditional twisted group completions
----------------------------------------------------------------
The calculations done earlier in the section validate the following bold conjecture.
\[conj:trace\] Let $\Ga$ be a torsion-free group such that $(\Gamma, \sigma)$ satisfies Conjecture \[TwistedBost\], and such that the classifying space $B\Gamma$ is a smooth, compact, oriented manifold.\
[(1) (Even dimensional case)]{} Suppose that $B\Gamma$ is of dimension $2n$. If $a_1(j), \dots, a_{b_{2j}}(j)$ are generators of $H^{2j}(B\Gamma, \Z) \cap H^{2j}(B\Gamma, \bbR) $, where $b_{2j} = \dim H^{2j}(B\Gamma, \bbR) $, then the range of the trace map is $$\label{evenD}
[{\operatorname{tr}}] (K_0 (\cA(\Gamma, \sigma)) ) = \Z + \Z\theta + \sum_{j=1}^{n-1}\sum_{k=1}^{b_{2j}} \Z r_{k, j, n} \theta_k(j),$$ where $\theta_k(j) = \langle [ \omega^{n-j}\cup a_k(j)], [B\Gamma]\rangle$ for $k=1, \dots, b_{2j}$, $2(2\pi)^n\theta = \langle[\omega^n], [B\Gamma]\rangle$ and $r_{k, j, n} $ are universal constants.\
[(2) (Odd dimensional case)]{} Suppose that $B\Gamma$ is of dimension $2n-1$. If $a_1(j), \dots, a_{b_{2j-1}}(j)$ are generators of $H^{2j-1}(B\Gamma, \Z) \cap H^{2j-1}(B\Gamma, \bbR) $, where $b_{2j-1} = \dim H^{2j-1}(B\Gamma, \bbR) $, then the range of the trace map is $$\label{oddD}
[{\operatorname{tr}}] (K_0 (\cA(\Gamma, \sigma)) ) = \Z +
\sum_{j=1}^{n-1}
\sum_{k=1}^{b_{2j-1}} \Z r_{k, j, n}' \theta_k(j),$$ where $\theta_k(j) = \langle [ \omega^{n-j}\cup a_k(j)], [B\Gamma]\rangle$ for $k=1, \dots, b_{2j-1}$ and $r_{k, j, n} '$ are universal constants.
In section \[3&4\], we have a more explicit form of Conjecture \[conj:trace\], whenever the dimension of $B\Gamma$ is less than or equal to 4.
Generators for $K_1$ of twisted group algebra completions of surface groups
===========================================================================
In this section we shall focus on the degree one part of conjecture \[TwistedBost\], and more precisely we shall identify the generators for $K_1$ of a fundamental group of a Riemann surface. We assume that all the groups are torsion-free. We shall more generally obtain partial results in low-dimensional homology ($B\Gamma$ of dimension less than or equal to 2). In this case, there exist (as shown by Natsume in [@Natsume] in the untwisted case) natural homomorphisms $$\begin{aligned}
\beta_t:{H_1(\Gamma,{\Z})}=\Gamma^{\rm ab} & \to & {K_1(B\Gamma)}\\
\beta_a^\sigma: {H_1(\Gamma,{\Z})}=\Gamma^{\rm ab} & \to &
{K_1(\cA(\Gamma, \sigma))}\end{aligned}$$ such that $\beta_a^\sigma =\mu_1^\sigma\circ\beta_t$. Here $\cA(\Gamma, \sigma)$ is any good unconditional completion of the twisted group algebra $\bbC(\Gamma, \sigma)$. Defining $\beta_t:\Gamma^{\rm ab}\to
K_1(B\Gamma)$ has been done by Valette in [@Valette], and we recall here the construction. Since $\pi_1(B\Gamma)=\Gamma$, an element $\gamma\in\Gamma$ can be viewed as a pointed continuous map $\gamma:S^1\to B\Gamma$, inducing a map in $K$-homology, $$\gamma_*:K_1(S^1)\to K_1(B\Gamma).$$ The generator of $K_1(S^1)\simeq{\bbZ}$ can be described by the class of the cycle $(\pi,D)$ where $\pi$ is the representation of $C(S^1)$ on $L^2(S^1)$ by pointwise multiplication and $$D=-i\frac{t}{dt}.$$ An element $\gamma\in\Gamma$ gets then mapped to the class of the cycle $\gamma_*(\pi,D)=(\gamma_*\pi,D)$, where for $X$ a compact subset of $B\Gamma$ containing $\gamma(S^1)$ and $f\in C(X)$, $\gamma_*\pi(f)=\pi(f\circ\gamma)$ is the pointwise multiplication by $f\circ\gamma$ on $L^2(S^1)$. In other terms one defines $$\begin{aligned}
\tilde{\beta}_t:\Gamma & \to & K_1(B\Gamma)\\
\gamma & \mapsto & [(\gamma_*\pi,D)].\end{aligned}$$ According to Valette in [@Valette], the map $\tilde{\beta}_t:\Gamma\to K_1(B\Gamma)$ is a group homomorphism and hence factors through $$\beta_t:\Gamma^{\rm ab}\to K_1(B\Gamma).$$ To define $\beta_a^\sigma:\Gamma^{\rm ab}\to{K_1(\cA(\Gamma,
\sigma))}$, simply map a representative $[\gamma]$ of $\Gamma^{\rm ab}$ to the class $[T_{[\gamma]}]$ of the invertible operator $T_{[\gamma]}$ in $\cA(\Gamma, \sigma)$. The following is then an adaptation of a result due to Natsume [@Natsume], and the proof we give here is an easy twist of the one given in [@Valette], taken from [@Bettaieb-Matthey]. It explains how the map $\beta_t$ is related to the maps $\mu_1^{\sigma}$ and $\beta_a^{\sigma}$.
\[K1twist\] For $\sigma$ a multiplier on $\Gamma$ with $\delta(\sigma)=0$, we have that $\beta_a^{\sigma}=\mu_{\sigma}^{\cA}\circ\beta_t$.
It is enough to see that $$\tilde{\beta}_a^{\sigma}=\mu_{\sigma}^{\cA}\circ\tilde{\beta}_t:\Gamma
\to
K_1(\cA(\Gamma,\sigma)).$$ For $\gamma\in\Gamma$, denote by $\gamma$ the (unique) homomorphism ${\bbZ}\to\Gamma$ such that $\gamma(1)=\gamma$. Consider then the diagram $$\xymatrix{
{\bbZ}\ar[rrr]_{\gamma}\ar[rd]^{\beta_a^{\sigma}}\ar[dd]_{\beta_t} &
& &
{\Gamma}\ar[dl]^{\tilde{\beta}_a^{\sigma}}\ar[dd]^{\tilde{\beta}_t}\\
& {K_1(\cA({\bbZ},\sigma))}\ar[r]_{\gamma_*} &
{K_1(\cA(\Gamma,\sigma))} & \\
{K_1(S^1)}\ar[ur]_{\mu_{\sigma}^{\cA}}
\ar[rrr]_{\gamma_*}
& & & {K_1(B\Gamma)}\ar[ul]_{\mu_{\sigma}^{\cA}}
}$$ where by abuse of notation $\sigma$ denotes also the multiplier $\sigma$ restricted to ${\bbZ}$. That $\tilde{\beta}_a^{\sigma}\circ\gamma=\gamma_*\circ\beta_a^{\sigma}$ is a simple computation, $\tilde{\beta}_t\circ\gamma=\gamma_*\circ\beta_t$ by definition of $\tilde{\beta}_t$, and $\gamma_*\circ\mu_1^{\sigma}=\mu_1^{\sigma}\circ\gamma_*$ by naturality of the twisted assembly map (see [@Ma]). That $\beta_a=\mu_1^{\bbZ}\circ\beta_t$ follows from the proof of the isomorphism \[TwistedBC\] for ${\bbZ}$ (see [@Ma]) and we conclude the proof by a diagram chase.
\[thm:rs\] Let $\Gamma_g$ be the fundamental group of a compact Riemann surface of genus $g\ge 1$. Then the map $\beta_a^\sigma$ is an isomorphism.
It is well known that $\beta_t$ is an isomorphism in this case. Also $\Gamma_g$ is in class ${\mathcal C}'$ and has property RD, so that $\mu_\sigma^\cA$ is an isomorphism. The result now follows from Proposition \[K1twist\].
This corollary shows that we have obtained in particular the explicit generators for $K_1(C^*_r(\Gamma_g,\sigma))$, since $\Gamma_g$ has property RD. More explicitly, consider the standard presentation of $\Gamma_g$ in terms of generators and relations, namely, $$\Gamma_g = \left\{ a_j, b_j : \prod_{j=1}^g [a_j, b_j] = 1\right\}.$$ Then by Corollary \[thm:rs\], the unitary operators $\{T_{a_j}, T_{b_j} \in U(\ell^2(\Gamma)):
j=1,\ldots g\}$ form a natural set of generators for $K_1(C^*_r(\Gamma,\sigma))$ over $\mathbb Z$. The corollary at the same time gives explicit generators for $K_1(\cA(\Gamma_g,\sigma))$ for any good unconditional completion.
$K$-homology
------------
Thanks to [@CHMM], $K_1(C^*_r(\Gamma,\sigma)) \cong \mathbb Z^{2g}$ and $K_0(C^*_r(\Gamma,\sigma)) \cong \mathbb Z^{2}$ whenever $\Gamma =
\Gamma_g$ as above. Moreover, we have determined a natural set of generators for $K_1(C^*_r(\Gamma_g,\sigma))$. It was also shown in [@CHMM] that $C^*_r(\Gamma,\sigma)$ is a $K$-amenable $C^*$-algebra. These two facts together with the universal coefficient theorem [@RS] enable us to also compute the $K$-homology groups as $$K^1(C^*_r(\Gamma,\sigma)) \cong \mathbb Z^{2g},$$ and also $$K^0(C^*_r(\Gamma,\sigma)) \cong \mathbb Z^{2},$$ where the $K$-homology groups $K^i(C^*_r(\Gamma,\sigma)) $ are defined as usual as the Kasparov groups $KK^i(C^*_r(\Gamma,\sigma), \mathbb C)$ for $i=0,1$.
Appendix: Twisted Rapid Decay {#RD .unnumbered}
=============================
by Indira Chatterji,
Department of Mathematics, Cornell University, Ithaca NY 14853, USA.\
email: [indira@math.cornell.edu]{} Throughout this appendix, $\Ga$ is a finitely generated group, endowed with a length function $\ell$, and $\sigma$ is a multiplier on $\Ga$. We adopt the notations used in the first paragraph of the paper.
\[sRD\] We will say that the group $\Gamma$ has *$\sigma$-twisted Rapid Decay property (with respect to the length $\ell$)* if $$H^{\infty}_{{\ell}}(\Gamma,\sigma)\subseteq C^*_r(\Gamma,\sigma).$$ We just say that the group $\Gamma$ has the *Rapid Decay property (with respect to the length $\ell$)*, if it has the $\sigma$-twisted Rapid Decay property (with respect to the length $\ell$) for the constant multiplier 1. For short, we shall say that a group $\Gamma$ has *property $\sigma$-RD* if there esists a length function $\ell$ with respect to which $\Gamma$ has the $\sigma$-twisted Rapid Decay property.
In the context of noncommutative geometry, the reduced $C^*$-algebra $C^*_r(\Gamma,\sigma)$ represents the space of [*continuous*]{} functions on a noncommutative manifold, and $H^{\infty}_{{\ell}}(\Gamma,\sigma)$ the space of of [*smooth*]{} functions on the same noncommutative manifold. This comes from the abelian case, where using Fourier transforms, one easily sees that $C^*_r(\bbZ^n) \cong C(\mathbb T^n)$ and that $H^{\infty}_{\ell}(\bbZ^n) \cong C^\infty(\mathbb T^n)$ (for the word length associated to the generating set $S=\{(\pm 1,0,\dots),\dots,(0,\dots,\pm 1)\}$ of $\bbZ^n$). The ($\sigma$-twisted) Rapid Decay property can be rephrased as the desirable property that every smooth function on the noncommutative manifold is also a continuous function.
\[prop:RDequiv\] Let $\sigma$ be a multiplier on $\Gamma$ and $\ell$ be a length function on $\Gamma$. The following are equivalent:
- $\Gamma$ has $\sigma$-twisted Rapid Decay (with respect to the length $\ell$).
- There exist constants $C,s>0$ such that for any $f\in\bbC(\Gamma,\sigma)$ $$\|f\|_{op}\leq C\|f\|_s.$$
- There exists a polynomial $P$ such that for any $f\in\bbC(\Gamma,\sigma)$ and $f$ supported in a ball of radius $r$ $$\|f\|_{op}\leq P(r)\|f\|_{\ell^2\Gamma}.$$
- There exists a polynomial $P$ such that for any $f,g\in\bbC(\Gamma,\sigma)$ and $f$ supported in a ball of radius $r$ $$\|f*_{\sigma}g\|_{\ell^2\Gamma}\leq P(r)\|f\|_{\ell^2\Gamma}\|g\|_{\ell^2\Gamma}.$$
$(1)\Leftrightarrow (2)$ As in the case of untwisted Rapid Decay, the inclusion $H^{\infty}_{{\ell}}(\Gamma,\sigma)\subseteq C^*_r(\Gamma,\sigma)$ is continuous since both inclusions $H^{\infty}_{{\ell}}(\Gamma,\sigma)\subseteq\ell^2\Gamma$ and $C^*_r(\Gamma,\sigma)\subseteq\ell^2\Gamma$ are continuous. Since $H^{\infty}_{{\ell}}(\Gamma,\sigma)$ is a Fréchet space, the continuity of the inclusion $H^{\infty}_{{\ell}}(\Gamma,\sigma)\subseteq C^*_r(\Gamma,\sigma)$ rephrases as the statement of (2). The converse is obvious since $H^{s+1}_{{\ell}}(\Ga)\subseteq H^s_{{\ell}}(\Ga)$.
$(2)\Rightarrow(3)\Rightarrow(4)$ Take $f\in\bbC(\Gamma,\sigma)$ supported in a ball of radius $r$, then $$\|f\|_{op}\leq
C\|f\|_s=C\sqrt{\sum_{\gamma\in\Gamma}|f(\gamma)|^2(1+\ell(\gamma))^{2s}
}\leq C(1+r)^s\|f\|_{\ell^2\Gamma}.$$ Hence (3) follows. Since $\|f\|_{op}=\sup\{\frac{\|f*_{\sigma}g\|_{\ell^2\Gamma}}{\|g\|_{\ell^2\Gamma}}|{0\not=g\in\ell^2\Gamma}\}$ we deduce (4) as well. That (4) implies (3) is by definition of the operator norm.
$(3)\Rightarrow(2)$ For $n\in\bbN$, denote by $S_n=\{\gamma\in\Gamma|n\leq\ell(\gamma)<n+1\}$ the sphere of radius $n$. For $f\in\bbC(\Gamma,\sigma)$ we have: $$\|f\|_{op}=\|\sum_{n=0}^{\infty}\lambda_{\sigma}(f|_{S_n})\|_{op}\leq\sum_{n=0}^{\infty}\|f|_{S_n}\|_{op},$$ so that using (3) we get the following bound $$\begin{aligned}
\|f\|_{op}&\leq
&\sum_{n=0}^{\infty}P(n+1)\|f|_{S_n}\|_{\ell^2\Gamma}\leq\sum_{n=0}^{\infty}C(n+1)^k\
|f|_{S_n}\|_{\ell^2\Gamma}\\
&\leq &
C\sqrt{\sum_{n=0}^{\infty}(n+1)^{-
2}}\sqrt{\sum_{n=0}^{\infty}(n+1)^{2k+2}\|f|_{S_n}\|_{\ell^2\Gamma}^2}\leq
C'\|f\|_{k+1}\end{aligned}$$ where $C'$ is some constant bigger than $C\pi/6$.
The following proposition was known by Ji and Schweitzer [@tuile], but the proof we give here might be shorter.
\[1=>sigma\]Let $\ell$ be a length function on $\Gamma$. If $\Gamma$ has Rapid Decay (with respect to the length $\ell$), then $\Gamma$ has $\sigma$-twisted Rapid Decay (with respect to the length $\ell$) for any multiplier $\sigma$.
Take $\gamma\in\Gamma$, then: $$|f*_{\sigma}g(\gamma)|=|\sum_{\mu\in\Gamma}f(\gamma^{-
1}\mu)g(\mu)\sigma(\gamma^{-
1}\mu,\mu)|\leq\sum_{\mu\in\Gamma}|f(\gamma^{-
1}\mu)|\,|g(\mu)|=|f|*|g|(\gamma)$$ so that summing and squaring over $\gamma\in\Gamma$ yields $$\|f*_{\sigma}g\|_{\ell^2\Gamma}\leq\||f|*|g|\|_{\ell^2\Gamma}\leq P(r)\|f\|_{\ell^2\Gamma}\|g\|_{\ell^2\Gamma}$$ and we conclude that $\Gamma$ has $\sigma$-twisted Rapid Decay using the previous proposition.
The following corollary is the first part of Proposition 2.1 in [@LaffRD] with an obvious modification.
\[remarque\] Let $\ell$ be a length function on $\Gamma$. If $\Gamma$ has Rapid Decay (with respect to the length $\ell$), then there is a constant $S$ sufficiently large such that for any multiplier $\sigma$ on $\Gamma$ and any $s\geq S$, $H^s_{\ell}(\Gamma,\sigma)$ is a Banach algebra such that $H^s_{\ell}(\Gamma,\sigma) \subseteq C^*_r(\Gamma, \sigma)$.
Let $s$ be bigger than the degree of the polynomial of point (3) in Proposition \[prop:RDequiv\]. We first have to show that there is a constant $K=K(s)$ such that for any $f,g\in\bbC(\Gamma,\sigma)$, $\|f*_{\sigma}g\|_s\leq
K\|f\|_s\|g\|_s$. But this is true since $\|f*_{\sigma}g\|_s\leq\||f|*|g|\|_s$ and $\||f|*|g|\|_s\leq K'\|f\|_s\|g\|_s$ by Proposition 2.1 part (a) in [@LaffRD] (see also Proposition 8.15 in [@Valette]) since we assumed that $\Gamma$ has Rapid Decay (with respect to the length $\ell$). Therefore $H^s_{\ell}(\Gamma,\sigma)$ is a Banach algebra. By Lemma \[1=>sigma\], we know that since $\Gamma$ has property RD, $\Gamma$ has property $\sigma$-twisted RD for any multiplier $\sigma$ on $\Gamma$, and hence $H^s_{\ell}(\Gamma,\sigma) \subseteq C^*_r(\Gamma, \sigma)$ follows from Proposition \[prop:RDequiv\] part (2).
In the context of noncommutative geometry, Corollary \[remarque\] can be viewed as the analog of the [Sobolev Embedding Theorem]{} for a compact manifold $M$, a simplified version of which saying that any function in the Sobolev space $W^{s,2}(M)$ for $s> \dim M/2$ is actually continuous. Indeed, using Fourier transforms, on can see that $W^{s,2}({\mathbb T}^n)\simeq H^{s}_{{\ell}}(\bbZ^n)$ for the word length associated to the generating set $S=\{(\pm 1,0,\dots),\dots,(0,\dots,\pm 1)\}$ of $\bbZ^n$, and that $C^*_r(\mathbb Z)\simeq C(\mathbb T^n)$.
Groups having Rapid Decay notably include: Polynomial growth groups (Jolissaint [@Jolissaint]), free groups (Haagerup [@Haagerup]) and more generally Gromov hyperbolic groups (Jolissaint-de la Harpe [@Harpe]), cocompact lattices in $SL_3(F)$ where $F$ is the $p$-adic field $\bbQ_p$, $\bbR,\bbC,\bbH$ or $E_{6(-26)}$, as well as finite products of rank one Lie groups (see Rammagge-Robertson-Steger [@RRS], Lafforgue [@LaffRD] and [@indira]) and all lattices in a rank one Lie group, see [@withKim].
[**Question**]{}: Is it possible to find a group $\Gamma$ which doesn’t have Rapid Decay, but which has $\sigma$-twisted Rapid Decay for some multiplier $\sigma$ on $\Gamma$ (or does the converse of Lemma \[1=>sigma\] hold)?
The following is the second part of Proposition 1.2 of [@LaffRD] with a trivial change. But we still recall Lafforgue’s proof below for the sake of completeness.
\[KtheoryIso\] Let $\ell$ be a length function on $\Gamma$. If $\Gamma$ has Rapid Decay (with respect to the length $\ell$), then for any multiplier $\sigma$ on $\Gamma$ and for $s$ sufficiently large (and also for $s=\infty$), the inclusion $H^{s}_\ell (\Gamma,\sigma)\hookrightarrow C^*_r(\Gamma,\sigma)$ induces an isomorphism in $K$-theory.
The idea of the proof is as follows. By Corollary \[remarque\], there exists $S>0$ and finite such that for any $s\geq S$, $H^s_\ell(\Gamma,\sigma) \subseteq C^*_r(\Gamma, \sigma)$, and since $\bbC(\Gamma, \sigma)\subseteq H^{s}_\ell(\Gamma,\sigma)$, it follows that $H^{s}_\ell(\Gamma,\sigma)$ is a dense $*$-subalgebra of $C^*_r(\Gamma, \sigma)$. All we have to show is that the inclusion $H^s_\ell(\Gamma,\sigma) \subseteq C^*_r(\Gamma, \sigma)$ is spectral, it then follows (see e.g. Proposition 8.14 of [@Valette]) that the inclusion $H^{s}_\ell(\Gamma,\sigma)\hookrightarrow C^*_r(\Gamma,\sigma)$ induces an isomorphism in $K$-theory.
Now, for two number $s,t$ such that $S<t<s$ the first step is to show that $H^s_\ell(\Gamma,\sigma)$ is stable by holomorphic functional calculus in $H^t_\ell(\Gamma,\sigma)$. To do so, and since $H^s_\ell(\Gamma,\sigma)$ is dense in $H^t_\ell(\Gamma,\sigma)$, it is enough (see Remark 8.13 in [@Valette]) to prove that the spectral radius $\rho_s(f)$ of $f\in H^s_\ell(\Gamma,\sigma)$ is the same as $\rho_t(f)$, the one of $f\in H^t_\ell(\Gamma,\sigma)$, namely that $$\label{rayonSpectral}\lim_{n\to\infty}\|f^{*_{\sigma}n}\|_s^{1/n}=\lim_{n\to\infty}\|f^{*_{\sigma}n}\|_t^{1/n},$$ where for $n\in\bbN$ we set $f^{*_{\sigma}n}=\underbrace{f*_{\sigma}f*_{\sigma}\dots *_{\sigma}f}_{n}$. Notice that since $t<s$, then $\|\ \|_t\leq\|\ \|_s$ and hence $\rho_t(f)\leq\rho_s(f)$, so we only need to prove the other inequality. For $\gamma\in\Gamma$, using the triangle inequality one sees that $$\begin{aligned}
|f^{*_{\sigma}n}(\gamma)|&\leq&\sum_{\gamma_1,\dots,\gamma_{n-1}\in\Gamma}|f(\gamma\gamma_1^{-1})||f(\gamma_1\gamma_2^{-1})|\dots |f(\gamma_{n-2}\gamma_{n-1}^{-1})| |f(\gamma_{n-1})|\\
&=&\sum_{\gamma_1\dots\gamma_{n}=\gamma}|f(\gamma_1)|\dots |f(\gamma_n)|\end{aligned}$$ Therefore, using that $(1+\ell(\gamma))^{s-t}\leq n^{s-t}\sum_{i=1}^n(1+\ell(\gamma_i))^{s-t}$ if $\gamma_1\dots\gamma_{n}=\gamma$ (which follows easily from Lemma 1.1.4 (3) in [@Jolissaint]) we deduce that $$\|f^{*_{\sigma}n}\|_s=\|(1+\ell)^{s-t}f^{*_{\sigma}n}\|_t\leq n^{s-t+1}K^{n-1}\|f\|_s\|f\|_t^{n-1},$$ where $K=K(t)$ is the constant in the proof of Corollary \[remarque\]. Taking the $n$-th root and the limit shows that $\lim_{n\to\infty}\|f^{*_{\sigma}n}\|_s^{1/n}\leq K\|f\|_t$. Replacing $f$ by $f^{*_{\sigma}m}$ in the previous inequlity, taking the $m$-th root and the limit shows $\rho_s(f)\leq\rho_t(f)$. We can now show that $H^s_\ell(\Gamma,\sigma) \subseteq C^*_r(\Gamma, \sigma)$ is spectral, namely that for $f\in H^s_\ell(\Gamma,\sigma)$, its spectral radius $\rho_s(f)$ equals $\rho_*(f)$, its spectral radius as an element of $C^*_r(\Gamma, \sigma)$. If $\rho_s(f)=0$ it is clear because $\rho_*(f)\leq\rho_s(f)$. Otherwise, Hölder’s inequality shows that $$\|f\|_t\leq\|f\|_s^{\frac{t}{s}}\|f\|_{\ell^2\Gamma}^{\frac{1-t}{s}},$$ and hence $$|f^{*_{\sigma}n}\|_{op}\geq\|f^{*_{\sigma}n}\|_{\ell^2\Gamma}\geq\|f^{*_{\sigma}n}\|^{\frac{s}{s-t}}_t\|f^{*_{\sigma}n}\|_s^{-\frac{t}{s-t}},$$ so that we conclude using equality (\[rayonSpectral\]).
[CHMMM]{} M. F. Atiyah. Astérisque [**32/33**]{} [(1976)]{}, 43–72.
M. F. Atiyah, I. M. Singer. *The index of elliptic operators. III.* Ann. of Math. (2) [**87**]{}, [(1968)]{}, 546–604.
P. Baum, A. Connes. *$K$-theory for Lie groups and foliations.* Enseign. Math. (2) [**46**]{} [(2000)]{}, no. 1-2, 3–42.
P. Baum and R. Douglas. *$K$-homology and index theory.* Proceedings of Symposia in Pure Mathematics, [**38**]{}, Part 1 (1982), 117–173. H. Bettaieb, M. Matthey, A. Valette. *Low-dimensional group homology and the Baum-Connes assembly map.* Preprint *1999*.
O. Bratteli, D. Robinson. Operator algebras and quantum statistical mechanics. 2. Equilibrium states. Models in quantum statistical mechanics. Second edition. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1997.
K. Brown. *Cohomology of groups.* Graduate texts in Mathematics, vol. [**87**]{}, Springer-Verlag, New York, 1982. J. Brüning and T. Sunada. *On the spectrum of gauge-periodic elliptic operators.* Méthodes semi-classiques, Vol. 2 (Nantes, 1991). [*Astérisque*]{} [**210**]{} (1992), 65–74. [*eidem.*]{} On the spectrum of periodic elliptic operators,[*Nagoya Math. J.*]{} [**126**]{} (1992), 159–171. A. Carey, K. Hannabuss, V. Mathai and P. McCann, *Quantum Hall Effect on the hyperbolic plane.* Commun. Math. Phys. [**190**]{} No. 3 (1998) 629–673. I. Chatterji. *Property (RD) for uniform lattices in products of rank one Lie groups with some rank two Lie groups.* Geometria Dedicata [**96**]{} (2003) 161-177. I. Chatterji, K. Ruane. *Some geometric groups with Rapid Decay* .
P. -A. Cherix, M. Cowling, P. Jolissaint, P. Julg, A. Valette. *Groups with the Haagerup property (Gromov’s a-T-menability).* Birkäuser. Progress in Math. 197, 2001. A. Connes, *An analogue of the Thom isomorphism for crossed products of a $C^*$ algebra by an action of $\mathbb R$,* (1981), 31-55.
A. Connes, H. Moscovici. *Cyclic cohomology, the Novikov conjecture and hyperbolic groups.* Topology [**29**]{} (1990), no. 3, 345–388. P. Greiner. *An asymptotic expansion for the heat equation.* Arch. Ration. Mech. and Anal., [**41**]{} (1971), 163-218. M. Gromov. *Volume and bounded cohomology.* Publ. IHES., [**56**]{} (1983) 213-307.
U. Haagerup. *An example of a nonnuclear C\*-algebra which has the metric approximation property.* Invent. Math. [**50**]{} (1979) 279–293. P. de la Harpe. *Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolissaint.* C. R. Acad. Sci. Paris Sér. I [**307**]{} [(1988)]{}, 771–774. F. Hirzebruch. *Topological methods in algebraic geometry.* Die Grundlehren der Mathematischen Wissenschaften, Band 131 Springer-Verlag New York, Inc., New York 1966.
R. Ji. *Smooth dense subalgebras of reduced group $C^*$-algebras, Schwartz cohomology of groups, and cyclic cohomology.* J. Funct. Anal [**107**]{} (1992), 1–33.
R. Ji, L. B. Schweitzer. *Spectral invariance of smooth crossed products, and rapid decay locally compact groups.* $K$-Theory [**10**]{} (1996), no. 3, 283–305. P. Jolissaint. *Rapidly decreasing functions in reduced C\*-algebras of groups.* Trans. Amer. Math. Soc. [**317**]{} [(1990)]{}, 167–196.
G. Kasparov. *$K$-theory, group C\*-algebras, and higher signatures (Conspectus).* Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), 101–146, London Math. Soc. Lecture Note Ser., 226, Cambridge Univ. Press, 1995.
G. Kasparov. *Operator $K$-theory and its applications: elliptic operators, group representations, higher signatures, $C^*$-extensions.* Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 987–1000, PWN, Warsaw, 1984.
Yu. Kordyukov, $L^p$-theory of elliptic differential operators on manifolds of bounded geometry, [*Acta Appl. Math.*]{}, [**23**]{} (1991), [223–260]{}.
V. Lafforgue. *$K$-Théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes.* Invent. Math. [**149**]{} (2002), no. 1, 1–95. V. Lafforgue. *A proof of property (RD) for discrete cocompact subgroups of $SL_3({\bf R})$ and $SL_3({\bf C})$.* Journal of Lie Theory [**10**]{}, [(2000)]{}, 255–267. V. Mathai. *$K$-theory of twisted group $C^*$-algebras and positive scalar curvature.* Contemp. Math. [**231**]{} (1999), 203–225. V. Mathai. *On positivity of the Kadison constant and noncommutative Bloch theory.* Tohoku Mathematical Publications, [**20**]{} (2001) 107-124. M. Marcolli, V. Mathai. *Twisted index theory on good orbifolds, I: noncommutative Bloch theory.* Communications in Contemporary Mathematics, [**1**]{} (1999) 553-587.
I. Mineyev, G. Yu. *The Baum-Connes conjecture for hyperbolic groups.* Invent. Math. [**149**]{} (2002) 1-95. T. Natsume. *The Baum-Connes conjecture, the commutator theorem and Rieffel projections.* C. R. Math. Rep. Acad. Sci. Canad. [**1**]{} [(1988)]{}, 13–18.
J. Packer, I. Raeburn, Twisted cross products of $C^*$-algebras, [*Math. Proc. Camb. Phil. Soc.*]{} [**106**]{} (1989), 293-311.
M. Pimsner and D. Voiculescu. *Exact sequences for $K$-groups and Ext-groups of certain cross-product $C\sp{*}$-algebras.* J. Operator Theory, [**4**]{} (1980), no. 1, 93-118.
J. Ramagge, G. Robertson, T. Steger. *A Haagerup inequality for $\widetilde A\sb
1\times\widetilde A\sb 1$ and $\widetilde A\sb 2$ buildings.* Geom. Funct. Anal. [**8** ]{}[(1998)]{}, no. [**4**]{}, 702–731.
M. Rieffel. *$C^*$-algebras associated with irrational rotations.* Pac. J. Math. [**93**]{} (1981), 415-429.
J. Rosenberg, C. Schochet, *The K" unneth theorem and the universal coefficient theorem for Kasparov’s generalized $K$-functor.* Duke Math. J. [**55**]{} (1987), no. 2, 431-474.
A. Valette. *Introduction to the Baum-Connes Conjecture.* Notes taken by Indira Chatterji. With an appendix by Guido Mislin. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2002.
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=1
Introduction
============
In this paper, we study the classification problem of equations of the form $$\begin{gathered}
u_{xy} = f(u,u_x,u_y) \label{u_xy}\end{gathered}$$ over the ring of complex-valued variables. Such equations have applications in many fields of mathematics and physics. Liouville [@Liouville], Bäcklund [@Backl1], Darboux [@Darb] and other authors [@Bianchi; @Tzitz] studying the surfaces of constant negative curvature discovered the first examples of integrable nonlinear hyperbolic equations. In the 1970s, one of the fundamental methods of mathematical physics, the inverse scattering method, was introduced. After that, since hyperbolic equations have many applications in physics (continuum mechanics, quantum field theory, theory of ferromagnetic materials etc.), many important studies were published.
Existence of higher symmetries is a hallmark of integrability of an equation. Drinfel’d, Sokolov and Svinolupov [@DrSvSok; @Sv] showed that symmetries can be effectively used for classification of evolution equations. Zhiber and Shabat [@ZhiberShabat] obtained the complete list of the Klein–Gordon equations $$\begin{gathered}
v_{xy} = F(v) \label{v_xy}\end{gathered}$$ with higher symmetries. However, the symmetry method for the classification of equations of form faces particular difficulties. Therefore, here we use differential substitutions to solve the classification problem.
Before going further, let us give some definitions. Let $u$ be a solution of equation . All the mixed derivatives of $u$ $$\begin{gathered}
\label{kuzn_variations}
u_x, \qquad u_{y}, \qquad u_{xx}, \qquad u_{yy}, \qquad\dots\end{gathered}$$ will be expressed through equation with differential consequences of this equation. Here $u$ and variables will be regarded as independent.
We begin with an important notion of (infinitesimal) symmetry of equation . Denote the operators of total derivatives with respect to $x$ and $y$ by $D$ and $\bar{D}$, respectively.
The symmetry of equation of order $(n,m)$ is the function $g = g(u, u_1, \dots, u_n$, $\bar{u}_1, \dots, \bar{u}_m)$, $g_{u_n} \neq 0$, $g_{\bar{u}_m} \neq 0$, satisfying the equation $$(D \bar{D} - f_{u_1} D - f_{\bar{u}_1} \bar{D} - f_u)g = 0.$$ Here $u_i = \frac{\partial^i u}{ \partial x^i}$ and $\bar{u}_i = \frac{\partial^i u}{ \partial y^i}$, $i \in \mathbb{N}$. If $n \leq 1$ and $m \leq 1$ then the function $g$ is called a classical symmetry, otherwise we have a higher symmetry.
Assume that $g$ is a symmetry of equation . It is easy to check that the derivatives $g_{u_n}$ and $g_{\bar{u}_m}$ satisfy the so-called characteristic equations $\bar{D} (g_{u_n}) = 0$ and $D (g_{\bar{u}_m}) = 0$, respectively. It actually can be shown that $g_{u_n}$ depends only on the variables $u, u_1, \dots, u_n$, while $g_{\bar{u}_m}$ is a function of the variables $u, \bar{u}_1, \dots, \bar{u}_m$.
The function $\omega(u, u_1, u_2, \dots, u_n)$, $\omega_{u_n} \neq 0$, is called an $x$-integral of order $n$ of equation if $\bar{D}(\omega) =0$. Similarly, the $y$-integral of order $m$ is the function $\bar{\omega}(u, \bar{u}_1, \bar{u}_2, \dots$, $\bar{u}_m)$, $\bar{\omega}_{\bar{u}_m} \ne 0$ which satisfies $D(\bar{\omega}) =0$.
Another important notion is the sequence of the Laplace invariants of equation .
The main generalized Laplace invariants of equation are the functions $H_0$ and $H_1$ given by the formulae $$H_1 = - D \left( \frac{\partial f}{\partial u_1}\right) + \frac{\partial f}{\partial u_1} \frac{\partial f}{\partial \bar{u}_1} + \frac{\partial f}{\partial u}, \qquad
H_0 = - \bar{D} \left( \frac{\partial f}{\partial \bar{u}_1}\right) + \frac{\partial f}{\partial u_1} \frac{\partial f}{\partial \bar{u}_1} + \frac{\partial f}{\partial u}.$$
Other Laplace invariants can be found recurring in the relation $$D \bar{D} (\ln H_i) = -H_{i+1} - H_{i-1} + 2H_i, \qquad i \in \mathbb{Z}.$$ Sokolov and Zhiber [@ZhSok] showed that the functions $H_1$ and $H_0$ are invariants of equation under the point transformations $u \rightarrow \zeta(x,y,u)$. Generalized Laplace invariants play a significant role in the investigation of integrability of equations. Namely, Anderson and Kamran [@Anderson2], Zhiber, Sokolov and Startsev [@ZhSokSt] proved that an equation has nontrivial $x$- and $y$-integrals if and only if the Laplace sequence of invariants terminates on both sides ($H_r = H_s \equiv 0$ for some values $r$ and $s$), which is indeed a definition of the (Darboux) integrability of an equation. Equations satisfying the last condition are called Liouville type equations. Using this definition for linear equations $V_{xy} + a(x,y)V_x + b(x,y)V_y + c(x,y)V = 0$, one can obtain equations with the finite Laplace sequence studied in detail by Goursat [@Goursat].
It should be noted that symmetries of Liouville type equations have two arbitrary functions, while the equations integrable by the inverse scattering method (for instance, the sine-Gordon equation) have a countable set of symmetries.
The main notion of the paper is the notion of differential substitutions.
The relation $$\begin{gathered}
\label{kuzn_subs}
v = \varphi \left( u, \frac{\partial u}{\partial x}, \dots,\frac{\partial^n u}{\partial x^n}, \frac{\partial u}{\partial y}, \dots, \frac{\partial^m u}{\partial y^m} \right)\end{gathered}$$ is called a differential substitution from equation to the equation $$\begin{gathered}
\label{def_v_xy}
v_{xy} = g(v, v_x, v_y)\end{gathered}$$ if function satisfies equation for every solution $u(x,y)$ of equation .
Before proceeding, let us briefly mention some works related to differential substitutions. Sokolov [@Sok] showed that substitutions can be used in the study of integrability of nonlinear differential equations. There exist various different definitions of exact integrable hyperbolic equations. Sokolov and Zhiber [@ZhSok] presented one of the most comprehensive reviews of such equations. As mentioned before, existence of higher symmetries is a hallmark of integrability of an equation. Meshkov and Sokolov [@MeshSok] presented the complete list of one-field hyperbolic equations with generalized integrable $x$- and $y$-symmetries of the third order. One can find many examples of nonlinear equations and differential substitutions in [@MeshSok; @ZhSok]. Startsev [@St2; @St1] described properties of generalized Laplace invariants of nonlinear equations with differential substitutions. Bäcklund transformations and, in particular cases, differential substitutions were studied by Khabirov [@Kh2]. Kuznetsova [@Kuzn1] described coupled equations for which linearizations are related by Laplace transformations of the first and the second orders. A Bäcklund transformation was constructed for such pairs.
Although we know a considerable amount of nonlinear equations which are connected with one another by differential substitutions, the problem of classifying differential substitutions and Bäcklund transformations was solved only for evolution equations.
Recently, Zhiber and Kuznetsova [@Kuzn2] have applied differential substitutions to classify equations. Namely, all equations of form are transformed into equations of form by differential substitutions of the special form $
v = \varphi(u, u_x)
$ were described. All these equations are contained in the following list: $$\begin{aligned}
{4}
&u_{xy} = u F' \bigl( F^{-1} (u_x) \bigr), \qquad &&v_{xy} = F(v), \qquad &&v = F^{-1}(u_x); &\\[-0.5ex]
&u_{xy} = \sin u \sqrt{1 - u^2_x}, \qquad &&v_{xy} = \sin v, \qquad &&v = u + \arcsin u_x;& \\[-0.5ex]
&u_{xy} = \exp u \sqrt{1 + u^2_x}, \qquad &&v_{xy} = \exp v, \qquad &&v = u + \ln \left( u_x + \sqrt{1 + u^2_x} \right);&\\[-1ex]
&u_{xy} = \frac{\sqrt{2 u_y}}{s'(u_x)}, \qquad &&v_{xy} = F(v), \qquad &&v = s(u_x),&
\intertext{where the functions $s$ and $f$ satisfy $s'(u_x) F( s(u_x) ) = 1$;}
& u_{xy} = \frac{c - u_y \varphi_u(u, u_x)}{\varphi_{u_x}(u, u_x)}, \qquad && v_{xy} = 0, \qquad && v = \varphi(u, u_x); &
\\
& u_{xy} = u_x \bigl( \psi(u, u_y) - u_y \alpha'(u) \bigr), \qquad && v_{xy} = \exp v, \qquad && v = \alpha(u) + \ln u_x, &
\intertext{where $\psi_u + \psi \psi_{u_y} - \alpha' u_y \psi_{u_y} = \exp \alpha$;}
& u_{xy} = u_x \bigl( \psi(u, u_y) - u_y \alpha'(u) \bigr), \qquad && v_{xy} = 0, \qquad && v = \alpha(u) + \ln u_x,&
\intertext{where $\psi_u + \psi \psi_{u_y} - \alpha' u_y \psi_{u_y} = 0$;}
& u_{xy} = u, \qquad && v_{xy} = v, \qquad && v = c_1 u + c_2 u_x; & \\
& u_{xy} = \delta(u_y), \qquad && v_{xy} = 1, \qquad && v = c_1 u + c_2 u_x, \qquad \delta(c_1 + c_2 \delta') = 1,&\end{aligned}$$
up to the point transformations $u \rightarrow \theta(u)$, $v \rightarrow \kappa(v)$, $x \rightarrow \xi x$, and $y \rightarrow \eta y$, where $\xi$ and $\eta$ are arbitrary constants. Here $c$ is an arbitrary constant, $c_1$ and $c_2$ are constants satisfying $(c_1,c_2)\ne(0,0)$, and the function $\psi$ satisfies $(\psi_u, \psi_{u_y}) \neq (0, 0)$.
Furthermore, all equations of form that can be transformed into equations of form by differential substitutions of the form $u = \psi(v, v_y)$ are given in the following list: $$\begin{aligned}
{6}
&v_{xy} = F(v), \qquad &&u_{xy} = F'\bigl( F^{-1}(u_x) \bigr)u, \qquad&& u=v_y;& \\
&v_{xy} = 1, \qquad &&u_{xy} = \frac{\psi''\bigl( \psi^{-1}(u) \bigr)u_y}{\psi' \bigl( \psi^{-1}(u) \bigr)}, \qquad&& u=\psi(v_y); &\\
&v_{xy} = 0, \qquad &&u_{xy} = 0, \qquad&& u = c v + \mu(v_y); &\\
&v_{xy} = 0, \qquad &&u_{xy} = - u_x \exp u, \qquad&& u = \ln v_y - \ln v; &\\
&v_{xy} = v, \qquad &&u_{xy} = u, \qquad&& u = c_1 v + c_2 v_y; &\\
&v_{xy} = 1, \qquad &&u_{xy} = 1, \qquad&& u = v + v_y,&\end{aligned}$$ up to the point transformations $u \rightarrow \theta(u)$, $v \rightarrow \kappa(v)$, $x \rightarrow \xi x$, and $y \rightarrow \eta y$, where $\xi$ and $\eta$ are arbitrary constants. Here $c$ is an arbitrary constant, $c_1$ and $c_2$ are constants satisfying $(c_1, c_2) \neq (0, 0)$.
Based on the above lists, Bäcklund transformations have been constructed for some pairs of equations. For instance, the equations $$\begin{gathered}
\label{Ex_B}
u_{xy} = F'\bigl( F^{-1}(u_x) \bigr)u,\qquad v_{xy} = F(v)\end{gathered}$$ are connected by the Bäcklund transformation $$v = F^{-1}(u_x), \qquad u = v_y.$$ Kuznetsova [@Kuzn1] showed that linearizations of equation are related by Laplace transformations of the first order. For example, we give the equations $$u_{xy} = \bigl( \lambda - \beta n b^{n-1}(u_x) \bigr) u , \qquad v_{xy} = \lambda v - \beta v^n, \qquad n > 0,$$ where $\lambda$ and $\beta$ are arbitrary constants, and the function $b$ satisfies the equation $\lambda b(u_x) - \beta b^n(u_x) = u_x$. The Bäcklund transformation is given by $$u = v_y, \qquad v = b(u_x).$$ Note that the equation $v_{xy} = \lambda v - \beta v^n$ is a version of the PHI-four equation [@SolAbdo]. The PHI-four equation and the corresponding Bäcklund transformation are obtained for $n=3$.
The purpose of this paper is to describe all equations of form that are transformed into equations of form by differential substitutions $$\begin{gathered}
v = \varphi(u,u_x,u_y), \qquad \varphi_{u_x} \varphi_{u_y} \neq 0, \label{phi}\end{gathered}$$ over the ring of complex-valued variables.
It should be noted that most of the differential substitutions which connect the well-known integrable equations have the form $v = \varphi(u, u_x, u_y)$ (see [@MeshSok; @ZhSok]). Therefore, we are interested just in this form of substitutions.
This paper is organized as follows. Section \[section2\] presents the complete list of equations that are transformed into the Klein–Gordon equations by differential substitutions of form . In Section \[section3\], the main theorem of the paper is proven. Section \[section4\] is devoted to the problem which is, in a sense, inverse to the original problem. Namely, equations are transformed into equations by differential substitutions of the form $$\begin{gathered}
\label{kuzn_psi}
u = \psi(v, v_y, v_x), \qquad \psi_{v_y} \psi_{v_x} \neq 0,\end{gathered}$$ over the ring of complex-valued variables.
Equations transformed into Klein–Gordon equations {#section2}
=================================================
In this section, we give all possible cases when equation is transformed into equation by a differential substitution of form . The main result of this paper is the following theorem.
\[theorem1\] Suppose that equation is transformed into the Klein–Gordon equation by differential substitution . Then equations , , and substitution take one of the following forms: $$\begin{aligned}
{6}
& u_{xy} = \sqrt{u^2_x + a}\sqrt{u^2_y + b}, \quad && v_{xy} = \tfrac{1}{2}\bigl( \exp v - a b \exp(-v) \bigr),\hspace*{-200mm} &&& \nonumber\\
&&& \raisebox{0pt}[0pt][0pt]{$v = \ln \left[ \Bigl( u_x + \sqrt{u^2_x + a} \Bigr) \Bigl( u_y + \sqrt{u^2_y + b} \Bigr)\right]$};\hspace*{-200mm} &&& \label{zhiber_eq2_1}\\[0.5ex]
& u_{xy} = \sqrt{u_x u_y}, \qquad && v_{xy} = \tfrac{1}{4}v, \qquad && v = \sqrt{u_x} + \sqrt{u_y}; & \label{zhiber_eq2_2} \\
& u_{xy} = \sqrt{u_x}, \qquad && v_{xy} = \tfrac{1}{2}, \qquad && v = \sqrt{u_x} + u_y;& \label{zhiber_eq2_3}\\
& u_{xy} = 1, \qquad && v_{xy} = 0, \qquad && v = u_x + u_y; & \label{zhiber_eq2_4}
\\
& u_{xy} = \frac{1}{\gamma'(u_y)}, \qquad && v_{xy} = 1, \qquad && v = u_x + \gamma(u_y) + u,& \label{zhiber_eq2_5}
\intertext{where the function $\gamma$ satisfies $1 - \frac{\gamma''}{\gamma'^2} = \gamma'$;}
& u_{xy} = 0, \qquad && v_{xy} = 0, \qquad && v = \beta(u_x) + \gamma(u_y) + c_3 u; & \label{zhiber_eq2_6}
\\
& u_{xy} = \mu(u) u_x u_y, \qquad && v_{xy} = 0, \qquad && v = c_1 \ln u_x + c_2 \ln u_y + \alpha(u), & \label{zhiber_eq2_7}
\intertext{where $\mu'(c_1 + c_2) + \mu^2 (c_1 + c_2) + \alpha'' + \alpha' \mu = 0$;}
& u_{xy} = \mu(u)u_x u_y, \qquad && v_{xy} = \exp v, \qquad && v = \ln (u_x u_y) + \alpha(u),& \label{zhiber_eq2_8}
\intertext{where $2 \mu' + 2 \mu^2 + \alpha'' + \alpha' \mu = \exp \alpha$;}
& u_{xy} = u, \qquad && v_{xy} = v, \qquad && v = c_1 u_y + c_2 u_x + c_3 u; & \label{zhiber_eq2_9} \\
& u_{xy} = \mu(u) (u_y + c)u_x,\qquad && v_{xy} = \exp v, \qquad && v = \ln(u_y + c) + \ln u_x + \alpha(u),\hspace*{-300mm}& \label{zhiber_eq2_10}
\intertext{where $2 \mu' + 2\mu^2 + \alpha'' + \alpha' \mu = \exp \alpha$, $2 \mu^2 + \mu' + \alpha' \mu = \exp \alpha$;}
& u_{xy} = \mu(u)(u_y + c) u_x, \qquad && v_{xy} = 0, \qquad && v = c_2 \ln(u_y\! + c)\! + c_1 \ln u_x \!+ \alpha(u),\hspace*{-300mm} &\label{zhiber_eq2_11}
\intertext{where $(\mu' + \mu^2)(c_1 + c_2) + \alpha'' + \alpha' \mu = 0$, $c_1 \mu' + \mu^2 (c_1 + c_2) + \alpha' \mu = 0$;}
&u_{xy} = \mu(u)u_x, \qquad && v_{xy} = 0, \qquad && v = u_y - \ln u_x + \alpha(u), & \label{zhiber_eq2_12}
\intertext{where $\alpha'' + \mu' = 0$, $\mu^2 - \mu' + \alpha' \mu = 0$;}
& u_{xy} = \frac{\mu(u) u_x}{\gamma'(u_y)}, \qquad && v_{xy} = 0, \qquad && v = \ln u_x + \gamma(u_y) + \alpha(u), &\label{zhiber_eq2_13}
\intertext{where $c_3 + \frac{\gamma''}{\gamma'^2} + c_4 \gamma' u_y = 0$, $\alpha'' + \mu' + c_4 \mu^2 = 0$, and $c_3 \mu^2 + \mu' + \mu^2 + \alpha' \mu =0$;}
& u_{xy} = \frac{u_x}{(au + b) \gamma'(u_y)}, \qquad && v_{xy} = \exp v, \qquad && v = \ln u_x + \gamma(u_y) - 2 \ln(au + b),\hspace*{-300mm} & \label{zhiber_eq2_14}
\intertext{where $c_3 + \frac{\gamma''}{\gamma'^2} + c_4 \gamma' u_y = - \gamma' \exp \gamma$, $c_3 + 1 -3a = 0$, and $c_4 + 2a^2 - a = 0$;}
& u_{xy} = - \frac{1}{u \beta'(u_x) \gamma'(u_y)}, \qquad && v_{xy} = 0, \qquad && v = \beta(u_x) + \gamma(u_y), & \label{zhiber_eq2_15}
\intertext{where $\frac{\beta''}{\beta'^2} = u_x \beta' + c_1$, $\frac{\gamma''}{\gamma'^2} = u_y \gamma' - c_1$;}
& u_{xy} = \frac{\mu(u)}{\beta'(u_x) \gamma'(u_y)}, \qquad && v_{xy} = \exp v, \qquad && v = \beta(u_x) + \gamma(u_y) + \alpha(u),\hspace*{-300mm}& \label{zhiber_eq2_16}
\intertext{where $u_x + \frac{1}{\beta'(u_x)} = \exp (\beta)$, $u_y + \frac{1}{\gamma'(u_y)} = \exp \gamma$, $\alpha'' =\exp \alpha$, and $\mu = (\exp \alpha) / \alpha'$;}
& u_{xy} = \frac{\mu(u)}{\beta'(u_x) \gamma'(u_y)}, \qquad && v_{xy} = \exp v, \qquad && v = \beta(u_x) + \gamma(u_y) + \alpha(u),\hspace*{-300mm} & \label{zhiber_eq2_17}
\intertext{where $2 u_x + \frac{1}{\beta'(u_x)} = \exp \beta$, $2 u_y + \frac{1}{\gamma'(u_y)} = \exp \gamma$, $\alpha' \mu - 2 \mu^2 = \exp \alpha$, and $\alpha'^2 = 8 \exp \alpha$;}
& u_{xy} = s(u) \sqrt{1 - u^2_x}\sqrt{1 - u^2_y}, \quad && v_{xy} = c \sin v, \qquad &&&\nonumber\\
&&& v = \arcsin u_x + \arcsin u_y + p(u),\hspace*{-300mm} &&& \label{zhiber_eq2_18}
\intertext{where $s'' - 2 s^3 + \lambda s = 0$, $p'^2 = 2 s' - 2 s^2 + \lambda$;}
& u_{xy} = s(u) b(u_x) \bar{b}(u_y), \qquad && v_{xy} = c_1 \exp v + c_2 \exp (-2v),\hspace*{-300mm} &&& \nonumber \\
&&& \label{zhiber_eq2_19} v = - \tfrac{1}{2} \ln(u_x - b(u_x)) - \tfrac{1}{2} \ln(u_y - \bar{b}(u_y)) + p(u), \hspace*{-50mm} &&&
\intertext{where $(u_x - b(u_x))(b(u_x) + 2u_x)^2 = 1$, $(u_y - \bar{b}(u_y))(\bar{b}(u_y) + 2 u_y)^2 = 1$, $s'' - 2 s s' - 4 s^3 = 0$, and
$p'^2 - 2 s p' - 3 s' - 2 s^2 = 0$;}
& u_{xy} = \frac{\nu(u) - q_u(u,u_y)}{q_{u_y}(u,u_y)}u_x, \qquad && v_{xy} = c_3 \exp v, \qquad && v = \ln u_x + q(u,u_y), & \label{zhiber_eq2_20}\end{aligned}$$ where $$\frac{\nu - q_u}{q_{u_y}}\left( \nu - \frac{\nu - q_u}{q^2_{u_y}} q_{u_y u_y} - 2\frac{q_{u u_y}}{q_{u_y}} \right) + \frac{\nu'}{q_{u_y}} - \frac{q_{uu}}{q_{u_y}} + \nu' u_y = c_3 \exp q, \qquad q_{uu_y} \neq 0,$$ up to the point transformations $u \rightarrow \theta(u)$, $v \rightarrow \kappa(v)$, $x \rightarrow \xi x$, and $y \rightarrow \eta y$, and the substitution $u + \xi x + \eta y \rightarrow u$, where $\xi$ and $\eta$ are arbitrary constants. Here $c_3$ and $c_4$ are arbitrary constants, $a$ and $b$ are constants satisfying $(a, b) \neq (0, 0)$, and $c$, $c_1$, and $c_2$ are nonzero constants; in cases and the function $\gamma$ satisfies the condition $\bigl( \gamma'' / \gamma'^2 \bigr)' \neq 0$; in cases – the functions $\beta$ and $\gamma$ satisfy the conditions $\bigl( \beta'' / \beta'^2 \bigr)' \neq 0$ and $\bigl( \gamma'' / \gamma'^2 \bigr)' \neq 0$ accordingly, the function $\mu$ satisfies $\mu' \neq 0$, and $\mu \neq 0$ in all cases.
Now, let us analyze some of the above equations in detail. Consider with $a b \neq 0$. Using the point transformations $\sqrt{a} x \rightarrow x$, $\sqrt{b} y \rightarrow y$, and $v - \ln(ab)^{1/2} \rightarrow v$, we obtain $$\begin{gathered}
\label{zhiber_eq2_21}
u_{xy} = \sqrt{u^2_x + 1} \sqrt{u^2_y + 1}.\end{gathered}$$ Equation is transformed into the sine-Gordon equation $$v_{xy} = \frac{1}{2} \bigl( \exp v - \exp(-v) \bigr)$$ by the differential substitution $$v = \ln \Big[\Bigl( u_x + \sqrt{u^2_x + 1} \Bigr)\Bigl( u_y + \sqrt{u^2_y + 1} \Bigr)\Big].$$ Equation is a $S$-integrable and possesses symmetries of the third order (see [@MeshSok]). Note that applying the point transformations $v \rightarrow i v$, $i x \rightarrow x$, $i y \rightarrow y$, and using the formula $\ln \big( \sqrt{1 - u^2_x} - i u_x \big) = - i \arcsin u_x$ we can also convert the above equations into $$u_{xy} = \sqrt{1 - u^2_x1 - u^2_y} \sqrt{1 - u^2_y}, \qquad v_{xy} = - \sin v, \qquad v = \arcsin u_x + \arcsin u_y.$$ Now, assume that $a = 0$. Under the transformations $v - \ln 2 \rightarrow v$, $\sqrt{b} y \rightarrow y$, and $v - \ln \sqrt{b} \rightarrow v$ equations take the form $$\begin{gathered}
\label{zhiber_eq2_24}
u_{xy} = u_x \sqrt{u^2_y + 1},
\qquad
v_{xy} = \exp v, \qquad v = \ln u_x + \ln \Bigl( u_y + \sqrt{u^2_y + 1} \Bigr).\end{gathered}$$ Applying the transformation $i y \rightarrow y$ to the above equations we arrive at $$u_{xy} = u_x \sqrt{1 - u^2_y}, \qquad v_{xy} = -i \exp v, \qquad v = - i \arcsin u_y + \ln u_x.$$ As shown in [@MeshSok], equation (\[zhiber\_eq2\_24\]$_1$) has symmetries of the third order. In [@MeshSok] the $x$- and $y$-integrals and the general solution of equation (\[zhiber\_eq2\_24\]$_1$) were presented.
Note that the equation (\[zhiber\_eq2\_2\]$_1$) is the Goursat equation. Its symmetries of the third order can be found, for instance, in [@MeshSok].
The equation (\[zhiber\_eq2\_3\]$_1$) has symmetries of the third order [@MeshSok]. The $x$- and $y$-integrals of this equation are given by $$\omega = \frac{u_{xx}}{\sqrt{u_x}}, \qquad \bar{\omega} = u_{yyy}.$$
Consider cases and . The equation $u_{xy} = \mu(u)u_x u_y $ possesses the $x$- and $y$-integrals of the first order, $\omega = \ln u_x - \sigma(u)$, $\bar{\omega} = \ln u_y - \sigma(u)$. Here $\sigma' = \mu$.
The equation $
u_{xy} = \mu(u)(u_y + c)u_x
$ in cases and possess the $y$-integral of the first order $\bar{\omega} = \ln (u_y + c) - \sigma(u),$ where $\sigma' = \mu$. The $x$-integral in case is $$\omega = \frac{u_{xxx}}{u_x} - \frac{3}{2} \frac{u^2_{xx}}{u^2_x} - \frac{1}{2} \bigl( \mu^2(u) + 2 \mu(u) \alpha'(u) + \alpha'^2(u) \bigr)u^2_x,$$ and in case we get the $x$-integral $$\omega = c_2 \mu(u) u_x + c_1 \frac{u_{xx}}{u_x} + \alpha'(u) u_x.$$
The equation (\[zhiber\_eq2\_14\]$_1$) possesses the $y$-integral of the first order and the $x$-integral of the third order $$\bar{\omega} = \gamma(u_y) - \frac{1}{a} \ln(au + b), \qquad
\omega = \frac{u_{xxx}}{u_x} - \frac{3}{2} \frac{u^2_{xx}}{u^2_{x}} + \frac{u^2_x (2a - 1)}{2 (au + b)^2}.$$
Now, we consider the equation which appears in and . The equation (\[zhiber\_eq2\_16\]$_1$) is transformed into the equation presented in [@ZhSok] by a point transformation and has the integrals of the second order $$\omega = \beta'(u_x) u_{xx} - \frac{ \mu'(u)}{\mu(u) \beta'(u_x)}, \qquad \bar{\omega} = \gamma'(u_y) u_{yy} - \frac{\mu'(u)}{\mu(u) \gamma'(u_y)}.$$ On the other hand, equation (\[zhiber\_eq2\_17\]$_1$) can be transformed into the equation given in [@ZhSok] $$\begin{gathered}
\label{zhiber_eq2temp2}
u_{xy} = \frac{1}{u} B(u_x) \bar{B}(u_y).\end{gathered}$$ Here $B(u_x)B'(u_x) + B(u_x) - 2u_x = 0$, $\bar{B}(u_y) \bar{B}'(u_y) + \bar{B}(u_y) - 2 u_y = 0$. The integrals of equation are [@ZhSok] $$\begin{gathered}
\omega = \frac{u_{xxx}}{B} + \frac{2(B - u_x)}{B^3} u^2_{xx} + \frac{2(2u_x + B)}{u B} + \frac{B(u_x + B)}{u^2}, \\
\bar{\omega} = \frac{u_{yyy}}{\bar{B}} + \frac{2(\bar{B} - u_y)}{\bar{B}^3} u^2_{yy} + \frac{2(2u_y + \bar{B})}{u \bar{B}} + \frac{\bar{B}(u_y + \bar{B})}{u^2}.\end{gathered}$$
The equation (\[zhiber\_eq2\_20\]$_1$) possesses the $y$-integral of the first order $
\bar{\omega} = q(u, u_y) - \sigma(u).
$ Here $\sigma' = \nu$. If $c_3 \neq 0$ then we obtain the $x$-integral of the third order $$\omega = \frac{u_{xxx}}{u_x} - \frac{3}{2} \frac{u^2_{xx}}{u^2_x} + \nu'(u) u^2_x - \frac{1}{2} \nu^2(u) u^2_x.$$ If $c_3 = 0$ then we have the $x$-integral of the second order $$\omega = \frac{u_{xx}}{u_x} + \nu(u) u_x.$$
Note that equations in and are well-known equations, which are integrable by the inverse scattering method (see [@ZhSok]).
All of the previously mentioned equations possessing $x$- and $y$-integrals are contained in the list of Liouville type equations given in [@ZhSok].
Now we will show how to obtain a solution of an equation from a solution of another one by applying differential substitutions. As an example, we consider case with specifying $\mu(u) = 1$, $\alpha(u) = \ln 2$. So we have $$u_{xy} = u_x u_y, \qquad v = \ln (2 u_x u_y), \qquad v_{xy} = \exp v.$$ The equation $u_{xy} = u_x u_y$ has the $x$-integral $\omega(x) = \exp(-u)u_x$. Integrating this equation with respect to $x$ and redenoting $\int \omega(x) dx$ by $\omega(x)$ we obtain $$\exp(-u) = \omega(x) + \bar{\omega}(y).$$ Hence $$u = -\ln \bigl(\omega(x) + \bar{\omega}(y) \bigr).$$ Substituting the function $u$ into the equation $v = \ln(2 u_x u_y)$ we get the general solution of the Liouville equation $v_{xy} = \exp v$ as $$v(x,y) = \ln \left( \frac{2 \omega'(x) \bar{\omega}'(y)}{\bigl( \omega(x) + \bar{\omega}(y) \bigr)^2} \right).$$
Proof of the main theorem {#section3}
=========================
In this section we prove Theorem \[theorem1\]. In order to do that we determine the functions $f$, $F$, and $\varphi$ in , and . By substituting function into equation and using equation we get $$\begin{gathered}
\varphi_u f + u_x \bigl( \varphi_{uu} u_y + \varphi_{u u_x} f + \varphi_{u u_y} u_{yy} \bigr) + u_{xx} \bigl( \varphi_{u_x u} u_y + \varphi_{u_x u_x} f + \varphi_{u_x u_y} u_{yy} \bigr) \nonumber\\
\qquad{} + \varphi_{u_x}\bigl( f_u u_x + f_{u_x} u_{xx} + f_{u_y} f\bigr) + \varphi_{u_y} \bigl( f_u u_y + f_{u_x} f + f_{u_y} u_{yy} \bigr) \nonumber\\
\qquad{}+ f \left( \varphi_{u_y u} u_y + \varphi_{u_y u_x} f + \varphi_{u_y u_y} u_{yy} \right) = F(\varphi). \label{zhiber4}\end{gathered}$$ Since the function $F(\varphi)$ depends only on $u$, $u_x$, and $u_y$, the coefficients at $u_{xx}$, $u_{yy}$, and $u_{xx}u_{yy}$ are equal to zero, i.e. $$\begin{gathered}
\varphi_{u_x u_y} = 0, \qquad
\varphi_{u u_x} u_y + \varphi_{u_x u_x} f + \varphi_{u_x} f_{u_x} = 0,\qquad
\varphi_{u u_y} u_x + \varphi_{u_y} f_{u_y} + f \varphi_{u_y u_y} = 0.\end{gathered}$$ Integration of these equations leads to $$\begin{gathered}
\varphi = p(u, u_x) + q(u, u_y), \label{firstcondition}\\
\varphi_u u_y + \varphi_{u_x} f = A(u, u_y), \label{secondcondition}\\
\varphi_u u_x + \varphi_{u_y} f = B(u, u_x). \label{thirdcondition}\end{gathered}$$ The remaining terms in give $$\begin{gathered}
\label{fourthcondition}
f \bigl( \varphi_u\! + u_x \varphi_{u u_x} \!+ \varphi_{u_x} f_{u_y}\! + \varphi_{u_y} f_{u_x}\! + u_y \varphi_{u u_y} \bigr)\! + \varphi_{u u} u_x u_y\! + \big( u_x \varphi_{u_x}\! + u_y \varphi_{u_y} \big) f_u = F(\varphi).\!\!\!\!\end{gathered}$$ Hence, the original classification problem is reduced to the analysis of equations –. Eliminating the function $f$ from equations and we obtain the relation $$\begin{gathered}
\bigl( A - u_y \varphi_u \bigr) \varphi_{u_y} = \bigl( B - u_x \varphi_u \bigr) \varphi_{u_x}. \label{zhiber12}\end{gathered}$$ Applying the operator $\frac{\partial^2}{\partial u_x \partial u_y}$ to equation we arrive at the equation $$\begin{gathered}
\bigl( u_y \varphi_{u_y} \bigr)_{u_y} \varphi_{u u_x} = \bigl( u_x \varphi_{u_x} \bigr)_{u_x} \varphi_{u u_y}. \label{zhiber13}\end{gathered}$$ Relation is satisfied if one of the following conditions hold: $$\begin{gathered}
\varphi_{u u_x} = 0, \qquad \varphi_{u u_y} = 0, \label{zhiber14}\\
\varphi_{u u_x} = 0, \qquad \bigl( u_x \varphi_{u_x} \bigr)_{u_x} = 0, \label{zhiber15}\\
( u_y \varphi_{u_y} )_{u_y} = 0, \qquad \varphi_{u u_y} = 0, \label{zhiber16}\\
( u_y \varphi_{u_y} )_{u_y} = 0, \qquad ( u_x \varphi_{u_x} )_{u_x} = 0, \label{zhiber17}\\
\frac{( u_y \varphi_{u_y} )_{u_y}}{\varphi_{u u_y}} = \frac{( u_x \varphi_{u_x} )_{u_x}}{\varphi_{u u_x}} = \lambda(u), \qquad \lambda(u) \neq 0. \label{zhiber18}\end{gathered}$$
First, let us analyze equation . By substituting the function $\varphi$ given by into equation we get $$\begin{gathered}
\label{zhiber18.5}
( u_y q_{u_y} )_{u_y} = \lambda(u) q_{u u_y}, \qquad ( u_x p_{u_x} )_{u_x} = \lambda(u) p_{u u_x}.\end{gathered}$$ Now we integrate the first equation of with respect to $u_y$ and the second one with respect to $u_x$. This gives $$u_y q_{u_y} = \lambda(u) q_u + C(u), \qquad u_x p_{u_x} = \lambda(u) p_u + E(u).$$ The general solutions of these equations are $$q = \Phi_1 ( u_y \kappa(u) ) + \epsilon(u), \qquad p = \Phi_2 ( u_x \kappa(u) ) + \mu(u),$$ where $
\kappa(u) = \lambda(u) \kappa'(u)$, $\lambda(u) \epsilon'(u) + C(u) = 0$, $\lambda(u) \mu'(u) + E(u) = 0$. Therefore, the function $\varphi$ defined by takes the form $$\varphi = \Phi(u) + \Phi_1 ( u_y \kappa(u) )+ \Phi_2 ( u_x \kappa(u) ).$$ Here $\Phi(u) = \epsilon(u) + \mu(u)$. Furthermore, if we use the point transformation $
\int{\kappa(u) du} \rightarrow u
$ in the above formula, we obtain $$\begin{gathered}
\varphi = \alpha(u) + \beta(u_x) + \gamma(u_y). \label{zhiber19}\end{gathered}$$ Clearly, function satisfying also takes form .
Assume that condition holds. In this case, the substitution of the functions $\varphi$ defined by into yields $$p_{u u_x} = 0, \qquad ( u_x p_{u_x} )_{u_x} = 0,$$ which gives $$p = \alpha(u) + c \ln u_x.$$ Here $c$ is an arbitrary constant. Hence, function takes the form $
\varphi = \alpha(u) + c \ln u_x + q(u, u_y)
$. Replacing $\alpha(u) + q(u, u_y)$ by $q(u, u_y)$ in this equation we get $$\begin{gathered}
\varphi = c \ln u_x + q(u, u_y). \label{zhiber20}\end{gathered}$$ Recall that $\varphi_{u_x} \varphi_{u_y} \neq 0$. This property implies $c \neq 0$. Clearly, case coincides with up to the permutation of $x$ and $y$.
It remains to consider the case when $\varphi$ satisfies . Based on , we rewrite as $$( u_y q_{u_y} )_{u_y} = 0, \qquad ( u_x p_{u_x} )_{u_x} = 0.$$ By integrating these equations we get the functions $q$ and $p$, $$q = \mu(u) \ln u_y + \epsilon(u), \qquad p = \kappa(u) \ln u_x + \delta(u).$$ Consequently, the function $\varphi$ defined by formula takes the form $$\begin{gathered}
\varphi = \alpha(u) + \kappa(u) \ln u_x + \mu(u) \ln u_y. \label{zhiber21}\end{gathered}$$ Thus, to solve the original classification problem it is sufficient to consider three cases: , , and .
Case $\boldsymbol{\varphi = \alpha(u) + \beta(u_x) + \gamma(u_y)}$
------------------------------------------------------------------
When we substitute into equation , we obtain $$\bigl( A(u, u_y) - u_y \alpha'(u) \bigr) \gamma'(u_y) = \bigl( B(u, u_x) - u_x \alpha'(u) \bigr) \beta'(u_x).$$ Since $u_x$ and $u_y$ are regarded as independent variables, the above equation is equivalent to the system $$\begin{gathered}
\bigl( A(u, u_y) - u_y \alpha'(u) \bigr) \gamma'(u_y) = \mu(u),\qquad
\bigl( B(u, u_x) - u_x \alpha'(u) \bigr) \beta'(u_x)= \mu(u).\end{gathered}$$ From this system we find the functions $A$ and $B$ as $$A = \frac{\mu}{\gamma'} + u_y \alpha', \qquad B = \frac{\mu}{\beta'} + u_x \alpha'.$$ By substituting $A$ and $B$ into equations and we determine $f$ as follows $$\begin{gathered}
f = \frac{\mu(u)}{\beta'(u_x) \gamma'(u_y)}. \label{zhiber22}\end{gathered}$$ Using we transform equation into $$\begin{gathered}
\frac{\alpha' \mu}{\beta' \gamma'} - \mu^2 \left( \frac{\gamma''}{\gamma'^2} + \frac{\beta''}{\beta'^2} \right) \frac{1}{\beta' \gamma'} + \alpha'' u_x u_y + \mu' \left( \frac{u_x}{\gamma'} + \frac{u_y}{\beta'} \right) = F(\alpha + \beta + \gamma). \label{zhiber23}\end{gathered}$$ Applying the operators $\frac{\partial}{\partial u_x}$ and $\frac{\partial}{ \partial u_y}$ to equation we obtain $$\begin{gathered}
-\alpha' \mu \frac{\beta''}{\beta'^2 \gamma'} -\mu^2 \left( -\frac{\gamma''}{\gamma'^3} \frac{\beta''}{\beta'^2} + \frac{1}{\gamma'} \left( \frac{\beta''}{\beta'^3} \right)' \right) + \alpha'' u_y + \mu' \left( \frac{1}{\gamma'} - u_y \frac{\beta''}{\beta'^2} \right) = F'(\alpha + \beta + \gamma) \beta',\\
-\alpha' \mu \frac{\gamma''}{\beta' \gamma'^2} - \mu^2 \left( \left( \frac{\gamma''}{\gamma'^3} \right)' \frac{1}{\beta'} - \frac{\beta''}{\beta'^3} \frac{\gamma''}{\gamma'^2} \right) + \alpha'' u_x + \mu' \left( -u_x \frac{\gamma''}{\gamma'^2} + \frac{1}{\beta'} \right) = F'(\alpha + \beta + \gamma) \gamma'.\end{gathered}$$ By eliminating $F'$ from these equations we get $$\begin{gathered}
-\alpha' \mu \frac{\beta''}{\beta'^2} - \mu^2 \left( \frac{\beta''}{\beta'^3} \right)' + \alpha'' u_y \gamma' - \mu' u_y \gamma' \frac{\beta''}{\beta'^2}
\nonumber\\
\qquad{} = -\alpha' \mu \frac{\gamma''}{\gamma'^2} - \mu^2 \left( \frac{\gamma''}{\gamma'^3} \right)' + \alpha'' u_x \beta' - \mu' u_x \beta' \frac{\gamma''}{\gamma'^2}.
\label{zhiber24}\end{gathered}$$ Under the action of the operator $\frac{\partial^2}{\partial u_x \partial u_y}$, equation takes the form $$\mu' \left( (u_x \beta')' \left( \frac{\gamma''}{\gamma'^2} \right)' - (u_y \gamma')' \left( \frac{\beta''}{\beta'^2} \right)' \right) = 0.$$ It can be easily seen that the above equation is true if one of the following conditions is met: $$\begin{gathered}
\mu'(u) = 0, \label{zhiber26}\\
(u_x \beta')' = 0, \qquad (u_y \gamma')' = 0, \label{zhiber27}\\
(u_x \beta')' = 0, \qquad \left( \frac{\beta''}{\beta'^2} \right)' = 0, \label{zhiber28}\\
\left( \frac{\gamma''}{\gamma'^2} \right) ' = 0, \qquad (u_y \gamma')' = 0, \label{zhiber29}\\
\left( \frac{\gamma''}{\gamma'^2} \right)' = 0, \qquad \left( \frac{\beta''}{\beta'^2} \right)' = 0, \label{zhiber30}\\
\frac{(u_x\beta')' }{\left( \displaystyle \frac{\beta''}{\beta'^2} \right)'} = \frac{ (u_y\gamma')' }{ \left( \displaystyle \frac{ \gamma''}{\gamma'^2} \right)' } \neq 0. \label{zhiber31}\end{gathered}$$ It should be noted that $\mu' \neq 0$ in cases –.
To analyze cases – in a unified manner we begin by giving the following lemma.
\[lemma1\] By condition , equations , , and substitution take one of the following forms: $$\begin{aligned}
{6}
& u_{xy} = 0, \qquad && v_{xy} = \exp v, \qquad && v = \alpha(u) + \ln (u_x u_y), & \label{zhiber32}
\intertext{where the function $\alpha$ satisfies $\alpha'' = \exp \alpha$;}
& u_{xy} = u_x u_y, \qquad && v_{xy} = \exp v, \qquad && v = \alpha(u) + \ln (u_x u_y),& \label{zhiber33}
\intertext{where $\alpha'' + \alpha' + 2 = \exp \alpha$;}
& u_{xy} = - u_x u_y, \qquad && v_{xy} = 0, \qquad && &\nonumber\\
&&& v = \exp u + (a_1 + b_1)u + a_1 \ln u_x + b_1 \ln u_y;\hspace*{-300mm} &&& \label{zhiber34}\\
& u_{xy} = c \sqrt{u^2_x + a_2} \sqrt{u^2_y + b_2}, \qquad && v_{xy} = \frac{c^2}{2} \bigl( \exp v - a_2 b_2 \exp(-v) \bigr),\hspace*{-300mm} &&& \nonumber \\
&&& v = \ln \left[ \left( u_x + \sqrt{u^2_x + a_2} \right) \left( u_y + \sqrt{u^2_y + b_2} \right) \right];\hspace*{-300mm} &&& \label{zhiber35}
\\
& u_{xy} = c \sqrt{1 - u^2_x} \sqrt{1 - u^2_y}, \qquad && v_{xy} = -c^2 \sin v, \qquad && v = \arcsin u_x + \arcsin u_y;\hspace*{-100mm} & \label{zhiber36}\\
& u_{xy} = c \sqrt{u_xu_y}, \qquad && v_{xy} = \frac{c^2 v}{4}, \qquad && v = \sqrt{u_x} + \sqrt{u_y};& \label{zhiber37}\\
& u_{xy} = c \sqrt{u_x}, \qquad && v_{xy} = \frac{c^2}{2}, \qquad && v = \sqrt{u_x} + u_y; & \label{zhiber38}\\
& u_{xy} = c, \qquad && v_{xy} = 0, \qquad && v = u_x + u_y;& \label{zhiber39}\\
& u_{xy} = c u_y \sqrt{1 - u^2_x}, \qquad && v_{xy} =- \mathrm{i} c^2 \exp v, \qquad && v = -\mathrm{i} \arcsin u_x + \ln u_y;\hspace*{-100mm} & \label{zhiber40}\\
& u_{xy} = \frac{a_1}{\gamma'(u_y)}, \qquad && v_{xy} = b_1, \qquad && v = u_x + \gamma(u_y) + u,& \label{zhiber41}
\intertext{where $a_1 - a^2_1 \frac{\gamma''}{\gamma'^2} = b_1 \gamma'$;}
& u_{xy} = a(u_x + c_7)(u_y + c_9), \qquad && v_{xy} = 0, \qquad && \nonumber\\
&&& v = a_1 \ln(u_x + c_7) + b_1 \ln (u_y + c_9) + u,\hspace*{-300mm} &&& \label{zhiber42}
\intertext{where $aa_1 + ab_1 + 1 = 0$;}
& u_{xy} = 0, \qquad && v_{xy} = 0, \qquad && v = \beta(u_x) + \gamma(u_y) + u, & \label{zhiber43}\end{aligned}$$ up to the point transformations $u \rightarrow \theta(u)$, $v \rightarrow \kappa(v)$, $x \rightarrow \xi x$, and $y \rightarrow \eta y$ and the substitution $u + \xi x + \eta y \rightarrow u$, where $\xi$ and $\eta$ are arbitrary constants. Here $\alpha''$, $\alpha'$, and $1$ are linearly independent functions, $c$, $c_1$, $c_2$, $c_7$, $c_9$, $a_1 \neq 0$, $b_1 \neq 0$, $a \neq 0$, $b_2$, and $a_2$ are arbitrary constants.
If condition holds then $\mu(u) = c$, where $c$ is an arbitrary constant. Rewriting we obtain $$\begin{gathered}
c \alpha'(u) \frac{\beta''(u_x)}{\beta'^2(u_x)} + c^2 \left( \frac{\beta''(u_x)}{\beta'^3(u_x)} \right)' + \alpha''(u) u_x \beta'(u_x) \\
\qquad{} = c \alpha'(u) \frac{\gamma''(u_y)}{\gamma'^2(u_y)} + c^2 \left( \frac{\gamma''(u_y)}{\gamma'^3(u_y)} \right)' + \alpha''(u) u_y \gamma'(u_y).\end{gathered}$$ Since we regard the variables $u_x$, $u_y$ as independent, this equation is equivalent to the equations $$\begin{gathered}
c \alpha'(u) \frac{\beta''(u_x)}{\beta'^2(u_x)} + c^2 \left( \frac{\beta''(u_x)}{\beta'^3(u_x)} \right)' + \alpha''(u) u_x \beta'(u_x) = \sigma(u),\\
c \alpha'(u) \frac{\gamma''(u_y)}{\gamma'^2(u_y)} + c^2 \left( \frac{\gamma''(u_y)}{\gamma'^3(u_y)} \right)' + \alpha''(u) u_y \gamma'(u_y) = \sigma(u).\end{gathered}$$ By the same fact that the variables $u_x$, $u_y$ are considered as independent we define the function $\sigma$ as $
\sigma(u) = A_1 \alpha'(u) + B_1 \alpha''(u) + C_1 $. According to this we rewrite the above equations as $$\begin{gathered}
\alpha' \left( c \frac{\beta''}{\beta'^2} - A_1 \right) + \alpha'' \left( u_x \beta' - B_1\right) = C_1 - c^2 \left( \frac{\beta''}{\beta'^3} \right)', \nonumber\\
\alpha' \left( c \frac{\gamma''}{\gamma'^2} - A_1 \right) + \alpha'' \left( u_y \gamma' - B_1 \right) = C_1 - c^2 \left( \frac{\gamma''}{\gamma'^3} \right)'.
\label{zhiber46}\end{gathered}$$ Here $A_1$, $B_1$, and $C_1$ are constants.
Let us assume that $1$, $\alpha'$, and $\alpha''$ are linearly independent functions. Clearly, equations imply $$\begin{gathered}
c \frac{\beta''}{\beta'^2} = A_1, \qquad u_x \beta' = B_1, \qquad C_1 - c^2 \left( \frac{\beta''}{\beta'^3} \right)' = 0, \\
c \frac{\gamma''}{\gamma'^2} = A_1, \qquad u_y \gamma' = B_1, \qquad C_1 - c^2 \left( \frac{\gamma''}{\gamma'^3} \right)' = 0.\end{gathered}$$ From these equations we get $$\beta' = \frac{B_1}{u_x}, \qquad \gamma' = \frac{B_1}{u_y}, \qquad -\frac{c}{B_1} = A_1, \qquad C_1 + \frac{c^2}{B^2_1} = 0.$$ Using the above equations we transform equation into the equation $$\begin{gathered}
u_x u_y \left( \frac{c \alpha'}{B^2_1} + \frac{2 c^2}{B^3_1} + \alpha'' \right) = F(\alpha + \beta + \gamma). \label{zhiber47}\end{gathered}$$ Since $1$, $\alpha'$, and $\alpha''$ are linearly independent functions, the left-hand side of equation does not vanish. Then $F \neq 0$. By differentiating with respect to $u_x$ and using $\beta' = B_1 / u_x$ we get the equation $1 = F'(z)B_1/F(z)$, where $z = \alpha + \beta + \gamma$. Its general solution is given by $$\begin{gathered}
F(z) = C_1 \exp(z/B_1). \label{zhiber48}\end{gathered}$$ Substituting function into equation and using $\beta = B_1\ln u_x + C_2$, $\gamma = B_1 \ln u_y + C_3$ we obtain $$\frac{c \alpha'}{B^2_1} + \frac{2c^2}{B^3_1} + \alpha'' = C_1 \exp\left( \frac{\alpha}{B_1} + \frac{C_2}{B_1} + \frac{C_3}{B_1} \right).$$ Thus, equations , , and have the following forms $${\displaystyle u_{xy} = \frac{c u_x u_y}{B^2_1}, \qquad v_{xy} = C_1 \exp(v/B_1), \qquad v = \alpha(u) + B_1 \ln( u_x u_y) + C_2 + C_3,}$$ where $${\displaystyle c \frac{\alpha'}{B^2_1} + 2c^2 \frac{1}{B^3_1} + \alpha'' = C_1 \exp\left( \frac{\alpha + C_2 + C_3}{B_1} \right).}$$ We redenote $(\alpha + C_2 + C_3) / B_1$ by $\alpha$. Under the point transformation $v \rightarrow B_1 v$ the above equations take the forms $${\displaystyle u_{xy} = \frac{c u_x u_y}{B^2_1}, \qquad v_{xy} = \frac{C_1}{B_1} \exp v, \qquad v = \alpha(u) + \ln (u_x u_y),}$$ where $${\displaystyle c \frac{\alpha'}{B^2_1} + 2 c^2 \frac{1}{B^4_1} + \alpha'' = \frac{C_1}{B_1} \exp \alpha.}$$ The multiplier $C_1/B_1$ can be eliminated by the shift $v \rightarrow v + \ln(B_1/C_1)$. Finally, redenoting $\alpha - \ln(B_1/C_1)$ by $\alpha $ and $c/B^2_1$ by $c$ we get $$u_{xy} = c u_x u_y, \qquad v_{xy} = \exp v, \qquad v = \alpha(u) + \ln (u_x u_y),$$ where $\alpha'' + c\alpha' + 2c^2 = \exp \alpha$. If $c = 0$ then these equations take the form . Otherwise, applying the point transformation $u \rightarrow u/c$ and redenoting $\alpha$ by $\alpha+\ln c^2$ we can reduce the above equations to form .
Let us assume that $1$, $\alpha'$, and $\alpha''$ are linearly dependent functions. It means that $$C_1 \alpha'' + C_2 \alpha' + C_3 = 0, \qquad (C_1, C_2, C_3) \neq (0, 0, 0).$$ If $C_1 = 0$ then $C_2 \neq 0$ and we get $
\alpha' = c.
$ Otherwise, $
\alpha'' = c_1 \alpha' + c_2. \label{zhiber53}
$ Case $\alpha' = c$ is a subcase of $\alpha'' = c_1 \alpha' + c_2$. This equation has two families of solutions $$\alpha = c_3 u^2 + c_4 u + c_5, \qquad \alpha = \frac{1}{c_1} \exp (c_1u) + c_6 u + c_7.$$ The constants $c_5$, $c_7$ can be eliminated by $\beta + c_5 \rightarrow \beta$, $\beta + c_7 \rightarrow \beta$ in equation . So there are two possibilities $$\begin{gathered}
\alpha = c_2 u^2 + c_3 u \label{zhiber54}\end{gathered}$$ and $$\alpha = \bigl( \exp c_1 u \bigr)/ c_1 + c_4 u,$$ which takes the form $$\begin{gathered}
\alpha = \exp (c_1u) + c_4 u, \qquad c_1 \neq 0 \label{zhiber55}\end{gathered}$$ under the shifts $u \rightarrow u + (\ln c_1) / c_1 $ and $\alpha \rightarrow \alpha + c_4 (\ln c_1) / c_1$.
Now, let us concentrate on case , taking into account the fact that $\mu(u) = c$. Equation can be rewritten as $$\begin{gathered}
\frac{c (c_1 \exp (c_1u) + c_4)}{\beta' \gamma'} - c^2 \left( \frac{\gamma''}{\gamma'^2} + \frac{\beta''}{\beta'^2} \right) \frac{1}{\beta' \gamma'} + c^2_1 \exp (c_1 u) u_x u_y = F(\alpha + \beta + \gamma). \label{zhiber56}\end{gathered}$$ Applying $\frac{\partial}{\partial u}$ to equation we obtain $$\frac{c c^2_1 \exp (c_1 u)}{\beta' \gamma'} + c^3_1 \exp( c_1 u) u_x u_y = F'(\alpha + \beta + \gamma) (c_1 \exp (c_1u) + c_4).$$ Therefore, $$\frac{c c^2_1}{\beta' \gamma'} + c^3_1 u_x u_y = (c_1 + c_4 \exp(-c_1u)) F'(\alpha + \beta + \gamma).$$ Next, by applying the differentiation $\frac{\partial}{\partial u}$ to both sides of this equation, we get $$-c_1 c_4 \exp(-c_1 u) F'(\alpha + \beta + \gamma) + (c_1 + c_4 \exp(-c_1 u)) (c_1 \exp (c_1 u) + c_4) F''(\alpha + \beta + \gamma) = 0.$$ It is not difficult to see that the above equation implies $$(c_1 \exp (c_1 u) + c_4)^2 F''(\alpha + \beta + \gamma) = c_1 c_4 F'(\alpha + \beta + \gamma).$$ Consequently, we have two possibilities $$\begin{gathered}
F'(\alpha + \beta + \gamma) = 0, \label{zhiber57}\\
\frac{F''(\alpha + \beta + \gamma)}{F'(\alpha + \beta + \gamma)} = \frac{c_1 c_4}{(c_1 \exp (c_1 u) + c_4)^2}. \label{zhiber58}\end{gathered}$$
Equation yields $F = c_5$, where $c_5$ is an arbitrary constant. In this case by using we obtain $$\frac{c c_1}{\beta' \gamma'} + c^2_1 u_x u_y = 0, \qquad \frac{c c_4}{\beta' \gamma'} - c^2 \left( \frac{\gamma''}{\gamma'^2} + \frac{\beta''}{\beta'^2} \right) \frac{1}{\beta' \gamma'} = c_5.$$ According to the fact that $u_x$ and $u_y$ are considered as independent variables we have $$\beta'(u_x) = \frac{c c_1}{c_6 u_x}, \qquad \gamma'(u_y) = -\frac{c_6}{c^2_1 u_y}, \qquad cc_4 - c^2 \left( \frac{c^2_1}{c_6} - \frac{c_6}{c_1c} \right) = c_5 \beta'(u_x) \gamma'(u_y).$$ Moreover, since $\beta' \gamma' \neq 0$ we get $c_5 = 0$, hence $F \equiv 0$. Consequently, equations , , and take the following forms $$u_{xy} = -c_1 u_x u_y, \qquad v_{xy} = 0, \qquad v = \exp c_1u + c_4 u + \frac{cc_1}{c_6} \ln u_x - \frac{c_6}{c^2_1} \ln u_y + c_7,$$ where $$c_4 - \frac{cc^2_1}{c_6} + \frac{c_6}{c_1} = 0, \qquad c_1 \neq 0.$$ Using the point transformations $ u \rightarrow u / c_1$, $v \rightarrow v - cc_1 \ln (c_1) / c_6 + c_6 \ln( c_1) / c^2_1 + c_7$, and redenoting $cc_1/c_6$ by $a_1$, $-c_6 / c^2_1$ by $b_1$ we get equation .
Now, suppose that is true. Applying $\frac{\partial}{\partial u_x}$ to both sides of equation we get $$\left( \frac{F''}{F'} \right)' \beta' = 0.$$ Recall that $\beta' \neq 0$, therefore $F'' / F' = 0$. This equation has two families of solutions. Namely, $F(z) = c_6 \exp c_5 z + c_7$, $c_5 c_6 \neq 0,
$ which turns into $$\begin{gathered}
F(z) = \exp c_5 z + c_7, \qquad c_5 \neq 0 \label{zhiber60}\end{gathered}$$ by the shift $z \rightarrow z - \left( \ln c_6 \right)/c_5$, and $$\begin{gathered}
F(z) = c_6 z + c_7, \qquad c_6 \neq 0. \label{zhiber61}\end{gathered}$$
Now consider equation . In this case, equation takes the form $$\begin{gathered}
\frac{c(c_1 \exp (c_1 u) + c_4)}{\beta' \gamma'} - c^2 \left( \frac{\gamma''}{\gamma'^2} + \frac{\beta''}{\beta'^2} \right) \frac{1}{\beta' \gamma'} + c^2_1 \exp (c_1u) u_x u_y \\
\qquad{} = \exp \bigl( c_5(\exp c_1u + c_4 u) \bigr) \exp( c_5 \beta ) \exp (c_5 \gamma).\end{gathered}$$ This equation is not satisfied because $c_5 c_1 \neq 0$.
Let us focus on equation . Equation can be written as $$\begin{gathered}
\frac{c (c_1 \exp (c_1 u) + c_4)}{\beta' \gamma'} - c^2 \left( \frac{\gamma''}{\gamma'^2} + \frac{\beta''}{\beta'^2} \right) \frac{1}{\beta' \gamma'} + c^2_1 \exp(c_1 u ) u_x u_y \\
\qquad{} = c_6 (\exp (c_1u) + c_4 u + \beta + \gamma) + c_7.\end{gathered}$$ Applying the operator $\frac{\partial}{\partial u}$ to the above equation gives $$\frac{c c^2_1 \exp (c_1 u)}{\beta' \gamma'} + c^3_1 u_x u_y \exp (c_1 u) = c_6 (c_1 \exp c_1 u + c_4).$$ Collecting the coefficients at $\exp(c_1 u)$ and rewriting the remaining terms we obtain $$\frac{cc^2_1}{\beta' \gamma'} + c^3_1 u_x u_y = c_6 c_1, \qquad c_6 c_4 = 0.$$ Since $u_x$ and $u_y$ are considered as independent, the first equation is true if and only if $c_6 = 0$. In this case, it is clear that we obtain the equations .
Assume that the function $\alpha$ satisfies equation . Using and $\mu(u) = c$ we transform equation into $$\frac{c}{\beta' \gamma'} (2c_2u + c_3) - c^2 \left( \frac{\gamma''}{\gamma'^2} + \frac{\beta''}{\beta'^2} \right) \frac{1}{\beta' \gamma'} + 2c_2 u_x u_y = F(c_2 u^2 + c_3 u + \beta + \gamma).$$ Differentiating this equation with respect to $u$ and denoting $c_2 u^2 + c_3 u + \beta + \gamma$ by $z$ we obtain $$\begin{gathered}
2c_2 \frac{c}{\beta' \gamma'} = F'(z) (2c_2u + c_3). \label{zhiber62}\end{gathered}$$ Now we should analyze equation . First, we suppose that $c_2 = c_3 = 0$. The function $\alpha$ described by equation vanishes. Equations can be written as $$c_1 u_x - c^2 \frac{\beta''}{\beta'^3} = a_1, \qquad c_1 u_y - c^2 \frac{\gamma''}{\gamma'^3} = b_1.$$ Here $a_1$, $b_1$ are arbitrary constants. The above equations imply $$\begin{gathered}
\beta'(u_x) = \sqrt{-c^2} \frac{1}{\sqrt{c_1 u^2_x - 2a_1u_x + 2 a_2}}, \qquad
\gamma'(u_y) = \sqrt{-c^2} \frac{1}{\sqrt{c_1 u^2_y - 2b_1 u_y + 2 b_2 }}.\end{gathered}$$ Integrating these equations we obtain distinct formulae which determine the functions $\beta$ and $\gamma$. Uniting these formulae in pairs we arrive at –.
Furthermore, we must consider equation if $c_2 \neq 0$, $c_3 = 0$, and $c_2 c_3 \neq 0$. Taking the logarithm of both sides of equation leads to $$\ln \left( 2c_2 \frac{c}{\beta' \gamma'} \right) = \ln F'(z) + \ln (2c_2 u + c_3).$$ To eliminate $\beta'(u_x)$ and $\gamma'(u_y)$ we differentiate this equation with respect to $u$, $$\begin{gathered}
0 = \frac{F''}{F'} (2c_2u + c_3) + \frac{2c_2}{2c_2u+c_3}. \label{zhiber77}\end{gathered}$$ Applying $\frac{\partial}{\partial u_x}$ to both sides of equation we get $(F''/ F')' = 0$, which means that $F''/F = c_4$. By virtue of this, equation is written as[$$c_4 (2 c_2u + c_3)^2 + 2c_2 = 0.$$ Hence $c_2 = 0$. This contradicts $c_2 \neq 0$.]{}
It remains to discuss the case if $c_2 = 0$, $c_3 \neq 0$. It is clear that we have $F(z) = c_4$ from equation . Here $c_4$ is an arbitrary constant. Rewriting with $\alpha = c_3 u$, $\mu = c$ we get $$\begin{gathered}
c_3 c - c^2 \left( \frac{\gamma''}{\gamma'^2} + \frac{\beta''}{\beta'^2} \right) = c_4 \beta' \gamma'. \label{zhiber78}\end{gathered}$$ The equation $$-c^2 \left( \frac{\beta''}{\beta'^2} \right)' = c_4 \beta'' \gamma',$$ arises when we apply $\frac{\partial}{\partial u_x}$ to both the sides of equation .
Suppose that $\beta'' = 0$. Determining the function $\beta$ as $\beta(u_x) = c_5 u_x + c_6$, we transform equation into an ordinary differential equation $$c_3 c - c^2 \frac{\gamma''}{\gamma'^2} = c_4 c_5 \gamma'.$$ Thus, we find equations of forms , , and , $$u_{xy} = \frac{c}{c_5 \gamma'(u_y)}, \qquad v_{xy} = c_4, \qquad v = c_5 u_x + \gamma(u_y) + c_3 u,$$ where $c_3 c - c^2 \gamma''/ \gamma'^2 = c_4 c_5 \gamma'$, $c_5 \neq 0$. We use the transformations $x / c_5 \rightarrow x$, $v / c_3 \rightarrow v$. Then we redenote $c_4 c_5 $ by $c_2$, $\gamma / c_3$ by $\gamma$. To obtain we apply the transformation $c_3 x \rightarrow x$ once again. Finally, we redenote $c / c^2_3$ by $a_1$, $c_2 / c^2_3$ by $b_1$.
Let us assume that $\beta'' \neq 0$. This assumption enables us to rewrite equation in the form $$-c^2 \frac{1}{\beta''(u_x)} \left( \frac{\beta''(u_x)}{\beta'^2(u_x)} \right)' = c_4 \gamma'(u_y).$$ Since $u_x$, $u_y$ are regarded as independent variables, the above equation is equivalent to the system $$\begin{gathered}
-c^2 \frac{1}{\beta''} \left( \frac{\beta''}{\beta'^2} \right)' = c_5, \qquad c_4 \gamma' = c_5. \label{zhiber80}\end{gathered}$$ If $c_4 = 0$ then $c_5 = 0$, which yields $c = 0$ or $\beta'' / \beta'^2 = - c_6 \neq 0$. The last equation implies $$\beta(u_x) = \frac{1}{c_6} \ln(c_6 u_x + c_7).$$ Substituting this function into equation and using $c_4 = 0$ we can define the function $\gamma$ as $$\gamma(u_y) = \frac{1}{c_8} \ln(c_8 u_y + c_9),$$ and the following equations result in $$\begin{gathered}
u_{xy} = c(c_6u_x + c_7)(c_8 u_y + c_9), \qquad v_{xy} = 0, \\
v = \frac{1}{c_6} \ln(c_6 u_x + c_7) + \frac{1}{c_8} \ln (c_8 u_y + c_9) + c_3u,\end{gathered}$$ where $cc_6 + cc_8 + c_3 =0$, $c_3 \neq 0$. We use the transformations $v/c_3 \rightarrow v$, $x / c_6 \rightarrow x$, and $y / c_8 \rightarrow y$. Replacing $1 / (c_3c_6)$ by $a_1$, $1/ (c_3c_8)$ by $b_1$, and $cc_6c_8$ by $a$, we get . If $c = 0$ then $c_5 = c_4 = 0$, and we obtain .
Let us turn back to the system . Given the assumption $c_4 \neq 0$, this enables us to find the function $\gamma$, $$\gamma(u_y) = \frac{c_5}{c_4} u_y + c_6.$$ We also have an ordinary differential equation defining the function $\beta$, $$-c^2 \frac{\beta''}{\beta'^2} = c_5 \beta' + c_7.$$ Rewriting equation by using these equations we get $c_7 + c c_3 = 0$ and, therefore, $$u_{xy} = \frac{c c_4}{c_5 \beta'(u_x)}, \qquad v = \frac{c_5}{c_4} u_y + \beta(u_x) + c_3 u, \qquad v_{xy} = c_4,$$ where $ -c^2 \beta'' / \beta'^2 = c_5 \beta' + c_7$, $ c_7 + c c_3 = 0$, and $c_4 c_5 \neq 0$. Clearly, this case coincides with equation up to the permutation of $x$ and $y$.
\[lemma2\] Assume that is satisfied and $\mu'(u) \neq 0$. Then equations , , and take one of the following forms: $$\begin{aligned}
{6}
& u_{xy} = \mu(u) u_x u_y, \qquad && v_{xy} = 0, \qquad && v = c_1 \ln u_x + c_2 \ln u_y + \alpha(u), & \label{zhiber81}
\intertext{where the functions $\mu$ and $\alpha$ satisfy
$\mu'(c_1 + c_2) + \mu^2 (c_1 + c_2) + \alpha'' + \alpha' \mu = 0$, $\mu' \neq 0$;}
& u_{xy} = \mu(u)u_x u_y, \qquad && v_{xy} = \exp v, \qquad && v = \ln (u_x u_y) + \alpha(u), & \label{zhiber82}\end{aligned}$$ where $\mu$ and $\alpha$ satisfy $ 2 \mu' + 2 \mu^2 + \alpha'' + \alpha' \mu = \exp \alpha$, up to the point transformations $u \rightarrow \theta(u)$, $v \rightarrow \kappa(v)$, $x \rightarrow \xi x$, and $y \rightarrow \eta y$, where $\xi$ and $\eta$ are arbitrary constants. Here $c_1$ and $c_2$ are nonzero constants.
Condition allows us to determine the functions $\beta$ and $\gamma$ as $$\beta(u_x) = c_1 \ln u_x, \qquad \gamma(u_y) = c_2 \ln u_y.$$ Using these equations can be written in the form $$\begin{gathered}
\mu'(u) u_x u_y \left( \frac{1}{c_2} + \frac{1}{c_1} \right) + \frac{\mu^2(u)u_x u_y}{c_1 c_2} \left( \frac{1}{c_2} + \frac{1}{c_1} \right) + \alpha''(u) u_x u_y + \alpha'(u) \mu(u) \frac{u_x u_y}{c_1 c_2} \nonumber\\
\qquad {} = F \bigl( c_1 \ln u_x + c_2 \ln u_y + \alpha(u) \bigr).\label{zhiber83}\end{gathered}$$ If we apply the operator $\frac{\partial}{\partial u_x}$ to both sides of equation , we obtain $$\mu'(u) u_y \left( \frac{1}{c_2} + \frac{1}{c_1} \right) + \frac{\mu^2(u) u_y}{c_1 c_2} \left( \frac{1}{c_2} + \frac{1}{c_1} \right) + \alpha''(u) u_y + \alpha'(u) \mu(u) \frac{ u_y}{c_1 c_2} = \frac{F' c_1}{u_x}.$$ Comparing the above equation with equation we notice that $F = c_1 F'$. Similarly, differentiating equation with respect to $u_y$ we deduce that $F = c_2F'$. These equations yield $F' = 0$ or $c_2 = c_1$.
If $F' = 0$, equation takes the form $$u_x u_y \left( \mu'(u) \left( \frac{1}{c_2} + \frac{1}{c_1} \right) + \frac{\mu^2}{c_1 c_2} \left( \frac{1}{c_2} + \frac{1}{c_1} \right) + \alpha'' + \frac{\alpha' \mu}{c_1 c_2} \right) = c.$$ Since $u$, $u_x$, and $u_y$ are regarded as independent variables and the functions $\mu$ and $\alpha$ are functions depending on $u$, we conclude that $c = 0$. Consequently, we obtain the equations $$u_{xy} = \frac{\mu(u) u_x u_y}{c_1 c_2}, \qquad v_{xy} = 0, \qquad v = c_1 \ln u_x + c_2 \ln u_y + \alpha(u),$$ where $$\mu' \left( \frac{1}{c_2} + \frac{1}{c_1} \right) + \frac{\mu^2}{c_1 c_2} \left( \frac{1}{c_2} + \frac{1}{c_1} \right) + \alpha'' + \frac{\alpha' \mu}{c_1 c_2} = 0.$$ Finally, replacing $\mu / c_1 c_2$ by $\mu$ we get equation .
If we replace $c_2$ with $c_1$, we determine $F = c_3 \exp (v/ c_1)$. Equation turns into $$\frac{2 \mu' u_x u_y}{c_1} + \frac{2 \mu^2 u_x u_y}{c^3_1} + \alpha'' u_x u_y + \frac{\alpha' \mu u_x u_y}{c^2_1} = c_3 u_x u_y \exp( \alpha(u) / c_1).$$ Thus, the following equations appear $$u_{xy} = \frac{1}{c^2_1} \mu(u) u_x u_y, \qquad v_{xy} = c_3 \exp(v / c_1), \qquad v = c_1 \ln u_x u_y + \alpha(u),$$ where $$\frac{2 \mu'}{c_1} + \frac{2 \mu^2}{c^3_1} + \alpha'' + \frac{\alpha' \mu}{c^2_1} = c_3 \exp(\alpha / c_1).$$ First, we redenote $\mu / c^2_1$ by $\mu$ and $\alpha / c_1$ by $\alpha $. Second, use the transformation $v \rightarrow c_1 v$ and then the shift $v \rightarrow v - \ln c$. Finally, replace $\alpha + \ln c$ by $\alpha$, $c_3 / c$ by $c_1$, and obtain the equations .
\[lemma3\] Assume that condition is satisfied but and are not. Then equations , , and take one of the following forms: $$\begin{aligned}
{6}
& u_{xy} = u, \qquad && v_{xy} = v, \qquad && v = c_1 u_y + c_2 u_x + c_3 u; & \label{zhiber84}\\
& u_{xy} = \mu(u) (u_y + b)u_x,\qquad && v_{xy} = \exp v, \qquad && v = \ln(u_y + b) + \ln u_x + \alpha(u), & \label{zhiber85}
\intertext{where the functions $\mu$ and $\alpha$ satisfy $2 \mu' + 2\mu^2 + \alpha'' + \alpha' \mu = \exp \alpha$, $2 \mu^2 + \mu' + \alpha' \mu = \exp \alpha$;}
& u_{xy} = \mu(u)(u_y + b) u_x, \qquad && v_{xy} = 0, \qquad && v = c_2 \ln(u_y + b) + c_1 \ln u_x + \alpha(u),& \label{zhiber86}
\intertext{where $\mu$ and $\alpha$ satisfy $(\mu' + \mu^2)(c_1 + c_2) + \alpha'' + \alpha' \mu = 0$, $c_1 \mu' + \mu^2 (c_1 + c_2) + \alpha' \mu = 0$;}
& u_{xy} = \mu(u)u_x, \qquad && v_{xy} = 0, \qquad && v = u_y - \ln u_x + \alpha(u), & \label{zhiber87}\end{aligned}$$ where $\mu$ and $\alpha$ satisfy $\alpha'' + \mu' = 0$, $\mu^2 - \mu' + \alpha' \mu = 0$, up to the point transformations $u \rightarrow \theta(u)$, $v \rightarrow \kappa(v)$, $x \rightarrow \xi x$ and $y \rightarrow \eta y$, where $\xi$ and $\eta$ are arbitrary constants. Here $c_3$ is an arbitrary constant, $c_1$, $c_2$, and $b$ are nonzero constants.
Condition implies the following three possibilities for functions $\beta$ and $\gamma$ $$\begin{aligned}
{3}
& \gamma(u_y) = c_1 u_y + c_2, \qquad && \beta(u_x) = c_3 u_x + c_4, & \label{zhiber88}\\
& \gamma(u_y) = - \frac{1}{c_1} \ln (a_1 u_y + b_1), \qquad && \beta(u_x) = -\frac{1}{c_2} \ln (a_2 u_x + b_2), & \label{zhiber89}\\
& \gamma(u_y) = c_1 u_y + c_2, \qquad && \beta(u_x) = -\frac{1}{c_3} \ln (a u_x + b). & \label{zhiber90}\end{aligned}$$ According to , equation can be written as $$\begin{gathered}
\frac{\mu'(u) u_y}{c_3} + \frac{\mu'(u) u_x}{c_1} + \alpha''(u) u_x u_y + \frac{\alpha'(u) \mu(u)}{c_1 c_3} = F(c_1 u_y + c_3 u_x + \alpha(u)). \label{zhiber91}\end{gathered}$$ Applying the operators $\frac{\partial}{\partial u_x}$ and $\frac{\partial}{\partial u_y}$ to both sides of gives $$\frac{\mu'}{c_1} + \alpha'' u_y = F' c_3, \qquad \frac{\mu'}{c_3} + \alpha'' u_x = F' c_1. \label{zhiber92}$$ Eliminating $F'$ from the above equations we obtain $\alpha'' (c_1 u_y - c_3 u_x) = 0$. Clearly, we have $\alpha'' = 0$, hence $\alpha = c_2 u + c_4$. Furthermore, by using any of the above equations we obtain $F' = \mu' / c_1 c_3$. Consequently, $$F(z) = \frac{c_5}{c_1 c_3} z + c_7, \qquad z=c_1u_y+c_3u_x+\alpha(u).$$ The equation $$\frac{\mu'' u_y}{c_3} + \frac{\mu'' u_x}{c_1} + \frac{c_2 \mu'}{c_1 c_3} = F' c_2$$ arises after the differentiation of equation with respect to $u$. Substituting $F' = \mu' / c_1 c_3$ into this equation yields $\mu(u) = c_5 u + c_6$. Therefore, the equation is equivalent to $$\frac{c_2 c_6}{c_1 c_3} = \frac{c_5 c_4}{c_1 c_3} + c_7.$$ Thus, we find that equations , , and the substitution have the forms $$u_{xy} = \frac{c_5 u + c_6}{c_1 c_3}, \qquad v_{xy} = \frac{c_5}{c_1 c_3} v + c_7, \qquad
v = c_1 u_y + c_3 u_x + c_2 u + c_4.$$ Using the transformations $u + c_6 / c_5 \rightarrow cu$, $v + c_1 c_3c_7 /c_5 \rightarrow cv$ and replacing $c_5 / c_1$ by $c_3$ we get .
Let us discuss the case when the functions $\gamma$ and $\beta$ are of form . It turns out that equation takes the form $$\begin{gathered}
-c_2 \mu' u_y \frac{a_2 u_x + b_2}{a_2} - \mu^2 \frac{c_1 c^2_2}{a_1 a_2} (a_2 u_x + b_2) (a_1 u_y + b_1) - \mu^2 \frac{c^2_1 c_2}{a_1 a_2}(a_1 u_y + b_1)(a_2u_x + b_2) \nonumber\\
\qquad{}- c_1 \mu' u_x \frac{a_1 u_y + b_1}{a_1} + \alpha'' u_x u_y + \alpha' \mu \frac{c_1 c_2}{a_1 a_2} (a_2 u_x + b_2)(a_1 u_y + b_1) \nonumber\\
\qquad\quad{}= F \left( -\frac{1}{c_1} \ln(a_1 u_y + b_1) - \frac{1}{c_2} \ln(a_2 u_x + b_2) + \alpha(u) \right).\label{zhiber93}\end{gathered}$$ Applying the operator $\frac{\partial}{\partial u_x}$ to both sides of equation leads to $$\begin{gathered}
-c_2 \mu' u_y - \mu^2 \frac{c_1 c^2_2}{a_1} (a_1 u_y + b_1) - c_1 \mu' \frac{a_1 u_y + b_1}{a_1} - \mu^2 \frac{c^2_1 c_2}{a_1} (a_1 u_y + b_1) \\
\qquad{}+ \alpha'' u_y + \alpha' \mu \frac{c_1 c_2}{a_1} (a_1 u_y + b_1) = F' \left( -\frac{1}{c_2} \right) \frac{a_2}{a_2 u_x + b_2}.\end{gathered}$$ The last equation and equation imply $$F' \left( -\frac{1}{c_2} \right) - F = -c_1 \mu' \frac{a_1 u_y + b_1}{a_1} \frac{b_2}{a_2} + \alpha'' u_y \frac{b_2}{a_2}.$$ Similarly, differentiating equation with respect to $u_y$ we obtain $$F' \left( -\frac{1}{c_1} \right) - F = -c_2 \mu' \frac{a_2 u_x + b_2}{a_2} \frac{b_1}{a_1} + \alpha'' u_x \frac{b_1}{a_1}.$$ To eliminate $u_x$ and $u_y$ we apply the operators $\frac{\partial}{\partial u_x}$ and $\frac{\partial}{\partial u_y}$ to the two above equations, respectively. We get $$F'' \left( - \frac{1}{c_2} - F' \right) = 0, \qquad F'' \left( -\frac{1}{c_1} - F' \right) = 0,$$ therefore $F''(c_2 - c_1) = 0$.
Assuming that $c_1 = c_2 =c$ we define $F$ as follows $$F(z) = -\frac{1}{c} \exp(-c z + c_7) + c_8.$$ Substituting the above function $F$ into equation we get $$\begin{gathered}
-\mu' u_y c \left( u_x + \frac{b_2}{a_2} \right) - 2 \mu^2 \frac{c^3}{a_1 a_2} (a_2 u_x + b_2) (a_1 u_y + b_1) - \mu' u_x c \left( u_y + \frac{b_1}{a_1} \right) + \alpha'' u_x u_y \\
\qquad{} + \alpha' \mu \frac{c^2}{a_1 a_2} (a_2 u_x + b_2) (a_1 u_y + b_1) = - \frac{1}{c} (a_2 u_x + b_2) (a_1 u_y + b_1) \exp ( -c \alpha + c_7 ) + c_8.\end{gathered}$$ Since $u$, $u_x$, and $u_y$ are considered as independent variables, the above equation is equivalent to the following system
\[zhiber97\] $$\begin{aligned}
& \displaystyle - 2c \mu' - 2 \mu^2 c^3 + \alpha'' + \alpha' \mu c^2 = -\frac{a_1 a_2}{c} \exp (-c \alpha + c_7), \\
& \displaystyle -2\mu^2 \frac{c^3}{a_1}b_1 - \mu' c \frac{b_1}{a_1} + \alpha' \mu c^2 \frac{b_1}{a_1} = - \frac{1}{c} a_2 b_1 \exp (-c \alpha + c_7), \\
& \displaystyle -\mu' c \frac{b_2}{a_2} - 2 \mu^2 c^3 \frac{b_2}{a_2} + \alpha' \mu c^2 \frac{b_2}{a_2} = - \frac{1}{c} a_1 b_2 \exp(-c \alpha + c_7),
\\
& \displaystyle -2 \mu^2 c^3 \frac{b_1 b_2}{a_1 a_2} + \alpha' \mu c^2 \frac{b_1 b_2}{a_1 a_2} = - \frac{1}{c} b_1 b_2 \exp(-c \alpha + c_7) + c_8.\end{aligned}$$
Note that $(b_1, b_2) \neq (0, 0)$. Otherwise, condition is true, which contradicts the assumption of the lemma. If $b_2 = 0$, $b_1 \neq 0$ then $c_8 = 0$ and $$\begin{gathered}
\label{zhiber97.5}
u_{xy} = \frac{\mu(u)c^2}{a_1} (a_1 u_y + b_1)u_x, \qquad v_{xy} = -\frac{1}{c} \exp(-cv+c_7),\nonumber \\
v = -\frac{1}{c} \ln (a_1 u_y + b_1) - \frac{1}{c} \ln (a_2 u_x) + \alpha(u),\end{gathered}$$ where the functions $\mu$ and $\alpha$ satisfy the following equations $$\begin{gathered}
-2c\mu' - 2\mu^2 c^3 + \alpha'' + \alpha' \mu c^2 = -\frac{a_1 a_2}{c} \exp(-c \alpha + c_7), \\
-2 \mu^2 c^3 - \mu' c + \alpha' \mu c^2 = - \frac{a_1 a_2}{c} \exp(-c \alpha + c_7).\end{gathered}$$ Applying the transformation $ -cv + c_7\rightarrow v $ and redenoting $-c \alpha + c_7 + \ln (a_1 a_2)$ by $\alpha$, $\mu c^2$ by $\mu$ and $b_1 / a_1$ by $b$, we transform into . It is not hard to prove that system has no solutions if $b_1b_2 \neq 0$.
Let us suppose that $F'' = 0$, hence $F(z) = cz + p$, where $c$ and $p$ are arbitrary constants. In this case equation is represented as $$\begin{gathered}
-c_2 \mu' u_y \left( u_x + \frac{b_2}{a_2} \right) - \mu^2 c_1 c^2_2 \left( u_x + \frac{b_2}{a_2} \right) \left( u_y + \frac{b_1}{a_1} \right) - c_1 \mu' u_x \left( u_y + \frac{b_1}{a_1} \right) + \alpha'' u_x u_y\\
\qquad{} - \mu^2 c^2_1 c_2 \left( u_x + \frac{b_2}{a_2} \right) \left( u_y + \frac{b_1}{a_1} \right) + \alpha' \mu c_1 c_2 \left( u_x + \frac{b_2}{a_2} \right) \left( u_y + \frac{b_1}{a_1} \right) \\
\qquad\quad{} = c \left( - \frac{1}{c_1} \ln (a_1 u_y + b_1) - \frac{1}{c_2} \ln(a_2 u_x + b_2) + \alpha(u) \right) + p.\end{gathered}$$ It is clear that the coefficients at $\ln (a_1 u_y + b_1)$ and $\ln(a_2 u_x + b_2)$ are equal to zero, i.e. $c = 0$. Since $u$, $u_x$, and $u_y$ are regarded as independent variables, the above equation is equivalent to the system $$\begin{gathered}
-c_2 \mu' - \mu^2 c_1 c^2_2 - c_1 \mu' - \mu^2 c^2_1 c_2 + \alpha'' + \alpha' \mu c_1 c_2 = 0, \\
-\mu^2 c_1 c^2_2 \frac{b_1}{a_1} - c_1 \mu' \frac{b_1}{a_1} - \mu^2 c^2_1 c_2 \frac{b_1}{a_1} + \alpha' \mu c_1 c_2 \frac{b_1}{a_1} = 0,\\
-c_2 \mu' \frac{b_2}{a_2} - \mu^2 c_1 c^2_2 \frac{b_2}{a_1} - \mu^2 c^2_1 c_2 \frac{b_2}{a_1} + \alpha' \mu c_1 c_2 \frac{b_2}{a_2} = 0, \\
{-\mu^2 c_1 c^2_2 - \mu^2 c^2_1 c_2 + \alpha' \mu c_1 c_2} \frac{b_1 b_2}{a_1 a_2} = p.\end{gathered}$$ Note that $(b_1,b_2) \neq (0, 0)$. Otherwise, condition is satisfied, which contradicts the assumption of the lemma. If $b_2 = 0$, $b_1 \neq 0$ then $p = 0$ and $$\begin{gathered}
u_{xy} = \mu(u) \frac{c_1 c_2}{a_1} (a_1 u_y + b_1)u_x, \qquad v_{xy} = 0, \\
v = -\frac{1}{c_1}\ln(a_1 u_y + b_1) - \frac{1}{c_2} \ln(a_2 u_x) + \alpha(u),\end{gathered}$$ where the functions $\mu$ and $\alpha$ satisfy the equations $$\begin{gathered}
\mu'(c_1 + c_2) + \mu^2 c_1 c_2 (c_1 + c_2) - \alpha'' - \alpha' \mu c_1 c_2 = 0,
\\
c_1 \mu' + \mu^2 c_1 c_2 (c_1 + c_2) - \alpha' \mu c_1 c_2 = 0.\end{gathered}$$ We replace $c_1 c_2 \mu$ by $\mu$, $-c_1 c_2 \alpha + c_2 \ln a_1 + c_1 \ln a_2$ by $\alpha$. Using the transformation $ v \rightarrow -v/ (c_1 c_2)$ and redenoting $b_1 / a_1$ by $b$ we transform the above equations into . If $b_1 b_2 \neq 0$ then the last system has no solutions.
Let us suppose that the functions $\gamma$ and $\beta$ are given by . We rewrite equation using , $$\begin{gathered}
-\frac{c_2}{a} \mu' u_y (au_x + b) + \frac{c^2_2}{a c_1} (a u_x + b) + \frac{1}{c_1} \mu' u_x + \alpha'' u_x u_y + \alpha' \mu (a u_x + b) \left( - \frac{c_2}{c_1 a} \right) \nonumber\\
\qquad{} = F \left( c_1 u_y - \frac{1}{c_2} \ln (au_x + b) + \alpha(u) \right).\label{zhiber98}\end{gathered}$$ Applying the operators $\frac{\partial}{\partial u_x}$ and $\frac{\partial}{\partial u_y}$ to both sides of equation we obtain $$\begin{gathered}
\label{zhiber99}
-c_2 \mu' u_y + \frac{c^2_2}{c_1} \mu^2 + \frac{1}{c_1} \mu' + \alpha'' u_y - \frac{c_2}{c_1} \alpha' \mu = F' \left( - \frac{1}{c_2} \right) \frac{a}{a u_x + b}, \\
-\frac{c_2}{a} \mu' (au_x + b) + \alpha'' u_x = F' c_1. \label{zhiber100}\end{gathered}$$
If $F' = 0$ then we obviously get $F = c_3$ and $$-c_2 \mu' + \alpha'' = 0, \qquad c^2_2 \mu^2 + \mu' - c_2 \alpha' \mu = 0, \qquad \mu' b = 0.$$ We analyze equation based on these equations and find that $c_3 = 0$. It allows us to determine equations , , and as follows $$u_{xy} = -\frac{c_2}{c_1} \mu(u) u_x, \qquad v_{xy} = 0 , \qquad v = c_1 u_y - \frac{1}{c_2} \ln (au_x) + \alpha(u),$$ where the functions $\mu$ and $\alpha$ satisfy $$\alpha'' = c_2 \mu', \qquad c^2_2 \mu^2 + \mu' - c_2 \alpha' \mu =0.$$ Point transformations enable us to represent the above equations in form .
Assuming that $F' \neq 0$ we can eliminate $F'$ from equations and $$\begin{gathered}
c^2_2 \left( \frac{a u_x + b}{a} \right) \mu' u_y - \frac{c^3_2}{c_1} \left( \frac{a u_x + b}{a} \right) \mu^2 - \frac{c_2}{c_1} \left( \frac{a u_x + b}{a} \right) \mu' - c_2 \left( \frac{au_x + b}{a} \right) \alpha'' u_y \\
\qquad{} + \frac{c^2_2}{c_1} \alpha' \mu \left( \frac{a u_x + b}{a} \right) = -\frac{c_2}{c_1 a} (a u_x + b) \mu' + \frac{\alpha''}{c_1}u_x.\end{gathered}$$ Recall that variables $u$, $u_x$, and $u_y$ are considered as independent. Hence, the above equation is equivalent to the system
\[zhiber101\] $$\begin{gathered}
c^2_2 \mu' - c_2 \alpha'' = 0,\\
-\frac{c^3_2}{c_1} \mu^2 - \frac{c_2}{c_1} \mu' + \frac{c^2_2}{c_1} \alpha' \mu = - \frac{c_2}{c_1} \mu' + \frac{\alpha''}{c_1}, \\
\frac{c^2_2 b}{a} \mu' - c_2 \frac{b}{a} \alpha'' =0,\\
- \frac{c^3_2}{c_1} \frac{b}{a} \mu^2 + \frac{c^2_2}{c_1} \alpha' \mu \frac{b}{a} = 0.\end{gathered}$$
If $b = 0$, we transform equation into $$-c_2 \mu' u_x u_y + \frac{c^2_2}{c_1} \mu^2 u_x + \frac{1}{c_1} \mu' u_x + \alpha'' u_x u_y - \frac{c_2}{c_1} \alpha' \mu u_x = F \left( c_1 u_y - \frac{1}{c_2} \ln (a u_x) + \alpha(u) \right).$$ Differentiating this equation with respect to $u_x$ we obtain $$\frac{c^2_2}{c_1} \mu^2 - c_2 \mu' u_y + \frac{1}{c_1} \mu' + \alpha'' u_y - \frac{c_2}{c_1} \alpha' \mu = -\frac{1}{c_2} F' \frac{1}{u_x}.$$ One can notice that these two equations imply $F + F'/c_2 = 0$ or $F(z) = c_3 \exp (-c_2 z)$. Consequently, we get $$-c_2 \mu' u_x u_y + \frac{c^2_2}{c_1} \mu^2 u_x + \frac{1}{c_1} \mu' u_x + \alpha'' u_x u_y - \frac{c_2}{c_1} \alpha' \mu u_x = c_3 \exp(-c_2 c_1 u_y) a u_x \exp(\alpha).$$ This equation is not realized because of the given assumptions $c_3 \neq 0$ and $a \neq 0$.
Now, it remains only to consider the case when $b \neq 0$. System takes the form $$c_2 \mu' - \alpha'' = 0, \qquad -c^3_2 \mu^2 + c^2_2 \alpha' \mu = \alpha'', \qquad -c_2 \mu^2 + \alpha' \mu = 0.$$ These equations imply that $\mu' = 0$, which contradicts the given assumptions of the lemma.
\[lemma4\] Suppose that condition holds but , , and do not. Then equations , , and take one of the following forms: $$\begin{aligned}
{6}
& u_{xy} = \frac{\mu(u) u_x}{\gamma'(u_y)}, \qquad && v_{xy} = 0, \qquad && v = \ln u_x + \gamma(u_y) + \alpha(u), & \label{zhiber102}
\intertext{where $c_3 + \frac{\gamma''}{\gamma'^2} + c_4 \gamma' u_y = 0$, $\alpha'' + \mu' + c_4 \mu^2 = 0$, and $c_3 \mu^2 + \mu' + \mu^2 + \alpha' \mu =0$;}
& u_{xy} = \frac{u_x}{(au + b) \gamma'(u_y)}, \qquad && v_{xy} = \exp v,\qquad &&&\nonumber\\
&&& v = \ln u_x + \gamma(u_y) - 2 \ln(au + b) + \ln(-c_5),\hspace*{-200mm} &&& \label{zhiber103}\end{aligned}$$ where $c_3 + \frac{\gamma''}{\gamma'^2} + c_4 \gamma' u_y = c_5 \gamma' \exp \gamma$, $c_3 + 1 -3a = 0,$ and $c_4 + 2a^2 - a = 0$, up to the point transformations $u \rightarrow \theta(u)$, $v \rightarrow \kappa(v)$, $x \rightarrow \xi x$, and $y \rightarrow \eta y$, where $\xi$ and $\eta$ are arbitrary constants. Here $c_3$, $c_4$ are arbitrary constants, $c_5 \neq 0$, and $(a, b) \neq (0, 0)$.
According to , the function $\beta$ is of the form $\beta = c_1 \ln u_x + c_2$. Without loss of generality, we may set $\beta = c_1 \ln u_x$. Substituting $\beta$ into equation we obtain $$\begin{gathered}
\frac{\alpha'\mu u_x}{c_1 \gamma'} - \frac{\mu^2 u_x}{c_1 \gamma'} \left( \frac{\gamma''}{\gamma'^2} - \frac{1}{c_1} \right) + \alpha'' u_x u_y + \mu' \left( \frac{u_x}{\gamma'} + \frac{u_x u_y}{c_1} \right) = F(\alpha + \beta + \gamma). \label{zhiber105}\end{gathered}$$ Applying the operator $\frac{\partial}{\partial u_x}$ to both sides of leads to $$\begin{gathered}
\frac{\alpha' \mu}{c_1 \gamma'} - \frac{\mu^2}{c_1 \gamma'} \left( \frac{\gamma''}{\gamma'^2} - \frac{1}{c_1} \right) + \alpha'' u_y + \mu' \left( \frac{1}{\gamma'} + \frac{u_y}{c_1} \right) = F' \left( \frac{c_1}{u_x} \right). \label{zhiber106}\end{gathered}$$ From equations and it follows that $F = F' c_1/u_x$, hence $F(z) = c_2 \exp(z / c_1)$. By substituting $F$ into equation we get $$u_x \left( \frac{\alpha' \mu}{c_1 \gamma'} - \frac{\mu^2}{c_1 \gamma'}\left( \frac{\gamma''}{\gamma'^2} - \frac{1}{c_1}\right) + \alpha'' u_y + \mu' \left( \frac{1}{\gamma'} + \frac{u_y}{c_1} \right) \right) = c_2 u_x \exp(\gamma / c_1) \exp(\alpha / c_1).$$ This equation can be written in the form $$\mu' c_1 + \alpha' \mu + \frac{\mu^2}{c_1} - \mu^2 \frac{\gamma''}{\gamma'^2} + (\alpha'' c_1 + \mu') \gamma' u_y = c_2 c_1 \gamma' \exp(\gamma / c_1) \exp(\alpha / c_1).$$ Having the fixed value of $u$ we can determine $\gamma$ as a solution of the ordinary differential equation $$c_3 + \frac{\gamma''}{\gamma'^2} + c_4 \gamma' u_y = c_1 c_5 \gamma' \exp(\gamma / c_1).$$ Moreover, based on this equation we get $$\begin{gathered}
\alpha' \mu + \frac{\mu^2}{c_1} + c_1 \mu' + c_3 \mu^2 + \gamma u_y \big(c_1 \alpha'' + \mu' + c_4 \mu^2\big)\\
\qquad{}- c_1 \gamma' \exp(\gamma / c_1) (c_5 \mu^2 + c_2 \exp(\alpha / c_1)) = 0.\end{gathered}$$ Note that if $u_y = \kappa \exp(\gamma / c_1)$ then $\gamma = c_1 \ln (u_y / \kappa)$ and $(\gamma' u_y)' = 0$. Since the last equation contradicts the assumption of the lemma, we obtain that $u_y$ and $\exp(\gamma / c_1)$ are linearly independent and that is why $$\begin{gathered}
c_1 \alpha'' + \mu' + c_4 \mu^2 = 0, \qquad c_5 \mu^2 + c_2 \exp(\alpha / c_1) = 0, \qquad c_3 \mu^2 + c_1 \mu' + \frac{\mu^2}{c_1} + \alpha' \mu = 0.\end{gathered}$$
In order to find equations , , and we first set $c_5 = 0$, hence $c_2 = 0$ and $$u_{xy} = \frac{\mu(u)}{\beta'(u_x)\gamma'(u_y)}, \qquad v_{xy} = 0, \qquad v = \beta(u_x) + \gamma(u_y) + \alpha(u),$$ where the functions $\beta$ and $\gamma$ are solutions of the ordinary differential equations $$\beta' = \frac{c_1}{u_x}, \qquad c_3 + \frac{\gamma''}{\gamma'^2} + c_4 \gamma' u_y = 0,$$ and the functions $\mu$ and $\alpha$ satisfy the equations $$c_1 \alpha'' + \mu' + c_4 \mu^2 = 0, \qquad c_3 \mu^2 + c_1 \mu' + \frac{\mu^2}{c_1} + \alpha' \mu = 0.$$ We use the transformation $v \rightarrow c_1 v$. Next, we redenote $\alpha / c_1$ by $\alpha$, $\gamma / c_1$ by $\gamma$, and $\mu / c^2_1 $ by $ \mu$. Finally, after replacing $c_4 c^2_1$ by $c_4$ and $c_1 c_3$ by $c_3$, is obtained.
If $c_5 \neq 0$ then we get $$u_{xy} = \frac{\mu(u)}{\beta'(u_x) \gamma'(u_y)}, \qquad v_{xy} = c_2 \exp(v / c_1), \qquad v = \beta(u_x) + \gamma(u_y) + \alpha(u),$$ where the functions $\beta$ and $\gamma$ are the solutions of the ordinary differential equations $$\beta' = \frac{c_1}{u_x}, \qquad c_3 + \frac{\gamma''}{\gamma'^2} + c_4 \gamma' u_y = c_1 c_5 \gamma' \exp(\gamma / c_1),$$ and the functions $\alpha$ and $\mu$ are given by the equations $$\begin{gathered}
\alpha = 2 c_1 \ln(-2c_1) - 2c_1 \ln \left( -\frac{2}{3} \sqrt{- \frac{c_2}{c_5}} \frac{c_3 c_1 + 1}{c_1} u + c_6 \right), \\
\mu = \sqrt{-\frac{c_2}{c_5}} \left( \frac{-2c_1}{-\frac{2}{3} \sqrt{-\frac{c_2}{c_5}} \left( \frac{c_3 c_1 + 1}{c_1} \right)u + c_6} \right),\\
\frac{2}{9} \left( \frac{c_3c_1 +1}{c_1} \right)^2 - \frac{1}{3} \left( \frac{c_3 c_1 + 1}{c^2_1} \right) + c_4 = 0.\end{gathered}$$ After point transformations we get .
\[lemma5\] Suppose that condition holds but – do not. Then equations , , and take one of the following forms: $$\begin{aligned}
{6}
& u_{xy} = - \frac{1}{u \beta'(u_x) \gamma'(u_y)}, \qquad && v_{xy} = 0, \qquad && v = \beta(u_x) + \gamma(u_y), & \label{zhiber108}
\intertext{where $\frac{\beta''}{\beta'^2} = u_x \beta' + c_1$, $\frac{\gamma''}{\gamma'^2} = u_y \gamma' - c_1$;}
& u_{xy} = \frac{\mu(u)}{\beta'(u_x) \gamma'(u_y)}, \qquad && v_{xy} = \exp v, \qquad && v = \beta(u_x) + \gamma(u_y) + \alpha(u),\hspace*{-50mm}& \label{zhiber109}
\intertext{where $ u_x + \frac{1}{\beta'(u_x)} = \exp (\beta)$, $u_y + \frac{1}{\gamma'(u_y)} = \exp \gamma$, $\alpha'' =\exp \alpha$, and $\mu = (\exp \alpha) / \alpha'$;}
& u_{xy} = \frac{\mu(u)}{\beta'(u_x) \gamma'(u_y)}, \qquad && v_{xy} = \exp v, \qquad && v = \beta(u_x) + \gamma(u_y) + \alpha(u),\hspace*{-50mm}& \label{zhiber110}
\intertext{where
$-c u_x + \frac{1}{\beta'(u_x)} = \exp \beta$, $-c u_y + \frac{1}{\gamma'(u_y)} = \exp \gamma$,
$\alpha' \mu + 2 \mu^2 (c+1) = \exp \alpha$, $\alpha'^2 = 2 c^2 \exp \alpha$, $c = - \frac{1}{2}, -2$;}
& u_{xy} = \frac{\mu(u)}{\beta'(u_x)\gamma'(u_y)}, \qquad && v_{xy} = \exp v + \exp (-v), \qquad && v = \beta(u_x) + \gamma(u_y) + \alpha(u),\hspace*{-50mm}& \label{zhiber111}
\intertext{where
$A_1 \exp \beta + B_1 \exp (-\beta) = u_x$, $A_2 \exp \gamma + B_2 \exp(-\gamma) = u_y$,
$\alpha'' = \frac{1}{4} \left( \frac{\exp(-\alpha)}{B_1 B_2} + \frac{\exp \alpha}{A_1 A_2} \right)$, $\mu = \frac{\alpha''}{\alpha'}$;}
& u_{xy} = \frac{\mu(u)}{\beta'(u_x) \gamma'(u_y)}, \qquad && v_{xy} = \exp v + \exp(-2 v), \qquad && v = \beta(u_x) + \gamma(u_y) + \alpha(u),\hspace*{-50mm}& \label{zhiber112}\end{aligned}$$ where $A_1 \exp \beta + B_1 \exp (-2 \beta) = u_x$, $A_2 \exp \gamma + B_2 \exp (-2 \gamma) = u_y$, $\alpha'^2 = \frac{2}{9} \left( \frac{4 \exp \alpha}{A_1 A_2} - \frac{1}{2} \frac{\exp(-2 \alpha)}{B_1 B_2} \right)$, $-2\mu^2 + \alpha' \mu - \frac{1}{9} \left( \frac{\exp \alpha}{ A_1 A_2} + \frac{\exp(-2\alpha)}{B_1 B_2} \right) = 0$, up to the point transformations $u \rightarrow \theta(u)$, $v \rightarrow \kappa(v)$, $x \rightarrow \xi x$, and $y \rightarrow \eta y$, where $\xi$ and $\eta$ are arbitrary constants. Here $A_1$, $A_2$, $B_1$, and $B_2$ are nonzero constants.
Considering that $u_x$ and $u_y$ are independent variables, equation yields $$\frac{ (u_x\beta')' }{\left( \dfrac{\beta''}{\beta'^2} \right)'} = c, \qquad \frac{ (u_y\gamma')' }{ \left( \dfrac{\gamma''}{\gamma'^2} \right)' } = c, \qquad c \neq 0.$$ Integrating these equations we obtain $$\begin{gathered}
\frac{\beta''}{\beta'^2} = c u_x \beta' + c_1, \qquad \frac{\gamma''}{\gamma'^2} = c u_y \gamma' + c_2. \label{zhiber113}\end{gathered}$$ According to , equation is rewritten in the form $$\begin{gathered}
\label{zhiber114}
\frac{1}{\beta'} \left(\! \frac{\alpha' \mu}{ \gamma'} - \frac{\mu^2}{\gamma'} \big(c_1 + c u_y \gamma' + c_2\big) + \mu' u_y \right)
+ u_x \!\left(\! -\frac{c \mu^2}{\gamma'} + \alpha'' u_y + \frac{\mu'}{\gamma'} \right) = F(\alpha + \beta + \gamma).\!\!\!\!\end{gathered}$$ Having fixed values of $u$ and $u_y$ we can define that $F(\beta + c_3) = c_4 u_x + c_5 / \beta'$. Without loss of generality, we redenote $\beta + c_3$ by $\beta$, therefore $$\begin{gathered}
F(\beta) = c_4 u_x + \frac{c_5}{\beta'}. \label{zhiber115}\end{gathered}$$ Applying the operator $\frac{\partial}{\partial u_x}$ to both sides of equation and using we obtain $$F'(\beta) = -cc_5 u_x + \frac{c_4 - c_1 c_5}{\beta'}.$$ We differentiate this equation with respect to $u_x$, $$F''(\beta) = -c(c_4 - c_1 c_5) u_x - \frac{cc_5 +c_1 (c_4 - c_1 c_5)}{\beta'}.$$ The above three equations allow us to establish that the function $F$ satisfies the ordinary differential equation $$\begin{gathered}
F'' = c_7 F' + c_8 F. \label{zhiber116}\end{gathered}$$ Equation possesses two families of solutions $$F(v) = A_1 \exp(\sigma_1 v) + B_1 \exp(\sigma_2 v), \qquad \sigma_1 \neq \sigma_2,$$ and $$F(v) = (A_2 + B_2 v) \exp(\sigma v).$$ Setting definite values of the constants $A_i$, $B_i$, where $i=1,2$, we obtain that the function $F$ can take only one of the following forms $$\begin{gathered}
F(v) = 0, \label{zhiber117}\\
F(v) = 1, \label{zhiber118}\\
F(v) = v, \label{zhiber119}\\
F(v) = v \exp v, \label{zhiber120}\\
F(v) = \exp v, \label{zhiber121}\\
F(v) = \exp v + 1, \label{zhiber122}\\
F(v) = \exp v + \exp (\sigma v). \label{zhiber123}\end{gathered}$$ From equation by setting different values of $u$ and $u_y$ we obtain a set of equations $$\begin{gathered}
\alpha_i u_x + \frac{\beta_i}{\beta'(u_x)} = F\left( \beta(u_x) + \gamma_i \right). \label{zhiber124}\end{gathered}$$ Here $\alpha_i$, $\beta_i$, and $\gamma_i$ are constants, $i= 1, 2, \dots, n$. Thus, we will focus on .
Let us assume that $(\alpha_i, \beta_i)$ are linearly dependent vectors. This means that a set of numbers $\mu_i$ satisfying $$(\alpha_i, \beta_i) = \mu_i (\alpha_1, \beta_1), \qquad \mu_1 = 1,$$ exists. Using this equation we rewrite as $$\begin{gathered}
\mu_i \left( \alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} \right) = F(\beta + \gamma_i). \label{zhiber125}\end{gathered}$$ Now, we will deal with equations –.
We begin with . In this case we have $$\begin{gathered}
\mu_i \left( \alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} \right) = 0 \label{zhiber126}\end{gathered}$$ from the equation . Suppose that $\alpha_1 = \beta_1 = 0$. In equation , we find $$\begin{gathered}
\mu' - c \mu^2 + \alpha'' u_y \gamma' = 0, \qquad \alpha' \mu - \mu^2(c_1 + c_2 + c u_y \gamma') + \mu' u_y \gamma' = 0. \label{zhiber127}\end{gathered}$$
If $\alpha'' = 0$ then $\alpha = \epsilon u + \delta$, hence from we have $$\mu(u) = - \frac{1}{cu + \kappa}, \qquad
\frac{\epsilon}{cu + \kappa} + \frac{c_1 + c_2}{(cu + \kappa)^2} = 0.$$ Clearly, the last equation requires $\epsilon =0$ and $c_2 = - c_1$. Thus, we determine equations , , and as follows $$u_{xy} = \frac{\mu(u)}{\beta'(u_x) \gamma'(u_y)}, \qquad v_{xy} = 0,\qquad v = \beta(u_x) + \gamma(u_y) + \alpha(u),$$ where $$\begin{gathered}
\mu(u) = - \frac{1}{c u + \kappa}, \qquad \alpha(u) = \delta, \qquad \frac{\beta''}{\beta'^2} = c u_x \beta' + c_1, \qquad \frac{\gamma''}{\gamma'^2} = c u_y \gamma' - c_1.\end{gathered}$$ We replace $\beta$ by $a \beta$, $ \gamma$ by $a \gamma$. Take the constant $a$ so that $a^2 c \rightarrow 1$. Using the transformations $u + \kappa / c \rightarrow u$, $v - \delta \rightarrow av$ and redenoting $a c_1 \rightarrow c_1$ obtain equation .
Now, assume that $\alpha'' \neq 0$. The equation $$u_y \gamma'(u_y) = \frac{c \mu^2 - \mu'}{\alpha''}$$ arises from . Since $u$ and $u_y$ are regarded as independent variables, the last equation leads to $u_y \gamma'(u_y) = \kappa$, where $\kappa$ is a constant. This contradicts the assumption of the lemma. Consider the case where $\alpha_1 \beta_1 \neq 0$. We have the equation $\beta'(u_x) = - \beta_1 / (\alpha_1 u_x)$ which results from , and it contradicts the assumptions of the lemma.
Let us discuss the case where $F$ is determined by . Rewriting we have $$\mu_i \left( \alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} \right) = 1.$$ This equation must be true for every $i=1, 2, \dots$. This requirement implies that $\mu_i = 1$, $\alpha_i = \alpha_1$, and $\beta_i = \beta_1$ for every $i$. Taking this into account we define $\beta'$ as follows: $$\begin{gathered}
\beta'(u_x) = \frac{\beta_1}{1 - \alpha_1 u_x}. \label{zhiber128}\end{gathered}$$ Rewriting by using we see that this case is not realized.
Now, we assume that $F$ is described by . Equations , are presented in the forms $$\alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} = \beta(u_x) + \gamma_1, \qquad \mu_i \left( \alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} \right) = \beta(u_x) + \gamma_i.$$ Consequently, $$\beta(u_x) (\mu_i - 1) + \gamma_1 \mu_i - \gamma_i = 0.$$ It is clear that $\mu_i = 1$, $\gamma_i = \gamma_1$. Hence, $\alpha_i = \alpha_1$, $\beta_i = \beta_1$ for every $i$. So we have $$\beta' = \frac{\beta_1}{\beta(u_x) - \alpha_1 u_x + \gamma_1}.$$ Trying to simplify by using this equation gives a contradiction to the assumption of the lemma. Concentrate on the case when $F$ satisfies . We can rewrite equations , as $$\alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} = (\beta + \gamma_1) \exp (\beta + \gamma_1), \qquad \mu_i \left( \alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} \right) = (\beta + \gamma_i) \exp(\beta + \gamma_i).$$ Comparing these equations we conclude that $$\left( \beta (\exp \gamma_i - \mu_i \exp \gamma_1) + \gamma_i \exp \gamma_i - \mu_i \gamma_1 \exp \gamma_1 \right)\exp \beta = 0.$$ Recall that $\beta$ depends on the variable $u_x$, while the remaining terms of the above equations are constants. Hence, we have $$\exp \gamma_i - \mu_i \exp \gamma_1 = 0, \qquad \gamma_i \exp \gamma_i - \mu_i \gamma_1 \exp \gamma_1 = 0.$$ From these equations we obtain $\gamma_i \exp \gamma_i - \gamma_1 \exp \gamma_i = 0$, hence $\gamma_i = \gamma_1$ for all $i$. By we determine that $\alpha(u) + \gamma(u_y) = \gamma_1$, where $\gamma_1$ is an arbitrary constant. This equation contradicts $\gamma_{u_y} \neq 0$.
Let the function $F$ be defined by . From we obtain $$\begin{gathered}
\alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} = \exp(\beta + \gamma_1). \label{zhiber129}\end{gathered}$$ Note that $\beta_1 \neq 0$, otherwise $(\beta' u_x)' = 0$. Redenoting $\beta + \gamma_1$ by $\beta$ we rewrite equation in the form $$\begin{gathered}
\alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} = \exp \beta. \label{zhiber130}\end{gathered}$$ From equations and we find that $c = -\alpha_1 / \beta_1$, $c_1 = -1-c$. Now, we rewrite equation based on equation $$\begin{gathered}
\frac{1}{\beta'} \exp \beta \left( \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'} \big(c_1 + c u_y \gamma' + c_2\big) + \mu' u_y \right) + u_x \left( -\frac{c \mu^2}{\gamma'} + \alpha'' u_y + \frac{\mu'}{\gamma'} \right) \\
\qquad{} - \frac{\alpha_1}{\beta_1} u_x \left( \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'}\big(c_1 + c u_y \gamma' + c_2\big) + \mu' u_y \right)
= \exp(\alpha + \gamma) \exp \beta.\end{gathered}$$ Since $(\beta' u_x)' \neq 0$, $\exp \beta$ and $u_x$ are linearly independent, the above equation is equivalent to the system $$\begin{gathered}
\frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'} \big(c_1 + c u_y \gamma' + c_2\big) + \mu' u_y = \exp(\alpha + \gamma) \beta_1,\\
-\frac{\alpha_1}{\beta_1} \left( \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'}\big(c_1 + c u_y \gamma' + c_2\big) + \mu' u_y \right) + \left( -\frac{c \mu^2}{\gamma'} + \alpha'' u_y + \frac{\mu'}{\gamma'} \right) = 0.\end{gathered}$$ Hence, we get $$\begin{gathered}
{\displaystyle u_{xy} = \frac{\mu(u)}{\beta'(u_x)\gamma'(u_y)}, \qquad v_{xy} = \exp v, \qquad v = \beta(u_x) + \gamma(u_y) + \alpha(u),}\label{zhiber131}\end{gathered}$$ where $$\begin{gathered}
{\displaystyle \alpha_1 u_x + \frac{\beta_1}{\beta'} = \exp \beta, \qquad \frac{\beta''}{\beta'^2} = c u_x \beta' + c_1, \qquad c_1 = - 1 -c, \qquad c \beta_1 = - \alpha_1}, \nonumber \\
{\displaystyle \frac{\gamma''}{\gamma'^2} = c u_y \gamma' + c_2, \qquad \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'} (c u_y \gamma' + c_1 + c_2) + \mu' u_y = \exp(\alpha + \gamma) \beta,} \nonumber \\
{\displaystyle -\alpha_1 \exp(\alpha + \gamma) + \alpha'' u_y + \frac{\mu' - c \mu^2}{\gamma'} = 0.}\end{gathered}$$
Now, consider case . Equations and can be rewritten in the forms $$\alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} = \exp(\beta + \gamma_1) + 1, \qquad \mu_i \left( \alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} \right) = \exp(\beta + \gamma_i) + 1.$$ It is not hard to show that $$\exp \beta \left( \mu_i \exp \gamma_1 - \exp \gamma_i \right) + \mu_i - 1 = 0.$$ The dependence of $\beta$ only on the variable $u_x$ implies that $\mu_i = 1$ and $\gamma_i = \gamma_1$ for every $i$. This gives $\alpha(u) + \gamma(u_y) = \gamma_1$, where $\gamma_1$ is a constant, which contradicts the assumption $\gamma_{u_y} \neq 0$.
It remains to consider the case when $F$ is given by to complete the analysis in the case when $(\alpha_i, \beta_i)$ are linearly dependent vectors. Using we transform equations and into $$\begin{gathered}
\alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} = \exp(\beta + \gamma_1) + \exp (\sigma(\beta + \gamma_1)),\\
\mu_i \left( \alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} \right) = \exp(\beta + \gamma_i) + \exp (\sigma (\beta + \gamma_i)).\end{gathered}$$ Consequently, we get $$\exp \beta \left( \mu_i \exp \gamma_1 - \exp \gamma_i \right) + \exp(\sigma \beta) \left( \mu_i \exp(\sigma \gamma_1) - \exp(\sigma \gamma_i) \right) = 0.$$ Recall that $\sigma \neq 1$. Collecting coefficients at $\exp \beta$ and $\exp( \sigma \beta) $ yields $$\mu_i \exp \gamma_1 = \exp \gamma_i, \qquad \mu_i \exp(\sigma \gamma_1) = \exp (\sigma \gamma_i).$$ The above equations provide $\mu_i \exp(\sigma \gamma_1) (\mu^{\sigma - 1}_i - 1) = 0$, hence $\mu_i = 1$. It follows that $\gamma_i = \gamma_1$ for every $i$. By we find that $\alpha(u) + \gamma(u_y) = \gamma_1$. This equation contradicts $\gamma_{u_y} \neq 0$.
Now, we must deal with the case when $\alpha_i$, $\beta_i$, $i=1,2$, satisfying $\alpha_1 \beta_2 - \beta_1 \alpha_2 \neq 0$ exist. Setting definite values of $u$, $u_y$ in $\eqref{zhiber114}$ we obtain the system $$\alpha_1 u_x + \frac{\beta_1}{\beta'(u_x)} = F(\beta(u_x) + \gamma_1), \qquad \alpha_2 u_x + \frac{\beta_2}{\beta'(u_x)} = F (\beta(u_x) + \gamma_2).$$ Because of the given assumption $(u_x \beta')_{u_x} \neq 0$ we get $$\begin{gathered}
\kappa_1 F(\beta + \gamma_1) - \kappa_2 F(\beta + \gamma_2) = u_x, \qquad \kappa_3 F(\beta + \gamma_1) - \kappa_4 F(\beta + \gamma_2) = \frac{1}{\beta'}. \label{zhiber132}\end{gathered}$$ We use $$\begin{gathered}
\kappa_1 = \frac{\beta_2}{\alpha_1 \beta_2\! - \alpha_2 \beta_1}, \qquad \kappa_2 = \frac{\beta_1}{\alpha_1 \beta_2\! - \alpha_2 \beta_1}, \qquad \kappa_3 = \frac{\alpha_2}{\beta_1 \alpha_2\! - \beta_2 \alpha_1}, \qquad \kappa_4 = \frac{\alpha_1}{\beta_1 \alpha_2\! - \beta_2 \alpha_1}.\end{gathered}$$
Let us analyze equation taking into account conditions –.
Consider the case when $F$ is given by . It is not hard to show that equation implies $u_x = 0$. Thus, this case is not realized. Next, based on we obtain that $u_x$ is a constant. So it is also not possible.
If is true then system can be written as follows $$\kappa_1 (\beta + \gamma_1) - \kappa_2 (\beta + \gamma_2) = u_x, \qquad \kappa_3 (\beta + \gamma_1) - \kappa_4 (\beta + \gamma_2) = \frac{1}{\beta'}.$$ It is not hard to verify that $$\beta'(\kappa_1 - \kappa_2) = 1, \qquad \beta (\kappa_3 - \kappa_4) + \gamma_1 \kappa_3 - \gamma_2 \kappa_4 = \kappa_1 - \kappa_2.$$ Note that we used the properties $\kappa_1 - \kappa_2 \neq 0$, $\kappa_1 - \kappa_2 \neq 0$, which result from $\alpha_1 \beta_2 - \beta_1 \alpha_2 \neq 0$. Further, since $\kappa_3 - \kappa_4 \neq 0$, $\beta$ is a constant. This contradicts $\beta_{u_x} \neq 0$.
Let us discuss the case when the function $F$ is defined by . Rewriting we get $$\begin{gathered}
\kappa_1 (\beta + \gamma_1) \exp(\beta + \gamma_1) - \kappa_2 (\beta + \gamma_2) \exp(\beta + \gamma_2) = u_x, \\
\kappa_3(\beta + \gamma_1) \exp(\beta + \gamma_1) - \kappa_4 (\beta + \gamma_2) \exp(\beta + \gamma_2) = \frac{1}{\beta'}.\end{gathered}$$ Setting $A = \kappa_1 \exp \gamma_1 - \kappa_2 \exp \gamma_2$ and $B = \kappa_1 \gamma_1 \exp \gamma_1 - \kappa_2 \gamma_2 \exp \gamma_2$ we obtain $$\begin{gathered}
u_x = A \beta \exp \beta + B \exp \beta. \label{zhiber133}\end{gathered}$$ It is not difficult to determine that equations , lead to $$\begin{gathered}
(A\!+\!B)\!\left( \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'}\big(c_1\!+\!c u_y \gamma'\!+\!c_2\big) + \mu' u_y \right) +B\!\left( \alpha'' u_y + \frac{\mu'\!-\!c\mu^2}{\gamma'} \right) = (\alpha\!+\!\gamma)\exp(\alpha + \gamma),
\\
A \left( \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'} \big(c_1 + c u_y \gamma' + c_2\big) + \mu' u_y + \alpha'' u_y + \frac{\mu' - c \mu^2}{\gamma'} \right) = \exp(\alpha + \gamma).\end{gathered}$$ Rewriting by using we find that $c = 1$, $c_1 = -2$. Thus, we obtain the equations $$\begin{gathered}
\label{zhiber134}
{\displaystyle u_{xy} = \frac{\mu(u)}{\beta'(u_x)\gamma'(u_y)}, \qquad v_{xy} = v \exp v, \qquad v = \alpha(u) + \beta(u_x) + \gamma(u_y),}\end{gathered}$$ herewith $$\begin{gathered}
{(A\!+\!B)\!\left( \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'}(u_y \gamma' -2 + c_2) + \mu' u_y \right) + B\!\left( \alpha'' u_y + \frac{\mu' - \mu^2}{\gamma'} \right) = (\alpha + \gamma) \exp(\alpha + \gamma),} \nonumber \\
{A \left( \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'} (u_y \gamma' -2 + c_2) + \mu' u_y + \alpha'' u_y + \frac{\mu' - \mu^2}{\gamma'}\right) = \exp(\alpha + \gamma),} \nonumber \\
{u_x = A \beta \exp \beta + B \exp \beta, \qquad \frac{\beta''}{\beta'^2} = u_x \beta' -2,\frac{\gamma''}{\gamma'^2} = u_y \gamma' + c_2.}\end{gathered}$$ Note that case yields the equations .
Next, assume that the function $F$ is defined by . Hence, we write as $$\begin{gathered}
\kappa_1 \left( \exp(\beta + \gamma_1) + 1 \right) - \kappa_2 \left( \exp(\beta + \gamma_2) + 1 \right) = u_x,\\
\kappa_3 \left( \exp(\beta + \gamma_1) +1 \right) - \kappa_4 \left( \exp(\beta + \gamma_2) + 1 \right) = \frac{1}{\beta'}.\end{gathered}$$ Eliminating $\beta'$ from the last equation we get $$\exp \beta(\kappa_3 \exp \gamma_1 - \kappa_4 \exp \gamma_2 - \kappa_1 \exp \gamma_1 + \kappa_2 \exp \gamma_2) + \kappa_3 - \kappa_4 = 0.$$ It is easy to show from this equation that $\beta$ is a constant. This contradicts $\beta_{u_x} \neq 0$.
Assuming that holds, we can write as $$\begin{gathered}
\exp \beta(\kappa_1 \exp \gamma_1 - \kappa_2 \exp \gamma_2) + \exp (\sigma \beta) (\kappa_1 \exp( \sigma \gamma_1) - \kappa_2 \exp (\sigma \gamma_2)) = u_x,
\nonumber \\
\exp \beta (\kappa_3 \exp \gamma_1 - \kappa_4 \exp \gamma_2) + \exp (\sigma \beta) (\kappa_3 \exp (\sigma \gamma_1) - \kappa_4 \exp (\sigma \gamma_2)) = \frac{1}{\beta'}.\label{zhiber135}\end{gathered}$$ And further, from based on we obtain $$\begin{gathered}
(1 + c + c_1)(\kappa_1 \exp \gamma_1 - \kappa_2 \exp \gamma_2) \exp \beta \nonumber\\
\qquad{}+\big(\sigma^2 + c + c_1 \sigma\big)\big(\kappa_1 \exp (\sigma \gamma_1) - \kappa_2 \exp (\sigma \gamma_2)\big) \exp \sigma \beta = 0.\label{zhiber136}\end{gathered}$$ From using again we get $$\begin{gathered}
(\kappa_1 \exp \gamma_1 - \kappa_2 \exp \gamma_2) \left( \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'}(c u_y \gamma' + c_1 + c_2) \right. \nonumber\\
\left.\qquad{} + \mu' u_y + \frac{\mu' - c \mu^2}{\gamma'} + \alpha'' u_y \right) = \exp(\alpha + \gamma),\label{zhiber137_0}
\\
(\kappa_1 \exp \sigma \gamma_1 - \kappa_2 \exp \sigma \gamma_2) \left( \sigma \left( \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'}(c u_y \gamma' + c_1 + c_2) + \mu' u_y \right) \right.\nonumber\\
\left. \qquad{}+ \frac{\mu' - c \mu^2}{\gamma'} + \alpha'' u_y \right) = \exp\sigma(\alpha + \gamma). \label{zhiber137_1}\end{gathered}$$ Note that if $\kappa_1 \exp (\sigma \gamma_1) - \kappa_2 \exp (\sigma \gamma_2) = 0$ then equations and imply that $\exp \sigma(\alpha + \gamma) = 0$. Consequently, the equalities $1+ c + c_1 = 0$ and $\sigma^2 + c_1 \sigma + c = 0$ arise from equation . The solution of the last equation is found as $\sigma = c$, where $c = -1 - c_1$. Thus, denoting $A = \kappa_1 \exp \gamma_1 - \kappa_2 \exp \gamma_2$, $B = \kappa_1 \exp (\sigma \gamma_1) - \kappa_2 \exp( \sigma \gamma_2)$ we obtain $$\begin{gathered}
\label{zhiber138}
{\displaystyle u_{xy} = \frac{\mu(u)}{\beta'(u_x) \gamma'(u_y)}, \qquad v_{xy} = \exp v + \exp (\sigma v), \qquad v = \alpha(u) + \beta(u_x) + \gamma(u_y),}\end{gathered}$$ where $$\begin{gathered}
A \exp \beta + B \exp (\sigma \beta) = u_x, \qquad \frac{\beta''}{\beta'^2} = \sigma u_x \beta' -1-\sigma, \qquad \frac{\gamma''}{\gamma'^2} = \sigma u_y \gamma' + c_2 , \nonumber \\
A \left( \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'}\big(\sigma u_y \gamma'+c_2-1\big) + \mu' u_y + \frac{\mu' }{\gamma'} + \alpha'' u_y \right) = \exp(\alpha + \gamma), \nonumber \\
B \left( \sigma \left( \frac{\alpha' \mu}{\gamma'} - \frac{\mu^2}{\gamma'}\big(\sigma u_y \gamma' + c_2 - \sigma\big) + \mu' u_y \right) +
\frac{\mu'}{\gamma'} + \alpha'' u_y \right) = \exp\sigma(\alpha + \gamma).\end{gathered}$$
Let us discuss the results obtained. We should analyze the equations and conditions for the parameters found in cases – and use the fact that functions and are invariant under the permutation of $\beta(u_x)$ and $\gamma(u_y)$.
In case we obtained . By interchanging $\beta(u_x)$ and $\gamma(u_y)$ we get $$\begin{gathered}
\alpha_2 u_y + \frac{\beta_2}{\gamma'} = \exp \gamma, \qquad \frac{\gamma''}{\gamma'^2} = c u_y \gamma' + c_2, \qquad c_2 = -1-c, \qquad c \beta_2 = - \alpha_2,\nonumber\\
\frac{\alpha' \mu}{\beta'} - \frac{\mu^2}{\beta'} (c u_x \beta' + c_1 + c_2) + \mu' u_x = \exp (\alpha + \beta) \beta_2, \nonumber\\
-\alpha_2 \exp(\alpha+ \beta) + \alpha'' u_x + \frac{\mu' - c \mu^2}{\beta'} = 0. \label{zhiber139}\end{gathered}$$ We substitute $\gamma$ satisfying the conditions for the parameters listed for equation into . At the same time we substitute $\beta$ satisfying the conditions for the parameters listed for equation into . As a result, we obtain the system $$\begin{gathered}
\frac{1}{\beta_2} (\exp \gamma - \alpha_2 u_y)\big(\alpha' \mu + 2 \mu^2 (1+c)\big) + \mu' u_y = \exp(\alpha+ \gamma) \beta_1,\\
-\alpha_1 \exp(\alpha+\gamma) + \alpha'' u_y + \big(\mu' - c \mu^2\big) \frac{1}{\beta_2}(\exp \gamma - \alpha_2 u_y) = 0,\\
\frac{1}{\beta_1}(\exp \beta - \alpha_1 u_x)\big(\alpha' \mu + 2 \mu^2(1+c)\big) + \mu' u_x = \exp(\alpha + \beta) \beta_2,\\
- \alpha_2 \exp(\alpha+ \beta) + \alpha'' u_x + \big(\mu' - c \mu^2\big) \frac{1}{\beta_1}(\exp \beta - \alpha_1 u_x) = 0.\end{gathered}$$ Since $\exp \gamma$ and $u_y$, $\exp \beta$ and $u_x$ are independent, equations , , and take the following forms: $$u_{xy} = \frac{\mu(u)}{\beta'(u_x)\gamma'(u_y)}, \qquad v_{xy} = \exp v, \qquad v = \alpha(u) + \beta(u_x) + \gamma(u_y),$$ where $\alpha$ and $\beta$ are solutions of the ordinary differential equations $$\alpha_1 u_x + \frac{\beta_1}{\beta'} = \exp \beta, \qquad \alpha_2 u_y + \frac{\beta_2}{\gamma'(u_y)} = \exp \gamma, \qquad -\frac{\alpha_1}{\beta_1} = - \frac{\alpha_2}{\beta_2} = c,$$ and the functions $\mu$ and $\alpha$ satisfy $$\begin{gathered}
\alpha' \mu + 2 \mu^2 (c + 1) = \beta_1 \beta_2 \exp \alpha, \qquad c \beta_1 \beta_2 \exp \alpha + \mu' - c \mu^2 = 0, \qquad \alpha'' + c(\mu' - c \mu^2) = 0.\end{gathered}$$ Analyzing the last system we obtain cases , . It is easy to verify that case is not possible.
Based on we get . Interchanging $\beta(u_x)$ and $\gamma(u_y)$ implies $$\begin{gathered}
\label{zhiber141}
\begin{split}
&{A_2 \exp \gamma + B_2 \exp \sigma \gamma = u_y, \qquad \frac{\gamma''}{\gamma'^2} = \sigma u_y \gamma' + c_2, \qquad c_2 = -1-c,}\\
&{\displaystyle \frac{A_2}{\beta'}\left( \alpha' \mu - \mu^2(c_1 - 1) + \mu' \right) + A_2 u_x \left( -\sigma \mu^2 + \mu' + \alpha'' \right) = \exp(\alpha + \beta),}\\
&{\displaystyle \frac{B_2}{\beta'}\left( \sigma(\alpha' \mu - \mu^2 (c_2 - \sigma)) + \mu' \right) +u_x B_2 \left( \sigma( -\sigma \mu^2 + \mu') + \alpha'' \right) = \exp\sigma(\alpha+\beta).}
\end{split}\end{gathered}$$ Similarly, we substitute $\beta$ satisfying the conditions for the parameters listed for equation into and obtain $$\begin{gathered}
(A \exp \beta + B \sigma \exp \sigma \beta)A_2(\alpha' \mu + \mu^2(2+\sigma) + \mu') \\
\qquad{} +(A \exp \beta + B \exp \sigma \beta)A_2(\mu' - \sigma \mu^2 + \alpha'') = \exp(\alpha + \beta),
\\
(A \exp \beta + B \sigma \exp(\sigma \beta))B_2 \left( \sigma(\alpha' \mu + \mu^2(1 + 2\sigma) + \mu') \right) \\
\qquad{} +(A \exp \beta + B \exp \sigma \beta) B_2 \left( \sigma(\mu' - \sigma \mu^2) + \alpha'' \right) = \exp \sigma(\alpha + \beta).\end{gathered}$$ Taking into account the fact that $\exp \beta$, $\exp \sigma \beta$ are independent, we get $$\begin{gathered}
A A_2 \left( \alpha' \mu + \mu^2(2 + \sigma) + 2 \mu' - \sigma \mu^2 + \alpha'' \right) = \exp \alpha, \\
\sigma \alpha' \mu + \sigma (\sigma + 1) \mu^2 + (\sigma + 1)\mu' + \alpha'' = 0, \\
B B_2 \left( \sigma^2 \alpha' \mu + 2 \sigma^3 \mu^2 + 2 \sigma \mu' + \alpha'' \right) = \exp (\sigma \alpha).\end{gathered}$$ Solving the above system we obtain cases and .
Case $\boldsymbol{\varphi = c \ln u_x + q(u, u_y)}$
---------------------------------------------------
We have the following statement in this case.
\[lemma6\] Suppose that is satisfied. Then equations , , and take the following forms $$\begin{gathered}
\label{zhiber143}
u_{xy} = \frac{\mu(u) - q_u(u,u_y)}{q_{u_y}(u,u_y)}u_x, \qquad v_{xy} = c_2 \exp v, \qquad v = \ln u_x + q(u,u_y),\end{gathered}$$ where $$\frac{\mu - q_u}{q_{u_y}}\left( \mu - \frac{\mu - q_u}{q^2_{u_y}} q_{u_y u_y} - 2\frac{q_{u u_y}}{q_{u_y}} \right) + \frac{\mu'}{q_{u_y}} - \frac{q_{uu}}{q_{u_y}} + \mu' u_y = c_2 \exp q, \qquad q_{uu_y} \neq 0,$$ up to the point transformations $u \rightarrow \theta(u)$, $v \rightarrow \kappa(v)$, $x \rightarrow \xi x$, and $y \rightarrow \eta y$, where $\xi$ and $\eta$ are arbitrary constants.
Substituting function into equation we obtain $$A(u,u_y) q_{u_y}(u,u_y) - q_u(u, u_y) q_{u_y}(u, u_y) u_y + c q_u(u, u_y) = B(u,u_x)\frac{c}{u_x}.$$ Recall that $u_x$, $u_y$ are considered as independent variables. Hence, the above equation is equivalent to the system $$\displaystyle A q_{u_y} - q_u q_{u_y} u_y + c q_u = \mu(u), \qquad \frac{B c}{u_x} = \mu(u).$$ From these equations we find the functions $A$ and $B$, $$B = \frac{\mu u_x}{c}, \qquad A = \frac{\mu + q_u q_{u_y} u_y - c q_u}{q_{u_y}}.$$ By using these equations in each of equations , we determine the function $f$ of equation as $$f = \frac{\mu - c q_u}{c q_{u_y}} u_x.$$ Substituting the functions and $f$ into we have $$u_x \left(
\frac{\mu - cq_u}{cq_{u_y}}\left( \frac{\mu}{c} - \frac{\mu - cq_u}{q^2_{u_y}} q_{u_y u_y} - 2c\frac{q_{u u_y}}{q_{u_y}} \right) + \frac{\mu'}{q_{u_y}} - \frac{q_{uu}}{q_{u_y}} + \frac{\mu' u_y}{c} \right) = F(c \ln u_x + q).$$ It is not difficult to prove by differentiating this equation with respect to $u_x$ that $c F' = F$. Consequently, $
F(z) = c_2 \exp(z/c).
$ Here $c_2$ is an arbitrary constant. Thus, equations , , and are of the forms $$u_{xy} = \frac{\mu(u) - c q_u(u,u_y)}{c q_{u_y}(u,u_y)} u_x, \qquad v_{xy} = c_2 \exp(v/c), \qquad v = c \ln u_x + q(u,u_y),$$ where $$\frac{\mu - cq_u}{cq_{u_y}}\left( \frac{\mu}{c} - \frac{\mu - cq_u}{q^2_{u_y}} q_{u_y u_y} - 2c\frac{q_{u u_y}}{q_{u_y}} \right ) + \frac{\mu'}{q_{u_y}} - c\frac{q_{uu}}{q_{u_y}} + \frac{\mu' u_y}{c} = c_2 \exp (q/c).$$ Finally, the transformations $v \rightarrow c v$, $q \rightarrow c q$, $\mu \rightarrow c^2 \mu$, and $c_2 / c \rightarrow c_2$ transform these equations into .
Case $\boldsymbol{\varphi = \alpha(u) + \kappa(u) \ln u_x + \mu(u) \ln u_y}$
----------------------------------------------------------------------------
By substituting into we obtain $$\begin{gathered}
\left( A(u,u_y) -(\kappa'(u)\ln u_x + \mu'(u) \ln u_y + \alpha'(u)) u_y \right) \frac{\mu(u)}{u_y} \\
\qquad{} = \left( B(u, u_y) - (\kappa'(u)\ln u_x + \mu'(u) \ln u_y + \alpha'(u)) u_x \right) \frac{\kappa(u)}{u_x},\end{gathered}$$ which can be written as $$\begin{gathered}
\frac{B(u,u_x) \kappa(u)}{u_x} + \left( \kappa'(u) \ln u_x + \alpha'(u) \right) \left( \mu(u) - \kappa(u) \right) \\
\qquad{}= \frac{A(u, u_y) \mu(u)}{u_y} - \mu'(u) \ln u_y \left( \mu(u) - \kappa(u) \right).\end{gathered}$$ Since $u_x$ and $u_y$ are regarded as independent variables, the above equation is equivalent to the system $$\begin{gathered}
\frac{B(u,u_x) \kappa(u)}{u_x} + \left( \kappa'(u) \ln u_x + \alpha'(u) \right) \left( \mu(u) - \kappa(u) \right) = \lambda(u), \\
\frac{A(u, u_y) \mu(u)}{u_y} - \mu'(u) \ln u_y \left( \mu(u) - \kappa(u) \right) = \lambda(u).\end{gathered}$$ The formulae $$B = \frac{\left( \lambda - (\mu - \kappa)(\kappa' \ln u_x + \alpha') \right)u_x}{\kappa}, \qquad A= \frac{\left( \lambda + \mu' (\mu - \kappa) \ln u_y \right) u_y}{\mu}$$ thereby immediately follow. Substituting $A$ and $B$ into equations and we find $f$, $$\begin{gathered}
f = \frac{\lambda - \kappa \mu' \ln u_y - \mu \kappa' \ln u_x - \mu \alpha'}{\kappa \mu} u_x u_y. \label{zhiber147}\end{gathered}$$ We apply the operator $\frac{\partial}{\partial u_x}$ to both sides of equation and use the equations obtained. So we get $F' \kappa = F$, while applying $\frac{\partial}{\partial u_y}$ implies $F' \mu = F$. This requires $\mu(u) = \kappa(u) = c$. Thus $\varphi$ takes the form $\varphi = \alpha(u) + c \ln( u_x u_y)$, and case is reduced to case considered earlier.
Theorem \[theorem1\] follows from Lemmas \[lemma1\]–\[lemma6\].
Differential substitutions of the form $\boldsymbol{u=\psi(v,v_x,v_y)}$ {#section4}
=======================================================================
In this section we consider the problem which is, in a sense, inverse to the original problem. The aim is to describe equations of form which are transformed into equations of form by differential substitutions .
\[theorem2\] Suppose that equation is transformed into equation by differential substitution . Then equations , and substitution take one of the following forms: $$\begin{aligned}
{6}
& v_{xy} = v, \qquad && u_{xy} = u, \qquad && u = c_1 u_x + c_2 u_y + c_3 u; & \\
& v_{xy} = 0, \qquad && u_{xy} = 0, \qquad && u = \beta(v_x) + \gamma(v_y) + c_3 v; &\\
& v_{xy} = 0, \qquad && u_{xy} = \exp(u) u_y, \qquad && u = \ln \left( - \frac{p'(v) v_x}{\mu(v_y) + p(v)} \right), &
\intertext{where $p'(v) = \exp(cv)$;}
& v_{xy} = 1, \qquad && u_{xy} = c_1 (u_x - c_2), \qquad && u = \exp(c_1 v_x) + c_2 v_y; & \\
& v_{xy} = \exp v, \qquad && u_{xy} = u u_x, \qquad && u = v_y + \mu(v_x) \exp v, &
\intertext{where $2 \mu' = \mu^2$;}
& v_{xy} = 0, \qquad && u_{xy} = \exp u, \qquad && u = \ln (v_x v_y) + \delta(v), &
\intertext{where $\delta''(v) = \exp \delta(v)$;}
& v_{xy} = 1, \qquad && u_{xy} = c_1 u_x + c_2 u_y - c_1 c_2 u, \qquad && u = \exp(c_1 v_x) + \exp(c_2 v_y) &\end{aligned}$$ up to the point transformations $u \rightarrow \theta(u)$, $v \rightarrow \kappa(v)$, $x \rightarrow \xi x$, and $y \rightarrow \eta y$ and the substitution $u + \xi x + \eta y \rightarrow u$, where $\xi$ and $\eta$ are arbitrary constants. Here $c$ is an arbitrary constant, $c_1$ and $c_2$ are nonzero constants.
Note that symmetries, $x$- and $y$-integrals, and the general solutions of the equations $u_{xy} = u u_x$ and $u_{xy} = \exp(u) u_y$ were given in [@MeshSok]. The transformation connecting the Liouville equation to the wave equation is well known (see [@ZhSok]).
Here we just give the outline of the proof.
Substituting the function $\psi$ given by into equation and we obtain $$\begin{gathered}
\psi_v F + \psi_{v_x} F' v_x + \psi_{v_y} F' v_y + v_x \bigl( \psi_{vv} v_y + \psi_{v v_x} F + \psi_{v v_y} v_{yy} \bigr) \nonumber\\
\qquad{} +v_{xx} \bigl( \psi_{v_x v} v_y + \psi_{v_x v_x} F + \psi_{v_x v_y} v_{yy} \bigr) + \bigl( \psi_{v_y v} v_y + \psi_{v_y v_x} F + \psi_{v_y v_y} v_{yy}\bigr) F \nonumber\\
\qquad\quad{} = f \bigl( \psi, \psi_v v_x + \psi_{v_x} v_{xx} + \psi_{v_y}F, \psi_v v_y + \psi_{v_x} F + \psi_{v_y} v_{yy} \bigr).\label{zhiber_eq4_8}\end{gathered}$$ Denote the arguments of the function $f$ by $a$, $b$, and $c$. Recall that we have $\psi_{v_x} \psi_{v_y} \neq 0$. The equality $f''_{bb} = f''_{cc} = 0$ thereby immediately follows from equation . Hence, equation takes the form $$u_{xy} = \alpha(u) + \beta(u) u_x + \gamma(u) u_y + \epsilon(u) u_x u_y.$$ After the point transformation $u \rightarrow A(u)$ with $A'' - \epsilon A'^2 = 0$ the above equation takes the form $$u_{xy} = f = \alpha(u) + \beta(u) u_x + \gamma(u) u_y.$$ Next, taking into account the last equality which defines the function f we can rewrite equation as follows $$\begin{gathered}
\psi_v F + \psi_{v_x} F' v_x + \psi_{v_y} F' v_y + v_x \bigl( \psi_{vv} v_y + \psi_{v v_x} F + \psi_{v v_y} v_{yy} \bigr) \\
\qquad{} + v_{xx} \bigl( \psi_{v_x v} v_y + \psi_{v_x v_x} F + \psi_{v_x v_y} v_{yy} \bigr) + \bigl( \psi_{v_y v} v_y + \psi_{v_y v_x} F + \psi_{v_y v_y} v_{yy}\bigr) F \\
\qquad\quad{} = \alpha(\psi) + \beta(\psi) \bigl( \psi_v v_x + \psi_{v_x} v_{xx} + \psi_{v_y}F \bigr) + \gamma(\psi) \bigl( \psi_v v_y + \psi_{v_x} F + \psi_{v_y} v_{yy} \bigr).\end{gathered}$$ Since $v_{xx}$ and $v_{yy}$ are independent variables, this equation is equivalent to the system $$\begin{gathered}
\psi_{v_x v_y} = 0, \\
\psi_{v_x v} v_y + \psi_{v_x v_x} F = \beta(\psi) \psi_{v_x}, \\
\psi_{v_y v} v_x + F \psi_{v_y v_y} = \gamma(\psi) \psi_{v_y}, \\
\psi_v F + \psi_{v_x} F' v_x + \psi_{v_y} F' v_y + \psi_{vv} v_x v_y + v_x \psi_{v v_x} F + v_y \psi_{v v_y} F + F^2 \psi_{v_y v_x} \\
\qquad{}= \alpha(\psi) + \beta(\psi) \bigl( \psi_v v_x + \psi_{v_x} v_{xx} + \psi_{v_y}F \bigr) + \gamma(\psi) \bigl( \psi_v v_y + \psi_{v_x} F + \psi_{v_y} v_{yy} \bigr).\end{gathered}$$ Consequently, we have $$\begin{gathered}
\psi = A(v, v_x) + B(v, v_y), \\
A_{v v_x} v_y + A_{v_x v_x} F = \beta(A+B) A_{v_x}, \\
B_{v v_y} v_x + B_{v_y v_y} F = \gamma(A+B) B_{v_y}, \\
(A_v + B_v) F + A_{v_x} F' v_x + B_{v_y} F' v_y + (A_{vv} + B_{vv}) v_x v_y + v_x A_{v v_x} F + v_y B_{v v_y} F \\
\qquad{}= \alpha(A+B) + \beta(A+B) \bigl( v_x(A_v + B_v) + F B_{v_y} \bigr) + \gamma(A+B) \bigl( v_y (A_v + B_v) + A_{v_x} F \bigr).\end{gathered}$$ By using the above equations we prove Theorem \[theorem2\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work is partially supported by the Russian Foundation for Basic Research (RFBR) (Grants 11-01-97005-Povolj’ie-a, 12-01-31208 mol-a).
[99]{}
Anderson I.M., Kamran N., The variational bicomplex for hyperbolic second-order scalar partial differential equations in the plane, [*Duke Math. J*](http://dx.doi.org/10.1215/S0012-7094-97-08711-1) **87** (1997), 265–319.
Bäcklund A.V., Einiges über Curven und Flächen Transformationen, *Lund Universitëts Arsskrift* **10** (1874), 1–12.
Bianchi L., Ricerche sulle superficie elicoidali e sulle superficie a curvatura costante, *Ann. Scuola Norm. Sup. Pisa Cl. Sci.* **2** (1879), 285–341.
Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. II, Gauthier-Villars, Paris, 1889.
Drinfel’d V.G., Svinolupov S.I., Sokolov V.V., Classification of fifth-order evolution equations having an infinite series of conservation laws, *Dokl. Akad. Nauk Ukrain. SSR Ser. A* (1985), no. 10, 8–10.
Goursat E., Leçon sur l’intégration des équations aux dérivées partielles du second ordre á deux variables indépendantes, I, II, Hermann, Paris, 1896.
Khabirov S.V., Infinite-parameter families of solutions of nonlinear differential equations, [*Sb. Math.*](http://dx.doi.org/10.1070/SM1994v077n02ABEH003442) **77** (1994), 303–311.
Kuznetsova M.N., Laplace transformation and nonlinear hyperbolic equations, *Ufa Math. J.* **1** (2009), no. 3, 87–96.
Kuznetsova M.N., On nonlinear hyperbolic equations related with the Klein–Gordon equation by differential substitutions, *Ufa Math. J.* **4** (2012), no. 3, 86–103.
Liouville J., Sur l’equation aux différences partielles $\partial^2 \log
\lambda /\partial u\partial v \pm \lambda /(aa^2)=0$, *J. Math. Pures Appl.* **18** (1853), 71–72.
Meshkov A.G., Sokolov V.V., Hyperbolic equations with third-order symmetries, [*Theoret. Math. Phys.*](http://dx.doi.org/10.1007/s11232-011-0004-3) **166** (2011), 43–57.
Sokolov V.V., On the symmetries of evolution equations, [*Russian Math. Surveys*](http://dx.doi.org/10.1070/RM1988v043n05ABEH001927) **43** (1988), no. 5, 165–204.
Soliman A.A., Abdo H.A., New exact solutions of nonlinear variants of the RLN, the PHI-four and Boussinesq equations based on modified extended direct algebraic method, *Int. J. Nonlinear Sci.* **7** (2009), 274–282, [arXiv:1207.5127](http://arxiv.org/abs/1207.5127).
Startsev S.Ya., Hyperbolic equations admitting differential substitutions, [*Theoret. Math. Phys.*](http://dx.doi.org/10.1023/A:1010359808044) **127** (2001), 460–470.
Startsev S.Ya., Laplace invariants of hyperbolic equations linearizable by a differential substitution, [*Theoret. Math. Phys.*](http://dx.doi.org/10.1007/BF02557408) **120** (1999), 1009–1018.
Svinolupov S.I., Second-order evolution equations with symmetries, [*Russian Math. Surveys*](http://dx.doi.org/10.1070/RM1985v040n05ABEH003693) **40** (1985), no. 5, 241–242.
Tzitzéica G., Sur une nouvelle classe de surfaces, *C. R. Acad. Sci.* **144** (1907), 1257–1259.
Zhiber A.V., Shabat A.B., Klein–Gordon equations with a nontrivial group, *Soviet Phys. Dokl.* **24** (1979), 607–609.
Zhiber A.V., Sokolov V.V., Exactly integrable hyperbolic equations of [L]{}iouville type, [*Russian Math. Surveys*](http://dx.doi.org/10.1070/rm2001v056n01ABEH000357) **56** (2001), no. 1, 61–101.
Zhiber A.V., Sokolov V.V., Startsev S.Ya., Darboux integrable nonlinear hyperbolic equations, *Dokl. Math.* **52** (1995), 128–130.
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abstract: 'We report electron microscopy observations of the surface plastic flow in polishing of rough metal surfaces with a controlled spherical asperity structure. We show that asperity–abrasive sliding contacts exhibit viscous behavior, where the material flows in the form of thin fluid-like layers. Subsequent bridging of these layers among neighboring asperities result in progressive surface smoothening. Our study provides new phenomenological insights into the long-debated mechanism of polishing. The observations are of broad relevance in tribology and materials processing.'
author:
- 'Ashif S. Iquebal'
- Dinakar Sagapuram
- 'Satish Bukkapatnam[^1]'
bibliography:
- 'references\_arXiv\_submission\_main.bib'
title: Surface plastic flow in polishing of rough surfaces
---
Mechanical interactions between severely rubbing surfaces have long been of fundamental interest for understanding friction in a wide range of domains including tribology, materials processing and geophysics. An important practical application of such interactions is in polishing of materials where rubbing action of fine abrasives is utilized to obtain smooth surfaces for application in optics, microscopy and mechanical instrumentation.
The practice of polishing to impart solid surfaces with smooth, lustrous finish has been known for centuries. The use of hard abrasives such as corundum and diamond for polishing in fact dates back to the Neolithic period [@lu2005earliest] and Leonardo da Vinci is credited with the earliest systematic design of a polishing machine [@pedretti1978codex]. It might be surprising then to know that the mechanism of polishing—how surface irregularities are smoothened out by abrasive particles—is still unsettled. Excellent account of the history and theories of polishing can be found in [@cornish1961mechanism; @rabinowicz1968polishing; @archard1985mechanical]. However, it may suffice to note that mainly two lines of thought for the polishing mechanism have prevailed: that of abrasion and surface flow. Early theories by Hooke and Newton [@newton1979opticks], followed by those of Herschel [@archard1985mechanical] and Rayleigh [@bulsara1998mechanics] viewed polishing essentially as an abrasion or a grinding process at a very fine scale where surface irregularities are removed by cutting action of the abrasives. The work by Samuels [@samuels2003metallographic] presented irrefutable evidence for this mechanism and showed how abrasives act as planing tools and result in the generation of well-defined chips as they slide past a surface. The alternative theory emerges from the work by Beilby [@beilby1921aggregation] who proposed surface smoothening occurring via surface flow and material redistribution. Here, it is believed that the material from surface peaks ‘flows’ to fill up the valleys and forms a thin vitreous surface layer, generally referred to as the “Beilby layer”. Bowden and Hughes [@bowden1937physical] further developed this theory and proposed that surface flow is in fact mediated by local melting at the surface–abrasive contacts. Electron diffraction measurements of polished surfaces have been presented as indirect evidence for the Beilby layer formation, but these observations were later proved to be inconclusive. To our knowledge, no conclusive evidence for the surface flow or melting has been provided to date. Other theories of polishing also exist, among which noteworthy is the molecular level material removal mechanism put forward by Rabinowicz [@rabinowicz1968polishing] based on energy considerations.
![Surface morphology characteristics of Ti-6Al-4V alloy sample prepared using electron beam melting. (a) Scanning electron micrograph showing the spherical asperity structure of the sample surface; (b) and (c) are respectively the distribution plots for the asperity height and diameter as measured using white light interferometry.[]{data-label="fig:im1"}](im1v1){width="80.00000%"}
In this study, we report direct experimental observations of surface plastic flow in polishing of an idealized rough metal surface having spherical asperities. Our electron microscopy observations of polished surfaces reveal viscous flow at the asperity–abrasive sliding contacts, involving surface material flow towards the asperity sides in the form of thin fluid-like layers. The subsequent stages of polishing involve bridging of these layers among different asperities to result in a smooth finish. Our study, besides confirming many hypotheses of the surface flow theory, provides new phenomenological insights into various stages of surface plastic flow in polishing of rough surfaces.
Ti-6Al-4V samples of $\diameter$50 mm and 7 mm thickness with controlled surface topography consisting of spherical asperity structure were prepared using the electron beam melting process. Details of the processing conditions for generating this asperity structure are provided in the Supplemental Material. Scanning electron micrograph of the representative surface morphology is shown in Fig. \[fig:im1\](a). The distributions of asperity height and diameter are shown in Figs. \[fig:im1\](b) and \[fig:im1\](c), respectively. The asperity height as well as the diameter exhibits a Weibull distribution with an average value of 72 $\mu$m and 64.5 $\mu$m, respectively and a standard deviation of $\sim15$ $\mu$m. It may be noted that the idealization of surfaces as a collection of spherical asperities (with Gaussian and Weibull distribution of heights) has been the basis for many prior theoretical analyses of elastic–plastic contacts between rough surfaces [@Greenwood300; @whitehouse1970properties; @hutchings1992tribology].
[The disk samples were polished on a Buehler Metaserv Grinder-Polisher (model 95-C2348-160) using silicon carbide (SiC) polishing pads ($\diameter$203 mm), in stages, with progressively smaller abrasives ranging from 30 $\mu$m to 5 $\mu$m under dry conditions. A steady nominal down pressure of $\sim0.5$ kPa was maintained and the polisher speed was fixed at 500 rpm. The workpiece sample was manually subject to a quasi-random orbital motion. The final polishing step involved the use of alumina abrasives ($< 1$ $\mu$m), suspended in an aqueous solution (20% by wt., pH $\approx7.5$) for 20 minutes to impart a specular finish to the surface.]{} The polishing was interrupted at every 90 s intervals to observe the surface morphology changes and asperity structure evolution using scanning electron microscopy (SEM). Quantitative details pertaining to the surface finish including surface roughness $(S_a)$ and volume of inter-asperity “valleys" ($S_v$) were measured using white light interferometry. Inter-asperity valleys were characterized by the surface heights lying below $10^{\text{th}}$ percentile on the bearing area curve (i.e., the cumulative distribution of surface profile) [@hutchings1992tribology]. To ensure that observations and measurements were made at the same surface location during different polishing steps, the sample surface was initially indented with a $2\times2$ mm square grid. The vertices of this grid enabled us to image the same surface location after each interrupted test. To facilitate better observations of the plastic flow patterns at asperity surfaces, the sample was tilted by $70^{\circ}$ in the scanning electron microscope.
Electron microscopy of the surface asperities enabled us to capture key phenomenological details of the polishing mechanism. Figures \[fig:im2\](b) and \[fig:im2\](c) show typical asperity structures after 90 s of polishing. Severe shear of the asperity surface and accumulation of the material towards asperity edges (see at arrow) is evident from Fig. \[fig:im2\](a). This flow pattern is reminiscent of plastic sliding between surfaces oriented at shallow angles, such as in sliding indentation or ‘machining’ under highly negative rake angles [@Challen161; @komanduri1971some]. Furthermore, the sheared surface material is often seen to flow to the lateral sides of the asperity as thin layers (Fig. \[fig:im2\](b)). Interestingly, the flow is seen to be quite symmetric around the periphery of sheared surface, with deposited material layer showing a molten-like appearance. The sliding direction between the asperity and abrasive particle can be inferred from the sliding marks in Fig. \[fig:im2\](b). This omni-directional flow at the surface, coupled with the observation of rheological flow features at the asperity edges (Fig. \[fig:im2\](a)), suggests fluid-like behavior of the surface plastic flow in polishing.
To explore the possible origin for this flow behavior, we estimated the “flash” temperature at the asperity–abrasive sliding contacts using the circular moving heat source model [@carslaw1959conduction], where the abrasive particle was treated as a semi-infinite moving body over which a stationary heat source acts. The heat source intensity was taken as the heat dissipation due to plastic shearing of the asperity at the sliding asperity–abrasive contact. Details of the flash temperature calculations are presented in the Supplemental Material. The analysis showed that the temperature rise at the sliding contacts monotonically increases with the circular contact area. The calculated temperatures for $\sim30$% of the sliding contacts were above 700 K. While these temperatures are well below the melting temperature ($T_m=$ 1925 K) of Ti-6Al-4V, they are in the typical dynamic recrystallization temperature range ($700-900$ K) for this alloy where significant flow softening occurs [@liao1998adiabatic]. At such temperatures, rate-dependent viscous plastic flow is not uncommon in metals [@ashby1982deformation]. Similar fluid-like flow phenomenon in metals have been also noted previously in other sliding configurations [@sundaram2012mesoscale; @trent2000metal] and shear bands [@sagapuram2016geometric; @healy2015shear; @spaepen2006metallic].
Our observations of the polished surfaces provide evidence for the surface flow theory in that the surface smoothening is mediated by material redistribution more so than material removal. Figure \[fig:im2\](c) shows the progression of the plastic flow at the asperity surface on continued polishing (beyond 90 s). Apparently, the repeated shearing at the asperity surface upon encountering a sliding abrasive results in stacking of multiple thin layers on the lateral sides of the asperity (see at arrow). In effect, this results in a radial increase in the flattened area of the asperity.
Figure \[fig:im3\] illustrates the surface morphology characteristics at 180 s. As seen from Fig. \[fig:im3\](a), individual asperity surfaces are unresolvable by this stage, and the surface can be described as an interconnected network of flat islands. Interspersed among these regions are the unfilled depressions. A closer inspection of the flattened regions reveals that their formation is mediated by bridging of the smeared surface material between the neighboring asperities, as shown in Fig. \[fig:im3\](b) (see at arrow). Indeed, this “welding” between the asperities may be expected given the occurrence of severe plastic flow and temperatures at the asperity surfaces. In our experiments, this bridging phenomenon was noted only when the distance between the edges of two neighboring asperities was below a critical value of $\sim 30$ $\mu$m. For asperities separated by larger distances, lateral flow of the material (Fig. \[fig:im2\]) was seen to continue until the effective distance between the asperities approached the critical value. Continued polishing causes complete bridging of individual asperities, resulting in a nominally smooth surface (for example, see top row in Fig. \[fig:im5\](a)). The elimination of microscale depressions during final stages of polishing again seems to occur as a result of material flow from neighboring flat regions. A series of SEM images taken at successions of 90 s, and showing the closure of a surface depression, is presented in Fig. \[fig:im4\]. An important consequence of repeated plastic flow at the surface is the microstructure refinement at the surface and associated increase in the strength. Indeed, hardness measurements (Vickers indentation, load $500$ g) showed the surface to be characterized by a higher hardness (375 kg/mm$^2$) compared to the base material (350 kg/mm$^2$).
{width="100.00000%"}
Figure \[fig:im5\] summarizes the surface morphology evolution during the entire duration of polishing. The micrographs show the surface flow and bridging among the asperities (Fig. \[fig:im5\](a), top row), together with the gradual reduction of the volume of inter-asperity valleys (light regions). This results in a strongly connected network of flat areas (dark regions) that eventually evolve to form a uniformly smooth surface (with average roughness, $S_a \sim 30$ nm). The bridging process can in fact be treated as an evolving random graph $G = (V,E)$, where the nodes $(V)$ denote the asperities, and the edges $(E)$ are the probabilities $p(i,j)$ for a bridge to exist between nodes $i$ and $j$ $\forall i,j\in V$ (see Fig.\[fig:im5\](a), bottom row). A spectral graph measure called the Fiedler number, $\lambda_2$ [@chung1997spectral], serves as a natural quantifier to capture the effects of polishing on the surface morphology, particularly during the bridging process [@rao2015graph]. For example, $\lambda_2=0$ indicates complete absence of bridge formation; in contrast, $\lambda_2= 0.23$ suggests a high degree of bridging where every node is connected to at least six other neighboring nodes (representative of close packing of spherical asperities). Details related to the calculation of $\lambda_2$ are presented in the Supplemental Material. The micrograph patterns as well as the corresponding $\lambda_2$ values presented in Fig. \[fig:im5\](a) suggest that as polishing ensues and the asperity diameters grow, the propensity of neighboring asperities to bridge (i.e., $p(i,j)$) progressively increases. Quantitatively speaking, the initial value of $\lambda_2=0.068$ (see Fig. \[fig:im5\](a), bottom row) indicates little bridging (average number of bridges connecting a node or the “degree" is $<1$) as reflected in $p(i,j)$ being close to zero between almost all asperities. Specifically, the edges connecting the neighboring nodes are almost absent initially, and low probability edges (red) connect only a sparse set of neighboring nodes. After 450 s of polishing, $\lambda_2$ increases to 0.130, suggesting a higher degree of bridging among all neighboring asperities (degree $\geq4$), and high $p(i,j)$ values.
The corresponding temporal evolution of $S_{v}$ and $S_a$, captured using surface interferometry, are given in Figs. \[fig:im5\](b) and \[fig:im5\](c), respectively. While both $S_v$ and $S_a$ decrease monotonically with time, $S_v$ drops sharply from $\sim2\times10^5$ $\mu$m^3^ to $2.1 \times10^3$ $\mu$m^3^ between 90 s and 180 s (Fig. \[fig:im5\](b)). This corresponds to the time interval where bridging of the asperities is predominant (see Fig. \[fig:im3\]). Unlike $S_v$, the $S_a$ continues to decrease even after 180 s, likely because of surface smoothening via reduction in microscale surface depressions during the final stages of polishing (see Fig. \[fig:im4\]).
While this study has focused on a Ti-based alloy system, the current findings are likely to be more generic to polishing of a range of other material systems. In fact, surface flow profiles at the asperities similar to that in Figs. \[fig:im2\] and \[fig:im3\] were also observed during polishing of tantalum oxide (Ta$_2$O$_5$; see Fig. S4 in the Supplemental Material). These observations in oxide materials, while at first surprising given their inherent brittle behavior, can be explained by the high asperity–abrasive contact pressures that typically exceed the workpiece material’s hardness. These high contact pressures can in turn promote plastic flow even in highly brittle materials [@bridgman1952studies; @marsh1964plastic]. Additionally, the asperity–abrasive contact temperature calculations for polishing of Ta[$_2$]{}O[$_5$]{} showed that the flash temperatures can be a significant fraction ($\sim0.4T_m$) of its melting temperature, which could potentially enhance the propensity for viscous-type flow at asperity surfaces.
In closing, this letter presents direct experimental evidence for the surface flow mechanism of polishing, and reports new phenomenological observations pertaining to plastic flow aspects in smoothening of rough surfaces. These involve material flow from the asperity contact surfaces to the lateral sides in the form of thin viscous layers, bridging of neighboring asperities, and eventual filling-up of the small surface depressions by material flow from the smooth surface regions. While these observations are consistent with the Beilby–Bowden’s material redistribution theory of polishing, several important distinctions are in order. First, no evidence for surface melting or amorphization was noted in contrast to the original hypotheses [@beilby1921aggregation; @bowden1937physical], although the microscopy observations of the surface flow profiles, together with the temperature calculations of the asperity–abrasive sliding contacts, strongly suggest the occurrence of viscous flow. Second, as demonstrated in Fig. \[fig:im2\], the material redistribution is facilitated by the material flow as thin layers that make self-contact with the asperity sides. This is again at variance with the original ideas where the surface valleys are believed to be filled purely via mechanical deformation (as in compression or indentation plastic flows) of the asperities. Lastly, bridging among asperities is seen to be an important mechanism by which neighboring asperities merge to form a smooth surface network. This has not been accounted for in any of the prior studies. Besides polishing, our observations are also of relevance to a range of other engineering and physical systems where micro-scale asperity contacts, characterized by high pressures, are of intrinsic interest, e.g., tribological systems, erosion and earthquakes. The well-known observations of the folded-layer structures in metamorphic rocks [@hopgood1999general], which bear striking resemblance to the thin-layer stacking profiles in Fig. \[fig:im2\](c), alludes to the possibility of similar viscous flow phenomena playing a role also on a much larger scale in geophysical formations.
**Acknowledgments**: The authors would sincerely like to acknowledge Dr. Alex Fang, Texas A&M University, for providing access to the lapping machine and the National Science Foundation (CMMI- 1538501) for their kind support of this research.
seccntformat\#1[\#1ignorethe\#1]{} gobbletwo @numberline \#1[\#1@numberline[\#1]{}]{}
**Supplemental Material**
[ p[5em]{} p[18em]{} ]{}\
$z$ & asperity heights from the reference plane\
$a$ & contact radius of the asperity–abrasive contact area\
$R$ & asperity radius\
$\Delta T_{max}$ & maximum temperature rise at the asperity–abrasive interface\
$H$ & workpiece surface hardness\
$V$ & polishing speed\
$q$ & total heat flux at the asperity–abrasive interface\
$q_1,q_2$ & heat flux at the asperity and abrasive surfaces, respectively\
$P_{e1}, P_{e2}$ & Peclet number for the asperity and abrasive body\
$\mu$ & coefficient of friction at the asperity–abrasive interface\
$k_1,k_2$ & thermal conductivity of asperity and abrasive, respectively\
$K_2$ & thermal diffusivity of the abrasive\
$\rho_2$ & density of abrasive\
$C_2$ & specific heat of abrasive\
[ M[8em]{} M[10em]{} M[7em]{}]{} Material & Thermal conductivity (W/m/K) & Hardness (GPa)\
Ti-6Al-4V & 7.2 – 11.2 \[S1\] & 3.5 – 3.75 \[S1\]\
$\text{Ta}_2\text{O}_5$ & 0.9 – 4 \[S2\] & 1.46 – 4.21 \[S3\]\
SiC & 60 \[S4\] & 25 \[S5\]\
[**S1: Electron beam melting process parameters**]{}
Ti-6Al-4V cylindrical disks ($\diameter$50 mm and 7 mm thickness) were prepared using an ARCAM electron beam melting machine operating at a vacuum of $\sim2$ Pa and accelerating voltage of $\sim60$ kV. The process involved raking a $50$ $\mu$m layer of Ti-6Al-4V powder of average $\diameter72$ $\mu$m (see Fig. S3) for the distribution of radius of Ti-6Al-4V particles) using a focused beam of 3 mA, scanning at a speed of 10 m/s. The resulting surface consists of granular Ti-6Al-4V particles with a unique spherical asperity structure. Such a controlled asperity structure is ideal for systematic investigation of the surface flow behavior during polishing.
[**S2: Calculation of flash temperatures at the asperity–abrasive contacts**]{}
![Schematic showing contact (solid line) between the workpiece surface consisting of spherical asperities and the polishing pad at a distance $S_z$ (average asperity heights) from the workpiece reference plane (dotted line). Here, the asperity height, $z$, is measured with respect to the workpiece reference plane.[]{data-label="fig:suppim1"}](suppim1){width="75.00000%"}
![Moving circular heat source model for the contact between asperity and abrasive to calculate the temperature rise during polishing. Here, the abrasive is considered as the semi-infinite moving heat source and the asperity acts as a stationary heat source.[]{data-label="fig:suppim2"}](suppim2){width="75.00000%"}
For a given asperity height $(z)$ distribution, only the asperities for which $z>S_z$ and $z\leq S_z+2R$ are involved in the polishing process, as schematically shown in Fig. S1. Here, the asperity height, $z$, is measured with respect to the workpiece reference plane (dotted line in the schematic in Fig. S1). We assume that the clearance between the workpiece reference plane and the polishing pad (solid line) is equal to the average surface asperity heights, $S_z$, of the workpiece. The diameter of asperity–abrasive contact ($2a$) can then be calculated for a given value of $S_z$, asperity radius ($R$) and height ($z$) distribution.\
Given the radius of contact, we calculate flash temperature by treating the contact as a moving circular heat source (Fig. S2). The heat source intensity is taken as the heat dissipation due to plastic shearing of the metal asperity at the sliding interface. The heat partition between the asperity and the abrasive particle is determined by setting equal the maximum (quasi-steady state) temperatures of the asperity and abrasive particle within the contact, according to Blok’s postulate \[S6\]. Here, we treat the abrasive as a semi-infinite moving body (with velocity $V$) over which a stationary heat source (with uniform heat flux) acts. The steady state flash temperature occurring at the contact center can accordingly be given by the first order approximation to Jaegar’s circular moving heat source model \[S7, S8\] as: $$\Delta T_{max}\big|_{abrasive}=\frac{2q_2a}{k_2\sqrt{(\pi(P_{e2}+1.273))}}$$ where, Peclet number, $P_{e2}=Va/2K_2$ and $K_2={k_2}/{\rho_2 C_2} \approx 4\times10^{-5}$ m$^2$/s. For $V=5$ m/s and contact radius $a$, we have $P_{e2}=6.25\times10^5a$. For the asperity (which is treated as a stationary source), we have: $$\Delta T_{max}\big|_{asperity}=\frac{q_1a}{k_1}$$
![(a) Flash temperature map for Ti-6Al-4V as a function of asperity radius and height, both of which follow a truncated Weibull distribution with average at 36 $\mu$m and 64.5 $\mu$m, respectively, and a standard deviation $\sim15$ $\mu$m. $S_z$ corresponds to the average asperity height. []{data-label="fig:suppim3"}](suppim3){width="100.00000%"}
Assuming adiabatic conditions, where plastic dissipation at the interface is completely converted into heat, the total heat flux, $q$, at the circular contact is given by: $$q =q_1+q_2=\mu H V$$ By equating the maximum temperatures at the asperity and abrasive surface, we have: $$\Delta T_{max}=\frac{\mu HVa}{k_1}\left(1+\frac{k_2}{2k_1}\sqrt{\pi(P_{e2}+1.273)} \right)^{-1}$$
We solve for $\Delta T_{max}$ for Ti-6Al-4V using the values in Table 1, and the corresponding flash temperature map as a function of asperity height and radius is shown in Fig. S3(a). Any asperity for which $z<S_z$ or $z\geq S_z+2R$ would not be involved in the polishing process as it would either make no contact with the abrasive or lie outside the asperity–abrasive contact region (solid line in Fig. S1). These two cases are marked as “**p**" and “**q**" in Fig. S3(a). Elsewhere, we notice that larger values of $R$ and $z$ result in higher flash temperatures.
[While the assumption of abrasive as a semi-infinite plane maybe reasonable during the initial stages of polishing, the configuration is reversed as polishing process progresses. During the intermediate and final stages, polishing maybe represented as individual abrasive particles sliding across a semi-infinite workpiece surface. For this latter configuration, we assume abrasive particles as sliding conical indenters plastically deforming the workpiece surface. Again for this case, the problem is that of a moving semi-infinite body (workpiece surface) over which stationary heat source (abrasive-workpiece surface contact) acts. The maximum flash temperature rise at the contact in this case is given as: $$\Delta T_{max}=\frac{\mu HVa}{k_2}\left(1+\frac{k_1}{2k_2}\sqrt{\pi(P_{e1}+1.273)} \right)^{-1}$$ The calculated sliding temperatures for this configuration are slightly larger than those in the earlier configuration where abrasive was taken as a semi-infinite plane (Fig. S2). The difference between temperature estimates for these two configurations is within 20% (at a contact radius of $\sim$40 $\mu$m) for the contact areas considered here.]{} In both the configurations, for $\sim30\%$ of the sliding contacts, maximum flash temperatures are above the dynamic recrystallization temperature of the alloy ($\sim700$ K).
Similar calculations for $\text{Ta}_2\text{O}_5$ showed the flash temperature to be in the range of 750 K. In this case, the average radius of the asperity–abrasive contact area was inferred from Fig. S4 as $\sim15$ $\mu$m. $V$ was taken as 5 m$/$s, as for Ti-6Al-4V polishing. Again the calculated flash temperatures at the asperity–abrasive contacts are high enough, $\sim0.4T_m$, where viscous-like flow may be expected.\
![Scanning electron micrographs showing surface morphological changes in $\text{Ta}_2\text{O}_5$: (a) before and (b) after polishing. []{data-label="fig:suppim4"}](suppim4){width="75.00000%"}
{width="\textwidth"}
**S3: Graph representation of topological evolution**
The process of the merger (connectivity) among the asperities is analyzed as an evolving random graph [$G=(V,E)$]{} whose nodes ($V$) are the asperities, and the edges ($E$) are given by the probability, $p_{ij}$, of the existence of a bridge connecting the asperities $i,j\in V$. The probability of existence of an edge ($p_{ij} \forall i,j\in V$) is inversely proportional to the inter-asperity distance and is calculated using the radial basis function as $p_{ij}=(1+\exp(||V_i-V_j||))^{-1}$. The radial basis function assigns lower $p_{ij}$ to the edges as the physical distance between connecting node increases. We first determine the normalized Laplacian, $\mathcal{L}$, from the graph, $G$ as $\mathcal{L=D}^{-\frac{1}{2}}\times L\times \mathcal{D}^{-\frac{1}{2}}$ where $L$ is the combinatorial Laplacian defined as $L\overset{\Delta}{=}\mathcal{D-S}$. Here, $\mathcal{D}$ is the dianognal matrix representing the degree of each node and is given as $\mathcal{D}=\left( \begin{matrix}
\sum_{j=1}^{N}p_{1j}&\sum_{j=1}^{N}p_{2j}&...&\sum_{j=1}^{N}p_{Nj} \end{matrix}\right) $, and $\mathcal{S}$ being the similarity matrix. It has been established that the second largest eigenvalue of $\mathcal{L}$ captures the algebraic connectivity in the graph, also called the Fiedler number ($\lambda_2$) \[S9, S10\].
The lower bound on $\lambda_2$ is calculated using the geometric embedding of planar graph on a unit sphere as presented in \[S11\], where each of the nodes are represented by non-overlapping semi-spherical caps of radius $r_i, i\in V$. For the micrograph in Fig. S5, $|V|=160$. A strongly connected network of asperities can be assumed as an ideal close packing of uniform spheres such that each node is connected to at most 6 nearest neighbors. Under such conditions it can be shown that $0.23\leq\lambda_2\leq0.3$ holds. The initial value of $\lambda_2=0.068$ (see Fig. S5, bottom row) indicates that the degree of each node is $<1$. After 450 s of polishing, $\lambda_2$ increases to 0.130 suggesting a minimum degree of 4 among all neighboring asperities. The network structure along with the corresponding $\lambda_2$ values is summarized in Fig. S5. Additionally, the linear increase in the value of $\lambda_2$ suggests that there are significant topological changes in the surface even during the final stages of polishing process which otherwise are not reflected in the $S_a$ or $S_v$ measurements (see Fig. 5 in the main text).
(-1.30,0) – (1.30,0); (-0.90,0) – (0.90,0); (-0.5,0) – (0.5,0);
1. G. Welsch, R. Boyer, and E. Collings, *Materials Properties Handbook: Titanium Alloys* (ASM International, Materials Park, OH, 1993).
2. C. D. Landon, R. H. Wilke, M. T. Brumbach, G. L. Brennecka, M. Blea-Kirby, J. F. Ihlefeld, M. J. Marinella, and T. E. Beechem, Applied Physics Letters **107**, 023108 (2015).
3. O. Shcherbina, M. Palatnikov, and V. Efremov, Inorganic Materials **48**, 433 (2012).
4. Q. Liu, H. Luo, L. Wang, and S. Shen, Journal of Physics D: Applied Physics (2016).
5. Y. Ahn, S. Chandrasekar, and T. N. Farris, Journal of Tribology 119, 163 (1997).
6. H. Blok, in *Proceedings of the general discussion on lubrication and lubricants*, Vol. 2 (London: IMechE, 1937) pp. 222-235.
7. H. S. Carslaw and J. C. Jaeger, *Conduction of Heat in Solids* (Clarendon Press, Oxford, 1959).
8. X. Tian and F. E. Kennedy, Journal of Tribology **116**, 167 (1994).
9. F. R. K. Chung, *Spectral Graph Theory*, Vol. 92 (American Mathematical Society, RI, 1997).
10. P. K. Rao, O. F. Beyca, Z. Kong, S. T. Bukkapatnam, K. E. Case, and R. Komanduri, IIE Transactions **47**, 1088 (2015).
11. D. A. Spielman and S. H. Teng, Linear Algebra and its Applications **421**, 284 (2007).
[^1]: \[Corresponding author\]Corresponding author
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---
abstract: 'NELIOTA is a new ESA activity launched at the National Observatory of Athens in February 2015 aiming to determine the distribution and frequency of small near-earth objects (NEOs) via lunar monitoring. The project involves upgrading the 1.2m Kryoneri telescope of the National Observatory of Athens, buliding a two fast-frame camera instrument, and developing a software system, which will control the telescope and the cameras, process the images and automatically detect NEO impacts. NELIOTA will provide a web-based user interface, where the impact events will be reported and made available to the scientific community and the general public. The objective of this 3.5 year activity is to design, develop and implement a highly automated lunar monitoring system, which will conduct an observing campaign for 2 years in search of NEO impact flashes on the Moon. The impact events will be verified, characterised and reported. The 1.2m telescope will be capable of detecting flashes much fainter than current, small-aperture, lunar monitoring telescopes. NELIOTA is therefore expected to characterise the frequency and distribution of NEOs weighing as little as a few grams.'
---
Near-Earth Objects (NEOs) are ubiquitous in the space environment. They are thought to originate from fragments created during asteroid collisions, asteroids diverted from the asteroid belt through the gravitational influence of planets, or cometary debris. NEOs have orbits crossing into the inner Solar System and intersecting the Earth’s trajectory, posing a threat to artificial satellites, spacecraft, and astronauts. The atmosphere of the Earth offers protection from all but the largest NEO impacts, which do not completely burn up as they enter the atmosphere, at speeds of tens of km s$^{-1}$. However, the surface of the Moon remains susceptible to impacts by small NEOs and can be used to study their properties.
NEO lunar impacts are observed as bright flashes of light. The impacting meteoroids travel at large speeds (20 to 50 km s$^{-1}$), and thus contain tremendous kinetic energy that causes the rocks and soil on the lunar surface to heat up and glow. Ground-based observers detect flashes lasting from a fraction of a second to several seconds, with light curves showing a sharp rise and an exponential fading tail. Surveys show a peak in impacts during meteor showers, as the Earth-Moon system passes through relatively dense clouds of meteoroids, when crossing the orbits of comets, however, impacts are detected continuously, without them necessarily exhibiting a connection to comet debris or a meteor shower.
In order to quantify the frequency and characteristics of NEOs, several campaigns are underway, such as the Lunar Impact Monitoring at NASA’s Marshall Space Flight Center (Suggs et al. 2014), the MIDAS project (Madiedo et al. 2014, 2015), and the ILIAD Network (Ait Moulay Larbi et al. 2015). Suggs et al. (2014) reported over 300 impacts down to $R=
10.2$ mag, while surveying an area of 3.8 x 10$^6$ km$^2$ over 7 years. The analysis of 126 flashes that were detected during photometric conditions, yielded a survey completeness limit of $R= 9$ mag. The association of certain impacts with meteor streams provided constraints on the impact speeds and thus their kinetic energy.
Lunar monitoring surveys for NEO impacts typically involve small, 30-50 cm telescopes, tracking at the lunar rate, that are equipped with video cameras recording at a rate of 30 frames per second (fps). The dark portion of the lunar surface is monitored during the phases corresponding to 10-50% illumination. The aim is to maximize the number of lunar impacts detected, by maximizing the lunar surface observed, while avoiding the illuminated surface of the Moon. The goal of such surveys is to measure the distribution of sizes and masses of objects impacting the Moon, as well as their flux, and detect a significant number of impacts from which to obtain statistical results on their characteristics.
NELIOTA aims to increase the number of detected faint lunar impacts, and therefore increase the statistics to obtain their size distribution, speeds, frequency, and characterize the impact ejecta. Using the 1.2m Cassegrain reflector telescope at Kryoneri Observatory, manufactured and installed in 1975 by the British company Grubb Parsons Co., Newcastle (Figure \[fig1\]), we aim to push the detection limit for the first time to $V= 12$ mag. Note, that the surface brightness of the earthshine ranges between 12 m$_V$ arcsec$^{-2}$ (New Moon) and 17 m$_V$ arcsec$^{-2}$ (near Full Moon), with variations on the timescale of hours of the order of 0.25 m$_V$ arcsec$^{-2}$ due to terrestrial meteorology (Montanes-Rodriguez et al. 2007). Given the expected power law size distribution of NEOs, we anticipate providing significant numbers of small NEOs by detecting faint flashes. These data would be valuable for characterizing the meteor environment and providing guidelines to spacecraft manufacturers for protection of their vehicles, as well as for future space mission planning.
The objective of NELIOTA is to design, develop and implement a highly automated lunar monitoring system using existing facilities at the National Observatory of Athens, Greece. For the first phase of the project, DFM Engineering, Inc. will be retrofiting and upgrading the electronics and mechanical parts of the 1.2m Kryoneri telescope, located in the Northern Peloponnese, in Greece. A dual imaging instrument, also designed and manufactured by DFM Engineering, Inc., along with two Andor Zyla 5.5 sCMOS fast-frame cameras recording at 30 fps, will be used to simultaneously monitor the non-illuminated lunar surface for impact flashes and to reject cosmic rays. Our setup will provide a field-of-view $\sim$17’x14’. Specialised software is being developed to control the telescope and cameras, as well as to process the resulting images to detect the impacts automatically. The NELIOTA system will then publish the data on the web so it can be made available to the scientific community and the general public. Following a 2 month commissioning phase, there will be a 22 month observing campaign for NEO impact flashes on the Moon. The impact events will be verified, characterised and recorded. The 1.2m Kryoneri telescope will be capable of detecting flashes far fainter than telescopes currently monitoring the Moon.
![The 1.2 m Cassegrain reflector telescope at Kryoneri Observatory, which is being retrofit and upgraded to detect lunar impacts in the framework of the NELIOTA project.[]{data-label="fig1"}](Tel.pdf){width="11cm"}
2015, *Earth, Moon, and Planets*, 115, 1
2014, *MNRAS*, 439, 2364
2015, *A&A*, 577, 118
2007, *AJ*, 134, 1145
2014, *Icarus*, 238, 23
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---
abstract: 'We report the discovery of optical emission from the non-radiative shocked ejecta of three young Type Ia supernova remnants (SNRs): SNR 0519-69.0, SNR 0509-67.5, and N103B. Deep integral field spectroscopic observations reveal broad and spatially resolved \[Fe<span style="font-variant:small-caps;">xiv</span>\] 5303Å emission. The width of the broad line reveals, for the first time, the reverse shock speeds. For two of the remnants we can constrain the underlying supernova explosions with evolutionary models. SNR 0519-69.0 is well explained by a standard near-Chandrasekhar mass explosion, whereas for SNR 0509-67.5 our analysis suggests an energetic sub-Chandrasekhar mass explosion. With \[S<span style="font-variant:small-caps;">xii</span>\], \[Fe<span style="font-variant:small-caps;">ix</span>\], and \[Fe<span style="font-variant:small-caps;">xv</span>\] also detected, we can uniquely visualize different layers of the explosion. We refer to this new analysis technique as “supernova remnant tomography”.'
author:
- 'Ivo R. Seitenzahl'
- Parviz Ghavamian
- 'J. Martin Laming'
- 'Frédéric P. A. Vogt'
bibliography:
- 'bibliography.bib'
title: Optical tomography of chemical elements synthesized in Type Ia supernovae
---
Type Ia Supernovae (SNe Ia) are the thermonuclear explosions of white dwarf stars. In spite of their importance as distance indicators in Cosmology [@riess1998a; @perlmutter1999a] and their major contribution to nucleosynthesis [@seitenzahl2017a], no consensus has been reached on their explosion mechanism(s) and progenitor system(s) [@hillebrandt2013a]. Even for the well-studied, nearby SN 2011fe, a comparison of the observations and synthetic spectral time series of the two leading explosion models has failed to produce a clear winner: the “single degenerate” delayed-detonation model of a ${\sim}1.4\,{\ensuremath{\mathrm{M}_\odot}}$ WD [@seitenzahl2013a] and the “double-degenerate” merger with a ${\sim}1.1\,{\ensuremath{\mathrm{M}_\odot}}$ [@pakmor2012a] primary WD explain the observations nearly equally well [@roepke2012a].\
An alternative approach to solving the SN Ia progenitor problem is via multi-wavelength observations of supernova remnants (SNRs). Following the thermonuclear incineration of a white dwarf, the freshly synthesized heavy elements are ejected at high velocity. The supersonic expansion drives a forward shock into the surrounding interstellar medium and a reverse shock back into the remains of the supernova explosion, eventually heating the ejecta to X-ray emitting temperatures [@reynolds2008a]. The most important parameters governing the evolution of SNRs are the chemical composition, kinetic energy and mass of the ejecta, as well as the ambient medium density [@truelove1999a], all of which are closely linked to the explosion mechanism. As the supernova ejecta progressively ionize behind the reverse shock, zones of higher and higher atomic ionization are produced in succession behind this shock. Optical forbidden line emission from low-lying atomic transitions of these highly-ionized atoms is expected. Many of these lines were first seen in the solar corona and are hence referred to as “coronal” lines.\
The coronal \[Fe<span style="font-variant:small-caps;">xiv</span>\] magnetic dipole transition 3s$^{2}$3p$^{2}$ (P$_{1/2} - $ P$_{3/2}$) produces a green emission line at 5302.8Å with an emissivity that peaks in ionization equilibrium at temperatures near $2\times10^6$K [@bryans2006a] and is produced over the range $7.0 < \log(\mathrm{T}) < 7.5$ in the shock models presented below. Earlier detections of \[Fe<span style="font-variant:small-caps;">xiv</span>\] in SNRs were from “radiative” cloud shocks in ISM material (${\sim}300 - 500\,\mathrm{km}\,\mathrm{s}^{-1}$, where the postshock gas undergoes thermal instability and the shock dynamics are strongly affected by radiative cooling), such as those detected in Puppis A [@lucke1979a; @teske1987a], N49 [@dopita2016a], and 1E0102.2-7219 [@vogt2017a], following model predictions [@allen2008a]. In these cases, the sensitivity of the detectors has been the limiting factor in detecting optical \[Fe<span style="font-variant:small-caps;">xiv</span>\] from the much faster non-radiative shocks ($>2000\,\mathrm{km}\,\mathrm{s}^{-1}$, no thermal instability) in both the swept up interstellar gas and reverse shocked ejecta. As we show in this paper, the superior sensitivity of the Multi Unit Spectroscopic Explorer (MUSE) Integral field spectrograph on the European Southern Observatory (ESO) Very Large Telescope (VLT) and the larger light gathering area of its 8.2m mirror have now enabled the detection of faint optical coronal line emission in non-radiative shocks. Using public MUSE data from the ESO archive, we have discovered \[Fe<span style="font-variant:small-caps;">xiv</span>\] 5303Å emission from the reverse shocks of the three youngest Type Ia supernova remnants in the Large Magellanic Cloud (LMC) [@hughes1995a]: SNR 0519-69.0, SNR 0509-67.5, and N103B (SNR 0509-68.7). For further details on the observations, data reduction and processing see the Supplemental Information.
{width="\textwidth"}
{width="\textwidth"}
To our knowledge this is the first detection of optical emission from the non-radiatively shocked ejecta of any Type Ia supernova remnant. As expected, we find the peak of the \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission, which appears as a narrow band in the interiors of the SNRs, immediately interior to the peak of the Fe K X-ray emission detected by the Chandra X-ray Observatory (see Fig. 1), since the reverse shock is propagating inwards in a Lagrangian sense. The detection of optical coronal line emission from pure non-radiative ejecta shocks of Type Ia SNRs opens a long sought window into the kinematic study of young Type Ia SNRs. In the case of SNR 0519-69.0 (hereafter 0519) and SNR 0509-67.5 (hereafter 0509), the \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission appears as a nearly circular shell (see Fig. 1(A) and (B)). For N103B (Fig. 1(C)), the signal is contaminated by residuals from superimposed bright stars and the \[Fe<span style="font-variant:small-caps;">xiv</span>\] behind the reverse shock appears much more asymmetric. This is likely a consequence of the strong interaction of this SNR with high-density material on its western side [@williams2014a]. The observed morphology of three nearly concentric shells of Balmer-line emission from the forward shock (blue) on the outside, with X-ray emission (red) from the hot, reverse shocked ejecta just inside the Balmer filaments, and coronal \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission (green) inside of the X-ray emitting ejecta, is a beautiful confirmation of the extant theory of SNR evolution. To probe the kinematics of the iron-rich ejecta in each SNR, we have extracted \[Fe<span style="font-variant:small-caps;">xiv</span>\] line profiles (see Fig. 2) from selected regions (indicated in Fig. 1) of the three SNRs. Fitting single Gaussians and a linear continuum to the line profiles, we obtain velocity widths of $2460\pm100\,\mathrm{km}\,\mathrm{s}^{-1}$ for 0519, $4370\pm100\,\mathrm{km}\,\mathrm{s}^{-1}$ for 0509, and $3290\pm100\,\mathrm{km}\,\mathrm{s}^{-1}$ for N103B.
The near spherical symmetry of 0519 and 0509 allows us to model them in 1D (whereas the strongly asymmetric morphology of the \[Fe<span style="font-variant:small-caps;">xiv</span>\] in N103B does not), so in the remainder of this report we focus on these two SNRs for a quantitative analysis. While the approximate location of the reverse shock can be inferred from X-ray observations of the shocked ejecta [@kosenko2010a], the resolved line width of the \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission presented here allows us for the first time to directly determine the reverse shock speed a new observational constraint. The radius of the peak of the \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission, modeled as a spherical shell, is $2.86\pm0.10\,$pc for 0509 and $2.36\pm0.18\,$pc for 0519, respectively. To provide estimates of the total line fluxes we integrated the broad \[Fe<span style="font-variant:small-caps;">xiv</span>\] line over the full extent of the emission in each SNR and fit a single Gaussian to each line profile after subtracting a linear continuum. Corrected for extinction and reddening by dust, we obtain estimates of total line fluxes of $1.1\times10^{-14}\,\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$ for 0519 and $0.9\times 10^{-14}\,\mathrm{erg}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$ for 0509.
A valuable constraint on the interpretation of our \[Fe<span style="font-variant:small-caps;">xiv</span>\] measurements is found in the time evolution of observed light echoes the reflections of supernova light by interstellar dust sheets. Modeling of the light echoes \[19\] allowed for an explosion model and SNR evolution independent determination of the SNR ages. These models placed 0519 at $600\pm200$yr and 0509 at $400\pm120$yr [@rest2005a]. Further, since these two SNRs are located in the LMC, their distances are reliably known to be 50kpc, with an uncertainty of only 2 per cent [@pietrzynski2013a]. This allows us to accurately relate angular size to physical size. The forward shock position and velocity can be inferred from the broad Balmer-line emission [@hovey2015a; @hovey2018a]. With reliable observational constraints on the age, forward shock position and velocity, as well as reverse shock position and velocity, we are now in a position to limit explosion model parameters commensurate with the observational constraints.
----------------------------------------------------- ------------------------------ ---------- ------------------------------- ------------
SNR
observation model observation model
Age (yr) $600\pm200$ [@rest2005a] 750 $400\pm120$ [@rest2005a] 310
$310\pm35$ [@hovey2015a]
$v_f (\mathrm{km}\,\mathrm{s}^{-1}$) $2770\pm500$ [@kosenko2010a] 2516 $6500\pm200$ [@hovey2015a] 6539
2650 [@hovey2018a]
$R_f$ (pc) $4.0\pm0.3$ [@kosenko2010a] 4.07 3.636 [@hovey2015a] 3.64
$v_r (\mathrm{km}\,\mathrm{s}^{-1}$) 1887 4766
$R_r$ (pc) 2.16 2.74
$v_{exp} (\mathrm{km}\,\mathrm{s}^{-1}$) 4057 5170
$n_{e}t$ $3.8\pm0.3$ [@kosenko2010a] 3.7 0.85 3.4 [@badenes2008a] 0.315
$(10^{10}\,\mathrm{cm}^{-3}\,\mathrm{s})$ 1.4 1.6 [@warren2004a]
$T_e$ (K) 3.2e7 [@kosenko2010a] 5.1e7 $3.6\pm0.6$e7 [@badenes2008a] 1.97e7
4.6 5.8e7 [@warren2004a]
$R_{Fe\textsc{xiv}}$ 2.18 2.55 2.8 2.9 2.76 2.96 2.81 2.85
(pc)
$W_{Fe\textsc{xiv}}$ $2460\pm94$ 3600 $4365\pm107$ 5117
($\mathrm{km}\,\mathrm{s}^{-1}$)
$F_{Fe\textsc{xiv}}$ 1.1e-14 0.9e-14
$(\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1})$
$M_{Fe\textsc{xiv}}$ 0.03 0.015
$(M_{\odot})$
$M_{Fetot}$ 0.38 0.515
$(M_{\odot})$
----------------------------------------------------- ------------------------------ ---------- ------------------------------- ------------
: \[table1\]Explosion energies $E_{51}$ ($10^{51}\,\mathrm{erg}$), ejecta masses $M_{ej}$ (solar masses), ISM densities $n_{ISM}$ ($\mathrm{amu}\,\mathrm{cm}^{-3}$), assumed age, modeled forward and reverse shock velocities and radii, and for 0519 ejecta ionization age and electron temperature, compared with observational values from literature references. Modeled and observed radii, width, \[Fe<span style="font-variant:small-caps;">xiv</span>\] 5303Å flux, Fe mass associated with \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission (includes \[Fe<span style="font-variant:small-caps;">x</span>\] also for 0509), and estimate of total SNR Fe mass.
Table 1 gives a summary of our SNR models based on [@truelove1999a] for 0519 and 0509, designed to match forward and reverse shock velocities ($v_f$ and $v_r$) and radii ($R_f$ and $R_r$) at the current epoch. The ejecta mass is $M_{ej}$, $E_{51}$ is the explosion kinetic energy in $10^{51}\,\mathrm{erg}$, and $n_{ISM}$ is the ambient density in $\mathrm{amu}\,\mathrm{cm}^{-3}$. For 0519 we take $M_{ej} = 1.4\,{\ensuremath{\mathrm{M}_\odot}}$, E$_{51} = 1$, and a chemical composition 33% O, 12% Si and 55% Fe mass, to match the results of X-ray analysis [@kosenko2010a]. With $n_{ISM} = 1.5\,\mathrm{cm}^{-3}$ taken to match the forward shock, we also get good agreement for the ejecta ionization age and electron temperature. Currently the reverse shock in 0519 has passed through approximately 95% of the ejecta (mass coordinate 0.05), and the \[Fe<span style="font-variant:small-caps;">xiv</span>\]-emitting plasma is near mass coordinate 0.2, expanding with $v_{exp} = 1887\,\mathrm{km}\,\mathrm{s}^{-1}$. For further details on our SNR hydrodynamical evolution model and ionization structure calculations see the Supplemental Information.
In the case of 0509, while adopting an explosion energy $E_{51} = 1.5$ and $n_{ISM} = 0.4\,\mathrm{cm}^{-3}$ allows the forward shock radius and velocity to be matched as well as the emission measure of shocked ISM, a similar ejecta mass and composition to 0519 do not allow the Fe to ionize as far as $\mathrm{Fe}^{13+}$. However, a smaller ejecta mass $M_{ej}$ = 1[$\mathrm{M}_\odot$]{} allows the reverse shock to reach the ejecta core-envelope boundary, where the maximum ionization age occurs, earlier in the SNR evolution. This produces sufficient $\mathrm{Fe}^{13+}$ and $\mathrm{Fe}^{14+}$ here to generate brighter \[Fe<span style="font-variant:small-caps;">xiv</span>\] 5303Å than \[Fe<span style="font-variant:small-caps;">xi</span>\] 7892Å or \[Fe<span style="font-variant:small-caps;">x</span>\] 6376Å, neither of which are unambiguously detected. We note that this high explosion energy is realistic and can be readily obtained from detonation of a 1[$\mathrm{M}_\odot$]{} white dwarf with a 0.85[$\mathrm{M}_\odot$]{} core consisting of 60% carbon and 40% oxygen (by mass) surrounded by a 0.15[$\mathrm{M}_\odot$]{} shell of helium. Burning 0.5[$\mathrm{M}_\odot$]{} of the core to iron-group elements (using the binding energy of $^{56}$Ni) and the remainder of the star to intermediate mass elements (using the binding energy of $^{28}$Si) gives a kinetic energy of $1.5\times10^{51}\,\mathrm{erg}$, after accounting for the gravitational binding energy $E_{g} = -4.6\times 10^{50}\,\mathrm{erg}$ and the internal energy $E_{int} = 2.9\times 10^{50}\,\mathrm{erg}$.
For the 1[$\mathrm{M}_\odot$]{} ejecta model, the reverse shock in 0509 has passed through approximately 74% of the ejecta (mass coordinate 0.26) at the present time, and the \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission originates from mass coordinates ${\sim}0.5 - 0.7$, expanding with $v_{exp} = 4766\,\mathrm{km}\,\mathrm{s}^{-1}$. Table 1 gives a summary of parameters connected with the \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission for both remnants. There is good agreement between predicted and observed radii, with the observations giving a wider range of values. Presumably this arises partly from simple projection effects and partly from deviations of the SNR geometry from spherical symmetry. The line widths, however, are over-predicted by about 10 20%. The theoretical prediction is directly connected to the speed of the reverse shock and is possibly affected by the parametrization of the ejecta density profile by a uniform density core, or by clumping of the ejecta, which would slow down the reverse shock.
In Table 1, the de-reddened fluxes in \[Fe<span style="font-variant:small-caps;">xiv</span>\] are given for the two remnants, with an estimate of the Fe mass in all charge states associated with the \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission, coming from our ionization balance calculations. The final row of Table 1 gives an estimate of the total Fe in each remnant. To the Fe associated with \[Fe<span style="font-variant:small-caps;">xiv</span>\], we add the mass of currently unshocked ejecta ($0.18 \times 1.4 = 0.25\,{\ensuremath{\mathrm{M}_\odot}}\ $ for 0519, $0.5 \times 1.0 = 0.5\,{\ensuremath{\mathrm{M}_\odot}}\ $ for 0509), assumed all Fe, and for 0519 we add estimates of the shocked Fe mass seen in X-rays \[18\]. For further details on the Fe mass estimate from the observed line flux see the Supplemental Information.
The characteristic velocity, distance, and time in our models depend on $(E_{51}/M_{ej})^{1/2}$, $(M_{ej}/n_{ISM})^{1/3}$, and $M_{ej}^{5/6}E_{51}^{-1/2}n_{ISM}^{-1/3}$, respectively, so in Table 1 only $n_{e}t$ and $T_e$ change if $E_{51}$, $M_{ej}$, and $n_{ISM}$ vary by the same factor. A factor of ${\sim}4$ increase in $n_{e}t$ is required to improve the agreement between predicted and measured $n_{e}t$ for 0509, which conflicts with established Type Ia SN theory. If we solely increase $M_{ej}$ and the age for 0509, $n_{e}t$ and $T_{e}$ increase somewhat, but simultaneously $v_f$ decreases and $R_f$ increases, worsening the prediction of the forward shock trajectory. A modest increase in $M_{ej}$ by ${\sim}0.2 - 0.4\,{\ensuremath{\mathrm{M}_\odot}}\ $ is allowable but would require the Fe to be embedded in He-rich ejecta to achieve the necessary degree of ionization. Such a near Chandrasekhar-mass scenario with unburned helium in the ejecta seems unlikely, but we cannot firmly rule out a near Chandrasekhar-mass explosion as for example in [@badenes2008a]. The larger mass makes the reverse shock slower, brings the \[Fe<span style="font-variant:small-caps;">xiv</span>\] width into better agreement with observations, and increases our estimate for the total Fe mass because the slower reverse shock has not propagated as far through the ejecta. However, the most satisfactory explanation for the $n_{e}t$ values is that the strong Si, S, Ar, and Ca emission seen in X-rays [@warren2004a] arises from ejecta clumps, with densities locally enhanced by a factor of ${\sim}4$. This gives a predicted $n_{e}t$ of order $10^{10}\,\mathrm{cm}^{-3}\,\mathrm{s}$. Using an electron density of $4\,\mathrm{cm}^{-3}$ to interpret the emission measures given in [@warren2004a] then yields masses of clumped ejecta of 0.068, 0.035, 0.007, and 0.003[$\mathrm{M}_\odot$]{} for Si, S, Ar, and Ca, respectively, implying that a total of about 0.11[$\mathrm{M}_\odot$]{} out of a total shocked ejecta mass of about 0.74[$\mathrm{M}_\odot$]{} is clumped by a factor of about 4. Approximately 0.2[$\mathrm{M}_\odot$]{} of the shocked ejecta mass is then visible in \[Fe<span style="font-variant:small-caps;">xiv</span>\] and \[Fe<span style="font-variant:small-caps;">xv</span>\]. Therefore, we favor the low mass – high explosion energy scenario.
A remaining question is why 0509 exhibits clumpy ejecta while 0519 apparently does not. Aside from being more than twice as old as 0509, 0519 is in a significantly more advanced evolutionary state due to its higher ambient density. Presumably, all ejecta clumps in 0519 have been destroyed by instabilities following reverse shock passage [@pittard2016a; @orlando2010a], whereas this has not yet occurred in 0509. Kelvin-Helmholtz and Richtmyer-Meshkov instabilities typically destroy clumps on a timescale of a few clump shock crossing times. Clumping of Fe in 0509 would remove the need for a He-dominated composition in the 1.4[$\mathrm{M}_\odot$]{} model for explaining the Fe ionization, but poses problems in that clumping of SN ejecta is usually assumed to occur as a result of the inflation of Fe-Co-Ni bubbles by radioactivity. Fe should therefore be under-dense, though [@wang2001a] interpret Fe knots as being due to $^{54}$Fe.
In addition to the ubiquitous \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission, we also find three additional broad lines in 0509, which we identify as coronal \[S<span style="font-variant:small-caps;">xii</span>\] 7613.1Å, \[Fe<span style="font-variant:small-caps;">ix</span>\] 8236.8Å, (Fig. 3A) and \[Fe<span style="font-variant:small-caps;">xv</span>\] 7062.1Å (Fig. 3B). We also detect \[Fe<span style="font-variant:small-caps;">xv</span>\] 7062.1Å in N103B. The presence of these further coronal lines in addition to \[Fe<span style="font-variant:small-caps;">xiv</span>\] opens the door to a new field of study: supernova remnant tomography, the study of spatially resolved, optical coronal line emission from non-radiative reverse shocks in Type Ia supernova ejecta. The energetics of SNRs means that most of the emission from shocked ejecta is radiated at X-ray frequencies, observed with relatively poor spectral and spatial resolution due to technical limitations on the available instrumentation. Study of the optical coronal line profiles allows for the measurement of Doppler shifts and broadening. Furthermore, since the emission arises from much closer to the reverse shock than the X-ray emission, it is more sensitive to shock and pre-shock parameters. In contrast, the X-ray observations probe only the clumped ejecta, providing a less accurate picture of the spatial distribution of explosion products than the optical \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission.
In the cases discussed here, the best match to the forward and reverse shocks pushes the SNR age to one end or the other of the uncertainty range coming from the light echoes and constrains the ejecta masses to around 1.4[$\mathrm{M}_\odot$]{} for 0519 and likely to significantly below the Chandrasekhar mass for 0509 (${\sim}1.0\,{\ensuremath{\mathrm{M}_\odot}}$). In the absence of such information, the SNR age is much less constrained, with corresponding greater uncertainties in ejecta mass and explosion energy. Our dynamical models give a good match to the spectral properties of 0519 and 0509, with some clumping of the ejecta required for the latter SNR. Last, we note that the observed light echo spectra enabled [@rest2008a] to assign the supernova that gave rise to 0509 to the spectroscopic sub-class of 1991T-like SNe Ia. Taking our explosion mass constraint at face value, this indicates that 1991T-like SNe Ia originate from detonations of sub-Chandrasekhar mass white dwarfs.
Acknowledgements
================
This research has made use of the following PYTHON packages: MATPLOTLIB [@hunter2007a], ASTROPY [@astropy2013a; @astropy2018a], a community-developed core PYTHON package for Astronomy APLPY [@robitaille2012a], an open-source plotting package for PYTHON, ASTROQUERY [@ginsburg2017a], a package hosted at https://astroquery.readthedocs.io which provides a set of tools for querying astronomical web forms and databases, STATSMODEL [@seabold2010a] and BRUTIFUS [@vogt2019a], a PYTHON module to process data cubes from integral field spectrographs. This research has also made use of the ALADIN [@bonnarel2000a] interactive sky atlas, of SAOIMAGE DS9 [@joye2003a] developed by Smithsonian Astrophysical Observatory, of NASA’s Astrophysics Data System, and of NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. IRS was supported by the Australian Research Council through grant number FT160100028. PG acknowledges support from the rector funded visiting fellowship scheme at the University of New South Wales in Canberra. JML was supported by the NASA ADAP Program Grant NNH16AC24I and by Basic Research Funds of the Chief of Naval Research. FPAV acknowledges an ESO fellowship.
Author contributions:
=====================
Conceptualisation: IRS; PG; Formal Analysis: IRS, JML; Investigation: IRS, PG, JML; Methodology: IRS, PG, JML; Project administration: IRS; Resources: FPAV; Software: IRS, JML, FPAV; Visualization: IRS, JML, FPAV; Writing – original draft: IRS, PG, JML, FPAV; Writing – review & editing: IRS, PG, JML, FPAV.
\
MUSE observations and data reduction
====================================
SNR 0519-69.0 was observed with the Multi-Unit Spectroscopic Explorer (MUSE) on UT4 at the Very Large Telescope (VLT), under P.Id. 096.D-0352\[A\] (P.I: Leibundgut), on 17.01.2016 and 18.01.2016, for a total $6\times900\,\mathrm{s}$ on-source. SNR 0509-67.5 was observed with MUSE, under P.Id. 0101.D-0151\[A\] (P.I.: Morlino), on 21.11.2017, 15.12.2017, and 22.01.2018, for a total of $16 \times 701$s on-source. SNR N103B was observed with MUSE, under P.Id. 096.D-0352\[A\] (P.I: Leibundgut), on 12.12.2016 and 17.12.2016, for a total of $16 \times 900$s on-source. We downloaded the 6, 16, and 16 raw MUSE frames for SNR 0519-69.0, SNR 0509-67.5, and SNR N103B (respectively) from the ESO Science Archive Facility, together with the associated raw calibration selected via calselector. Each raw frame was reduced individually using the MUSE pipeline 2.4.2 [@weilbacher2015a] via its workflow in Reflex v2.9.1 [@freudling2013a]. The image quality was measured manually from stars in the field-of-view, in all the individual datacubes, at 7000Å. For SNR 0519-69.0 and SNR N103B, all the individual observations have an image quality in the range 0.8$^{\prime\prime}$ 0.9$^{\prime\prime}$ and 0.5$^{\prime\prime}$ 0.8$^{\prime\prime}$ (respectively, so that we used them all to assemble the final combined cubes, which have an image quality of 0.8$^{\prime\prime}$ and 0.6$^{\prime\prime}$ for a total of 5,400s and 14,400s on-source, respectively. For SNR 0509-67.5, the first 8 exposures have an image quality in the range 1.0$^{\prime\prime}$ 1.5$^{\prime\prime}$, whereas the last 8 exposures, acquired in January 2018, have an image quality in the range 0.7$^{\prime\prime}$ 0.9$^{\prime\prime}$. To facilitate the removal of the stellar continuum in the field (see below) and maximize our ability to spatially resolve the shell structure of the remnant, we only use the 8 sharpest MUSE exposures to assemble the combined datacube, with a final image quality of 0.8$^{\prime\prime}$.
The lack of dedicated sky observations for all 3 targets led us to skip the sky subtraction step in the data reduction cascade, given the crowding of the fields and the underlying photo-ionized gaseous emission from the LMC. Instead, we use brutifus a Python package to process MUSE datacubes (https://fpavogt.github.io/brutifus/) to subtract the sky emission extracted from a handful of regions selected by hand in each cube (see Supplemental Figs. 4-6). These regions were chosen a) to avoid any bright star in the white-light image of the cube, b) to be located away from the extend of the SNRs, and c) to avoid the brightest region of nebular emission from the LMC ISM.
We correct all three combined datacubes for Galactic extinction along the line-of-sight using another dedicated brutifus routine. We assume a Fitzpatrick 1999 reddening law [@fitzpatrick1999a] with $R_v = 3.1$, $A_B= 0.272$ and $A_V=0.206$ (for all three systems), derived via NED from a recalibration [@schlafly2011a] of the infrared-based dust map of [@schlegel1998a]. The resulting flux correction is shown in Supplemental Fig. 7.
We also rely on brutifus to perform a crude subtraction of the stellar and nebular continuum of the three combined cubes. In practice, brutifus relies on the Locally Weighted Scatterplot Smoothing (LOWESS) algorithm [@cleveland1979a] to perform a non-parametric fit on a spaxel-by-spaxel basis. The advantage of this technique is that it is a) robust against the presence of emission lines, and b) can handle any type of smoothly varying continuum equally well, as illustrated by [@vogt2017b].
Hydrodynamics and Ionization Structure
======================================
We model the Fe coronal line forbidden emission using the method originally pioneered by Hamilton and Sarazin [@hamilton1984a]. Within a framework of analytic hydrodynamics describing the SNR evolution [@truelove1999a], we integrate equations for the time dependent ionization balance between the forward or reverse shocks. Our full method is described in [@laming2003a; @hwang2012a] who coined the acronym BLASt Propagation in Highly EMitting EnviRonment (BLASPHEMER).
Here we concentrate on Type Ia SNRs expanding into a uniform density interstellar medium (ISM). We take a core-envelope ejecta density profile, where the uniform density ejecta core is surrounded by an envelope with density proportional to r$^{-7}$. In all cases, the most highly ionized ejecta are found at the core-envelope boundary. We assume collisionless electron heating to $10^{6}$K ahead of the shock, following [@ghavamian2007a], followed by heating by adiabatic compression and Coulomb equilibration with the ions.
0519: We assume an ejecta composition of 33% O, 12% Si and 55% Fe by mass, following [@kosenko2010a]. Ahead of the reverse shock, O is 50% O$^{+}$ and 50% O$^{++}$, while Si and Fe are 25% singly ionized, 50% doubly ionized, and 25% triply ionized. Supplemental Figure 8 shows the results for 0519 on the left panels. The top panel shows the predicted radial extent of the Fe$^{9+}$, Fe$^{10+}$ and Fe$^{13+}$ ions. The reverse shock is predicted to be at 2.03 pc, so it can be seen the Fe$^{9+}$ comes up first, before Fe$^{10+}$ and Fe$^{13+}$ as expected, with Fe$^{13+}$ closest to the core-envelope boundary and hence the brightest due to being in the highest density. [@kosenko2010a] see low and high ionization Fe ejecta in X-ray emission, which we locate in even higher density ejecta, with the high ionization Fe ejecta located close to the core-envelope boundary with ionization age $n_{e}t = 3.9 \times 10^{10}\,\mathrm{cm}^{-3}\,\mathrm{s}$. The middle panel shows the electron density profile with radius, with a strong “spike” at a radius of 2.92 pc, corresponding to the core-envelope boundary. The bottom panel shows the time after explosion of reverse shock passage for ejecta at the different radii. The radius of the contact discontinuity is overestimated. [@truelove1999a] give no guidance on this so this has been taken from [@chevalier1982a], which is more appropriate for the earlier phases on SNR evolution.
0509: A similar ejecta composition and mass to 0519 do not ionize Fe as far as Fe$^{13+}$, and would predict significantly higher relative intensity in \[Fe <span style="font-variant:small-caps;">x</span>\] and \[Fe<span style="font-variant:small-caps;">xi</span>\] than is actually present. The simplest modification is to take a smaller ejecta mass $M_{ej} = 1\,\mathrm{M}_{\odot}$ that allows the reverse shock to reach the ejecta core-envelope boundary, where the maximum ionization age occurs, earlier in the SNR evolution, allowing ionization of Fe as far as Fe$^{13+}$ and Fe$^{14+}$, with \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission more intense compared to \[Fe<span style="font-variant:small-caps;">x</span>\] and \[Fe<span style="font-variant:small-caps;">xi</span>\]. The right hand panels of Supplemental Figure 5 show the same plots as previously for 0519. In the top panel, Fe$^{13+}$ and Fe$^{14+}$ are seen around the core-envelope boundary, while Fe$^{9+}$ and Fe$^{10+}$ exist at lower net regions surrounding it in the envelope and core. The density profile in the middle panel is not so strongly peaked as in 0519, suggesting stronger \[Fe<span style="font-variant:small-caps;">x</span>\] and \[Fe<span style="font-variant:small-caps;">xi</span>\] emission with respect to \[Fe<span style="font-variant:small-caps;">xiv</span>\]. For completeness, the time since explosion of reverse shock passage is similarly plotted in the bottom right.
Line Emission and Fe Mass Estimates
===================================
The Fe coronal forbidden lines are emitted from temperatures of order $ 1 - 2 \times 10^{7}\,\mathrm{K}$, well above those where they are emitted in conditions of ionization equilibrium. We take the emissivity in these lines due to electron impacts from the CHIANTI code. To this we add an emissivity due to impacts by heavy ions in the plasma, calculated using a generalization of methods in [@laming1996a]. The line flux, f, in photons cm$^{-2}\,\mathrm{s}^{-1}$, is given by $f = (C_{e}n_e + C_in_i) n_{Fe}V / 4 \pi d^2$, where $C_en_e + C_in_i$ is the collisional excitation rate due to scattering electrons, $n_e$, and ions $n_i$, with excitation rate coefficients $C_{e}$ and $C_i$ respectively, $n_{Fe}$ is the target ion density, V is the volume of emitting plasma, and d = 50 kpc is the distance to the SNR. The total target ion mass is $$Vn_{Fe}m_{Fe} = 4\pi d^2 f m_{Fe} / (C_{e}n_e + C_in_i)$$
where $m_{Fe}$ is the target ion mass. The Fe mass connected with the \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission is calculated from equation 1, with $n_e$, and $n_i$ coming from the hydrodynamics and ionization evolution calculations, $C_{e}$ and $C_i$ given in Supplemental Table 2, $d = 50\,\mathrm{kpc}$ and $f$ measured as described above. Finally, we correct for the Fe ionization balance, also coming from the hydrodynamics and ionization evolution, to give a mass in all Fe charge states in the region from where \[Fe<span style="font-variant:small-caps;">xiv</span>\] photons are emitted.
In 0519 we take the low and high ionization Fe emission measures $EM_{Fe} = Vn_en_i$ quoted by [@kosenko2010a] and convert to masses according to $M_{Fe} = EM_{Fe} m_{Fe} / n_e$. Finally in both remnants, we assume the ejecta interior to the \[Fe<span style="font-variant:small-caps;">xiv</span>\] emission is dominated by Fe, and multiply the relevant ejecta mass coordinates (0.18 for 0519, 0.5 for 0509) by 1.4 M$_{\odot}$ (for 0519) and 1.0 M$_{\odot}$ (for 0509) to complete the mass estimate.
----------------------------------------------------- --------------------- --------------------- --------------------- --------------------- ---------------------
Transition $e^-$ 1.6e7 K p 5e8 K $\alpha$ 2e9 K $C^{4+}$ 6e9 K $O^{6+}$ 8e9 K
Fe<span style="font-variant:small-caps;">xiv</span> $6.0\times 10^{-9}$ $1.4\times 10^{-9}$ $2.1\times 10^{-8}$ $3.1\times 10^{-8}$ $3.4\times 10^{-8}$
Fe<span style="font-variant:small-caps;">x</span> $3.7\times 10^{-9}$ $1.1\times 10^{-9}$ $7.7\times 10^{-9}$ $9.7\times 10^{-9}$ $1.3\times 10^{-8}$
----------------------------------------------------- --------------------- --------------------- --------------------- --------------------- ---------------------
: Electron and ion impact excitation rates for \[Fe<span style="font-variant:small-caps;">xiv</span>\] 5303Å and \[Fe<span style="font-variant:small-caps;">x</span>\] 6376Å transitions.
|
---
abstract: 'We study the dynamical evolution of globular clusters using our Hénon-type Monte Carlo code for stellar dynamics including all relevant physics such as two-body relaxation, single and binary stellar evolution, Galactic tidal stripping, and strong interactions such as physical collisions and binary mediated scattering. We compute a large database of several hundred models starting from broad ranges of initial conditions guided by observations of young and massive star clusters. We show that these initial conditions very naturally lead to present day clusters with properties including the central density, core radius, half-light radius, half-mass relaxation time, and cluster mass, that match well with those of the old Galactic globular clusters. In particular, we can naturally reproduce the bimodal distribution in observed core radii separating the “core-collapsed" vs the “non core-collapsed" clusters. We see that the core-collapsed clusters are those that have reached or are about to reach the equilibrium “binary burning" phase. The non core-collapsed clusters are still undergoing gravo-thermal contraction.'
title: 'Understanding the Dynamical State of Globular Clusters: Core-Collapsed vs Non Core-Collapsed'
---
\[firstpage\]
Galaxy: kinematics and dynamics – Galaxies: star clusters: general – globular clusters: general – Methods: numerical
Introduction
============
Studying the evolution of dense star clusters, such as old globular clusters (GCs), is of great interest for a variety of branches of astronomy and astrophysics. The high central densities and high masses of GCs make them hotbeds for strong dynamical interactions facilitating formation of many exotic sources (e.g., X-ray binaries, millisecond radio pulsars, type Ia supernovae, and blue straggler stars). GCs are important targets for extragalactic Astronomy. Detailed observations of their spatial distribution in a galaxy can constrain, for example, the potential of the dark matter halo, and give clues to the assembly history of the galaxy. The old ages of GCs provide a direct window into the major star-formation episodes in the early universe.
One long-standing question regarding GCs concerns the nature of their progenitors. The observed young massive star clusters seem to be potential progenitors of the current GCs. The masses, typical sizes, and inferred dissolution timescales for the so-called “super star clusters", as observed, for example, in M51 , make them especially good candidates. Interestingly, similar to the Galactic GCs (GGC), the sizes of observed super star clusters are typically a few parsecs independent of the cluster mass . However, the link between these super star clusters and the old evolved GGCs as well as GCs in other galaxies remains speculative.
The main difficulty in establishing a link between the two populations is that almost a Hubble time of evolution separates them. Although the individual qualitative effects of each physical process in a GC has been known from decades of numerical studies [e.g., @2003gmbp.book.....H], it is impossible to estimate the collective effect of these interdependent processes unless a self-consistent simulation is done including them all. For example, two-body relaxation leads to a slow energy diffusion from the core to the halo in a GC leading to a slow and steady contraction of the core until the gravo-thermal instability occurs and the core collapses under gravity. Since, due to equipartition of energy, the low-mass stars are scattered to higher velocities relative to their high-mass counterparts, two-body relaxation also leads to mass segregation in the cluster. The high-mass stars sink to the core and the low-mass stars escape to the halo and preferentially get stripped through the tidal boundary of the cluster. Because of this preferential loss of low-mass stars, the stellar MF changes with time. The stellar MF, in turn, determines the fraction of low and high-mass stars in a cluster as well as the average stellar mass, affecting the mass-segregation timescale [@2004ApJ...604..632G]. The contraction of the core via two-body relaxation increases its density. The core density determines the interaction rate in the core. These rates affect the binary-single (BS) and binary-binary (BB) interaction probabilities at a given time. The BS and BB interactions in turn generate energy in the core and can support the core stopping further contraction, thus, affecting the central density. The BS and BB interactions also change the orbital properties of the binaries taking part in these scattering interactions. These changes in the binary orbits can alter the evolution pathways that would be taken by a given binary which in turn changes, for example, the rate of formation of exotic stellar populations such as X-ray binaries and blue straggles stars (BSS). Due to this complexity in the evolution of dense star clusters, the only way to learn more about the possible initial properties of the observed GCs is to perform numerical simulations including all of the above physical processes in tandem with reasonable accuracy for a large enough $N$. The study of GCs has a somewhat long and varied history during which numerical simulations and observations have complemented each other [@2003gmbp.book.....H]. In particular, understanding the physical processes in the cores of GGCs has been of prime interest since the evolution of these dense clusters are driven mainly by the energy generation in the core and the transport of this energy from the core. The GGCs are observed to show a clear bimodal distribution of the core sizes that separates the so-called “core-collapsed" clusters from the non core-collapsed clusters [@1996AJ....112.1487H; @2005ApJS..161..304M]. The core-collapsed clusters in general show a power-law slope in their density profiles near the center. In contrast, the density profiles of the non core-collapsed clusters are well described by a King profile [@1966AJ.....71...64K] and show a clear flat part near the center. Theoretical analysis based on the current estimated relaxation times for the GGCs indicate that the majority of the GGCs should have had a deep collapse [@1993ASPC...50..373D; @1996AJ....112.1487H]. Thus, before it was found that all GGCs contain dynamically significant numbers of binaries, theoretical studies focused on understanding the process of core-collapse via the balance of outward diffusion of energy from the core due to two-body relaxation and post-collapse evolution due to dynamical formation of binaries [e.g., @2003gmbp.book.....H]. After it was observed in the early 1990s that all GGCs contain sufficient number of binaries such that they must have been born with substantial primordial binary populations, theoretical studies focused on properties of clusters in the “binary-burning" phase in which the outward diffusion of energy via two-body relaxation is balanced by production of energy via dynamical hardening of binaries [e.g., @1994ApJ...431..231V; @2007ApJ...658.1047F]. It was also realized that even a small primordial binary fraction can support the core from deep collapse for more than a Hubble time [@2007ApJ...658.1047F]. However, comparison between theoretical predictions and observations show that the theoretically predicted core radii during the binary-burning phase for these clusters are at least an order of magnitude smaller than the observed core radii for the bulk of the GGCs [@1994ApJ...431..231V]. Additional energy sources were proposed to explain these large core sizes [e.g., @2007MNRAS.379L..40M; @2008MNRAS.tmp..374M; @2008IAUS..246..151C; @2007MNRAS.374..857T], however, these scenarios need rather special conditions and seem unlikely to be satisfied by most of the GGCs. Very basic questions remain. Does the bimodal distribution for the GGC core sizes separating the core-collapsed and non core-collapsed clusters indicate a physical difference between these clusters? What dynamical stage are the core-collapsed and non core-collapsed clusters in today? Can the large core sizes of the non core-collapsed (majority of the Galactic population) be explained without any special conditions simply as a result of $\approx 12\,{\rm{Gyr}}$ of evolution from realistic initial conditions?
In this study we present the results from a large number of computer simulations (224) with initial conditions drawn from a multidimensional grid, spanning all relevant parameter ranges as suggested by observations of young star clusters, with the goal of reproducing a population of old clusters similar to the GGCs. We use CMC, a [Hénon]{}-type Monte Carlo code including all physical processes such as single and binary stellar evolution, two-body relaxation, strong encounters comprising physical collisions and binary-mediated scattering, and tidal stripping due to the Galactic tidal field. CMC has been extensively tested and the results from CMC show excellent agreement with those from direct $N$-body simulations [e.g., @2007ApJ...658.1047F; @2010ApJ...719..915C; @2012ApJ...750...31U]. Our goal is to find whether starting from realistic initial conditions (including $N$, stellar mass function, central density, compactness parameter, binary fraction, and cluster size) typical of the observed young massive star clusters and without any special treatment clusters similar in properties to the observed old GGCs are naturally obtained after $\approx 12\,{\rm{Gyr}}$ of evolution. In particular, we focus on understanding the bimodal distribution of the observed GGC core radii to identify the dynamical stages of the so called core-collapsed and non core-collapsed clusters.
In Section \[sec:numerics\] we briefly explain our code and introduce working definitions for key structural parameters of a cluster. In Section \[sec:initial\_conditions\] we describe the multidimensional grid of initial parameters explored in this study. In Section \[sec:results\] we present our key results. In Section \[sec:core-collapsed\] we show our results identifying the dynamical evolutionary state for the observed core-collapsed GGCs. Finally we conclude in Section \[sec:conclusion\].
Numerical Method {#sec:numerics}
================
We use our [Hénon]{}-type Cluster Monte Carlo code (CMC) to numerically model star clusters with single and binary stars including all physical processes relevant in globular clusters such as two-body relaxation, single and binary stellar evolution, strong encounters including physical collisions and binary mediated strong scattering encounters. This code has been developed and rigorously tested over the past decade [@2000ApJ...540..969J; @2001ApJ...550..691J; @2003ApJ...593..772F; @2007ApJ...658.1047F; @2010ApJ...719..915C; @2012ApJ...750...31U; @2012arXiv1206.5878P]. Using a large grid of initial conditions over the range of values typical for observed young massive clusters we create over $200$ detailed star-by-star models and evolve them for $12\,\rm{Gyr}$. A handful (about 6) of these models reach a very deep collapse phase. For these models the CMC time steps become minuscule and the code grinds to a halt. We stop our simulations at that point for these clusters. In reality, the deep-collapse phase is halted via formation of the so-called “three-body" binaries and the cluster enters into the gravo-thermal oscillation phase. Since in CMC we do not yet include the possibility of creating new binaries via three body encounters, we do not address this phase at this stage. This however, is not a serious limitation. All clusters that reach a very deep collapse stage before the integration is stopped at $12\,{\rm{Gyr}}$, had zero primordial binaries which is not realistic [e.g., @2008AJ....135.2155D]. Here, we only include these clusters in our analysis as limiting cases. Even a small non-zero primordial binary fraction ($f_b$; lowest $f_b = 5\%$ is used in our simulations) can stop the cluster core from collapsing through the dynamical hardening of binaries preventing very deep core collapse. At this stage the core size remains more or less constant, which is also commonly referred to as “binary-burning" stage. A more detailed description of the code, and the various qualitatively different dynamical stages for a cluster’s evolution is presented in @2010ApJ...719..915C.
Since one key goal for this study is to compare the properties of our simulated models at about $12\,{\rm{Gyr}}$ with the properties of observed GGCs, we have to make sure the same parameter definitions are used. In particular there are different definitions for the core radius and the cluster size commonly used by theoreticians and observers. The three dimensional core radius $r_c$ widely used in $N$-body simulations is a density-weighted measure related to the virial radius in the core [@1985ApJ...298...80C]. In contrast, the core radius for observed clusters, ${r_{c, \rm{obs}}}$, is often defined as the distance from the center where the projected surface brightness profile drops to half its central value. Alternatively, when observations with sufficiently high resolution are available, the core radius can also be based on star-counts representing the radius where the projected surface number density is half the corresponding central value. If the core radius is calculated using the stellar number density profile, we will call it ${r^{N}_{c, \rm{obs}}}$, while the core radius based on the brightness, or luminosity, density profile is denoted by ${r^{L}_{c, \rm{obs}}}$. In real clusters the density profiles can have large scatter, especially close to the centre due to Poisson noise and the presence of bright giants that dominate the light but are only few in number. Hence, it is common practice to exclude the light from the brightest giants when calculating the luminosity density profile for real clusters to reduce noise [e.g., @2006AJ....132..447N]. For the purpose of this study we define ${r^{L, \rm{cut}}_{c, \rm{obs}}}$ as the core radius obtained from a luminosity density profile excluding giants with $L_\star > 20\,{L_\odot}$. Since the estimated value of ${r_{c, \rm{obs}}}$ is strongly dependent on the estimate of the peak luminosity density at the centre of a cluster, values of ${r^{L}_{c, \rm{obs}}}$ and ${r^{L, \rm{cut}}_{c, \rm{obs}}}$ can differ significantly. For real clusters even when star counts are available, only stars above a certain brightness can be counted due to completeness issues. Hence, we define ${r^{N, \rm{cut}}_{c, \rm{obs}}}$ calculated using a number density profile including only stars that are on the main-sequence or on the giant branch with masses $> 0.2\,{M_\odot}$, representing the currently achievable completeness limit [e.g., @2012MNRAS.422.1592L]. We find that the ${r^{N, \rm{cut}}_{c, \rm{obs}}}$ values and the ${r^{N}_{c, \rm{obs}}}$ values are not much different for our simulated clusters.
The three dimensional half-mass radius $r_h$ is defined as the radius that includes half of the total mass of the cluster. However, for real clusters this quantity is not directly accessible, rather, the two dimensional, projected distance enclosing half of the total light of the cluster is calculated and is defined as the half-light radius $r_{hl}$. We further define ${r^{\rm{cut}}_{hl, \rm{obs}}}$ for the half-light radius for our simulated models by calculating the same quantity but excluding bright giants with $L_\star > 20\,{L_\odot}$. We find that the ${r_{hl}}$ and ${r^{\rm{cut}}_{hl, \rm{obs}}}$ values show only minor differences.
Initial Conditions {#sec:initial_conditions}
==================
Our choice of initial conditions is based on observations of super star clusters. All our simulated clusters have initial virial radii between $r_v = 3$ – $4\,{\rm{pc}}$ independent of other cluster parameters. This range corresponds to initial three-dimensional half mass radii, $r_h$, ranging from $2$ – $3\,{\rm{pc}}$. These values are in agreement with observations of young, massive clusters that indicate that the effective cluster sizes are rather insensitive to the cluster mass, as well as metallicity and have a median value of $\approx 3\,\rm{pc}$. In addition, observations of old massive LMC clusters, old GCs in NGC 5128, old clusters in M51, as well as the GGCs indicate that the effective cluster radii show only a weak correlation with the distance from the galactic center .
To restrict the huge parameter space to a certain extent we place all our simulated clusters in a circular orbit at a moderate Galactocentric distance of $r_{G}=8.5\,\rm{kpc}$. We avoid modeling GCs very close to the Galactic center, where the Galactic field is so strong that the tidal stellar loss dominates the cluster’s evolution. Due to the assumption of spherical symmetry, Monte Carlo codes cannot directly model the tear-drop shaped tidal boundary of a star cluster. Instead these codes use some prescription based method [see a detailed discussion and calibration in @2010ApJ...719..915C]. If tidal dissolution is the dominant driver of the GC’s evolution, the approximate method may not be accurate enough. Choosing a circular orbit for the simulated clusters is a simplification; however, the results should still be valid for eccentric orbits with some effective Galactocentric distance ($>8.5\,\rm{kpc}$) [e.g., @2003MNRAS.340..227B]. The Galactic tidal field, and consequently the initial tidal radius, ($r_t$), for the clusters is calculated following Equation \[eq:rt\] [@2003MNRAS.340..227B], $$\label{eq:rt}
r_t = \left( \frac{GM_{\rm{cl}}}{2V_G^2}\right)^{1/3} R_G^{2/3}\, ,$$ using a Galactic rotation speed $V_G = 220\,\rm{km}\,\rm{s}^{-1}$. Here the cluster mass is denoted by $M_{\rm{cl}}$.
For our set of runs we vary the initial number of stars, $N_i$, between $4 -10\times10^5$, encompassing the bulk of the GGCs [@2005ApJS..161..304M]. The initial positions and velocities are sampled from a King model distribution function with dimensionless potential, $W_0$, in the range $4-7.5$. We vary the initial binary fraction, $f_b$, between $0-0.3$. The stellar masses of the stars, or primaries in case of a binary, are chosen from the IMF presented in @2001MNRAS.322..231K [their Equations $1$ and $2$] in the stellar mass range $0.1-100\,\rm{M_\odot}$. Secondary binary companion masses are sampled from a uniform distribution of mass ratios in the range $0.1\,\rm{M_\odot}-m_p$, where $m_p$ is the mass of the primary. The semi-major axes, $a$, for stellar binaries are chosen from a distribution flat in log within physical limits, namely between $5\times$ the physical contact of the components and the local hard-soft boundary. Although initially all binaries in our models are hard at their respective positions, some of these hard binaries can become soft during the evolution of the cluster. The cluster contracts under two-body relaxation and the velocity dispersion increases making initially hard binaries soft. Moreover, binaries sink to the core due to mass segregation where the velocity dispersion is higher than that at the initial binary positions. We include these soft binaries in our simulations until they are naturally disrupted via strong encounters in the cluster.
Basic Structural Parameters of Our Model Clusters {#sec:results}
=================================================
Here we present the evolution of some structural properties of the simulated clusters and compare them with the same properties of the observed GGCs. For each of these comparison plots, the evolution of a certain cluster property is shown together with the distribution of the corresponding observationally derived values for the GGC population. Since all our simulated models are at a Galactocentric distance of $8.5\,{\rm{kpc}}$, we also show the observed distributions for a subset of GGCs satisfying $7 \leq R_{G} \leq 10\,{\rm{kpc}}$. Our goal here is to simply test whether, starting from observationally motivated initial conditions typical for massive, young clusters, the final properties of the cluster ensemble naturally attain ranges of values as observed in the GGCs. We do not intend to reproduce the present day distribution for these properties since this would require to introduce the distribution of cluster initial conditions as another parameter, and, consequently, significantly more simulations which is beyond the scope of this study.
![Evolution of the star cluster mass for all simulated models. The solid (black), dotted (red), short-dashed (blue), long-dashed (green), and dot-dashed (magenta) lines denote models with primordial $f_b = 0$, $0.05$, $0.1$, $0.2$, and $0.3$, respectively. The histograms show the mass distributions of the observed GGCs. The solid histogram is for GGCs with Galactocentric distances between $7$ and $10\,{\rm{kpc}}$. The dashed histogram is for all observed GGCs. The masses of the observed GGCs are derived from their integrated V-band magnitudes [@1996AJ....112.1487H] using Equation \[eq:M/L\]. []{data-label="plot:t_mass"}](plots/t_mass_RG_Sol_fixed_rvir_all.ps){width="90.00000%"}
Figure \[plot:t\_mass\] shows the evolution of the total mass of our simulated clusters. The initial sharp decrease in the cluster mass ($M_{\rm{cl}}$) is because of the high mass loss rate via winds and compact object formation of the massive stars in the GC. Later on, $M_{\rm{cl}}$ decreases at a slower, nearly constant rate caused by a steady stellar mass loss through the tidal boundary of the GC [@2003MNRAS.340..227B; @2010MNRAS.tmp..844D]. The histograms on the right show the distribution of observed GGC masses. The observed GGC masses are estimated from the absolute visual magnitudes ($M_v$) given in @1996AJ....112.1487H using Equation \[eq:M/L\] assuming a uniform mass to light ratio ${M_{\rm{cl}}/L_{\rm{cl}}}= 2\,{M_\odot/L_\odot}$ for all clusters. $$M_{\rm{cl}} = 10^{ - (M_v - 4.75)/2.5 + 0.30103}.
\label{eq:M/L}$$ This is an approximation. The ${M_{\rm{cl}}/L_{\rm{cl}}}$ can vary from cluster to cluster and can also depend on the evolutionary stage of the cluster in question . However, since here we are only interested in reproducing the cluster mass ranges for most of the GGCs and do not try to model a particular cluster, this estimate should be appropriate. Figure \[plot:t\_mass\] shows that our model clusters have final masses typical for the bulk of the observed GGCs. Since these clusters are modeled with the total $N$ and masses typical of the observed GGCs the model properties can be directly compared with the overall GGC population without any need for scaling.
![Same as Figure \[plot:t\_mass\], but for the central density $\rho_c$. The $\rho_c$ values for the observed GGCs are also shown in histograms. The solid and the dashed histograms are from GGC populations selected as in Figure \[plot:t\_mass\]. []{data-label="plot:t_rhoc"}](plots/t_rho0_RG_Solar_fixed_rvir.ps){width="90.00000%"}
Figure \[plot:t\_rhoc\] shows the evolution of the central density in our simulated models. As a dynamically important GC property, the central density ($\rho_c$) determines the interaction cross-sections for strong scattering inside the core, for example, for BB and BS interactions, and stellar collisions. These strong interactions in turn modify the properties of the core, through, e.g., binary burning. The $\rho_c$ values sharply decrease during the first $\sim 1\,{\rm{Gyr}}$ of evolution as the high-mass stars lose mass via stellar winds, and compact object formation. Followed by the sharp decrease $\rho_c$ increases almost linearly over time during the gravo-thermal core contraction stage. The histograms show the central densities of the observed GGCs. Here we convert the bolometric luminosity densities presented in @1996AJ....112.1487H and assume ${M_{\rm{cl}}/L_{\rm{cl}}}= 2\,{M_\odot/L_\odot}$. The range of central densities for our collection of simulated models is compatible with the range for observed GGCs.
![Evolution of the three dimensional core radius $r_c$ for all simulated clusters with primordial binary fraction $f_b = 0$–$0.3$. The line colors and styles have the same meaning as in Figure \[plot:t\_mass\]. A few ($6$) clusters with no primordial binaries reach the deep collapse phase within the a Hubble time. We stop integrations for those clusters when this phase is reached. The histograms show the observed observed core radii for the GGCs. The solid and dashed histograms show GGCs with Galactocentric distances between $7$ – $10\,{\rm{kpc}}$ and all GGCs. []{data-label="plot:t_rc"}](plots/t_rc_phys_RG_Sol_fixed_rvir.ps){width="90.00000%"}
Figure \[plot:t\_rc\] shows the evolution of the density weighted three dimensional core radius $r_c$. The $r_c$ for all clusters sharply increases initially for up to about $1\,{\rm{Gyr}}$, because of mass loss due to stellar evolution. This mass loss happens mainly at the deepest part of the cluster potential since the high-mass stars reside near the cluster center due to mass segregation and are affected by mass loss the most. The resulting loss of gravitational binding energy expands the cluster core. Once the rate of mass loss goes down, the core contracts via diffusive energy transport from the core to the outside through two-body relaxation. The gravo-thermal contraction ceases when this energy flow is balanced by the production of energy via dynamical hardening of binaries, the binary-burning phase. In our collection of models about 20 clusters show clear binary-burning end stages exhibited by a near constant core radius at late stages ($t>8\,{\rm{Gyr}}$). The bulk of our models are still contracting at $t_{cl} = 12\,{\rm{Gyr}}$. A few of our models with $f_b = 0$ do not show the binary-burning stage and enter deep-collapse after gravo-thermal contraction. This is because we do not account for dynamical creation of binaries. However, as mentioned before, $f_b = 0$ is a limiting case and not representative for observed GCs. Even with $f_b = 5\%$, we find that the clusters that complete the slow contraction phase do not suffer deep collapse, rather reach a steady binary-burning energy equilibrium phase.
![Same as Figure \[plot:t\_rc\], but for the ratio of the three dimensional core radius to the three dimensional half mass radius ($r_c/r_h$). The $r_c/r_h$ values for the observed GGCs are also shown in histograms. The solid and the dashed histograms are from GGC populations selected as in Figure \[plot:t\_rc\]. []{data-label="plot:t_rcoverrh"}](plots/t_rcoverrh_RG_Sol_fixed_rvir.ps){width="90.00000%"}
Another reliably measured and frequently used parameter reflecting the dynamical state of the evolution of GCs is the ratio between the core radius $r_c$ and the half-mass radius $r_h$ [e.g., @2007ApJ...658.1047F]. Figure \[plot:t\_rcoverrh\] shows the evolution of $r_c/r_h$ for all our clusters. The range of final $r_c/r_h$ values of the simulated clusters agree well with the observed ones in the GGC population, producing values at $12\,\rm{Gyr}$ close to the peak of the observed distribution.
![Histogram of the ratio of the three dimensional code defined core radius $r_c$ to the observed two dimensional projected core radius $r_{c, \rm{obs}}$ for all simulated clusters. The observed core radii $r_{c, \rm{obs}}$ are estimated by using the distance from the center of the cluster where the density is half of the maximum density at the center. The different histograms show the distribution of values based on different methods to calculate the density distribution. The thin lines are for observed core radii values calculated using a density distribution including all stars in the cluster. Thick lines are for the same but here the density distribution is calculated using a subset of stars satisfying $0.01 {L_\odot}< L_\star < 20\,{L_\odot}$. These values ensure that only main-sequence stars $> 0.2\,{M_\odot}$ are included and very bright giants are excluded. The solid (black) and dashed (red) histograms are for observed core radii values calculated from the stellar luminosity density, and number density distributions, respectively. []{data-label="plot:rc_rcobs_hist"}](plots/rcoverrcobs_hist.ps){width="90.00000%"}
We should remind the readers, however, that these $r_c$ and $r_h$ values are not exactly the quantities observed directly. A more careful comparison between these structural parameters in our models and the observed values in the GGCs requires “observing" the models and defining the model quantities such as $r_c$ and $r_h$ as the observers define them for a real cluster (Section \[sec:numerics\]). For example, at $12\,{\rm{Gyr}}$ our model [run12]{} has the $N$-body defined $r_c = 2.6\,{\rm{pc}}$. If the same model is observed and the core radius is calculated using the surface number density distribution using all main-sequence stars satisfying $M_\star \geq 0.2\,{M_\odot}$ and low-luminosity ($L_\star < 20\,{L_\odot}$) giant stars, the core radius ${r^{N, \rm{cut}}_{c, \rm{obs}}}= 2.0$ (Table \[tab:list\]). Figure \[plot:rc\_rcobs\_hist\] shows the distribution of values for the ratio $r_{c, \rm{obs}} / r_c$ using $r_{c, \rm{obs}}$ values obtained in 4 different ways. The values for the different definitions of $r_{c, \rm{obs}}$ and the resulting ratios follow different distributions. Thus, if one wants to estimate what the three dimensional $N$-body defined $r_c$ value is from the observed value of $r_{c, \rm{obs}}$ and vice versa, one should be careful about how $r_{c, \rm{obs}}$ for that cluster was calculated. In general, $r_{c, \rm{obs}}$ calculated using the number density profile of the cluster show lower levels of errors. Moreover, the distributions do not change significantly based on which stars were included in the sample to calculate the surface number density profile. The code defined $r_c$ is typically between $1$ – $2$ times the $r^N_{c, \rm{obs}}$ for all cluster models in our collection. The luminosity density profiles are subject to a lot more noise compared to the number density profiles. As a result the $r_c/r^L_{c, \rm{obs}}$ calculated using the luminosity density profiles show a larger spread. In addition, which stars were included in calculating the luminosity density profile matters in this calculation significantly. Typically, the code-defined $r_c$ values are between $0.25$ – $1.5$ times and $0.5$ – $2.5$ times the $r_{c, \rm{obs}}^{L}$ and ${r^{L, \rm{cut}}_{c, \rm{obs}}}$, respectively.
![Histogram for the ratio of the three dimensional code defined half-mass radius $r_h$ to the two dimensional observed half light radius $r_{hl, \rm{obs}}$. The thin and thick lines have the same meaning as in Figure \[plot:rc\_rcobs\_hist\]. []{data-label="plot:rh_rhlobs_hist"}](plots/rhoverrhl_hist.ps){width="90.00000%"}
Similarly, only the half-light radius projected on the sky ($r_{hl}$) is observed in reality and not the three dimensional half mass radius ($r_h$). For example, for the simulated cluster [run12]{} the sky projected half light radius including all stars is $r_{hl, \rm{obs}} = 4.1\,\rm{pc}$. If a luminosity cut-off is used for the same cluster the half light radius becomes ${r^{\rm{cut}}_{hl, \rm{obs}}}= 4.4\,\rm{pc}$. The theoretically calculated three dimensional half-mass radius for the same cluster at the same age is $r_h = 7.0\,\rm{pc}$ (Table \[tab:list\]). This difference is not simply due to the projection effect. This difference depends intricately on the positions of the bright giant stars that dominate the light but are few in number. Figure \[plot:rh\_rhlobs\_hist\] shows the distribution of the ratio $r_h/r_{hl}$ calculated using two separate samples of stars, i.e., with and without applying a luminosity cut-off. Typically, the three dimensional half mass radius $r_h$ is between $1.5$ – $2$ times the observed ${r^{\rm{cut}}_{hl, \rm{obs}}}$ if the luminosity cut described in Section \[sec:numerics\] is used. Note that this range includes the expected geometric factor of $\sqrt{3/2}$ due to projection effect. The spread in values is moderately larger for $r_h/r_{hl, \rm{obs}}$. This is due to the increased statistical fluctuations created by the bright and low number of giants with $L_\star > 20\,{L_\odot}$.
We are not the first to point out this discrepancy in definitions. For instance, @2007MNRAS.379...93H and @2010ApJ...708.1598T already found that the $N$-body definition of $r_c/r_h$ can differ from an observed $r_c/r_h$ by a factor of a few, and our results confirm that. However, Figures \[plot:rc\_rcobs\_hist\] and \[plot:rh\_rhlobs\_hist\] are expected to be useful for observers and theorists modeling GCs since the three dimensional $r_c$ or $r_h$ can be be easily estimated using the presented distributions if the observed values for these are known.
![The observed half-light radius $r_{hl, \rm{obs}}^{\rm{cut}}$ vs the central density ($\rho_c$) for all simulated clusters. Histograms on either side show the $r_{hl, \rm{obs}}$ and $\rho_c$ distributions for the observed GGCs. The $r_{hl, \rm{obs}}$ and $\rho_c$ distributions in our models agree well with the bulk of the observed GGC values. The solid and the dashed histograms have the same meaning as in Figure \[plot:t\_rc\]. []{data-label="plot:rh_rhoc"}](plots/rhobs_rhocobs.ps){width="90.00000%"}
We now compare the values of the structural parameters for our simulated models calculated in a similar way as they are calculated for observed GGCs. Figure \[plot:rh\_rhoc\] shows the $\rho_c$ as a function of $r^{\rm{cut}}_{hl, \rm{obs}}$. The observed values for all GGCs and GGCs in the Solar neighborhood are also shown in the histograms along the respective axes. The two clusters of points show $r^{\rm{cut}}_{hl, \rm{obs}}$ vs $\rho_c$ values for clusters with initial $r_v = 4$ and $3\,{\rm{pc}}$, respectively. Our collection of models shows very similar half-light radii and central densities compared to those properties for the bulk of the GGCs. Note that the final ${r^{\rm{cut}}_{hl, \rm{obs}}}$ values are strongly dependent on the initial $r_v$ for a given Galactocentric distance ($r_{G}$) as evidenced in the clustering of the points in Figure \[plot:rh\_rhoc\]. Interestingly, for GCs at $r_{G}\approx8.5\,{\rm{kpc}}$ the half-light radii remain close to the initial $r_v$ according to our models. Thus, the present observed half-light radii can be used as an effective indicator for what the initial $r_v$ was for these clusters. The other parts of the observed histograms for ${r^{\rm{cut}}_{hl, \rm{obs}}}$ can be easily populated if more simulations are performed using smaller initial $r_v$. In addition, using smaller $r_{G}$ will also lead to smaller final ${r^{\rm{cut}}_{hl, \rm{obs}}}$ values for these clusters as they cannot expand further once they fill their tidal radii.
![${r^{N, \rm{cut}}_{c, \rm{obs}}}$ vs $\rho_c$. The solid and dashed histograms have the same meaning as in Figure \[plot:t\_rc\]. []{data-label="plot:rc_rhoc"}](plots/rcobs_rhocobs.ps){width="90.00000%"}
Figure \[plot:rc\_rhoc\] compares the final values of ${r^{N, \rm{cut}}_{c, \rm{obs}}}$ and $\rho_c$ for all our simulated models with the observed core radius distribution for the GGCs. As expected, the smaller the ${r^{N, \rm{cut}}_{c, \rm{obs}}}$, the higher the $\rho_c$. Our simulated models populate a very similar range in ${r^{N, \rm{cut}}_{c, \rm{obs}}}$ values as is observed in the bulk of the GGCs. This is also true when other definitions of ${r_{c, \rm{obs}}}$ are used for the comparison.
![Ratio of the observed core radius to the observed half light radius ${r^{N, \rm{cut}}_{c, \rm{obs}}}/ {r^{\rm{cut}}_{hl, \rm{obs}}}$ vs the central density $\rho_c$. The histograms have the same meaning as in Figure \[plot:rh\_rhoc\]. []{data-label="plot:rcoverrh_rhoc"}](plots/rcoverrh_rhoc_obs.ps){width="90.00000%"}
Figure \[plot:rcoverrh\_rhoc\] shows the scatter plot for the ${r^{N, \rm{cut}}_{c, \rm{obs}}}/{r^{\rm{cut}}_{hl, \rm{obs}}}$ and $\rho_c$ for our models together with the corresponding histograms for the observed GGCs. The values for this dynamically important and dimensionless measure for the compactness of the cluster show excellent agreement with the values typically found in all observed GGCs.
![Comparison of the half-mass relaxation time $t_{r,h}$ for our models with the observed GGCs. Solid (black) and dashed (black) histograms show the distributions for the $t_{r,h}$ values of the GGCs in the Solar neighborhood ($7 \leq r_G \leq 10\,{\rm{kpc}}$), and all GGCs, respectively. The dash-dotted (red) histogram is for the $t_{r,h}$ values from our models calculated being consistent with the assumptions made in estimating $t_{r,h}$ values in the Harris catalog of GGCs [see text; @1996AJ....112.1487H]. The dotted histogram (green) shows the $t_{r,h}$ values from our models if they are calculated using the actual values of $M_{\rm{cl}}$, $<M_\star>$, and $r_h$. The bin at $\log(t_{r,h}/\rm{Myr})$ just above 2 for the histogram of the Solar neighborhood clusters consists of a single rather unusual sparse cluster E3. The model $t_{r,h}$ values show good agreement with the $t_{r,h}$ values of the bulk of the Solar neighborhood GGCs. []{data-label="plot:trh_comp"}](plots/trh_compare_Djorgovski_sim.ps){width="90.00000%"}
Figure \[plot:trh\_comp\] shows the distributions of relaxation times at $r_h$ ($t_{r,h}$) for our models, all GGCs, and a subsample of the GGCs in the solar neighborhood ($7 \leq r_G \leq 10\,{\rm{kpc}}$). The relaxation time is a very important dynamical quantity because it is the time scale on which the global cluster properties evolve. However, it is difficult to derive this quantity accurately for observed clusters since it depends on several dynamical cluster properties that are not directly observable. Traditionally, multiple assumptions are made to calculate $t_{r,h}$ for observed GGCs [@1996AJ....112.1487H see the bibliography in the online database]. To remain consistent with the derived values in @1996AJ....112.1487H we adopt the same assumptions to calculate the $t_{r,h}$ values for our models. At the end of the simulation ($\approx 12\,{\rm{Gyr}}$) we calculate the total luminosity ($L_{\rm{cl}}$) of the model cluster, and compute its total mass using $M_{\rm{cl}}/L_{\rm{cl}}=2\, {M_\odot}/{L_\odot}$ (actually can have values between $1.2$ – $2.2\,{M_\odot/L_\odot}$ in our models). The total number of stars is then calculated assuming an average stellar mass $<M_\star> = 1/3\,{M_\odot}$ (actually can have values between $0.35$ – $0.39\,{M_\odot}$ in our models). We use ${r^{\rm{cut}}_{hl, \rm{obs}}}$ as a proxy for $r_h$. We estimate the $t_{r,h}$ values for our models using Equation 11 of @1993ASPC...50..373D with the corrected coefficient as mentioned in @1996AJ....112.1487H. The $t_{r,h}$ values for our models calculated this way agree well for the GGCs in the Solar neighborhood. The bin showing an unusually low $t_{r,h}$ value just above $\log(t_{r,h}/\rm{Myr}) = 2$ contains a single cluster E3. A quick search indicates that E3 is a sparse unusual cluster. Again we remind the readers that here we are only interested in comparing the ranges of $t_{r,h}$ from our models and the observed GGCs. Note that the values for $t_{r,h}$ based on the actual cluster parameters can be a few times higher depending on the particular cluster properties. The differences between the actual dynamical $t_{r,h}$ values and those derived for the observed clusters come essentially from the many assumptions listed above.
A long standing puzzle has been the apparent discrepancy between the theoretically predicted $r_c/r_h$ values and the values observed for the GGCs. Early numerical simulations as well as analytical studies expected that most GGCs are in the binary-burning stage. However, the simulated $r_c/r_h$ values resulting from binary burning have been found to be about an order of magnitude smaller than that for the bulk of the observed population [e.g., @1994ApJ...431..231V; @2007ApJ...658.1047F] and it was already known that this amount of discrepancy cannot be entirely coming from differences in definitions [e.g., @2007MNRAS.379...93H; @2010ApJ...708.1598T]. Consequently, additional energy sources in the core to expand $r_c$ have been investigated. Several studies proposed different additional energy generation mechanisms to explain the large observed $r_c/r_h$ values [e.g. @2007MNRAS.374..857T; @2008IAUS..246..151C; @2008ApJ...673L..25F; @2008MNRAS.tmp..374M]. However, these additional energy sources require rather special conditions. For example, high central densities are required to create a population of high-mass stars via physical collisions that can then suffer expedited mass loss via compact object formation [@2008IAUS..246..151C]. On the other hand, presence of an intermediate mass black hole also cannot be common for the GGCs [@2007MNRAS.374..344T]. Our models, in contrast, are generated without any special assumptions using observationally motivated initial conditions, and they naturally create a population of model clusters with properties in excellent agreement with the bulk of the observed GGC properties. These results indicate that the progenitors of today’s GGCs were very similar in properties to the young massive clusters observed, for example, in M 51 . Our results also indicate that the majority of today’s GGCs have not yet reached the binary-burning, energy-equilibrium stage, and are still contracting under energy transport via two-body relaxation.
What is a “core-collapsed" cluster? {#sec:core-collapsed}
===================================
![Examples for a binary-burning cluster (left) and a core-contracting cluster (right) from our models chosen randomly. The top panels show the evolution of $r_c/r_h$ for each clusters. The bottom panels show the surface luminosity density profiles calculated excluding the bright ($L_\star > 20\,{L_\odot}$) giants to reduce noise. The errorbars are estimated $1\sigma$ Poisson errors for each bin. The binary-burning model shows a clear power-law slope until the data is too noisy. In contrast, the core-contracting model shows a clear King density profile. []{data-label="plot:example"}](plots/core_collapsed_vs_non_core_collapsed_sbp.ps){width="90.00000%"}
A lot of early theoretical work was devoted to understand the gravo-thermal collapse and subsequent evolution of the core and the cluster as a whole due to hardening of primordial binaries. However, analytical results as well as numerical simulations showed that the $r_c$ values in the binary-burning phase are about an order of magnitude too small compared to the bulk of the GGC cores.
Our simulation results presented in Section \[sec:results\] show that this is simply because the bulk of the observed GGCs are [*not*]{} in binary-burning equilibrium stage. Rather they are still contracting. Now we focus on understanding the clearly bimodal distribution of the core radii of the GGCs. Depending on the shape of the observed cluster density profiles, all GGCs are divided into two categories, namely core-collapsed and non core-collapsed clusters [e.g., @1996AJ....112.1487H; @2005ApJS..161..304M]. The so called core-collapsed clusters exhibit a power-law increase in the density profile until the limit of resolution of the observation. The non core-collapsed clusters show a very clear flattening of the density profile and the profile for these clusters are well fitted with a King profile [@1966AJ.....71...64K]. Is there a distinct difference dynamically between the two categories of clusters?
Figure \[plot:example\] shows examples of two representative clusters chosen randomly from our large collection of models, one is in a clearly binary-burning stage and the other is still contracting. The evolution of $r_c/r_h$ is shown for the two clusters as well as their respective surface density profiles. The binary-burning cluster can be clearly identified by the near constant value of $r_c/r_h$ starting at about $8\,{\rm{Gyr}}$. The core contracting cluster shows a constant rate of contraction until $12\,{\rm{Gyr}}$. The surface density profiles for the two clusters are very different. The surface density profile for the binary-burning cluster looks very much like a so called core-collapsed cluster and exhibits a clearly power-law slope to very small radius below which the profile is noisy. In contrast, the surface density profile for the core contracting cluster shows a clear King profile with a distinct flat part near the center below a few parsecs.
![Distribution of $r_{c, \rm{obs}}^{N, \rm{cut}}$/$r_{hl, \rm{obs}}^{\rm{cut}}$. The thick and thin lines denote values from our simulated models and observed GGCs, respectively. The solid and dashed histograms show distributions for model clusters in binary-burning energy equilibrium and in gravo-thermal contraction stages, respectively. The solid and dashed histograms for the observed GGCs denote core-collapsed clusters and non core-collapsed clusters, respectively. The ratio between the core and half-light radius for binary-burning clusters have very similar values compared to the values for the core-collapsed population among the GGCs. On the other hand the bulk of the GGCs have values for this ratio similar to the model clusters that are still in the gravo-thermal contraction stage at integration stopping time ($\approx 12\,{\rm{Gyr}}$). []{data-label="plot:core-collapsed"}](plots/core_collapsed_non_core_collapsed_rcoverrh.ps){width="90.00000%"}
We divide the full sample of our simulations in two subgroups: 1) binary-burning: models showing a clear binary-burning stage before integration is stopped, and 2) core-contracting: models still contracting due to two-body relaxation until integration stopping time. Figure \[plot:core-collapsed\] shows the distributions for ${r^{N, \rm{cut}}_{c, \rm{obs}}}/{r^{\rm{cut}}_{hl, \rm{obs}}}$ for the two subgroups of our theoretical models. The distributions peak at different values, with the binary-burning clusters having much lower core radii. The distributions for ${r_{c, \rm{obs}}}$ for the observed core-collapsed and non core-collapsed GGCs are also shown. Note that the values for the binary-burning clusters from our models and the core-collapsed clusters from the observed GGCs show a very similar range of values. Similarly, ${r^{N, \rm{cut}}_{c, \rm{obs}}}/{r^{\rm{cut}}_{hl, \rm{obs}}}$ for the contracting clusters in our models show very similar values compared to those of the observed non core-collapsed GGCs. In addition, most of the core-contracting models with small ${r^{N, \rm{cut}}_{c, \rm{obs}}}< 0.1\,{\rm{pc}}$ values are in fact about to start binary-burning, but was classified by contracting since the constant $r_c/r_h$ stage of evolution was not clearly seen. The observed GGCs show larger ranges for the core radii values compared for both categories of clusters compared to our models. This is simply because we are forced to limit the ranges of the grid of initial conditions to constrain the number of required cluster calculations to a tractable amount. We remind the readers that the heights of the histograms between the model clusters and the observed GGCs are not compared here, since that depends directly on the distributions of initial cluster parameters, determination of which is beyond the scope of this study, and are chosen arbitrarily. Only ranges in values are compared.
Conclusions {#sec:conclusion}
===========
We have presented a large ($\sim 200$) collection of cluster models created with the Northwestern group’s [Hénon]{}-type Monte Carlo code in star-by-star detail. We start our models with initial parameters including the stellar mass spectrum, cluster size, concentration, and primordial binary fraction $f_b$ over large ranges (Section \[sec:initial\_conditions\]) guided by observed young massive clusters . We find that these initial clusters very naturally produce a population of model clusters with structural properties including cluster mass, $\rho_c$, $r_c$, $r_h$, and $t_{r,h}$ in excellent agreement with the bulk of the GGC properties after about $12\,{\rm{Gyr}}$ of evolution without any special considerations or fine tuning (e.g., very high density to aid collisional expedited stellar mass loss via compact object formation; @2008IAUS..246..151C; or intermediate mass black holes; @2007MNRAS.374..857T). We pay attention to the various different commonly used definitions of the structural parameters $r_c$ and $r_h$ and calculate these quantities from our models as an observer would for real clusters. These parameters are then compared and found to agree well with the ranges from observed GGCs. Using our large collection of models we also show the distribution of the ratio of the three dimensional code-defined $r_c$ and $r_h$ to the corresponding “observed" values (Figures \[plot:rc\_rcobs\_hist\], \[plot:rh\_rhlobs\_hist\]). We expect that these distributions of ratios for the $r_c$ and $r_h$ values will be valuable for observers and theorists alike to convert the values of these parameters from one set of definitions to another.
From the evolution of the code-defined three-dimensional structural parameters of all our models, we find that all qualitatively different evolutionary stages are observed, in particular, the initial expansion due to stellar evolution driven mass loss, core contraction driven by two-body relaxation, and the binary-burning equilibrium stage (for clusters with $f_b>0$) driven by a balance between energy production via dynamical hardening of binaries in the core and outward diffusion of energy from the core due to two-body relaxation. Our results indicate that the progenitors of today’s GGCs were very similar in properties to the present day young massive clusters . Of course, the metallicities of these progenitors must have been much lower compared to today’s massive young clusters.
After establishing that our collection of cluster models are representative of the observed GGCs we investigate the apparently bimodal distribution of the observed core radii of the GGCs created by the core collapsed clusters and non core-collapsed clusters. In particular, we answer the question if the core collapsed clusters are dynamically different from the non core collapsed clusters. We find that the surface brightness profile for the binary-burning cluster shows a prominent power-law slope near the center (Figure \[plot:example\]). The core collapsed clusters are observationally defined as the clusters that show this distinct feature in their surface brightness profiles [e.g., @2005ApJS..161..304M]. In contrast, a model cluster that is still contracting at $12\,{\rm{Gyr}}$ shows a surface brightness profile that has a clear flat central part and is fitted well by a King profile [@1966AJ.....71...64K]. We further divide our models into two subsets, one containing clusters in the binary-burning stage, and the other containing clusters that are still contracting at $12\,{\rm{Gyr}}$. We compare the ratio between the core and half light radii of our binary-burning and contracting clusters with those for the core-collapsed and non core-collapsed clusters in the observed GGCs, respectively. We find that the binary-burning ${r^{N, \rm{cut}}_{c, \rm{obs}}}/{r^{\rm{cut}}_{hl, \rm{obs}}}$ values in our models are in agreement with those of the core-collapsed GGCs. Similarly, the contracting ${r^{N, \rm{cut}}_{c, \rm{obs}}}/{r^{\rm{cut}}_{hl, \rm{obs}}}$ values in our models are in agreement with those of the non core-collapsed GGCs (Figure \[plot:core-collapsed\]). Thus our results clearly indicate that the so called core-collapsed GGCs are in fact at the binary-burning stage whereas, the non core-collapsed GGCs are still contracting under two-body relaxation. This also indicates that the majority of the GGCs (since most GGCs are non core-collapsed) are not in energy equilibrium as was expected by some earlier theoretical models [e.g., @2007ApJ...658.1047F]. One key implication for this finding is that analytical estimates of interaction rates in a GGC must take into account the fact that the present day observed structural parameters including $\rho_c$ has not been constant and is still evolving. Hence, to calculate a correct estimate one must integrate the time dependent cross-sections (based on the changing values of these parameters) over an appropriate length of time as has been done in, e.g., @2008ApJ...673L..25F.
Acknowledgement: We thank the anonymous referee for his help rectifying some mistakes and useful suggestions. This work was supported by NASA ATP Grant NNX09AO36G at Northwestern University. SC acknowledges support from the Theory Postdoctoral Fellowship from UF Department of Astronomy and College of Liberal Arts and Sciences.
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& 4 & 2.5 & 4 & 1.6 & 3.3 & 12.2 & 0.00 & 0.00 & 0.84 & 12 & 1.4 & 3 & 2.0 & 1.7 & 7.1 & 4.2 & 1.0 & 0.00 & 0.00 & 1.5\
[run2]{} & 4 & 3.8 & 6 & 1.6 & 3.3 & 16.2 & 0.00 & 0.00 & 0.84 & 12 & 2.1 & 5 & 2.3 & 1.9 & 7.0 & 4.3 & 1.1 & 0.00 & 0.00 & 1.5\
[run3]{} & 4 & 5.1 & 8 & 1.6 & 3.3 & 23.3 & 0.00 & 0.00 & 0.84 & 12 & 2.8 & 7 & 2.4 & 1.9 & 6.9 & 4.4 & 1.3 & 0.00 & 0.00 & 1.5\
[run4]{} & 4 & 6.4 & 10 & 1.6 & 3.3 & 29.0 & 0.00 & 0.00 & 0.84 & 12 & 3.5 & 9 & 2.5 & 2.9 & 6.8 & 4.4 & 1.5 & 0.00 & 0.00 & 1.5\
[run5]{} & 4 & 2.6 & 4 & 1.6 & 3.3 & 12.3 & 0.05 & 0.05 & 0.84 & 12 & 1.4 & 3 & 2.1 & 2.3 & 7.2 & 4.2 & 0.9 & 0.05 & 0.07 & 1.4\
[run6]{} & 4 & 3.9 & 6 & 1.6 & 3.3 & 17.3 & 0.05 & 0.05 & 0.84 & 12 & 2.1 & 5 & 2.3 & 2.1 & 7.1 & 4.4 & 1.1 & 0.05 & 0.07 & 1.5\
[run7]{} & 4 & 5.3 & 8 & 1.6 & 3.3 & 24.7 & 0.05 & 0.05 & 0.84 & 12 & 2.9 & 7 & 2.4 & 2.2 & 7.0 & 4.4 & 1.2 & 0.05 & 0.07 & 1.5\
[run8]{} & 4 & 6.6 & 10 & 1.6 & 3.3 & 30.2 & 0.05 & 0.05 & 0.84 & 12 & 3.6 & 9 & 2.5 & 3.3 & 6.9 & 4.4 & 1.3 & 0.05 & 0.06 & 1.5\
[run9]{} & 4 & 2.7 & 4 & 1.6 & 3.3 & 12.4 & 0.10 & 0.10 & 0.84 & 12 & 1.4 & 3 & 2.2 & 2.1 & 7.3 & 4.4 & 0.8 & 0.09 & 0.14 & 1.4\
[run10]{} & 4 & 4.0 & 6 & 1.6 & 3.3 & 17.7 & 0.10 & 0.10 & 0.84 & 12 & 2.2 & 5 & 2.3 & 1.9 & 7.2 & 4.4 & 1.1 & 0.09 & 0.14 & 1.5\
[run11]{} & 4 & 5.4 & 8 & 1.6 & 3.3 & 25.3 & 0.10 & 0.10 & 0.84 & 12 & 2.9 & 7 & 2.5 & 2.0 & 7.1 & 4.4 & 1.2 & 0.09 & 0.13 & 1.5\
[run12]{} & 4 & 6.8 & 10 & 1.6 & 3.3 & 30.9 & 0.10 & 0.10 & 0.84 & 12 & 3.7 & 9 & 2.6 & 2.0 & 7.0 & 4.4 & 1.3 & 0.09 & 0.13 & 1.5\
[run13]{} & 4 & 2.8 & 4 & 1.6 & 3.3 & 14.3 & 0.20 & 0.20 & 0.84 & 12 & 1.5 & 3 & 2.3 & 2.5 & 7.5 & 4.5 & 0.8 & 0.18 & 0.26 & 1.4\
[run14]{} & 4 & 4.3 & 6 & 1.6 & 3.3 & 18.3 & 0.20 & 0.20 & 0.84 & 12 & 2.3 & 5 & 2.4 & 2.1 & 7.3 & 4.3 & 1.0 & 0.18 & 0.26 & 1.5\
[run15]{} & 4 & 5.7 & 8 & 1.6 & 3.3 & 27.6 & 0.20 & 0.20 & 0.84 & 12 & 3.0 & 7 & 2.6 & 2.2 & 7.2 & 4.5 & 1.1 & 0.18 & 0.25 & 1.5\
[run16]{} & 4 & 7.1 & 10 & 1.6 & 3.3 & 33.4 & 0.20 & 0.20 & 0.84 & 12 & 3.8 & 9 & 2.7 & 2.1 & 7.2 & 4.5 & 1.2 & 0.18 & 0.24 & 1.5\
[run17]{} & 5 & 2.5 & 4 & 1.5 & 3.3 & 14.2 & 0.00 & 0.00 & 1.0 & 12 & 1.4 & 3 & 1.9 & 1.2 & 7.2 & 4.3 & 1.4 & 0.00 & 0.00 & 1.5\
[run18]{} & 5 & 3.8 & 6 & 1.5 & 3.3 & 18.8 & 0.00 & 0.00 & 1.0 & 12 & 2.1 & 5 & 2.1 & 1.7 & 7.0 & 4.4 & 1.4 & 0.00 & 0.00 & 1.5\
[run19]{} & 5 & 5.1 & 8 & 1.5 & 3.3 & 27.2 & 0.00 & 0.00 & 1.0 & 12 & 2.8 & 7 & 2.3 & 1.5 & 6.9 & 4.4 & 1.5 & 0.00 & 0.00 & 1.5\
[run20]{} & 5 & 6.4 & 10 & 1.5 & 3.3 & 33.3 & 0.00 & 0.00 & 1.0 & 12 & 3.5 & 9 & 2.3 & 2.4 & 6.9 & 4.4 & 1.7 & 0.00 & 0.00 & 1.5\
[run21]{} & 5 & 2.6 & 4 & 1.5 & 3.3 & 14.3 & 0.05 & 0.05 & 1.0 & 12 & 1.4 & 3 & 2.0 & 1.4 & 7.3 & 4.4 & 1.2 & 0.05 & 0.08 & 1.5\
[run22]{} & 5 & 3.9 & 6 & 1.5 & 3.3 & 20.1 & 0.05 & 0.05 & 1.0 & 12 & 2.1 & 5 & 2.2 & 1.4 & 7.1 & 4.4 & 1.2 & 0.05 & 0.07 & 1.5\
[run23]{} & 5 & 5.3 & 8 & 1.5 & 3.3 & 28.9 & 0.05 & 0.05 & 1.0 & 12 & 2.9 & 7 & 2.3 & 2.0 & 7.1 & 4.4 & 1.4 & 0.05 & 0.07 & 1.5\
[run24]{} & 5 & 6.6 & 10 & 1.5 & 3.3 & 34.7 & 0.05 & 0.05 & 1.0 & 12 & 3.6 & 9 & 2.4 & 2.6 & 7.0 & 4.4 & 1.7 & 0.05 & 0.07 & 1.5\
[run25]{} & 5 & 2.7 & 4 & 1.5 & 3.3 & 14.4 & 0.10 & 0.10 & 1.0 & 12 & 1.4 & 3 & 2.0 & 1.3 & 7.4 & 4.4 & 1.1 & 0.09 & 0.14 & 1.5\
[run26]{} & 5 & 4.0 & 6 & 1.5 & 3.3 & 20.5 & 0.10 & 0.10 & 1.0 & 12 & 2.2 & 5 & 2.2 & 1.4 & 7.3 & 4.4 & 1.2 & 0.09 & 0.14 & 1.5\
[run27]{} & 5 & 5.4 & 8 & 1.5 & 3.3 & 29.6 & 0.10 & 0.10 & 1.0 & 12 & 2.9 & 7 & 2.4 & 2.0 & 7.2 & 4.3 & 1.4 & 0.09 & 0.13 & 1.5\
[run28]{} & 5 & 6.8 & 10 & 1.5 & 3.3 & 35.4 & 0.10 & 0.10 & 1.0 & 12 & 3.7 & 9 & 2.4 & 2.4 & 7.1 & 4.5 & 1.6 & 0.09 & 0.13 & 1.5\
[run29]{} & 5 & 2.8 & 4 & 1.5 & 3.3 & 16.8 & 0.20 & 0.20 & 1.0 & 12 & 1.5 & 3 & 2.2 & 2.1 & 7.6 & 4.4 & 0.9 & 0.18 & 0.26 & 1.4\
[run30]{} & 5 & 4.3 & 6 & 1.5 & 3.3 & 21.2 & 0.20 & 0.20 & 1.0 & 12 & 2.3 & 5 & 2.3 & 1.9 & 7.4 & 4.5 & 1.1 & 0.18 & 0.26 & 1.5\
[run31]{} & 5 & 5.7 & 8 & 1.5 & 3.3 & 32.3 & 0.20 & 0.20 & 1.0 & 12 & 3.0 & 7 & 2.5 & 2.2 & 7.3 & 4.4 & 1.2 & 0.18 & 0.24 & 1.5\
[run32]{} & 5 & 7.1 & 10 & 1.5 & 3.3 & 38.4 & 0.20 & 0.20 & 1.0 & 12 & 3.8 & 9 & 2.5 & 2.0 & 7.2 & 4.4 & 1.6 & 0.18 & 0.25 & 1.5\
[run33]{} & 6 & 2.5 & 4 & 1.4 & 3.2 & 17.2 & 0.00 & 0.00 & 1.3 & 12 & 1.4 & 3 & 1.7 & 0.9 & 7.3 & 4.3 & 1.8 & 0.00 & 0.00 & 1.5\
[run34]{} & 6 & 3.8 & 6 & 1.4 & 3.3 & 22.4 & 0.00 & 0.00 & 1.3 & 12 & 2.1 & 5 & 1.9 & 1.2 & 7.1 & 4.3 & 1.9 & 0.00 & 0.00 & 1.6\
[run35]{} & 6 & 5.1 & 8 & 1.4 & 3.3 & 32.5 & 0.00 & 0.00 & 1.3 & 12 & 2.8 & 7 & 2.1 & 1.6 & 7.0 & 4.3 & 2.0 & 0.00 & 0.00 & 1.5\
[run36]{} & 6 & 6.4 & 10 & 1.4 & 3.3 & 40.0 & 0.00 & 0.00 & 1.3 & 12 & 3.5 & 9 & 2.2 & 2.1 & 7.0 & 4.4 & 2.2 & 0.00 & 0.00 & 1.6\
[run37]{} & 6 & 2.6 & 4 & 1.4 & 3.2 & 17.3 & 0.05 & 0.05 & 1.3 & 12 & 1.4 & 3 & 1.8 & 1.5 & 7.4 & 4.4 & 1.5 & 0.05 & 0.08 & 1.5\
[run38]{} & 6 & 3.9 & 6 & 1.4 & 3.3 & 24.0 & 0.05 & 0.05 & 1.3 & 12 & 2.1 & 5 & 2.0 & 1.7 & 7.3 & 4.4 & 1.6 & 0.05 & 0.07 & 1.5\
[run39]{} & 6 & 5.3 & 8 & 1.4 & 3.3 & 34.6 & 0.05 & 0.05 & 1.3 & 12 & 2.8 & 7 & 2.2 & 1.7 & 7.2 & 4.4 & 1.7 & 0.05 & 0.07 & 1.5\
[run40]{} & 6 & 6.6 & 10 & 1.4 & 3.3 & 41.6 & 0.05 & 0.05 & 1.3 & 12 & 3.6 & 9 & 2.2 & 1.7 & 7.1 & 4.4 & 2.1 & 0.05 & 0.07 & 1.5\
[run41]{} & 6 & 2.7 & 4 & 1.4 & 3.2 & 17.4 & 0.10 & 0.10 & 1.3 & 12 & 1.4 & 3 & 1.9 & 1.5 & 7.5 & 4.4 & 1.5 & 0.09 & 0.15 & 1.5\
[run42]{} & 6 & 4.0 & 6 & 1.4 & 3.3 & 24.5 & 0.10 & 0.10 & 1.3 & 12 & 2.2 & 5 & 2.1 & 1.3 & 7.4 & 4.4 & 1.5 & 0.09 & 0.14 & 1.5\
[run43]{} & 6 & 5.4 & 8 & 1.4 & 3.3 & 35.4 & 0.10 & 0.10 & 1.3 & 12 & 2.9 & 7 & 2.2 & 2.0 & 7.3 & 4.5 & 1.7 & 0.09 & 0.13 & 1.5\
[run44]{} & 6 & 6.8 & 10 & 1.4 & 3.3 & 42.5 & 0.10 & 0.10 & 1.3 & 12 & 3.7 & 9 & 2.3 & 1.9 & 7.2 & 4.5 & 1.9 & 0.09 & 0.13 & 1.6\
[run45]{} & 6 & 2.8 & 4 & 1.4 & 3.2 & 20.3 & 0.20 & 0.20 & 1.3 & 12 & 1.5 & 3 & 2.1 & 1.1 & 7.8 & 4.5 & 1.1 & 0.18 & 0.27 & 1.4\
[run46]{} & 6 & 4.3 & 6 & 1.4 & 3.3 & 25.2 & 0.20 & 0.20 & 1.3 & 12 & 2.3 & 5 & 2.2 & 1.7 & 7.6 & 4.5 & 1.3 & 0.18 & 0.26 & 1.5\
[run47]{} & 6 & 5.7 & 8 & 1.4 & 3.3 & 38.7 & 0.20 & 0.20 & 1.3 & 12 & 3.0 & 7 & 2.4 & 1.8 & 7.4 & 4.4 & 1.4 & 0.18 & 0.24 & 1.5\
[run48]{} & 6 & 7.1 & 10 & 1.4 & 3.3 & 46.0 & 0.20 & 0.20 & 1.3 & 12 & 3.8 & 9 & 2.4 & 2.4 & 7.3 & 4.5 & 1.8 & 0.18 & 0.25 & 1.5\
[run49]{} & 7 & 2.5 & 4 & 1.3 & 3.2 & 21.6 & 0.00 & 0.00 & 1.5 & 12 & 1.4 & 3 & 1.5 & 1.0 & 7.4 & 4.4 & 3.0 & 0.00 & 0.00 & 1.6\
[run50]{} & 7 & 3.8 & 6 & 1.3 & 3.2 & 27.6 & 0.00 & 0.00 & 1.5 & 12 & 2.1 & 5 & 1.7 & 1.6 & 7.3 & 4.4 & 2.6 & 0.00 & 0.00 & 1.6\
[run51]{} & 7 & 5.1 & 8 & 1.3 & 3.3 & 40.2 & 0.00 & 0.00 & 1.5 & 12 & 2.8 & 7 & 1.8 & 1.5 & 7.2 & 4.4 & 3.0 & 0.00 & 0.00 & 1.6\
[run52]{} & 7 & 6.4 & 10 & 1.3 & 3.2 & 49.6 & 0.00 & 0.00 & 1.5 & 12 & 3.5 & 9 & 1.9 & 1.2 & 7.1 & 4.4 & 3.1 & 0.00 & 0.00 & 1.6\
[run53]{} & 7 & 2.6 & 4 & 1.3 & 3.2 & 21.7 & 0.05 & 0.05 & 1.5 & 12 & 1.4 & 3 & 1.5 & 1.2 & 7.6 & 4.5 & 2.5 & 0.05 & 0.08 & 1.6\
[run54]{} & 7 & 3.9 & 6 & 1.3 & 3.2 & 29.7 & 0.05 & 0.05 & 1.5 & 12 & 2.1 & 5 & 1.8 & 1.5 & 7.4 & 4.5 & 2.4 & 0.05 & 0.08 & 1.6\
[run55]{} & 7 & 5.3 & 8 & 1.3 & 3.3 & 43.0 & 0.05 & 0.05 & 1.5 & 12 & 2.8 & 7 & 1.9 & 1.4 & 7.3 & 4.4 & 2.6 & 0.05 & 0.07 & 1.6\
[run56]{} & 7 & 6.6 & 10 & 1.3 & 3.2 & 51.5 & 0.05 & 0.05 & 1.5 & 12 & 3.6 & 9 & 2.0 & 1.7 & 7.2 & 4.5 & 2.9 & 0.05 & 0.07 & 1.6\
[run57]{} & 7 & 2.7 & 4 & 1.3 & 3.2 & 21.7 & 0.10 & 0.10 & 1.5 & 12 & 1.4 & 3 & 1.7 & 1.3 & 7.7 & 4.5 & 1.8 & 0.09 & 0.16 & 1.5\
[run58]{} & 7 & 4.0 & 6 & 1.3 & 3.2 & 30.3 & 0.10 & 0.10 & 1.5 & 12 & 2.2 & 5 & 1.9 & 1.5 & 7.5 & 4.5 & 2.0 & 0.09 & 0.14 & 1.5\
[run59]{} & 7 & 5.4 & 8 & 1.3 & 3.3 & 44.0 & 0.10 & 0.10 & 1.5 & 12 & 2.9 & 7 & 2.0 & 1.4 & 7.4 & 4.5 & 2.5 & 0.09 & 0.14 & 1.6\
[run60]{} & 7 & 6.8 & 10 & 1.3 & 3.2 & 52.5 & 0.10 & 0.10 & 1.5 & 12 & 3.6 & 9 & 2.1 & 1.4 & 7.3 & 4.5 & 2.5 & 0.09 & 0.13 & 1.6\
[run61]{} & 7 & 2.8 & 4 & 1.3 & 3.2 & 25.5 & 0.20 & 0.20 & 1.5 & 12 & 1.5 & 3 & 1.9 & 1.4 & 7.9 & 4.6 & 1.4 & 0.18 & 0.27 & 1.5\
[run62]{} & 7 & 4.3 & 6 & 1.3 & 3.2 & 31.2 & 0.20 & 0.20 & 1.5 & 12 & 2.3 & 5 & 2.1 & 1.5 & 7.7 & 4.5 & 1.6 & 0.18 & 0.26 & 1.5\
[run63]{} & 7 & 5.7 & 8 & 1.3 & 3.3 & 48.2 & 0.20 & 0.20 & 1.5 & 12 & 3.0 & 7 & 2.1 & 1.9 & 7.6 & 4.5 & 2.1 & 0.18 & 0.26 & 1.5\
[run64]{} & 7 & 7.1 & 10 & 1.3 & 3.2 & 56.9 & 0.20 & 0.20 & 1.5 & 12 & 3.8 & 9 & 2.2 & 1.7 & 7.5 & 4.5 & 2.1 & 0.18 & 0.24 & 1.5\
[run65]{} & 8 & 2.5 & 4 & 1.2 & 3.2 & 28.5 & 0.00 & 0.00 & 1.9 & 12 & 1.4 & 3 & 1.1 & 0.8 & 7.7 & 4.5 & 7.1 & 0.00 & 0.00 & 1.7\
[run66]{} & 8 & 3.8 & 6 & 1.2 & 3.2 & 35.7 & 0.00 & 0.00 & 1.9 & 12 & 2.1 & 5 & 1.4 & 0.8 & 7.5 & 4.5 & 4.7 & 0.00 & 0.00 & 1.7\
[run67]{} & 8 & 5.1 & 8 & 1.2 & 3.2 & 52.8 & 0.00 & 0.00 & 1.9 & 12 & 2.8 & 7 & 1.6 & 1.2 & 7.4 & 4.5 & 4.7 & 0.00 & 0.00 & 1.7\
[run68]{} & 8 & 6.4 & 10 & 1.2 & 3.2 & 64.3 & 0.00 & 0.00 & 1.9 & 12 & 3.5 & 9 & 1.6 & 1.2 & 7.3 & 4.5 & 5.1 & 0.00 & 0.00 & 1.7\
[run69]{} & 8 & 2.6 & 4 & 1.2 & 3.2 & 28.6 & 0.05 & 0.05 & 1.9 & 12 & 1.4 & 3 & 1.3 & 1.0 & 7.8 & 4.5 & 4.0 & 0.05 & 0.08 & 1.6\
[run70]{} & 8 & 3.9 & 6 & 1.2 & 3.2 & 38.5 & 0.05 & 0.05 & 1.9 & 12 & 2.1 & 5 & 1.5 & 1.1 & 7.7 & 4.6 & 4.1 & 0.05 & 0.08 & 1.6\
[run71]{} & 8 & 5.3 & 8 & 1.2 & 3.2 & 56.6 & 0.05 & 0.05 & 1.9 & 12 & 2.8 & 7 & 1.6 & 1.3 & 7.5 & 4.6 & 4.1 & 0.05 & 0.08 & 1.6\
[run72]{} & 8 & 6.6 & 10 & 1.2 & 3.2 & 66.5 & 0.05 & 0.05 & 1.9 & 12 & 3.6 & 9 & 1.8 & 1.4 & 7.4 & 4.5 & 4.2 & 0.05 & 0.07 & 1.6\
[run73]{} & 8 & 2.7 & 4 & 1.2 & 3.2 & 28.6 & 0.10 & 0.10 & 1.9 & 12 & 1.4 & 3 & 1.4 & 1.1 & 7.9 & 4.6 & 3.4 & 0.09 & 0.16 & 1.6\
[run74]{} & 8 & 4.0 & 6 & 1.2 & 3.2 & 39.3 & 0.10 & 0.10 & 1.9 & 12 & 2.2 & 5 & 1.6 & 1.0 & 7.8 & 4.6 & 3.2 & 0.09 & 0.15 & 1.6\
[run75]{} & 8 & 5.4 & 8 & 1.2 & 3.2 & 57.9 & 0.10 & 0.10 & 1.9 & 12 & 2.9 & 7 & 1.8 & 1.5 & 7.6 & 4.6 & 3.3 & 0.09 & 0.14 & 1.6\
[run76]{} & 8 & 6.8 & 10 & 1.2 & 3.2 & 67.9 & 0.10 & 0.10 & 1.9 & 12 & 3.6 & 9 & 1.8 & 1.4 & 7.5 & 4.6 & 3.9 & 0.09 & 0.14 & 1.6\
[run77]{} & 8 & 2.8 & 4 & 1.2 & 3.2 & 33.7 & 0.20 & 0.20 & 1.9 & 12 & 1.5 & 3 & 1.7 & 1.2 & 8.2 & 4.6 & 2.1 & 0.18 & 0.28 & 1.5\
[run78]{} & 8 & 4.3 & 6 & 1.2 & 3.2 & 40.4 & 0.20 & 0.20 & 1.9 & 12 & 2.3 & 5 & 1.8 & 1.5 & 8.0 & 4.7 & 2.7 & 0.18 & 0.27 & 1.6\
[run79]{} & 8 & 5.7 & 8 & 1.2 & 3.2 & 63.4 & 0.20 & 0.20 & 1.9 & 12 & 3.0 & 7 & 1.9 & 1.5 & 7.8 & 4.6 & 2.9 & 0.18 & 0.26 & 1.6\
[run80]{} & 8 & 7.1 & 10 & 1.2 & 3.2 & 73.6 & 0.20 & 0.20 & 1.9 & 12 & 3.8 & 9 & 2.0 & 1.4 & 7.7 & 4.7 & 3.2 & 0.18 & 0.25 & 1.6\
[run81]{} & 6.5 & 2.5 & 4 & 1.1 & 3.2 & 40.1 & 0.00 & 0.00 & 1.4 & 12 & 1.4 & 3 & 0.5 & 0.7 & 8.0 & 4.7 & 64.9 & 0.00 & 0.00 & 2.1\
[run82]{} & 6.5 & 3.8 & 6 & 1.1 & 3.2 & 49.3 & 0.00 & 0.00 & 1.4 & 12 & 2.1 & 5 & 1.0 & 0.6 & 7.8 & 4.6 & 14.7 & 0.00 & 0.00 & 1.8\
[run83]{} & 6.5 & 5.1 & 8 & 1.1 & 3.2 & 73.0 & 0.00 & 0.00 & 1.4 & 12 & 2.8 & 7 & 1.2 & 0.8 & 7.6 & 4.7 & 11.7 & 0.00 & 0.00 & 1.8\
[run84]{} & 6.5 & 6.4 & 10 & 1.1 & 3.2 & 88.8 & 0.00 & 0.00 & 1.4 & 12 & 3.5 & 9 & 1.3 & 0.8 & 7.5 & 4.6 & 10.0 & 0.00 & 0.00 & 1.8\
[run85]{} & 6.5 & 2.6 & 4 & 1.1 & 3.2 & 40.3 & 0.05 & 0.05 & 1.4 & 12 & 1.4 & 3 & 0.9 & 0.5 & 8.1 & 4.7 & 14.0 & 0.05 & 0.10 & 1.8\
[run86]{} & 6.5 & 3.9 & 6 & 1.1 & 3.2 & 53.4 & 0.05 & 0.05 & 1.4 & 12 & 2.1 & 5 & 1.2 & 0.8 & 7.9 & 4.7 & 8.4 & 0.05 & 0.08 & 1.7\
[run87]{} & 6.5 & 5.3 & 8 & 1.1 & 3.2 & 78.5 & 0.05 & 0.05 & 1.4 & 12 & 2.8 & 7 & 1.3 & 0.8 & 7.8 & 4.8 & 9.0 & 0.05 & 0.08 & 1.7\
[run88]{} & 6.5 & 6.6 & 10 & 1.1 & 3.2 & 91.7 & 0.05 & 0.05 & 1.4 & 12 & 3.6 & 9 & 1.4 & 1.0 & 7.7 & 4.7 & 8.7 & 0.05 & 0.08 & 1.7\
[run89]{} & 6.5 & 2.7 & 4 & 1.1 & 3.2 & 40.2 & 0.10 & 0.10 & 1.4 & 12 & 1.4 & 3 & 1.1 & 0.8 & 8.2 & 4.7 & 6.1 & 0.09 & 0.17 & 1.7\
[run90]{} & 6.5 & 4.0 & 6 & 1.1 & 3.2 & 54.4 & 0.10 & 0.10 & 1.4 & 12 & 2.2 & 5 & 1.3 & 0.9 & 8.0 & 4.7 & 5.8 & 0.09 & 0.16 & 1.7\
[run91]{} & 6.5 & 5.4 & 8 & 1.1 & 3.2 & 80.2 & 0.10 & 0.10 & 1.4 & 12 & 2.9 & 7 & 1.4 & 0.9 & 7.9 & 4.8 & 7.3 & 0.09 & 0.15 & 1.7\
[run92]{} & 6.5 & 6.8 & 10 & 1.1 & 3.2 & 93.6 & 0.10 & 0.10 & 1.4 & 12 & 3.6 & 9 & 1.4 & 0.9 & 7.8 & 4.8 & 8.8 & 0.09 & 0.15 & 1.7\
[run93]{} & 6.5 & 2.8 & 4 & 1.1 & 3.2 & 47.4 & 0.20 & 0.20 & 1.4 & 12 & 1.5 & 3 & 1.4 & 1.1 & 8.4 & 4.8 & 4.0 & 0.18 & 0.30 & 1.6\
[run94]{} & 6.5 & 4.3 & 6 & 1.1 & 3.2 & 55.8 & 0.20 & 0.20 & 1.4 & 12 & 2.2 & 5 & 1.5 & 0.7 & 8.3 & 4.9 & 4.5 & 0.18 & 0.28 & 1.6\
[run95]{} & 6.5 & 5.7 & 8 & 1.1 & 3.2 & 88.0 & 0.20 & 0.20 & 1.4 & 12 & 3.0 & 7 & 1.6 & 1.3 & 8.1 & 4.8 & 4.7 & 0.18 & 0.27 & 1.6\
[run96]{} & 6.5 & 7.1 & 10 & 1.1 & 3.2 & 101.4 & 0.20 & 0.20 & 1.4 & 12 & 3.8 & 9 & 1.6 & 1.3 & 8.0 & 4.8 & 6.2 & 0.18 & 0.26 & 1.7\
[run97]{} & 7 & 2.5 & 4 & 0.9 & 3.2 & 61.0 & 0.00 & 0.00 & 1.5 & 11 & 1.4 & 3 & 0.3 & 0.9 & 8.2 & 4.6 & 249.4 & 0.00 & 0.00 & 2.3\
[run98]{} & 7 & 3.8 & 6 & 0.9 & 3.3 & 73.4 & 0.00 & 0.00 & 1.5 & 11 & 2.1 & 5 & 0.3 & 0.6 & 8.1 & 4.8 & 320.0 & 0.00 & 0.00 & 2.3\
[run99]{} & 7 & 5.1 & 8 & 0.9 & 3.3 & 110.3 & 0.00 & 0.00 & 1.5 & 12 & 2.8 & 7 & 0.4 & 0.9 & 8.0 & 4.5 & 281.9 & 0.00 & 0.00 & 2.2\
[run100]{} & 7 & 6.4 & 10 & 0.9 & 3.3 & 134.5 & 0.00 & 0.00 & 1.5 & 12 & 3.5 & 9 & 0.5 & 0.9 & 7.9 & 4.7 & 204.6 & 0.00 & 0.00 & 2.2\
[run101]{} & 7 & 2.6 & 4 & 0.9 & 3.2 & 61.1 & 0.05 & 0.05 & 1.5 & 12 & 1.4 & 3 & 0.7 & 0.3 & 8.5 & 5.0 & 28.3 & 0.05 & 0.10 & 1.9\
[run102]{} & 7 & 3.9 & 6 & 0.9 & 3.3 & 79.7 & 0.05 & 0.05 & 1.5 & 12 & 2.1 & 5 & 0.8 & 0.3 & 8.4 & 5.0 & 29.1 & 0.05 & 0.10 & 1.9\
[run103]{} & 7 & 5.3 & 8 & 0.9 & 3.3 & 118.9 & 0.05 & 0.05 & 1.5 & 12 & 2.8 & 7 & 0.9 & 0.5 & 8.2 & 5.0 & 33.7 & 0.05 & 0.10 & 1.9\
[run104]{} & 7 & 6.6 & 10 & 0.9 & 3.3 & 138.3 & 0.05 & 0.05 & 1.5 & 12 & 3.6 & 9 & 0.9 & 0.5 & 8.0 & 4.9 & 31.2 & 0.04 & 0.09 & 1.9\
[run105]{} & 7 & 2.7 & 4 & 0.9 & 3.2 & 60.8 & 0.10 & 0.10 & 1.5 & 12 & 1.4 & 3 & 1.0 & 0.8 & 8.7 & 5.0 & 12.0 & 0.09 & 0.18 & 1.8\
[run106]{} & 7 & 4.0 & 6 & 0.9 & 3.3 & 81.2 & 0.10 & 0.10 & 1.5 & 12 & 2.1 & 5 & 0.9 & 0.4 & 8.5 & 5.1 & 31.0 & 0.09 & 0.18 & 1.9\
[run107]{} & 7 & 5.4 & 8 & 0.9 & 3.3 & 121.3 & 0.10 & 0.10 & 1.5 & 12 & 2.9 & 7 & 1.0 & 0.4 & 8.3 & 5.0 & 25.0 & 0.09 & 0.17 & 1.8\
[run108]{} & 7 & 6.8 & 10 & 0.9 & 3.3 & 141.0 & 0.10 & 0.10 & 1.5 & 12 & 3.6 & 9 & 1.1 & 0.5 & 8.1 & 5.0 & 23.5 & 0.09 & 0.16 & 1.8\
[run109]{} & 7 & 2.8 & 4 & 0.9 & 3.2 & 72.1 & 0.20 & 0.20 & 1.5 & 12 & 1.5 & 3 & 1.2 & 0.5 & 8.8 & 5.0 & 6.2 & 0.18 & 0.32 & 1.7\
[run110]{} & 7 & 4.3 & 6 & 0.9 & 3.3 & 83.1 & 0.20 & 0.20 & 1.5 & 12 & 2.2 & 5 & 1.1 & 0.5 & 8.7 & 5.2 & 13.2 & 0.18 & 0.30 & 1.8\
[run111]{} & 7 & 5.7 & 8 & 0.9 & 3.3 & 133.2 & 0.20 & 0.20 & 1.5 & 12 & 3.0 & 7 & 1.3 & 0.6 & 8.5 & 5.1 & 10.4 & 0.18 & 0.29 & 1.7\
[run112]{} & 7 & 7.1 & 10 & 0.9 & 3.3 & 152.5 & 0.20 & 0.20 & 1.5 & 12 & 3.8 & 9 & 1.3 & 0.8 & 8.3 & 5.0 & 10.7 & 0.18 & 0.27 & 1.8\
[run113]{} & 7.5 & 2.5 & 4 & 0.7 & 3.3 & 103.6 & 0.00 & 0.00 & 1.7 & 9 & 1.4 & 3 & 0.3 & 0.4 & 8.3 & 4.9 & 341.4 & 0.00 & 0.00 & 2.3\
[run114]{} & 7.5 & 3.8 & 6 & 0.7 & 3.4 & 119.9 & 0.00 & 0.00 & 1.7 & 10 & 2.1 & 5 & 0.3 & 0.6 & 8.2 & 4.8 & 455.5 & 0.00 & 0.00 & 2.3\
[run115]{} & 7.5 & 5.1 & 8 & 0.7 & 3.4 & 181.1 & 0.00 & 0.00 & 1.7 & 10 & 2.8 & 7 & 0.4 & 0.5 & 8.1 & 4.8 & 359.2 & 0.00 & 0.00 & 2.2\
[run116]{} & 7.5 & 6.4 & 10 & 0.7 & 3.4 & 220.4 & 0.00 & 0.00 & 1.7 & 10 & 3.5 & 9 & 0.4 & 0.2 & 8.0 & 4.9 & 394.2 & 0.00 & 0.00 & 2.3\
[run117]{} & 7.5 & 2.6 & 4 & 0.7 & 3.3 & 103.6 & 0.05 & 0.05 & 1.7 & 11 & 1.4 & 3 & 0.9 & 0.4 & 8.7 & 4.9 & 14.2 & 0.05 & 0.10 & 1.8\
[run118]{} & 7.5 & 3.9 & 6 & 0.7 & 3.4 & 130.7 & 0.05 & 0.05 & 1.7 & 10 & 2.1 & 5 & 0.7 & 0.8 & 8.6 & 4.8 & 59.1 & 0.05 & 0.10 & 2.0\
[run119]{} & 7.5 & 5.3 & 8 & 0.7 & 3.4 & 195.8 & 0.05 & 0.05 & 1.7 & 10 & 2.8 & 7 & 0.7 & 0.6 & 8.4 & 5.0 & 83.9 & 0.05 & 0.10 & 2.0\
[run120]{} & 7.5 & 6.6 & 10 & 0.7 & 3.4 & 225.8 & 0.05 & 0.05 & 1.7 & 10 & 3.6 & 9 & 0.7 & 0.5 & 8.2 & 5.0 & 76.1 & 0.05 & 0.09 & 2.0\
[run121]{} & 7.5 & 2.7 & 4 & 0.7 & 3.3 & 103.0 & 0.10 & 0.10 & 1.7 & 12 & 1.4 & 3 & 0.9 & 0.4 & 9.0 & 5.3 & 13.5 & 0.09 & 0.18 & 1.8\
[run122]{} & 7.5 & 4.0 & 6 & 0.7 & 3.3 & 133.1 & 0.10 & 0.10 & 1.7 & 11 & 2.1 & 5 & 0.8 & 0.4 & 8.8 & 5.2 & 39.5 & 0.09 & 0.18 & 1.9\
[run123]{} & 7.5 & 5.4 & 8 & 0.7 & 3.4 & 199.3 & 0.10 & 0.10 & 1.7 & 11 & 2.9 & 7 & 0.9 & 0.8 & 8.5 & 5.0 & 39.7 & 0.09 & 0.17 & 1.9\
[run124]{} & 7.5 & 6.8 & 10 & 0.7 & 3.4 & 230.1 & 0.10 & 0.10 & 1.7 & 11 & 3.6 & 9 & 0.8 & 0.5 & 8.4 & 5.0 & 58.5 & 0.09 & 0.17 & 2.0\
[run125]{} & 7.5 & 2.8 & 4 & 0.7 & 3.3 & 121.9 & 0.20 & 0.20 & 1.7 & 12 & 1.4 & 3 & 1.0 & 0.6 & 9.2 & 5.4 & 9.8 & 0.18 & 0.32 & 1.7\
[run126]{} & 7.5 & 4.3 & 6 & 0.7 & 3.4 & 135.8 & 0.20 & 0.20 & 1.7 & 12 & 2.2 & 5 & 0.9 & 0.7 & 9.1 & 5.2 & 24.0 & 0.18 & 0.32 & 1.8\
[run127]{} & 7.5 & 5.7 & 8 & 0.7 & 3.4 & 218.7 & 0.20 & 0.20 & 1.7 & 12 & 3.0 & 7 & 1.0 & 0.5 & 8.9 & 5.3 & 23.6 & 0.18 & 0.30 & 1.8\
[run128]{} & 7.5 & 7.1 & 10 & 0.7 & 3.4 & 248.9 & 0.20 & 0.20 & 1.7 & 12 & 3.8 & 9 & 1.0 & 0.3 & 8.8 & 5.4 & 29.8 & 0.18 & 0.29 & 1.9\
[run129]{} & 4 & 3.0 & 4 & 1.6 & 3.3 & 14.6 & 0.30 & 0.30 & 0.84 & 12 & 1.6 & 3 & 2.4 & 2.3 & 7.7 & 4.4 & 0.7 & 0.28 & 0.36 & 1.4\
[run130]{} & 4 & 4.5 & 6 & 1.6 & 3.3 & 19.7 & 0.30 & 0.30 & 0.84 & 12 & 2.4 & 5 & 2.5 & 2.6 & 7.5 & 4.4 & 0.9 & 0.27 & 0.35 & 1.5\
[run131]{} & 4 & 6.0 & 8 & 1.6 & 3.3 & 28.8 & 0.30 & 0.30 & 0.84 & 12 & 3.2 & 7 & 2.6 & 2.5 & 7.4 & 4.5 & 1.1 & 0.27 & 0.34 & 1.4\
[run132]{} & 4 & 7.5 & 10 & 1.6 & 3.3 & 34.7 & 0.30 & 0.30 & 0.84 & 12 & 4.0 & 9 & 2.8 & 3.1 & 7.3 & 4.5 & 1.1 & 0.27 & 0.33 & 1.5\
[run133]{} & 5 & 3.0 & 4 & 1.5 & 3.3 & 17.0 & 0.30 & 0.30 & 1.0 & 12 & 1.6 & 3 & 2.3 & 2.1 & 7.8 & 4.5 & 0.7 & 0.27 & 0.36 & 1.4\
[run134]{} & 5 & 4.5 & 6 & 1.5 & 3.3 & 22.8 & 0.30 & 0.30 & 1.0 & 12 & 2.4 & 5 & 2.4 & 2.4 & 7.6 & 4.5 & 1.0 & 0.27 & 0.36 & 1.5\
[run135]{} & 5 & 6.0 & 8 & 1.5 & 3.3 & 33.7 & 0.30 & 0.30 & 1.0 & 12 & 3.2 & 7 & 2.6 & 2.4 & 7.5 & 4.5 & 1.1 & 0.27 & 0.34 & 1.5\
[run136]{} & 5 & 7.5 & 10 & 1.5 & 3.3 & 39.8 & 0.30 & 0.30 & 1.0 & 12 & 4.0 & 9 & 2.6 & 2.7 & 7.3 & 4.4 & 1.5 & 0.27 & 0.34 & 1.5\
[run137]{} & 6 & 3.0 & 4 & 1.4 & 3.2 & 20.6 & 0.30 & 0.30 & 1.3 & 12 & 1.6 & 3 & 2.2 & 2.1 & 7.9 & 4.4 & 1.0 & 0.28 & 0.37 & 1.4\
[run138]{} & 6 & 4.5 & 6 & 1.4 & 3.3 & 27.1 & 0.30 & 0.30 & 1.3 & 12 & 2.4 & 5 & 2.3 & 1.9 & 7.7 & 4.5 & 1.1 & 0.27 & 0.36 & 1.5\
[run139]{} & 6 & 6.0 & 8 & 1.4 & 3.3 & 40.4 & 0.30 & 0.30 & 1.3 & 12 & 3.2 & 7 & 2.5 & 2.2 & 7.6 & 4.5 & 1.3 & 0.27 & 0.34 & 1.5\
[run140]{} & 6 & 7.5 & 10 & 1.4 & 3.3 & 47.6 & 0.30 & 0.30 & 1.3 & 12 & 4.0 & 9 & 2.5 & 2.5 & 7.5 & 4.5 & 1.6 & 0.27 & 0.34 & 1.5\
[run141]{} & 7 & 3.0 & 4 & 1.3 & 3.2 & 25.8 & 0.30 & 0.30 & 1.5 & 12 & 1.5 & 3 & 2.0 & 1.5 & 8.1 & 4.6 & 1.1 & 0.28 & 0.37 & 1.5\
[run142]{} & 7 & 4.5 & 6 & 1.3 & 3.2 & 33.4 & 0.30 & 0.30 & 1.5 & 12 & 2.4 & 5 & 2.2 & 1.7 & 7.9 & 4.5 & 1.5 & 0.27 & 0.36 & 1.5\
[run143]{} & 7 & 6.0 & 8 & 1.3 & 3.3 & 50.3 & 0.30 & 0.30 & 1.5 & 12 & 3.2 & 7 & 2.1 & 1.7 & 7.7 & 4.6 & 2.1 & 0.27 & 0.36 & 1.5\
[run144]{} & 7 & 7.5 & 10 & 1.3 & 3.2 & 58.8 & 0.30 & 0.30 & 1.5 & 12 & 4.0 & 9 & 2.4 & 2.0 & 7.6 & 4.6 & 1.8 & 0.27 & 0.34 & 1.5\
[run145]{} & 8 & 3.0 & 4 & 1.2 & 3.2 & 34.0 & 0.30 & 0.30 & 1.9 & 12 & 1.5 & 3 & 1.8 & 1.3 & 8.3 & 4.7 & 1.6 & 0.28 & 0.39 & 1.5\
[run146]{} & 8 & 4.5 & 6 & 1.2 & 3.2 & 43.2 & 0.30 & 0.30 & 1.9 & 12 & 2.4 & 5 & 1.9 & 1.7 & 8.1 & 4.7 & 2.1 & 0.27 & 0.37 & 1.5\
[run147]{} & 8 & 6.0 & 8 & 1.2 & 3.2 & 66.3 & 0.30 & 0.30 & 1.9 & 12 & 3.2 & 7 & 2.0 & 1.7 & 7.9 & 4.7 & 2.5 & 0.27 & 0.36 & 1.6\
[run148]{} & 8 & 7.5 & 10 & 1.2 & 3.2 & 76.0 & 0.30 & 0.30 & 1.9 & 12 & 4.0 & 9 & 2.1 & 1.6 & 7.8 & 4.7 & 2.7 & 0.27 & 0.35 & 1.6\
[run149]{} & 6.5 & 3.0 & 4 & 1.1 & 3.2 & 47.8 & 0.30 & 0.30 & 1.4 & 12 & 1.5 & 3 & 1.5 & 1.1 & 8.6 & 4.8 & 2.8 & 0.28 & 0.40 & 1.6\
[run150]{} & 6.5 & 4.5 & 6 & 1.1 & 3.2 & 59.5 & 0.30 & 0.30 & 1.4 & 12 & 2.3 & 5 & 1.8 & 1.3 & 8.4 & 4.8 & 2.7 & 0.27 & 0.37 & 1.6\
[run151]{} & 6.5 & 6.0 & 8 & 1.1 & 3.2 & 92.1 & 0.30 & 0.30 & 1.4 & 12 & 3.2 & 7 & 1.7 & 1.3 & 8.2 & 4.9 & 4.3 & 0.27 & 0.37 & 1.6\
[run152]{} & 6.5 & 7.5 & 10 & 1.1 & 3.2 & 104.6 & 0.30 & 0.30 & 1.4 & 12 & 4.0 & 9 & 1.7 & 1.4 & 8.1 & 4.8 & 5.1 & 0.27 & 0.37 & 1.6\
[run153]{} & 7 & 3.0 & 4 & 0.9 & 3.2 & 72.5 & 0.30 & 0.30 & 1.5 & 12 & 1.5 & 3 & 1.4 & 0.8 & 9.0 & 5.1 & 3.9 & 0.28 & 0.42 & 1.6\
[run154]{} & 7 & 4.5 & 6 & 0.9 & 3.3 & 88.3 & 0.30 & 0.30 & 1.5 & 12 & 2.3 & 5 & 1.4 & 0.9 & 8.8 & 5.1 & 6.2 & 0.27 & 0.40 & 1.7\
[run155]{} & 7 & 6.0 & 8 & 0.9 & 3.3 & 139.3 & 0.30 & 0.30 & 1.5 & 12 & 3.1 & 7 & 1.4 & 0.9 & 8.6 & 5.0 & 7.2 & 0.27 & 0.38 & 1.7\
[run156]{} & 7 & 7.5 & 10 & 0.9 & 3.3 & 157.0 & 0.30 & 0.30 & 1.5 & 12 & 3.9 & 9 & 1.6 & 1.0 & 8.5 & 5.0 & 6.7 & 0.27 & 0.37 & 1.7\
[run157]{} & 7.5 & 3.0 & 4 & 0.7 & 3.3 & 122.3 & 0.30 & 0.30 & 1.7 & 12 & 1.5 & 3 & 1.1 & 0.4 & 9.4 & 5.4 & 10.0 & 0.28 & 0.43 & 1.7\
[run158]{} & 7.5 & 4.5 & 6 & 0.7 & 3.3 & 143.3 & 0.30 & 0.30 & 1.7 & 12 & 2.3 & 5 & 1.1 & 0.5 & 9.3 & 5.4 & 11.6 & 0.27 & 0.42 & 1.8\
[run159]{} & 7.5 & 6.0 & 8 & 0.7 & 3.4 & 229.1 & 0.30 & 0.30 & 1.7 & 12 & 3.1 & 7 & 1.2 & 0.6 & 9.1 & 5.4 & 13.3 & 0.27 & 0.40 & 1.8\
[run160]{} & 7.5 & 7.5 & 10 & 0.7 & 3.4 & 255.8 & 0.30 & 0.30 & 1.7 & 12 & 3.9 & 9 & 1.2 & 0.7 & 9.0 & 5.4 & 15.5 & 0.27 & 0.39 & 1.8\
[run161]{} & 4 & 2.6 & 4 & 1.2 & 2.5 & 29.1 & 0.05 & 0.05 & 0.84 & 12 & 1.4 & 3 & 0.9 & 0.6 & 6.1 & 3.3 & 12.1 & 0.05 & 0.10 & 1.8\
[run162]{} & 4 & 3.9 & 6 & 1.2 & 2.5 & 41.0 & 0.05 & 0.05 & 0.84 & 12 & 2.1 & 5 & 1.2 & 0.8 & 5.8 & 3.4 & 8.5 & 0.05 & 0.08 & 1.7\
[run163]{} & 4 & 5.3 & 8 & 1.2 & 2.5 & 58.5 & 0.05 & 0.05 & 0.84 & 12 & 2.8 & 7 & 1.4 & 1.1 & 5.7 & 3.3 & 7.2 & 0.05 & 0.08 & 1.7\
[run164]{} & 4 & 6.6 & 10 & 1.2 & 2.5 & 71.7 & 0.05 & 0.05 & 0.84 & 12 & 3.6 & 9 & 1.5 & 1.2 & 5.6 & 3.3 & 6.9 & 0.04 & 0.08 & 1.7\
[run165]{} & 4 & 2.7 & 4 & 1.2 & 2.5 & 29.3 & 0.10 & 0.10 & 0.84 & 12 & 1.4 & 3 & 1.0 & 0.6 & 6.2 & 3.3 & 11.2 & 0.09 & 0.17 & 1.8\
[run166]{} & 4 & 4.0 & 6 & 1.2 & 2.5 & 41.9 & 0.10 & 0.10 & 0.84 & 12 & 2.2 & 5 & 1.2 & 1.0 & 6.0 & 3.4 & 7.5 & 0.09 & 0.16 & 1.7\
[run167]{} & 4 & 5.4 & 8 & 1.2 & 2.5 & 60.0 & 0.10 & 0.10 & 0.84 & 12 & 2.9 & 7 & 1.5 & 1.0 & 5.8 & 3.3 & 6.0 & 0.09 & 0.15 & 1.7\
[run168]{} & 4 & 6.8 & 10 & 1.2 & 2.5 & 73.2 & 0.10 & 0.10 & 0.84 & 12 & 3.6 & 9 & 1.5 & 0.9 & 5.7 & 3.3 & 6.2 & 0.09 & 0.14 & 1.7\
[run169]{} & 4 & 2.8 & 4 & 1.2 & 2.5 & 33.9 & 0.20 & 0.20 & 0.84 & 12 & 1.5 & 3 & 1.2 & 0.8 & 6.4 & 3.4 & 4.9 & 0.18 & 0.29 & 1.7\
[run170]{} & 4 & 4.3 & 6 & 1.2 & 2.5 & 43.3 & 0.20 & 0.20 & 0.84 & 12 & 2.3 & 5 & 1.4 & 1.1 & 6.1 & 3.4 & 5.6 & 0.18 & 0.29 & 1.7\
[run171]{} & 4 & 5.7 & 8 & 1.2 & 2.5 & 65.4 & 0.20 & 0.20 & 0.84 & 12 & 3.0 & 7 & 1.6 & 1.4 & 6.0 & 3.4 & 4.8 & 0.18 & 0.27 & 1.7\
[run172]{} & 4 & 7.1 & 10 & 1.2 & 2.5 & 79.2 & 0.20 & 0.20 & 0.84 & 12 & 3.8 & 9 & 1.6 & 1.3 & 5.8 & 3.4 & 5.3 & 0.18 & 0.26 & 1.7\
[run173]{} & 4 & 3.0 & 4 & 1.2 & 2.5 & 34.6 & 0.30 & 0.30 & 0.84 & 12 & 1.5 & 3 & 1.4 & 0.9 & 6.6 & 3.5 & 3.8 & 0.27 & 0.40 & 1.6\
[run174]{} & 4 & 4.5 & 6 & 1.2 & 2.5 & 46.6 & 0.30 & 0.30 & 0.84 & 12 & 2.4 & 5 & 1.5 & 1.3 & 6.3 & 3.4 & 4.3 & 0.27 & 0.38 & 1.6\
[run175]{} & 4 & 6.0 & 8 & 1.2 & 2.5 & 68.3 & 0.30 & 0.30 & 0.84 & 12 & 3.2 & 7 & 1.6 & 1.3 & 6.1 & 3.3 & 4.7 & 0.27 & 0.37 & 1.7\
[run176]{} & 4 & 7.5 & 10 & 1.2 & 2.5 & 82.2 & 0.30 & 0.30 & 0.84 & 12 & 4.0 & 9 & 1.8 & 1.5 & 6.0 & 3.4 & 4.5 & 0.27 & 0.35 & 1.7\
[run177]{} & 5 & 2.6 & 4 & 1.1 & 2.4 & 41.1 & 0.05 & 0.05 & 1.0 & 12 & 1.4 & 3 & 0.6 & 0.6 & 6.3 & 3.4 & 36.4 & 0.05 & 0.10 & 1.9\
[run178]{} & 5 & 3.9 & 6 & 1.1 & 2.5 & 56.8 & 0.05 & 0.05 & 1.0 & 12 & 2.1 & 5 & 1.0 & 0.6 & 6.1 & 3.5 & 14.9 & 0.05 & 0.09 & 1.8\
[run179]{} & 5 & 5.3 & 8 & 1.1 & 2.5 & 82.0 & 0.05 & 0.05 & 1.0 & 12 & 2.8 & 7 & 1.2 & 0.8 & 5.9 & 3.4 & 10.9 & 0.04 & 0.08 & 1.8\
[run180]{} & 5 & 6.6 & 10 & 1.1 & 2.5 & 98.6 & 0.05 & 0.05 & 1.0 & 12 & 3.6 & 9 & 1.2 & 1.0 & 5.8 & 3.4 & 12.2 & 0.04 & 0.08 & 1.8\
[run181]{} & 5 & 2.7 & 4 & 1.1 & 2.4 & 41.2 & 0.10 & 0.10 & 1.0 & 12 & 1.4 & 3 & 0.8 & 0.4 & 6.5 & 3.6 & 20.1 & 0.09 & 0.18 & 1.9\
[run182]{} & 5 & 4.0 & 6 & 1.1 & 2.5 & 58.0 & 0.10 & 0.10 & 1.0 & 12 & 2.2 & 5 & 1.0 & 0.6 & 6.2 & 3.4 & 12.8 & 0.09 & 0.17 & 1.8\
[run183]{} & 5 & 5.4 & 8 & 1.1 & 2.5 & 84.0 & 0.10 & 0.10 & 1.0 & 12 & 2.9 & 7 & 1.2 & 1.0 & 6.0 & 3.5 & 9.9 & 0.09 & 0.15 & 1.8\
[run184]{} & 5 & 6.8 & 10 & 1.1 & 2.5 & 100.7 & 0.10 & 0.10 & 1.0 & 12 & 3.6 & 9 & 1.3 & 1.2 & 5.9 & 3.4 & 10.3 & 0.09 & 0.15 & 1.8\
[run185]{} & 5 & 2.8 & 4 & 1.1 & 2.4 & 48.2 & 0.20 & 0.20 & 1.0 & 12 & 1.5 & 3 & 1.0 & 0.5 & 6.7 & 3.6 & 9.9 & 0.18 & 0.31 & 1.8\
[run186]{} & 5 & 4.3 & 6 & 1.1 & 2.5 & 59.8 & 0.20 & 0.20 & 1.0 & 12 & 2.3 & 5 & 1.2 & 0.9 & 6.4 & 3.5 & 8.8 & 0.18 & 0.29 & 1.7\
[run187]{} & 5 & 5.7 & 8 & 1.1 & 2.5 & 91.8 & 0.20 & 0.20 & 1.0 & 12 & 3.0 & 7 & 1.4 & 0.9 & 6.2 & 3.5 & 6.7 & 0.18 & 0.27 & 1.7\
[run188]{} & 5 & 7.1 & 10 & 1.1 & 2.5 & 109.0 & 0.20 & 0.20 & 1.0 & 12 & 3.8 & 9 & 1.4 & 0.9 & 6.1 & 3.4 & 8.6 & 0.18 & 0.27 & 1.8\
[run189]{} & 5 & 3.0 & 4 & 1.1 & 2.4 & 48.9 & 0.30 & 0.30 & 1.0 & 12 & 1.5 & 3 & 1.2 & 0.9 & 6.8 & 3.5 & 5.9 & 0.27 & 0.41 & 1.7\
[run190]{} & 5 & 4.5 & 6 & 1.1 & 2.4 & 64.2 & 0.30 & 0.30 & 1.0 & 12 & 2.3 & 5 & 1.4 & 1.2 & 6.6 & 3.5 & 5.9 & 0.27 & 0.39 & 1.7\
[run191]{} & 5 & 6.0 & 8 & 1.1 & 2.5 & 95.9 & 0.30 & 0.30 & 1.0 & 12 & 3.2 & 7 & 1.5 & 1.1 & 6.3 & 3.4 & 5.8 & 0.27 & 0.37 & 1.7\
[run192]{} & 5 & 7.5 & 10 & 1.1 & 2.5 & 112.8 & 0.30 & 0.30 & 1.0 & 12 & 4.0 & 9 & 1.5 & 0.9 & 6.2 & 3.4 & 6.8 & 0.27 & 0.36 & 1.7\
[run193]{} & 6 & 2.6 & 4 & 0.9 & 2.4 & 67.8 & 0.05 & 0.05 & 1.3 & 11 & 1.4 & 3 & 0.5 & 0.4 & 6.7 & 3.5 & 79.2 & 0.05 & 0.11 & 2.1\
[run194]{} & 6 & 3.9 & 6 & 0.9 & 2.4 & 91.3 & 0.05 & 0.05 & 1.3 & 12 & 2.1 & 5 & 0.7 & 0.6 & 6.5 & 3.5 & 53.2 & 0.05 & 0.10 & 2.0\
[run195]{} & 6 & 5.3 & 8 & 0.9 & 2.4 & 134.1 & 0.05 & 0.05 & 1.3 & 12 & 2.8 & 7 & 0.7 & 0.4 & 6.3 & 3.6 & 81.8 & 0.04 & 0.10 & 2.0\
[run196]{} & 6 & 6.6 & 10 & 0.9 & 2.4 & 157.7 & 0.05 & 0.05 & 1.3 & 12 & 3.6 & 9 & 0.8 & 0.5 & 6.2 & 3.6 & 56.5 & 0.04 & 0.09 & 2.0\
[run197]{} & 6 & 2.7 & 4 & 0.9 & 2.4 & 67.8 & 0.10 & 0.10 & 1.3 & 12 & 1.4 & 3 & 0.6 & 0.4 & 6.9 & 3.9 & 49.9 & 0.09 & 0.18 & 2.0\
[run198]{} & 6 & 4.0 & 6 & 0.9 & 2.4 & 93.2 & 0.10 & 0.10 & 1.3 & 12 & 2.2 & 5 & 0.8 & 0.4 & 6.6 & 3.7 & 36.1 & 0.09 & 0.18 & 1.9\
[run199]{} & 6 & 5.4 & 8 & 0.9 & 2.4 & 137.2 & 0.10 & 0.10 & 1.3 & 12 & 2.9 & 7 & 0.8 & 0.5 & 6.4 & 3.7 & 35.0 & 0.09 & 0.17 & 1.9\
[run200]{} & 6 & 6.8 & 10 & 0.9 & 2.4 & 161.0 & 0.10 & 0.10 & 1.3 & 12 & 3.6 & 9 & 1.0 & 0.6 & 6.3 & 3.6 & 27.8 & 0.09 & 0.16 & 1.9\
[run201]{} & 6 & 2.8 & 4 & 0.9 & 2.4 & 79.8 & 0.20 & 0.20 & 1.3 & 12 & 1.5 & 3 & 0.7 & 0.4 & 7.1 & 3.9 & 33.7 & 0.18 & 0.33 & 1.9\
[run202]{} & 6 & 4.3 & 6 & 0.9 & 2.4 & 95.7 & 0.20 & 0.20 & 1.3 & 12 & 2.2 & 5 & 0.9 & 0.6 & 6.8 & 3.8 & 18.3 & 0.18 & 0.31 & 1.8\
[run203]{} & 6 & 5.7 & 8 & 0.9 & 2.4 & 150.3 & 0.20 & 0.20 & 1.3 & 12 & 3.0 & 7 & 1.0 & 0.6 & 6.6 & 3.7 & 21.7 & 0.18 & 0.29 & 1.8\
[run204]{} & 6 & 7.1 & 10 & 0.9 & 2.4 & 174.4 & 0.20 & 0.20 & 1.3 & 12 & 3.8 & 9 & 1.2 & 0.8 & 6.4 & 3.6 & 16.5 & 0.18 & 0.28 & 1.8\
[run205]{} & 6 & 3.0 & 4 & 0.9 & 2.4 & 80.5 & 0.30 & 0.30 & 1.3 & 12 & 1.5 & 3 & 1.0 & 0.7 & 7.3 & 3.8 & 10.2 & 0.27 & 0.42 & 1.8\
[run206]{} & 6 & 4.5 & 6 & 0.9 & 2.4 & 102.4 & 0.30 & 0.30 & 1.3 & 12 & 2.3 & 5 & 1.1 & 0.4 & 7.0 & 3.7 & 11.9 & 0.27 & 0.40 & 1.8\
[run207]{} & 6 & 6.0 & 8 & 0.9 & 2.4 & 157.1 & 0.30 & 0.30 & 1.3 & 12 & 3.1 & 7 & 1.2 & 1.1 & 6.7 & 3.6 & 10.8 & 0.27 & 0.39 & 1.8\
[run208]{} & 6 & 7.5 & 10 & 0.9 & 2.4 & 180.2 & 0.30 & 0.30 & 1.3 & 12 & 3.9 & 9 & 1.3 & 0.9 & 6.6 & 3.7 & 11.9 & 0.27 & 0.37 & 1.8\
[run209]{} & 7 & 2.6 & 4 & 0.7 & 2.4 & 144.9 & 0.05 & 0.05 & 1.5 & 9 & 1.4 & 3 & 0.7 & 0.4 & 7.0 & 3.8 & 34.5 & 0.05 & 0.11 & 1.9\
[run210]{} & 7 & 3.9 & 6 & 0.7 & 2.4 & 189.0 & 0.05 & 0.05 & 1.5 & 10 & 2.1 & 5 & 0.6 & 0.5 & 6.8 & 3.7 & 69.0 & 0.05 & 0.10 & 2.0\
[run211]{} & 7 & 5.3 & 8 & 0.7 & 2.5 & 281.9 & 0.05 & 0.05 & 1.5 & 9 & 2.9 & 7 & 0.6 & 0.7 & 6.5 & 3.5 & 91.5 & 0.05 & 0.10 & 2.0\
[run212]{} & 7 & 6.6 & 10 & 0.7 & 2.5 & 327.7 & 0.05 & 0.05 & 1.5 & 9 & 3.6 & 9 & 0.7 & 0.6 & 6.4 & 3.6 & 88.1 & 0.04 & 0.09 & 2.0\
[run213]{} & 7 & 2.7 & 4 & 0.7 & 2.4 & 144.2 & 0.10 & 0.10 & 1.5 & 10 & 1.4 & 3 & 0.8 & 0.4 & 7.2 & 4.0 & 21.7 & 0.09 & 0.19 & 1.9\
[run214]{} & 7 & 4.0 & 6 & 0.7 & 2.4 & 192.6 & 0.10 & 0.10 & 1.5 & 9 & 2.2 & 5 & 0.8 & 0.3 & 6.9 & 3.8 & 38.9 & 0.09 & 0.18 & 1.9\
[run215]{} & 7 & 5.4 & 8 & 0.7 & 2.5 & 287.6 & 0.10 & 0.10 & 1.5 & 10 & 2.9 & 7 & 0.7 & 0.3 & 6.7 & 3.8 & 76.3 & 0.09 & 0.18 & 2.0\
[run216]{} & 7 & 6.8 & 10 & 0.7 & 2.5 & 334.2 & 0.10 & 0.10 & 1.5 & 10 & 3.7 & 9 & 0.7 & 0.7 & 6.6 & 3.7 & 69.2 & 0.09 & 0.17 & 2.0\
[run217]{} & 7 & 2.8 & 4 & 0.7 & 2.4 & 171.0 & 0.20 & 0.20 & 1.5 & 11 & 1.5 & 3 & 0.9 & 0.7 & 7.5 & 3.9 & 18.6 & 0.18 & 0.32 & 1.8\
[run218]{} & 7 & 4.3 & 6 & 0.7 & 2.4 & 197.0 & 0.20 & 0.20 & 1.5 & 10 & 2.3 & 5 & 0.9 & 0.4 & 7.2 & 4.0 & 26.9 & 0.18 & 0.31 & 1.9\
[run219]{} & 7 & 5.7 & 8 & 0.7 & 2.5 & 315.7 & 0.20 & 0.20 & 1.5 & 11 & 3.0 & 7 & 0.8 & 0.6 & 7.0 & 3.8 & 42.3 & 0.18 & 0.31 & 1.9\
[run220]{} & 7 & 7.1 & 10 & 0.7 & 2.5 & 361.5 & 0.20 & 0.20 & 1.5 & 11 & 3.8 & 9 & 0.8 & 0.5 & 6.9 & 3.9 & 51.0 & 0.18 & 0.29 & 2.0\
[run221]{} & 7 & 3.0 & 4 & 0.7 & 2.4 & 171.8 & 0.30 & 0.30 & 1.5 & 12 & 1.5 & 3 & 1.0 & 0.6 & 7.9 & 4.3 & 12.4 & 0.27 & 0.42 & 1.8\
[run222]{} & 7 & 4.5 & 6 & 0.7 & 2.4 & 209.2 & 0.30 & 0.30 & 1.5 & 12 & 2.3 & 5 & 0.9 & 0.5 & 7.6 & 4.1 & 23.9 & 0.27 & 0.42 & 1.9\
[run223]{} & 7 & 6.0 & 8 & 0.7 & 2.5 & 330.2 & 0.30 & 0.30 & 1.5 & 11 & 3.1 & 7 & 0.9 & 0.6 & 7.3 & 4.0 & 26.5 & 0.27 & 0.41 & 1.9\
[run224]{} & 7 & 7.5 & 10 & 0.7 & 2.4 & 372.1 & 0.30 & 0.30 & 1.5 & 11 & 3.9 & 9 & 1.0 & 0.6 & 7.1 & 3.9 & 31.2 & 0.27 & 0.39 & 1.9\
P., [Lamers]{} H. J. G. L. M., [Baumgardt]{} H., 2009, [[A&A]{}]{}, 502, 817
K. M., [Zepf]{} S. E., 2001, [[AJ]{}]{}, 122, 1888
H., [Makino]{} J., 2003, [[MNRAS]{}]{}, 340, 227
S., [Hut]{} P., 1985, [[ApJ]{}]{}, 298, 80
S., [Fregeau]{} J. M., [Rasio]{} F. A., 2008, in IAU Symposium Vol. 246 of IAU Symposium, [Effects of Stellar Collisions on Star Cluster Evolution and Core Collapse]{}. pp 151–155
S., [Fregeau]{} J. M., [Umbreit]{} S., [Rasio]{} F. A., 2010, [[ApJ]{}]{}, 719, 915
D. S., [Richer]{} H. B., [Anderson]{} J., [Brewer]{} J., [Hurley]{} J., [Kalirai]{} J. S., [Rich]{} R. M., [Stetson]{} P. B., 2008, [[AJ]{}]{}, 135, 2155
O., [Clarke]{} C. J., [Freitag]{} M., 2010, [[MNRAS]{}]{}, p. 844
S., 1993, in [S. G. Djorgovski & G. Meylan]{} ed., Structure and Dynamics of Globular Clusters Vol. 50 of Astronomical Society of the Pacific Conference Series, [Physical Parameters of Galactic Globular Clusters]{}. p. 373
J. M., 2008, [[ApJ]{}]{}, 673, L25
J. M., [G[ü]{}rkan]{} M. A., [Joshi]{} K. J., [Rasio]{} F. A., 2003, [[ApJ]{}]{}, 593, 772
J. M., [Rasio]{} F. A., 2007, [[ApJ]{}]{}, 658, 1047
M. A., [Freitag]{} M., [Rasio]{} F. A., 2004, [[ApJ]{}]{}, 604, 632
H. C., [Harris]{} G. L. H., [Hesser]{} J. E., [MacGillivray]{} H. T., 1984, [[ApJ]{}]{}, 287, 185
W. E., 1996, [[AJ]{}]{}, 112, 1487
D., [Hut]{} P., 2003, [The Gravitational Million-Body Problem: A Multidisciplinary Approach to Star Cluster Dynamics]{}. Cambridge University Press, 2003
J. E., [Harris]{} H. C., [van den Bergh]{} S., [Harris]{} G. L. H., 1984, [[ApJ]{}]{}, 276, 491
P. W., 1962, [[PASP]{}]{}, 74, 248
J. A., [Faber]{} S. M., [Shaya]{} E. J., [Lauer]{} T. R., [Groth]{} J., [Hunter]{} D. A., [Baum]{} W. A., [Ewald]{} S. P., [Hester]{} J. J., [Light]{} R. M., [Lynds]{} C. R., [O’Neil]{} Jr. E. J., [Westphal]{} J. A., 1992, [[AJ]{}]{}, 103, 691
J. R., 2007, [[MNRAS]{}]{}, 379, 93
N., [Lee]{} M. G., 2008, [[AJ]{}]{}, 135, 1567
K. J., [Nave]{} C. P., [Rasio]{} F. A., 2001, [[ApJ]{}]{}, 550, 691
K. J., [Rasio]{} F. A., [Portegies Zwart]{} S., 2000, [[ApJ]{}]{}, 540, 969
I. R., 1966, [[AJ]{}]{}, 71, 64
P., 2001, [[MNRAS]{}]{}, 322, 231
N., [Umbreit]{} S., [Sills]{} A., [Knigge]{} C., [de Marchi]{} G., [Glebbeek]{} E., [Sarajedini]{} A., 2012, [[MNRAS]{}]{}, 422, 1592
A. D., [Wilkinson]{} M. I., [Davies]{} M. B., [Gilmore]{} G. F., 2007, [[MNRAS]{}]{}, 379, L40
A. D., [Wilkinson]{} M. I., [Davies]{} M. B., [Gilmore]{} G. F., 2008, [[MNRAS]{}]{}, p. 374
M., 1987, [[ApJ]{}]{}, 323, L41
D. E., [van der Marel]{} R. P., 2005, [[ApJS]{}]{}, 161, 304
B. W., [Whitmore]{} B. C., [Schweizer]{} F., [Fall]{} S. M., 1997, [[AJ]{}]{}, 114, 2381
E., [Gebhardt]{} K., 2006, [[AJ]{}]{}, 132, 447
B., [Umbreit]{} S., [Liao]{} W.-K., [Choudhary]{} A., [Kalogera]{} V., [Memik]{} G., [Rasio]{} F. A., 2012, ArXiv e-prints
R. A., [Gieles]{} M., [Haas]{} M. R., [Bastian]{} N., [Larsen]{} S. S., 2009, [The Radii of Thousands of Star Clusters in M51 with HST/ACS]{}. p. 103
R. A., [Haas]{} M. R., [Gieles]{} M., [Bastian]{} N., [Larsen]{} S. S., [Lamers]{} H. J. G. L. M., 2007, [[A&A]{}]{}, 469, 925
M., [Ardi]{} E., [Mineshige]{} S., [Hut]{} P., 2007, [[MNRAS]{}]{}, 374, 857
M., [Heggie]{} D. C., [Hut]{} P., 2007, [[MNRAS]{}]{}, 374, 344
M., [Vesperini]{} E., [Pasquato]{} M., 2010, [[ApJ]{}]{}, 708, 1598
S., [Fregeau]{} J. M., [Chatterjee]{} S., [Rasio]{} F. A., 2012, [[ApJ]{}]{}, 750, 31
S., [Morbey]{} C., [Pazder]{} J., 1991, [[ApJ]{}]{}, 375, 594
E., [Chernoff]{} D. F., 1994, [[ApJ]{}]{}, 431, 231
B. C., [Schweizer]{} F., 1995, [[AJ]{}]{}, 109, 960
\[lastpage\]
|
---
abstract: 'Double detonations in double white dwarf (WD) binaries undergoing unstable mass transfer have emerged in recent years as one of the most promising Type Ia supernova (SN Ia) progenitor scenarios. One potential outcome of this “dynamically driven double-degenerate double-detonation” (D$^6$) scenario is that the companion WD survives the explosion and is flung away with a velocity equal to its $>\unit[1000]{km \, s^{-1}}$ pre-SN orbital velocity. We perform a search for these hypervelocity runaway WDs using *Gaia*’s second data release. In this paper, we discuss seven candidates followed up with ground-based instruments. Three sources are likely to be some of the fastest known stars in the Milky Way, with total Galactocentric velocities between $1000$ and $\unit[3000]{km \, s^{-1}}$, and are consistent with having previously been companion WDs in pre-SN Ia systems. However, although the radial velocity of one of the stars is $>\unit[1000]{km \, s^{-1}}$, the radial velocities of the other two stars are puzzlingly consistent with 0. The combined five-parameter astrometric solutions from *Gaia* and radial velocities from follow-up spectra yield tentative 6D confirmation of the D$^6$ scenario. The past position of one of these stars places it within a faint, old SN remnant, further strengthening the interpretation of these candidates as hypervelocity runaways from binary systems that underwent SNe Ia.'
author:
- 'Ken J. Shen'
- Douglas Boubert
- 'Boris T. Gänsicke'
- 'Saurabh W. Jha'
- 'Jennifer E. Andrews'
- Laura Chomiuk
- 'Ryan J. Foley'
- Morgan Fraser
- Mariusz Gromadzki
- James Guillochon
- 'Marissa M. Kotze'
- Kate Maguire
- 'Matthew R. Siebert'
- Nathan Smith
- Jay Strader
- Carles Badenes
- 'Wolfgang E. Kerzendorf'
- Detlev Koester
- Markus Kromer
- Broxton Miles
- Rüdiger Pakmor
- Josiah Schwab
- Odette Toloza
- Silvia Toonen
- 'Dean M. Townsley'
- 'Brian J. Williams'
title: '****'
---
Introduction {#sec:intro}
============
Type Ia supernovae (SNe Ia) are one of the most common types of SNe in the local Universe. They are best known for their utility as cosmological standardizable candles [@ries98; @perl99] and also play a crucial role in galactic chemical evolution [@tww95]. There is general agreement that the exploding star is a carbon/oxygen white dwarf (C/O WD) and that a companion star triggers runaway nuclear fusion in the WD, leading to a SN Ia powered by the decay of radioactive $^{56}$Ni [@pank62a; @cm69; @maoz14a]. However, despite decades of focused effort, there is no consensus regarding the nature of the companion or the mechanism by which the WD explodes, or even more fundamentally, whether one or multiple progenitor scenarios are responsible.
In “single-degenerate” scenarios, the companion is a non-degenerate hydrogen- or helium-burning star, while in “double-degenerate” scenarios, the companion is another WD. These companions may trigger an explosion in the primary WD in a variety of ways that, in some cases, can be shared among different companion types. For example, models in which the growth of the primary WD leads to the ignition of convective carbon burning, causing a deflagration, detonation, and subsequent SN Ia, have been proposed for hydrogen-rich single-degenerate donors [@wi73; @nomo82a], helium-rich single degenerates [@yoon03a], and double-degenerate merger remnants [@it84; @webb84]. The double-detonation mechanism, in which a helium shell detonation sets off a carbon core detonation [@taam80b; @livn90; @shen14a], has been proposed for helium single degenerates [@nomo82b; @wtw86] and for double degenerates that undergo either stable [@bild07; @fhr07; @fink10; @sb09b] or unstable mass transfer [@guil10; @dan11; @rask12; @pakm13a; @dan15a]. Note that the secondary WDs in these double-degenerate double-detonation systems do not have to be helium core WDs, because C/O WDs are born with significant surface helium layers.
In all progenitor scenarios except for the subclasses of double-degenerate scenarios in which the companion WD is completely destroyed [@it84; @webb84; @pakm12b; @papi15a], the companion star will survive the explosion of the primary WD. This surviving companion will fly away from the site of the explosion with the orbital velocity it had prior to the explosion. The impact of the SN ejecta will also strip material from the companion, deposit shock energy, and possibly pollute the remaining surface layers. These effects lead to observable peculiarities of varying degree, depending on the nature of the companion [@mbf00; @pakm08a; @pan13a; @shap13a; @shen17a].
Searches for surviving companions within the remnants of historical SNe have predominantly focused on the relatively slow ejection velocities and small search radii implied by single-degenerate scenarios (e.g., @ruiz04b [@sp12; @kerz14c]). These studies have failed to conclusively discover any surviving companions, which may not be surprising given other mounting evidence that single-degenerate scenarios cannot be responsible for the bulk of SNe Ia (e.g., @leon07 [@kase10; @gb10; @li11; @bloo12; @olli15a; @magu16a; @wood17a]).
Only one study by [@kerz18a] has covered a large enough search region to probe the $\unit[1000-2500]{km \, s^{-1}}$ runaway velocities expected for surviving WD companions. In their examination of the remnant of SN 1006, no bright, blue sources resembling those predicted by [@shen17a] were found within $8.5'$ of the remnant’s center. However, if the companion WD has cooled significantly since the SN, it may appear as a more typical-looking WD, which would be difficult to distinguish among the other normal stars within SN 1006’s remnant. Furthermore, if significant iron-line blanketing occurs, or if the SN otherwise dramatically alters the companion’s appearance, the surviving WD would not be as blue as predicted by [@shen17a]; such redder sources do exist within the remnant but have yet to be systematically followed up.
In this work, we use *Gaia*’s second data release (DR2; @gaia16a [@gaia18a]) to perform an all-sky search for hypervelocity surviving companion WDs. In Section \[sec:motandprop\], we make predictions for the expected state and number of surviving WDs that may be detected by *Gaia*. In Section \[sec:gaia\], we describe our search for hypervelocity WDs in DR2. We detail our findings and our follow-up efforts in Section \[sec:followup\], and we conclude in Section \[sec:conc\].
Motivation for and properties of a surviving companion WD {#sec:motandprop}
=========================================================
In this section, we motivate the possibility that a companion WD might survive the SN Ia and describe its expected properties following the explosion. We also calculate the expected number of such sources in *Gaia* DR2 under the assumption that all SNe Ia yield a surviving companion WD. We note here that we distinguish between companion WDs that survive normal SNe Ia and the kicked bound WD remnants of explosions that may lead to the peculiar class of SNe Iax [@jord12a; @fole13a; @long14a; @fink14a]. While extremely interesting, the surviving WD primaries of SN Iax explosions are not expected to reach velocities higher than $\unit[1000]{km \, s^{-1}}$ and are not the subject of discussion in this section. We also draw a distinction here between the hypervelocity ($> \unit[1000]{km \, s^{-1}}$) WDs we discuss in this work and the $\ll \unit[1000]{km \, s^{-1}}$ WDs predicted in previous studies that result from the evolution of non-degenerate survivors of single-degenerate SN Ia progenitor scenarios [@hans03a; @just09a].
Motivation
----------
Mass transfer between two WDs can be dynamically stable or unstable, depending on the donor’s response to mass loss, whether or not the mass transfer is conservative, and the degree to which the angular momentum of the transferred mass can be converted back into orbital angular momentum [@mns04]. For large enough mass ratios $\gtrsim 0.2$, the accretion stream directly impacts the more massive WD, and no accretion disk is formed, which implies inefficient angular momentum transfer to the orbit. Furthermore, the mass transfer can be super-Eddington and thus non-conservative. Both of these effects destabilize the binary, and thus most double WD binaries are expected to undergo dynamically unstable mass transfer.
Double WD binaries with extreme mass ratios $\lesssim 0.2$ can have sub-Eddington accretion rates and form accretion disks, both of which help to stabilize mass transfer. However, [@shen15a] points out that even these systems can be driven to unstable mass transfer. The accumulation of the donor’s hydrogen- and helium-rich layers on the accretor lead to classical-nova-like events in which the accreted shell expands relatively slowly due to thermonuclear burning. Dynamical friction between the donor and the expanding envelope causes the binary separation to decrease, which increases the mass transfer rate into the super-Eddington regime and destabilizes even these extreme mass ratio binaries. This theoretical possibility is borne out by the relatively low birth rate of AM CVn systems [@brow16b], providing evidence that all double WD systems eventually undergo runaway mass transfer.
In the original double-degenerate scenario [@it84; @webb84], such double WD systems undergoing dynamically unstable mass transfer coalesce to form a single merger remnant that explodes as a SN Ia after $\sim \unit[10^4]{yr}$. However, recent studies have suggested that the SN Ia can be triggered during the coalescence itself due to the presence of helium in the surface layers of the companion WD, which initiates the SN Ia explosion via the double-detonation mechanism [@guil10; @dan11; @rask12; @pakm13a; @dan15a]. We term this combination of explosion mechanism, companion star, and mode of mass transfer the “dynamically driven double-degenerate double-detonation” (D$^6$) scenario. In fact, when all appropriate nuclear reactions are accounted for [@shen14b], helium shell detonations, and subsequent carbon core detonations, may even occur during the relatively quiescent initial phases of dynamical mass transfer, before the complete tidal disruption of the companion WD [@pakm13a; @shen17a]. Thus, there is the exciting possibility that the D$^6$ scenario yields a smoking gun in the form of a surviving companion WD.
While theoretical confirmation of the existence of a surviving companion WD awaits future detailed simulations, several observables suggest it is a necessary outcome if double WD binaries make up the bulk of SN Ia progenitors. If fully disrupted, the companion WD would form a thick torus surrounding the primary WD [@ggi04; @dan14a], which would impart significant asymmetry to the ejecta when the explosion occurred [@rask14a]. However, such asymmetry is at odds with the low levels of polarization measured in SNe Ia [@wang08a; @bull16a]. Furthermore, much of the disrupted WD material would remain at low velocities throughout the evolution of the SN [@pakm12b]. In the late-time nebular phase, when the SN ejecta becomes optically thin, oxygen in the disrupted WD material may become visible as strong, narrow emission features, but these are almost never detected (although see @krom13a and @taub13a for the case of SN 2010lp). Given these observational constraints and theoretical motivation, we proceed under the presumption that most, if not all, SNe Ia leave an intact companion WD.
Luminosity {#sec:lum}
----------
An estimate of the number of runaway WDs that *Gaia* will detect requires knowledge of their brightness following the SN Ia explosion. As a lower limit, we can assume that the evolution up to the explosion and its aftermath have no effect on the companion, so that its luminosity is the same as that of a WD that has cooled in isolation since its birth. Using publicly available DB WD cooling tables[^1] [@holb06a; @kowa06a; @trem11a; @berg11a], we find that a $0.6 \, (1.0) { \, M_\sun }$ WD that has cooled for $\unit[1]{Gyr}$ since its birth has an absolute visual magnitude of $ 12.9 \, (12.8)$. However, this declines to $ 14.3 \, (14.0) $ by $\unit[3]{Gyr}$. Given *Gaia*’s magnitude limit of $ 21$, this lower limit to the luminosity implies a $3 \, {\rm Gyr}$-old $0.6 { \, M_\sun }$ WD can only be seen out to $\unit[200]{pc}$, while a $1 \, {\rm Gyr}$-old $1.0 { \, M_\sun }$ WD will be detected out to $\unit[400]{pc}$.
However, there are several processes that will increase the luminosity of the companion above this lower limit prior to and just after the SN Ia explosion. The most important of these is likely tidal heating. As the orbit of the two WDs decays due to gravitational wave emission, the binary separation shrinks and the less massive companion begins to feel the tidal field of the primary. If the companion becomes and remains tidally locked to the primary, viscous dissipation yields a luminosity at the onset of mass transfer ranging from $0.15-1000 \, L_\odot$, depending on the component masses [@iben98a], far higher than that of an old, isolated WD.
These values represent upper limits to the effects of tidal heating, as the companion will not be able to maintain complete synchronicity with the orbit. Recent work has found that tidal effects are dominated by the excitation of gravity waves within the companion, which deposit their energy and angular momentum near the surface of the WD [@fl11; @burk13a]. These surface layers can be kept near synchronous rotation all the way to the onset of mass transfer, but since the energy is not deposited deep within the core, the timescale for this excess heat to be radiated away is shorter than the typical $>\unit[10^6]{yr}$ between the explosion and the present day for nearby runaway WDs (Section \[sec:numest\]), and thus the WD would cool and approach the luminosity of a dim, isolated WD.
However, as argued by [@burk13a], the strength of the WD’s fossil magnetic field should be more than adequate to maintain solid body rotation between the outer layers, where angular momentum is deposited, and the core, especially if the field is wound up during the evolution towards solid body rotation. As the interior of the WD is spun up, it will also be heated by small scale turbulence at a comparable level to the dissipation necessary to maintain complete synchronicity with the orbit. In this case, the physical picture approaches that assumed by [@iben98a], and the luminosities at the point of Roche lobe overflow (RLOF) will approach their values. Such luminosities $ \geq 0.1 \, L_\odot$ will be visible to $\unit[1]{kpc}$, a distance at which *Gaia*’s faint end parallax errors become the primary limitation (Section \[sec:gaia\]). Thus, when calculating the expected number of runaway WDs *Gaia* will find in Section \[sec:numest\], we limit our search volume to a sphere around the Sun with a radius of $\unit[1]{kpc}$.
In addition to tidal heating, there are several other possible mechanisms that may change the appearance of a surviving companion WD, all of which occur after the explosion of the primary. The first is due to the impact of the SN ejecta on the companion, which will deposit shock energy and may also ablate some material from the surface. While similar processes have been modeled for single-degenerate companions (e.g., @mbf00 [@pakm08a; @liu12a; @pan14a]), detailed calculations of the post-impact state of a surviving WD have not yet been performed, so quantitative predictions cannot be made (but see @papi15a for a preliminary investigation of some of these effects). However, given the $>\unit[10^6]{yr}$ average delay between the SN and the present day for local runaway WDs (Section \[sec:numest\]) and the much shorter thermal time at the relatively shallow depths where this shock energy should be deposited, it seems unlikely that these effects will still be observable for all but the youngest WDs within historical SN remnants.
A second mechanism that may heat the surviving WD is due to the rapid expansion of the exploding primary, which causes the tidal field felt by the companion WD to change abruptly. Dissipation during the subsequent relaxation to a new hydrostatic equilibrium will deposit heat throughout the WD. However, as in the first case, most of the tidal deformation occurs near the surface of the WD, so that most of the dissipation will also be preferentially located at shallow depths where the thermal timescale is relatively short. As above, given the large average age of the local surviving WDs, we expect this excess heat to be negligible except for ex-companions to historical SNe.
A final mechanism to increase the surviving companion’s post-SN luminosity concerns the capture and accretion of $^{56}$Ni from the SN ejecta by the surviving companion. As discussed in [@shen17a], much of the high-entropy $^{56}$Ni remains fully ionized as it settles onto the WD’s surface and cannot decay via standard bound electron captures until it cools. Thus, the radioactive decay of this accreted $^{56}$Ni can keep the companion WD relatively bright centuries after the SN has faded.
As mentioned above, detailed calculations of the SN ejecta’s interaction with the surviving WD have not yet been performed. This means that stellar evolution calculations, like those in [@shen17a], are forced to rely on simple estimates for the amount of radioactive material captured and its initial thermal state. As such, accurate quantitative predictions cannot yet be made. However, similarly to shock heating by the SN ejecta and tidal relaxation, it is likely that the delayed radioactive decays only affect the outer layers of the companion WD, limiting the time when they can contribute to the luminosity and alter the colors of the WD to centuries. These effects are important for runaway WDs from historical SNe, such as Tycho, Kepler, SN 185, and SN 1006 [@kerz18a], but may be negligible for the much older runaway WDs that should form the bulk of our candidates. For most of the hypervelocity WDs *Gaia* is likely to detect, we expect that the energy deposited much deeper in the WD’s interior from tidal heating prior to the SN Ia will likely determine its present-day luminosity.
Surface abundances {#sec:abun}
------------------
As discussed in [@shen17a], the surviving WD will capture some of the lowest velocity SN ejecta, primarily composed of $^{56}$Ni. The energy released by the slowly decaying $^{56}$Ni blows a wind from the WD’s surface, ejecting much of the accreted SN ejecta, but some of the material should remain bound and might be detectable with follow-up high resolution spectra. Unfortunately, as before, the lack of relevant hydrodynamic simulations makes accurate predictions of the surface abundances of a surviving WD difficult.
A surviving WD companion should be hydrogen-free, because the hydrogen layer that most WDs are born with will have been transferred to the primary and ejected from the system in the $\sim \unit[1000]{yr}$ prior to the SN [@kbs12; @shen13a; @shen15a]. Given the long cooling timescales of the expected *Gaia* WDs within $\unit[1]{kpc}$, sedimentation may cause heavy metals to sink [@paqu86b; @dupu92a], leaving only helium or carbon and oxygen in the atmosphere, depending on the initial composition of the companion, but anomalous abundances could potentially still be observable for the young runaway WDs from Tycho, Kepler, SN 185, and SN 1006. However, this statement depends on the existence and depth of a surface convection zone, the thermal profile, and, if the surface layers remain $\gtrsim \unit[2{\times10^{4}}]{K}$, the competing effect of radiative levitation [@chay95a; @chay95b]. We regard the expected surface abundances to be uncertain and thus one of the motivations for follow-up spectroscopy.
Velocity
--------
Given the uncertainties in the observational characteristics discussed in the previous sections, the clearest and most obvious distinguishing feature of a surviving companion WD is its hypervelocity. The orbital velocity of a companion star during Roche lobe overflow (RLOF), which will be its runaway velocity once the primary WD explodes,[^2] is a function of its mass, $M_2$, radius, $R_2$, and the mass of the exploding WD, $M_1$: $$\begin{aligned}
v_{\rm runaway} = \sqrt{ \frac{ GM_1^2 }{ \left( M_1+M_2 \right) a } } ,\end{aligned}$$ where the binary separation during RLOF, $a$, is approximated by [@eggl83] as $$\begin{aligned}
a = \frac{R_2}{0.49} \left\{ 0.6 + \left( \frac{M_1}{M_2} \right)^{2/3} \ln \left[ 1+ \left( \frac{M_2}{M_1} \right)^{1/3} \right] \right\} .\end{aligned}$$
![The companion’s orbital velocity vs. mass at RLOF. The upper boundary of each region corresponds to a $1.1 { \, M_\sun }$ primary WD; the lower corresponds to a $0.85 { \, M_\sun }$ primary.[]{data-label="fig:mvsv"}](mvsv){width="\columnwidth"}
We calculate simple mass-radius relationships for isolated helium and C/O WDs, as well as for hydrogen- and helium-burning non-degenerate companions with the stellar evolution code `MESA` [@paxt11; @paxt13; @paxt15a; @paxt18a]. The resulting runaway velocities vs. companion masses are shown in Figure \[fig:mvsv\]. We assume the mass of the exploding WD ranges from $0.85 { \, M_\sun }$ (lower boundary of each region) to $1.1 { \, M_\sun }$ (upper boundary), as motivated by WD detonation calculations [@sim10; @blon17a; @shen18a].
Both non-degenerate and degenerate SN Ia companions undergo processes that increase their radii as compared to isolated stars. As discussed in Section \[sec:lum\], tidal effects in the D$^6$ scenario lead to dissipation throughout the companion WD, including near its surface where it may cause radial expansion. In single-degenerate scenarios, the long pre-explosion phase of mass transfer pushes the donor out of thermal equilibrium and leads to “thermal bloating” [@knig11a]. Thus, the actual companion radii when the SN Ia explosion occurs may be slightly larger than the radii of isolated stars we find with `MESA`. However, since the companion’s orbital velocity scales as the square root of its radius, a 10% increase in the radius, e.g., only corresponds to a 5% decrease in the velocities shown in Figure \[fig:mvsv\].
It is clear that the runaway velocities of surviving companion WDs will be markedly different from those of any other possible companions. A fiducial $0.6 { \, M_\sun }$ companion to a $1.0 { \, M_\sun }$ primary WD will have a runaway velocity of $\unit[1800]{km \, s^{-1}}$, while the companion of a near-equal mass ratio $1.0+1.0 { \, M_\sun }$ binary will have a velocity of $\unit[2200]{km \, s^{-1}}$. At a distance of $\unit[1]{kpc}$, these velocities translate to proper motions of $0.4-0.5'' \, {\rm yr^{-1}}$ if the velocities are in the plane of the sky, easily detectable by *Gaia* and below the very high proper motions $\geq 0.6'' \, {\rm yr^{-1}}$ where completeness becomes an issue [@gaia16a; @gaia18a].
Estimated number of runaway WDs {#sec:numest}
-------------------------------
As discussed in the previous sections, *Gaia* will only be able to detect surviving companion WDs that are relatively nearby ($ \leq \unit[1]{kpc}$) or that were reheated by historical SNe and have not yet dimmed significantly. The SN remnant most likely to host a surviving WD observable with *Gaia* is SN 1006, due to its proximity ($\simeq \unit[2.2]{kpc}$), low extinction, and lack of crowding [@wink03a]. Although [@kerz18a] did not detect any young WDs on the WD cooling track within this remnant, it is possible that the accreted iron-group elements have shifted the spectral energy distribution redward so that the surviving WD no longer sits on the WD cooling track. Indeed, some sources $ \lesssim \unit[1]{mag}$ redder than the cooling track do exist within the remnant; we will specifically target these and other sources within SN 1006’s remnant in Section \[sec:gaia\], as well as performing searches within the remnants of SN 185 (RCW 86) and Tycho’s and Kepler’s SNe.
In order to estimate the number of runaway WDs unassociated with historical SNe within $\unit[1]{kpc}$ of the Sun, we require a model for the stellar density as a function of position within the Milky Way. We assume all the stars reside in the thin disk, with an exponential scale length of $\unit[2.6]{kpc}$, an exponential scale height of $\unit[300]{pc}$, and a normalization that yields a total stellar mass of $5{\times10^{10}} { \, M_\sun }$ [@blan16a]. We take the Sun’s Galactocentric distance to be $\unit[8.2]{kpc}$, its height from the midplane to be $\unit[25]{pc}$, and the Milky Way’s specific SN Ia rate to be $10^{-13} \, {\rm SNe} \, {\rm yr^{-1}} \, M_\odot^{-1}$ [@li11c].
With these values and additional assumptions that the Galactic potential is negligible and that the runaway WDs all have the same velocity of $1800 \, (2200) \, {\rm km \, s^{-1}}$, we find the expected number of nearby runaway WDs to have a Poissonian distribution centered at $28 \, (23)$, with a 95% confidence interval of $17-39 \, (14-33)$. Of these, $22 \, (18)$ will be more than $10^\circ$ off the plane of the Milky Way and thus will not be subject to significant extinction or crowding. These numbers are significantly higher than the $\lesssim 1$ expected hypervelocity stars ejected from the Galactic center currently passing within $\unit[1]{kpc}$ [@hill88a; @brow15a], so we do not expect contamination to be a concern. Even if the rate of hypervelocity ejections from the Galactic center is much higher than previously thought, luminosities, colors, spectra, and whether or not their orbits intersect the Galactic center should allow us to easily differentiate between runaway WDs and hypervelocity stars.
![Cumulative distribution functions of the time between the SN Ia event that ejected the runaway WD and the present day. Solid lines represent CDFs for the whole volume within $\unit[1]{kpc}$; dashed lines represent runaway WDs $>10^\circ$ off the Galactic plane.[]{data-label="fig:agecdf"}](agecdf){width="\columnwidth"}
The cumulative distribution function (CDF) of delay times, $t_{\rm delay}$, between the SN explosion and the present for these local runaway WDs is shown in Figure \[fig:agecdf\]. The average elapsed time for runaway WDs moving at $1800 \, (2200) \, {\rm km \, s^{-1}}$ is $2.3{\times10^{6}} \, (1.5{\times10^{6}}) \, {\rm yr}$. As discussed in Section \[sec:lum\], this value is longer than the thermal timescale at the depth where gravity waves dissipate tidal energy, but it is shorter than the thermal timescale from the center of the WD to the surface. Thus, as long as significant tidal heating is deposited deep within the companion, these runaways should be visible to at least $\unit[1]{kpc}$ for *Gaia*’s limiting magnitude of $21$.
![Cumulative distribution functions of the runaway WD velocity in the plane of the sky. Solid lines represent CDFs for the whole volume within $\unit[1]{kpc}$; dashed lines represent runaway WDs $>10^\circ$ off the Galactic plane.[]{data-label="fig:vcdf"}](vcdf){width="\columnwidth"}
A similar CDF of proper motion velocities, $v_{\rm proper}$, is shown in Figure \[fig:vcdf\]. Geometric effects imply that the velocity of the WDs in the plane of the sky will be smaller than their total velocities, but over $80\%$ of the nearby runaway WDs will still have $v_{\rm proper} >\unit[1000]{km \, s^{-1}}$. At a distance of $\unit[1]{kpc}$, $\unit[1000]{km \, s^{-1}}$ translates to a proper motion of $0.2'' \, {\rm yr^{-1}}$, which is much larger than the expected proper motion errors at *Gaia*’s magnitude limit. Furthermore, while $\sim 20\%$ of stars with proper motions $ \geq 0.6'' \, {\rm yr}$ may be missing from DR2 [@gaia16a], only the $\sim 1$ runaway WD expected to be closer than $ \unit[300]{pc}$ will be affected.
Candidate selection in *Gaia* DR2 {#sec:gaia}
=================================
*Gaia* was launched in December 2013 and has been recording the precise astrometry of billions of stars since July 2014 [@gaia16a]. When its nominal five-year science mission is complete, it will have measured parallaxes, $\varpi$, and proper motions, $\mu$, of most of the stars brighter than $G \simeq 21$. In this section, we outline our search strategy and describe the hypervelocity WD candidates found in *Gaia*’s second data release (DR2), which occurred on 25 April 2018 and provided astrometric parameters of $\simeq 1.3{\times10^{9}}$ stars.
Search strategy
---------------
The relatively small number ($\sim 30$) of expected local hypervelocity runaway WDs within $\unit[1]{kpc}$ is equivalent to the number of $6$-$\sigma$ outliers in a normally distributed set of samples as large as *Gaia*’s dataset. We must therefore exercise caution in our search strategy in order to avoid being overwhelmed by false positives.
We begin by restricting the $1.3{\times10^{9}}$ sources to those with proper motions above a conservative limit equivalent to $\unit[1000]{km \, s^{-1}}$ at $\unit[1]{kpc}$. For such proper motions $\mu \geq \unit[211]{mas \, yr^{-1}}$, the fractional errors are $<0.01$, so we neglect them for simplicity. A $\varpi > 3 \sigma_\varpi$ cut is also applied, although two sources with $\varpi < 3 \sigma_\varpi$ (O1 and O3 in Table \[tab:cand\]) were followed up prior to the implementation of this cut. The sources are then rank-ordered by $v_{3 \sigma} \equiv \mu/(\varpi + 3 \sigma_\varpi)$, where $\sigma_\varpi$ is the parallax error. By including the $3 \sigma_\varpi$ term, we are effectively calculating the proper motion velocity using the $3$-$\sigma_\varpi$ upper bound for the parallax.
Since the fractional parallax errors can be $\sim 1$, we apply a Bayesian framework to the top 500 entries to better quantify the posterior probability that the candidates are actually hypervelocity WDs. We use an exponentially decreasing space density distance prior [@astr16a],[^3] assume the likelihoods of the parallaxes are normally distributed, and calculate posteriors $$\begin{aligned}
P(r | \varpi, \sigma_\varpi) = \frac{ P( \varpi | r, \sigma_\varpi) P(r)}{ \int P( \varpi | r, \sigma_\varpi) P(r) } .
\label{eqn:post}\end{aligned}$$ Finally, we integrate the posterior with appropriate limits to find the probability, $P_{1000}$, that the proper motion velocity is larger than $\unit[1000]{km \, s^{-1}}$, and the probability, $P_{1000-3000}$, that it is bounded between 1000 and $\unit[3000]{km \, s^{-1}}$.
Note that the calculated probabilities, $P_{1000}$ and $P_{1000-3000}$, should not be taken literally, as they may be strongly influenced by the non-Gaussianity of the parallax distribution’s tails above $\sim 4$-$\sigma_\varpi$ [@luri18a]. We merely use these probabilities as a guide for ranking our sources for follow-up.
We slightly alter our strategy to additionally search for surviving WDs within the four Galactic remnants of suspected SNe Ia: the remnant of SN 185 (RCW 86), at a distance of $\unit[2.0-3.0]{kpc}$ [@held13a], the remnant of SN 1006 ($\unit[2.1-2.3]{kpc}$; @wink03a), Tycho’s SN remnant ($\unit[3-5]{kpc}$; @haya10a), and Kepler’s SN remnant ($\unit[3-6]{kpc}$; @sank05a [@chio12a]). We apply the same search strategy as above, but we relax our proper motion cuts to the equivalents of $\unit[1000]{km \, s^{-1}}$ at the upper limits for the remnants’ distances. We also restrict our search regions to circles around the geometric centers of the remnants with radii corresponding to proper motion velocities of $\unit[4000]{km \, s^{-1}}$ at the lower limits for the distances. This high proper motion velocity accounts for the combination of the surviving companion WD’s velocity and the initial velocity of the exploding WD and its remnant.
List of hypervelocity candidates
--------------------------------
The *Gaia* source IDs of seven of the candidates that we were able to follow up with ground-based instruments are shown in Table \[tab:cand\], along with their nicknames, which we will use hereafter, astrometric parameters, values for $v_{3 \sigma}$, and probabilities, $P_{1000}$ and $P_{1000-3000}$. The associated photometry from *Gaia*, PS1 [@cham16a], and Skymapper [@wolf18a] and the telescopes used to follow up the sources are shown in Table \[tab:phot\], along with comments about the individual stars. None of the searches within the Galactic SN Ia remnants revealed any obvious hypervelocity candidates.
[cc|cccccccc]{} 5805243926609660032 & D6-1 & 249.3819752 & -74.3434986 & $0.471 \pm 0.102 $ & $-80.3 \pm 0.1$ & $-195.9 \pm 0.2$ & 1293 & 1.00 & 0.79\
1798008584396457088 & D6-2 & 324.6124885 & 25.3737115 & $1.052 \pm 0.109$ & $98.4 \pm 0.2$ & $240.4 \pm 0.2$ & 894 & 0.98 & 0.98\
2156908318076164224 & D6-3 & 283.0078540 & 62.0361675 & $0.427 \pm 0.126$ & $9.0 \pm 0.2$ & $211.5 \pm 0.3$ & 1247 & 1.00 & 0.57\
2050179518946705152 & O1 & 291.3306894 & 36.4500600 & $0.237 \pm 0.317$ & $-137.6 \pm 0.6 $ & $-214.7 \pm 0.6 $ & 1018 & 1.00 & 0.17\
4396109004117478656 & O2 & 237.5476987 & -7.8881665 & $2.186 \pm 0.225$ & $-359.7 \pm 0.5 $ & $-228.4 \pm 0.3 $ & 706 & 0.37 & 0.37\
5884527618445501056 & O3 & 238.9823874 & -55.4937974 & $0.360 \pm 0.571$ & $90.0 \pm 0.8 $ & $-268.9 \pm 0.8 $ & 649 & 1.00 & 0.20\
1820931585123817728 & O4 & 296.3894478 & 17.2130727 & $0.574 \pm 0.076$ & $-82.4 \pm 0.1$ & $-149.5 \pm 0.1$ & 1010 & 1.00 & 1.00\
[c|ccccccccccccccc]{} D6-1 & $17.4$ & $17.6$ & $ 17.1$ & — & — & — & — & — & 18.4 & 17.6 & 17.4 & 17.4 & — & SALT & D$^6$ WD candidate\
D6-2 & 17.0 & 17.1 & 16.7 & 17.1 & 17.0 & 17.1 & 17.2 & 17.2 & — & — & — & — & — & NOT & D$^6$ WD candidate\
D6-3 & 18.3 &18.4 & 18.0 & 18.6 & 18.3 & 18.3 & 18.4 & 18.5 & — & — & — & — & — & NOT & D$^6$ WD candidate\
O1 & 17.0 & 17.6 & 16.0 & 18.0 & 17.0 & 16.5 & 16.2 & 16.1 & — & — & — & — & — & NOT & Ordinary star\
O2 & 18.0 & 18.8 & 17.1 & 19.1 & 19.1 & 18.1 & 17.5 & 17.2 & — & — & — & — & — & Bok & Ordinary star\
O3 & 17.8 & — & — & — & — & — & — & — & — & 16.7 & 16.0 & — & — & SALT & Ordinary star\
O4 & 16.0 & 16.5 & 15.2 & — & — & — & — & — & — & — & — & — & — & Shane & Ordinary star\
Analysis of the hypervelocity candidates {#sec:followup}
========================================
The seven candidates listed in Tables \[tab:cand\] and \[tab:phot\] were followed up at the Bok telescope (2.3-m), the Nordic Optical Telescope (NOT, 2.5-m), the Shane telescope at the Lick Observatory (3.0-m), and the Southern African Large Telescope (SALT, 9.8-m). Of these seven candidates, the spectra show that four (nicknamed O1-O4) are ordinary-looking hydrogen-rich stars whose true parallaxes are likely much larger than measured, implying they are nearby stars with more pedestrian proper motion velocities. We do not discuss these stars here further. We now turn to the three remaining candidates.
Spectroscopic follow-up
-----------------------
{width="90.00000%"}
The SALT data for D6-1 were obtained through Director’s Discretionary Time (proposal 2017-2-DDT-005, PI: Jha) and made use of the Robert Stobie Spectrograph (RSS), with a $1.5''$ wide longslit and the PG0900 grating, resulting in a spectral resolution $\lambda/\Delta\lambda \approx 900$ over the wavelength range $\unit[392-713]{nm}$. The data were reduced with a custom pipeline that incorporates routines from PyRAF and PySALT [@craw10a].
D6-2 and D6-3 were observed using the ALFOSC spectrograph on the Nordic Optical Telescope, located at Roque de Los Muchachos on La Palma. All observations were taken using Gr4, which covers the region from $3400 - 9000 \, {\rm \AA}$ at low resolution, and a $1''$ slit oriented at the parallactic angle. Weather conditions were excellent, with seeing below 1. Spectra were reduced using the dedicated pipeline ALFOSCGUI. Overscan and bias subtraction and flat fielding were performed before 1D spectra were optimally extracted. The dispersion solution for the spectra was obtained from arc lamps taken with the same configuration as the science spectra at the start of the night. In addition, a linear wavelength shift was applied to each spectrum based on sky emission lines. Telluric absorptions have been corrected for using observations of a spectrophotometric standard. The resolution of the spectra (as measured from narrow sky lines in the spectra) was $\sim15 \, $Å, while the S/N ratio for D6-2 and D6-3 was $\sim 25$.
Radial velocities (RVs) were measured via cross-correlation against the MILES library, comprising nearly 1000 stellar templates [@sanc06a; @falc11a]. We used the `rvsao` [@kurt98a] implementation of the [@tonr79a] algorithm. We find over one hundred good matches in the template library for each of D6-1, D6-2, and D6-3, based on the reported $r$ statistic (ranging from $r =$ 4 to 8). We average the resulting RVs and take the scatter across templates to be indicative of the scale of potential systematic uncertainties. We note that cross-correlation among D6-1, D6-2, and D6-3 themselves results in a much higher $r \simeq 15$, showing that these spectra are much more similar to each other than they are to entries in the template library. The resulting RV shifts are shown in Table \[tab:vel\].
Figure \[fig:specWD\] shows the spectra of D6-1, D6-2, and D6-3. An RV shift of $\unit[1200]{km \, s^{-1}}$ is applied to D6-1, while the spectra of D6-2 and D6-3 are unshifted. The spectra contain a multitude of absorption features. No transitions of hydrogen or helium are detected, ruling out canonical atmosphere compositions [@klei13a]. Comparison with unusual WDs identified by SDSS [@gaen10a; @gent15a; @kepl16a] clearly reveals strong features of carbon, oxygen, magnesium, and calcium in the three candidates. Based on their $G_\textrm{BP}-G_\textrm{RP}$ colors, which are slightly redder than those of GD492, SDSS1102+2054, and SDSS1140+1824 (see Section \[sec:cmd\]), we estimate effective temperatures of $\simeq 8000$K. Determining accurate atmospheric parameters for stars with such unusual compositions will require higher-quality spectroscopy.
As discussed in Section \[sec:abun\], the absence of hydrogen is expected for a surviving D$^6$ WD, as it would have been transferred stably to the primary WD and ejected from the system prior to the explosion. The non-detection of helium is unconstraining due to the relatively low surface temperatures. In fact, in their modeling of GD 492, a hypervelocity star spectroscopically similar to our three candidates, [@radd18a] find significant helium is required even though it is not directly observable.
![Zoom-in of the spectra for D6-1, D6-2, and D6-3. Observed wavelengths have been transformed to the heliocentric frame but are otherwise uncorrected for RV shifts. The Ca [ii]{} H&K lines for D6-1 are clearly shifted by $\unit[1200]{km \, s^{-1}}$, as shown by the red lines, while the same features are consistent with their rest wavelength values for D6-2 and D6-3.[]{data-label="fig:zoomspecWD"}](zoomspec){width="\columnwidth"}
Figure \[fig:zoomspecWD\] shows a zoomed-in portion of the D6-1, D6-2, and D6-3 observed wavelength spectra, highlighting the region near the Ca [ii]{} H&K lines. It is clear that D6-1 has a radial velocity shift of $\unit[1200]{km \, s^{-1}}$, as shown by the red lines. However, D6-2 and D6-3 have RVs consistent with being $< \unit[100]{km \, s^{-1}}$. These very low RVs cast doubt on the interpretation of these stars as hypervelocity stars: it seems unlikely that these stars would have a combination of very high proper motion velocities but very low RVs. As a check, all the *Gaia* proper motions were confirmed through examination of the fields of these candidates in epochs 1 and 2 of the Digitized Sky Survey, and also the Sloan Digital Sky Survey [@alam15a] for D6-2. In all cases the long baseline proper motions are consistent with *Gaia*’s values. If we assume that the measured parallaxes of D6-2 and D6-3 are instead systematically incorrect and that the transverse velocities of D6-2 and D6-3 are a more typical $\unit[100]{km \, s^{-1}}$, then the implied distances are $\sim \unit[100]{pc}$. This would suggest absolute magnitudes of $\emph{G}=12-13$. Thus, a possible interpretation is that these two objects are faint, nearby white dwarfs and that the parallax measurements have extremely large systematic uncertainties.
On the other hand, the *Gaia* noise values for D6-2 and D6-3 imply clean measurements. Moreover, D6-1’s extremely high RV makes its proper motion velocity credible.[^4] Finally, the fact that all three candidates, selected for their extreme proper motion velocities, are similar to each other and to GD 492, another hypervelocity star [@venn17a; @radd18a; @radd18b], suggests that the proper motion velocities of D6-1, D6-2, and D6-3 are indeed very high.
Posterior velocity distributions and orbital solutions
------------------------------------------------------
![Posterior probabilities of total Galactocentric velocities for D6-1, D6-2, and D6-3. An exponentially decreasing space density distance prior is used, and the parallax, proper motion, and RV errors are assumed to be normally distributed.[]{data-label="fig:vpost"}](vpost){width="\columnwidth"}
[c|ccc]{} D6-1 & $1200 \pm 40$ &$2200$ \[$1400-6800$\] & $2300$ \[$1600-6600$\]\
D6-2 & $20 \pm 60$ &$1200$ \[$700-1500$\] & $1300$ \[$1000-1900$\]\
D6-3 & $-20 \pm 80$ &$2400$ \[$1700-11100$\] & $2400$ \[$1400-9000$\]\
To emphasize the extreme nature of D6-1, D6-2, and D6-3, Figure \[fig:vpost\] shows the posterior distributions of their total Galactocentric velocities. These posteriors were derived by sampling the distance posteriors (equation \[eqn:post\]), as well as sampling the RVs and proper motions within their assumed Gaussian uncertainties, and applying a heliocentric to Galactocentric coordinate transformation with `astropy` [@astr18a].
Further velocity information is listed in Table \[tab:vel\]. D6-1, D6-2, and D6-3 have high probabilities of being three of the fastest known stars in the Milky Way, possibly only surpassed by pulsars kicked from core collapse SNe and the stars in close orbit around the Galactic center.
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
Samples of the orbital solutions for the candidate hypervelocity stars are shown in Figures \[fig:orbD6-1\], \[fig:orbD6-2\], and \[fig:orbD6-3\], calculated with `galpy` [@bovy15a] and assuming an MWPotential2014 gravitational potential and an exponentially decreasing space density distance prior as before. It is clear that all three candidates are unbound from the Milky Way and that almost none of the orbital solutions passes near the Galactic center. Taken as a group, it is highly unlikely that the hypervelocity nature of these stars is due to Galactic center ejection.
Color-magnitude diagram and possible interpretation {#sec:cmd}
---------------------------------------------------
![Color-magnitude diagram of the three hypervelocity candidates, GD 492, US 708, and three chemically peculiar WDs (colored symbols). Black circles and colored regions show reliably measured stars from *Gaia*.[]{data-label="fig:cmd"}](cmd){width="\columnwidth"}
In Figure \[fig:cmd\], we show a color-magnitude diagram using *Gaia*’s $G_{\rm BP}-G_{\rm RP}$ color and the absolute *Gaia* $G$-band magnitude, assuming the measured values of the parallax. Black points and colored regions show a random sub-sample of $5.7{\times10^{6}}$ stars from *Gaia* DR2 with accurate parallaxes ($\varpi/\sigma_\varpi>30$), clean astrometry (`astrometric_excess_noise` $<1$) and low reddening ($|b|>30^\circ$). Our three hypervelocity candidates, GD 492 [@venn17a], US 708 [@hirs05a; @just09a; @geie15a], and the three peculiar SDSS WDs from Figure \[fig:specWD\] are shown as colored symbols.
D6-1, D6-2, and D6-3 form a relatively tight grouping in color-magnitude space that includes GD 492. These four stars currently have radii between typical WDs and main sequence stars, similar to subdwarf stars; GD 492’s radius is estimated to be $0.2 \, R_\odot$ [@radd18b]. However, if they had previously been helium-rich subdwarf companions of exploding WDs in SN Ia systems, Figure \[fig:mvsv\] shows that their velocities would be well below $\unit[1000]{km \, s^{-1}}$. There have been some suggestions that GD 492 is instead the bound WD remnant of a SN Iax explosion that has received a large kick due to asymmetric mass ejection or due to the disruption of the binary system upon instantaneous mass loss [@venn17a; @radd18a; @radd18b]. However, the predicted kicks in these systems due to asymmetric mass ejection range from only ten to several hundred $\unit[]{km \, s^{-1}}$ [@fink14a; @long14a], far below the observed velocities of the D$^6$ stars. The orbital velocity of a Chandrasekhar-mass WD in a tight binary with a helium star can approach GD 492’s velocity, but it cannot explain the even higher velocities of D6-1, D6-2, and D6-3.
These three stars, and possibly also GD 492, may instead be the surviving companion WDs of D$^6$ SNe Ia. While they are clearly not typical WDs now, mechanisms discussed in Section \[sec:lum\] that were active during the phase of dynamical mass transfer prior to the SN Ia and the post-explosion evolution may have deposited enough energy to lift the degeneracy of the outer layers and cause the WDs to temporarily appear as subdwarf stars just after the explosion.
Our expectation was that most of the excess energy would be deposited near the surface of the WD, where it would be radiated away on relatively short timescales, so that the stars would quickly return to being dim, typical-looking WDs. This could indeed be true, making it difficult to observe the majority of the runaway WDs in the Solar neighborhood. However, a small fraction of these stars will have experienced SNe Ia much more recently than the average runaway WD, possibly rendering them still bright enough to observe.
A similar calculation to the one described in Section \[sec:numest\] yields $300-400$ runaway WDs within the $\sim \unit[65]{kpc^3}$ volume in which we observed D6-1, D6-2, and D6-3. If we include GD 492, we have observed 1% of the potential nearby runaway WDs, which corresponds to stars ejected more recently than $ \sim \unit[4{\times10^{4}}]{yr}$ (Fig. \[fig:agecdf\]). Thus, D6-1, D6-2, D6-3, and possibly GD 492 may just represent the small portion of runaway WDs that have been violently altered by SNe Ia so recently they have yet to evolve back into typical-looking WDs.
US 708, a hypervelocity helium-rich subdwarf, sits blueward of this group. It is possible that, as stars like D6-1, D6-2, D6-3, and GD 492 radiate the deposited energy and contract, the unseen helium that [@radd18a] require in their best-fit models of GD 492’s spectrum becomes directly observable as the photosphere becomes hotter. Thus, these stars could evolve to appear like US 708 on their way back to the WD cooling track. Future observations and detailed stellar evolution calculations and spectral modeling will help to test all of these intriguing possibilities.
D6-2’s association with SN remnant G70.0-21.5
---------------------------------------------
Motivated by the possibility that these four stars may have been ejected from the sites of SNe Ia $\sim \unit[4{\times10^{4}}]{yr}$ ago, we search for existing SN remnants along their past orbital solutions up to their positions $\sim\unit[10^5]{yr}$ ago. We use the online catalogs of D.A. Green[^5] and G. Ferrand[^6], as well as the Open Supernova Catalog[^7] [@guil17a]. Additionally, we searched data from the [*ROSAT*]{} All-Sky Survey for SN remnant sources that emit soft X-rays.
For stars D6-1, D6-3, and GD 492 we find no SN remnant candidate sources along the past velocity vectors. We note that this does not rule out their association with a SN, because SN remnants can dissipate into the interstellar medium on timescales ranging from a few thousand years up to a few hundred thousand years, depending on the remnant. Furthermore, the regions around these three stars lack high-quality H$\alpha$ imaging, so limits on the non-existence of remnants are not constraining.
{width="\textwidth"}
However, as shown in Figure \[fig:g70\], D6-2’s position $\unit[9{\times10^{4}}]{yr}$ in the past places it from the approximate geometric center of the faint, old SN remnant G70.0-21.5. This remnant, first identified by @fese15a, consists of a shell of H$\alpha$ filaments, along with several other spectral lines only associated with slowly moving, radiative shocks from an old SN remnant. Additionally, @fese15a report faint [*ROSAT*]{} X-ray emission from near the center, another indication of a SN remnant.
Of the known remnants, G70.0-21.5 lies furthest from the Galactic plane, suggesting it was produced by a SN Ia. The distance to G70.0-21.5 inferred from its shock velocities is $\unit[1-2]{kpc}$, consistent with D6-2’s parallax-measured distance of $ \unit[1.0 \pm 0.1 ]{kpc} $. [@fese15a] also conclude that the remnant is quite old, perhaps between several $10^4$ and $\unit[10^5]{yr}$. The probability of a chance alignment between D6-2’s past position and the projected center of an unassociated SN remnant that is at a consistent distance and has a consistent age is likely very small, especially given the lack of any other obvious SN remnants in the VTSS images of this region. However, this probability should be quantified in future work.
At a distance of $\unit[1]{kpc}$, G70.0-21.5’s height below the Galactic plane is $\unit[400]{pc}$, matching D6-2’s height $\unit[10^5]{yr}$ ago. D6-1 and D6-3 were farther from the plane at that time: $700$ and $\unit[1200]{pc}$, respectively. Such offsets are reasonable since SNe Ia can occur in old stellar populations and the thick disk’s scale height is $\unit[900]{pc}$ [@blan16a]. Furthermore, D$^6$ survivors that were recently ejected from closer to the Galactic plane would suffer higher extinction and could be unobservable. Thus, these three D$^6$ candidates may represent more than just 1% of the $300-400$ expected survivors within $\unit[2.5]{kpc}$, after accounting for those that remain obscured at low Galactic latitudes.
While D6-2 did not pass exactly through G70.0-21.5’s center as reported by [@fese15a], the approximate determination of the center of this aspherical remnant makes such a comparison difficult. Moreover, G70.0-21.5’s advanced age means that its appearance is heavily influenced by the structure of the inhomogeneous surrounding interstellar medium, further complicating determination of the site of the explosion. We thus do not regard the $\unit[0.9]{deg}$ offset between D6-2’s past position and [@fese15a]’s reported center as evidence against the association.
It is not possible to rule out other SN remnants present in this field that are even older, fainter, and more diffuse than G70.0-21.5. At some point, as SN remnants age, they simply dissolve away into the interstellar medium, below any possible threshold of detection. Nonetheless, the close association between D6-2 and a known SN remnant, G70.0-21.5, is quite tantalizing. Though the remnant is at a very advanced evolutionary stage, it may be possible to follow up the X-ray emission with deeper observations to determine the ejecta abundances, and thus, the SN type. We note that while none of the other hypervelocity runaway candidates are associated with known SN remnants, they also all lack high-quality H$\alpha$ imaging, which is how G70.0-21.5 was discovered. Future observations will ascertain if any of the remaining candidates’ remnants are observable but are so faint they have been missed by previous lower-quality searches.
Conclusions {#sec:conc}
===========
We have searched *Gaia*’s second data release for hypervelocity runaway WDs that survived dynamically driven double-degenerate double-detonation (D$^6$) SNe Ia. We followed up seven candidates with ground-based telescopes. Of these, three are consistent with being hypervelocity runaway stars that were previously the WD companions to primary WDs that exploded as SNe Ia. One candidate is also closely associated with a faint, old SN remnant at a distance consistent with the candidate’s measured parallax.
One lingering puzzle is the very low RV measured for two of the hypervelocity stars. However, given these two stars’ close photometric and spectroscopic match to the third star, which does have a large RV, and the association of one of the low RV stars with a SN remnant, the peculiar RVs may just be a statistical fluctuation due to small numbers. Future detection and characterization of more D$^6$ survivors will hopefully ease this tension.
While the candidates are much brighter and have larger radii than expected, plausible mechanisms exist that may have changed the appearance of these hypervelocity runaway stars for a short time following the SN Ia explosion. If validated by future high-resolution spectroscopy and detailed stellar evolution calculations and atmospheric modeling, these stars would confirm the success of the D$^6$ scenario in producing SNe Ia. Future study of these candidate D$^6$ survivors would also shed important insight on the physics of double WD binaries and the aftermath of SN Ia explosions, including tidal heating, stellar ablation, and ejecta deposition.
The increased luminosity of the three D$^6$ candidates raises the possibility that such survivors could be observable within relatively close and young SN remnants such as Tycho, Kepler, and SN 1006. However, previous searches within these remnants, including one designed to look for hot surviving WD companions, and our search using *Gaia* DR2 have failed to find any convincing candidates. It is possible that D$^6$ survivors do exist within these remnants but are presently too faint or too red or blue to be easily disentangled from the other stars. Future analysis and modeling will help to discover or constrain the existence of surviving WDs within these remnants.
Several candidates, undiscussed here, remain to be followed up. Furthermore, given the large parallax uncertainties near the magnitude limit of DR2, it is likely that more such stars exist but are hiding in the data. Dedicated work and future *Gaia* data releases may be able to tease out more candidates in the coming years.
We thank Josh Bloom, Brian Metzger, Alison Miller, Peter Nugent, and Eliot Quataert for discussions, and the referee for helpful comments. K.J.S. is supported by NASA through the Astrophysics Theory Program (NNX17AG28G). D.B. thanks the UK Science and Technology Facilities Council for supporting his PhD. We thank Encarni Romero Colmenero and Petri Vaisanen for the Director’s Discretionary Time award on SALT. S.W.J. is supported in part by NSF award AST-1615455. The UCSC group is supported in part by NASA grant NNG17PX03C, NSF grant AST-1518052, the Gordon & Betty Moore Foundation, the Heising-Simons Foundation, and by fellowships from the Alfred P. Sloan Foundation and the David and Lucile Packard Foundation to R.J.F. M.F. is supported by a Royal Society - Science Foundation Ireland University Research Fellowship. K.M. is supported by STFC through an Ernest Rutherford fellowship. W.E.K. is supported by an ESO Fellowship and the Excellence Cluster Universe, Technische Universit[ä]{}t M[ü]{}nchen, Boltzmannstrasse 2, D-85748 Garching, Germany. Support for J.S. is provided by NASA through Hubble Fellowship grant \#HST-HF2-51382.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. We acknowledge the “Observational Signatures of Type Ia Supernova Progenitors III” workshop at the Lorentz Center, where some of the preparatory work for this study was performed.
This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} (<https://www.cosmos.esa.int/gaia>), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement.
The NOT data was obtained through NUTS, which is supported in part by the Instrument Center for Danish Astrophysics (IDA). Support also comes from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 320964 (WDTracer). Some of the observations reported in this paper were obtained with the Southern African Large Telescope (SALT). PyRAF is a product of the Space Telescope Science Institute, which is operated by AURA for NASA.
The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen’s University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.
The national facility capability for SkyMapper has been funded through ARC LIEF grant LE130100104 from the Australian Research Council, awarded to the University of Sydney, the Australian National University, Swinburne University of Technology, the University of Queensland, the University of Western Australia, the University of Melbourne, Curtin University of Technology, Monash University and the Australian Astronomical Observatory. SkyMapper is owned and operated by The Australian National University’s Research School of Astronomy and Astrophysics. The survey data were processed and provided by the SkyMapper Team at ANU. The SkyMapper node of the All-Sky Virtual Observatory (ASVO) is hosted at the National Computational Infrastructure (NCI). Development and support the SkyMapper node of the ASVO has been funded in part by Astronomy Australia Limited (AAL) and the Australian Government through the Commonwealth’s Education Investment Fund (EIF) and National Collaborative Research Infrastructure Strategy (NCRIS), particularly the National eResearch Collaboration Tools and Resources (NeCTAR) and the Australian National Data Service Projects (ANDS).
natexlab\#1[\#1]{}\[1\][[\#1](#1)]{} \[1\][doi: [](http://doi.org/#1)]{} \[1\][[](http://ascl.net/#1)]{} \[1\][[](https://arxiv.org/abs/#1)]{}
, S., [Albareti]{}, F. D., [Allende Prieto]{}, C., [et al.]{} 2015, , 219, 12,
, T. L., & [Bailer-Jones]{}, C. A. L. 2016, , 832, 137,
, P., [Wesemael]{}, F., [Dufour]{}, P., [et al.]{} 2011, , 737, 28,
, L., [Shen]{}, K. J., [Weinberg]{}, N. N., & [Nelemans]{}, G. 2007, , 662, L95,
, J., & [Gerhard]{}, O. 2016, , 54, 529,
, S., [Dessart]{}, L., [Hillier]{}, D. J., & [Khokhlov]{}, A. M. 2017, , 470, 157,
, J. S., [Kasen]{}, D., [Shen]{}, K. J., [et al.]{} 2012, , 744, L17,
, J. 2015, , 216, 29,
, W. R. 2015, , 53, 15,
, W. R., [Kilic]{}, M., [Kenyon]{}, S. J., & [Gianninas]{}, A. 2016, , 824, 46,
, M., [Sim]{}, S. A., [Pakmor]{}, R., [et al.]{} 2016, , 455, 1060,
, J., [Quataert]{}, E., [Arras]{}, P., & [Weinberg]{}, N. N. 2013, , 433, 332,
, K. C., [Magnier]{}, E. A., [Metcalfe]{}, N., [et al.]{} 2016, arXiv:1612.05560.
, P., [Fontaine]{}, G., & [Wesemael]{}, F. 1995, , 99, 189,
, P., [Vennes]{}, S., [Pradhan]{}, A. K., [et al.]{} 1995, , 454, 429,
, A., [Schure]{}, K. M., & [Vink]{}, J. 2012, , 537, A139,
, S. A., & [McKee]{}, C. 1969, , 157, 623,
, S. M., [Still]{}, M., [Schellart]{}, P., [et al.]{} 2010, in , Vol. 7737, Observatory Operations: Strategies, Processes, and Systems III, 773725
, M., [Guillochon]{}, J., [Br[ü]{}ggen]{}, M., [Ramirez-Ruiz]{}, E., & [Rosswog]{}, S. 2015, , 454, 4411,
, M., [Rosswog]{}, S., [Br[ü]{}ggen]{}, M., & [Podsiadlowski]{}, P. 2014, , 438, 14,
, M., [Rosswog]{}, S., [Guillochon]{}, J., & [Ramirez-Ruiz]{}, E. 2011, , 737, 89,
, B., [Simonetti]{}, J. H., & [Topasna]{}, G. A. 1998, , 15, 147,
, J., [Fontaine]{}, G., [Pelletier]{}, C., & [Wesemael]{}, F. 1992, , 82, 505,
, P. P. 1983, , 268, 368,
, J., [S[á]{}nchez-Bl[á]{}zquez]{}, P., [Vazdekis]{}, A., [et al.]{} 2011, , 532, A95,
, R. A., [Neustadt]{}, J. M. M., [Black]{}, C. S., & [Koeppel]{}, A. H. D. 2015, , 812, 37,
, M., [Hillebrandt]{}, W., & [R[ö]{}pke]{}, F. K. 2007, , 476, 1133,
, M., [R[ö]{}pke]{}, F. K., [Hillebrandt]{}, W., [et al.]{} 2010, , 514, A53,
, M., [Kromer]{}, M., [Seitenzahl]{}, I. R., [et al.]{} 2014, , 438, 1762,
, R. J., [Challis]{}, P. J., [Chornock]{}, R., [et al.]{} 2013, , 767, 57,
, J., & [Lai]{}, D. 2011, , 412, 1331,
, [Brown]{}, A. G. A., [Vallenari]{}, A., [et al.]{} 2018, , accepted (arXiv:1804.09365).
, [Prusti]{}, T., [de Bruijne]{}, J. H. J., [et al.]{} 2016, , 595, A1,
, B. T., [Koester]{}, D., [Girven]{}, J., [Marsh]{}, T. R., & [Steeghs]{}, D. 2010, Science, 327, 188,
, S., [F[ü]{}rst]{}, F., [Ziegerer]{}, E., [et al.]{} 2015, Science, 347, 1126,
, N. P., [G[ä]{}nsicke]{}, B. T., & [Greiss]{}, S. 2015, , 448, 2260,
, M., & [Bogd[á]{}n]{}, [Á]{}. 2010, , 463, 924,
, J., [Garc[í]{}a-Berro]{}, E., & [Isern]{}, J. 2004, , 413, 257,
, J., [Dan]{}, M., [Ramirez-Ruiz]{}, E., & [Rosswog]{}, S. 2010, , 709, L64,
, J., [Parrent]{}, J., [Kelley]{}, L. Z., & [Margutti]{}, R. 2017, , 835, 64,
, B. M. S. 2003, , 582, 915,
, A., [Yamaguchi]{}, H., [Tamagawa]{}, T., [et al.]{} 2010, , 725, 894,
, E. A., [Vink]{}, J., [Bamba]{}, A., [et al.]{} 2013, , 435, 910,
, J. G. 1988, , 331, 687,
, H. A., [Heber]{}, U., [O’Toole]{}, S. J., & [Bresolin]{}, F. 2005, , 444, L61,
, J. B., & [Bergeron]{}, P. 2006, , 132, 1221,
, Jr., I., & [Tutukov]{}, A. V. 1984, , 54, 335,
, Jr., I., [Tutukov]{}, A. V., & [Fedorova]{}, A. V. 1998, , 503, 344,
, IV, G. C., [Perets]{}, H. B., [Fisher]{}, R. T., & [van Rossum]{}, D. R. 2012, , 761, L23,
, S., [Wolf]{}, C., [Podsiadlowski]{}, P., & [Han]{}, Z. 2009, , 493, 1081,
, D. L., [Bildsten]{}, L., & [Steinfadt]{}, J. D. R. 2012, , 758, 64,
, D. 2010, , 708, 1025,
, S. O., [Koester]{}, D., & [Ourique]{}, G. 2016, Science, 352, 67,
, W. E., [Childress]{}, M., [Scharw[ä]{}chter]{}, J., [Do]{}, T., & [Schmidt]{}, B. P. 2014, , 782, 27,
, W. E., [Strampelli]{}, G., [Shen]{}, K. J., [et al.]{} 2018, , 479, 192,
, S. J., [Kepler]{}, S. O., [Koester]{}, D., [et al.]{} 2013, , 204, 5,
, C., [Baraffe]{}, I., & [Patterson]{}, J. 2011, , 194, 28,
, P. M., & [Saumon]{}, D. 2006, , 651, L137,
, M., [Fink]{}, M., [Stanishev]{}, V., [et al.]{} 2013, , 429, 2287,
, M. J., & [Mink]{}, D. J. 1998, , 110, 934,
, D. C. 2007, , 670, 1275,
, W., [Chornock]{}, R., [Leaman]{}, J., [et al.]{} 2011, , 412, 1473,
, W., [Bloom]{}, J. S., [Podsiadlowski]{}, P., [et al.]{} 2011, , 480, 348,
, Z. W., [Pakmor]{}, R., [R[ö]{}pke]{}, F. K., [et al.]{} 2012, , 548, A2,
, E. 1990, , 354, L53,
, M., [Jordan]{}, IV, G. C., [van Rossum]{}, D. R., [et al.]{} 2014, , 789, 103,
, X., [Brown]{}, A. G. A., [Sarro]{}, L. M., [et al.]{} 2018, , accepted (arXiv:1804.09376).
, K., [Taubenberger]{}, S., [Sullivan]{}, M., & [Mazzali]{}, P. A. 2016, , 457, 3254,
, D., [Mannucci]{}, F., & [Nelemans]{}, G. 2014, , 52, 107,
, E., [Burrows]{}, A., & [Fryxell]{}, B. 2000, , 128, 615,
, T. R., [Nelemans]{}, G., & [Steeghs]{}, D. 2004, , 350, 113,
, K. 1982, , 253, 798,
—. 1982, , 257, 780,
, R. P., [Mushotzky]{}, R., [Shaya]{}, E. J., [et al.]{} 2015, , 521, 332,
, R., [Kromer]{}, M., [Taubenberger]{}, S., [et al.]{} 2012, , 747, L10,
, R., [Kromer]{}, M., [Taubenberger]{}, S., & [Springel]{}, V. 2013, , 770, L8,
, R., [R[ö]{}pke]{}, F. K., [Weiss]{}, A., & [Hillebrandt]{}, W. 2008, , 489, 943,
, K.-C., [Ricker]{}, P. M., & [Taam]{}, R. E. 2013, , 773, 49,
—. 2014, , 792, 71,
, Jr., T. 1962, PhD thesis, Howard University
, O., [Soker]{}, N., [Garc[í]{}a-Berro]{}, E., & [Aznar-Sigu[á]{}n]{}, G. 2015, , 449, 942,
, C., [Pelletier]{}, C., [Fontaine]{}, G., & [Michaud]{}, G. 1986, , 61, 197,
, B., [Bildsten]{}, L., [Dotter]{}, A., [et al.]{} 2011, , 192, 3,
, B., [Cantiello]{}, M., [Arras]{}, P., [et al.]{} 2013, , 208, 4,
, B., [Marchant]{}, P., [Schwab]{}, J., [et al.]{} 2015, , 220, 15,
, B., [Schwab]{}, J., [Bauer]{}, E. B., [et al.]{} 2018, , 234, 34,
, S., [Aldering]{}, G., [Goldhaber]{}, G., [et al.]{} 1999, , 517, 565,
, R., [Hollands]{}, M. A., [Gaensicke]{}, B. T., [et al.]{} 2018, , submitted (arXiv:1804.09677).
, R., [Hollands]{}, M. A., [Koester]{}, D., [et al.]{} 2018, , accepted (arXiv:1803.07564).
, C., [Kasen]{}, D., [Moll]{}, R., [Schwab]{}, J., & [Woosley]{}, S. 2014, , 788, 75,
, C., [Scannapieco]{}, E., [Fryer]{}, C., [Rockefeller]{}, G., & [Timmes]{}, F. X. 2012, , 746, 62,
, A. G., [Filippenko]{}, A. V., [Challis]{}, P., [et al.]{} 1998, , 116, 1009,
, P., [Comeron]{}, F., [M[é]{}ndez]{}, J., [et al.]{} 2004, , 431, 1069,
, P., [Peletier]{}, R. F., [Jim[é]{}nez-Vicente]{}, J., [et al.]{} 2006, , 371, 703,
, R., [Blair]{}, W. P., [Delaney]{}, T., [et al.]{} 2005, Advances in Space Research, 35, 1027,
, B. E., & [Pagnotta]{}, A. 2012, , 481, 164,
, B. J., [Kochanek]{}, C. S., & [Stanek]{}, K. Z. 2013, , 765, 150,
, K. J. 2015, , 805, L6,
, K. J., & [Bildsten]{}, L. 2009, , 699, 1365,
—. 2014, , 785, 61,
, K. J., [Guillochon]{}, J., & [Foley]{}, R. J. 2013, , 770, L35,
, K. J., [Kasen]{}, D., [Miles]{}, B. J., & [Townsley]{}, D. M. 2018, , 854, 52,
, K. J., & [Moore]{}, K. 2014, , 797, 46,
, K. J., & [Schwab]{}, J. 2017, , 834, 180,
, S. A., [R[ö]{}pke]{}, F. K., [Hillebrandt]{}, W., [et al.]{} 2010, , 714, L52,
, R. E. 1980, , 242, 749,
, S., [Kromer]{}, M., [Pakmor]{}, R., [et al.]{} 2013, , 775, L43,
, [Price-Whelan]{}, A. M., [Sip[ő]{}cz]{}, B. M., [et al.]{} 2018, arXiv:1801.02634.
, F. X., [Woosley]{}, S. E., & [Weaver]{}, T. A. 1995, , 98, 617,
, J., & [Davis]{}, M. 1979, , 84, 1511,
, P.-E., [Bergeron]{}, P., & [Gianninas]{}, A. 2011, , 730, 128,
, S., [Nemeth]{}, P., [Kawka]{}, A., [et al.]{} 2017, Science, 357, 680,
, L., & [Wheeler]{}, J. C. 2008, , 46, 433,
, R. F. 1984, , 277, 355,
, J., & [Iben]{}, I. J. 1973, , 186, 1007,
, P. F., [Gupta]{}, G., & [Long]{}, K. S. 2003, , 585, 324,
, C., [Onken]{}, C. A., [Luvaul]{}, L. C., [et al.]{} 2018, , 35, e010,
, T. E., [Ghavamian]{}, P., [Badenes]{}, C., & [Gilfanov]{}, M. 2017, Nature Astronomy, 1, 800,
, S. E., [Taam]{}, R. E., & [Weaver]{}, T. A. 1986, , 301, 601,
, S.-C., & [Langer]{}, N. 2003, , 412, L53,
[^1]: <http://www.astro.umontreal.ca/~bergeron/CoolingModels>
[^2]: In principle, the runaway velocity is somewhat smaller than the pre-SN orbital velocity due to the portion of the exploding WD that remains at velocities $\lesssim v_{\rm runaway}$. However, in practice, this mass is always $<0.05 { \, M_\sun }$ [@shen18a] and will have a negligible effect on the runaway velocity.
[^3]: Other priors, such as a uniform distance prior, were also tried, but they do not significantly affect the ordering of the candidate list.
[^4]: D6-1’s RV was confirmed on a subsequent night, ruling out the possibility that the high RV is due to orbital motion in a tight binary system.
[^5]: <https://www.mrao.cam.ac.uk/surveys/snrs>
[^6]: <http://www.physics.umanitoba.ca/snr/SNRcat>
[^7]: <https://sne.space>
|
---
abstract: 'The three bio-inspired strategies that have been used for balance recovery of biped robots are the ankle, hip and stepping Strategies. However, there are several cases for a biped robot where stepping is not possible, e. g. when the available contact surfaces are limited. In this situation, the balance recovery by modulating the angular momentum of the upper body (Hip-strategy) or the Zero Moment Point (ZMP) (Ankle strategy) is essential. In this paper, a single Model Predictive Control (MPC) scheme is employed for controlling the Capture Point (CP) to a desired position by modulating both the ZMP and the Centroidal Moment Pivot (CMP). The goal of the proposed controller is to control the CP, employing the CMP when the CP is out of the support polygon, and/or the ZMP when the CP is inside the support polygon. The proposed algorithm is implemented on an abstract model of the SURENA III humanoid robot. Obtained results show the effectiveness of the proposed approach in the presence of severe pushes, even when the support polygon is shrunken to a point or a line.'
author:
- 'Milad Shafiee-Ashtiani,$^{1}$ Aghil Yousefi-Koma,$^{1}$ Masoud Shariat-Panahi,$^{2}$ and Majid Khadiv$^{3}$ [^1][^2][^3]'
bibliography:
- 'Master.bib'
title: '**Push Recovery of a Humanoid Robot Based on Model Predictive Control and Capture Point** '
---
INTRODUCTION
============
The main destination of humanoid robots research is realizing a robot that is able to work in real environments. Because of unstable nature of the biped robots, the ability of recovering from unexpected external disturbances is essential. In recent years, several attempts have been made by researchers to generate robust locomotion of biped robots ([@kajita2003biped; @wieber2006trajectory; @pratt2006capture; @herdt2010online; @stephens2010push; @aftab2012ankle; @koolen2012capturability]). A common criterion for ensuring dynamic balance during walking is to maintain the Zero Moment Point (ZMP) or the Center of Pressure (CoP) within the support polygon of the contact points. The main approaches that have been used for balancing and walking of humanoid robots in the presence of disturbances are based on the Model Predictive Control (MPC) or controlling the Capture Point (CP) [@kajita2003biped; @wieber2006trajectory; @pratt2006capture; @herdt2010online; @stephens2010push; @aftab2012ankle; @koolen2012capturability; @englsberger2015three; @krause2012stabilization; @yun2011momentum].
Kajita et al.[@kajita2003biped] introduced preview control of ZMP and paved a way for robust walking pattern generation. This method was expressed more generally as an MPC problem by Wieber et al. [@wieber2006trajectory]. To increase the robustness of the gaits, the MPC formulation in [@wieber2006trajectory] has been modified to adapt the step locations [@herdt2010online; @stephens2010push]. However, the upper-body angular momentum has not been employed in these works. As a result, Aftab et. al [@aftab2012ankle] proposed a single MPC that uses all the ankle, hip, and stepping strategies for balance recovery of humanoid robots. In all of these works, the CoM has been considered as the state of the system. However, relating the problem in this way constrains both divergent and convergent components of motion [@englsberger2015three].
Pratt et al. [@pratt2006capture; @koolen2012capturability] introduced the CP by splitting the Center of Mass (CoM) dynamics into stable and unstable components. The state variable related to the unstable part of the CoM dynamics has been named the Capture Point (CP). The CP specifies when and where a humanoid must step to in order to maintain balance, however it requires a controller for stabilizing unstable nature of dynamic of the CP. To this end, Englsberger et al. [@englsberger2015three; @krause2012stabilization] developed a controller for CP tracking without using the effect of upper-body angular momentum (CMP modulating) and by guiding the CP only by CoP modulation. The effect of upper-body angular momentum plays a key role for balance recovery especially in the situation that stepping is not possible or contact surface is small [@yun2011momentum; @kiemel2012balance; @wiedebach2016walking].
In this paper, in order to utilize the usefulness of the two mentioned approaches, the CP concept is used in an MPC. To do so, an effective MPC scheme is developed for push recovery by manipulating the CoP when the CP is within the support polygon, and employing the CMP modulation when the CP is out of the support polygon. The main goal of this controller is to maintain the CP, CMP and CoP on the center of support polygon. The proposed algorithm is capable of dealing with severe pushes while the contact surface is a line or a point. The remainder of this paper is organized as follows. The CoM dynamics, and the CP formulations are reviewed in Sec [slowromancap2@]{}. The proposed push recovery controller is presented in Sec [slowromancap3@]{}. In [slowromancap4@]{}, the obtained simulations results are presented and discussed. Finally, Section [slowromancap5@]{} concludes the findings.
Center of Mass Dynamics
=======================
Linear Inverted Pendulum
------------------------
Using the full nonlinear dynamics of a humanoid robot for gait planning makes the corresponding optimization problem non-convex [@khadiv2015optimal]. However, the dynamics of a biped robot can be approximated by the Linear Inverted Pendulum Model (LIPM) [@kajita20013d]. This model is a good dynamic approximation of a biped robot, particularly for the standing posture. The LIPM uses the following assumptions [@kajita20013d]:
- The rate of change of angular momentum is zero,
- The CoM height remains constant
Based on the mentioned assumptions and Fig.\[fig1\], the equation of Motion of the LIPM can be expressed as follows:
$$\ddot x_{c}= \omega_{n}^{2}( x_{c}-p_{x})
\label{eq:2}$$
where $m$ is the robot mass, the CoM position is given by $P_c=[x_c,y_c,z_c]^T$, $ P_{zmp}=[p_x, p_y, 0]^T$ is the position of the ZMP and $\omega_{n}=\sqrt{(g/z_c )}$ is the natural frequency of the LIPM. The Ground Reaction Force (GRF) intersects with the CoM because the base joint of the pendulum is torque-free and the rate of change of angular momentum is zero. As shown in Fig.\[fig1\], $F_z$ is the vertical component of the GRF. It compensates the gravitational force $F_g$ acting on the CoM. The inertial force $ F_r=m\ddot x _c$ completes the equilibrium of forces in $P_c$. The equation of motion in frontal plane and sagittal plane are independent. By adding the external force (Disturbance) to the dynamics of the LIPM, the equations of motion can be modified and written as: $$\begin{aligned}
\ddot x_{c}= \omega_n^{2}( x_{c}-p_{x}) +\frac{F_{ext,x}}{m}\hspace{1.1cm}\\
\ddot y_{c}= \omega_n^{2}( y_{c}-p_{y}) +\frac{F_{ext,y}}{m}\hspace{1.1cm}\\
\end{aligned}
\label{eq:3}$$
The effect of angular momentum of the upper-body, especially the torso and arms, can play an important role in push recovery. These joints can be used to apply a torque about the CoM. The CMP, is equal to the CoP in the case of zero torque about the CoM such as the LIPM. For a non-zero moment about the COM, however, the CMP can be out of the support polygon, while the COP still remains inside the support polygon. This effect can be embedded by considering the upper body as a flywheel that can be actuated directly as shown by Pratt [@pratt2006capture]. In other words, the CMP is the point where a line parallel to the ground reaction force and passing through the COM intersects the ground. Therefore, by adding this effect to the LIPM dynamics, the equations of motion can be written as:
$$\begin{aligned}
\ddot x_{c}= \omega_n^{2}( x_{c}-p_{x}) -\frac{\dot H_{y}}{mz}+\frac{F_{ext,x}}{m}\hspace{1.1cm}\\
\ddot y_{c}= \omega_n^{2}( y_{c}-p_{y}) +\frac{\dot H_{x}}{mz}+\frac{F_{ext,y}}{m}\hspace{1.1cm}\\
\end{aligned}
\label{eq4}$$
where $\dot{H}$ is the rate of upper-body angular momentum that can be handled by the torque of arm and trunk joints. The relation between the ZMP and the CMP can be written as: [@khadiv2015optimal]:
$$\begin{aligned}
CMP_{x} =p_{x}+\frac{\dot H_{y}}{F_z}\hspace{1.1cm}\\
CMP_{y} =p_{y}-\frac{\dot H_{x}}{F_z}\hspace{1.1cm}\\
\end{aligned}
\label{eq5}$$
As a result, combining (\[eq4\]) and (\[eq5\]), we obtain: $$\begin{aligned}
\ddot x_{c}= \omega_n^{2}( x_{c}-CMP_{x}) +\frac{F_{ext,x}}{m}\hspace{1.1cm}\\
\ddot y_{c}= \omega_n^{2}( y_{c}-CMP_{y}) +\frac{F_{ext,y}}{m}\hspace{1.1cm}\\
\end{aligned}
\label{eq6}$$
When the moment about the CoM is non-zero, such as when a disturbance is applied, the CMP and ZMP will diverge and CMP can leave the support polygon for controlling the CP, when the CP is outside of the support polygon.
Capture Point Dynamics
----------------------
The unstable part of the LIPM dynamics has been called the CP and can be defined as follows [@pratt2006capture; @koolen2012capturability; @englsberger2015three]: $$\begin{aligned}
{\xi_x}={x_c}+\frac{{{ \accentset{\mbox{\bfseries .}}{x}}_c}}{{\omega_n}} \hspace{1.1cm}\\
{\xi_y}={y_c}+\frac{{{ \accentset{\mbox{\bfseries .}}{y}}_c}}{{\omega_n}} \hspace{1.1cm}\\
\end{aligned}
\label{eq7}$$ From (\[eq7\]), the CoM dynamics is given by:
$$\begin{aligned}
{\dot x_c}={{\omega_n}} ({\xi}-{ x_c} )\hspace{1.1cm}\\
{\dot y_c}={{\omega_n}} ({\xi}-{ y_c} )\hspace{1.1cm}\\
\end{aligned}
\label{eq8}$$
By differentiating (\[eq8\]) and substituting (\[eq6\]) the CP dynamics is given by:
$$\begin{aligned}
{ \accentset{\mbox{\bfseries .}}{\xi}}_x=\omega_n({\xi_x}-{CMP}_{x})+\frac{F_{ext,x}}{m\omega_n}\hspace{1.1cm}\\
{ \accentset{\mbox{\bfseries .}}{\xi}}_y=\omega_n({\xi_y}-{CMP}_{y})+\frac{F_{ext,y}}{m\omega_n}\hspace{1.1cm}\\
\end{aligned}
\label{eq9}$$
\[7em\] \[8em\] \[8em\]
\[7em\] \[8em\] \[8em\]
As it is obvious in (\[eq9\]), the CMP can push the CP. In order to recover the balance of a humanoid robot, the CP should be controlled. When the CP is located within the support polygon, it can be controlled by the CoP [@englsberger2015three], and when it is located out of the support polygon it can be controlled by the CMP or stepping.
Using the concept of CP we can determine when and where to take a step to recover from a push [@pratt2006capture]. If the CP is located within the support polygon, the robot is able to recover from the push without having to take a step. In order to stop in one step, the support polygon must have an intersection with the capture region as it shown on Fig.\[fig2\].(b), [@pratt2006capture]. The robot will fail to recover from a severe push in one step, if the capture region does not intersect with the kinematic workspace of the swing foot. In the next sections we will discuss how to use the CP in Push recovery controller based on the MPC scheme.
Human-Inspired Balancing Strategies
-----------------------------------
The response of a human to progressively increasing disturbances can be categorized into three basic strategy: (1) ankle strategy, (2) hip strategy (3) and stepping strategy. Humans tend to use the ankle strategy in case of small pushes to bring back the CP to its desired position as depicted in Fig.\[fig3\](a). However, the contact between the foot and floor is a unilateral constraint and if the ankle torque becomes too large, the CoP locates on the edge of the support polygon and the foot starts to rotate. Angular momentum of the upper body can be generated in the direction of the disturbance by applying a torque on the hip joint or arm joint as shown in Fig.\[fig3\](b). This strategy also called CMP Balancing. With increasing the disturbance the useful strategy will be stepping Fig.\[fig3\](c). However, there are several situations might occur where stepping is not possible as shown in Fig.\[fig4\]. In this situation the balance recovery by Hip-Ankle strategy is necessary [@kiemel2012balance].
Moreover in the situations that contact surface is small such as right side of Fig.\[fig4\], generating upper body angular momentum for balance recovery is unavoidable. In this paper, the Hip-Ankle strategy is used in a single MPC scheme that will be presented in the following section.
PUSH RECOVERY CONTROLLER
========================
discrete state-space form of LIPM+flywheel dynamics
---------------------------------------------------
We discretize the LIPM dynamics in the sagittal plane, while the procedure for the other direction is similar: $$\begin{aligned}
x_{c,t+1}= (1-\omega_n T) x_{c,t}+\omega_n T \xi_{x,t}\hspace{3.25cm}\\
\xi_{x,t+1}= (1+\omega_n T)\xi_{x,t}-\omega_n T (p_{x,t}+\frac{\dot H_{y,t}}{mg}) +\frac{F_{ext,x}}{m\omega_n}\hspace{.79cm}\\
p_{x,t+1}= p_{x,t} +\dot p_{x,t}T\hspace{4.95cm}\\
\dot H_{y,t+1}= \dot H_{y,t} +\ddot H_{y,t}T\hspace{4.9cm}\\
\end{aligned}
\label{eq5}$$
This system can be re-written in discrete state-space form:
$$\mathbf{X}_{t+1}={A}_t \mathbf{X}_{t}+{B}\mathbf{U}_{t}
\label{eq11}$$
where $ \mathbf {X_t}=[x_{c,t},{\xi}_{x,t} ,p_{x,t}, \dot H_{y,t}, F_{ext,x}] $ is the vector of state variables and $ U_t=[\dot p_{x,t},\ddot H_{y,t}]$ specifies the control inputs. The last state variable $ F_{EXT}$ is activated in the step time that a push is exerted by defining $\mu$. Therefore, when a push is exerted we have $\mu=1$ and in the other step times $\mu$ is equal to zero: $$A_{t}=\begin{bmatrix}
(1-\omega_n T) & \omega_n T& 0 & 0 & 0\\[\verticaldistance]
0 & (1+\omega_n T) & -\omega_n T &\frac{-\omega_n T}{mg} & \frac{1}{m\omega_n} \\[\verticaldistance]
0 &0 &1 &0 & 0 \\[\verticaldistance]
0 &0 &0 &1 & 0 \\[\verticaldistance]
0 &0 &0 &0 & \mu
\end{bmatrix}$$
$$B=
\begin{bmatrix}
0 &0 \\
0 & 0 \\
T & 0 \\
0 & T \\
0 & 0
\end{bmatrix}$$
Given a sequence of control inputs $\mathbf {\hat U}$, the linear model in (\[eq11\]) can be converted into a sequence of states $\mathbf {\hat X}$, for the whole prediction horizon:
$$\begin{aligned}
\mathbf{\hat X}=\hat {\mathbf A}\mathbf{X}_{t}+\hat{\mathbf B}\mathbf{\hat U}\hspace{2cm}\\
\mathbf {\hat X}=[\mathbf {X^T_{t+1}}, \mathbf {X^T_{t+2}}, ..... , \mathbf { X^T_{t+N}}]\hspace{1cm}\\
\mathbf {\hat U}=[\mathbf {U^T_{t}}, \mathbf {U^T_{t+1}}, ..... , \mathbf {U^T_{t+N-1}}]\hspace{0.9cm}\\
\end{aligned}
\label{eq12}$$
where $\mathbf {\hat A}$ and $\mathbf {\hat B}$ are defined recursively from (\[eq11\]). The control inputs are the rate of change of ZMP position and the rate of upper-body angular momentum. As a result, the core of the Proposed MPC is based on combined hip and ankle strategies.
Model Predictive Control (MPC)
------------------------------
We present an MPC Controller that uses the concept of hip and ankle strategies in its core by modulating the ZMP and CMP as control inputs, considering future constraints on the CP . Using the LIPM+Flywheel, the trajectory optimization is simplified to a Quadratic Programming (QP) problem. The LIPM+Flywheel has a linear dynamics and the corresponding optimization problem is linear and can be solved in real-time. The push recovery control objective is simplified to optimize control inputs subject to terminal constraints on the CP, CMP and change of angular momentum. Constraints will be discussed in the next subsection. The objective function used in this paper is as follows: $$\begin{aligned}
J= \sum_{\mathclap{k=1}}^{N} \alpha_{1}\| \xi_{k+1}-\xi^{ref}_{k+1} \|_{x}^2 +\alpha_{2}\| { \accentset{\mbox{\bfseries .\hspace{-0.25ex}.}}{H}}_{k} \|_{x}^2+ \alpha_{3}\| { \accentset{\mbox{\bfseries .}}{cop}}_{k} \|_{x}^2 + \alpha_{4}\| { \accentset{\mbox{\bfseries .}}{H}}_{k} \|_{y}^2\\ +\alpha_{5}\| \xi_{k+1}^{ref}- \xi_{k+1} \|_{y}^2 +\alpha_{6}\| { \accentset{\mbox{\bfseries .\hspace{-0.25ex}.}}{H}}_{k} \|_{y}^2+\hspace{0.2cm}
\alpha_{7}\| { \accentset{\mbox{\bfseries .}}{cop}}_{k} \|_{y}^2 +
\alpha_{8}\| { \accentset{\mbox{\bfseries .}}{H}}_{k} \|_{x}^2
\hspace{0.01cm}
\\[10pt]
\end{aligned}
\label{eq13}$$ where $ \dot P_x ,\dot P_y, \ddot H_y$ and $ \ddot H_x$ are vectors of control inputs over the next $ N$ time steps. The first term minimizes distance between the desired and actual CP. The second and third terms are considered for modulating the ZMP and CMP in order to control the CP. The forth term is used for minimizing the rate of change of angular momentum. The $\alpha_i$ are the weights each term that can be regulated in different situations. This proposed cost function consider both rotational and linear dynamics of biped robots. The proposed objective function can be converted to the following standard quadratic form: $$\begin{aligned}
J=\frac{1}2 \hspace{.1cm}\mathbf {\hat U}^TH\hspace{0.1cm}\mathbf {\hat U} +\hspace{.1cm} \mathbf {\hat U}^T f\\
st. \hspace{1.7cm}\\
C \hspace{.1cm} \mathbf {\hat U}+D=0 \hspace{1.1cm}\\
E\hspace{.1cm} \mathbf {\hat U}+F \leq0 \hspace{1.1cm}
\end{aligned}
\label{eq14}$$ where $A, B, C$ and $D$ are coefficient matrices, with $H$ and $f$ being the Hessian matrix and gradient vector of the objective function respectively.
Constraints
-----------
The real power of MPC is the consideration of future constraints. Our goal in push recovery controller is to maintain the ZMP inside the support polygon and controlling the CP by modulating the ZMP and CMP. Furthermore, we need to coincide the CP with the ZMP in the support polygon center at the end of motion, while the rate of change of upper-body angular momentum is zero. This means we have the following constraints: $$\begin{aligned}
{\xi}_{x,N}={\xi_{ref,x}}\hspace{1.1cm}\\
{x}_{c,N}={\xi_{ref,x}}\hspace{1.1cm}\\
{p}_{x,N}={\xi_{ref,x}}\hspace{1.1cm}\\
{{ \accentset{\mbox{\bfseries .}}{H}}}_{y,N}=0 \hspace{1.1cm}\\
{p}_{x ,i} \in Support Polygon
\end{aligned}
\label{eq15}$$ where ${\xi_{ref,x}}$ is the reference CP that is located on the center of support polygon. The first four constraints are equality constraints for the last step time of motion. The last equation is an inequality constraint that enforces the ZMP to remain inside the support polygon. Similar equations can be derived for the lateral direction. Using the objective function of (\[eq13\]) and adding the constraints of (\[eq15\]), control inputs can be optimized during push recovery by the QP.
SIMULATION AND DISCUSSION
=========================
To verify the performance of the push recovery controller, we performed simulations using MATLAB. The proposed controller is implemented on an abstract model of the SURENA [slowromancap3@]{} humanoid robot. Parameters that have been used in the simulation is shown in Table.\[Tab1\] The time of balance recovery is considered 1.5 s. The allowable rate of upper-body angular momentum that can be applied is 190 N.m during 1.5 s according to [@aftab2012ankle].
Variable Symbol Value
--------------- -------------- ---------------------------------------------
Height - 190 cm
CoM Height $z_c$ 75 cm
Mass m 98 kg
Foot Length - 25 cm
Foot Width - 15 cm
Trunk Inertia - $ 8 \, \, \text{Kg.m}^2$
Arms Inertia - $ 3 \, \, \text{Kg.m}^2$
Step-time T $ 0.05 \,\, \text{s}$
MPC gain $\alpha_{1}$ $1 \, \, {\text{m}^{-1}}$
MPC gain $\alpha_{2}$ $3\, \, {s}.{(\text{N.m})}^{-1}$
MPC gain $\alpha_{3}$ $10^{-6} \, \, {\text{s}}.{\text{m}^{-1}}$
MPC gain $\alpha_{4}$ $10^{-3} \, \, {(\text{N.m})}^{-1}$
MPC gain $\alpha_{5}$ $1\, \, {\text{m}^{-1}}$
MPC gain $\alpha_{6}$ $1\, \, {\text{s}}.{(\text{N.m})}^{-1}$
MPC gain $\alpha_{7}$ $10^{-6} \, \, {\text{s}}.{\text{m}^{-1}}$
MPC gain $\alpha_{8}$ $10^{-3} \, \, {(\text{N.m})}^{-1}$
: Variable used in the simulation(Based on SURENA [slowromancap3@]{})[]{data-label="table_example"}
\[Tab1\]
Simulation Results
------------------
In the first scenario, a push with the magnitude of 360 N in sagittal direction, and another one with the magnitude of 140 N in frontal plane are exerted on the CoM of the robot. As we expected, the large push throws the CP out of the support polygon, and the ZMP cannot navigate it. Therefore, the angular momentum is generated by the MPC to move the CMP outside the support polygon for controlling the CP. The maximum flywheel torque for push recovery is about 50 N.m that is realizable on our considered robot. The trajectory of CP, CoP, CMP and CoM during balance recovery is shown in Fig.\[fig5\].
In the second scenario, the robot stands on one leg, while the contact surface is shrunken to a line or a point. Two examples for this situation are standing on lumber and standing on rock. In this case, the CMP modulation recovers the robot from the disturbance, because the support polygon is too small and the ankle strategy is not helpful anymore. In this situation, the CP leaves the support polygon and the CoP remains on the bounds of the support polygon, while the CMP pushes the CP to the desired position. As shown in Fig.\[fig6\], in the first case, pushes with magnitude of 350 N in sagittal and 100 N in lateral direction are exerted on the CoM, while the surface contact is a line. In the second case, pushes with magnitude of 140 N in sagittal and 100 N in frontal plane are exerted on the CoM, while the surface contact is a point[^4].
As shown in Fig.\[fig5\], \[fig6\], in all simulations the angle of the hip pitch joint is smaller than 1.5 rad that is allowable [@aftab2012ankle].
Based on the simulation results, the regulation of angular momentum is so beneficial during push recovery, especially in the standing on small contact surfaces or in the situations where stepping is not possible. Based on presented results the proposed method has the following features:
- The presented MPC scheme is capable of generating human-like response to external disturbances; for example, when the exerted force is small, it uses the ankle strategy for balance recovery. Furthermore, in the presence of large disturbances, it generates angular momentum and uses hip-ankle strategy simultaneously.
- The proposed push recovery controller can compensate the severe pushes, when the robot stands on small contacts such as a line or a point and also is capable of saving the robot from falling in the situations that stepping is not possible.
[2]{}
[2]{}
CONCLUSION AND FUTURE WORK
==========================
In this paper, a push recovery controller based on the CP concept and through an MPC framework is developed. The core of the proposed MPC is based on a combined hip and ankle strategies by modulating the CMP and ZMP to control the CP. The results showed that this controller is capable of rejecting severe pushes, even in the case where the support polygon is limited to a line or a point, and stepping is not allowed. The effectiveness of the proposed MPC scheme was demonstrated by simulating an abstract model of the SURENA III humanoid robot.
Despite all above advantages, this controller is implemented only in simulation. Implementing on the experimental setup has more practical challenges[@wiedebach2016walking]. For example, accurate state estimation to obtain the CP position, the saturation of actuators especially in the case where the support polygon is a point or a line, foot slipping and bringing the upper-body back into an upright position are some of main challenges of experimental implementation that will be discussed in the future works.
[^1]: $^{1}$Center of Advanced Systems and Technologies (CAST) School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran. ( [shafiee.a@ut.ac.ir]{}) ( [aykoma@ut.ac.ir]{})
[^2]: $^{2}$School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran. ( [mshariatp@ut.ac.ir]{})
[^3]: $^{3}$Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran ( [mkhadiv@mail.kntu.ac.ir]{})
[^4]: A summary of the simulation scenarios is available on https://youtu.be/bDPafm-6CLk
|
---
abstract: 'The Double Pulsar, PSR J$0737$–$3039$A/B, is a unique system in which both neutron stars have been detected as radio pulsars. As shown in Ferdman et al., there is no evidence for pulse profile evolution of the A pulsar, and the geometry of the pulsar was fit well with a double-pole circular radio beam model. Assuming a more realistic polar cap model with a vacuum retarded dipole magnetosphere configuration including special relativistic effects, we create synthesized pulse profiles for A given the best-fit geometry from the simple circular beam model. By fitting synthesized pulse profiles to those observed from pulsar A, we constrain the geometry of the radio beam, namely the half-opening angle and the emission altitude, to be $\sim$$30\degr$ and $\sim$10 neutron star radii, respectively. Combining the observational constraints of PSR J0737–3039A/B, we are able to construct the full three-dimensional orbital geometry of the Double Pulsar. The relative angle between the spin axes of the two pulsars ($\Delta_{\rm S}$) is estimated to be $\sim$$(138\degr\pm5\degr)$ at the current epoch and will likely remain constant until tidal interactions become important in $\sim$85 Myr, at merger.'
author:
- 'B. B. P. Perera, C. Kim, M. A. McLaughlin, R. D. Ferdman, M. Kramer, I. H. Stairs, P. C. C. Freire, and A. Possenti'
bibliography:
- 'psrrefs.bib'
- 'modrefs.bib'
- 'journals.bib'
- '0737Ack.bib'
title: 'Realistic Modeling of the Pulse Profile of PSR J0737–3039A'
---
Introduction
============
PSR J$0737$–$3039$ is the only neutron star - neutron star (NS-NS) binary in which both NSs have been detected as radio pulsars [@bdp+03; @lbk+04]. This unique system provides an opportunity to determine the beam geometry of the individual pulsars, allowing us to construct the full three dimensional orbital and spin geometry of the binary. This information is vital in order to understand binary formation/evolution involving supernova natal kicks [@plp+04; @fkl+11] and to study gravitational wave (GW) signals and outcomes from NS-NS mergers [e.g., @plp+13; @ahl+08; @it00]. This system also provides the most precise test to date of general relativity in the strong-field regime [@ksm+06].
The first-born, recycled pulsar of the system, PSR J0737–3039A (hereafter pulsar A), has a spin period ($P_{\rm s}$) of 22.7 ms. The second-born and slower PSR J0737–3039B (hereafter pulsar B) rotates every 2.8 s. The two pulsars orbit each other in a tight ($P_{\rm orb}=2.4$ hrs) and moderately eccentric ($e \sim$$0.088$) orbit [@ksm+06]. Relativistic spin precession is expected from such binary systems, and the geodetic spin precession rates of the two pulsars are theoretically predicted to be $4\fdg8$ yr$^{-1}$ and $5\fdg1$ yr$^{-1}$ for pulsars A and B, respectively [e.g., @bo75b]. Relativistic precession has been measured for two binary systems based on observation: for pulsar B from a detailed study of pulsar A eclipses [@bkk+08] and for the NS-NS binary PSR B1534$+$12 from the observed secular and periodic variations in pulse profiles due to spin precession and aberration [@sta04]. A long-term pulse profile analysis of pulsar B reveals that the relativistic spin precession results in a dramatic evolution in the pulse profile, finally culminating in the radio emission disappearance with respect to our line-of-sight [@pmk+10]. By fitting an elliptical emission beam model, the geometry of pulsar B, including the emission altitudes, was constrained [@plg+12].
Although the spin precession rates of the A and B pulsars are comparable, there is no evidence for secular variation in the pulse profile of A [@mkp+05; @fsk+08; @fsk+13]. The most plausible explanation for this stable pulse profile is that pulsar A’s spin misalignment, $\delta_{\rm A}$, from the orbital normal is very small. Assuming the two pulse components in A’s profile are formed from the two radio beams corresponding to each magnetic pole, @fsk+08 determined its geometry with a double-pole circular beam model. Using six years of data, @fsk+13 found that pulse profiles show no evidence for shape evolution. By fitting their model to measured pulse profile widths at different intensity levels (30%, 35%, 40%, 45%, and 50% with respect to the peak) individually, they constrained the magnetic misalignment and the spin colatitude of the pulsar to be $\alpha_{\rm A} = 90\degr\pm 16\degr$ and $\delta_{\rm A} < 2\fdg3$ at a 68% confidence level, respectively. This geometry supports the orthogonal configuration of pulsar A.
Recent [*Fermi*]{} observations of the Double Pulsar revealed pulsed gamma-ray emission from pulsar A [@gkj+13]. The gamma-ray emission is explained by outer magnetosphere models: the “outer gap” (OG) model [@chr86a; @rom96a] in which the emission is generated within the gap region between the null-charge surface and the light cylinder, or the “two-pole caustic” (TPC) model [@dr03] in which the emission is generated within the gap region extending from the NS surface to the light cylinder. @gkj+13 found that the peaks of the gamma-ray and the radio profiles are not aligned in pulsar A’s spin phase. This implies that the radio and gamma-ray emission are generated in different locations in pulsar A’s magnetosphere. Therefore, we strongly believe that pulsar A’s radio emission originates within the inner magnetosphere. They also constrained pulsar A’s geometric angles $\alpha_{\rm A}$ and $\zeta_{\rm E}$ by fitting OG and TPC models separately to observed gamma-ray light curves combining a single-altitude hollow cone radio beam model to observed radio pulse profiles. In contrast, we use only radio observation in this study and model the radio pulse profile by using a radio beam model in which the emission is active across its entire beam. Both outer magnetosphere models derived geometry given in @gkj+13 also favor an orthogonal configuration for pulsar A.
The main purpose of this work is to develop a realistic model of the radio beam geometry of pulsar A. We first repeat the analysis of @fsk+13, but analyze pulse widths including lower intensity levels such as 5% in order to incorporate any subtle changes in pulsar A’s pulse profiles. Then, for a more realistic model, we consider a polar cap (PC) beam model that involves the dipole magnetic field structure to fully describe the shape of the pulsar magnetosphere. We assume that pulsar radio emission is generated at a lower altitude within the PC region, where the open field lines are located. In general, pulsar magnetosphere models are constructed at the following two limits: (a) a vacuum limit [@deu55] and (b) a force-free magnetohydrodynamics (MHD) limit with a plasma-filled magnetosphere [@spi06]. However, a true magnetosphere operates between these two limits [see @lst+12; @kkh+12]. The MHD solutions are considered to be more realistic, but are computationally expensive to implement [see @hsd+08]. Further, @hdm+11 found the rotating dipole magnetosphere in vacuum provides better fits to observed high-energy pulse profiles than the force-free magnetosphere. Therefore, we incorporate a semi-analytic, widely used, vacuum retarded dipole magnetic field structure [e.g., @yad97; @dh04; @bs10] with a PC radio emission model to construct the beam geometry of pulsar A. This allows us to estimate the half-opening angle and the emission height for pulsar A.
In Section \[data\], we present the data and the profile width variation of pulsar A. In Section \[circular\], we constrain the pulsar geometry using the same simple circular beam model given in @fsk+13 with more pulse width measurements. In Section \[advanced\], we describe our analysis with the PC model based on a vacuum retarded dipole configuration. We then compare pulsar A’s beam geometry obtained from the circular and PC beam models. Combining our results with those from @plg+12, we determine the orbital geometry of the Double Pulsar, including the relative angle of the spin axes of two pulsars in Section \[full\_geo\]. Finally, we discuss results in Section \[dis\].
Pulse Profiles of Pulsar A {#data}
==========================
In this section, we describe our analysis of pulsar A’s pulse profiles. We use the same data set that @fsk+13 used in their analysis (from 2005 June (MJD 53524) to 2011 June (MJD 55721) at an observing frequency of 820 MHz). All pulse profiles are constructed with 2048 bins across the spin phase, resulting in a time resolution of $\sim$$10$ $\mu$s. As shown in Figure 1 therein, pulsar A’s pulse profile has not significantly changed within that time span.
The brightest component (P1) of A’s pulse profile is narrower than the secondary component (P2) (see Figure \[profile\]). We note that some studies defined P2 as the brightest component [e.g., @fsk+13; @gkj+13]. At each epoch, we calculate pulse widths of P1 and P2, separately, at different intensity levels (5%, 25%, 45%, and 65%) relative to each component’s peak height. The uncertainties of these widths are calculated from the off-pulse root-mean-square deviation. We selected these particular intensity levels in order to reflect the width evolution. For instance, the 5% width includes any subtle changes of the profile at lower intensity, and the other intensity levels were chosen with 20% increments between 5% and 65% to avoid noise properties such as the plateau region around 10% and the feature around 70% of component P2 (see Figure \[profile\]). Figure \[width\] shows the pulse widths obtained from P1 (left panel) and P2 (middle panel) at the 5% intensity level. The root-mean-square deviations of the measured widths for P1 and P2 are $0\fdg5$ and $0\fdg1$, respectively, which are smaller than the typical uncertainties of these measurements. In addition, the least-square fits show that the 5% widths of P1 and P2 decrease with a rate of $0\fdg1(1)$ yr$^{-1}$ and $0\fdg01(3)$ yr$^{-1}$, respectively. Together, these indicate that there is no significant variation in pulse widths over time. This is consistent with the results obtained in previous studies [@mkp+05; @fsk+13]. Note also that P1 and P2 are separated by almost $180(4)\degr$ at the 5% level (see Figure \[width\]), supporting the assumption that A is an orthogonal rotator and the two pulse components are due to seeing a radio beam from each pole of the NS.
Beam Geometry of Pulsar A with a Simple Circular Beam {#circular}
=====================================================
In this section, we repeat the analysis of @fsk+13 using the same simple circular radio beam model and measured pulse widths from P1 and P2 at different intensity levels (5%, 25%, 45%, and 65%) to constrain $\alpha_{\rm A}$ and $\delta_{\rm A}$ independently of the line-of-sight. The only difference between our analysis and that of @fsk+13 is that they calculated pairs of ($\alpha_{\rm A}$, $\delta_{\rm A}$) for each intensity level, while we obtain a single pair of ($\alpha_{\rm A}$ and $\delta_{\rm A}$) considering all four intensity levels. This results in values with smaller error bars. Once we obtain $\alpha_{\rm A}$ and $\delta_{\rm A}$, we fix both angles in order to estimate the half-opening angle of the beam $\rho_{\rm A}$ at the 5% intensity level. We assume the 5% intensity level is roughly the boundary of the beam.
Detecting a stable pulse profile over time implies that our line-of-sight always observes nearly the same cross section of pulsar A’s radio beam. Therefore, as long as one is only concerned with global geometric angles such as $\alpha_{\rm A}$ and $\delta_{\rm A}$, the circular radio beam is a simple, yet valid, choice. Thus, we do not investigate other complex shapes such as an elliptical beam which was used for pulsar B where our line-of-sight cuts through significantly different parts of the beam in a short-timescale of years [@plg+12].
Following the analysis described in @fsk+13, we calculate the model-estimated pulse profile width $w_{\rm j}(t)$ at a given epoch $t$ for a given $\alpha_{\rm A}$, $\delta_{\rm A}$, $\rho_{\rm A,j}$, and $T_0$ and then fit the observed pulse width at the same epoch $t$ to the model-estimated width. Here, [*j*]{} represents different intensity levels and $T_0$ is a reference epoch when the spin axis is in the plane of the line-of-sight and the orbital normal axis. We follow this method for measured pulse profiles at all epochs. By using a likelihood analysis combined with a grid search as described in @pmk+10 and @plg+12, we obtain $\alpha_{\rm A}$ and $\delta_{\rm A}$. During the fitting procedure, we use a single $\delta_{\rm A}$ and $T_0$ for both pulse components P1 and P2, assuming A is an orthogonal rotator.
We assume that the two radio beams of the pulsar are circular and have independent beam sizes: this model is denoted as CBM. Therefore, we vary the north ($\rho_{\rm A,N}$) and south ($\rho_{\rm A,S}$) beams for each parameter combination ($\alpha_{\rm A}$, $\delta_{\rm A}$) until we get the maximum likelihood. After searching the entire parameter space, we obtain the best-fit values as follows: $\alpha_{\rm A} = 88\fdg1^{+3\fdg0}_{-0\fdg6}$, $\delta_{\rm A} \leq 2\fdg8$ with a best-fit of $0\fdg9$, and $T_0 = 61800$ (see Table 1). The beam sizes $\rho_{\rm A,N}$ and $\rho_{\rm A,S}$ at the 5% intensity are $27\degr \pm 1\degr$ and $32\degr \pm 1\degr$ for P1 and P2, respectively. Note that, our best-fit $\alpha_{\rm A}$ and $\delta_{\rm A}$ are consistent with the results reported in @fsk+13 within their 68% uncertainties.
---------------- --------------------- --------- -------- -------------------- ------- ---------------- ---------------- --------------
($\degr$ yr$^{-1}$) () () () (MJD) () (R$_{\rm NS}$)
Pulsar A: from
CBM 4.8 88(3) 1(2) \[87.8, 89.6\]$^a$ 61800 27(1), 32(1) 10(2), 11(2) this work
TPC$^b$ - 80(9) 0$^c$ 86(14) – 32(1), 32(1) 11(2), 11(2) this work, 1
OG$^b$ - 88(17) 0$^c$ 74(14) – 33(2), 38(1) 12(2), 15(2) this work, 1
RVM$^b$ - 99(8) 0$^c$ 96(13) – 30(1), 33(1) 10(2), 12(3) this work, 1
Pulsar A - 90(8) $<$6.1 – – $<90$ – 2
- 90(16) $<$2.3 – – 12$^d$, 19$^d$ – 3
Pulsar B 5.1 61(8) 138(5) \[50, 133\] 57399 14.3 \[15, 38\]$^e$ 4
B1913$+$16 1.2 153(8) 22(8) \[130, 154\] 98296 9 – 5
B1534$+$12 0.5 103(1) 25(4) \[52, 102\] – 4.9 – 6,7
J1141$-$6545 1.4 160(16) 93(16) \[20, 166\] 53000 – – 8
J1906$+$0746 2.2 81(66) 89(85) – – – – 9
---------------- --------------------- --------- -------- -------------------- ------- ---------------- ---------------- --------------
Beam Geometry of Pulsar A with a Retarded Vacuum Dipole PC Model {#advanced}
================================================================
The circular radio beam model discussed in the previous section provides the information about the geometry ($\alpha_{\rm A}$, $\delta_{\rm A}$) of the pulsar with beam size $\rho_{\rm A}$. In order to estimate the radio emission altitudes in detail, we need to account for the magnetic field line structure. In this section, we investigate the radio emission beam of A by applying a dipole magnetosphere configuration and assume that the 5% intensity levels of the profile, or wings, are generated from the radio emission near the boundary of the last open and closed field lines.
Following what was derived in @deu55 and used in @yad97, @dh04, and @bs10, we model pulsar A’s magnetosphere by a vacuum dipole field at retarded time $t_{\rm r} = t - r/c$. For a pulsar rotating around the z-axis with an angular velocity of $\Omega$ and magnetic inclination $\alpha_{\rm A}$, the time dependent magnetic moment is given as $\vec{\mu}(t) = \mu(\sin\alpha_{\rm A}\cos\Omega t \hat{x} + \sin\alpha_{\rm A}\sin\Omega t \hat{y} + \cos\alpha_{\rm A} \hat{z})$ in Cartesian coordinates. Then the magnetic field of the retarded dipole can be written as
$$\vec{B}_{\rm ret} = -\left[ \frac{\vec{\mu}(t)}{r^3} + \frac{\dot{\vec{\mu}}(t)}{cr^2} + \frac{\ddot{\vec{\mu}}(t)}{c^2r} \right] + \vec{r}\cdot \left[ 3\frac{\vec{\mu}(t)}{r^3} + 3\frac{\dot{\vec{\mu}}(t)}{cr^2} + \frac{\ddot{\vec{\mu}}(t)}{c^2r} \right]\vec{r}~,$$
where $r=|\vec{r}|$ is the radial distance and $c$ is the speed of light [see @bs10]. As shown in @dh04, we can write the Cartesian components of $\vec{B_{\rm ret}}$ as follows
$$\begin{aligned}
\label{bfield}
B_{\rm ret,x} = \frac{\mu}{r^5} ( 3xz\cos\alpha_{\rm A} + \sin\alpha_{\rm A} ( [(3x^2 - r^2) + 3xyr_{\rm n} + (r^2-x^2)r_{\rm n}^2] \cos(\Omega t - r_{\rm n}) \nonumber \\
+ [3xy-(3x^2-r^2)r_{\rm n} - xyr_{\rm n}^2] \sin(\Omega t - r_{\rm n}) ) ) \nonumber \\
B_{\rm ret,y} = \frac{\mu}{r^5} ( 3yz\cos\alpha_{\rm A} + \sin\alpha_{\rm A}([3xy + (3y^2-r^2)r_{\rm n} -xyr_{\rm n}^2 ] \cos(\Omega t - r_{\rm n}) \nonumber \\
+ [(3y^2-r^2) - 3xyr_{\rm n} + (r^2-y^2)r_{\rm n}^2 ] \sin(\Omega t - r_{\rm n}))) \\
B_{\rm ret,z} = \frac{\mu}{r^5} ( (3z^2-r^2)\cos\alpha_{\rm A} + \sin\alpha_{\rm A}[(3xz+3yzr_{\rm n} - xzr_{\rm n}^2) \cos(\Omega t - r_{\rm n}) \nonumber \\
+ (3yz - 3xzr_{\rm n} - yzr_{\rm n}^2) \sin(\Omega t - r_{\rm n})])~. \nonumber\end{aligned}$$
Here $r_{\rm n} \equiv r/R_{\rm LC}$, where $R_{\rm LC}$ is the light cylinder radius. Using pulsar A’s spin period ($P_{\rm s}=22.7$ ms), we fix pulsar A’s light cylinder radius to be $R_{\rm LC} = cP_{\rm s}/2\pi = 1100$ km in the calculation. Then the ratio $r_{\rm n}$ is small in the vicinity of the NS surface and the retarded field configuration is almost the same as the static field configuration in the ‘near’ zone (i.e., $r << R_{\rm LC}$). As explained in @dh04, the location of any corotating point within the magnetosphere does not depend on time. In other words, the retarded magnetic dipole field configuration is fixed in space and time in corotating frame. However, the field line structure rotates around the rotation axis as a whole with the pulsar spin.
As shown in @fsk+13, and also in Section \[circular\], pulsar A’s spin colatitude $\delta_{\rm A}$ is almost zero. Thus, to simplify the model, we assume that pulsar A’s spin axis is aligned with the orbital angular momentum ($\delta_{\rm A}=0\degr$). In order to determine the magnetic field lines, we use Equation (\[bfield\]) with the fourth-order Runge-Kutta integration method. Two angles ($\theta_{\rm m}$, $\phi_{\rm m}$) are used to define the footpoint of the magnetic field line on the NS surface, where we assume a NS radius of R$_{\rm NS}=$10 km. Here, $\theta_{\rm m}$ is the colatitude angle from the magnetic axis and $\phi_{\rm m}$ is the azimuth of the field line footpoint. Then we determine the field line which starts from this initial point. First, we determine the last closed field lines, by varying $\theta_{\rm m}$ for a given $\phi_{\rm m}$ (i.e., bisection in $\theta_{\rm m}$ at fixed $\phi_{\rm m}$) until the field line becomes tangent to the light cylinder. We then define the PC region by calculating the footpoint of these last closed field lines on the NS surface. The shape of the PC region predicted by a retarded magnetic field is typically not symmetric around the magnetic axis and is dependent on $\alpha_{\rm A}$ [see Figure 2 in @dh04]. As pointed out in @dhr04, we use bisection in $\phi_{\rm m}$ at fixed $\theta_{\rm m}$ around the ‘notch’ region to correct the PC rim. A field line with a smaller $\theta_{\rm m}$ is open with respect to the light cylinder and referred as an open field line. We then model these open field lines which are located within the PC region. In order to do that, we define a set of field line footpoint rings within the PC region with a fixed colatitude ratio of $\theta_{\rm m} / \theta_{\rm rim}$, where $\theta_{\rm rim}$ is the colatitude of the PC rim of a given $\phi_{\rm m}$. We calculate footpoints with a fixed $4\degr$ increment in the azimuthal direction and obtain 90 field lines in each ring. Then, we define several sets of footpoint rings according to the colatitude ratio from $0.1$ to $1$ with an increment of $0.05$. We note that increasing the number of footpoints in a ring and the number of rings within the PC would smooth the modeled pulse profile and the shape variation becomes negligible. By testing several values, we found that the above given increments on these two parameters were sufficient for this analysis. Starting from these footpoints, we draw the open field lines using numerical integration.
Although the exact radio emission mechanism is not well understood, we believe that the charged particles stream along magnetic field lines and emit radiation tangential to the local magnetic field line at the emitting point. Therefore, we first determine the photon emission direction at any given emission point on a field line in the PC region. In order to do that, we perform numerical integration with a fixed step size. By using a smaller step size, we can safely assume that the unit vector of the field line segment at a given point is indeed tangent to the field line. This guarantees that the unit vector of emitted photons ($\hat{\eta}'$) are also tangential to the field line at this point. The unit vector of photons are represented by two angles: the colatitude of the tangent from the rotation axis or the viewing angle $\zeta$ and the azimuth angle or the spin longitude $\phi$. Here, we consider the inertial observer frame, where the direction of the photon is not $\hat{\eta}'$. In order to get the photon direction correctly in this frame ($\hat{\eta}$), we use the aberration formula [see Equation (1) in @dr03] that accounts for the local corotational velocity with respect to the inertial observer frame as follows
$$\hat{\eta} = \frac{\hat{\eta}' + [\gamma+(\gamma-1)(\vec{\beta}\cdot\hat{\eta}')/\beta^2]\vec{\beta}}{\gamma(1+\vec{\beta}\cdot\hat{\eta}')}~,$$
where $\gamma = (1-\beta^2)^{-1/2}$ is the Lorentz factor and $\vec{\beta} = (\vec{\Omega} \times \vec{r})/c$ is the local corotation velocity in units of the speed of light at the emission point $\vec{r}$. Due to aberration, we observe emission slightly earlier in time, or in spin phase. The aberration is in particular important when the emission point $\vec{r}$ is large, i.e., the maximum aberration occurs close to the light cylinder radius with the maximum corotation velocity. Since pulsar A is nearly orthogonal, the corotation velocity of charged particles is important and hence, we include aberration in the model. This is the aberration correction method given in many previous studies. However, @bs10 pointed out that this method leads to an inconsistency since the retarded dipole field was traced in the lab frame, but the aberration was computed treating the field in the instantaneous corotating frame. They showed that this can be corrected by taking a coordinate transformation first, and then correct for the aberration. However, this is a second-order correction in $r/R_{\rm LC}$, so that it is not important for low-altitude radio emission and does not affect our results. Therefore, we do not include this second order correction in our model.
The next step is to include the photon propagation time delay between low- and high-altitude emission reaching the observer. This delay is given as $\vec{r}\cdot\hat{\eta}/R_{\rm LC}$ and is added to the aberration corrected azimuth $\phi$ of the emission point to get the correct phase of each photon [see @dr03]. One of the important observable consequences of the delay is that trailing photons are piled up at a particular spin phase, giving rise to large number of photons (‘caustic’ regions) and producing emission peaks at the line-of-sight as the pulsar rotates. Both aberration and propagation time delay are most important in outer magnetospheric models that describe high-altitude emission, but they cannot be neglected at low altitudes.
We then map aberration and propagation delay corrected photons in a parameter space of $\zeta$ versus $\phi$, which is usually called a sky map. We use a bin size of $1\degr$ in both $\zeta$ and $\phi$ directions. We assume the coherent radio emission is generated at a particular height above the PC region (see Figure \[diagram\]). We further assume that the emissivity of the photon emission is constant across this region. The modeled pulse profile is generated by limiting the photon emission to a particular region at this height above the PC (see Section \[fitting\] for more detail). Then, a horizontal cut of the sky map at a given viewing angle $\zeta$ returns the model pulse profile. By fitting the model pulse profile to pulsar A’s observed profile, we can determine the radio emission altitude $h$ and the size of the radio beam $\rho_{A}$ based on the last closed field lines. This $\rho_{A}$ is an independent estimate for pulsar A’s beam size from what we determined in Section \[circular\] through a simple circular beam model.
Fitting Pulse Profiles of Pulsar A {#fitting}
----------------------------------
As the PC region is bounded by the last closed field lines, we assume the outer edges (i.e., 5% intensity levels), or wings, of a pulse profile are generated from the emission within a thickness of $\Delta h$ along these last closed magnetic field lines at a emission altitude $h$ (see Figure \[diagram\]). The inner part of the pulse component is assumed to be generated from the emission within the same thickness of $\Delta h$ along open field lines above the NS surface. If we fit the entire pulse profile including outer and the inner parts of the pulse component at once, the region around the peak of the profile dominates the result, providing unrealistically large beam sizes and emission altitudes. Thus, we fit pulse profiles in two steps to avoid this issue.
The first step (Step One) is to estimate the emission altitude $h$ and the emission width $\Delta h$ which correspond to the profile wings at 5% intensity level. In order to do this, we map the photon emission from the last closed field lines by varying $h$ and $\Delta h$. Then we fit the modeled profile wings to the observed profile wings and obtain the best-fit $h$ and $\Delta h$ by a maximum likelihood method that we used in Section \[circular\]. We determine the half-opening angle or the beam size $\rho_{\rm A}$ of the radio beam from the direction of the photon emission at this best-fit $h$.
The second step (Step Two) is to model the entire region of the open magnetic field lines fixing the emission altitude to be the best-fit $h$ and $\Delta h$ obtained from Step One. We then compare the entire model pulse profile with the observed profile. However, using a single emission height from the edge of the beam is unrealistic as the emission is not necessarily generated at one particular altitude across the entire open field line region [@lm88]. Therefore, we investigate different emission altitudes across the beam in addition to a constant emission altitude. In this model, we simply assume the radio emission altitude falls off exponentially with height towards the center of the beam from the edge (see Figure \[diagram\]). Then we write an expression for the emission altitude at any point across the beam as $r=h\exp(-(\rho_{\rm r} - \rho_{\rm A})^2 /2\sigma^2)$. Again we assume that emission is generated within the thickness of $\Delta h$ at altitude $r$. We emphasize that $h$ is the emission height at the edge of the beam where the pulse profile wing is formed and is obtained from Step One. The parameter $\rho_{\rm r}$ is the colatitude of the photon at $r$ with respect to the pulsar’s magnetic axis and can be obtained from the direction of the photon emission. This definition implies an inequality relation: $\rho_{\rm r} \leq \rho_{\rm A}$. The parameter $\sigma$ determines the shape of the cross section of the emission thickness and can be written as $\sigma = \rho_{\rm A} / \sqrt{-2\ln (r_0/h)}$, assuming $\rho_{\rm r} = 0\degr$ along the magnetic axis at lower altitude. The height $r_0$ is the emission altitude at the magnetic axis. Once $h$ and $\rho_{\rm A}$ are obtained from Step One, we vary $r_0$ and fit the full model pulse profile to the observed profile as explained in Step Two and estimate the best-fit $r_0$. Instead of assuming two identical beams, we assume the emission altitudes at wings of north ($h_{\rm N}$) and south ($h_{\rm S}$) beams can be different and calculate each separately. Likewise, we define the emission altitude at the magnetic axis from the two beams as $r_{\rm 0,N}$ and $r_{\rm 0,S}$.
Below is a summary of our prescription for the pulse profile fitting using the PC model: (1) model the field line structure based on a retarded vacuum dipole for a given magnetic inclination $\alpha_{\rm A}$ (see Equation (\[bfield\])); (2) model the photon emission from the open and last closed field lines including the effects of aberration and light propagation delay; (3) assume the radio emission is generated from these field lines within a thickness of $\Delta h$ at a given altitude above the PC region, map the photon emission from this region in the space of $\zeta$ vs $\phi$; (4) obtain a model pulse profile for the viewing angle of the line-of-sight $\zeta_{\rm E}$; (5) fit the model pulse profile to the observed one and constrain the radio beam geometry.
In this work, we obtain four sets of ${ h, \rho_{\rm A}, {\rm and}~ r_{\rm 0}}$ using results from the model CBM and constraints from @gkj+13 combined with the PC model. The results are described in the next subsection, where subscripts of $N$ and $S$ denote the north and south poles.
Results
-------
The geometry of CBM yields that $\zeta_{\rm E}$ is consistent with being constant in time, due to an aligned or nearly aligned spin axis ($\delta_{\rm A} \leq 2\fdg8$). Therefore, a choice of MJD does not affect the model pulse profile significantly, if at all. As shown in previous studies and again confirmed in Section \[data\], the pulse profiles of pulsar A do not show a significant time evolution. Therefore, we consider the observed pulse profile on MJD 53861 (Figure \[profile\]) as the time-independent observed pulse profile of A and obtained $h$ and $\Delta h$ by fitting the model pulse profile to this one. According to the geometry, the best-fit radio beam parameters are estimated to be $h_{\rm N} = 10^{+1}_{-2}$ R$_{\rm NS}$, $h_{\rm S} = 11\pm2$ R$_{\rm NS}$, $\Delta h = 1\pm1$ R$_{\rm NS}$, $r_{0\rm,N} = 2^{+7}_{-1}$ R$_{\rm NS}$, and $r_{\rm 0,S} = 5^{+6}_{-4}$ R$_{\rm NS}$. The beam half-opening angles are $\rho_{\rm N} = 31\degr \pm 1\degr$ and $\rho_{\rm S} = 33\degr \pm 1\degr$. Figure \[profile\_part1\] presents the best-fit pulse profile obtained from the PC model with the geometry determined by CBM, overlaid with A’s pulse profile (solid).
Due to the lack of evolution in A’s pulse profile, @gkj+13 set $\delta_{\rm A} = 0\degr$ and fit TPC and OG emission models to gamma-ray light curves, separately, in order to obtain the geometry of the pulsar (see Table 1). The best-fit results from the TPC model are $\alpha_{\rm A} = 80(9)\degr$ and $\zeta_{\rm E} = 86(14)\degr$. With these parameters, we apply our PC beam model to search for the radio emission altitude and the radio beam size. The emission heights are estimated to be $h_{\rm N} = 11^{+1}_{-2}$ R$_{\rm NS}$ and $h_{\rm S} = 11^{+2}_{-1}$ R$_{\rm NS}$ with $\Delta h = 1.2^{+0.6}_{-0.9}$ R$_{\rm NS}$, $r_{0\rm ,N} = 2^{+8}_{-1}$ R$_{\rm NS}$, and $r_{\rm 0,S} = 1$ R$_{\rm NS}$ (i.e., on the NS surface) with an upper bound error of 1 R$_{\rm NS}$. The half-opening angles are $\rho_{\rm N} = 32\degr \pm 1\degr$ and $\rho_{\rm S} = 32\degr \pm 1\degr$. The OG model (i.e., $\alpha_{\rm A}=88(17)\degr$ and $\zeta_{\rm E}=74(14)\degr$) gives the best-fit parameters as follows: $h_{\rm N} = 12^{+2}_{-1}$ R$_{\rm NS}$ and $h_{\rm S} = 15\pm2$ R$_{\rm NS}$ with $\Delta h = 2\pm2$ R$_{\rm NS}$, $r_{\rm 0,N} = 2^{+9}_{-1}$ R$_{\rm NS}$, and $r_{\rm 0,S} = 1$ R$_{\rm NS}$ (i.e., on the NS surface) with an upper bound error of 12 R$_{\rm NS}$. The half-opening angles with the OG model are estimated to be $\rho_{\rm N} = 33\degr \pm 2\degr$ and $\rho_{\rm S} = 38\degr \pm 1\degr$. Figure \[profile\_part2\] shows the comparison between A’s pulse profile (solid) with the best-fit pulse profiles obtained from the PC model with the geometry estimated by the TPC and OG magnetosphere models, respectively. By comparing our PC beam model pulse profile with the hollow-cone beam model profile given in @gkj+13 (see Figure 2 therein), it is clearly seen that the PC beam model fits the pulse profile wings of both pulse components better. The brightest pulse of the modeled pulse profiles from both models are consistent with the observed profile. However, due to the hollow-cone nature of the radio beam model given in @gkj+13, the shape of P2 from the TPC-model-resulted pulse profile has a broad double peak structure. In contrast, our PC model results in a single peak structure for P2 and is more similar to the observed pulse component than the double peaked structure.
Lastly, if the constraints from polarization measurements provided by @gkj+13, $\alpha_{\rm A} = 99(8)\degr$ and $\zeta_{\rm E}=96(13)\degr$, are included with the PC model, we obtain the emission heights to be $h_{\rm N} = 10^{+1}_{-2}$ R$_{\rm NS}$ and $h_{\rm S} = 12^{+3}_{-1}$ R$_{\rm NS}$ with $\Delta h = 1.25\pm 1.0$ R$_{\rm NS}$, $r_{\rm 0,N} = 2\pm 1$ R$_{\rm NS}$, and $r_{\rm 0,S} = 1$ R$_{\rm NS}$ with an upper bound error of 1 R$_{\rm NS}$. The half-opening angles of the two beams are estimated to be $\rho_{\rm N} = 30\degr \pm 1\degr$ and $\rho_{\rm S} = 33\degr \pm 1\degr$, respectively. Figure \[profile\_part2\] shows the best-fit pulse profile obtained from the PC model with the geometry estimated by the polarization observation. As shown in @gkj+13, the polarimetric profile for pulsar A is complicated. They fit their model only to the rapid polarization angle sweep seen in the linearly polarized region around the peak of each pulse component and then constrained the geometry, and the emission altitudes based on the phase lag between the polarization angle sweep and the magnetic axis. Their results showed that the emission altitude corresponding to the central region of the second brightest pulse component (i.e., P2 in this work) is $\sim$1 R$_{\rm NS}$, which is in agreement with our result (i.e., $r_{\rm 0,S} = 1$ R$_{\rm NS}$). However, their best-fit emission altitude corresponding to the central region of the brightest pulse component is $\sim$12 R$_{\rm NS}$, which is greater than our best-fit value of $r_{\rm 0,N} = 2$ R$_{\rm NS}$. We note that the geometry and the emission altitudes constrained by @gkj+13 were meaningful only for small sections of the pulse profile, close to the peak of each pulse component. However, we fit the PC model for the entire pulse profile assuming this geometry. This inconsistency could be the reason for the differences in resulting emission altitudes for P1 from the two models.
The Orbital Geometry of the Double Pulsar {#full_geo}
=========================================
When the beam geometry of A and B pulsars are known, it is possible to calculate the relative angle ($\Delta_{\rm S}$) between the spin axes of the two pulsars and to fully configure the orbital geometry of the Double Pulsar. The main results for A’s geometry are summarized in Table 1. @pmk+10 [@plg+12] presented the geometry of pulsar B and @kpm13 provided constraints on the beaming fraction.
Utilizing the results described in earlier sections, we can write $\Delta_{\rm S}(t)$ as follows
$$\label{re_angle}
\cos{(\Delta_{\rm S}(t))} = \cos\delta_{\rm A} \cos\delta_{\rm B} + \sin\delta_{\rm A} \sin\delta_{\rm B} \cos(\Delta \phi_{\rm prec}(t)),$$
where $\Delta_{\rm S}(t)$ is the relative angle between the spin axes of A and B at time $t$. The angles $\delta_{\rm A}$ and $\delta_{\rm B}$ are spin misalignment angles of A and B with respect to the orbital angular momentum. The angle $\Delta \phi_{\rm prec}(t)$ is the relative spin precession angle and is defined by $\Delta \phi_{\rm prec}(t) = \phi_{\rm prec,A}(t) - \phi_{\rm prec,B}(t)$, where $\phi_{\rm prec,i}(t) = \Omega_{\rm prec,i}(t-T_0)$ is the spin precession phase and $\Omega_{\rm prec,i}$ is the spin precession rate for ${\rm i}=$ A and B pulsars. Note that the angle $\Delta_{\rm S}(t)$ is not affected by the details of our assumptions on the pulsar radio beams or magnetic misalignment. Geodetic precession of the two pulsars would cause $\Delta_{\rm S}$ to change over time.
With the particular geometric framework, the minimum and maximum $\Delta_{\rm S}(t)$ are given by $\delta_{\rm B}-\delta_{\rm A}$ and $\delta_{\rm B}+\delta_{\rm A}$, respectively. Based on our results and @plg+12, $\delta_{\rm B} > \delta_{\rm A}$ (Table 1). At the current epoch, $\Delta_{\rm S}(t)$ for CBM is $138(5)\degr$. Considering the 2$\sigma$ uncertainties of $\delta_{\rm A}$ and $\delta_{\rm B}$ given in Table 1, we estimate the uncertainty of $\Delta_{\rm S}(
t)$ to be $\pm$$6\degr$. If $\delta_{\rm A}$ does not equal 0, then $\Delta_{\rm S}$ will show a variation with time with a period of 1385 years, based on the precession periods of pulsar A (75 yrs) and pulsar B (71 yrs). The evolution of $\Delta_{\rm S}$ of the Double Pulsar is expected to follow Equation (\[re\_angle\]) until tidal interactions become important in $\sim$85 Myr at merger phase [see @kpm13].
Discussion and Conclusions {#dis}
==========================
In this work, we constrain A’s beam geometry assuming a double-pole geometry to estimate the magnetic misalignment angle ($\alpha_{\rm A}$) and the colatitude of the spin axis ($\delta_{\rm A}$). @fsk+13 estimated these angles to be $\alpha_{\rm A} = 90\degr \pm 16\degr$ and $\delta_{\rm A} \le 2\fdg3$ at 68% confidence. Our results have smaller errors, but are consistent with these (see Table 1). As shown in @mkp+05, @fsk+08 [@fsk+13] and Section \[data\] in this work, the orthogonal geometry of the pulsar provides no secular variation in the observed pulse profiles. In the studies of other pulsar binaries given in Table 1, similar geometrical frameworks have been used to constrain the geometry based on the observed pulse profile variations due to spin precession. The pulse profile evolution of these systems are all detectable due to their large spin misalignment. However, the evolution is somewhat long-term due to the much smaller spin precession rates compared to that of the double pulsar; see @kra98 for the details of the long-term profile evolution of PSR B$1913+16$. As shown for pulsar A, we note that the pulsar geometry, and the spin precession rate, both play an important role in the profile evolution of binary pulsars.
The recent [*Fermi*]{} detection of pulsed gamma-ray emission from pulsar A revealed that the peaks of the high-energy and radio profiles are not aligned in spin phase [@gkj+13]. This implies that high- and low-energy emission is produced at two different locations in the magnetosphere. @gkj+13 used the OG and TPC models to describe the high-energy emission and gamma-ray light curves from pulsar A. The RVM, based on the radio polarization measurement, is used to constrain A’s beam geometry. We incorporate $\alpha_{\rm A}$ and $\zeta_{\rm E}$ from these models and the model given in Section \[circular\] in our PC magnetosphere beaming model to synthesize the pulse profile of A and then estimate the radio beam size and the emission altitude. All our model pulse profiles are qualitatively in agreement with A’s observed profile, but none of them are perfect fits. We find that including the aberration effects results in a better fit to the leading step-like part of the P1 component, but the peak of P2 component does not fit well with observation. However, the edges of P2 fit reasonably well with the observed pulse component. Thus, a high photon density towards the leading edge of the south emission beam compared to its trailing edge can result in more photons around the leading edge of P2 component. This may move the peak of P2 to lower spin phases and fit with the observed peak better. In reality, different beam shapes can exist; two asymmetric beams may provide a better fit for observation. Further, a partially filled patchy beam structure can be another option in pulsar beam modeling. However, these beam structures are not easily modeled due to the large number of free parameters associated with them.
Using pulsar A’s spin period, period derivative, and the observing frequency of 820 MHz in the expression derived by @kg03a, we obtain A’s emission altitude ($h_{\rm A}\sim$$10$ R$_{\rm NS}$). Empirical fits to canonical, non-recycled, pulsars imply a correlation between the radio beam size and the pulsar’s spin period: $\rho = 5\fdg4 P_{\rm s}^{-0.5}$ [@kxl+98]. It is not clear whether the recycled pulsars follow this empirical relation, but if we assume this, the beam size of A is estimated to be $\sim38\degr$. In this case, $h_{\rm A}$ and $\rho_{\rm A}$ obtained from the PC beam model for pulsar A are consistent with those predicted by the above relations. Regardless of assumption on the $\rho-P_{\rm s}$ relation, the emission altitudes for recycled pulsars J0437–4715 ($P_{\rm s}=5.75$ ms) and B1913$+$16 ($P_{\rm s}=59$ ms) were estimated to be $h<$9 R$_{\rm NS}$ [@gt06] and $h<$20 R$_{\rm NS}$ [@kg97], respectively. Our results on pulsar A’s emission altitudes ($<$15 R$_{\rm NS}$) are consistent with those of above two pulsars and the differences in estimates are associated with different magnetosphere sizes ($\equiv cP_{\rm s}/2\pi$), based on their pulse periods.
Our measurement of $\Delta_{\rm S}=138^{\circ}$, at the current moment, is the first measurement of the spin orientation in a NS-NS binary. This information is useful in studying the final evolution of NS-NS binaries. Since the detection of gravitational waves from merging binary neutron stars is a major goal of gravitational-wave observatories such as Advanced LIGO [@har10] and Advanced Virgo [@daa+13], constraining this parameter for known or new NS-NS binaries in the future is important. Recently, @brown12 showed that including spin effects in GW search templates could provide increases in sensitivity. They also discussed that GWs from all known NS-NS binaries in the Galactic disk can be described with non-spinning waveforms. Even for the Double Pulsar, with the fastest spinning recycled pulsar among the known NS-NS binaries, the aligned spin with respect to the orbital normal and the long spin period of B make a non-spinning waveform a suitable template to search for GWs. However, spin effects are likely to be more important for NS-NS binaries with two faster-spinning neutron stars, so that our measurement provides a useful path to study such complicated systems.
BBPP, MAM, and CK are supported through the Research Corporation. CK is supported in part by the National Research Foundation Grant funded by the Korean Government (No. NRF-2011-220-C00029). CK thank Hee-Il Kim and Alex Nielsen for useful discussions. Pulsar research at UBC is supported by an NSERC Discovery grant.
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abstract: 'Loop-weighted walk with parameter $\lambda\geq 0$ is a non-Markovian model of random walks that is related to the loop $O(N)$ model of statistical mechanics. A walk receives weight $\lambda^{k}$ if it contains $k$ loops; whether this is a reward or punishment for containing loops depends on the value of $\lambda$. A challenging feature of loop-weighted walk is that it is not purely repulsive, meaning the weight of the future of a walk may either increase or decrease if the past is forgotten. Repulsion is typically an essential property for lace expansion arguments. This article circumvents the lack of repulsion and proves, for any $\lambda>0$, that loop-weighted walk is diffusive in high dimensions by lace expansion methods.'
author:
- 'Tyler Helmuth[^1] [^2]'
bibliography:
- 'refs1.bib'
title: 'Loop-Weighted Walk'
---
[ oldtitletitle title[oldtitle]{}]{} [ @oldtitletitle title[@oldtitle]{}]{}
Introduction and Main Results {#sec:LWW-Intro-Results}
=============================
*Loop-weighted walk with parameter $\lambda$*, abbreviated *$\lambda$-LWW*, is a model of self-interacting walks that can be informally defined as follows. Formal definitions will be given in . Let $\omega$ be a walk on a graph. A walk is called a *loop* if $\omega$ begins and ends at the same vertex. The *loop erasure* ${\mathrm{LE}}(\omega)$ is formed by chronologically removing loops from $\omega$. If $n_{L}(\omega)$ denotes the number of loops removed, the $\lambda$-LWW weight of a walk $\omega$ is $$\label{eq:LWW-Weight-Intro}
w_{\lambda}(\omega) = \lambda^{n_{L}(\omega)}.$$
Throughout this article it will be assumed that $\lambda\geq 0$, so defines a non-negative weight on walks. In particular, $w_{\lambda}$ defines a probability measure on $n$-step walks that begin at a fixed vertex of a graph by defining the probability of $\omega$ to be proportional to $w_{\lambda}(\omega)$. If $0\leq \lambda<1$ the effect of the weight is to discourage walks from containing loops, and for this parameter range $\lambda$-LWW interpolates between the uniform measure on $n$-step self-avoiding walks ($0$-LWW) and the uniform measure on all $n$-step walks ($1$-LWW). If $\lambda>1$ the weight encourages the existence of loops: walks are rewarded for returning to vertices that have been visited in the past. Note that $\lambda$-LWW for $\lambda\neq 1$ is not a Markovian model of walks.
In addition to being an interesting model of self-interacting random walks that encompasses the well-known models of self-avoiding and simple random walk, $\lambda$-LWW also has connections with spin models in statistical mechanics. The description of these connections will be deferred until after the results of the article are described, see .
This article consists of a lace expansion analysis of $\lambda$-LWW. The lace expansion, originally introduced by Brydges and Spencer [@BrydgesSpencer1985], is a powerful tool for proving mean-field behaviour in high dimensions [@Slade2006]. With few exceptions, see the discussion at the end of , walk models that have been successfully studied with the lace expansion have been *purely repulsive*. A walk model being purely repulsive means that the weight $w$ on walks that defines the model satisfies the inequality $$\label{eq:LWW-Repulsive-Intro}
w(\omega\circ\eta) \leq w(\omega) w(\eta),$$ where $\omega\circ\eta$ is the concatenation of two walks $\omega$ and $\eta$. For example, self-avoiding walk is purely repulsive. In general $\lambda$-LWW is *not* purely repulsive if $\lambda\neq 0,1$. See .
(s0) at (1,0) ; (s2) at (2,0) ; (s3) at (3,0) ; (s4) at (4,0) ; (s5) at (5,0) ; (s6) at (6,0) ; (s0) to (6,0); (6,0) to\[out=0, in=0\] (5.5,1) to\[out=180,in=90\] (s5); (s5) to\[out=270, in= 0\] (4.5,-1) to\[out=180, in=270\] (s4); (s4) to\[out=90, in=0\] (3.5,1) to\[out=180,in=90\] (s3); (s3) to\[out=270, in= 0\] (2.5,-1) to\[out=180, in=270\] (s2); (s2) to (2,1.5);
at (s0) ; at (6.06,0) ; at (2.0,1.56) ; at (s0) \[below\] [$\omega_{0}$]{}; at (s6) \[below right\] [$\!\!\omega_{n}=\omega^{\prime}_{0}$]{}; at (2.05,1.55) \[right\] [$\omega^{\prime}_{n^{\prime}}$]{};
(s0) at (1,0) ; (s2) at (2,0) ; (s3) at (3,0) ; (h3) at (3,1) ; (h4) at (4,1) ; (h5) at (5,1) ; (s6) at (6,0) ; (s0) to (6,0); (6,0) to\[out=0, in=0\] (5.5,1) to\[out=180,in=0\] (h3) to\[out=180,in=90\] (s2); (s2) to\[out=270, in= 180\] (2.5,-1) to\[out=0, in=270\] (s3); (s3) to\[out=90, in=270\] (h3); (h3) to\[out=90, in=180\] (3.5,1.5) to\[out=0, in=90\] (h4); (h4) to\[out=270, in=180\] (4.5,.5) to\[out=0, in=270\] (h5); (h5) to (5,1.5);
at (s0) ; at (6.06,0) ; at (5.0,1.56) ; at (s0) \[below\] [$\omega_{0}$]{}; at (s6) \[below right\] [$\!\!\omega_{n}=\omega^{\prime}_{0}$]{}; at (5.05,1.55) \[right\] [$\omega^{\prime}_{n^{\prime}}$]{};
The most significant step required to analyze $\lambda$-LWW with the lace expansion is therefore a technique to overcome the lack of repulsion. This is done by resumming $\lambda$-LWW to obtain a model of self-interacting and self-avoiding walks. The particular form of the $\lambda$-LWW weight leads to a very explicit description of the self-interaction in terms of a generalization of the loop measure of [@LawlerLimic2010], and this explicit description makes it clear that the self-interaction is repulsive. This enables a lace expansion to be performed. Further details about the proof follow after the statement of .
Some notation will be needed to state the results. Let ${\langle \cdot \rangle}_{n}^{\lambda}$ denote expectation with respect to the measure on $n$-step walks associated to $w_{\lambda}$. Let $c_{n}^{\lambda}$ be the normalizing factor for the expectation, i.e., the sum over all $n$ step walks weighted by $\lambda^{n_{L}(\omega)}$ as in . Let $\chi_{\lambda}(z) =
\sum_{n}c_{n}^{\lambda}z^{n}$, and let $z_{c}(\lambda)$ be the radius of convergence of $\chi_{\lambda}(z)$. The main result of this article can be summarized as saying that, in high dimensions, $\lambda$-LWW has mean field behaviour at criticality.
\[thm:LWW-Main-Intro\] Fix $\lambda\geq 0$ and consider $\lambda$-LWW on ${{\mathbb{Z}}}^{d}$. There exists $d_{0}=d_{0}(\lambda)$ such that for $d\geq
d_{0}$ there are constants $A$ and $D$ such that
1. The susceptibility diverges linearly: $\chi_{\lambda}(z) \sim
Az_{c} (z_{c}-z)^{-1}$ as $z\nearrow z_{c}$,
2. $c_{n}^{\lambda} = A{\left(z_{c}(\lambda)\right )}^{-n}(1+O(n^{-\delta}))$ for any $\delta<1$, and
3. $\lambda$-LWW is diffusive: ${\langle {\left\vert\omega_{n}\right\vert}^{2} \rangle}^{\lambda}_{n} = Dn(1+O(n^{-\delta}))$ for any $\delta<1$.
For $\lambda=0$ has been proven with $d_{0}=5$ by Hara and Slade [@HaraSlade1992]. It is worth emphasizing that holds for $\lambda>1$ when $\lambda$-LWW is attractive in the sense that the formation of loops is encouraged.
\[rem:LWW-Dimension\] No attempt has been made to track the value of $d_{0}$ that is required, and the proof presented in this article requires $d_{0}\gg
9$. The true behaviour of $d_{0}(\lambda)$ is an interesting question for future study.
Let us say a few more words about the proof of . As described earlier, the key step is a resummation of $\lambda$-LWW into a self-interacting self-avoiding walk. The self-interaction of the self-avoiding walk is a many-body interaction, and this leads to a hypergraph-based lace expansion instead of the graph-based lace expansion that is used for self-avoiding walk. We stress that hypergraphs are merely an organizational tool, and no prior knowledge of hypergraphs is needed to understand the expansion. Once the lace expansion has been performed the various self-interacting self-avoiding walk quantities can be re-expressed in terms of $\lambda$-LWW. The diagrams that occur in analyzing the expansion generalize the diagrams for self-avoiding walk, and when $\lambda=0$ they reduce to the diagrams for self-avoiding walk. With some effort it is possible to analyze the diagrams for $\lambda>0$ with existing methods. Once the analysis of the diagrams is completed it is possible to apply established techniques to analyze $\lambda$-LWW, namely the trigonometric approach to the convergence of the lace expansion [@Slade2006] and complex analytic methods for studying asymptotics.
In fact holds in greater generality. Let $\lambda_{\ell}\geq 0$ be the weight of the loop $\ell$. Replace the weight $\lambda$ per loop in with the product of $\lambda_{\ell}$ over the set of loops $\ell$ that are erased when performing loop erasure on $\omega$. Assume the set of weights $\{\lambda_{\ell}\}$ satisfy a mild symmetry hypothesis, see , and are uniformly bounded above. Then the results of continue to hold.
The remainder of the introduction is as follows. describes an important connection between $\lambda$-LWW and the loop $O(N)$ model of statistical physics. gives a formal definition of $\lambda$-LWW, relates $\lambda$-LWW to a self-interacting and self-avoiding walk, and outlines how this enables a lace expansion analysis. Lastly, establishes a few conventions used in the remainder of the article.
Motivation from Statistical Mechanics {#sec:LWW-Intro-SM}
-------------------------------------
For $N\in {{\mathbb{N}}}$ the *$O(N)$ model* on a graph finite $G = (V,E)$ is a generalization of the Ising model. To each vertex $x\in V$ is associated a *spin* $\vec s_{x}$ taking values in the unit sphere in ${{\mathbb{R}}}^{N}$. The probability of a spin configuration is defined by $$\label{eq:LWW-Intro-SM-1}
{{\mathbb{P}}}{\left( \{\vec s_{x}\}_{x\in V}\right )} \propto \exp ( \beta \sum_{x\sim y}
\vec s_{x}\cdot \vec s_{y}),$$ where $\beta$ is a real parameter and the summation is over all edges $\{x,y\}\in E$. In [@DomanyMukamelNienhuisSchwimmer1981] a simplification of the $O(N)$ model known as the *loop $O(N)$ model* was introduced. The loop $O(N)$ model is defined in terms of subgraph configurations on $G$. In the special case of a graph with vertex degree bounded by $3$, the loop $O(N)$ model configurations are subgraphs that are disjoint unions of cycles of length at least $3$, and the probability of a subgraph $H$ is given by $$\label{eq:LWW-Intro-SM-2}
{{\mathbb{P}}}(H) \propto z^{{\left\vertE(H)\right\vert}} N^{\# H},$$ where $\# H$ denotes the number of connected components of $H$. Note that the probability in may be negative if $N<0$: defines a signed measure in general.
The definition of the loop $O(N)$ model on an arbitrary graph $G$ involves noncyclic subgraphs, see for example [@ChayesPryadkoShtengel2000]. The noncyclic subgraphs are predicted by non-rigorous renormalization group arguments to be irrelevant [@Nienhuis2010], at least for ${\left\vertN\right\vert}\leq 2$ on planar graphs. Call the model whose configurations are disjoint unions of cyclic subgraphs the $O(N)$ cycle gas. For $N\in{{\mathbb{N}}}$ this model has previously appeared in the physics literature as a model for melting transitions [@HelfrichRys1982].
As described in , $\lambda$-LWW is a walk representation of the $O(N)$ cycle gas. The two-point function of $\lambda$-LWW corresponds to a two-point correlation in the $O(N)$ cycle gas for $N=-2\lambda$. In other words, $\lambda$-LWW yields a *probabilistic* interpretation of the $O(N)$ cycle gas for $N<0$. This is an example of a “negative activity isomorphism theorem”: an equivalence between a statistical mechanics model at negative activity ($N<0$) and a probability model. An important previous example of such a theorem is the Brydges–Imbrie isomorphism between branched polymers in ${{\mathbb{R}}}^{d+2}$ and the hard-core gas in ${{\mathbb{R}}}^{d}$ [@BrydgesImbrie2003]. In the present work the isomorphism allows results about $\lambda$-LWW to be transferred to the $O(N)$ cycle gas for $N<0$. For example, the isomorphism theorem combined with immediately implies the following corollary.
\[cor:LWW-CYCLE\] For $d$ sufficiently large the susceptibility of the $O(N)$ cycle gas on ${{\mathbb{Z}}}^{d}$ for $N<0$ diverges linearly at the critical point.
This section may be summarized as saying that $\lambda$-LWW can be viewed as a random walk representation of an approximation of the $O(N)$ model. Thus $\lambda$-LWW fits into a long history of random walk representations of spin models [@Aizenman1982; @BrydgesFrohlichSpencer1982; @FernandezFrohlichSokal1992] inspired by the pioneering work of Symanzik [@Symanzik1969].
Introduction to the Loop-Weighted Walk Model {#sec:LWW-Basics}
--------------------------------------------
The rest of the paper will be concerned with ${{\mathbb{Z}}}^{d}$, the simple cubic lattice in $d$ dimensions. Edges $\{x,y\}$ will often be abbreviated $xy$. Two vertices $x$ and $y$ will be called *adjacent*, written $x\sim y$, if $xy$ is an edge in ${{\mathbb{Z}}}^{d}$. Let $\Omega = \{ y\in {{\mathbb{Z}}}^{d} \mid y\sim 0\}$, so ${\left\vert\Omega\right\vert} = 2d$ is the number of vertices adjacent to the origin $0$.
### Model Definition {#sec:LWW-Definition}
The next paragraphs establish some conventions about walks. An *$n$-step walk* $\omega = (\omega_{0}, \omega_{1}, \dots,
\omega_{n})$ is a sequence of $n+1$ adjacent vertices in ${{\mathbb{Z}}}^{d}$. Given a walk $\omega$, ${\left\vert\omega\right\vert}$ will denote the number of steps in $\omega$. A walk is a *loop* if $\omega_{{\left\vert\omega\right\vert}} = \omega_{0}$, *self-avoiding* if $\omega_{i}=\omega_{j}$ implies $i=j$, and a *self-avoiding polygon* if $\omega_{i}=\omega_{j}$ and $i\neq j$ implies $\{i,j\} =
\{0,{\left\vert\omega\right\vert}\}$.
A walk $\omega$ *begins* at $\omega_{0}$ and *ends* at $\omega_{{\left\vert\omega\right\vert}}$. Let $\omega\colon x\to y$ denote the set of walks beginning at $x$ and ending at $y$. Let ${{\Omega}_\mathrm{SAW}}(x,y) = \{
\omega\colon x\to y \mid \textrm{$\omega$ self-avoiding}\}$; if $x=y$ this is taken to be the set of self-avoiding polygons beginning at $x$. Let ${{\Omega}_\mathrm{SAP}}= \cup_{x}{{\Omega}_\mathrm{SAW}}(x,x)$ and ${{\Omega}_\mathrm{SAW}}=
\cup_{x}\cup_{y}{{\Omega}_\mathrm{SAW}}(x,y)$. If $\omega^{(i)} = (\omega^{(i)}_{0},
\dots, \omega^{(i)}_{k_{i}})$ for $i=1,2$ and $\omega^{(1)}_{k_{1}} =
\omega^{(2)}_{0}$ the *concatenation* $\omega^{(1)}\circ
\omega^{(2)}$ of $\omega^{(1)}$ with $\omega^{(2)}$ is defined by $$\label{eq:LWW-Concatenation}
\omega^{(1)}\circ\omega^{(2)} = (\omega^{(1)}_{0}, \dots,
\omega^{(1)}_{k_{1}}, \omega^{(2)}_{1}, \dots,
\omega^{(2)}_{k_{2}}).$$
To define $\lambda$-LWW precisely requires an explicit description of the loop erasure of a walk $\omega$. Define $$\begin{aligned}
\tau_{\omega} &= \min {\left\{ i \mid \exists\, j<i \textrm{ such that }
\omega_{i}=\omega_{j}\right\}}, \\
\tau_{\omega}^{\star} &= \min {\left\{j \mid \omega_{j} =
\omega_{\tau_{\omega}}\right\}}.\end{aligned}$$ If $\omega$ is a self-avoiding walk, define $\tau_{\omega} =
\tau_{\omega}^{\star}=\infty$. The time $\tau_{\omega}$ is the first time a walk visits a vertex twice.
\[def:LWW-Erasure\] Let $\omega$ be a walk of length $n$. The *single loop erasure ${\mathrm{LE}}^{1}(\omega)$* of $\omega$ is given by $$\label{eq:LWW-Single-Erase}
{\mathrm{LE}}^{1}(\omega) = (\omega_{0}, \dots,
\omega_{\tau_{\omega}^{\star}\wedge n}, \omega_{\tau_{\omega}+1},
\dots, \omega_{n}),$$ where $a\wedge b$ denotes the minimum of $a$ and $b$. The walk $(\omega_{\tau_{\omega}^{\star}}, \omega_{\tau_{\omega}^{\star}+1},
\dots, \omega_{\tau_{\omega}})$ is the *loop removed by loop erasure*. The *loop erasure* ${\mathrm{LE}}(\omega)$ of $\omega$ is the result of iteratively applying ${\mathrm{LE}}^{1}$ until $\tau_{\omega}=\infty$.
By construction, each loop removed from a walk by loop erasure is a self-avoiding polygon.
\[def:LWW-Loop-Vector\] The *loop vector $n_{L}(\omega)$* of $\omega$ is the vector with coordinates $$\label{eq:LWW-Loop-Vector}
n_{L}^{\eta}(\omega) = \textrm{\# of times $\eta$
is removed by loop erasure applied to $\omega$}, \qquad \eta\in{{\Omega}_\mathrm{SAP}}.$$
In what follows $\lambda$ will denote a vector of activities $\lambda_{\eta}\geq 0$ for $\eta\in{{\Omega}_\mathrm{SAP}}$. Inequalities with respect to $\lambda$ are to be interpreted pointwise in $\eta\in{{\Omega}_\mathrm{SAP}}$. Define $$\label{eq:LWW-Variable-Loop-Weight}
\lambda^{n_{L}(\omega)} = \prod_{\eta}\lambda_{\eta}^{n_{L}^{\eta}(\omega)}.$$
\[def:LWW-LRW\] Let $\lambda\geq 0$, $z\geq 0$. The weight $w_{\lambda,z}$ of $\lambda$-LWW at activity $z$ is given by $$\label{eq:LWW-LRW}
w_{\lambda,z}(\omega) = z^{{\left\vert\omega\right\vert}}\lambda^{n_{L}(\omega)}.$$
\[def:LWW-Susceptibility\] The *susceptibility $\chi_{\lambda}(z)$* of $\lambda$-LWW is $$\label{eq:LWW-Susceptibility}
\chi_{\lambda}(z) = \sum_{x\in {{\mathbb{Z}}}^{d}}\sum_{\omega\colon 0 \to x} w_{\lambda,z}(\omega).$$ The *critical point $z_{c}(\lambda)$* of $\lambda$-LWW is defined to be the radius of convergence of $\chi_{\lambda}(z)$.
If $0\leq\lambda\leq 1$ then $\chi_{\lambda}(z) \leq \chi_{1}(z)$, and hence $\chi_{\lambda}(z)$ converges for $z<{\left\vert\Omega\right\vert}^{-1}$. The next proposition gives a mild condition under which the critical point is nontrivial.
\[prop:LWW-Trivial-G-Bound\] Let $\bar\lambda = \sup_{\eta}\lambda_{\eta}>1$. If $z<({\left\vert\Omega\right\vert}\sqrt{\bar\lambda})^{-1}$ then $\chi_{\lambda}(z)$ is finite.
An $n$-step walk contains at most $\lfloor n/2\rfloor$ loops, and weighting each loop by $\bar\lambda$ yields an upper bound for $\chi_{\lambda}(z)$. Cancelling the factors of $\sqrt{\bar\lambda}$ gives the claim, as the resulting sum is $\chi_{1}(\bar z)$ for some $\bar z < {\left\vert\Omega\right\vert}^{-1}$.
If ${\mathcal{R}}$ is an isometry of ${{\mathbb{Z}}}^{d}$, and $A\subset {{\mathbb{Z}}}^{d}$, let ${\mathcal{R}} A = \{{\mathcal{R}}a \mid a\in A\}$.
\[ass:LWW-Symmetry\] Assume that $\lambda_{\eta} = \lambda_{{\mathcal{R}} \eta}$ for any isometry ${\mathcal{R}}$ and any $\eta\in{{\Omega}_\mathrm{SAP}}$. Further assume that $\lambda_{\eta} =
\lambda_{\tilde \eta}$ if $\eta$ and $\tilde \eta$ are self-avoiding polygons that differ only in terms of initial vertex and orientation.
\[ass:LWW-Bounded\] Assume $\sup_{\eta\in{{\Omega}_\mathrm{SAP}}}\lambda_{\eta}<\infty$.
\[thm:LWW-Main\] Fix $\lambda\geq 0$. If and hold, then there exists $d_{0}=d_{0}(\lambda)$ such that for $d\geq d_{0}$ there are constants $A$ and $D$ such that the conclusions of hold.
is the special case of when the loop activities $\lambda$ are constant. The constants $A$ and $D$ have explicit expressions, see . For the remainder of the article it will be assumed that and hold.
### Aspects of Proof {#sec:LWW-Proof-Outline}
This section describes the basic facts about $\lambda$-LWW that allow for a lace expansion analysis, and gives an outline of the proof of .
\[def:LWW-LELWW\] The *loop-erased $\lambda$-LWW* weight $\bar w_{\lambda,z}$ on self-avoiding walks is $$\label{eq:LWW-LELWW}
\bar w_{\lambda,z}(\eta) = {{\mathbbm{1}}_{\left\{\eta\in{{\Omega}_\mathrm{SAW}}\right\}}}
\sum_{\omega \colon {\mathrm{LE}}(\omega)=\eta} w_{\lambda,z}(\omega).$$
Note that the definition of $\bar w_{\lambda,z}$ assigns non-zero weight only to self-avoiding walks. The definition of $\bar
w_{\lambda,z}$ implies that for any $x\in {{\mathbb{Z}}}^{d}$ $$\label{prop:LWW-2PT-Equivalence}
\sum_{\omega\colon 0\to x} \bar w_{\lambda,z}(\omega) =
\sum_{\omega\colon 0 \to x} w_{\lambda,z}(\omega),$$ as the left-hand side is just a reorganization of the right-hand side. This identity will be important in what follows.
\[def:LWW-range\] The *range*, ${\mathrm{range}(\omega)}$, of a walk $\omega$ is the set of vertices visited by $\omega$.
The *$\lambda$-LWW loop measure at activity $z$* of a closed walk $\omega$ is given by $w_{\lambda,z}(\omega) / {\left\vert\omega\right\vert}$. The next definition introduces a convenient shorthand for the loop measure of certain subsets of walks; note that $\mu_{\lambda,z}$ is not a measure.
\[def:LWW-LM\] Let $A,B\subset {{\mathbb{Z}}}^{d}$. The *$\lambda$-LWW loop measure* $\mu_{\lambda,z}(A;B)$ is $$\label{eq:LWW-LM}
\mu_{\lambda,z}(A;B) = \sum_{x} \mathop{\sum_{\omega\colon x\to
x}}_{{\left\vert\omega\right\vert} \geq 1}
{{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega)} \cap A\neq \emptyset\right\}}}
{{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega)} \cap B = \emptyset\right\}}}
\frac{ w_{\lambda,z}(\omega)}{{\left\vert\omega\right\vert}}.$$ Define $\mu_{\lambda,z}(A) = \mu_{\lambda,z}(A;\emptyset)$. For singleton sets $\{x\}, \{y\}$, let $\mu_{\lambda,z}(x;y) =
\mu_{\lambda,z}(\{x\};\{y\})$.
For the special case of $\lambda=1$ the next theorem is [@LawlerLimic2010 Proposition 9.5.1].
\[thm:LWW-LM-Rep\] The loop erased $\lambda$-LWW weight on self-avoiding walks can be written in terms of the $\lambda$-LWW loop measure: $$\label{eq:LWW-LM-Rep}
\bar w_{\lambda,z}(\eta) = \sum_{\omega\colon{\mathrm{LE}}(\omega)=\eta}
w_{\lambda,z}(\omega) = z^{{\left\vert\eta\right\vert}} \exp (\mu_{\lambda,z}({\mathrm{range}(\eta)})).$$
Deferred to .
A function $f$ on subsets of ${{\mathbb{Z}}}^{d}$ is said to be *weakly increasing* if $A\subset B$ implies $f(A) \leq
f(B)$, and *weakly decreasing* if $f(A)\geq f(B)$.
\[prop:LWW-LM-Properties\] Assume $z\geq 0$, $\lambda\geq 0$.
1. Let $A,B\subset {{\mathbb{Z}}}^{d}$. Then for any isometry ${\mathcal{R}}$ $$\label{eq:LWW-LM-Symmetries}
\mu_{\lambda,z}({\mathcal{R}} A;{\mathcal{R}} B) = \mu_{\lambda,z}(A;B),$$
2. $\mu_{\lambda,z}(A;B)$ is weakly increasing in $A$ and weakly decreasing in $B$.
The first item follows from the isometry invariance of $w_{\lambda,z}$, which follows from . The second follows as increasing $A$ (decreasing $B$) reduces (increases) the constraints on the set of walks that contribute to the defining sum, and $w_{\lambda,z}(\omega)\geq 0$.
If $\eta = \eta_{1}\circ \eta_{2}$ is self-avoiding then and the definition of the loop measure imply $$\label{eq:LWW-LELWW-Repulsive}
\bar w_{\lambda,z}(\eta) = z^{{\left\vert\eta_{1}\right\vert}} z^{{\left\vert\eta_{2}\right\vert}}
\exp( \mu_{\lambda,z}({\mathrm{range}(\eta_{1})})) \exp
(\mu_{\lambda,z}({\mathrm{range}(\eta_{2})} ; {\mathrm{range}(\eta_{1})})).$$ By the second statement of dropping the constraint in the second loop measure increases the weight, and hence *loop-erased $\lambda$-LWW* is purely repulsive. This enables a lace expansion analysis of $\lambda$-LWW as the two-point functions of $\lambda$-LWW and loop-erased $\lambda$-LWW coincide by . This is done as follows:
- derives a lace expansion for $\lambda$-LWW. This is done by manipulating the identity $$\label{eq:LWW-Intro-X-Gas}
\bar w_{\lambda,z} (\eta) = z^{{\left\vert\eta\right\vert}}{{\mathbbm{1}}_{\left\{\eta\in{{\Omega}_\mathrm{SAW}}\right\}}}
\prod_{X\in{\mathcal{X}}}(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{\ell(X) \cap {\mathrm{range}(\eta)}\neq \emptyset\right\}}}},$$ where $$\begin{aligned}
\label{eq:LWW-Specialization-1}
{\mathcal{X}} &= \cup_{x\in{{\mathbb{Z}}}^{d}}{\left\{ \omega\colon x\to x,
{\left\vert\omega\right\vert}\geq 1\right\}}, \\
\label{eq:LWW-Specialization-2}
\ell(\omega) &= {\mathrm{range}(\omega)}, \\
\label{eq:LWW-Specialization-3}
\alpha_{\omega} &= \exp {\left( \frac{ w_{\lambda,z}(\omega)}
{{\left\vert\omega\right\vert}}\right )} - 1.
\end{aligned}$$ In the condition ${\left\vert\omega\right\vert}\geq 1$ can be relaxed to ${\left\vert\omega\right\vert}>1$ as all closed walks have length at least $2$. Note that $\alpha_{\omega}\geq 0$ for any closed walk $\omega$ as $\lambda\geq 0$, and that the product in converges for $z$ sufficiently small by .
- expresses the results of in terms of $\mu_{\lambda,z}$, as opposed to the variables $\alpha_{\omega}$.
- and prove the convergence of the lace expansion at the critical point. The strategy is based on [@Slade2006].
- Lastly, proves the main theorem after establishing some further estimates on the lace expansion coefficients. The analysis is based on [@MadrasSlade2013].
Before carrying out the arguments outlined above, let us briefly comment on other relevant non-repulsive random walks that have been studied. Ueltschi [@Ueltschi2002] has given a lace expansion analysis of a self-avoiding walk with nearest neighbour attractions; the attraction means his model is not repulsive. The analysis in [@Ueltschi2002] overcomes the lack of repulsion by exploiting the self-avoiding nature of the walk. Implementing this idea requires technical assumptions that (i) the attraction is sufficiently weak and (ii) the self-avoiding walk can take steps of unbounded range. A second non-repulsive model that has been studied is excited random walk: the analysis of this model in [@vdHofstadHolmes2012] is essentially a lace expansion analysis. These results have a somewhat different flavour as the walk being studied has non-zero speed. Roughly speaking, the lack of repulsion is overcome by using the transience of the walk in the excited direction.
Notation and Conventions {#sec:LWW-Conventions}
------------------------
Let ${{\mathbbm{1}}_{\left\{A\right\}}}$ denote the indicator function of a set $A$. For notational ease we will occasionally also make use of the Kronecker delta $\delta_{x,y} = {{\mathbbm{1}}_{\left\{x=y\right\}}}$. The *single step distribution* $D(x)$ is defined by $D(x) = {\left\vert\Omega\right\vert}^{-1}
{{\mathbbm{1}}_{\left\{x\sim 0\right\}}}$, where we recall that ${\left\vert\Omega\right\vert}=2d$ and $x\sim 0$ indicates that $x$ is a nearest neighbour of $0$ in ${{\mathbb{Z}}}^{d}$.
The Fourier transform $\hat f\colon {{\left[-\pi,\pi\right ]}}^{d}\to {{\mathbb{C}}}$ of a function $f$ on ${{\mathbb{Z}}}^{d}$ is defined by $$\label{eq:LWW-FT}
\hat f(k) = \sum_{x\in {{\mathbb{Z}}}^{d}} e^{ik\cdot x}f(x)$$
Subwalks of a walk $\omega$ can be identified by specifying the subinterval that defines them. That is, for $0\leq a <b\leq
{\left\vert\omega\right\vert}$ define $\omega{\left[a,b\right ]} = (\omega_{a}, \dots,
\omega_{b})$, $\omega{\left[a,b\right )} = \omega{\left[a,b-1\right ]}$, $\omega{\left(a,b\right ]} =
\omega{\left[a+1,b\right ]}$, and $\omega{\left(a,b\right )} = \omega{\left[a+1,b-1\right ]}$. By convention ${\left[a,a\right ]}=\{a\}$, so $\omega{\left[a,a\right ]}=\omega_{a}$. To avoid some ungainly notation, let $\omega{\left[a{\colon\!}\right ]} =
\omega{\left[a,{\left\vert\omega\right\vert}\right ]}$.
By convention $\inf\emptyset=\infty$ and $\sup\emptyset =
-\infty$. The set $\{0,1,\dots, n\}$ will be denoted ${\left[n\right ]}$, and ${\left[\omega\right ]}$ will denote ${\left[{\left\vert\omega\right\vert}\right ]}$ when $\omega$ is a walk. Further, $c$ will denote a positive constant independent of the dimension $d$ and activity $z$; the precise value of $c$ may change from line to line.
A Lace Expansion {#sec:LWW-Lace}
================
\[rem:LWW-Memory-Expansion\] The lace expansion presented here can be derived by other means, e.g., the technique developed for self-interacting walks in [@vdHofstadHolmes2012].
Graphical Representations {#sec:LWW-Graph-Representation}
-------------------------
This section provides a representation of the weight $\bar
w_{\lambda,z}$ in terms of graphs. The utility of such a representation is that it allows recursive identities to be derived.
### Graph Representation of Self Avoidance {#sec:LWW-Timelike}
\[def:LWW-Graph\] Let $A$ be a set. For $s,t\in A$, $s\neq t$, the pair $\{s,t\}
\equiv st$ is called an *edge*. A *graph* $\Gamma$ on $A$ is a set of edges.
The condition $\omega\in{{\Omega}_\mathrm{SAW}}$ that a walk $\omega$ is self-avoiding can be expressed using graphs. $$\label{eq:LWW-Timelike}
{{\mathbbm{1}}_{\left\{\omega\in {{\Omega}_\mathrm{SAW}}\right\}}} = \prod_{0\leq s<t\leq{\left\vert\omega\right\vert}}
{{\mathbbm{1}}_{\left\{\omega_{s}\neq\omega_{t}\right\}}}
= \prod_{0\leq s<t\leq {\left\vert\omega\right\vert}}(1-{{\mathbbm{1}}_{\left\{\omega_{s}=\omega_{t}\right\}}})
= \sum_{\Gamma} \prod_{st\in\Gamma}
{\left(-{{\mathbbm{1}}_{\left\{\omega_{s}=\omega_{t}\right\}}}\right )},$$ The sum in the rightmost term is over all graphs $\Gamma$ on ${\left[\omega\right ]}$, where we recall the definition ${\left[\omega\right ]} =
{\left[{\left\vert\omega\right\vert}\right ]} = \{0,1,\dots, {\left\vert\omega\right\vert}\}$.
### Hypergraph Decomposition of LWW Weight {#sec:LWW-Spacelike}
A representation of the weight on self-avoiding walks due to the product over ${\mathcal{X}}$ in is less straightforward than the graph representation of self-avoidance. This is because the condition of self-avoidance involves two distinct times, while the condition that ${\mathrm{range}(\omega)}\cap \ell(X)\neq\emptyset$ involves many distinct times. This issue can be handled by using inclusion-exclusion. A convenient way to represent the results of inclusion-exclusion is in terms of hypergraphs. We emphasize, however, that no prior knowledge of hypergraphs are needed to understand the expansion – they are only used as a bookkeeping instrument.
\[lem:LWW-Hypergraph-Representation\] Let $\omega$ be a walk, and let $X\in {\mathcal{X}}$. Then $$\begin{aligned}
\label{eq:LWW-Hypergraphs-General-1}
(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{\ell(X)\cap
{\mathrm{range}(\omega)}\neq\emptyset\right\}}}}
&= \mathop{\prod_{J\subset {\left[\omega\right ]}\colon {\left\vertJ\right\vert}\geq 1}}(1+{F_{J,X}}(\omega)),\end{aligned}$$ where $$\label{eq:LWW-Hypergraphs-General-3}
{F_{J,X}}(\omega) \equiv
\begin{cases}
\phantom{-}\alpha_{X} \prod_{j\in J}{{\mathbbm{1}}_{\left\{\omega_{j}
\in \ell(X)\right\}}},
& {\left\vertJ\right\vert}\in 2{{\mathbb{N}}}+ 1\\
-\frac{\alpha_{X}}{1+\alpha_{X}} \prod_{j\in
J}{{\mathbbm{1}}_{\left\{\omega_{j} \in \ell(X)\right\}}} & {\left\vertJ\right\vert}\in 2{{\mathbb{N}}}.
\end{cases}$$ In $0$ is included in $2{{\mathbb{N}}}$.
Apply inclusion-exclusion to the condition $\ell(X)\cap
{\mathrm{range}(\omega)}\neq\emptyset$: $$\begin{aligned}
{{\mathbbm{1}}_{\left\{\ell(X)\cap {\mathrm{range}(\omega)}\neq\emptyset\right\}}} &=
1 - {{\mathbbm{1}}_{\left\{\ell(X)\cap {\mathrm{range}(\omega)}=\emptyset\right\}}} \\
&= 1 - \prod_{j=0}^{{\left\vert\omega\right\vert}}(1- {{\mathbbm{1}}_{\left\{\omega_{j}\in\ell(X)\right\}}}) \\
&= \mathop{\sum_{J\subset{\left[\omega\right ]}\colon{\left\vertJ\right\vert}\geq 1}}
(-1)^{{\left\vertJ\right\vert}+1} \prod_{j\in
J}{{\mathbbm{1}}_{\left\{\omega_{j}\in\ell(X)\right\}}}.
\end{aligned}$$ Then $$\begin{aligned}
(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{\ell(X)\cap
{\mathrm{range}(\omega)}\neq\emptyset\right\}}}} &= \prod_{J\subset
{\left[\omega\right ]}\colon {\left\vertJ\right\vert}\geq 1}
(1+\alpha_{X})^{(-1)^{{\left\vertJ\right\vert}+1}\prod_{j\in
J}{{\mathbbm{1}}_{\left\{\omega_{j}\in \ell(X)\right\}}}} \\
&= \mathop{\prod_{J\subset {\left[\omega\right ]}\colon {\left\vertJ\right\vert}\geq 1}}(1+{F_{J,X}}(\omega)),
\end{aligned}$$ where the weights ${F_{J,X}}$ arise from the identities $(1+\alpha)^{-{\mathbbm{1}}_{A}} = 1 - \frac{\alpha}{1+\alpha}{\mathbbm{1}}_{A}$ and $(1+\alpha)^{{\mathbbm{1}}_{A}} = 1+ \alpha{\mathbbm{1}}_{A}$.
\[def:LWW-Hypergraph\] A *hypergraph* $G$ on a countable set $A$ is a (possibly empty) finite subset of $A$. Each element of $G$ is called a *hyperedge*.
To connect this definition with the more familiar notion of a graph, consider the case when $A$ is $V^{2}\setminus \{ \{x,x\} \mid x\in
V\}$ for $V$ a finite set. A subset of $A$ is then the edge set of a graph on $V$.
If $F(a)$ is an indeterminate associated to the hyperedge $a$ then, as formal power series, $$\label{eq:LWW-Hypergraph-Product}
\prod_{a\in A}(1+ F(a))= \sum_{G} \prod_{a\in G}F(a),$$ where the sum on the right-hand side of is over all hypergraphs on $A$. In what follows we perform calculations in the sense of formal power series. We will ultimately find that our final expressions have interpretations as convergent objects.
To represent the product over ${\mathcal{X}}$ in in terms of hypergraphs take $A$ in to be $(2^{{\left[n\right ]}}\setminus
\emptyset) \times {\mathcal{X}}$. If $a\in A$ then $a=(J,X)$ for $J$ a non-empty subset of ${\left[n\right ]}$ and $X\in {\mathcal{X}}$. Define $F(a) =
F_{J,X}$. This implies $$\begin{aligned}
\label{eq:LWW-Hypergraph-Product-1}
\prod_{X\in {\mathcal{X}}}(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{\ell(X)\cap
{\mathrm{range}(\omega)}\neq \emptyset\right\}}}} &= \prod_{X\in {\mathcal{X}}} \mathop{\prod_{J\subset {\left[\omega\right ]}\colon {\left\vertJ\right\vert}\geq 1}}(1+F_{J,X}(\omega)) \\
\label{eq:LWW-Hypergraph-Product-2}
&= \sum_{G}\prod_{(J,X)\in G}F_{J,X}(\omega),\end{aligned}$$ where the sum in is over all hypergraphs.
The next corollary is a useful hypergraph representation of the weight carried by a subwalk.
\[cor:LWW-Remainder\] Let $\omega$ be an $n$-step walk. For $k\leq n$, $X\in {\mathcal{X}}$, $$\label{eq:LWW-Remainder}
(1+\alpha_{X})^{ {{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega{\left[0,k\right )})} \cap
\ell(X) = \emptyset\right\}}} {{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega{\left[k,n\right ]})}
\cap \ell(X)\neq \emptyset\right\}}}} =
\mathop{\prod_{J\subset {\left[n\right ]}\colon {\left\vertJ\right\vert}\geq 1,}}_{J\cap
{\left[k,n\right ]}\neq\emptyset}(1+{F_{J,X}}(\omega)).$$
Observe that ${{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega{\left[k,n\right ]})}\cap \ell(X)\neq
\emptyset\right\}}}{{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega{\left[0,k\right )})} \cap \ell(X) =
\emptyset\right\}}} $ can be rewritten as ${{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega)}\cap \ell(X)\neq\emptyset\right\}}} -
{{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega{\left[0,k\right )})}\cap \ell(X)\neq
\emptyset\right\}}}$. The corollary follows by applying to both $\omega$ and $\omega{\left[0,k\right )}$ and dividing.
### The Full Graphical Representation {#sec:LWW-Hypergraph-Representation}
Let $J\subset {\left[n\right ]}$ be non-empty and let $X$ denote an element of ${\mathcal{X}}\cup\{\emptyset\}$. A pair $(J,X)$ is *timelike* if ${\left\vertJ\right\vert}=2$, $X=\emptyset$. A pair is *spacelike* if $X\neq\emptyset$.
The use of spacelike and timelike as labels has no relation to the use of these terms in physics. Extend the definition of ${F_{J,X}}$ by defining ${F_{J,X}}$ via if $(J,X)$ is spacelike, and defining ${F_{st,\emptyset}} =
-{{\mathbbm{1}}_{\left\{\omega_{s}=\omega_{t}\right\}}}$ for timelike hyperedges $(st,\emptyset)$. Let ${{\mathcal{G}}}{\left[a,b\right ]}$ denote the set of hypergraphs whose hyperedges are pairs $(J,X)$ such that (i) $X\in{\mathcal{X}}\cup\{\emptyset\}$, (ii) $J\subset \{a,a+1,\dots, b\}$, ${\left\vertJ\right\vert}\geq
1$, and (iii) $X=\emptyset$ implies ${\left\vertJ\right\vert}=2$. Define ${{\mathcal{G}}}(n)\equiv
{{\mathcal{G}}}{\left[0,n\right ]}$. The decompositions of imply that $$\begin{aligned}
\label{eq:LWW-Hypergraph-Interaction}
c_{n}(0,x) &=\mathop{\sum_{\omega\colon 0 \to x}}_{{\left\vert\omega\right\vert}=n}
{{\mathbbm{1}}_{\left\{\omega\in{{\Omega}_\mathrm{SAW}}\right\}}} \prod_{X\in{\mathcal{X}}}(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{ \ell(X)\cap
{\mathrm{range}(\omega)} \neq \emptyset\right\}}}} \\
\label{eq:LWW-Hypergraph-Interaction-1}
&= \mathop{\sum_{\omega\colon 0 \to x}}_{{\left\vert\omega\right\vert}=n}
\sum_{G\in{{\mathcal{G}}}(n)} \prod_{(J,X)\in G}{F_{J,X}}(\omega).\end{aligned}$$
Lace Graphs {#sec:LWW-Laces}
-----------
\[def:LWW-Lace-Connectedness\] A graph $\Gamma$ on ${\left[a,b\right ]}$ is *(lace) connected* if (i) $b>a+1$, (ii) for all $a<j<b$ there is an edge $st\in\Gamma$ such that $s<j<t$ and (iii) there are $j_{1},j_{2}$ such that $aj_{1}$, $j_{2}b\in\Gamma$. Let ${G}{\left[a,b\right ]}$ (resp. ${G^{c}}{\left[a,b\right ]}$) denote the set of graphs (resp. lace connected graphs) on ${\left[a,b\right ]}$.
We caution the reader that the definition of lace connectedness is not the same as the graph theoretical definition of connectedness. The adjective lace will be dropped in what follows, as the graph-theoretic notion of connectedness is not relevant in this section.
A function $w$ on graphs on the discrete interval ${\left[a,b\right ]}$ is called *multiplicative* if $w(G) = \prod_{st\in E(G)}w(st)$. Note that a multiplicative function on graphs assigns the empty graph weight $1$. If $w$ is a multiplicative function on graphs on ${\left[a,b\right ]}$ define $$K{\left[a,b\right ]} = \sum_{G\in{G}{\left[a,b\right ]}}w(G), \qquad J{\left[a,b\right ]} =
\sum_{G\in{G^{c}}{\left[a,b\right ]}} w(G),$$ and let $K{\left[a,b\right ]}=J{\left[a,b\right ]}=0$ if $a>b$. For $a<b$ the observation that a graph on ${\left[a,b\right ]}$ either contains $a$ in a connected subgraph or does not and the definition of connectedness imply $$\label{eq:LWW-Connectedness-Recursion}
K{\left[a,b\right ]} = K{\left[a,a+1\right ]}K{\left[a+1,b\right ]} + \sum_{j\geq 2}J{\left[a,a+j\right ]} K{\left[a+j,b\right ]}.$$
\[def:lace-graph\] A graph is a *lace graph* if the removal of any edge results in a graph which is not connected.
A *labelled graph* is a graph where each edge is given a label of either spacelike or timelike; a labelled graph may contain both the edge $(st,\mathrm{spacelike})$ and the edge $(st,\mathrm{timelike})$. The definition of a lace graph applies to labelled graphs as the notion of connectedness does not depend on the labelling. The following procedure associates a unique lace $L_{\Gamma}$ to each labelled connected graph $\Gamma$ on ${\left[a,b\right ]}$. The labelled lace $L_{\Gamma}$ consists of the set of edges $s_{i}t_{i}$ along with their labellings, where $s_{i}t_{i}$ are determined by $s_{1}=a$, $t_{1} = \max \{v \colon s_{1}v \in \Gamma\}$, $t_{i+1} = \max\{ v \colon \exists\, \textrm{$s<t_{i}$ such that
$sv\in \Gamma$}\}$, and $s_{i+1} = \min \{s\colon st_{i+1}\in \Gamma\}$. If this does not uniquely specify $s_{i}t_{i}$ then $s_{i}t_{i}$ is chosen to have the label spacelike. The procedure terminates when $t_{i+1}=b$. See .
(0,0) – (26,0); (0,0) to\[out=90,in=90\] (10,0); (2,0) to\[out=90,in=90\] (5,0); decorate \[decoration=[zigzag, segment length=5, amplitude = .8]{}\] [(4,0) to\[out=90,in=90\] (6,0)]{}; (8,0) to\[out=90,in=90\] (15,0); decorate \[decoration=[zigzag, segment length=5, amplitude = .8]{}\] [(8,0) to\[out=45,in=135\] (15,0)]{}; (12,0) to\[out=90,in=90\] (25,0); (13,0) to\[out=90,in=90\] (19,0); decorate \[decoration=[zigzag, segment length=5, amplitude = .8]{}\] [(20,0) to\[out=90,in=90\] (22,0)]{}; (21,0) to\[out=90,in=90\] (23,0); decorate \[decoration=[zigzag, segment length=5, amplitude = .8]{}\] [(24,0) to\[out=90,in=90\] (26,0)]{};
(0,0) – (26,0); (0,0) to\[out=90,in=90\] (10,0); (8,0) to\[out=90,in=90\] (15,0); (12,0) to\[out=90,in=90\] (25,0); decorate \[decoration=[zigzag, segment length=5, amplitude = .8]{}\] [(24,0) to\[out=90,in=90\] (26,0)]{};
A labelled edge $st$ is said to be *compatible* with a lace $L$ if $L_{L\cup\{st\}} = L$, i.e., if the addition of the labelled edge $st$ does not alter the outcome of the above algorithm. Let ${\mathcal{L}}{\left[a,b\right ]}$ denote the set of labelled lace graphs on ${\left[a,b\right ]}$ and ${\mathcal{C}}(L)$ the set of compatible labelled edges for a lace $L\in{\mathcal{L}}$.
\[lem:LWW-Lace-Prescription\] Let $w$ be a weight on labelled edges $st$. Then $$\sum_{\Gamma\in {G^{c}}{\left[a,b\right ]}}\prod_{st\in \Gamma}w(st) =
\sum_{L\in{\mathcal{L}}{\left[a,b\right ]}} \prod_{st\in L}
w(st)\prod_{s^{\prime}t^{\prime}\in{\mathcal{C}}(L)}(1+w(s^{\prime}t^{\prime})),$$ where the sums are over labelled connected graphs and labelled laces, respectively.
The proof is the same as the proof for unlabelled graphs, see [@BrydgesSpencer1985], [@Slade2006], or [@Zeilberger1997].
\[rem:LWW-Connectedness\] is *not* the definition of lace connectedness typically used for self-avoiding walk, as the graph consisting of the single edge $\{a,a+1\}$ is not being considered connected. This change is entirely cosmetic for self-avoiding walk as graphs consisting of a single edge $\{a,a+1\}$ do not contribute.
Laces and Hypergraphs {#sec:LWW-Hyperlaces}
---------------------
This section obtains an analogue of for hypergraphs.
### Recursion Relation for Hypergraphs {#sec:LWW-Recursion}
\[def:LWW-Graph-Spans\] For a hyperedge $(J,X)$ define ${\mathrm{span}}{(J,X)} = \{\min J, \max
J\}$. If $(J,X)$ is spacelike label ${\mathrm{span}}{(J,X)}$ spacelike, and if $(J,X)$ is timelike label ${\mathrm{span}}{(J,X)}$ timelike. If $G$ is a hypergraph the labelled graph $\Gamma_{G}$ with labelled edges $\{{\mathrm{span}}{(J,X)} \mid (J,X)\in G\}$ will be called the *graph of spans* of $G$.
A hypergraph $G$ on ${\left[a,b\right ]}$ is *connected* if the graph of spans of $G$ is connected on ${\left[a,b\right ]}$. The set of connected hypergraphs on ${\left[a,b\right ]}$ is denoted ${{\mathcal{G}}}^{c}{\left[a,b\right ]}$.
The objects $\alpha$ and $\alpha_{0}$ in the next definition have interpretations in terms of the loop measure, but for now should be thought of as convenient shorthand.
\[def:LWW-Renormalized-Activity\] Let ${\mathcal{X}}_{0} = \{X\in{\mathcal{X}} \mid 0\in\ell(X)\}$ and let $y\in\Omega$ be a vertex adjacent to $0$. Define $$\label{eq:LWW-RA}
\alpha_{0} = \alpha_{0}({\mathcal{X}}) = \prod_{X\in{\mathcal{X}}_{0}}(1+\alpha_{X}), \qquad
\alpha = \alpha({\mathcal{X}}) = \prod_{X\in {\mathcal{X}}_{0}}(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{y\notin \ell(X)\right\}}}},$$ That $\alpha$ is independent of the vertex $y\in\Omega$ chosen follows from the isometry invariance of the loop-weighted walk weight.
By translation invariance $\alpha_{0}$ is also given by the product over $X\in {\mathcal{X}}$ such that any single fixed vertex is contained in $\ell(X)$, and hence $$\label{eq:LWW-Alpha0-Graphs}
\alpha_{0} = \sum_{G\in {{\mathcal{G}}}{\left[1,1\right ]}} w(G),$$ where $w(G) = \prod_{(J,X)\in G}{F_{J,X}}$. Using and the definition of connectedness for hypergraphs implies that for $n\geq 1$ $$\begin{gathered}
\label{eq:LWW-Hypergraph-Recursion-A}
\sum_{G\in
{{\mathcal{G}}}{\left[0,n\right ]}}w(G) =
\alpha_{0}^{-1}\sum_{G_{1}\in{{\mathcal{G}}}{\left[0,1\right ]}}
\sum_{G_{2}\in{{\mathcal{G}}}{\left[1,n\right ]}}w(G_{1}) w(G_{2}) \\+
\alpha_{0}^{-1}\sum_{j\geq 2}
\sum_{G_{1}\in{{\mathcal{G}}}^{c}{\left[0,j\right ]}} \sum_{G_{2}\in {{\mathcal{G}}}{\left[j,n\right ]}} w(G_{1}) w(G_{2}).\end{gathered}$$ The factor of $\alpha_{0}^{-1}$ multiplying the first term arises since the hypergraphs $G\in{{\mathcal{G}}}{\left[1,1\right ]}$ are double counted due to being present in both ${{\mathcal{G}}}{\left[0,1\right ]}$ and ${{\mathcal{G}}}{\left[1,n\right ]}$. The factor of $\alpha_{0}^{-1}$ multiplying the second factor arises similarly, due to double counting of the sum over ${{\mathcal{G}}}{\left[j,j\right ]}$; translation invariance implies this is the same as the sum over ${{\mathcal{G}}}{\left[1,1\right ]}$. The next lemma simplifies by computing the sum over ${{\mathcal{G}}}{\left[0,1\right ]}$.
\[lem:LWW-Hypergraph-Recursion\] Fix $n\geq 1$. Then $\sum_{G\in {{\mathcal{G}}}{\left[0,n\right ]}} \prod_{(J,X)\in
G}{F_{J,X}}$ is equal to $$\label{eq:LWW-Hypergraph-Recursion}
\alpha \sum_{G\in{{\mathcal{G}}}{\left[1,n\right ]}}\prod_{(J,X)\in G}{F_{J,X}}
+ \alpha_{0}^{-1}\sum_{j\geq 2}
\sum_{G_{1}\in{{\mathcal{G}}}^{c}{\left[0,j\right ]}}\sum_{G_{2}\in {{\mathcal{G}}}{\left[j,n\right ]}}
\prod_{(J,X)\in G_{1}} {F_{J,X}} \prod_{(J^{\prime},X^{\prime})\in G_{2}}
{F_{J^{\prime},X^{\prime}}}$$
Let $\omega$ be a walk. and imply that $$ \label{eq:LWW-HR-2}
\sum_{G\in{{\mathcal{G}}}{\left[0,1\right ]}}\prod_{(J,X)\in G}{F_{J,X}}(\omega) =
{{\mathbbm{1}}_{\left\{\omega_{0}\neq \omega_{1}\right\}}} \prod_{X\in{\mathcal{X}}}(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{\{\omega_{0},\omega_{1}\}\cap \ell(X)\neq \emptyset\right\}}}}.$$ The constraint that $\omega_{0}\neq \omega_{1}$ is irrelevant as $\omega_{j+1}\neq \omega_{j}$ for any walk. Using the representation of $\alpha_{0}$ in gives $$\label{eq:LWW-HR-3}
\frac{ \sum_{G\in{{\mathcal{G}}}{\left[0,1\right ]}} \prod_{(J,X)\in
G}{F_{J,X}}(\omega)} {\sum_{G\in {{\mathcal{G}}}{\left[1,1\right ]}}
\prod_{(J,X)\in G}{F_{J,X}} (\omega)} = \prod_{X\in {\mathcal{X}}}
(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{\omega_{1}\in\ell(X)\right\}}}{{\mathbbm{1}}_{\left\{\omega_{0}\notin
\ell(X)\right\}}}},$$ and this last quantity is $\alpha$ by . Using and to rewrite gives the claim.
### Laces for Hypergraphs and Weights on Lace Edges {#sec:LWW-Hypergraph-Lace}
The weight $w(G) = \prod {F_{J,X}}$ on hypergraphs can be pushed forward to a weight $w_{\star}^{\omega}(st)$ on labelled graphs; recall that labelled graphs were introduced following . Explicitly, the weight $w_{\star}^{\omega}(st)$ is defined by $$\begin{aligned}
\label{eq:LWW-Pushforward-Weight-1}
w_{\star}^{\omega}(st,\mathrm{timelike}) &\equiv
-{{\mathbbm{1}}_{\left\{\omega_{s}=\omega_{t}\right\}}} \\
\label{eq:LWW-Pushforward-Weight-2}
w_{\star}^{\omega}(st,\mathrm{spacelike}) &\equiv
(1-{{\mathbbm{1}}_{\left\{\omega_{s}=\omega_{t}\right\}}})\!\!\!\sum_{\{(J_{i},X_{i})\}\colon
{\mathrm{span}}(J_{i},X_{i})=st} \prod_{i}{F_{J_{i},X_{i}}}(\omega).\end{aligned}$$ The sum for a spacelike edge in is over all non-empty collections of hyperedges, each of whose span is the labelled edge $(st,\mathrm{spacelike})$. The factor $(1-{{\mathbbm{1}}_{\left\{\omega_{s}=\omega_{t}\right\}}})$ accounts for the possibility that a timelike hyperedge exists when the edge $st$ is given the label spacelike. Note that this weight neglects hyperedges $(J,X)$ with ${\left\vertJ\right\vert}=1$. For notational ease let ${F_{j,X}} =
{F_{\{j\},X}}$.
\[lem:LWW-Hypergraph-Lace-Lift\] The following identity holds for $a<b$: $$\label{eq:LWW-Hypergraph-Lace}
\sum_{G\in {{\mathcal{G}}}^{c}{\left[a,b\right ]}}\prod_{(J,X)\in G}{F_{J,X}} =
\mathop{\prod_{a\leq j\leq b}}_{X\in {\mathcal{X}}}(1+{F_{j,X}})
\sum_{L\in {\mathcal{L}}{\left[a,b\right ]}} \prod_{st\in L}w_{\star}(st)\!\!\!\!\!\!\!
\mathop{\prod_{(J^{\prime},X^{\prime})\colon}}_{{\mathrm{span}}(J^{\prime},X^{\prime})\in {\mathcal{C}}(L)}(1+{F_{J^{\prime},X^{\prime}}}).$$ The left-hand sum is over all connected hypergraphs on ${\left[a,b\right ]}$, while the right-hand sum is over labelled laces.
Apply with the weight $w_{\star}$, and take the product of this equation with the first term on the right-hand side of : $$\mathop{\prod_{a\leq j\leq b}}_{X\in {\mathcal{X}}}(1+{F_{j,X}})\!\!\! \sum_{\Gamma\in {G^{c}}{\left[a,b\right ]}}\prod_{st\in
\Gamma}w_{\star}(st) = \mathop{\prod_{a\leq j\leq b}}_{X\in {\mathcal{X}}}(1+{F_{j,X}})
\!\!\!\sum_{L\in{\mathcal{L}}{\left[a,b\right ]}} \prod_{st\in L}
w_{\star(st)}\!\!\!\!\!\!\! \prod_{s^{\prime}t^{\prime}\in{\mathcal{C}}(L)}(1+w_{\star}(s^{\prime}t^{\prime})).$$ Expanding the product over connected labelled graphs with weight $w_{\star}$ gives the left-hand side of as hyperedges of the form $(\{j\},X)$ play no role in connectivity, and for each $st$ the weight $w_{\star}$ is a sum of the possible collections of hyperedges whose span is $st$. Similarly, $1+w_{\star}(ij)$ for $ij\in {\mathcal{C}}(L)$ can be written in the product form used above, giving the right-hand side of .
The next definition and lemma simplifies the sum over laces in by resumming the contributions to the product over $st\in L$.
\[def:LWW-Walk-I2PF\] For $0\leq s<t$ define $I^{\omega}_{{\mathcal{X}}}(s,t) = 1$ if $\omega_{s}=\omega_{t}$, and if $\omega_{s}\neq
\omega_{t}$ define $$\label{eq:LWW-Walk-I2PF}
I^{\omega}_{{\mathcal{X}}}(s,t) =
1-\prod_{X\in{\mathcal{X}}}{\left(1 -
\frac{\alpha_{X}}{1+\alpha_{X}}{{\mathbbm{1}}_{\left\{\omega_{s}\in \ell(X)\right\}}}
{{\mathbbm{1}}_{\left\{\omega_{t}\in \ell(X)\right\}}} {{\mathbbm{1}}_{\left\{\ell(X)\cap
{\mathrm{range}(\omega{\left(s,t\right )})}=\emptyset\right\}}}\right )}.$$
\[lem:LWW-Span-Resummation\] Let $st$ be an edge. Then $$\label{eq:LWW-Span-Resummation-1}
w_{\star}^{\omega}(st,\mathrm{spacelike}) +
w_{\star}^{\omega}(st,\mathrm{timelike})
= -I^{\omega}_{{\mathcal{X}}}(s,t)$$
The case $\omega_{s}=\omega_{t}$ corresponds to the timelike edge. Consider the spacelike term. As any non-empty collection of spacelike hyperedges $\{(J_{i},X_{i})\}$ may be chosen in the equation can be rewritten as $$w_{\star}^{\omega}(st,\mathrm{spacelike}) =
(1-{{\mathbbm{1}}_{\left\{\omega_{s}=\omega_{t}\right\}}}){\left[
\mathop{\prod_{(J,X)\colon}}_{{\mathrm{span}}(J,X)=st}(1+{F_{J,X}}(\omega)) - 1\right ]}.$$ A hyperedge with span $st$ and second element $X$ is equivalent to a possibly empty subset $J$ of ${\left(s,t\right )}$. Using ${F_{J\cup\{ab\},X}} =
{{\mathbbm{1}}_{\left\{\omega_{a}\in\ell(X)\right\}}}
{{\mathbbm{1}}_{\left\{\omega_{b}\in\ell(X)\right\}}} {F_{J,X}}$ gives $$w_{\star}^{\omega}(st,\mathrm{spacelike}) =
{{\mathbbm{1}}_{\left\{\omega_{s}\neq\omega_{t}\right\}}}{\left(\prod_{X\in{\mathcal{X}}}\prod_{J\subset {\left(s,t\right )}}{\left(1+{{\mathbbm{1}}_{\left\{\omega_{s}\in
\ell(X)\right\}}}{{\mathbbm{1}}_{\left\{\omega_{t}\in \ell(X)\right\}}} {F_{J,X}}(\omega)\right )} - 1\right )},$$ where we recall that ${F_{\emptyset,X}}(\omega) =
-\alpha_{X}(1+\alpha_{X})^{-1}$. Putting the condition that $\omega_{s}$ and $\omega_{t}$ are in $\ell(X)$ into the product, separating the case $J=\emptyset$, and then applying yields $$\begin{aligned}
w_{\star}^{\omega}(st,\mathrm{spacelike}) &=
{{\mathbbm{1}}_{\left\{\omega_{s}\neq\omega_{t}\right\}}}{\left(\mathop{\prod_{X\in{\mathcal{X}}\colon}}_{\omega_{s},\omega_{t}\in\ell(X)}{\left[ (1-\frac{\alpha_{X}}{1+\alpha_{X}})
\mathop{\prod_{J\subset {\left(s,t\right )}}}_{{\left\vertJ\right\vert}\geq 1} (1+{F_{J,X}}(\omega))\right ]} - 1\right )} \\
&={{\mathbbm{1}}_{\left\{\omega_{s}\neq\omega_{t}\right\}}}{\left(
\mathop{\prod_{X\in{\mathcal{X}}\colon}}_{\omega_{s},\omega_{t}\in
\ell(X)}(1+\alpha_{X})^{-{{\mathbbm{1}}_{\left\{
{\mathrm{range}(\omega{\left(s,t\right )})}\cap\ell(X)=0\right\}}}} - 1\right )},
\end{aligned}$$ which is the second half of .
The Lace Expansion Equation {#sec:LWW-X-Gas-Expansion}
---------------------------
This section shows how the recursion for the interaction expressed in translates into a recursion for the $c_{n}$. By summing the resulting recursion over $n$ the desired lace expansion is obtained.
### Lace Expansion Equation {#sec:LWW-Expansion-Equation}
For $m\geq 2$ define $\pi^{(N)}_{m}(x)$ to be $$\label{eq:LWW-Pi-Definition}
z^{m} \alpha_{0}^{-1}
\mathop{\sum_{\omega\colon 0\to x}}_{{\left\vert\omega\right\vert}=m}
\sum_{L\in {\mathcal{L}}^{(N)}{\left[0,m\right ]}}
{\left(\prod_{st\in L} I^{\omega}_{{\mathcal{X}}}(s,t)\right )}
\prod_{{\mathrm{span}}(J,X)\in{\mathcal{C}}(L)}(1+{F_{J,X}}(\omega)) \mathop{\prod_{a\leq j\leq
b}}_{X^{\prime}\in {\mathcal{X}}}(1+{F_{j,X^{\prime}}}(\omega)),$$ where ${\mathcal{L}}^{(N)}{\left[0,m\right ]}$ is the set of laces with $N$ edges on the interval ${\left[0,m\right ]}$. Let $\pi_{m}$ denote $\sum_{N\geq 1} (-1)^{N}
\pi_{m}^{(N)}$. Define $c_{m}=0$ for $m<0$. combined with imply $$\label{eq:LWW-Lace-1}
z^{n}c_{n}(0,x) =
\begin{cases}
z\alpha\sum_{y\sim 0}z^{n-1}c_{n-1}(y,x) + \sum_{j\geq
2}\sum_{y}\pi_{j}(y)z^{n-j}c_{n-j}(y,x) & n \geq 1 \\
\alpha_{0}\delta_{0,x} & n=0.
\end{cases}$$ Let $G_{z}(x) = \sum_{n}z^{n}c_{n}(0,x)$. Summing over $n$, using the translation invariance of $G_{z}(x)$, and taking the Fourier transform yields $$\label{eq:LWW-Lace-2}
\hat G_{z}(k) = \alpha_{0} + \alpha z{\left\vert\Omega\right\vert}\hat D(k) \hat
G_{z}(k) + \hat \Pi_{z}(k) \hat G_{z}(k),$$ where $\Pi_{z}(x) = \sum_{m\geq 2}\pi_{m}(x)$.
The next two sections give expressions for $\pi_{m}^{(N)}(x)$ in terms of the quantities $\alpha_{X}$.
### Walk Representation of $\pi^{(N)}_{m}(x)$ for $N=1$ {#sec:LWW-X-Gas-pi-1}
If $N=1$ the lace consists of a single edge $0m$. If $x=0$ then $\omega_{0}=\omega_{m}$, $I^{\omega}_{{\mathcal{X}}}(0,m)=1$, and $$\label{eq:LWW-X-Gas-pi-1-timelike}
\pi^{(1)}_{m}(0) = z^{m}\alpha_{0}^{-1} \mathop{\sum_{\omega\colon 0
\to 0}}_{{\left\vert\omega\right\vert}=m} {{\mathbbm{1}}_{\left\{\omega\in{{\Omega}_\mathrm{SAP}}\right\}}}
\prod_{X}(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega)}\cap \ell(X) \neq\emptyset\right\}}}}.$$ If $x\neq 0$ the set of incompatible hyperedges are those that contain both $0$ and $m$. Let $m_{1} = m-1$. implies that for $\omega\in {{\Omega}_\mathrm{SAW}}$ $$\label{eq:Edit-1}
\prod_{(J,X)\in {\mathcal{C}}(0m)}(1+{F_{J,X}}(\omega)) =
\prod_{X\in {\mathcal{X}}} (1+\alpha_{X})^{ {{\mathbbm{1}}_{\left\{ {\mathrm{range}(\omega)}\cap
\ell(X)\neq\emptyset\right\}}} +{\mathbbm{1}}_{A}}$$ where $$\label{eq:Edit-2}
{\mathbbm{1}}_{A} = {{\mathbbm{1}}_{\left\{\omega_{0}\in \ell(X)\right\}}}
{{\mathbbm{1}}_{\left\{\omega_{m}\in \ell(X)\right\}}} {{\mathbbm{1}}_{\left\{
{\mathrm{range}(\omega{\left[1,m_{1}\right ]})} \cap \ell(X) = \emptyset\right\}}},$$ while if $\omega$ is not self-avoiding the right-hand side of is zero. To see these two claims, use to compute the products over hyperedges $(J,X)$ with (i) $J\subset {\left[1,m_{1}\right ]}$, (ii) $J\subset {\left[1,m\right ]}$ with $m\in J$, and (iii) $J\subset {\left[0,m_{1}\right ]}$ with $0\in J$. The product over compatible hyperedges is the product of these terms. The definition of $I^{\omega}_{{\mathcal{X}}}(0,m)$ when $\omega_{m}=x\neq 0$ then gives a formula for $\pi^{(1)}_{m}(x)$: $$\begin{aligned}
\label{eq:LWW-X-Pi-1}
\pi^{(1)}_{m}(x) = z^{m}\alpha_{0}^{-1}
&\mathop{\sum_{\omega\colon 0 \to x}}_{{\left\vert\omega\right\vert}=m}
{{\mathbbm{1}}_{\left\{\omega\in {{\Omega}_\mathrm{SAW}}\right\}}}
\prod_{X}(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega)} \cap
\ell(X)\neq \emptyset\right\}}}} \\
&{\left( \prod_{X\in {\mathcal{X}}}(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{\omega_{0}\in \ell(X)\right\}}}
{{\mathbbm{1}}_{\left\{\omega_{m}\in \ell(X)\right\}}}
{{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega{\left[1,m_{1}\right ]})}
\cap \ell(X) = \emptyset\right\}}}} - 1\right )}.\end{aligned}$$
### Walk Representation of $\pi^{(N)}_{m}(x)$ for $N\geq 2$ {#sec:LWW-X-Gas-pi-N}
For $N\geq 2$ the central observation is that the edges of a lace on the discrete interval ${\left[a,b\right ]}$ divides the interval ${\left[a,b\right ]}$ into $2N-1$ subintervals, see .
\[def:LWW-Valid\] Let $m\in {{\mathbb{N}}}$. A vector $\vec m$ with components $m_{1}, \dots,
m_{2N-1}$ is called *valid* if (i) $m_{1}\geq 1$, $m_{2N-1}\geq
1$, and $m_{2j}\geq 1$ for $1\leq j \leq N-1$, (ii) $m_{2j+1}\geq 0$ for $1\leq j\leq N-1$, and (iii) $\sum m_{i} = m$.
The lengths of the subintervals determined by a lace form valid vector $\vec m$. The restrictions on which $m_{i}$ are strictly positive arise from the definition of connectedness, see [@Slade2006 Section 3.3] for more details. The subintervals are given by $$\label{eq:LWW-UB-N.1}
\bar I_{1} = {\left[0,m_{1}\right ]}, \quad \bar I_{2} = {\left[m_{1},
m_{1}+m_{2}\right ]},\dots,
\bar I_{2N-1} = {\left[m_{1} + \dots m_{2N-2}, m_{1} + \dots m_{2N-1}\right ]}.$$ To each interval $\bar I_{k}$ associate a walk $\omega^{(k)}$, e.g.$\omega^{(2)} = (\omega_{m_{1}}, \omega_{m_{1}+1}, \dots,
\omega_{m_{1}+m_{2}})$. The walks $\omega^{(k)}$ interact with one another through the compatible edges.
To the $k^{\mathrm{th}}$ interval associate (i) all hyperedges whose span is contained in $\bar I_{k}$ and (ii) all compatible hyperedges $(J,X)$ such that ${\mathrm{span}}(J,X)$ is not contained in $\bar I_{k}$ with $\max
J\in\bar I_{k}$ *and* $\max J \neq \max \bar I_{k}$.
For the subinterval $2N-1$ omit the last condition. That is, if a hyperedge has $\max J=m$ associate this edge to $\bar
I_{2N-1}$. Subintervals $\bar I_{k}$ for $k<2N-1$ are missing hyperedges of the form $(\max \bar I_{k},X)$. Including them, and dividing by $\alpha_{0}$ to correct for this, shows the weight associated to the interval $\bar I_{k}$ is $$\label{eq:LWW-UB-N.2}
\alpha_{0}^{-1}\mathop{\prod_{(J,X)}}_{J\subset \bar I_{K}}(1+{F_{J,X}})
\prod_{{\mathrm{span}}(J^{\prime},X^{\prime})\in {\mathcal{C}}_{k}}(1+{F_{J^{\prime},X^{\prime}}}),$$ where the factor of $\alpha_{0}^{-1}$ for $k=2N-1$ comes from the prefactor $\alpha_{0}^{-1}$ in the definition of $\pi^{(N)}_{m}$.
The last two factors can be evaluated together. A compatible hyperedge must have its minimum index be at least the second index of either $\omega^{(k-2)}$ or $\omega^{(k-3)}$. Suppose the first case; the second is similar. implies the product in forces $\omega^{(k)}$ to be self-avoiding, $\omega^{(k)}$ to avoid $\omega^{(k-1)}$ and $\omega^{(k-2)}{\left[1{\colon\!}\right ]}$, and assigns $\omega^{(k)}$ the weight $$\label{eq:LWW-UB-N.3}
\alpha_{0}^{-1} \prod_{X\in {\mathcal{X}}}
(1+\alpha_{X})^{{{\mathbbm{1}}_{\left\{ {\mathrm{range}(\omega^{(k)})}\cap
\ell(X) \neq \emptyset\right\}}} {{\mathbbm{1}}_{\left\{
{\mathrm{range}(\omega^{(k-2)}\circ\omega^{(k-1)}{\left[1{\colon\!}\right ]})}\cap
\ell(X) = \emptyset\right\}}}},$$
(v0) at (0,0) ; (v1) at (2,0) ; (v0p) at (2,2) ; (v2) at (4,2) ; (v1p) at (4,0) ; (v3) at (6,0) ; (v2p) at (6,2) ; (v4) at (8,2) ; (v3p) at (8,0) ; (v5) at (10,0) ;
at (v1) \[below\] [$x_{1}$]{}; at (v0p) \[above\] [${x_{0}^{\prime}}$]{}; at (v2) \[above\] [$x_{2}$]{}; at (v1p) \[below\] [$x_{1}^{\prime}$]{}; at (v3) \[below\] [$x_{3}$]{}; at (v2p) \[above\] [$x_{2}^{\prime}$]{}; at (v4) \[above\] [$x_{4}$]{}; at (v3p) \[below\] [$x_{3}^{\prime}$]{};
at (v0) \[below left\] [$0$]{}; at (v5) \[below right\] [$x$]{};
(e1) at (1,0) ; (e2) at (2,1) ; (e3) at (3,2) ; (e4) at (4,1) ; (e5) at (5,0) ; (e6) at (6,1) ; (e7) at (7,2) ; (e8) at (8,1) ; (e9) at (9,0) ;
(v0) – (v1) – (v0p) – (v2) – (v1p) – (v3) – (v2p) – (v4) – (v3p) – (v5);
(v0) to\[out=30, in=240 \] (v0p); (v0) to\[out=60, in=210\] (v0p); (v1) to\[out=15, in=165\] (v1p); (v1) to\[out=-15, in=195\] (v1p); (v2) to\[out=15, in=165\] (v2p); (v2) to\[out=-15, in=195\] (v2p); (v3) to\[out=15, in=165\] (v3p); (v3) to\[out=-15, in=195\] (v3p); (v4) to\[out=330,in=120\] (v5); (v4) to\[out=300,in=150\] (v5);
at (e1) \[below\] [$m_{1}$]{}; at (e2) \[right\] [$m_{2}$]{}; at (e3) \[above\] [$m_{3}$]{}; at (e4) \[left\] [$m_{4}$]{}; at (e5) \[below\] [$m_{5}$]{}; at (e6) \[right\] [$m_{6}$]{}; at (e7) \[above\] [$m_{7}$]{}; at (e8) \[left\] [$m_{8}$]{}; at (e9) \[below\] [$m_{9}$]{};
As an explicit formula for $\pi^{(N)}_{m}$ detailing the constraints is unwieldy, let us explain the formula with a brief discussion of the diagrammatic representation of $\pi^{(N)}_{m}$ in . The solid lines represent a subdivision of a walk $\omega$ into subwalks; these subwalks are subject to self-avoidance constraints detailed below. Pairs of zigzag lines represent $I^{\omega}_{{\mathcal{X}}}(t_{i},t_{i+1})$, where $t_{i}$ is the time $x_{i}$ occurs in the walk $\omega = \omega^{(1)} \circ \dots
\circ \omega^{(2N-1)}$. Each walk $\omega^{(i)}$ has length $m_{i}$ and is self-avoiding. Further, each walk $\omega^{(i)}$ avoids some of the previous walks $\omega^{(j)}$ for $j<i$, excluding the endpoint of $\omega^{(i-1)}$. To be precise, $\omega^{(2)}$ avoids $\omega^{(1)}$, $\omega^{(2k+1)}$ avoids $\omega^{(2k-1)}$ and $\omega^{(2k)}$, and $\omega^{(2k+2)}$ avoids $\omega^{(2k-1)}$, $\omega^{(2k)}$, and $\omega^{(2k+1)}$. The walk $\omega^{(j)}$ is weighted by those closed walks in ${\mathcal{X}}$ that do not intersect the $\omega^{(j)}$ which $\omega^{(i)}$ is forbidden to intersect; for example, in the walk $\omega^{(k)}$ is being weighted by all closed walks that do not intersect $\omega^{(k-1)}$ or $\omega^{(k-2)}{\left[1{\colon\!}\right ]}$.
\[rem:LWW-X-Gas-Repulsion\] Since $\alpha_{X}\geq 0$ for each $X$, ignoring the constraint that some closed walks do not weight a subwalk gives an upper bound for the weight on the subwalks $\omega^{(i)}$. Ignoring the constraint of avoiding $\omega^{(j)}$ for some $j<i$ gives a further upper bound on $\pi^{(N)}_{m}(x)$.
Concrete Expressions for the Lace Expansion for $\lambda$-LWW {#sec:LWW-LM-Formulation}
=============================================================
Quantities such as $\alpha_{0}({\mathcal{X}})$ and $I_{{\mathcal{X}}}^{\omega}$ will be written as $\alpha_{0}(\lambda,z)$, $I_{\lambda,z}^{\omega}$ and similarly in what follows. The arguments $\lambda$ and $z$ may be omitted to lighten the notation. As emphasized earlier, $\lambda\geq 0$ and $z\geq 0$ implies $w_{\lambda,z}(\omega)\geq 0$, and hence $\alpha_{\omega}\geq 0$. In particular, by we can obtain upper bounds by ignoring constraints.
\[def:LWW-2-PT\] The *two point function* $G_{\lambda,z}(x,y)$ for $\lambda$-LWW is defined by $$\label{eq:LWW-WLAW-2PT}
G_{\lambda,z}(x,y) = \sum_{\omega\colon x \to y} w_{\lambda,z}(\omega).$$
By and the two-point function $G_{\lambda,z}$ of $\lambda$-LWW is given by the two-point function of self-avoiding walks weighted as in . For future reference we state a reformulation of as a proposition.
\[prop:LWW-Lace-Expansion-WLAW\] $$\label{eq:LWW-Lace-Expansion-WLAW}
\hat G_{\lambda,z}(k) = \frac{\alpha_{0}(\lambda,z)}
{1 - \alpha(\lambda,z){\left\vert\Omega\right\vert}\hat D(k) - \hat\Pi_{\lambda,z}(k)}.$$
To analyze the recursion it will be convenient to rewrite the equation in terms of $w_{\lambda,z}$ and the loop measure $\mu_{\lambda,z}$. The quantities $\alpha_{0}(\lambda,z)$ and $\alpha(\lambda,z)$ can be expressed as, for $y\sim 0 \in {{\mathbb{Z}}}^{d}$, $$\label{eq:LWW-Specalization-4}
\alpha_{0}(\lambda,z) = \exp {\left(\mu_{\lambda,z}(0)\right )}, \qquad
\alpha(\lambda,z) = \exp {\left(\mu_{\lambda,z}(0;y)\right )}.$$ Note that $\alpha_{0}\geq \alpha \geq 1$. Let $I^{\omega}_{\lambda,z}
= I^{\omega}_{{\mathcal{X}}}$. $I^{\omega}_{\lambda,z}(a,b)$ can be written in a loop measure like way: $$\label{eq:LWW-Specialization-I2PF}
I^{\omega}_{\lambda,z}(a,b) = {{\mathbbm{1}}_{\left\{\omega_{a} = \omega_{b}\right\}}} +
{{\mathbbm{1}}_{\left\{\omega_{a}\neq\omega_{b}\right\}}} {\left(1 - e^{-
\mu_{\lambda,z}( \omega_{a},\omega_{b}; {\mathrm{range}(\omega{\left(a,b\right )})})}\right )},$$ where $$\label{eq:LWW-Loop-Measure-Generalized}
\mu_{\lambda,z}(A,B;C) = \sum_{x} \mathop{\sum_{\omega\colon x\to
x}}_{{\left\vert\omega\right\vert} \geq 1}
\frac{w_{\lambda,z}(\omega)}{{\left\vert\omega\right\vert}}
{{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega)}\cap A \neq\emptyset\right\}}}
{{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega)} \cap B \neq \emptyset\right\}}}
{{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega)} \cap C = \emptyset\right\}}}.$$ As with the loop measure, define $\mu_{\lambda,z}(A,B) =
\mu_{\lambda,z}(A,B;\emptyset)$. The effect of this more complicated object is to require that both an element from $A$ *and* $B$ are in the range of the walk.
Convergence of the Lace Expansion I. Preliminaries {#sec:LWW-Convergence}
==================================================
This section establishes the basic facts used to prove the convergence of the lace expansion. The strategy is that of [@Slade2006], suitably adapted and modified for $\lambda$-LWW. An important role is played by the function $H_{\lambda,z}$ in the next definition.
\[def:LWW-Reduced-2PT-Function\] The *reduced two point function* $H_{\lambda,z}(x,y)$ is defined by $$\label{eq:LWW-WLAW-2PT-H}
H_{\lambda,z}(x,y) = (1-\delta_{x,y})G_{\lambda,z}(x,y).$$
A useful fact that will be used repeatedly is that $$\label{eq:LWW-G-H-Relation}
G_{\lambda,z}(x,y) = \delta_{x,y}\alpha_{0}(\lambda,z) +
H_{\lambda,z}(x,y).$$ The two-point functions $G_{\lambda,z}$ and $H_{\lambda,z}$ inherit the isometry invariance of the weight $w_{\lambda,z}$. By translation invariance $G_{\lambda,z}(x,y) = G_{\lambda,z}(0,y-x)$; it will be convenient to write $G_{\lambda,z}(x)$ for $G_{\lambda,z}(0,x)$.
Random Walk Quantities and Bounds {#sec:LWW-SRW-Versions}
---------------------------------
The *random walk $2$-point function* $C_{z}(x)$ and its Fourier transform $\hat C_{z}(k)$ are given by $$\label{eq:LWW-SRW-FT}
C_{z}(x) = \sum_{\omega\colon x\to x}z^{{\left\vert\omega\right\vert}}, \qquad
\hat C_{z}(k) = \frac{1}{1-z{\left\vert\Omega\right\vert}\hat D(k)}.$$
The following facts about the random walk two-point function will be useful. For notational clarity, let $\beta$ be a quantity that is $O({\left\vert\Omega\right\vert}^{-1})$. $\beta$ is to be thought of as being a small parameter.
\[lem:LWW-SL5.5\] Assume $d>4$. Then for $0\leq z \leq {\left\vert\Omega\right\vert}^{-1}$ $$\begin{aligned}
\label{eq:LWW-SL5.5.1}
\sup_{x}D(x) &\leq \beta \\
\label{eq:LWW-SL5.5.2}
{\|C_{z}\|}_{2}^{2} &\leq 1 + c\beta\\
\label{eq:LWW-SL5.5.3}
{\|(1-\cos(k\cdot x))C_{z}(x)\|}_{\infty} &\leq
5(1+c\beta)(1-\hat D(k))
\end{aligned}$$
\[prop:LWW-Small\] Let $r\in {{\mathbb{N}}}$. There is a constant $K$ independent of $d$ such that for $d>2r$. $$\label{eq:LWW-Small}
\int_{{{\left[-\pi,\pi\right ]}}^{d}} {\left(\frac{1}{1-\hat D(k)}\right )}^{r}\,
\frac{d^{d}k}{{(2\pi)^{d}}} \leq 1+c\beta.$$
This follows by the argument used in the proof of [@MadrasSlade2013 Lemma A.3].
Convergence Strategy and Basic Bounds {#sec:LWW-Convergence-Proof}
-------------------------------------
The proof of convergence is based on comparing the behaviour of simple random walk and $\lambda$-LWW. Define $p(z)$ by $$\label{eq:LWW-Rescaling-Definition}
\frac{\hat G_{\lambda,z}(0)}{\alpha_{0}(\lambda,z)} =
\frac{1}{1-p(z){\left\vert\Omega\right\vert}} = \hat C_{p(z)}(0).$$ Roughly speaking, the intuition is that $\lambda$-LWW should behave like simple random walk. The definition of $p(z)$ serves to determine the activity of the simple random walk that matches $\lambda$-LWW with activity $z$. The following bootstrap lemma is what enables conclusions to be drawn for $z<z_{c}(\lambda)$.
\[lem:LWW-Bootstrap\] Let $a<b$, let $f$ be a continuous function on the interval ${\left[z_{1},z_{2}\right )}$, and assume that $f(z_{1})\leq a$. Suppose for each $z\in {\left(z_{1},z_{2}\right )}$ that $f(z)\leq b$ implies $f(z)\leq a$. Then $f(z)\leq a$ for all $z\in {\left[z_{1},z_{2}\right )}$.
To describe the function $f$ used in applying some definitions are needed.
\[def:LWW-Delta-k\] Define $\Delta_{k}\hat A(\ell)$ by $$-\frac{1}{2} \Delta_{k} \hat A(\ell) = \hat A(\ell) - \frac{1}{2}
{\left( \hat A(\ell +k) + \hat A(\ell -k)\right )},$$ and define $$\begin{gathered}
\nonumber
U_{p(z)}(k,\ell) = 16\hat C_{p(z)}^{-1}(k) \bigg( \hat C_{p(z)}(\ell-k)
\hat C_{p(z)}(\ell) + \hat C_{p(z)}(\ell + k) \hat C_{p(z)}(\ell)\\+ \hat
C_{p(z)}(\ell - k)\hat C_{p(z)}(\ell + k)\bigg).
\end{gathered}$$
The quantity $U_{p(z)}$ is a convenient upper bound for $\frac{1}{2}
{\left\vert\Delta_{k}\hat C_{p(z)}(\ell)\right\vert}$: this can be seen by [@Slade2006 Lemma 5.7]. Define $f(z) = \max \{f_{1}(z),
f_{2}(z), f_{3}(z)\}$, where $$\label{eq:LWW-Comparison-Functions}
f_{1}(z) = z\alpha(\lambda,z){\left\vert\Omega\right\vert}, \quad f_{2}(z) =
\sup_{k\in{\left[-\pi,\pi\right ]}^{d}} \frac{ {\left\vert\hat G_{\lambda,z}(k)\right\vert}
}{\hat C_{p(z)}(k)}, \quad f_{3}(z) =
\sup_{k,\ell\in{\left[-\pi,\pi\right ]}^{d}} \frac{ \Delta_{k}\hat
G_{\lambda,z}(\ell)}{U_{p(z)}(k,\ell)}.$$
The next lemma will be useful for estimating $G_{\lambda,z}$.
\[lem:LWW-One-Step-SM\] Assume $y\neq x$. The following inequality holds: $$\label{eq:LWW-One-Step-SM}
G_{\lambda,z}(x,y) \leq
z\alpha(\lambda,z){\left\vert\Omega\right\vert}\sum_{u}D(u)G_{\lambda,z}(u,y).$$
This can be proven using the loop measure representation. For $\eta$ a walk beginning at $u\sim 0$, let $0\eta = (0,u)\circ\eta$. $$\begin{aligned}
G_{\lambda,z}(0,y) &= \sum_{\eta\colon 0\to y}
{{\mathbbm{1}}_{\left\{\eta\in {{\Omega}_\mathrm{SAW}}\right\}}}
z^{{\left\vert\eta\right\vert}} \exp {\left( \mu_{\lambda,z}({\mathrm{range}(\eta)})\right )} \\
&= \sum_{u\sim 0} \sum_{\eta\colon u\to y}
{{\mathbbm{1}}_{\left\{0\eta\in {{\Omega}_\mathrm{SAW}}\right\}}}
z \exp {\left( \mu_{\lambda,z}(0; {\mathrm{range}(\eta)})\right )} z^{{\left\vert\eta\right\vert}} \exp
{\left( \mu_{\lambda,z}({\mathrm{range}(\eta)})\right )} \\
&\leq z\alpha(\lambda,z){\left\vert\Omega\right\vert} \sum_{u}D(u) \sum_{\eta\colon
u\to y} {{\mathbbm{1}}_{\left\{\eta\in {{\Omega}_\mathrm{SAW}}\right\}}} z^{{\left\vert\eta\right\vert}} \exp
{\left( \mu_{\lambda,z}({\mathrm{range}(\eta)})\right )} \\
&= z\alpha(\lambda,z){\left\vert\Omega\right\vert} \sum_{u}D(u)G_{\lambda,z}(u,y),
\end{aligned}$$ The inequality follows as (a) implies $\mu_{\lambda,z}(0;{\mathrm{range}(\eta)})$ is bounded above by $\mu_{\lambda,z}(0;u) = \alpha_{0}$ and (b) ${{\mathbbm{1}}_{\left\{0\eta\in{{\Omega}_\mathrm{SAW}}\right\}}}$ is bounded above by ${{\mathbbm{1}}_{\left\{\eta\in{{\Omega}_\mathrm{SAW}}\right\}}}$.
\[prop:LWW-SL5.10\] Assume $d>4$. Fix $z\in {\left(0,z_{c}\right )}$ and assume $f(z)\leq K$. Then there is a constant $c_{K}$ independent of $z$ and $d$ such that $$\begin{aligned}
\label{eq:LWW-SL5.10.1}
{\|(1-\cos(k\cdot x))H_{\lambda,z}\|}_{\infty} &\leq
c_{K}(1+\beta) \hat C_{p(z)}(k)^{-1}, \\
\label{eq:LWW-SL5.10.2}
{\|H_{\lambda,z}\|}_{2}^{2} &\leq c_{K}\beta \\
\label{eq:LWW-SL5.10.3}
{\|H_{\lambda,z}\|}_{\infty} & \leq c_{K}\beta.
\end{aligned}$$
The general fact that ${\|g\|}_{\infty}\leq {\|\hat g\|}_{1}$ and the identity $$\sum_{x}\cos (k\cdot x) f(x) e^{i\ell\cdot x} = \frac{1}{2} {\left(\hat
f(\ell+k) + \hat f(\ell-k)\right )}$$ imply that $${\|(1-\cos(k\cdot x))H_{\lambda,z}(x)\|}_{\infty} =
{\|(1-\cos(k\cdot x))G_{\lambda,z}(x)\|}_{\infty}
\leq \frac{1}{2} {\|\Delta_{k}\hat G_{\lambda,z}(\ell)\|}_{1}.$$ The definition of $U$, the fact that $f_{3}\leq K$, and the Cauchy-Schwarz inequality then imply $${\|(1-\cos(k\cdot x))H_{\lambda,z}(x)\|}_{\infty} \leq 16K \hat
C_{p(z)}(k)^{-1} 3 {\|\hat C_{p(z)}\|}_{2}^{2},$$ which yields after using .
To estimate ${\|H_{\lambda,z}\|}_{2}^{2}$ note that implies $$H_{\lambda,z}(x) \leq z\alpha(\lambda,z){\left\vert\Omega\right\vert}
D\ast G_{\lambda,z}(x)$$ The factor $z\alpha{\left\vert\Omega\right\vert}$ is estimated using $f_{1}(z)\leq K$. To estimate $D\ast G_{\lambda,z}$ use Parseval’s identity, $f_{2}(z)\leq K$, and : $${\|H_{\lambda,z}\|}_{2}^{2} \leq K^{2}
{\|D\ast G_{\lambda,z}\|}_{2}^{2}
\leq K^{4} {\|\hat D \hat C_{{\left\vert\Omega\right\vert}^{-1}}\|}_{2}^{2} =
K^{4}({\|\hat C_{{\left\vert\Omega\right\vert}^{-1}}\|}_{2}^{2}-1)\leq
c K^{4} \beta.$$
For the last inequality use the fact that $\sup_{x}H_{\lambda,z}(x) = \sup_{x\neq 0}G_{\lambda,z}(x)$, , and then again. Using $f_{1}\leq K$ gives $$H_{\lambda,z}(x) \leq K\alpha_{0}(\lambda,z)D(x) +
K^{2}D\ast D\ast G_{\lambda,z}(x).$$ A little manipulation shows that ${\|D\ast
D \ast G_{\lambda,z}\|}_{\infty}\leq {\|\hat D^{2}\hat
C^{2}_{p(z)}\|}_{1}$, so implies $${\|D\ast D \ast G_{\lambda,z}\|}_{\infty} \leq c K\beta.$$ implies $D(x)\leq \beta$ so it suffices to show $\alpha_{0}(\lambda,z)$ is bounded above. This follows from $f_{2}\leq K$: $$\begin{aligned}
\alpha_{0} = \int_{{{\left[-\pi,\pi\right ]}}^{d}} \hat G_{\lambda,z}(k)\, \frac{d^{d}k}{{(2\pi)^{d}}}
\leq K \int_{{{\left[-\pi,\pi\right ]}}^{d}} \hat C_{p(z)}(k)\, \frac{d^{d}k}{{(2\pi)^{d}}} \leq K
{\|\hat C_{{\left\vert\Omega\right\vert}^{-1}}\|}_{1},
\end{aligned}$$ and this last integral is finite for $d\geq 3$, and decreases as the dimension $d$ increases.
Convergence of the Lace Expansion II. Diagrammatic Bounds and Convergence {#sec:LWW-Diagrammatic-Bounds}
=========================================================================
To control the lace expansion it is necessary to show that $\hat \Pi_{\lambda,z}$ is small. This is done by obtaining bounds on norms of $\Pi^{(N)}_{\lambda,z}$ in terms of $H_{\lambda,z}$, $G_{\lambda,z}$, and $I_{\lambda,z}$. These bounds are known as *diagrammatic bounds*. Coupled with diagrammatic bounds are what make the hypothesis $f(z)\leq K$ powerful.
Obtaining diagrammatic bounds requires bounding the weight of walks constrained to have $\omega_{j}=x$ in terms of unconstrained walks. This is best illustrated by an example. Consider obtaining a bound for $\frac{d}{dz}G_{\lambda,z}(0,x)$. For self-avoiding walk ($\lambda=0$) this is straightforward: the Leibniz rule implies the derivative is a sum over all self-avoiding walks from $0$ to $x$ together with a marked edge. Splitting the walk at the marked edge and using the fact that self-avoiding walk is purely repulsive yields $$\label{eq:LWW-DB-Intro-1}
\frac{d}{dz} G_{0,z}(0,x) \leq z^{-1}G_{0,z}\ast H_{0,z}(0,x).$$
For $\lambda>0$ a similar argument is possible, but the weight on the second half of the walk is not $w_{\lambda,z}$: memory of the first half of the walk is needed to know when loops are erased. derives identities for walks that play the role of for $\lambda>0$. uses these identities to derive the diagrammatic bounds necessary to apply .
Decompositions for $\lambda$-LWW {#sec:LWW-Decompositions}
--------------------------------
The formulas presented in this section are the result of tracking what happens when loop erasure is performed. The reader may find it helpful to draw examples while reading the text.
### Decompositions from Loop Erasure {#sec:LWW-Decomp-LE}
The loop erasure of a walk can be viewed as a last exit decomposition: if $\omega\colon x\to y$ then the second vertex in the loop erasure is the first vertex visited after the last visit to $x$. Iterating this implies the next proposition.
\[prop:LWW-LE-LE\] Let $\omega$ be a walk. Define $\ell_{0}=0$, and $\ell_{k} = \sup
\{j \mid \omega_{j}=\omega_{\ell_{k-1}} \} + 1$ for $k\in
{{\mathbb{N}}}$. Suppose there are $n+1$ finite values of $\ell_{k}$ such that $\ell_{k}\leq {\left\vert\omega\right\vert}$. Then $$\label{eq:LWW-LE-LE}
{\mathrm{LE}}(\omega) = (\omega_{\ell_{0}}, \omega_{\ell_{1}}, \dots, \omega_{\ell_{n}}).$$
In the restriction to finite values at most ${\left\vert\omega\right\vert}$ is due to the fact that there will be an $\ell_{k} =
{\left\vert\omega\right\vert}+1$, and then $\ell_{k+1} = -\infty$. The loop erasure of a walk $\omega$ induces a decomposition of $\omega$. Let $\eta
= {\mathrm{LE}}(\omega) = (\omega_{\ell_{0}}, \dots,
\omega_{\ell_{k}})$. Define, for $0\leq r< s\leq k$, $$\label{eq:LWW-LE-Pre}
\eta^{-1}{\left[r,s\right ]} = \omega{\left[\ell_{r},\ell_{s}-1\right ]},$$ where, recalling , $\ell_{k+1}={\left\vert\omega\right\vert}+1$. See .
It would be more accurate to write ${\mathrm{LE}}(\omega)^{-1}{\left[r,s\right ]}$ as the definition requires knowledge of the walk $\omega$ whose loop erasure is $\eta$. As the walk $\omega$ will be clear from context this will not cause any confusion.
(0.1,0.1) to\[out=45, in=270\] (1,1) to\[out=90, in=0\] (0,2) to\[out=180, in=90\] (-1,1) to\[out=270, in=135\] (0,0) to (1,0) to\[out=-45, in=90\] (2,-.5) to\[out=270,in=0\] (1,-1) to\[out=180,in=270\] (0,-.5) to\[out=90,in=225\] (.90,-.1); (1.1,0) to\[out=0,in=0\] (1,2) to\[out=180,in=180\] (2,0) to\[out=0,in=0\] (2,1) to\[out=180,in=180\] (1.95,.05); (2.1,0) to\[out=0,in=90\] (3,-1) to\[out=180,in=180\] (3,0) to\[out=0,in=270\] (4,1) to\[out=90,in=0\] (3,2) to\[out=180,in=180\] (2.9,0.1); at (0,0) [$0$]{}; at (1,0) [$t_{1}$]{}; at (2,0) [$t_{2}$]{}; at (3,0) [$t_{3}$]{}; at (0,0) ; at (1,0) ; at (2,0) ; at (3,0) ;
The following extension of the notion of the concatenation of two walks will be notationally convenient. If $\omega^{i}\colon x_{i}\to
y_{i}$ and $y_{1}\sim x_{2}$ write $\omega^{1}\diamond \omega^{2}$ for the walk that consists of $\omega^{1}$ followed by a step from $y_{1}$ to $x_{2}$ followed by the walk $\omega^{2}$.
Fix a walk $\omega$ whose loop erasure is $k$ steps long. A sequence of times $0=t_{0}<t_{1}<t_{2}<\dots <t_{n}=k$ induces a decomposition of $\omega$ by using : $$\label{eq:LWW-Segment-Split}
\omega = \eta^{-1}{\left[t_{0},t_{1}\right ]} \diamond \dots \diamond
\eta^{-1}{\left[t_{n-1},t_{n}\right ]}.$$ This decomposition has two notable features. First, the loop erasure of the segments of the decomposition yield $\eta{\left[t_{i},t_{i+1}-1\right ]}$. Second, each segment, barring perhaps the first segment, never returns to its starting vertex. See .
The next definitions serve to formalize the idea that given the loop erasure $\eta = {\mathrm{LE}}(\omega{\left[0,j\right ]})$ of a walk $\omega$ up to time $j$, the remainder of $\omega$ has the effect of erasing some of $\eta$, and then extending the remainder of $\eta$ to complete the formation of ${\mathrm{LE}}(\omega)$.
\[def:LWW-hitting\] Let $A\subset {{\mathbb{Z}}}^{d}$. The *hitting time $\tau_{\omega}(A)$* of $A$ by $\omega$ is $\tau_{\omega}(A) = \inf \{ j\geq 0 \mid
\omega_{j}\in A\}$.
\[def:LWW-Shrinking\] Let $\eta\colon x\to y$ be a self-avoiding walk, and let $\omega$ be a walk beginning at $y$. Let $\eta^{0} = \eta{\left[0,{\left\vert\eta\right\vert}\right )}$. For $k\geq 1$ inductively define $$\label{eq:LWW-Shrinking}
s^{k}_{\eta}(\omega) = \tau_{\omega}(\eta^{k-1}), \qquad
t^{k}_{\omega}(\eta) = \eta^{-1}(\omega_{s^{k}_{\eta}(\omega)}), \qquad
\eta^{k} = \eta{\left[0,t^{k}_{\omega}(\eta))\right )}.$$ The times $s^{k}_{\eta}(\omega)$ are the *shrinking times of $\eta$ by $\omega$*.
See for an illustration of shrinking times. The walks $\eta^{k}$ in the definition are decreasing in length, and it follows that the times $t^{k}_{\omega}(\eta)$ are decreasing in $k$.
(s1) at (2,0) ; (m12) at (3,1) ; (s2) at (4,0) ; (m23) at (5,1) ; (s23) at (5,0) ; (s3) at (6,0) ; (0,0) to\[out=3,in=177\] (s1) to\[out=-2,in=181\] (s2) to\[out=3,in=178\] (s3); (s3) to\[out=60,in=0\] (m23) to\[out=180,in=90\] (s2); (s2) to\[out=270,in=250\] (s23); (s23) to\[out=70,in=270\] (m23); (m23) to\[out=90,in=90\] (m12) to\[out=270, in=90\] (s1); at (s2) [$\tau_{\omega}(\eta^{1})$]{}; at (s1) [$\tau_{\omega}(\eta^{0})$]{};
### Expected Visits of $\lambda$-LWW {#sec:LWW-Visits}
The next proposition gives a formula for the expected number of visits of a closed $\lambda$-LWW to a given vertex $y$. We will first give an informal description of the formula. The number of visits by a walk $\omega$ to a vertex $y$ can be expressed as $$\label{eq:LWW-Bubble-Chain-Heuristic-1}
{\left\vert \{ j\geq 1 \mid \omega_{j}=y\}\right\vert} = \sum_{j\geq 1}
{{\mathbbm{1}}_{\left\{\omega_{j}=y\right\}}}.$$ Consider a walk with $\omega_{j}=y$. This naturally splits into two pieces: the walk $\omega^{(a)}$ up to time $j$, and the walk $\omega^{(b)}$ after time $j$. The splitting times introduced in then splits each of $\omega^{(a)}$ and $\omega^{(b)}$ into $k$ segments if there are $k$ splitting times. In the segments of $\omega^{(a)}$ are called $\omega^{(i)}$ for $i=1, \dots, k$, and the segments of $\omega^{(b)}$ are called $\omega^{(k+i)}$ for $i=1, \dots, k$. The conditions $A_{i}$ and $B_{i}$ are formalizations of the fact that these subwalks arise from splitting times.
\[prop:LWW-Bubble-Chain\] Fix $x,y\in {{\mathbb{Z}}}^{d}$, $y\neq x$. Then $$\begin{aligned}
\label{eq:LWW-Bubble-Chain}
\sum_{\omega\colon x\to x} &{\left\vert\{j\geq 1 \mid \omega_{j}=y\}\right\vert}
w_{\lambda,z}(\omega)
= \alpha_{0}\sum_{k\geq 1} \mathop{\sum_{x_{0}, \dots,
x_{k}}}_{\mathrm{distinct}} \sum_{i=1}^{k}{{\mathbbm{1}}_{\left\{x_{0}=x\right\}}}
{{\mathbbm{1}}_{\left\{x_{k}=y\right\}}} \lambda^{k} \\
&\!\!\!\!\!\!\!\!\mathop{\sum_{\omega^{(i)}\colon x_{i-1}\to
x_{i}}}_{\omega^{(k+i)}\colon x_{k-i+1}\to x_{k-i}}
{\left[\prod_{i=1}^{k} w_{\lambda,z}(\omega^{(i)})
{{\mathbbm{1}}_{\left\{\omega^{(i)}\in A_{i}\right\}}}\right ]}
{\left[\prod_{i=1}^{k} w_{\lambda,z}(\omega^{(k+i)})
{{\mathbbm{1}}_{\left\{\omega^{(k+i)}\in B_{i}\right\}}}\right ]}
\end{aligned}$$ where $A_{i}$ and $B_{i}$ are defined as follows. A walk $\omega$ is in $A_{i}$ if $\omega{\left[1{\colon\!}\right ]}$ does not hit ${\mathrm{LE}}(\omega^{(j)})$ for any $j<i$. A walk $\omega$ is in $B_{i}$ if
1. $\omega$ does not hit $\omega^{k-j}$ for $j>i+1$,
2. $\omega{\left[1{\colon\!}\right ]}$ hits $\omega^{k-i}$ at $\omega^{k-i}_{0}$,
3. $\omega$ hits $\omega^{k-i-1}$ at $\omega^{k-i}_{0}$, and $\omega$ does not hit ${\mathrm{LE}}(\omega^{k-i-1})\setminus \{
\omega^{k-i}_{0}\}$.
Rewrite ${\left\vert \{ j\geq 1 \mid \omega_{j}=y\}\right\vert}$ as $\sum_{j\geq 1}
{{\mathbbm{1}}_{\left\{\omega_{j}=y\right\}}}$. To prove the claim it suffices to show that walks with $\omega_{j}=y$ are in bijection with the summands such that ${\left\vert \omega^{(1)} \circ \dots \circ
\omega^{(k)}\right\vert} = j$.
Suppose $\omega_{j}=y$, and let $\eta = {\mathrm{LE}}(\omega{\left[0,j\right ]})$. Let $t^{\ell},s^{\ell}$ be $t^{\ell}_{\omega}(\eta)$ and $s^{\ell}_{\eta}(\omega)$, respectively. Assume there are $k$ shrinking times for the walk $\omega$. Observing that $\omega$ closed implies $t^{k}=0$, $s^{k}={\left\vert\omega\right\vert}$ implies $$\begin{aligned}
\label{eq:LWW-BC-1}
\omega{\left[0,j\right ]} &= \eta^{-1}{\left[t^{k},t^{k-1}\right ]} \diamond \dots\diamond
\eta^{-1}{\left[t^{2},t^{1}\right ]}
\diamond \eta^{-1}{\left[t^{1},{\left\vert\eta\right\vert}\right ]}\\
\label{eq:LWW-BC-2}
\omega{\left[j{\colon\!}\right ]} &= \omega{\left[j,s^{1}\right ]} \diamond \dots \diamond
\omega {\left[s^{k-1},s^{k}\right ]}.
\end{aligned}$$ Call the subwalks on the right-hand sides of and the *constituents* of $\omega{\left[0,j\right ]}$ and $\omega{\left[j{\colon\!}\right ]}$, respectively. Call a walk $\omega\colon x\to
x$ an *excursion* if the only occurrences of $x$ in $\omega$ are $\omega_{0}$ and $\omega_{{\left\vert\omega\right\vert}}$.
Separating any initial excursions from $x$ to $x$ from the first subwalk comprising $\omega{\left[0,j\right ]}$ gives the factor $\alpha_{0}$. To complete the claim, notice that any excursions immediately after a shrinking time that occur prior to the next hitting time of $\eta^{\ell}$ can be transferred to the previous subwalk comprising $\omega{\left[j{\colon\!}\right ]}$. In the case of the first constituent of $\omega{\left[j{\colon\!}\right ]}$ the excursions can be transferred to the last constituent of $\omega{\left[0,j\right ]}$.
The next proposition handles the case of visits to the initial vertex of a walk.
\[prop:LWW-BC-Diag\] $$\label{eq:LWW-BC-Diag}
\mathop{\sum_{\omega\colon x\to x}}_{{\left\vert\omega\right\vert}\geq 1}
{\left\vert\{j\geq 1 \mid \omega_{j}=x\}\right\vert}
w_{\lambda,z}(\omega) = \alpha_{0}(\alpha_{0}-1).$$
Write ${\left\vert\{j\geq 1\mid\omega_{j}=x\}\right\vert}$ as $\sum_{j\geq 1}
{{\mathbbm{1}}_{\left\{\omega_{j}=x\right\}}}$. Inserting this into the left-hand side of and split each walk $\omega$ at time $j$. Summing the remainder after time $j$ gives a factor $\alpha_{0}$. Summing over $j$ gives $\alpha_{0}-1$ as $j\geq 1$ implies the empty walk is excluded.
To avoid explicitly writing the cumbersome right-hand side of repeatedly it will be convenient to introduce a short-hand definition:
\[def:LWW-True-BC\] The *bubble chain ${\mathrm{BC}}_{\lambda,z}(x,y)$ from $x$ to $y$* is defined to be $\alpha_{0}(\alpha_{0}-1)$ if $x=y$ and the right-hand side of if $x\neq y$.
The next decomposition formula is the analogue of for walks $\omega$ that are not closed. Some notation will be needed: for $\eta$ a self-avoiding walk ending at $x$ define ${\mathrm{BC}}_{\lambda,z}^{\eta}(x,y)$ to be the bubble chain in ${{\mathbb{Z}}}^{d} \setminus \{\eta_{0}, \dots,
\eta_{{\left\vert\eta\right\vert}-1}\}$. See .
(s00) at (0,0) ; (s20) at (2,0) ; (s40) at (4,0) ; (s60) at (6,0) ; (s51) at (5,1) ; (s32) at (3,2) ; (s555) at (5.5,.5) ; (s22) at (2,2) ; (s00) to\[out=3,in=179\] (s20) to\[out=-1,in=180\] (s40); (4,0) to\[out=30,in=270\] (s51) to\[out=135,in=0\] (s32) to\[out=135,in=90\] (s22) to\[out=270,in=225\] (3,2) to\[out=-45,in=180\] (s51) to\[out=205,in=60\] (4,0); (4,0) to\[out=135,in=180\] (4,1) to\[out=0,in=180\] (s555); at (s00) [$\omega_{0}=x$]{}; at (5.5,.5) [$\omega_{{\left\vert\omega\right\vert}}=y$]{}; at (s22) [$\omega_{j}=b$]{}; at (s40) [$v$]{};
\[prop:LWW-Bubble-Chain-Split\] Fix $x,y,b\in{{\mathbb{Z}}}^{d}$, $x\neq y$, $b\neq x$. Then $$\begin{aligned}
\label{eq:LWW-Bubble-Chain-Split}
\sum_{\omega\colon x\to y}&
{{\mathbbm{1}}_{\left\{x\notin\omega{\left[1{\colon\!}\right ]}\right\}}} {\left\vert\{j\geq 1 \mid
\omega_{j}=b\}\right\vert} w_{\lambda,z}(\omega) = \\&\sum_{a\in
{{\mathbb{Z}}}^{d}} \mathop{\sum_{\omega^{(1)}\colon x\to a}}_{x\notin\omega{\left[1{\colon\!}\right ]}}
\mathop{\sum_{\omega^{(2)}\colon a\to
y}}_{\omega^{(2)}{\left[1{\colon\!}\right ]} \cap
{\mathrm{LE}}(\omega^{(1)})=\emptyset}\!\!\!\!\!\!\! {\left(\delta_{a,b} +
{\mathrm{BC}}_{\lambda,z}^{{\mathrm{LE}}(\omega^{(1)})}(a,b)\right )}
w_{\lambda,z}(\omega^{(1)}) w_{\lambda,z}(\omega^{(2)}).
\end{aligned}$$
This follows by writing ${\left\vert \{j\geq 1 \mid \omega_{j}=b\}\right\vert}$ as $\sum_{j\geq 1} {{\mathbbm{1}}_{\left\{\omega_{j}=b\right\}}}$ and noting that this splits, by applying with $\eta =
{\mathrm{LE}}(\omega{\left[0,j\right ]})$, a walk $\omega$ into (i) an initial segment $\omega^{(1)}$ whose loop erasure is the subset of ${\mathrm{LE}}(\omega{\left[0,j\right ]})$ that is contained in ${\mathrm{LE}}(\omega)$, (ii) a bubble chain from the endpoint of $\omega^{(1)}$ to $b$ whose walks do not hit ${\mathrm{LE}}(\omega^{(1)})$; if the endpoint of $\omega^{(1)}$ is $b$ then it is also possible this walk is null, and (iii) a walk $\omega^{(2)}$ from the endpoint of $\omega^{(1)}$ to $y$ that does not, after the first vertex, hit ${\mathrm{LE}}(\omega^{(1)})$.
The restriction in to walks $\omega$ that do not return to their initial vertex is simply because this is the type of sum that will occur most frequently in what follows.
### Two-Point Functions and Their Derivatives {#sec:LWW-2PT-Deriv}
The quantity $I^{\omega}_{\lambda,z}(a,b)$ defined in is inconvenient due to its dependence on the details of $\omega$; the next definition introduces a simple upper bound.
\[def:LWW-LRW-I2P\] The *interaction two-point function* $I_{\lambda,z}(x,y)$ is the function $$\label{eq:LWW-LRW-I2P}
I_{\lambda,z}(x,y) = {{\mathbbm{1}}_{\left\{x=y\right\}}} + {{\mathbbm{1}}_{\left\{x\neq
y\right\}}}{\left(1 - e^{- \mu_{\lambda,z}(x,y;\emptyset)}\right )}.$$
\[lem:LWW-Repulsion-I\] Let $\omega$ be a walk of length $n$, and let $0\leq a<b\leq n$. $$\label{eq:LWW-Repulsion-I}
I_{\lambda,z}^{\omega}(a,b) \leq I_{\lambda,z}(\omega_{a},\omega_{b}).$$
If $\omega_{a}=\omega_{b}$ then is an equality. If $\omega_{a}\neq \omega_{b}$ the inequality follows because the loop measure is decreasing in its final argument.
The important aspect of the next bound is that it is independent of $\eta$.
\[prop:LWW-I2P-DB\] Let $\eta\colon x\to y$ be a self-avoiding walk. Then $$\label{eq:LWW-I2P-DB}
\frac{d}{dz} I^{\eta}_{\lambda,z}(x,y) \leq {{\mathbbm{1}}_{\left\{x\neq
y\right\}}}z^{-1} \sum_{a\in {{\mathbb{Z}}}^{d}}
\mathop{\sum_{\omega\colon a\to a}}_{{\left\vert\omega\right\vert}\geq 1}
{{\mathbbm{1}}_{\left\{x\in\omega\right\}}} {{\mathbbm{1}}_{\left\{y\in\omega\right\}}}
w_{\lambda,z}(\omega).$$ Further, the right-hand side of is an upper bound for $\frac{d}{dz} I_{\lambda,z}(x,y)$ as well.
Differentiate, and then use $e^{-x}\leq 1$ for $x\geq 0$.
\[def:LWW-Reduced-2PT\] The *scaled two-point functions* $\bar G_{\lambda,z}(x,y)$ and $\bar H_{\lambda,z}(x,y)$ are defined by $$\bar G_{\lambda,z}(x,y) = \alpha_{0}(\lambda,z)^{-1}
G_{\lambda,z}(x,y), \qquad \bar H_{\lambda,z}(x,y) =
\alpha_{0}(\lambda,z)^{-1} H_{\lambda,z}(x,y).$$
Let $\bar B_{\lambda,z}(x) = \bar H_{\lambda,z}(x)^{2}$. An upper bound on ${\mathrm{BC}}_{\lambda,z}$ is obtained by dropping the constraints $A_{i}$ and $B_{i}$.
\[def:LWW-BC\] Define ${\mathrm{B^{\star}}}_{\lambda,z}(x,y)$ by $$\label{eq:LWW-BC}
{\mathrm{B^{\star}}}_{\lambda,z}(x,y) = \alpha_{0}
\begin{cases}
\sum_{k\geq 1} \lambda^{k}\underbrace{\bar B_{\lambda,z}\ast \dots
\ast \bar B_{\lambda,z}}_{\textrm{$k$ terms}} (x,y) &
x\neq y \\
\alpha_{0} - 1 & x=y
\end{cases}$$
\[prop:LWW-BC-Bound\] Let $\eta$ be any self-avoiding walk ending at $x$. Then $$\label{eq:LWW-BC-Bound}
{\mathrm{BC}}_{\lambda,z}^{\eta}(x,y) \leq
{\mathrm{BC}}_{\lambda,z}(x,y) \leq
{\mathrm{B^{\star}}}_{\lambda,z}(x,y).$$
The first inequality follows as the set of summands is increasing from left to right and all summands are non-negative. For the second inequality note that relaxing the conditions $A_{i}$ and $B_{i}$ increases the set of summands. Using $H_{\lambda,z}(x,y) =
H_{\lambda,z}(y,x)$, which follows from , to reverse the direction of the walks $\omega^{(k+i)}$ gives the upper bound ${\mathrm{B^{\star}}}_{\lambda,z}(x,y)$.
The next lemma shows that if a sum over walks satisfying some constraints is upper bounded by relaxing the constraints, an upper bound on the derivative is obtained by differentiating the upper bound. This will be used frequently.
\[lem:LWW-Derivative-UB\] Suppose $A,B$ are two sets of walks, and $A\subset B$. Then $$\frac{d}{dz}\sum_{\omega\in A} w_{\lambda,z}(\omega) \leq
\frac{d}{dz} \sum_{\omega\in B} w_{\lambda,z}(\omega).$$
Each summand is non-negative as the weight of a walk $\omega$ is proportional to $z^{{\left\vert\omega\right\vert}}$, and the set of summands on the right-hand side is larger.
The formulas of yield diagrammatic bounds on derivatives of two-point functions by applying the identity $$\label{eq:LWW-Walk-Leibniz}
{\left\vert\omega\right\vert} = \sum_{a\in {{\mathbb{Z}}}^{d}} {\left\vert \{ j\geq 1 \mid \omega_{j}=a\}\right\vert},$$ where $j=0$ is not included because there ${\left\vert\omega\right\vert}+1$ vertices in a walk.
\[prop:LWW-H-DB\] For $x\in {{\mathbb{Z}}}^{d}$, $x\neq 0$, $$\label{eq:LWW-H-DB}
\frac{d}{dz} \bar G_{\lambda,z}(x) =
\frac{d}{dz} \bar H_{\lambda,z}(x)
\leq z^{-1} (1+{\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}) \bar G_{\lambda,z}\ast
\bar H_{\lambda,z}(x).$$
The first equality is straightforward as $\bar G_{\lambda,z}(x) =
\delta_{0,x} + \bar H_{\lambda,z}(x)$ by . For the inequality observe that $$\frac{d}{dz} \bar H_{\lambda,z}(x) = z^{-1}
\mathop{\sum_{\omega\colon 0\to x}}_{0\notin \omega{\left[1{\colon\!}\right ]}}
{\left\vert\omega\right\vert} w_{\lambda,z}(\omega)$$ Applying and yields $$z^{-1}\sum_{b\in {{\mathbb{Z}}}^{d}} \sum_{a\in {{\mathbb{Z}}}^{d}}
\mathop{\sum_{\omega^{(1)}\colon 0\to
a}}_{0\notin\omega{\left[1{\colon\!}\right ]}}
\mathop{\sum_{\omega^{(2)}\colon a\to
x}}_{\omega^{(2)}{\left[1{\colon\!}\right ]} \cap
{\mathrm{LE}}(\omega^{(1)})=\emptyset} (\delta_{a,b} +
{\mathrm{BC}}_{\lambda,z}^{{\mathrm{LE}}(\omega^{(1)})}(a,b))
w_{\lambda,z}(\omega^{(1)}) w_{\lambda,z}(\omega^{(2)}).$$ By removing the restriction on the bubble chain gives an upper bound. The sum over $b$ then gives the factor $1+ {\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}$. Dropping the constraint that $\omega^{(2)}$ does not intersect ${\mathrm{LE}}(\omega^{(1)})$ gives the claim.
\[prop:LWW-Alpha-DB\] $$\label{eq:LWW-Alpha-DB}
\frac{d}{dz} \alpha_{0}(\lambda,z) = z^{-1}
{\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1} $$
As a zero step walk does not survive being differentiated, $$\frac{d}{dz} \alpha_{0}(\lambda,z) =
z^{-1}\mathop{\sum_{\omega\colon 0 \to 0}}_{{\left\vert\omega\right\vert} \geq 1}
{\left\vert\omega\right\vert} w_{\lambda,z}(\omega).$$ The proposition follows by (i) applying to rewrite ${\left\vert\omega\right\vert}$, (ii) using to recognize the resulting sum as the $1$-norm of the bubble chain, and (iii) using to upper bound the norm of the bubble chain.
\[prop:LWW-BC-DB\] $$\begin{aligned}
\label{eq:LWW-BC-DB}
\frac{d}{dz} {\|{\mathrm{BC}}_{\lambda,z}\|}_{1} &\leq
z^{-1}{\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}{\left( 3\alpha_{0} - 1 -
\alpha_{0}^{2} + {\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}\right )} \\ &+
2\alpha_{0}z^{-1}\lambda {\| \bar H_{\lambda,z}\cdot {\left(\bar
G_{\lambda,z}\ast \bar H_{\lambda,z}\right )}\|}_{1}{\left(1 +
{\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}\right )}^{3}.
\end{aligned}$$
By it suffices to obtain bounds on the derivative of ${\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}$. For the summand with $x=y$ the an upper bound is $z^{-1}{\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}(\alpha_{0}-1) +
z^{-1}\alpha_{0}{\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}$ by .
For $x\neq y$ differentiating and using gives an upper bound $z^{-1}{\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}\alpha_{0}^{-1}({\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}
- \alpha_{0}(\alpha_{0}-1))$ if the derivative is applied to $\alpha_{0}$. The factor of $\alpha_{0}^{-1}$ can be dropped to give an upper bound as $\alpha_{0}\geq 1$. When the derivative is not applied to $\alpha_{0}$ we have, using , the upper bound $$\begin{aligned}
\frac{d}{dz} {\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1} &= \alpha_{0}\frac{d}{dz}
\sum_{k\geq 1} \sum_{y} \lambda^{k} \underbrace{\bar
H_{\lambda,z}^{2} \ast \dots
\ast \bar H_{\lambda,z}^{2}}_{\textrm{$k$ terms}}(y) \\
&\leq 2\alpha_{0} \sum_{k\geq 1} \sum_{y} k\lambda^{k} {\left(\bar
H_{\lambda,z}\frac{d}{dz} \bar H_{\lambda,z}\right )}\ast
\underbrace{\bar H_{\lambda,z}^{2} \ast \dots \ast \bar
H_{\lambda,z}^{2}}_{\textrm{$k-1$ terms}}(y) \\
\nonumber
&= 2\alpha_{0}z^{-1}\lambda {\|\bar H_{\lambda,z} \cdot {\left(\bar G_{\lambda,z}\ast
\bar H_{\lambda,z}\right )}\|}_{1} (1+{\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1})^{3}.
\end{aligned}$$ Summing these upper bounds gives the result.
Diagrammatic Bounds {#sec:LWW-DB-Pi}
-------------------
The bounds derived in this section will be obtained under the assumption that $f(z)\leq K$ for $z<z_{c}(\lambda)$. In particular the results of hold. It will also be assumed that the dimension $d$ is sufficiently large, i.e., $\beta$ is sufficiently small.
### Initial Diagrammatic Bounds {#sec:LWW-DB-Auxiliary}
\[prop:LWW-UB-Alpha\] If $z<z_{c}$ and $f(z)\leq K$ then $\alpha_{0}(\lambda,z)\leq 1+ c\beta$.
By definition and $$\label{eq:LWW-UB-Alpha.1}
\alpha_{0}(\lambda,z) = \exp {\left(\mu_{\lambda,z}(0)\right )}
= 1 + \mathop{\sum_{\omega\colon 0 \to 0}}_{{\left\vert\omega\right\vert} \geq 1}
w_{\lambda,z}(\omega).$$ The walks contributing to the sum have their last vertex a neighbour of $0$, so $$\label{eq:LWW-UB-Alpha.2}
\mathop{\sum_{\omega\colon 0 \to 0}}_{{\left\vert\omega\right\vert} \geq 1} w_{\lambda,z}(\omega)
= z\lambda{\left\vert\Omega\right\vert}D\ast H_{\lambda,z}(0),$$ which is bounded by $z\lambda{\left\vert\Omega\right\vert}{\|H_{\lambda,z}\|}_{\infty}$. The claim follows from $z{\left\vert\Omega\right\vert}\leq f_{1}(z) \leq K$ and .
\[prop:LWW-BC-Geometric\] If $z<z_{c}$ and $f(z)\leq K$ then ${\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}\leq c\beta$.
Repeatedly using ${\|f\ast g\|}_{1}\leq {\|f\|}_{1}{\|g\|}_{1}$ implies $${\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1} \leq \alpha_{0}{\left( (\alpha_{0}-1) +
\sum_{k\geq 1} \lambda^{k} {\|\bar H_{\lambda,z}\|}_{2}^{2k}\right )},$$ The interchange of summations is valid as each term is non-negative. By $\alpha_{0}\leq 1 + c\beta$ so $\alpha_{0}-1 \leq c\beta$. Since $\alpha_{0}\geq 1$, ${\|\bar
H_{\lambda,z}\|}_{2}^{2}\leq {\|H_{\lambda,z}\|}_{2}^{2}$, so implies that for $\beta$ sufficiently small $$\sum_{k\geq 1}\lambda^{k}{\|\bar H_{\lambda,z}\|}_{2}^{2k} \leq
c\beta. \qedhere$$
\[prop:LWW-UB-I\] Let $I_{\lambda,z}(x) = I_{\lambda,z}(0,x)$. If $z<z_{c}$ and $f(z)\leq K$ then ${\|I_{\lambda,z}\|}_{1} \leq 1 + c\beta$.
The inequality $1-e^{-x}\leq x$ implies that $1+{\|{{\mathbbm{1}}_{\left\{x\neq 0\right\}}} \mu_{\lambda,z}(0,x)\|}_{1}$ is an upper bound for ${\|I_{\lambda,z}\|}_{1}$. The factor of $1$ is from the term ${{\mathbbm{1}}_{\left\{x=0\right\}}}$ in $I_{\lambda,z}$. Observe that ${\|{{\mathbbm{1}}_{\left\{x\neq 0\right\}}}\mu_{\lambda,z}(0,x)\|}_{1}$ is bounded by $$\label{eq:LWW-UB-I-1}
\sum_{x\neq 0} \sum_{y} \mathop{\sum_{\omega\colon y \to y}}_{{\left\vert\omega\right\vert}\geq 1}
{{\mathbbm{1}}_{\left\{0\in \omega\right\}}} {{\mathbbm{1}}_{\left\{x\in \omega\right\}}}
\frac{ w_{\lambda,z}(\omega)}{{\left\vert\omega\right\vert}}
\leq \sum_{y} \mathop{\sum_{\omega\colon y\to y}}_{{\left\vert\omega\right\vert}\geq
1} {{\mathbbm{1}}_{\left\{0\in\omega\right\}}} w_{\lambda,z}(\omega),$$ as $\sum_{x\neq 0}{{\mathbbm{1}}_{\left\{x\in\omega\right\}}} \leq
{\left\vert{\mathrm{range}(\omega)}\right\vert}\leq {\left\vert\omega\right\vert}$. By translation invariance this is $$\label{eq:LWW-UB-I-2}
\sum_{y} \mathop{\sum_{\omega\colon 0 \to 0}}_{{\left\vert\omega\right\vert}\geq 1}
{{\mathbbm{1}}_{\left\{-y\in \omega\right\}}} w_{\lambda,z}(\omega) =
{\|\mathop{\sum_{\omega\colon 0 \to 0}}_{{\left\vert\omega\right\vert}\geq 1}
{{\mathbbm{1}}_{\left\{y\in \omega\right\}}} w_{\lambda,z}(\omega)\|}_{1},$$ where the norm is with respect to $y$. To establish the proposition (i) bound ${{\mathbbm{1}}_{\left\{y\in\omega\right\}}}$ by ${\left\vert \{j\geq 1 \mid
\omega_{j}=y\}\right\vert}$, (ii) apply for the summands with $y=0$, (iii) apply and for the summands with $y\neq 0$, and (iv) observe that the sum of these two bounds is ${\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}$ and apply .
### Bounds for $\pi^{(1)}$ {#sec:LWW-Pi-DB-1.1}
\[prop:LWW-DB-1-Rep\] $$\label{eq:LWW-DB-1-Rep}
\pi^{(1)}(x) = \sum_{m}\pi^{(1)}_{m}
\begin{cases}
= z\lambda{\left\vert\Omega\right\vert} D\ast \bar H_{\lambda,z}(0) & x = 0 \\
\leq \bar H_{\lambda,z}(x)I_{\lambda,z}(0,x)
e^{\mu_{\lambda,z}(0,x)} & x\neq 0,
\end{cases}$$
For $x=0$ the claim follows from the identities in . For $x\neq 0$ use . Recall the loop measure representation of the second product, i.e., the expression for $I_{{\mathcal{X}}}^{\omega}$ given by . The desired bound follows by forgetting the constraint in the loop measure and the rearrangement $e^{\mu_{\lambda,z}(0,x)}-1 = e^{\mu_{\lambda,z}(0,x)}
I_{\lambda,z}(0,x)$.
\[prop:LWW-DB-1\] Suppose $f(z)\leq K$. The following bounds hold for $u=0,1$ and $k\in {{\left[-\pi,\pi\right ]}}^{d}$: $$\label{eq:LWW-DB-1}
{\|{\left\vertx\right\vert}^{2u}\pi^{(1)}\|}_{1} \leq
c\beta{\left({{\mathbbm{1}}_{\left\{u=0\right\}}}+ {\|{\left\vertx\right\vert}^{2u}
\bar H_{\lambda,z}\|}_{\infty}\right )},$$ and $$\label{eq:LWW-DB-1-Trig}
{\|(1-\cos k\cdot x)\pi^{(1)}(x)\|}_{1} \leq c\beta
{\|(1-\cos k \cdot x)\bar H_{\lambda,z}\|}_{\infty}.$$
The triangle inequality, ${\|f\ast g\|}_{1}\leq {\|f\|}_{\infty}
{\|g\|}_{1}$ with $g=I_{\lambda,z}$, and $1-\cos 0 =0$ imply $$\begin{aligned}
\label{eq:LWW-DB-1.1}
{\|{\left\vertx\right\vert}^{2u}\pi^{(1)}\|}_{1} &\leq {{\mathbbm{1}}_{\left\{u=0\right\}}}
z {\left\vert\Omega\right\vert} {\|\bar H_{\lambda,z}\|}_{\infty}
+ {\|I_{\lambda,z}(0,x)
e^{\mu_{\lambda,z}(0,x)}{{\mathbbm{1}}_{\left\{x\neq 0\right\}}}\|}_{1}
{\| {\left\vertx\right\vert}^{2u}\bar H_{\lambda,z}\|}_{\infty} \\
\label{eq:LWW-DB-1-Trig.1}
{\|(1-\cos k\cdot x)\pi^{(1)}(x)\|}_{1} & \leq
{\|I_{\lambda,z}(0,x) e^{\mu_{\lambda,z}(0,x)}{{\mathbbm{1}}_{\left\{x\neq 0\right\}}}\|}_{1}
{\|(1-\cos k \cdot x)\bar H_{\lambda,z}\|}_{\infty}.
\end{aligned}$$ Using $z{\left\vert\Omega\right\vert} \leq
f_{1}(z) \leq K$ by $f_{3}\leq K$ and $\sup_{x}e^{\mu_{\lambda,z}(0,x)} \leq \alpha_{0}(\lambda,z)$ implies $$\begin{aligned}
{\|I_{\lambda,z}(0,x)
e^{\mu_{\lambda,z}(0,x)}{{\mathbbm{1}}_{\left\{x\neq 0\right\}}}\|}_{1}
&\leq \alpha_{0}{\|I_{\lambda,z}(0,x){{\mathbbm{1}}_{\left\{x\neq 0\right\}}}\|}_{1} \\
&\leq \alpha_{0}{\left( {\|I_{\lambda,z}(0,x)\|}_{1} - 1\right )}.
\end{aligned}$$ The conclusion now follows from .
### Bounds for $\pi^{(N)}$, $N\geq 2$ {#sec:LWW-Pi-DB-N.1}
\[prop:LWW-Pi-Repulsive-Bound\] Let $m\geq 2$, $x\in {{\mathbb{Z}}}^{d}$, and $N\geq 2$. Let $x_{0}=0,
x_{N-1}^{\prime}=x$. Then $$\begin{aligned}
\label{eq:LWW-Pi-Repulsive-Bound}
{\left\vert\pi_{m}^{(N)}(x)\right\vert} \leq &\sum_{\vec m} \mathop{\sum_{x_{1},
\dots, x_{N-1}}}_{x_{0}^{\prime}, \dots, x_{N-2}^{\prime}}
\mathop{\sum_{\omega^{(1)}\colon 0 \to
x_{1}}}_{{\left\vert\omega^{(1)}\right\vert}=m_{1}}
\mathop{\sum_{\omega^{(2)}\colon x_{1} \to
x_{0}^{\prime}}}_{{\left\vert\omega^{(2)}\right\vert}=m_{2}} \dots
\mathop{\sum_{\omega^{(2N-2)}\colon x_{N-1} \to
x_{N-2}^{\prime}}}_{{\left\vert\omega^{(2N-2)}\right\vert}=m_{2N-2}}
\mathop{\sum_{\omega^{(2N-1)}\colon x_{N-2}^{\prime} \to
x}}_{{\left\vert\omega^{(2N-1)}\right\vert}=m_{2N-1}} \\
&\prod_{j=0}^{N-1}{\left\vert I_{\lambda,z}(x_{j},x_{j}^{\prime})\right\vert}
\prod_{k=1}^{2N-1}
\alpha_{0}^{-1}\exp{\left(\mu_{\lambda,z}({\mathrm{range}(\omega^{(k)})})\right )}
\end{aligned}$$ where the summation is over valid vectors $\vec m$ (recall ) of subinterval lengths such that $\sum m_{i}=m$.
This follows from . By the factors of $I^{\omega}_{\lambda,z}$ can be replaced by $I_{\lambda,z}$. In the language of an ${\mathcal{X}}$ gas, as $\alpha_{\omega}\geq 0$ for any walk $\omega$, the ${\mathcal{X}}$ gas for $\lambda$-LWW is repulsive in the sense of , and this proves the claim.
Upper bounds on ${\|\pi^{(N)}(x)\|}_{1}$ can be efficiently found by formulating in terms of multiplication and convolution operators. Let ${\mathcal{M}}_{g}$ and ${\mathcal{C}}_{g}$ denote multiplication and convolution by $g$, respectively: ${\mathcal{M}}_{g}f = gf$ and ${\mathcal{C}}_{g}f = g\ast f$.
\[lem:LWW-MC-Formulation\] Fix $N\geq 2$ and let $\bar H = \bar
H_{\lambda,z}$, $\bar G = \bar G_{\lambda,z}$, and $I =
I_{\lambda,z}$. Then $$\label{eq:LWW-MC-Bare}
\sum_{x}{\left\vert\pi^{(N)}(x)\right\vert} \leq {\| {\left({\mathcal{C}}_{\bar H\ast I} {\mathcal{M}}_{\bar H}\right )}{\left({\mathcal{C}}_{\bar G\ast I} {\mathcal{M}}_{\bar H}\right )}^{N-2} \bar H\ast I\|}_{\infty}.$$
The definition of a valid vector of lengths implies that summing over all valid vectors of lengths results in the sums over walks with indices $1$, $2j$, and $2N-1$ being replaced by $\bar H_{\lambda,z}$, and the remaining sums of walks are replaced by $\bar G_{\lambda,z}$. Consulting , this means that all horizontal solid lines except the leftmost and rightmost are weighted by $\bar
G_{\lambda,z}$, while the rest are weighted by $\bar
H_{\lambda,z}$. Formally, $$\begin{aligned}
\label{eq:LWW-Pi-2PT-Bound}
{\left\vert\pi^{(N)}(x)\right\vert} \leq &\mathop{ \sum_{x_{1}, \dots, x_{N-1}}
}_{x_{0}^{\prime}, \dots, x_{N-2}^{\prime}}
{\left(\prod_{j=0}^{N-1} I_{\lambda,z}(x_{j},x_{j}^{\prime})\right )}
\bar H_{\lambda,z}(x_{0},x_{1})\\&
{\left(\prod_{j=0}^{N-2} \bar H_{\lambda,z}(x_{j}^{\prime}, x_{j+1})\right )}
{\left(\prod_{j=0}^{N-3}\bar G_{\lambda,z}(x_{j}^{\prime},x_{j+2})\right )}
\bar H_{\lambda,z}(x_{N-2}^{\prime},x_{N-1}^{\prime}).
\end{aligned}$$ Replace the factor $I_{\lambda,z}(x_{0},x_{0}^{\prime})$ by $I_{\lambda, z}(y,x_{0}^{\prime})$ in and call the resulting function $F(x,y)$. As $\sum_{x}{\left\vertF(x,0)\right\vert} =
\sum_{x}{\left\vert\pi^{(N)}(x)\right\vert}$ the quantity $\sup_{y}\sum_{x}{\left\vertF(x,y)\right\vert}$ is an upper bound for the left-hand side of . The associativity of convolution implies $$\sum_{x}{\left\vertF(x,y)\right\vert} = {\left(({\mathcal{C}}_{\bar H} {\mathcal{C}}_{I}) {\mathcal{M}}_{\bar H} {\left( {\mathcal{C}}_{\bar G} {\mathcal{C}}_{I} {\mathcal{M}}_{\bar H}\right )}^{N-2} \bar H\ast I\right )}(y).$$ follows as ${\mathcal{C}}_{\bar G}{\mathcal{C}}_{I} = {\mathcal{C}}_{\bar
G\ast I}$ and ${\mathcal{C}}_{\bar H} {\mathcal{C}}_{I} = {\mathcal{C}}_{\bar H \ast I}$.
The right-hand side of can be easily estimated with the help of the next lemma.
\[lem:LWW-SL-Lp\] Given non-negative even functions $f_{0}, f_{1}, \dots, f_{2M}$ on ${{\mathbb{Z}}}^{d}$, define ${\mathcal{C}}_{j}$ and ${\mathcal{M}}_{j}$ to be the operations of convolution with $f_{2j}$ and multiplication by $f_{2j-1}$ for $j=1,
\dots, M$. Then for any $k\in \{0, \dots, 2M\}$, $$\label{eq:LWW-SL-Lp}
{\|{\mathcal{C}}_{M}{\mathcal{M}}_{M} \dots {\mathcal{C}}_{1} {\mathcal{M}}_{1}f_{0}\|}_{\infty}
\leq {\|f_{k}\|}_{\infty} \prod {\| f_{j} \ast f_{j^{\prime}}\|}_{\infty},$$ where the product is over disjoint consecutive pairs $j,j^{\prime}$ taken from $\{0, \dots, 2M\} \setminus \{k\}$.
The strange formatting of the bounds in the next proposition are strictly for typographic convenience; in applications we multiply through by the denominators of the left-hand sides.
\[prop:LWW-DB-N\] Let $N\geq 2$. Then for $z<z_{c}$ and $u\in\{0,1\}$ $$\begin{aligned}
\label{eq:LWW-Pi-UB-N.1}
\frac{{\|{\left\vertx\right\vert}^{2u}\pi^{(N)}(x)\|}_{1}}{(2N-1)^{u}}
&\leq {\|{\left\vertx\right\vert}^{2u}\bar H_{\lambda,z}\|}_{\infty}
(c\beta)^{N-2 + {{\mathbbm{1}}_{\left\{N=2\right\}}}}
(1+c\beta)^{N+{{\mathbbm{1}}_{\left\{N\geq 3\right\}}}} \\
\label{eq:LWW-Pi-UB-N.2}
\frac{{\|(1-\cos(k\cdot x)) \pi^{(N)}(x)\|}_{1}}{(4N-1)(2N-1)}
&\leq {\|(1-\cos (k\cdot x))\bar
H_{\lambda,z}(x)\|}_{\infty}
(c\beta)^{N-2+{{\mathbbm{1}}_{\left\{N=2\right\}}}}
(1+c\beta)^{N+{{\mathbbm{1}}_{\left\{N\geq 3\right\}}}}
\end{aligned}$$
Suppose that both $$\label{eq:LWW-DB-N.1}
\frac{{\|{\left\vertx\right\vert}^{2u}\pi^{(N)}(x)\|}_{1}}{(2N-1)^{u}} \leq
{\|{\left\vertx\right\vert}^{2u}\bar H_{\lambda,z}\|}_{\infty} {\|\bar
G_{\lambda,z} \ast \bar G_{\lambda,z}\|}_{\infty} {\|\bar
G_{\lambda,z}\ast \bar H_{\lambda,z}\|}^{N-2}_{\infty}
{\|I_{\lambda,z}\|}^{N}_{1}$$ and $$\begin{aligned}
\nonumber
\frac{{\|(1-\cos(k\cdot x)) \pi^{(N)}(x)\|}_{1}}{(4N-1)(2N-1)} \leq&
{\|(1-\cos (k\cdot x))\bar H_{\lambda,z}(x)\|}_{\infty}
{\|\bar G_{\lambda,z}\ast \bar G_{\lambda,z}\|}_{\infty}
\\ \label{eq:LWW-DB-N.2} &
\times{\|\bar G_{\lambda,z}\ast \bar H_{\lambda,z}\|}^{N-2}_{\infty}
{\|I_{\lambda,z}\|}^{N}_{1}.
\end{aligned}$$ Suppose further that if $N=2$ the same bounds hold with each term $\bar G_{\lambda,z}$ replaced by $\bar H_{\lambda,z}$. The claim then follows, as , the triangle inequality, Cauchy-Schwarz, and $\bar H_{\lambda,z}\leq H_{\lambda,z}$ imply $$\label{eq:LWW-Conv-Bound}
{\|\bar H_{\lambda,z}\ast \bar G_{\lambda,z}\|}_{\infty} = {\|
\bar H_{\lambda,z} + \bar H_{\lambda,z}\ast
\bar H_{\lambda,z}\|}_{\infty} \leq {\|\bar H_{\lambda,z}\|}_{\infty} +
{\|\bar H_{\lambda,z}\|}_{2}^{2} \leq c\beta,$$ and implies ${\|I_{\lambda,z}\|}_{1}\leq 1 +
c\beta$. The rest of the proof establishes .
First observe that the difference between $N=2$ and $N\geq 3$ is only that all two-point functions in are $\bar H_{\lambda,z}$ for $N=2$, while for $N\geq 3$ factors of $\bar
G_{\lambda,z}$ arise.
If $u=0$ follows by applying to the right-hand side of , putting the sup norm on the final $I_{\lambda,z}\ast \bar H_{\lambda,z}$, and using the inequality $${\|\bar G_{\lambda,z}\ast \bar H_{\lambda,z}\ast I_{\lambda,z}\|}_{\infty}
\leq
{\|\bar G_{\lambda,z}\ast \bar H_{\lambda,z}\|}_{\infty}
{\|I_{\lambda,z}\|}_{1}.$$
For $u=1$, note that $x = x_{1} + \dots x_{2N-1}$, where $x_{j}$ is the displacement along the $j^{\mathrm{th}}$ subwalk in a summand contributing to $\pi^{(N)}$. As ${\left\vertx\right\vert}^{2} \leq \sum
{\left\vertx_{i}\right\vert}^{2}$ it follows that an upper bound is given by $$\sum_{j=1}^{2N-1} {\| {\left({\mathcal{C}}_{\bar H\ast I} {\mathcal{M}}_{\bar H}\right )}{\left({\mathcal{C}}_{\bar G\ast I} {\mathcal{M}}_{\bar H}\right )}^{N-2} \bar H\ast I\|}_{\infty},$$ where the $j^{\mathrm{th}}$ two-point function $\bar G$ or $\bar H$ is replaced with ${\left\vertx\right\vert}^{2}\bar H$. The claim follows by (i) applying and putting the sup norm on the term involving the factor of ${\left\vertx\right\vert}^{2}$ (ii) noting that the resulting norms are of the form ${\|\bar H \ast \bar H \ast I\|}_{\infty}$, ${\|I\ast \bar G \ast \bar H\|}_{\infty}$, ${\|I\ast I \ast\bar
G \ast \bar G\|}_{\infty}$, or ${\|\bar H \ast I\|}_{\infty}$ and (iii) iterating ${\|f\ast g\|}_{\infty}\leq
{\|f\|}_{\infty}{\|g\|}_{1}$. The uniform upper bound follows by using ${\|\bar H\ast \bar H\|}_{\infty}\leq {\|\bar H \ast \bar
G\|}_{\infty}$.
To prove let $t= \sum_{j=1}^{n}
t_{j}$. Then (see [@Slade2006 Section 4.2.3]) $$\label{eq:LWW-SL-Path-Inequality}
(1-\cos t) \leq (2n+1)\sum_{j=1}^{n}(1-\cos t_{j}).$$ Letting $t_{j} = k\cdot x_{j}$ where $x_{j}$ is the displacement along the $j^{\mathrm{th}}$ subwalk the argument used to prove with $u=1$ can be applied to give . The prefactor $(4N-1)(2N-1)$ arises as for an $N$ edge lace there are $2N-1$ subwalks, so $n=2N-1$ in .
Completion of the Bootstrap {#sec:LWW-Bootstrap}
---------------------------
This section begins by using the diagrammatic bounds of to establish that $\Pi$ is small under the hypothesis $f(z)\leq K$.
\[lem:LWW-SL5.11\] Fix $z\in {\left(0,z_{c}\right )}$ and assume $d$ is sufficiently large. If $f(z) \leq K$, then there is a constant $\bar c_{K}$ independent of $z$ and $d$ such that $$\begin{aligned}
\label{eq:LWW-Pi-Small}
\sum_{x\in {{\mathbb{Z}}}^{d}}{\left\vert\Pi_{z}(x)\right\vert} &\leq \bar c_{K}\beta \\
\label{eq:LWW-Pi-Cos-Small}
\sum_{x\in {{\mathbb{Z}}}^{d}}(1-\cos(k\cdot x)){\left\vert\Pi_{z}(x)\right\vert} & \leq \bar
c_{K}\beta \hat C_{p(z)}(k)^{-1}.
\end{aligned}$$
This follows by combining the bounds of for $u=0$ with the bound ${\|(1-\cos k\cdot
x)H_{\lambda,z}(x)\|}_{\infty} \leq c_{K}(1+\beta)\hat
C_{p(z)}^{-1}(k)$ of .
The remainder of this section is devoted to verifying the hypothesis of for $z_{1}=0$, $z_{2} = z_{c}(\lambda)$, $a=4$ and $b=1+O(\beta)$.
\[lem:LWW-SL5.12\] The function $f$ obeys $f(0)=1$.
Clearly $f_{1}(0)=0$. The definition of $p(z)$ implies $p(0)=0$ as $\alpha_{0}(\lambda,0)=1$, so $f_{2}(0)=1$. Lastly, $f_{3}(0)=0$: $U_{0}=48$ while $\Delta_{k}\hat G_{\lambda,0} = 0$.
\[lem:LWW-SL5.14\] The function $f$ is continuous on ${\left[0,z_{c}\right )}$.
It suffices to show $f_{1},f_{2},f_{3}$ are continuous on ${\left[0,r\right ]}$ for any $r<z_{c}$. For $f_{1}$ this follows as $\alpha(\lambda,z) \leq
\alpha_{0}(\lambda,z)\leq \chi_{\lambda}(z)$, i.e., $\alpha(\lambda,z)$ has a convergent power series representation.
Recall (see [@Slade2006 Lemma 5.13]) that the supremum of an equicontinuous family of functions over a compact interval is a continuous function, provided this supremum is finite. It follows that it is enough to prove a bound uniform in $k$ on the derivative of $f_{2}$ (resp. $f_{3}$) with respect to $z$. Since equicontinuity of a family $\{{\left\vertg_{\alpha}\right\vert}\}$ is equivalent to equicontinuity of $\{g_{\alpha}\}$, the absolute value on $\hat
G_{\lambda,z}$ (resp. $\Delta_{k}\hat G_{\lambda,z}$) can be ignored. For $f_{2}$ the derivative is $$\frac{d}{dz} \frac{\hat G_{\lambda,z}(k)}{\hat C_{p(z)}(k)} =
\frac{1}{\hat C_{p(z)}(k)^{2}} {\left[ \hat C_{p(z)}(k) \frac{ d\hat
G_{\lambda,z}(k)}{dz} - \hat G_{\lambda,z}(k) \frac{ d\hat
C_{p(z)}(k)}{dp}|_{p=p(z)} \frac{dp(z)}{dz}\right ]}.$$ Now note: ${\left\vert\hat G_{\lambda,z}(k)\right\vert} \leq \chi_{\lambda}(r)$, ${\left\vert\frac{d}{dz} \hat G_{\lambda,z}(k)\right\vert} \leq
{\left\vert\frac{d}{dz}\chi_{\lambda}(r)\right\vert}$, ${\left\vert \partial_{p}\hat
C_{p}(k)\right\vert} \leq {\left\vert\Omega\right\vert} \chi_{\lambda}(r)^{2}$. Further, $$\begin{aligned}
{\left\vert\frac{d p(z)}{dz}\right\vert} &= {\left\vert\frac{d}{dz}{\left\vert\Omega\right\vert}^{-1}{\left(1 -
\frac{\alpha_{0}(\lambda,z)}{\chi_{\lambda}(z)}\right )}\right\vert} \\
&\leq {\left\vert\Omega\right\vert}^{-1}\alpha_{0}(\lambda,r)\frac{d}{dz}\chi_{\lambda,}(r)
\chi^{-2}_{\lambda}(0) + \chi_{\lambda}^{-1}(1)\frac{d}{dz}\alpha_{0}(\lambda,r),
\end{aligned}$$ and $\frac{d}{dz}\alpha_{0}(\lambda,r)$ is bounded above by $\frac{d}{dz}\chi_{\lambda}(r)$ by . A uniform bound on the derivative then follows from $$\frac{1}{2} \leq \hat C_{p(z)}(k) \leq \hat C_{p(z)}(0) =
\frac{\chi_{\lambda}(z)}{\alpha_{0}(\lambda,z)}
\leq \chi_{\lambda}(r),$$ where the second last equality follows from the definition of $p(z)$, and the last inequality from $\alpha_{0}(\lambda,z) \geq 1$.
For $f_{3}$ the calculation is essentially the same. Calculating the derivative shows that what is needed is upper bounds on ${\left\vert\hat
G_{\lambda,z}(k)\right\vert}$, ${\left\vert\frac{d}{dz} \hat G_{\lambda,z}(k)\right\vert}$, ${\left\vert \partial_{p} \hat C_{p}(k)\right\vert}$, and ${\left\vert\frac{d}{dz} p(z)\right\vert}$, along with upper and lower bounds on $\hat C_{p(z)}$. These bounds have already been obtained.
The next lemma completes the bootstrap argument.
\[lem:LWW-SL5.16\] Suppose $d$ is sufficiently large. Fix $z\in{\left(0,z_{c}\right )}$, and suppose that $f(z)\leq 4$. Then there is a constant $c$ independent of $z$ and $d$ such that $f(z)\leq 1 + c\beta$.
We prove $f_{j}(z)\leq 1 + c\beta$ for $j=1,2,3$ in sequence.
Since $\alpha_{0}(\lambda,z)$ and $\chi_{\lambda}(z)$ are both positive and finite it follows that $$\label{eq:LWW-Positive}
\frac{\alpha_{0}(\lambda,z)}{\chi_{\lambda}(z)} = 1 -
z\alpha(\lambda,z) {\left\vert\Omega\right\vert} - \hat \Pi_{\lambda,z}(0) > 0.$$ and together imply $$f_{1}(z) = z\alpha(\lambda,z){\left\vert\Omega\right\vert} \leq 1 + \hat \Pi_{\lambda,z}(0)
\leq 1 + \bar c_{4}\beta.$$
implies $\alpha_{0} \leq 1 + \bar c\beta$, so $f_{2}\leq 1+O(\beta)$ follows if $$\label{eq:LWW-C5.53}
\frac{\hat G_{\lambda,z}(k)}{\alpha_{0}(\lambda,z) \hat
C_{p(z)}(k)} = 1 + \frac{1 - p(z){\left\vert\Omega\right\vert}\hat D(k) - \hat
F_{\lambda,z}(k)}{\hat F_{\lambda,z}(k)}$$ is $1 + O(\beta)$, where $$\hat F_{\lambda,z}(k) \equiv \hat G_{\lambda,z}(k)^{-1} = 1 -
z\alpha(\lambda,z){\left\vert\Omega\right\vert}\hat D(k) -
\hat\Pi_{\lambda,z}(k).$$ By definition, $p(z){\left\vert\Omega\right\vert} = z\alpha(\lambda,z){\left\vert\Omega\right\vert} + \hat
\Pi_{\lambda,z}(0)$. Hence the numerator of the right-hand side of is $$\label{eq:LWW-C5.54}
1 - p(z){\left\vert\Omega\right\vert}\hat D(k) - \hat F_{\lambda,z}(k) = \hat
\Pi_{\lambda,z}(0) {\left( 1- \hat D(k)\right )} - {\left(
\hat \Pi_{\lambda,z}(0) - \hat \Pi_{\lambda,z}(k)\right )},$$ which is bounded above by $4\bar c_{4}\beta$. An alternative upper bound of the right hand side of follows from : $$\label{eq:LWW-SL5.55}
\hat
\Pi_{\lambda,z}(0) {\left( 1- \hat D(k)\right )} - {\left(
\hat \Pi_{\lambda,z}(0) - \hat \Pi_{\lambda,z}(k)\right )} \leq \bar
c_{4}\beta {\left(1-\hat D(k)\right )} + \bar c_{4}\beta {\left(1 - p(z)
{\left\vert\Omega\right\vert} \hat D(k)\right )}.$$ Since $$\label{eq:LWW-SL5.56}
{\left(1-\hat D(k)\right )}\hat C_{p(z)}(k) = 1 + \underbrace{\hat
D(k)}_{\leq 1} \underbrace{\frac{
p(z){\left\vert\Omega\right\vert} - 1}{1-p(z){\left\vert\Omega\right\vert}\hat D(k)}}_{\leq 1} \leq 2,$$ the numerator of is bounded by $$\label{eq:LWW-SL5.57}
3\bar c_{4}\beta {\left(1-p(z){\left\vert\Omega\right\vert}\hat D(k)\right )} \leq 3\bar
c_{4}\beta {\left( \hat F_{\lambda,z}(0) + {\left(1-\hat D(k)\right )}\right )}.$$
The denominator of is $$\begin{aligned}
\hat F_{\lambda,z}(k) &= \hat F_{\lambda,z}(0) + {\left( \hat
F_{\lambda,z}(k) - \hat F_{\lambda,z}(0)\right )} \\
\label{eq:LWW-SL5.58}
&=\hat F_{\lambda,z}(0) + z\alpha(\lambda,z){\left\vert\Omega\right\vert}{\left(1-\hat
D(k)\right )} + {\left( \hat \Pi_{\lambda,z}(0) - \hat \Pi_{\lambda,z}(k)\right )}.
\end{aligned}$$ Let $\bar \lambda = \sup_{\eta\in{{\Omega}_\mathrm{SAP}}}\lambda_{\eta}$, and $\lambda^{\star} = \max(1,\bar \lambda)$. For $z\leq
(2{\left\vert\Omega\right\vert}\sqrt{\lambda^{\star}})^{-1}$ (if $\lambda^{\star}>1$) or neglecting loops (if $\lambda^{\star}\leq 1$) implies $\hat
F_{\lambda,z}(0) \geq \hat C_{z\sqrt{\lambda^{\star}}}(0)^{-1} \geq
\frac{1}{2}$. Then $1-\hat D(k)\geq 0$ and imply $$\label{eq:LWW-SL5.59}
\hat F_{\lambda,z}(k) \geq \hat F_{\lambda,z}(0) - 2\bar
c_{4}\beta \geq \frac{1}{2} - 2\bar c_{4}\beta.$$
For $(2{\left\vert\Omega\right\vert}\lambda^{\star})^{-1} \leq z<z_{c}(\lambda)$ , $\hat F_{z}(0)>0$, and $\alpha(\lambda,z)\geq 1$ imply $$1 - p(z){\left\vert\Omega\right\vert} \hat D(k) = 1- (1-\hat F_{\lambda,z}(0))\hat
D(k) \leq 1 - \hat D(k) + \hat F_{\lambda,z}(0)$$ and hence $$\begin{aligned}
\hat F_{\lambda,z}(k) &\geq \hat F_{\lambda,z}(0) +
\frac{1}{2\sqrt{\lambda^{\star}}}
{\left(1 - \hat D(k)\right )} - \bar c_{4}\beta {\left(1-p(z){\left\vert\Omega\right\vert}\hat
D(k)\right )} \\
&\geq {\left(\frac{1}{2\sqrt{\lambda^{\star}}} - \bar c_{4}\beta\right )}
{\left(\hat F_{\lambda,z}(0) + {\left(1- \hat D(k)\right )}\right )}.
\end{aligned}$$ For $z\leq (2{\left\vert\Omega\right\vert}\lambda^{\star})^{-1}$ or $(2{\left\vert\Omega\right\vert}\lambda^{\star})^{-1} \leq z <z_{c}$ these lower and upper bounds combine to imply the right-hand side of is $1+O(\beta)$, and hence $f_{2}(z) = 1+
O(\beta)$.
Lastly consider $f_{3}(z)$. As for $f_{2}$, it suffices to prove the claim for $f_{3}/\alpha_{0}$. Let $\hat g_{\lambda,z}(k) =
z\alpha(\lambda,z) {\left\vert\Omega\right\vert} \hat D(k) + \hat
\Pi_{\lambda,z}(k)$, so $$\label{eq:LWW-SL5.62}
\frac{\hat G_{\lambda,z}(k)}{\alpha_{0}(\lambda,z)} =
\frac{1}{1-\hat g_{\lambda,z}(k)}.$$ The symmetry of $D(x)$ and $\Pi_{\lambda,z}(x)$ implies that $g_{\lambda,z}(x) = g_{\lambda,z}(-x)$, so applying Lemma 5.7 of [@Slade2006] (a general fact about even functions) gives $$\begin{aligned}
\label{eq:LWW-SL5.63}
\frac{1}{2}{\left\vert\Delta_{k}\hat G_{\lambda,z}(\ell)\right\vert} &\leq
\frac{1}{2} {\left( \hat G_{\lambda,z}(\ell-k) + \hat
G_{\lambda,z}(\ell + k)\right )} \hat G_{\lambda,z}(\ell) {\left( {\left\vert\hat
g_{\lambda,z}(0)\right\vert} - {\left\vert\hat g_{\lambda,z}(k)\right\vert}\right )} \\
&+ 4\hat G_{\lambda,z}(\ell-k) \hat G_{\lambda,z}(\ell) \hat
G_{\lambda,z}(\ell + k) {\left( {\left\vert\hat
g_{\lambda,z}(0)\right\vert} - {\left\vert\hat g_{\lambda,z}(k)\right\vert}\right )} {\left( {\left\vert\hat
g_{\lambda,z}(0)\right\vert} - {\left\vert\hat g_{\lambda,z}(\ell)\right\vert}\right )}.
\end{aligned}$$ Using $f_{2}(z) \leq 1+ O(\beta)$ bounds each factor of $\hat
G_{\lambda,z}$ by ${\left(1+O(\beta)\right )}\hat C_{p(z)}$. Further, $$\begin{aligned}
{\left\vert\hat g_{\lambda,z}(0)\right\vert} - {\left\vert\hat g_{\lambda,z}(k)\right\vert} &\leq
\sum_{x}{\left(1-\cos(k\cdot x)\right )}
{\left( z\alpha(\lambda,z) {\left\vert\Omega\right\vert} + {\left\vert\Pi_{z}(x)\right\vert}\right )} \\
&\leq z\alpha(\lambda,z){\left\vert\Omega\right\vert} {\left(1-\hat D(k)\right )} + \bar c_{4}
\beta \hat C_{p(z)}(k)^{-1} \\
&\leq {\left(2+O(\beta)\right )} \hat C_{p(z)}(k)^{-1},
\end{aligned}$$ where the second inequality is by and the third is by $f_{1}(z) \leq 1+ O(\beta)$ and . Combining the bounds and using the definition of $U_{p(z)}$ gives $f_{3}(z) \leq 1 + O(\beta)$.
\[cor:LWW-IRB\] For $d$ sufficiently large, $\lambda$-LWW satisfies a *$k$-space infrared bound*: there is a constant $K = 1+O(\beta)$ such that for $0\leq z\leq
z_{c}(\lambda)$ $$\hat G_{\lambda,z}(k) \leq K \hat C_{p(z)}(k).$$
The proof of showed that $f_{2}(z)\leq
1+O(\beta)$ without absolute values on $\hat G_{\lambda,z}$, uniformly for $z<z_{c}$. Taking a limit gives the claim.
The fact that the quantities $T_{\lambda,z}$ and $S_{\lambda,z}$ defined below are small will be important in what follows.
\[def:LWW-Bubble-Etc\] The *triangle diagram* $T_{\lambda,z}$ and *square diagram* $S_{\lambda,z}$ are the quantities $$\label{eq:LWW-Bubble-Etc}
T_{\lambda,z} = {\|\hat H_{\lambda,z}^{3}\|}_{1}, \qquad
S_{\lambda,z} = {\|\hat H_{\lambda,z}^{4}\|}_{1}.$$
\[cor:LWW-TS\] For $d$ sufficiently large and $z\leq z_{c}$ the triangle and square diagrams are bounded above by $c\beta$.
For notational convenience write $\bar H_{\lambda,z} =
\alpha_{0}^{-1} H_{\lambda,z}$, and similarly for $\bar
G_{\lambda,z}$. By $\alpha_{0}^{-1}\hat
H_{\lambda,z} = \alpha_{0}^{-1}\hat G_{\lambda,z}-1$. implies $\alpha_{0}^{-1}\hat G_{\lambda,z} \leq (1+O(\beta))\hat
C_{p(z)}$ since $\alpha_{0}\leq 1 + O(\beta)$. The claim follows from .
Proofs of the Main Results {#sec:LWW-Further}
==========================
To go beyond the $k$-space infrared bound of requires control of the derivatives of $G_{\lambda,z}$ and $\Pi_{\lambda,z}$ with respect to $z$. This control is established in . The remainder of the section establishes using arguments based on [@MadrasSlade2013 Chapter 6]. Throughout let $z_{c} =
z_{c}(\lambda)$.
Further Diagrammatic Bounds {#sec:LWW-Diagrammatic-Derivatives}
---------------------------
Having verified that the bounds of holds for $z<z_{c}$, the monotone convergence theorem implies they continue to hold at $z_{c}$.
\[prop:LWW-BC-D-DB\] For $d$ sufficiently large and $0<z\leq z_{c}$ $$\label{eq:LWW-BC-D-DB}
\frac{d}{dz} {\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1} \leq z_{c}^{-1}c\beta.$$
The left-hand side is a polynomial with positive coefficients, so it suffices to obtain an upper bound at $z=z_{c}$. By , $\alpha_{0}(\lambda,z_{c})\leq 1+ c\beta$, ${\|{\mathrm{B^{\star}}}_{\lambda,z_{c}}\|}_{1}\leq c\beta$, and , the claim follows.
\[prop:LWW-H-D-DB\] Let $d$ be sufficiently large, $0<z\leq z_{c}$, and $v = 1,2$. Then $$\label{eq:LWW-H-D-DB}
{\|\partial^{v}_{z} \bar G_{\lambda,z}\|}_{\infty} =
{\|\partial^{v}_{z} \bar H_{\lambda,z}\|}_{\infty} \leq
c\beta z_{c}^{-v}$$
As for it suffices to consider $z=z_{c}$. The equality of the first two terms follows from . implies $$\frac{d}{dz} \bar H_{\lambda,z} \leq
z^{-1}(1+{\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}) \bar H_{\lambda,z}
\ast \bar G_{\lambda,z}.$$ The claim follows for $v=1$ as ${\|\bar H_{\lambda,z} \ast \bar
G_{\lambda,z}\|}_{\infty} \leq c\beta$ by and ${\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}\leq c\beta$ by .
For $v=2$ apply . After computing the derivative and using the triangle inequality (i) argue as for $v=1$ for the term from differentiating $z^{-1}$ (ii) use when differentiating ${\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}$ and (iii) when differentiating either of the two-point functions use and $
{\|\bar H_{\lambda,z} \ast \bar H_{\lambda,z} \ast\bar
G_{\lambda,z}\|}_{\infty} \leq {\|\bar H_{\lambda,z} \ast \bar
H_{\lambda,z}\|}_{\infty} + {\|\bar H_{\lambda,z} \ast \bar
H_{\lambda,z} \ast \bar H_{\lambda,z}\|}_{\infty}$, and to see that this is bounded by $c\beta$. Each term is therefore bounded by $c\beta z_{c}^{-2}$.
\[prop:LWW-I-D-DB\] Let $d$ be sufficiently large, $0<z\leq z_{c}$, and $v=1,2$. Then $$\label{eq:LWW-I-D-DB}
{\|\partial^{v}_{z}I_{\lambda,z}\|}_{1} \leq c\beta z_{c}^{-v}$$
For $v=1$ note $$\frac{d}{dz} I_{\lambda,z} = \frac{d}{dz} (1 -
e^{-\mu_{\lambda,z}(0,x)}) \leq \frac{d}{dz} \mu_{\lambda,z}(0,x).$$ This bound is increasing in $z$, so considering $z_{c}$ is enough. Translation invariance, as in the proof of , implies this is equal to the derivative in $z$ of ${\|{\mathrm{B^{\star}}}_{\lambda,z}\|}_{1}$. The claim follows for $v=1$ from .
For $v=2$ it is enough to bound the derivative of the bound of . This is similar to the arguments already given; the only new terms that arise occur when differentiating ${\|\bar
H_{\lambda,z} \cdot \bar G_{\lambda,z} \ast \bar
H_{\lambda,z}\|}_{1}$, which is $\bar H_{\lambda,z}\ast \bar
G_{\lambda,z} \ast \bar H_{\lambda,z}(0)$. By after taking a derivative the result is, up to a factor of $(1+O(\beta))$, a square diagram $\bar H_{\lambda,z}\ast \bar G_{\lambda,z} \ast
\bar H_{\lambda,z} \ast \bar G_{\lambda,z}(0)$. Repeatedly using and shows this is at most $c\beta$.
\[prop:LWW-Derivative-Small\] For $d$ sufficiently large, $0<z<z_{c}$, and $v=1,2$ $$\label{eq:LWW-Derivative-Small}
{\|\partial^{v}_{z}\Pi_{\lambda,z}\|}_{1} \leq c\beta z_{c}^{-v}$$
The Leibniz rule and imply that the result of differentiating $\Pi$ is a sum of terms of the form of the bounds of , but where each term has one of the factors of $\bar G_{\lambda,z}$, $\bar H_{\lambda,z}$ or $I_{\lambda,z}$ differentiated. Given this, the argument is as in the proofs of and . Let us describe the proof for $N=2$. For $N=1$ the proof is similar as $e^{\mu_{\lambda,z}(0,x)}\leq \alpha_{0}$.
Consider $v=1$. There are $3N-1$ terms arising when differentiating $\pi^{(N)}_{\lambda,z}$. If $\bar G_{\lambda,z}$ or $\bar
H_{\lambda,z}$ is differentiated apply and place the sup norm on this term when applying , and then use to bound this norm. If $I_{\lambda,z}$ is differentiated use placing the $\sup$ norm on a term $H_{\lambda,z}$ and use to bound the one norm of the derivative of $I_{\lambda,z}$. This yields the claim as the factor of $3N-1$ is irrelevant for the convergence of the series.
If $v=2$ there are $(3N-1)^{2}$ terms. If both derivatives fall on a single factor proceed as in the previous paragraph and use or . If the derivatives fall on distinct factors, one factor being $I_{\lambda,z}$, proceed as before. For the remaining case, where two distinct factors of $\bar
H_{\lambda,z}$ (or $\bar G_{\lambda,z}$) are differentiated, place a sup norm on one term. The new term to bound when applying is of the form ${\|\bar H_{\lambda,z} \ast
\bar G_{\lambda,z} \ast \bar G_{\lambda,z} \ast
I_{\lambda,z}\|}_{\infty}$. It suffices to bound ${\|\bar H \ast
\bar G \ast \bar G\|}_{\infty}$, and this is bounded above by ${\|\bar H\ast \bar G\|}_{\infty} + {\|\bar H\ast \bar
H\|}_{\infty} + {\|\bar H \ast \bar H \ast \bar H\|}_{\infty}$, all of which are bounded by $c\beta$ by .
\[cor:LWW-Pi-D-NZ\] Let $d$ be sufficiently large and $0<z\leq z_{c}$. Then $-\frac{d}{dz}
\hat F_{z}(0)\geq c>0$.
The derivative is $$\label{eq:LWW-Derivative-F}
-\frac{d}{dz} \hat F_{\lambda,z}(0) = {\left\vert\Omega\right\vert}\alpha(\lambda,z)
+ z{\left\vert\Omega\right\vert}\frac{d}{dz}\alpha(\lambda,z) + \frac{d}{dz}\hat
\Pi_{\lambda,z}(0).$$ By ${\left\vert\frac{d}{dz}\hat
\Pi_{\lambda,z}(k)\right\vert}$ is bounded above by a constant since $z_{c}$ is bounded below by a term of order $\beta$ by . An argument as for shows the magnitude of the second term is bounded by a constant. As $\alpha(\lambda,z)\geq 1$ the first term dominates for $d$ sufficiently large.
### Derivatives of Moments {#sec:LWW-Derivative-Moments}
The next proposition (for $\lambda=0$) is [@Slade2006 Exercise 5.17].
\[lem:LWW-Pi-2M\] For $d$ sufficiently large and $0\leq z<z_{c}$ $$\label{eq:LWW-Pi-2M}
{\|{\left\vertx\right\vert}^{2}\Pi_{\lambda,z}(x)\|}_{1} \leq c\beta$$
This follows from $\hat C_{p(z)}^{-1} \leq 1-\hat D(k)$ and .
\[prop:LWW-Moment-Bounds\] For $0\leq z\leq z_{c}$ the following bounds hold: $$\begin{aligned}
\label{eq:LWW-Moment-Bounds-1}
{\|{\left\vertx\right\vert}^{2}H_{\lambda,z}(x)\|}_{\infty} &\leq c\beta, \\
\label{eq:LWW-Moment-Bounds-2}
{\|{\left\vertx\right\vert}^{2}H_{\lambda,z}(x)\|}_{2} &\leq c.
\end{aligned}$$
The proof relies on the identity $$\label{eq:LWW-MB-1}
{\left\vertx_{\mu}\right\vert}^{2}H_{\lambda,z}(x) = -
\int_{{{\left[-\pi,\pi\right ]}}^{d}} \partial^{2}_{k_{\mu}} \hat H_{\lambda,z}(k)
e^{-ik\cdot x}\, \frac{d^{d}k}{{(2\pi)^{d}}},$$ where $\mu$ is a unit basis vector of ${{\mathbb{Z}}}^{d}$. Omitting the subscripts $\lambda$ and $z$ and letting a subscript $\mu$ denote partial differentiation with respect to $k_{\mu}$ the derivative can be calculated: $$\label{eq:LWW-MB-2}
\hat G_{\mu,\mu}(k) = z\alpha{\left\vert\Omega\right\vert} \frac{\hat
D_{\mu,\mu}(k)}{\hat F^{2}(k)} + 2(z\alpha{\left\vert\Omega\right\vert})^{2}
\frac{ \hat D_{\mu}^{2}(k)}{\hat F^{3}(k)} + \frac{ \hat
\Pi_{\mu,\mu}(k)}{\hat F^{2}(k)} + 4z\alpha{\left\vert\Omega\right\vert}
\frac{\hat D_{\mu}(k)\hat \Pi_{\mu}(k)}{\hat F^{3}(k)} + 2
\frac{\hat \Pi^{2}_{\mu}(k)}{\hat F^{3}(k)}.$$ To obtain an estimate of ${\|{\left\vertx\right\vert}^{2}H_{\lambda,z}\|}_{\infty}$ take the absolute value of inside of the integral and estimate the resulting one norms. Using $z\alpha{\left\vert\Omega\right\vert}
\leq 1+ O(\beta)$ an upper bound for the first two terms is $$(1+O(\beta)){\left( {\| \frac{\hat D_{\mu,\mu}(k)} {(1-\hat
D(k))^{2}}\|}_{1} + 2{\| \frac{ \hat D_{\mu}^{2}(k)}
{(1-\hat D(k))^{3}}\|}_{1}\right )} \leq c\beta,$$ where the second inequality follows by estimating the integrals, see [@MadrasSlade2013 Appendix A].
For the remaining terms, ${\|{\left\vertx\right\vert}^{2}\Pi_{\lambda,z}\|}_{1}\leq
c\beta$ implies ${\|\hat \Pi_{\mu,\mu}\|}_{\infty}\leq
c\beta$. Since $\hat \Pi_{\mu}(k)=0$ when $k_{\mu}=0$ Taylor’s theorem and the above bound on ${\|\hat \Pi_{\mu,\mu}\|}_{\infty}$ imply ${\|\hat \Pi_{\mu}\|}_{\infty}\leq c\beta
{\left\vertk_{\mu}\right\vert}$. Lastly, ${\left\vert\hat D_{\mu}(k)\right\vert}_{\infty}\leq
c{\left\vertk_{\mu}\right\vert}$. These bounds combined with the $k$-space infrared bound imply each of the remaining terms are bounded by $c\beta$. This proves .
For ${\|{\left\vertx\right\vert}^{2}H_{\lambda,z}\|}_{2}$ use Parseval’s identity: ${\| \widehat{ {\left\vertx\right\vert}^{2}H_{\lambda,z}} \|}_{2} =
{\| \partial^{2}_{k}\hat H_{\lambda,z}\|}_{2}$. The previously described bounds for the numerators along with and imply that $\hat G_{\mu,\mu}(k)$ is square integrable in sufficiently high dimensions. This implies .
\[prop:LWW-Moment-Derivative-Small\] For $d$ sufficiently large and $0<z\leq z_{c}$ $$\label{eq:LWW-Moment-Derivative-Small}
{\|\partial^{v}_{z} {\left\vertx\right\vert}^{2} \Pi_{\lambda,z}\|}_{1} \leq
c\beta z_{c}^{-v}$$
Distribute the factor ${\left\vertx\right\vert}^{2}$ along the factors of $\bar
H_{\lambda,z}$ and $\bar G_{\lambda,z}$ as in the proof of . The proof is now essentially the same as for . For each term place the sup norm on the factor with the term ${\left\vertx\right\vert}^{2}$.
If a factor ${\left\vertx\right\vert}^{2}G_{\lambda,z}$ has been differentiated once or twice the resulting term whose norm must be estimated has the form of either $\bar
H_{\lambda,z}\ast \bar G_{\lambda,z}$ or $\bar H_{\lambda,z} \ast
\bar G_{\lambda,z} \ast \bar G_{\lambda,z}$. In either case the factor ${\left\vertx\right\vert}^{2}$ can again be split along the factors in the convolution. In the first case use , the triangle inequality, and Young’s inequality to obtain $${\|({\left\vertx\right\vert}^{2}\bar H_{\lambda,z}) \ast \bar
G_{\lambda,z}\|}_{\infty} \leq {\|{\left\vertx\right\vert}^{2}
\bar H_{\lambda,z}\|}_{\infty} + {\|{\left\vertx\right\vert}^{2}\bar
H_{\lambda,z}\|}_{2}{\|\bar H_{\lambda,z}\|}_{2},$$ and then use to see that this is bounded by $c\beta$. For the second case arguing similarly gives $$\begin{aligned}
\nonumber
{\|({\left\vertx\right\vert}^{2}\bar H_{\lambda,z}) \ast \bar G_{\lambda,z} \ast
\bar G_{\lambda,z}\|}_{\infty} \leq\, &
{\|({\left\vertx\right\vert}^{2} \bar H_{\lambda,z}) \ast \bar
G_{\lambda,z}\|}_{\infty} + {\|({\left\vertx\right\vert}^{2} \bar H_{\lambda,z})
\ast \bar H_{\lambda,z}\|}_{\infty} \\ &+ {\|{\left\vertx\right\vert}^{2} \bar
H_{\lambda,z}\|}_{2}{\|\bar H_{\lambda,z} \ast \bar
H_{\lambda,z}\|}_{2}.
\end{aligned}$$ The first case analysis implies the first two terms are bounded above by $c\beta$. Parseval’s identity combined with implies the last term is bounded by $c\beta$. The rest of the analysis of these terms is in the proof of .
The cases in which all derivatives fall on factors without the term ${\left\vertx\right\vert}^{2}$ can be handled in the same manner as in the proof of by using Young’s inequality, the triangle inequality, and .
Linear Divergence of $\chi_{\lambda}(z)$ as $z\nearrow z_{c}$ {#sec:LWW-Susceptibility-MF}
-------------------------------------------------------------
Before proving the linear divergence of the susceptibility it will be helpful to verify that it is only infinite at the critical point $z=z_{c}$ itself.
\[lem:LWW-Susceptibility-Infinite\] For $d$ sufficiently large and ${\left\vertz\right\vert}\leq z_{c}$ the inverse susceptibility $\hat F_{\lambda,z}(0)$ satisfies $$\label{eq:LWW-Susceptibility-Infinite}
{\left\vert\hat F_{\lambda,z}(0)\right\vert} \geq \frac{{\left\vert\Omega\right\vert}}{2} {\left\vertz_{c}-z\right\vert}.$$
As $\hat F_{\lambda,z_{c}}(0)=0$ the fundamental theorem of calculus implies $${\left\vertF_{\lambda,z}(0)\right\vert} = {\left\vert\int_{z_{c}}^{z}-\frac{d}{dz}\hat F_{z}(0)\,dz\right\vert}.$$ Using $\hat F_{\lambda,z_{c}}(0)=0$, , and integrating from $z_{c}$ to $z$ along the straight line $z_{t} = (1-t)z_{c} + tz$ implies $${\left\vert\hat F_{\lambda,z}(0)\right\vert} = {\left\vert\Omega\right\vert}{\left\vertz-z_{c}\right\vert} {\left\vert
\int_{0}^{1} \alpha(\lambda,z_{t}) + z_{t}\frac{d}{dz}\alpha(\lambda,z_{t})
+ {\left\vert\Omega\right\vert}^{-1}\frac{d}{dz}\hat \Pi_{\lambda,z_{t}}(0)\,
dt\right\vert}.$$ The last two terms are bounded by $c\beta$, see the proof of . The claim follows by taking the dimension sufficiently large as $\int \alpha = 1 + O(\beta)$.
Define constants $A=A(\lambda)$ and $D = D(\lambda)$ by $$\begin{aligned}
\label{eq:LWW-Constant-A}
A(\lambda) &= z_{c}^{-1}{\left(\alpha(\lambda,z_{c}){\left\vert\Omega\right\vert} +
z_{c}{\left\vert\Omega\right\vert} \frac{d}{dz}\alpha(\lambda,z_{c}) + \frac{d}{dz}\hat
\Pi_{\lambda,z_{c}}(0)\right )}^{-1}, \\
\label{eq:LWW-Constant-D}
D(\lambda) &= A(\lambda) {\left( - z_{c}{\left\vert\Omega\right\vert}\alpha(\lambda,z_{c})
\nabla^{2}_{k} \hat D(0) - \nabla^{2}_{k}\hat \Pi_{\lambda,z_{c}}(0)\right )}.\end{aligned}$$
\[thm:LWW-Susceptibility-MF\] For $d$ large enough, the susceptibility of $\lambda$-LWW diverges linearly as $z\nearrow z_{c}$: $$\label{eq:LWW-Susceptibility-MF}
\chi_{\lambda}(z) \sim \frac{Az_{c}}{z_{c}-z}.$$ The constant $A$ in is as in .
Recall $\hat F_{\lambda,z}(0) = \hat G_{\lambda,z}(0)^{-1}$ is zero at $z_{c}$ since $\chi_{\lambda}(z)\nearrow \infty$ as $z\nearrow
z_{c}$. $$\begin{aligned}
\chi_{\lambda}(z) &= \frac{1}{\hat F_{\lambda,z}(0) - \hat
F_{\lambda,z_{c}}(0)} \\
&= \frac{1}{z_{c}-z} {\left[ \alpha(\lambda,z_{c}){\left\vert\Omega\right\vert} +
z{\left\vert\Omega\right\vert} \frac{\alpha(\lambda,z_{c}) -
\alpha(\lambda,z)}{z_{c}-z} + \frac{\hat
\Pi_{\lambda,z_{c}}(0) - \hat
\Pi_{\lambda,z}(0)}{z_{c}-z}\right ]}^{-1}.
\end{aligned}$$ The claim follows from and combined with $\alpha_{0}\leq 1 + c\beta$ for $z\leq z_{c}$, which implies differentiability of $\alpha_{0}$ at $z_{c}$.
Growth Rate and Diffusive Scaling {#sec:LWW-Growth}
---------------------------------
To establish the growth rate of $\lambda$-LWW, as well as the diffusive scaling, a Tauberian type theorem is needed. The statement and proof of the next lemma in [@MadrasSlade2013] involve fractional derivatives of order $1+\epsilon$ for $0<\epsilon<1$, but the arguments apply without modification for two ordinary derivatives.
\[lem:LWW-Tauberian\] Let $$\label{eq:LWW-Taub-1}
f(z) = \frac{1}{\phi(z)} = \sum_{n=0}^{\infty}b_{n}z^{n},$$ where $\phi(z) = \sum_{n=0}^{\infty}a_{n}z^{n}$. Suppose that $$\label{eq:LWW-Taub-2}
\sum_{n=0}^{\infty}n^{2}{\left\verta_{n}\right\vert}R^{n}<\infty,$$ so in particular, $\phi(z)$, $\phi^{\prime}(z)$, and $\phi^{\prime\prime}(z)$ are finite when ${\left\vertz\right\vert}=R$. Assume in addition that $\phi^{\prime}(R)\neq 0$. Suppose that $\phi(R)=0$ and $\phi(z)\neq 0$ for ${\left\vertz\right\vert}\leq R$, $z\neq R$. Then $$\label{eq:LWW-Taub-3}
f(z) = \frac{1}{-\phi^{\prime}(R)}\frac{1}{R-z} + O(1)$$ uniformly in ${\left\vertz\right\vert}\leq R$, and $$\label{eq:LWW-Taub-4}
b_{n} = R^{-n-1}{\left[\frac{1}{-\phi^{\prime}(R)} + O(n^{-\alpha})\right ]}
\quad \textrm{as $n\to\infty$},$$ for every $\alpha<1$.
Recall that $c_{n}^{\lambda}$ is the total mass of $n$-step $\lambda$-LWW, i.e., $$c_{n}^{\lambda} = \sum_{x} \mathop{\sum_{\omega\colon 0
\to x}}_{{\left\vert\omega\right\vert}=n} \lambda^{n_{L}(\omega)}.$$
\[thm:LWW-Growth-Rate\] For $d$ sufficiently large and any $\delta<1$ $$c_{n}^{\lambda} = A(\lambda)z_{c}(\lambda)^{-n}(1+O(n^{-\delta})).$$
Apply to $\hat F_{\lambda,z}(0)$. The verification of the hypotheses of the theorem are the conclusions of , , and .
The proof of the next theorem is essentially the proof for self-avoiding walk in [@MadrasSlade2013] verbatim; it is reproduced here for the sake of completeness. The next lemma, which will be used several times, is stated here for the convenience of the reader.
\[lem:LWW-SL6.3.2\] Let $f(z) = \sum_{n=0}^{\infty} a_{n}z^{n}$. Let $R>0$, and suppose $f^{\prime}(R) = \sum_{n=0}^{\infty}n {\left\verta_{n}\right\vert}R^{n-1}<\infty$, so in particular $f(z)$ converges for ${\left\vertz\right\vert}\leq R$. Then for ${\left\vertz\right\vert}\leq R$ $${\left\vertf(z)- f(R)\right\vert} \leq f^{\prime}(R) {\left\vertR-z\right\vert}.$$ If $f^{\prime\prime}(z)(R)<\infty$, then for ${\left\vertz\right\vert}\leq R$ $${\left\vertf(z) - f(R) - f^{\prime}(R)(z-R)\right\vert} \leq \frac{1}{2}
f^{\prime\prime}(R) {\left\vertR-z\right\vert}^{2}.$$
\[thm:LWW-Diffusive\] For $d$ sufficiently large $\lambda$-LWW is diffusive: $$\label{eq:LWW-Diffusive}
{\langle {\left\vert\omega(n)\right\vert}^{2} \rangle}^{\lambda}_{n} = Dn(1+O(n^{-\delta}))$$ as $n\to\infty$ for any $\delta<1$. The constant $D$ is that of .
Let $\nabla^{2}_{k}$ denote the $k$-space Laplacian. Then $$\label{eq:LWW-Diffusive-1}
{\langle {\left\vert\omega(n)\right\vert}^{2} \rangle}_{\lambda,n} = -\frac{\nabla^{2}_{k}\hat
c_{n}^{\lambda}(0)}{ c_{n}^{\lambda}}.$$ Since $\hat c_{n}^{\lambda}(k)$ is the coefficient of $z^{n}$ in $\hat
G_{\lambda,z}(k)$ Cauchy’s formula implies $$\label{eq:LWW-Diffusive-2}
-\nabla^{2}_{k}\hat c_{n}^{\lambda}(0) = \frac{1}{2\pi i} \oint \frac{
\nabla^{2}_{k} \hat F_{\lambda,z}(0)}{ \hat
F_{\lambda,z}(0)^{2}} \frac{dz}{z^{n+1}},$$ where the integral is around a small origin centred circle. Define $E(z)$ by $$\label{eq:LWW-Diffusive-3}
\frac{\nabla^{2}_{k} \hat F_{\lambda,z}(0)}{ \hat
F_{\lambda,z}(0)^{2}} = \frac{ \nabla^{2}_{k}\hat F_{z_{c}}(0)}
{ {\left[ \frac{d}{dz}\hat F_{z_{c}}(0)\right ]}^{2}(z_{c}-z)^{2}} + E(z).$$ Making this substitution into and calculating the first integral implies $$\label{eq:LWW-Diffusive-4}
-\nabla^{2}_{k}\hat c_{n}^{\lambda}(0) = \frac{ \nabla^{2}_{k}\hat F_{z_{c}}(0)}
{ {\left[ \frac{d}{dz}\hat F_{z_{c}}(0)\right ]}^{2}}(n+1)z_{c}^{-n-2} +
\frac{1}{2\pi i} \oint E(z) \frac{dz}{z^{n+1}}.$$
Assuming the integral of $E(z)$ is $O(n^{\delta}z_{c}^{-n})$ for every $\delta>0$ implies the theorem by inserting the behaviour of $c_{n}^{\lambda}$ given by .
To verify the assumption it suffices by to prove ${\left\vertE(z)\right\vert} \leq \mathrm{const.} {\left\vertz_{c}-z\right\vert}^{-1}$ for all ${\left\vertz\right\vert}\leq z_{c}$. Split $E(z)$ as $E(z) = T_{1}(z) +
T_{2}(z)$ with $$\begin{aligned}
\label{eq:LWW-Diffusive-6}
T_{1}(z) &= {\left[\frac{d}{dz}\hat F_{\lambda,z_{c}}(0)\right ]}^{-2} \frac{
\nabla^{2}_{k} \hat F_{\lambda,z}(0) - \nabla^{2}_{k}\hat
F_{\lambda,z_{c}}(0)}{(z_{c}-z)^{2}} \\
\label{eq:LWW-Diffusive-7}
T_{2}(z) &= \frac{ - \nabla^{2}_{k} \hat F_{\lambda,z}(0) {\left[
\hat F_{\lambda,z}(0)^{2} - {\left[\frac{d}{dz}\hat
F_{\lambda,z_{c}}(0)\right ]}^{2}(z_{c}-z)^{2}\right ]}
}{{\left[\frac{d}{dz}\hat F_{\lambda,z_{c}}(0)\right ]}^{2} \hat
F_{\lambda,z}(0)^{2}(z_{c}-z)^{2}}.
\end{aligned}$$
The numerator of $T_{1}(z)$ is differentiable in $z$ by , so (i) of implies the numerator is bounded above by a constant times ${\left\vertz_{c}-z\right\vert}$. It follows that ${\left\vertT_{1}\right\vert}\leq
O({\left\vertz_{c}-z\right\vert}^{-1})$.
For $T_{2}$ note that $\hat F_{\lambda,z}(0)^{2} \geq
\textrm{const.}{\left\vertz_{c}-z\right\vert}^{2}$ by so $$\begin{aligned}
\label{eq:LWW-Diffusive-8}
{\left\vertT_{2}(z)\right\vert} \leq \mathrm{const.}{\left\vertz_{c}-z\right\vert}^{-4} {\left[ \hat
F_{\lambda,z}(0) + \frac{d}{dz}\hat
F_{\lambda,z_{c}}(0)(z_{c}-z)\right ]} {\left[ \hat F_{\lambda,z}(0)
- \frac{d}{dz} \hat F_{\lambda,z_{c}}(0)(z_{c}-z)\right ]},
\end{aligned}$$ as $\nabla^{2}_{k}\hat F_{\lambda,z}(0)$ is bounded by a constant by . By (ii) of , , and $\hat
F_{\lambda,z_{c}}(0) = 0$, the middle term is $O({\left\vertz_{c}-z\right\vert}^{2})$. Using (i) of for $\hat F_{\lambda,z}(0)$ in the last term shows the last term is $O({\left\vertz_{c}-z\right\vert})$. Thus ${\left\vertT_{2}(z)\right\vert}\leq O({\left\vertz_{c}-z\right\vert}^{-1})$, which proves the claim.
Loop Measure Representation of $\lambda$-LWW {#sec:LWW-Viennot}
============================================
The purpose of this appendix is to provide a proof of .
A fundamental property of $\lambda$-LWW is that it admits a loop measure representation. The representation follows from a theorem of Viennot [@Viennot1986] and is proved via the theory of heaps of pieces in .
\[rem:LWW-Use-of-Heaps\] The methods of [@LawlerLimic2010 Chapter 9] are sufficient to derive formulas that would suffice for the lace expansion analysis of $\lambda$-LWW. These methods have the benefit of brevity, but they do not reveal the connection with the loop $O(N)$ model. For this reason we have chosen to present a more scenic route here.
The rest of this section will take place in the context of an arbitrary graph $G$, as specializing to ${{\mathbb{Z}}}^{d}$ does not provide any simplification. The theory of heaps of pieces will be freely used; see [@Viennot1986] or [@Krattenthaler2006] for an introduction.
Viennot’s Theorem {#app:LWW-Viennot}
-----------------
\[def:LWW-Cycle\] A *trivial cycle* is a single edge of $G$. An *oriented cycle* is either (i) an oriented cyclic subgraph of $G$ or (ii) a trivial cycle in $G$.
An oriented cycle corresponds to an equivalence class of self-avoiding polygons, where a self-avoiding polygon $\omega = (\omega_{0}, \dots,
\omega_{k}=\omega_{0})$ is equivalent to any cyclic permutation $\tilde\omega = (\omega_{r}, \omega_{r+1}, \dots, \omega_{k},
\omega_{1}, \dots, \omega_{r})$. For example, a trivial cycle $\{x,y\}$ corresponds to the self-avoiding polygons $(x,y,x)$ and $(y,x,y)$, while an oriented 3-cycle corresponds to walks of the form $(x,y,z,x)$ and cyclic permutations thereof for $x,y,z$ distinct.
\[def:LWW-Heap-Cycles\] A *heap of (oriented) cycles* is a heap of pieces whose labels are oriented cycles. Two oriented cycles $C_{1}$, $C_{2}$ are concurrent if $V(C_{1})\cap V(C_{2}) \neq
\emptyset$, i.e., if the cycles share a vertex.
\[def:LWW-IH-Legal-Pair\] A pair $(\eta, H)$ where $\eta$ is a self-avoiding walk from $a$ to $b$ and $H$ is a heap of cycles whose maximal elements’ labels each contain a vertex in $\eta$ is called a *legal $(a,b)$ pair*. Let ${{\mathcal{V}}}(a,b)$ denote the set of legal $(a,b)$ pairs, and ${{\mathcal{V}}}$ denote the set of all legal pairs.
, the loop measure representation of $\lambda$-LWW, is a byproduct of the proof of the following theorem of Viennot.
\[thm:LWW-Bijection\] There is a bijection $\phi_{ab}$ from the set ${{\mathcal{V}}}(a,b)$ of legal $(a,b)$ pairs to the set of walks ${\Omega}(a,b)$ from $a$ to $b$. Further,
1. The multi-set of edges in a legal $(a,b)$ pair $(\eta,H)$ is the same as the multi-set of edges in the walk $\phi_{ab}((\eta,H))$.
2. The multi-set of oriented cycles $\{ \ell(x) \mid x\in H\}$ for a heap $(H,\ell,\preceq)$ is the same as the multi-set of oriented cycles that are erased by applying loop erasure to $\phi_{ab}(
(\eta,H))$.
is not proven in [@Viennot1986]. For the sake of completeness and the convenience of the reader a proof is given in . The remainder of this section consists of a heuristic description of the proof; see also which depicts the proof strategy.
[1.0]{}
in [0,...,6]{} in [0,...,5]{} () at (,) ;
(-1,-1) rectangle (7,6); (0,1) to (5,1) to (5,5) to (6,5) to (6,4) to (54); (54) to (2,4) to (2,5) to (3,5) to (34); (34) to (3,2) to (52); (52) to (6,2) to (6,3) to (53); (53) to (4,3) to (42); (42) to (41); (41) to (4,0);
in [0,...,6]{} in [0,...,5]{} () at (,) ; (-1,-1) rectangle (7,6);
(0,1) to (5,1) to (5,5) to (6,5) to (6,4) to (54); (54) to (2,4) to (2,5) to (3,5) to (34); (34) to (3,2) to (52); (52) to (6,2) to (6,3) to (53); (53) to (4,3) to (42); (42) to (41); (41) to (4,0);
[1.0]{}
in [0,...,6]{} in [0,...,5]{} () at (,) ;
(-1,-1) rectangle (7,6); (0,1) to (5,1) to (5,4) to (2,4) to (2,5) to (3,5) to (34); (34) to (3,2) to (52); (52) to (6,2) to (6,3) to (53); (53) to (4,3) to (42); (42) to (41); (41) to (4,0);
in [0,...,6]{} in [0,...,5]{} () at (,) ;
(-1,-1) rectangle (7,6); (5,4.75) to (5,5) to (6,5) to (6,4) to (54) to (5,4.75); at (54) ;
(-1,-1) rectangle (7,6); (0,1) to (5,1) to (5,4) to (2,4) to (2,5) to (3,5) to (34); (34) to (3,2) to (52); (52) to (6,2) to (6,3) to (53); (53) to (4,3) to (42); (42) to (41); (41) to (4,0);
[1.0]{}
in [0,...,6]{} in [0,...,5]{} () at (,) ;
(-1,-1) rectangle (7,6); (0,1) to (5,1) to (5,4) to (3,4) to (3,2) to (52); (52) to (6,2) to (6,3) to (53); (53) to (4,3) to (42); (42) to (41); (41) to (4,0);
in [0,...,6]{} in [0,...,5]{} () at (,) ;
(-1,-1) rectangle (7,6); (5,4.75) to (5,5) to (6,5) to (6,4) to (5,4) to (5,4.75); (2,4.75) to (2,5) to (3,5) to (34) to (2,4) to (2,4.75); at (34) ;
(0,1) to (5,1) to (5,4) to (3,4) to (3,2) to (52); (52) to (6,2) to (6,3) to (53); (53) to (4,3) to (42); (42) to (41); (41) to (4,0);
[1.0]{}
in [0,...,6]{} in [0,...,5]{} () at (,) ;
(-1,-1) rectangle (7,6); (0,1) to (5,1) to (5,2) to (6,2) to (6,3) to (5,3); (5,3) to (4,3) to (4,2); (4,2) to (41); (41) to (4,0);
in [0,...,6]{} in [0,...,5]{} () at (,) ;
(-1,-1) rectangle (7,6); (5,4.75) to (5,5) to (6,5) to (6,4) to (54) to (5,4.75); (2,4.75) to (2,5) to (3,5) to (34) to (2,4) to (2,4.75); (3,3) to (3,2) to (52) to (5,4) to (3,4) to (3,3); at (52) ;
(0,1) to (5,1) to (5,2) to (6,2) to (6,3) to (5,3); (5,3) to (4,3) to (4,2); (4,2) to (41); (41) to (4,0);
[1.0]{}
in [0,...,6]{} in [0,...,5]{} () at (,) ;
(-1,-1) rectangle (7,6); (0,1) to (4,1) to (4,0);
in [0,...,6]{} in [0,...,5]{} () at (,) ;
(-1,-1) rectangle (7,6); (5,4.75) to (5,5) to (6,5) to (6,4) to (54) to (5,4.75); (2,4.75) to (2,5) to (3,5) to (34) to (2,4) to (2,4.75); (3,3) to (3,2) to (42) to (52) to (53) to (5,4) to (3,4) to (3,3); (4,1.5) to (41) to (5,1) to (5,2) to (6,2) to (6,3) to (4,3) to (4,1.5); at (41) ;
(0,1) to (4,1) to (4,0);
Let $\omega$ be a walk from $a$ to $b$. Trace $\omega$ until the first time a vertex is visited twice. This identifies a first closed subwalk $C_{1} = (\omega_{\tau^{\star}_{\omega}}, \dots,
\omega_{\tau_{\omega}})$. Remove $C_{1}$ by performing a single loop erasure, and form a heap of pieces consisting of a single piece labelled $C_{1}$. The first time a vertex is visited twice by the walk ${\mathrm{LE}}^{1}(\omega)$ identifies a second closed subwalk, call this $C_{2}$. Remove $C_{2}$ and form a new heap of pieces by adding a second piece labelled $C_{2}$ to the heap consisting of $C_{1}$. Continuing in this manner removes all of the closed subwalks from $\omega$, resulting in a self-avoiding walk $\eta$ from $a$ to $b$. Each maximal piece in the heap is labelled by a cycle that shares a vertex with $\eta$. In other words, this procedure converts each walk from $a$ to $b$ into a legal pair $(\eta,H)$.
Conversely, consider a legal pair $(\eta,H)$. To invert the procedure what is required is a way to reduce the heap to the empty heap one piece at a time, while inserting the labels of the removed pieces into the (initially) self-avoiding walk $\eta$. This is relatively straightforward: the maximal pieces of the heap $H$ have labels that share a vertex with $\eta$, and hence the maximal pieces can be ordered by using the linear order on vertices in $\eta$. Take the maximal piece in this order, remove it from the heap to get a heap $H^{\prime}$, and glue the corresponding label into $\eta$ to get a walk $\eta^{\prime}$. The maximal elements of $H^{\prime}$ have labels that share a vertex with $\eta^{\prime}$, and hence this procedure can be iterated.
These operations are in fact inverses of one another. The next section makes the preceding discussion precise.
Proof of Viennot’s Theorem {#app:LWW-Viennot-Proof}
--------------------------
The theorem requires two algorithms, one which inserts oriented cycles into a given walk, and one which removes oriented cycles from a walk. Removing oriented cycles is achieved by loop erasure. The other algorithm is introduced now.
\[def:LWW-Loop-Insertion\] Let $\omega$ be a walk of length $n$, and let $C$ be an oriented cycle of length $k$. Assume that $C$ and $\omega$ have a vertex in common, and let $i$ be the minimal index such that $\omega_{i}$ is a vertex in $C$. Let $(c_{0}, \dots, c_{k})$ be the unique representative of $C$ such that $c_{0}=\omega_{i}$. The *loop insertion $\omega {\oplus}C$ of $C$ into $\omega$* is the walk $(\omega_{0}, \dots, \omega_{i-1}, c_{0}, \dots, c_{k}, \omega_{i+1},
\dots, \omega_{n})$.
In words, to insert a loop $C$ into a walk $\omega$ we find the first vertex $\omega_{i}$ in $\omega$ that is contained in $C$. $C$ is then rooted at $\omega_{i}$, $\omega$ is traversed until just before reaching $\omega_{i}$, $C$ is traversed, and then the remainder of $\omega$ is traversed.
\[lem:LWW-Remove-Insert\] Let $\omega$ be a walk, and let $C$ be the oriented cycle removed to create ${\mathrm{LE}}^{1}(\omega)$. Then ${\mathrm{LE}}(\omega){\oplus}C = \omega$.
The definition of $\tau^{\star}_{\omega}$ and the definition of loop erasure implies that the first vertex in common between ${\mathrm{LE}}^{1}(\omega)$ and $C$ is $\omega_{\tau^{\star}_{\omega}}$, and hence the closed self-avoiding walk representing $C$ that is inserted by loop insertion is $(\omega_{\tau^{\star}_{\omega}},
\dots, \omega_{\tau_{\omega}})$.
Given a collection of oriented cycles that intersect a walk it is necessary to determine the order in which the cycles should be inserted. The next definition gives the correct order for inverting loop erasure.
\[def:LWW-Walk-Order\] Let $\omega$ be a walk, and $C_{1},\dots, C_{k}$ a collection of oriented cycles that each share a vertex with $\omega$. Let $t_{j}=\min
\{i \mid \omega_{i}\in C_{j}\}$. The *walk order* on the oriented cycles is given by setting $C_{m}\geq C_{n}$ if $t_{m}\geq t_{n}$.
The following algorithm, called the *loop addition algorithm*, constructs a walk beginning at the vertex $a$ and ending at the vertex $b$ from a legal $(a,b)$ pair $(\eta,H)$.
1. Set $\omega^{0}=\eta$.
2. Suppose $H^{i-1}\neq\emptyset$. Set $\omega^{i} =
\omega^{i-1}{\oplus}C$, where $C$ is maximal in the walk order among the labels of the maximal pieces of $H^{i-1}$. Let $y$ be the piece whose label is $C$, and set $H^{i}=H^{i-1}\setminus \{y\}$.
3. If $H^{i-1}=\emptyset$, output $\omega=\omega^{i-1}$. Otherwise go to 2.
The algorithm is well-defined as the labels of the maximal pieces in a heap must be vertex disjoint, so the walk order is a strict total order on the maximal pieces of the heap. Note that at each step of the algorithm the walk $\omega^{i}$ begins at the vertex $a$ and ends at $b$, so $\omega$ is a walk from $a$ to $b$ as claimed.
\[lem:LWW-Insert-Remove\] Suppose $(\eta,H)\in{{\mathcal{V}}}$. Suppose the output of the loop addition algorithm is $\omega$. If $C$ is the last oriented cycle inserted, then the oriented cycle removed by loop erasure applied to $\omega$ is $C$.
The proof is by induction on the size of $H$. Suppose $C$ was the $(k+1)^{\mathrm{st}}$ oriented cycle added.
1. If $C$ was the label of a maximal element in $H^{k-1}$ then $C$ is vertex disjoint from the $k^{\mathrm{th}}$ added oriented cycle $C^{\prime}$. The definition of the walk order implies that the first vertex $C$ shares with $\omega^{k-1}$ occurs prior to the first vertex in $C^{\prime}$ because $C$ is disjoint from $C^{\prime}$. It follows that $C$ is the oriented cycle erased by loop erasure, as $C$ closes prior to $C^{\prime}$, which was previously (by induction) the first oriented cycle to close.
2. If $C$ was not the label of a maximal piece in $H^{k-1}$ then $C$ is the label of a piece that was below the $k^{\mathrm{th}}$ inserted piece. Suppose the $k^{\mathrm{th}}$ piece had label $C^{\prime}$. As $C$ intersects $C^{\prime}$, $C$ is inserted into the subwalk $C^{\prime}$ of $\omega^{k-1}$. By induction $C^{\prime}$ was the first oriented cycle to close in $\omega^{k-1}$, so $C$ is the first oriented cycle to close in $\omega^{k}$.
To construct a legal pair $(\eta,H)$ from a walk is fairly straightforward. By applying loop erasure oriented cycles are removed, and they naturally form a heap by using the heap composition operation. More precisely, we have the *(total) loop erasure algorithm*:
1. Set $\omega^{0}=\eta$, and $H^{0}=\emptyset$, where $\emptyset$ is the empty heap of oriented cycles.
2. If $\omega^{i-1}$ is not a self-avoiding walk, set $\omega^{i}={\mathrm{LE}}^{1}(\omega^{i-1})$, and if $C$ is the closed self-avoiding walk removed from $\omega^{i-1}$, let $H^{i}=H\circ \{\bar C\}$ where $\bar C$ is the oriented cycle corresponding to $C$.
3. If $\omega^{i-1}$ is a self-avoiding walk, output $(\omega^{i-1},H^{i-1})$. Otherwise go to 2.
Single loop erasure removes a subwalk of length at least 2 from any non-simple walk at each step, so iteratively applying ${\mathrm{LE}}^{1}$ stabilizes on a self-avoiding walk in a finite number of iterations. It follows that the total loop erasure is well defined.
\[lem:LWW-Erasure-Heap\] The output of the loop erasure algorithm applied to a walk $\omega =
(\omega_{0}, \dots, \omega_{n})$ is a pair $(\eta,H)\in{{\mathcal{V}}}(\omega_{0},\omega_{n})$.
At each step of the algorithm the maximal pieces of the heap $H^{i}$ share a vertex with the remaining walk $\omega^{i}$, and the algorithm only terminates once the remaining walk is self-avoiding. Removing a cycle cannot change the initial vertex of a walk, so $\eta_{0}=\omega_{0}$. If the final vertex of $\omega$ is removed it must be that visiting the final vertex completes a cycle, and hence $\eta$ ends at $\omega_{n}$.
We claim that loop erasure and loop addition are inverses of one another, and prove the claim by induction. Suppose the claim holds between walks whose loop erasure removes $k$ oriented cycles and pairs $(\eta,H)\in{{\mathcal{V}}}(a,b)$ whose heap $H$ has $k$ pieces.
On the one hand, inserting the final oriented cycle $C$ in the loop addition algorithm yields a walk, and $C$ is the first oriented cycle removed by loop erasure by . By induction it follows that loop erasure applied to the loop addition of a pair $(\eta,H)\in{{\mathcal{V}}}(a,b)$ returns $(\eta,H)$.
On the other hand when a single oriented cycle $C$ is removed from $\omega$ the cycle $C$ is minimal in the walk order and the removed oriented cycle is the label of a maximal piece. So the reconstruction of the heap formed by loop erasure proceeds as if the piece with label $C$ was not present, and hence (by induction) recreates ${\mathrm{LE}}^{1}(\omega)$ correctly. then implies that $\omega$ is the output of applying loop erasure and then loop addition.
Proof of {#app:LWW-LM-Rep}
---------
The proof of follows from two calculations. The first is a straightforward consequence of the fact that the bijection between walks and legal pairs is given by loop erasure. Let ${\mathcal{T}}$ denote the set of trivial heaps of oriented cycles, and ${\mathcal{H}}$ the set of heaps of oriented cycles. Let ${\vec{{\mathcal{C}}}}(\eta)$ denote the set of oriented cycles that *do not* share a vertex with the set $\eta$, and let ${\mathcal{H}}_{\eta}$ denote the set of heaps $H$ such that $(\eta,H)$ is a legal pair. The definition of $\lambda$-LWW, , and the heap theorem [@Viennot1986 Proposition 5.3] imply $$\begin{aligned}
\label{eq:LWW-LM-Triv-1}
\bar w_{\lambda,z}(\eta) &= \sum_{\omega\colon {\mathrm{LE}}(\omega)=\eta}
w_{\lambda,z}(\omega) \\
\label{eq:LWW-LM-Triv-2}
&= z^{{\left\vert\eta\right\vert}} \sum_{H\in{\mathcal{H}}_{\eta}} w_{\lambda,z}(H) \\
\label{eq:LWW-LM-Triv-3}
&= z^{{\left\vert\eta\right\vert}} \frac{ \sum_{T\in{\mathcal{T}}({\vec{{\mathcal{C}}}}(\eta))}
(-1)^{{\left\vertT\right\vert}}w_{\lambda,z}(T)} { \sum_{T\in {\mathcal{T}}}(-1)^{{\left\vertT\right\vert}}w_{\lambda,z}(T)},\end{aligned}$$ where $w_{\lambda,z}(H) = \prod_{x\in H}w_{\lambda,z}(\ell(x))$ for a heap $(H,\ell,\preceq)$. In particular note that this definition assigns a weight $z^{2}\lambda$ to a trivial cycle.
The second calculation is an expression for sums over trivial heaps of oriented cycles. follows by applying to the numerator and denominator of and cancelling common factors. This calculation is a calculation involving formal power series; to see that it holds as a relation between power series, note that for $z$ sufficiently small the final expressions are bounded by random walk quantities, which converge.
\[prop:LWW-SO\] $$\label{eq:LWW-SO}
\sum_{T\in{\mathcal{T}}({\vec{{\mathcal{C}}}}(A))} (-1)^{{\left\vertT\right\vert}}w_{\lambda,z}(T) = \exp {\left( -
\sum_{x\in {{\mathbb{Z}}}^{d}} \mathop{\sum_{\omega\colon x\to
x}}_{{\left\vert\omega\right\vert}\geq 1}
{{\mathbbm{1}}_{\left\{{\mathrm{range}(\omega)}\cap A = \emptyset\right\}}}
\frac{w_{\lambda,z}(\omega)}{{\left\vert\omega\right\vert}}\right )}$$
Let $\bar z = sz$. Then $w_{\lambda,z}(\omega)=w_{\lambda,\bar z}(\omega)$ when $s=1$. Using this observe that $$\sum_{T\in{\mathcal{T}}({\vec{{\mathcal{C}}}}(A))} (-1)^{{\left\vertT\right\vert}}w_{\lambda,z}(T) = \exp
\int_{0}^{1}\frac{d}{ds} \log \sum_{T\in{\mathcal{T}}({\vec{{\mathcal{C}}}}(A))}
(-1)^{{\left\vertT\right\vert}}w_{\lambda,\bar z}(T).$$ In calculating the derivative the Leibniz rule for differentiating $s^{k}$ can be interpreted as selecting one of the $k$ vertices contained in the cycles of a trivial heap. The selected vertex distinguishes a self-avoiding polygon. can be applied to transform this into a walk weighted by $w_{\lambda,z}$. The factor of $-1$ in the exponent arises from the application of , as the distinguished cycle carried a factor of $-1$. Lastly, the term ${\left\vert\omega\right\vert}^{-1}$ arises from the integration of $s^{{\left\vert\omega\right\vert}-1}$ from $0$ to $1$; the missing factor of $s$ is due to the differentiation which distinguished a vertex.
Relation to Correlations of the $O(N)$ Cycle Gas {#sec:LWW-Cycle-Correlation}
------------------------------------------------
Note that and imply that $G_{\lambda,z}(0,x)$ is given by a ratio of partition functions. The denominator is a sum over oriented mutually disjoint cyclic subgraphs, where the weight of a subgraph $H$ is $z^{{\left\vertE(H)\right\vert}}(-\lambda)^{\# H}$, where $\# H$ denotes the number of cyclic subgraphs contained in $H$. The numerator is a sum over self-avoiding walks from $0$ to $x$ along with disjoint cyclic subgraphs; the weight is the same as for the denominator except for the fact that the walk does not receive a factor of $\lambda$.
For each cycle of length at least $3$ summing over the possible orientations of the cycles results in a model of unoriented cycle, where each unoriented cycle has weight $-2\lambda$, except for trivial cycles, which have weight $-\lambda$. *Defining* the two-point correlation in the $O(N)$ cycle gas to be the ratio described in the previous paragraph gives the relation between the $O(N)$ cycle gas and $\lambda$-LWW. Note that if cycles of length two are assigned loop activity $0$ this yields a precise correspondence between $\lambda$-LWW and the $O(-2\lambda)$ cycle gas.
Acknowledgements {#sec:acknowledgements .unnumbered}
================
I would like to thank my PhD advisor, David Brydges, for many interesting and inspiring discussions related to this work, which formed a portion of my PhD thesis at the University of British Columbia. This paper was revised while I was at ICERM for the semester program *Phase Transitions and Emergent Properties*, and I would like to thank ICERM for their hospitality and support. Finally, I would like to thank Gordon Slade, Mark Holmes, and the referees for their helpful comments, critiques and references, which have greatly improved this article.
[^1]: Institute for Computational and Experimental Research in Mathematics, 121 South Main St. Providence, RI, 02903, USA.
[^2]: Current address: Department of Mathematics, 899 Evans Hall, Berkeley, CA, 94720-3840 USA. Email: jhelmt@math.berkeley.edu
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---
abstract: 'In the framework of the single-field slow-roll inflation, we derive the Hamiltonian of the linear primordial scalar and tensor perturbations in the form of time-dependent harmonic oscillator Hamiltonians. We find the invariant operators of the resulting Hamiltonians and use their eigenstates to calculate the adiabatic Berry phase for sub-horizon modes in terms of the Lewis-Riesenfeld phase. We conclude by discussing the discrepancy in the results of Pal et. al \[Class. Quant. Grav. [**30**]{}, 12 (2013)\] for these Berry phases, which is resolved to yield agreement with our results.'
author:
- 'Hamideh Balajany, Mohammad Mehrafarin'
title: 'Berry phase of primordial scalar and tensor perturbations in single-field inflationary models'
---
Introduction
============
Berry phase [@Berry] is a non-trivial geometric phase, distinct from the dynamical phase, that is picked up by a quantum system when it slowly traverses a closed path in the Hamiltonian parameter space. Because of the wide range of its applications, examples of Berry phase have appeared in many different areas of physics and astronomy [@Shapere; @Bohm; @Mehrafarin1; @Mehrafarin2; @Torabi1; @Torabi2; @Bakke; @Cai; @Cai:1990; @Corichi; @Mazur; @Dutta; @Melo; @Melo2; @Bakke2]. Of particular relevance to our work is the Berry phase of primordial cosmological perturbations, which are well accomodated in inflationary models [@Guth; @Bassett; @Mukhanov]. In single-field inflation, using the gauge invariant variable of Bardeen [@Bardeen:1980], the Berry phase has been obtained from the wave function of the perturbations by solving the associated Shrödinger equation [@Pal]. As the origin of our present universe, primordial perturbations have presumably left their mark to be traced in cosmological observations. In this regard, the Berry phase, as a footprint of the perturbations, can serve to probe the cosmological inflation [@Campo].
In this work, we obtain the Berry phase of the linear primordial perturbations in the single-field slow-roll inflation via a different approach. Our approach is based on reducing the problem to a time-dependent harmonic oscillator and, thereby, using the Lewis-Riesenfeld invariant operator method [@Lewis:1968; @Lewis; @Carvalho; @Pedrosa; @Pedrosa2] to obtain the Berry phase. This approach has been employed to obtain the Berry phase of relic gravitons in the FRW background [@Bakke]. Here, using the gauge invariant variables of Malik and Wands [@Malik], we derive the Hamiltonian of the scalar and tensor Fourier modes in the form of time-dependent harmonic oscillator Hamiltonians (Section 2). The Berry phase of a generalized harmonic oscillator has been derived in [@Monteoliva] using the Lewis-Riesenfeld invariant operator method. In the same manner, we find the invariant operators of the resulting Hamiltonians and use their eigenstates to calculate the adiabatic Berry phase for sub-horizon scalar and tensor modes as a Lewis-Riesenfeld phase (Section 3). Finally, we discuss the discrepancy in the results of [@Pal] for these Berry phases, which is resolved to yield agreement with our results.
The perturbation Hamiltonian
============================
In the single-field model, the universe is dominated by a scalar field ${\bar\varphi}$ with potential $V({\bar\varphi})$. The action is $$S=\int d^4x\sqrt{-g}\,\frac{1}{2}\left[R-g^{\mu\nu}\partial_\mu\bar\varphi\partial_\nu\bar\varphi-2V(\bar\varphi)\right] \label{A}$$ where units have been chosen such that $8 \pi G=\hbar=c=1$. The background universe is the flat FRW spacetime $$ds^2=-{N}^2(t)dt^2+a^2(t)\delta_{ij}\label{metric}dx^idx^j$$ where $a$ is the scale factor and $N$ depends on the choice of the time variable. (Conformal and cosmic time correspond to $N=a$ and $N=1$, respectively.) The background scalar field, which depends only on time, is $\varphi(t)$ with conjugate momentum $\Pi=\dot{\varphi}/{N}$. In the ADM formalism [@Arnowitt], where $$ds^2=-\bar {N}^2dt^2+\bar{h}_{ij}(dx^i+N^i dt)(dx^j+N^jdt)$$ the perturbed universe has $\bar h_{ij}=a^2e^{2\alpha}\delta_{ij}+\gamma_{ij}$, where $\alpha(t,\bm{x})$ is the scalar curvature perturbation and $\gamma_{ij}(t,\bm{x})$ is a divergence-less and traceless metric perturbation that represents transverse gravity waves.
Let us first consider the scalar perturbations. The linear scalar gauge invariant perturbation variable is constructed from the curvature and field perturbations ($\alpha$ and $\delta\bar\varphi$) according to [@Malik] $$\zeta(t,{\bm x})=\alpha-\frac{H}{\Pi} \delta\bar\varphi$$ where $H(t)=\dot{a}/Na$ is the background Hubble parameter. The first order slow-roll parameters are given by $$\eta(t)=\frac{1}{NH} \frac{\dot{\Pi}}{\Pi },\ \ \
\epsilon (t) = -\frac{\dot{H}}{NH^2}.$$ Working in the uniform energy density gauge, $\delta\bar\varphi=0$, action (\[A\]) to the second order in perturbation variable $\zeta$ is given by [@Tzavara] $$S_{\text{scalar}}=\int d^4x\, [a^{3}\frac{\epsilon}{N}{(\partial_t \zeta)}^2-a\epsilon N{(\partial_i\zeta)}^{2} ]. \label{E}$$ Choosing $t$ to be the conformal time $\tau$ by setting $N=a$, and defining the Mukhanov-type variable ${q}=-a\sqrt{2\epsilon}{\zeta}$, (\[E\]) becomes $$\begin{aligned}
\begin{array}{c}
S_{\text{scalar}}=\int d\tau d^{3}x\, \frac{1}{2} [ {- {({\partial_i q})}^{2}
+{q^\prime}^ 2+{{\bar{\mathscr{H}}}^2}{q^2}-2{\bar{\mathscr{H}}}qq^{\prime}}],\\ \bar{\mathscr{H}}=\mathscr{H}+\frac{\epsilon^\prime}{2\epsilon}=\mathscr{H} (1+\epsilon+\eta)\label{ac}
\end{array}\end{aligned}$$ where prime indicates conformal time derivative and $\mathscr{H}={a^{\prime}}/a=aH$ is the conformal Hubble parameter. Representing the Fourier transforms of $q$ by $q_{\bm k}$ and forming the row matrix ${\bm q}_{\bm k}^T=(q_{\bm k}^{(R)}\ \ q_{\bm k}^{(I)})$ from the real and imaginary parts of $q_{\bm k}$, (\[ac\]) can be written as $${S}_{\text{scalar}}=\int d\tau \frac{d^3k}{(2\pi)^3}\mathcal {L}_{\bm{k},\text{scalar}} \ \ , \ \mathcal {L}_{\bm{k},\text{scalar}}=\frac{1}{2}[({\bm q}_{{\bm k}}^{\prime}-\bar\mathscr{H}{\bm q}_{{\bm k}})^T({\bm q}_{{\bm k}}^{\prime}-\bar\mathscr{H}{\bm q}_{{\bm k}})-k^2{\bm q}_{{\bm k}}^{T}{\bm q}_{{\bm k}}].$$ The corresponding Hamiltonian is given by $${\mathcal {H}}_{\text{scalar}}=\int \frac{d^3k}{(2\pi)^3}{\mathcal {H}}_{\bm k,\text{scalar}}\ \ , \
{\mathcal {H}}_{\bm k,\text{scalar}}=\sum_m{{\bm p}}^T_{\bm k}{\bm q}^\prime_{\bm k}-{\mathcal{L}}_{\bm k}$$ with ${\bm p}_{\bm k}^T=\partial {\mathcal{L}}_{\bm k,\text{scalar}}/\partial {\bm q}^{\prime}_{\bm k}=(p_{\bm k}^{(R)}\ \ p_{\bm k}^{(I)})$. Thus, promoting the canonically conjugate variables to operators (denoted by hat), the matrices become matrix operators, and $$\hat{{\mathcal{\bm H}}}_{\bm k,\text{scalar}}=\frac{1}{2}[{\hat{\bm p}}^{T}_{\bm k}{{\hat{\bm p}}_{\bm k}}+\bar\mathscr{H}({{\hat{\bm p}}^{T}_{\bm k}}{{\hat{\bm q}}_{\bm k}}+{{\hat{\bm q}}^{T}_{\bm k}}{{\hat{\bm p}}_{\bm k}})+k^{2}{{\hat{\bm q}}^{T}_{\bm k}{\hat{\bm q}}_{\bm k}}]\label{H}$$ which represents a time-dependent harmonic oscillator of frequency $\omega_{k}(\tau)=\sqrt{{k^{2}-{\bar\mathscr{H}}^{2}}}$.
As for the linear tensor perturbations, the second order action calculated from (\[A\]) is [@Tzavara] $${S}_{\text{tensor}}= \int {d^{4}x}\, \frac{1}{2}\,[ \frac{a^{3}}{4N}(\partial_t\gamma_{ij})^{2}-\frac{a N}{4}{({\partial_k\gamma}_{ij})}^{2}].$$ Set $N=a$ and write the Fourier transforms $\gamma_{ij\bm k}$ in terms of the polarization tensors $\varepsilon_{ij}^{s}({\bm k})$ ($s=1,2$) as $\gamma_{ij\bm k}=\sum_s\frac{\surd 2}{a}\chi^s_{\bm k}\, \varepsilon_{ij}^{s}({\bm k})$. We similarly get $${S}_{\text{tensor}}=\int d\tau \frac{d^3k}{(2\pi)^3}\mathcal {L}_{\bm{k},\text{tensor}} \ \ , \ \mathcal {L}_{\bm{k},\text{tensor}}=\sum_{s=1}^2\frac{1}{2}[({\bm \chi}^{s\prime}_{\bm k}-\mathscr{H}{\bm \chi}^s_{\bm k})^{T}({\bm \chi}^{s\prime}_{\bm k}-\mathscr{H}{\bm \chi}^s_{\bm k})-k^2{\bm \chi}^{sT}_{\bm k}{\bm \chi}^s_{\bm k}]$$ where $\bm {\chi}_{\bm k}^{sT}=(\chi_{\bm{k}}^{s(R)}\ \ \chi_{\bm{k}}^{s(I)})$. Note that the summation over $s$ pertains only when both polarizatios are present in the gravitational wave. Hence, defining the conjugate momenta $\bm{\pi}_{\bm{k}}^{sT}=\partial {\mathcal{L}}_{\bm k, \text{tensor}}/\partial {\bm \chi}^{s\prime }_{\bm k}=(\pi_{\bm{k}}^{s(R)}\ \ \pi_{\bm{k}}^{s(I)})$ and promoting to operators, we find $$\hat{{\mathcal{\bm H}}}_{\bm k,\text{tensor}}=\sum_{s}\hat{{\mathcal{\bm H}}}^s_{\bm k,\text{tensor}}\ \ , \
\hat{{\mathcal{\bm H}}}^s_{\bm k,\text{tensor}}=\frac{1}{2}[ \hat{\bm \pi}^{sT}_{\bm k}\hat{\bm \pi}^s_{\bm k}+\mathscr{H}(\hat{\bm \pi}^{sT}_{\bm k}\hat{\bm \chi}^s_{\bm k}+\hat{\bm \chi}^{sT}_{\bm k}\hat{\bm \pi}^s_{\bm k})+k^{2}\hat{\bm \chi}^{sT}_{\bm k}\hat{\bm \chi}_{\bm k}^s]. \label{I}$$ Thus, the Hamiltonian for tensor modes also coincides with that of a harmonic oscillator of frequency $\Omega_{k}(\tau)=\sqrt{{k^{2}-{\mathscr{H}}^{2}}}$.
Berry phase of the scalar and tensor modes
==========================================
We use the invariant operator method [@Lewis:1968; @Lewis] to determine the dynamical invariants of the harmonic oscillator Hamiltonians (\[H\]) and (\[I\]). The Berry phase can then be obtained as a Lewis-Riesenfeld phase [@Monteoliva], which is constructed from the eigenstates of the invariant operator.
The invariant operator, by definition, satisfies the von Neumann equation. It has been derived for the generalized harmonic oscillator Hamiltonian in the form, $
\frac{1}{2}{[Z\hat{\bm p}^2+Y({{\hat{\bm p}}{\hat{\bm q}}}+{{\hat{\bm q}}{\hat{\bm p}}})+X\hat{\bm q}^2]},
$ where $X,Y,Z$ are time dependent [@Engineer]. This has the same form as Hamiltonians (\[H\]) and (\[I\]). Thence, for (\[H\]) the invariant takes the form $$\hat{I}_{\bm k,\text{scalar}}=\frac{1}{2}\bigg\lbrace{ \frac{1}{\rho_{k}^2}\hat{\bm q}_{\bm k}^T\hat{\bm q}_{\bm k}+[{\rho_{k}(\hat{\bm p}_{\bm k}+\bar\mathscr{H}\hat{\bm q}_{\bm k}) - \rho_{k}^\prime \hat{\bm q}_{\bm k}}]^T[{\rho_{k}(\hat{\bm p}_{\bm k}+\bar\mathscr{H}\hat{\bm q}_{\bm k})- \rho_{k}^\prime \hat{\bm q}_{\bm k}}]}\bigg\rbrace$$ where the auxiliary variable ${\rho}_{k}(\tau)$ is a time-periodic solution of the Milne-Pinney equation $${{\rho}^{{\prime}{\prime}}_{k}}+({\omega}_{k}^{2}-{\bar\mathscr{H}}^\prime){{\rho}_{k}}-{\rho}^{-3}_{k}=0.\label{Milne}$$
We define the raising and lowering matrix operators by $$\hat{\bm A}_{\bm k}^{(\pm)}=\frac{1}{\sqrt{2}}\bigg\lbrace\frac{1}{\rho_{k}}\hat{\bm q}_{\bm k}\pm i[{ \rho_{k}^\prime \hat{\bm q}_{\bm k}}-\rho_{k}(\hat{\bm p}_{\bm k}+\bar\mathscr{H}\hat{\bm q}_{\bm k})]\bigg\rbrace \label{L}$$ and write $\hat{\bm A}_{\bm k}^{(\pm)T}=(\hat{A}_{\bm{ k} 1}^{(\pm )}\ \ \hat{A}_{\bm{ k} 2}^{(\pm )})$. The components 1 and 2 are standard raising and lowering operators that satisfy $$\begin{aligned}
\begin{array}{c}
[\hat{A}_{\bm{k}1}^{(\pm )},\hat{A}^{(\pm )}_{\bm{k}2}]=0, \ \ [\hat{A}_{\bm{k}1}^{(-)},\hat{A}^{(+)}_{\bm{k}1}]=[\hat{A}_{\bm{k}2}^{(-)},\hat{A}^{(+)}_{\bm{k}2}]=1 \\
\hat{A}_{\bm{k}1,2}^{(-)}\vert n_{\bm{k}1,2}\rangle =\sqrt{n_{\bm{k}1,2}}\, \vert n_{\bm{k}1,2}-1\rangle, \ \
\hat{A}^{(+)}_{\bm{k}1,2}\vert n_{\bm{k}1,2}\rangle =\sqrt{n_{\bm{k}1,2}+1}\, \vert n_{\bm{k}1,2}+1\rangle \label{N}
\end{array}\end{aligned}$$ where $\vert{n_{{\bm k}1},n_{{\bm k}2}}\rangle$ is the eigenstate of $\hat{I}_{\bm k}^{\text{scalar}}=\hat{\bm A}_{\bm k}^{(+)T}\hat{\bm A}_{\bm k}^{(-)}+1$ with eigenvalue $n_{\bm{ k}1}+n_{\bm{ k}2}+1$.
The accumulated Berry phase over time period $\tau_0$ is derivable from the Lewis-Riesenfeld phase according to [@Monteoliva] $$\Gamma_{\bm k,\text{scalar}}({n}_{\bm {k}1},{n}_{\bm {k}2},\tau_0)={\int}^{\tau_0}_{0}\left\langle {n}_{\bm{ k}1},{n}_{\bm {k}2} \left\vert i\partial_\tau\right\vert {n}_{\bm {k}1},{n}_{\bm{ k}2}\right\rangle\, d\tau.$$ To calculate the integrand, we proceed as follows. From (\[N\]), differentiation with respect to $\tau$ yields $$\frac{1}{\sqrt{n_{\bm {k}1}}}\left\langle{n_{\bm{ k}1}}\left\vert\partial_\tau \hat{A}^{(+)}_{\bm {k}1}\right\vert{n_{\bm {k}1}-1}\right\rangle=\left\langle{n_{\bm{ k}1}}\left\vert\partial_\tau\right\vert{n_{\bm{ k}1}}\right\rangle-\left\langle{n_{\bm {k}1}-1}\left\vert\partial_\tau\right\vert{n_{\bm{ k}1}-1}\right\rangle$$ together with a similar expression with subscript $1$ replaced by $2$. It follows that $$\left\langle{\bar n_{\bm {k}1}}\left\vert\partial_\tau\right\vert{\bar n_{\bm {k}1}}\right\rangle-\left\langle 0\left\vert\partial_\tau\right\vert 0\right\rangle=\sum_{n_{\bm{ k}1}=1}^{\bar n_{\bm {k}1}}
\frac{1}{\sqrt{n_{\bm{ k}1}}}\left\langle{n_{\bm {k}1}}\left\vert\partial_\tau \hat{A}^{(+)}_{\bm{ k}1}\right\vert{n_{\bm {k}1}-1}\right\rangle.$$ By using (\[L\]), we can express $\partial_\tau \hat{A}^{(+)}_{\bm {k}1}$ in terms of the raising and lowering operators to find $$\left\langle{n_{\bm {k}1}}\left\vert\partial_\tau \hat{A}^{(+)}_{\bm{ k}1}\right\vert{n_{\bm {k}1}-1}\right\rangle =-\frac{i}{2}\,(\omega^2_k\rho_{k}^2-\rho_{k}^{-2}+
{\rho_{k}^\prime}^2)\sqrt{n_{\bm {k}1}}$$ and therefore $$\left\langle{n_{\bm {k}1}}\left\vert i\partial_\tau\right\vert{n_{\bm {k}1}}\right\rangle=\left\langle 0\left\vert i\partial_\tau\right\vert 0\right\rangle
+\frac{1}{2}\,(\omega^2_k\rho_{k}^2-\rho_{k}^{-2}+
{\rho_{k}^\prime}^2)\,n_{\bm{ k}1}.$$ Bearing in mind the same expression with subscript $1$ replaced by $2$, it follows that $$\left\langle{n}_{\bm {k}1}, n_{\bm{k}2}\left\vert i\partial_\tau\right\vert , n_{\bm{k}1}, n_{\bm{k}2}\right\rangle=2\left\langle 0\left\vert i\partial_\tau\right\vert 0\right\rangle
+\frac{1}{2}\,(\omega^2_k\rho_{k}^2-\rho_{k}^{-2}+
{\rho_{k}^\prime}^2)\,(n_{\bm{ k}1}+n_{\bm {k}2}).$$ Conveniently choosing the Lewis gauge [@Lewis] $$\left\langle{0}\left\vert{i\partial_\tau}\right\vert{0}\right\rangle
= \frac{1}{4}\, (\omega^2_k\rho_{k}^2-\rho_{k}^{-2}+
{\rho_{k}^\prime}^2)$$ we finally obtain $${\Gamma}_{\bm k,\text{scalar}}=\frac{1}{2}(n_{\bm{k}1}+n_{\bm{k}2}+1)\int_0^{\tau_0} (\omega^2_k\rho_{k}^2-\rho_{k}^{-2}+
{\rho_{k}^\prime}^2)\,d\tau. \label{phase}$$
In the adiabatic limit of slow time variation, we introduce the adiabatic parameter $\lambda$ ($\ll1$) and write $\eta=\lambda\tau$. Substituting in (\[Milne\]) gives ${\rho}_{k}^2=1/\sqrt{k^2-\mathscr{H}^2}+O(\lambda)$ as $\bar\mathscr{H}^\prime=\lambda d\mathscr{H}/d\eta+O(\lambda^2)$, $\epsilon$ and $\eta$ being first order in $\lambda$. Thus, on using (\[Milne\]), the integrand of (\[phase\]) becomes $$\bar\mathscr{H}^\prime\rho_k^2+{\rho_k^\prime}^2-\rho_k\rho_k^{\prime\prime}= \lambda \frac{d\mathscr{H}/d\eta}{\sqrt{k^2-\mathscr{H}^2}}+O(\lambda^2)\rightarrow\frac{\mathscr{H}^\prime}{\sqrt{k^2-\mathscr{H}^2}}.$$ Hence, in the adiabatic limit, $${\Gamma}_{\bm k,\text{scalar}}=\frac{1}{2}(n_{\bm{k}1}+n_{\bm{k}2}+1)\sin^{-1} \frac{\mathscr{H}_0}{k}$$ where $\mathscr{H}_0=\mathscr{H}(\tau_0)$. Note that ${k}\geq\mathscr{H}$, which means that the above result for Berry phase holds for sub-horizon modes that oscillate with real frequency. Thus, ${{\tau}_{0}}$ corresponds to the conformal time at which $\mathscr{H}=k$, i.e., $\mathscr{H}_0=k$, so that $${\Gamma}_{\bm k,\text{scalar}}= (n_{\bm{k}1}+n_{\bm{k}2}+1)\, \frac{\pi}{4}$$ which yields $\Gamma_{\text{scalar}}=\pi/4$ for the ground state. The adiabatic Berry phase is, thus, independent of the (conformal) Hubble parameter, in contrast to the general non-adiabatic Berry phase given by (\[phase\]).
For tensor modes, because of the identical form of the Hamiltonian for each polarization state, as given by (\[I\]) , we just have to introduce the polarization index $s$ in the above steps and make the correspondences $\omega_k\rightarrow\Omega_k$, $\bar\mathscr{H}\rightarrow\mathscr{H}$. Thus the invariant operator of $\hat{{\mathcal{\bm H}}}_{\bm k,\text{tensor}}$ is $\hat{I}_{\bm k,\text{tensor}}=\sum_s \hat{I}_{\bm k,\text{tensor}}^{s}$, where $\hat{I}_{\bm k,\text{tensor}}^{s}$ is the invariant of $\hat{{\mathcal{\bm H}}}^{s}_{\bm k,\text{tensor}}$. We have $$\hat{I}_{\bm k,\text{tensor}}^{s}=\hat{\bm A}_{\bm k}^{s(+)T}\hat{\bm A}_{\bm k}^{s(-)}+1, \ \ \ \hat{\bm A}_{\bm k}^{s(\pm)}=\frac{1}{\sqrt{2}}\bigg\lbrace\frac{1}{\rho_{k}}\hat{\bm \chi}^s_{\bm k}\pm i[{ \rho_{k}^\prime \hat{\bm \chi}^s_{\bm k}}-\rho_{k}(\hat{\bm \pi}^s_{\bm k}+\mathscr{H}\hat{\bm \chi}^s_{\bm k})]\bigg\rbrace$$ where $\rho_k$ satisfies $${{\rho}^{{\prime}{\prime}}_{k}}+({\Omega}_{k}^{2}-{\mathscr{H}}^\prime){{\rho}_{k}}-{\rho}^{-3}_{k}=0.$$ The eigenstate of $ \hat{I}_{\bm k,\text{tensor}}^{s}$ is $\vert\bm{n}^s_{\bm {k}}\rangle\equiv\vert n^s_{\bm{k}1},n^s_{\bm{k}2}\rangle$ with eigenvalue $ n^s_{\bm{k}1}+n^s_{\bm{k}2}+1$, so that the eigenstate of $\hat{I}_{\bm k,\text{tensor}}$ is $\vert\bm{n}_{\bm {k}}^1,\bm{n}^2_{\bm {k}}\rangle$. The Berry phase is, therefore, given by $$\Gamma_{\bm k,\text{tensor}}(\bm{n}^1_{\bm {k}},\bm{n}^2_{\bm {k}},\tau_0)={\int}^{\tau_0}_{0}\left\langle \bm{n}^1_{\bm{ k}},\bm{n}^2_{\bm {k}} \left\vert i\partial_\tau\right\vert\bm {n}^1_{\bm {k}},\bm{n}^2_{\bm{ k}}\right\rangle\, d\tau.$$ Noting that the integrand is equal to $\sum_s\left\langle{\bm{n}^s_{\bm k}}\left\vert i\partial_\tau\right\vert{\bm{n}^s_{\bm k}}\right\rangle$, we similarly obtain in place of (\[phase\]), $${\Gamma}_{\bm k,\text{tensor}}=\frac{1}{2}\sum_{s=1}^2(n_{\bm{k}1}^{s}+n_{\bm{k}2}^{s}+1)\int_0^{\tau_0} (\omega^2_k\rho_{k}^2-\rho_{k}^{-2}+
{\rho_{k}^\prime}^2)\,d\tau$$ and hence, in the adiabatic limit, $${\Gamma}_{\bm k,\text{tensor}}=\sum_s(n_{\bm{k}1}^{s}+n_{\bm{k}2}^{s}+1)\,\frac{\pi}{4}.$$ The summation pertains only when both polarizations are present in the gravitational wave. For the ground state, therefore, we have $\Gamma_{\text{tensor}}=\pi/4$ for each polarization.
discussion
==========
Considering linear primordial perturbations in the single-field slow-roll inflation, we have derived the Hamiltonian of the scalar and tensor modes in the form of time-dependent harmonic oscillator Hamiltonians. We obtained the invariant operators of the resulting Hamiltonians and used their eigenstates to calculate the adiabatic Berry phase for sub-horizon perturbations as a Lewis-Riesenfeld phase.
In conclusion, we ought to comment on the discrepancy in the results of [@Pal], where the scalar and tensor adiabatic Berry phases are obtained from the wave function of the perturbations. Their results for the ground state read as follows (in our notation): $$\Gamma_{\text{scalar}}=-\frac{\pi}{4}\frac{1+3\epsilon-\eta}{\sqrt{1+2(3\epsilon-\eta})}+O(\epsilon^2,\eta^2,\epsilon\eta),\ \ \
\Gamma_{\text{tensor}}=-\frac{\pi}{4}\frac{1+\epsilon}{\sqrt{1+2\epsilon}}+O(\epsilon^2,\eta^2,\epsilon\eta)$$ where $\Gamma_{\text{tensor}}$ pertains to each polarization. They also relate the Berry phases to observable parameters, viz spectral indices, through the slow roll parameters $\epsilon, \eta$. In accordance with the adiabatic requirement, the above expressions are claimed by the authors to be exact to first order in $\epsilon, \eta$. This is obviously incorrect because of the denominators. In fact, by a simple binomial expansion, the correct first order results are $\Gamma_{\text{scalar}}= \Gamma_{\text{tensor}}=-\pi/4$, which coincide with ours (up to an unimportant sign). Moreover, there is no relationship with spectral indices as far as the adiabatic approximation is concerned.
[widest-label]{} M. V. Berry, Proc. Roy. Soc. Lond. A [**392**]{}, 45 (1984). A. Shapere and F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989). A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, and J. Zwanziger, The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts, Applications in Molecular and Condensed Matter Physics (Springer, Berlin, 2013). M. Mehrafarin and H. Balajany, Phys. Lett. A [**374**]{}, 1608 (2010). M. Mehrafarin and R. Torabi, Phys. Lett. A [**373**]{}, 2114 (2009). R. Torabi and M. Mehrafarin, JETP Lett. [**95**]{}, 277 (2012). R. Torabi and M. Mehrafarin, JETP Lett. [**88**]{}, 590 (2008). K. Bakke, I. Pedrosa, and C. Furtado, J. Math. Phys. [**50**]{}, 113521 (2009). Y. Q. Cai and G. Papini, Mod. Phys. Lett. A [**4**]{}, 1143 (1989). Y. Q. Cai and G. Papini, Class. Quant. Grav. [**7**]{}, 269 (1990). A. Corichi and M. Pierrie, Phys. Rev. D [**51**]{}, 5870 (1995). P. O. Mazur, Phys. Rev. Lett. [**57**]{}, 929 (1986). D. P. Dutta, Phys. Rev. D [**48**]{}, 5746 (1993). J. Lemos de Melo, K. Bakke, and C. Furtado, Eur. Phys. J. Plus [**131**]{}, 165 (2016). J. Lemos de Melo, K. Bakke, and C. Furtado, EPL [**115**]{}, 20001 (2016). K. Bakke and C. Furtado, Phys. Rev. A [**87**]{}, 012130 (2013). A. H. Guth, Phys. Rev. D [**23**]{}, 347 (1981). B. A. Bassett, S. Tsujikawa, and D. Wands, Rev. Mod. Phys. [**78**]{}, 537 (2006). V. F. Mukhanov, H. A. Feldmann, and R. H. Brandenberger, Phys. Rep. [**215**]{}, 203 (1992). J. M. Bardeen, Phys. Rev. D [**22**]{}, 1882 (1980). B. K. Pal, S. Pal, and B. Basu, Class. Quant. Grav. [**30**]{}, 12 (2013). D. Campo and R. Parentani, Phys. Rev. D [**74**]{}, 025001 (2006). H. R. Jr. Lewis, J. Math. Phys. [**9**]{}, 1997 (1968). H. R. Jr. Lewis and W. B. Riesenfeld, J. Math. Phys. [**10**]{}, 1458 (1969). A. M. de M Carvalho, C. Furtado, and I. A. Pedrosa, Phys. Rev. D [**70**]{}, 123523 (2004). I.A. Pedrosa, C. Furtado, and A. Rosas, Phys. Lett. B [**651**]{}, 384 (2007). I. A. Pedrosa, Mod. Phys. Lett. B [**18**]{}, 1267 (2004). K. A. Malik and D. Wands, Class. Quant. Grav. [**21**]{}, 65 (2004). D. B. Monteoliva, H. J. Korsch, and J. A. Nunez, J. Phys. A [**27**]{}, 6897 (1994). R. Arnowitt, S. Deser, and C. Misner, Phys. Rev. [**116**]{}, 1322 (1959). E. Tzavara and B. Van Tent, JCAP [**2012**]{}, 023 (2012). M. H. Engineer and G. Ghosh, J. Phys. A [**21**]{}, L95 (1988).
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---
author:
- 'Claire Chainais-Hillairet and Maxime Herda'
bibliography:
- 'bib-dGdiscret.bib'
title: '$L^\infty$ bounds for numerical solutions of noncoercive convection-diffusion equations'
---
Introduction
============
[**The continuous problem.**]{} Let $\Om$ be an open bounded polygonal domain of $\R^p$ with $p=2$ or $3$. We denote by ${\rm m(\cdot)}$ both the Lebesgue and $p-1$ dimensional Hausdorff measure. We assume that $\pa \Omega= \Gamma^D\cup\Gamma^N$ with $\Gamma^D\cap\Gamma^N=\emptyset$ and ${\rm m}(\Gamma^D)>0$ and we denote by ${\mathbf n}$ the exterior normal to $\pa \Omega$. Let ${\mathbf U}\in C({\bar \Om})^2$ be a velocity field, $b\in \Linf(\Omega)$ assumed to be nonnegative, $f\in L^\infty(\Omega)$ a source term and $v^D\in \Linf (\Gamma^D)$ a boundary condition.
We consider the following convection-diffusion equation with mixed boundary conditions:
\[pb\_depart\] $$\begin{aligned}
&\Div (-\nabla v + {\mathbf U} v)+ bv=f&&\qquad \mbox{in }\Omega, \label{eq_v}\\
& (-\nabla v + {\mathbf U} v)\cdot {\mathbf n}=0 &&\qquad \mbox{on } \Gamma^N, \label{Neum_bc}\\
&v=v^D&&\qquad \mbox{on } \Gamma^D \label{Dir_bc}.\end{aligned}$$
This noncoercive elliptic linear problem has been widely studied by Droniou and coauthors, even with less regularity on the [data]{}, see for instance [@droniou_potan_2002; @DG_M2AN_2002; @Droniou_jnm_2003; @DGH_sinum_2003]. Nevertheless, up to our knowledge, the derivation of explicit $L^\infty$ bounds on numerical solutions has not been done in the literature.
[**The numerical scheme.**]{} The mesh of the domain $\Omega$ is denoted by $\cal M=(\T,\E,\cal P)$ and classically given by: $\mathcal T$, a set of open polygonal [ or polyhedral]{} control volumes; $\mathcal E$, a set of edges [or faces]{}; ${\mathcal P}=(x_K)_{K\in\mathcal T}$ a set of points. [In the following, we also use the denomination “edge” for a face in dimension $3$]{}. As we deal with a Two-Point Flux Approximation (TPFA) of convection-diffusion equations, we assume that the mesh is admissible in the sense of [@Eymard2000] (Definition 9.1).
We distinguish in $\E$ the interior edges, $\sigma =K|L$, from the exterior edges: $\E=\E_{int}\cup {\mathcal E}_{ext}$. Among the exterior edges, we distinguish the edges included in $\Gamma^D$ from the edges included in $\Gamma^N$: ${\mathcal E}_{ext}={\mathcal E}^D\cup {\mathcal E}^N$. For a given control volume $K\in{\mathcal T}$, we define ${\mathcal E}_K$ the set of its edges, which is also split into ${\mathcal E}_K={\mathcal E}_{K,int}\cup{\mathcal E}_{K}^D\cup{\mathcal E}_{K}^N$. For each edge $\sigma\in\E$, we pick one cell in the non empty set $\{K:\sigma\in\E_K\}$ and denote it by $K_\sigma$. In the case of an interior edge $\sigma=K|L$, $K_{\sigma}$ is either $K$ or $L$.
[ Let ${\rm d}(\cdot,\cdot)$ denote the Euclidean distance.]{} For all edges $\sigma\in{\mathcal E}$, we set ${\rm d}_{\sigma}={\rm d}(x_K,x_L)$ if $\sigma=K|L\in{\mathcal E}_{int}$ and ${\rm d}_{\sigma}={\rm d}(x_K,\sigma)$ if $\sigma\in{\mathcal E}_{ext}$ with $\sigma\in \E_K$ and the transmissibility coefficient is defined by $\tau_{\sigma}={\rm m}(\sigma)/{\rm d}_{\sigma}$, for all $\sigma\in{\mathcal E}$. We also denote by ${\mathbf n}_{K,\sigma}$ the normal to $\sigma\in{\mathcal E}_K$ outward $K$. We assume that the mesh satisfies the regularity constraint: $$\label{reg-mesh}
\exists \xi >0 \mbox{ such that } {\rm d}(x_K,\sigma)\geq \xi \, {\rm d}_{\sigma},\quad \forall K\in\T, \forall \sigma\in\E_K.$$ As a consequence, we obtain that $$\label{inegvol}
\sum_{\sigma\in\E_K} {\rm m}(\sigma)\dsig\leq \ds\frac{p}{\xi} {\rm m} (K)\quad \forall K\in\T.$$ The size of the mesh is defined by $h=\max\{\mbox{diam }(K)\,:\,K\in\T\}$.
Let us define $$\begin{aligned}
& f_K=\ds\frac{1}{{\rm m}(K)}\int_{K} f, \quad b_K=\ds\frac{1}{{\rm m}(K)}\int_{K} b \quad \forall K\in\T,\\
&U_{K,\sigma}=\ds\frac{1}{{\rm m}(\sigma)}\int_{\sigma} {\mathbf U}\cdot {\mathbf n}_{K,\sigma},\quad \forall K\in\T,\ \forall\sigma \in {\mathcal E}_K,\\
& v_\sigma^D=\ds\frac{1}{{\rm m}(\sigma)}\int_\sigma v^D,\quad \forall \sigma\in {\mathcal E}^D.
\end{aligned}$$ Given a Lipschitz-continuous function on $\R$ which satisfies $$\label{hyp_B}
B(0)=1,\quad\ B(s)>0\quad \mbox{ and }\quad B(s)-B(-s)=-s\quad\forall s\in\R,$$ we consider the B-scheme defined by $$\label{scheme}
\sum_{\sigma\in \E_K} {\mathcal F}_{K,\sigma}+ {\rm m}(K) b_K v_K= {\rm m}(K)f_K, \quad \forall K\in{\mathcal T},$$ where the numerical fluxes are defined by $$\label{numflux}
{\mathcal F}_{K,\sigma}=\left\{
\begin{aligned}
&0,\quad \forall K\in\T,\forall \sigma \in \E_K^N,\\
&\tau_{\sigma} \Bigl(B(-U_{K,\sigma}\dsig)v_K-B(U_{K,\sigma}\dsig)v_{K,\sigma}\Bigl),\quad\forall K\in\T, \forall \sigma\in\E_{K}\setminus \E_K^N,
\end{aligned}
\right.$$ with the convention $v_{K,\sigma}=v_L$ if $\sigma =K|L$ and $v_{K,\sigma}=v_\sigma^D$ if $\sigma \in\E_K^D$. Let us recall that the upwind scheme corresponds to the case $B(s)=1+s^-$ ($s^-$ is the negative part of $s$, while $s^+$ is its positive part) and the Scharfetter-Gummel scheme to the case $B(s)=s/(e^s-1)$. They both satisfy . The centered scheme which corresponds to $B(s)=1-s/2$ does not satisfy the positivity assumption. It can however be used if $|U_{K,\sigma}|\dsig\leq 2$ for all $K\in\T$ and $\sigma \in\E_K$. Thanks to the hypotheses , we notice that the numerical fluxes through the interior and Dirichlet boundary edges rewrite $$\label{numflux2}
{\mathcal F}_{K,\sigma}= \tau_\sigma B(|U_{K,\sigma}|\dsig)(v_K-v_{K,\sigma})+ {\rm m}(\sigma) \left(U_{K,\sigma}^+ v_K - U_{K,\sigma}^-v_{K,\sigma}\right) .$$
[**Main result.**]{} The scheme - defines a linear system of equations ${\mathbb M} {\mathbf v}={\mathbf S}$ whose unknown is ${\mathbf v}=(v_K)_{K\in\T}$; It is well-known that ${\mathbb M}$ is an M-matrix, which ensures existence and uniqueness of a solution to the scheme. Moreover, we may notice that, if $v^D$ and $f$ are nonnegative functions, then ${\mathbf S}$ has nonnegative values and therefore $v_K\geq 0$ for all $K\in\T$. Our purpose is now to establish $L^{\infty}$ bounds on ${\mathbf v}$ as stated in Theorem \[mainthm\].
\[mainthm\] Assume that ${\mathbf U}\in C({\bar \Om})^2$, $b\in \Linf(\Omega)$ with $b\geq 0$ [*a.e.*]{}, ${f\in L^\infty(\Omega)}$ and $v^D\in \Linf (\Gamma^D)$. There exists non-negative constants $\overline{M}$ (*resp.* $\underline{M}$) depending only on $\Omega$, $\xi$, the function $B$, $\|{\bf U}\|_{L^\infty}$, $\|f^+\|_{L^\infty}$ and $\|(v^D)^+\|_{L^\infty}$ (*resp.* $\|f^-\|_{L^\infty}$ and $\|(v^D)^-\|_{L^\infty}$) such that the solution ${\mathbf v}$ to the scheme - verifies $$-\underline{M}\ \leq\ v_K\ \leq\ \overline{M}, \quad \forall K\in\T.$$
The rest of this paper is dedicated to the proof of Theorem \[mainthm\]. It relies on a De Giorgi iteration method (see [@Vasseur_lectnotes] and references therein). In Section \[sec:particular\], we start by studying a particular case where the data is normalized. Then, we give the proof of the theorem in Section \[sec:proof\].
Let us mention that from the bounds of Theorem \[mainthm\], it is possible to establish global-in-time $L^\infty$ bounds for the corresponding evolution equation by using an entropy method (see [@chainais_2019_large Theorem 2.7]).
Study of a particular case {#sec:particular}
==========================
In this section, we consider the particular case where the source $f$ is non-negative and the boundary condition $v^D$ is non-negative and bounded by $1$.
Let us start with some notations. Given $m\geq 1$, we denote the $m$-th truncation threshold by $$\label{eq:trunc}
C_m=2(1-2^{-m})\,,$$ Then, we introduce the $m$-th energy $$\label{eq:energy}
E_m({\bf v})=\ds\sum_{\sigma\in\E_{int}\cup\E^D} \tau_\sigma \left[\log (1 +(v_{K,\sigma}-C_m)^+)-\log (1 +(v_{K}-C_m)^+)\right]^2.$$ When there is no ambiguity we write $E_m = E_m({\bf v})$. The first proposition is a fundamental estimate of the energy.
\[prop:fund\_ineq\] Assume that $f_K\geq 0$ for all $K\in\T$ and $v_\sigma^D \in [0,1]$ for all $\sigma \in \E^D$, so that the solution ${\mathbf v}$ to - satisfies $v_K\geq 0$ for all $K\in\T$. Then one has for all $m\geq1$ that $$\label{majEm}
E_m\ \leq\ \frac{4p}{\beta_{\mathbf U}^2}\left(\Vert {\mathbf U}\Vert_{L^\infty}^2+{\Vert f\Vert_{L^\infty}}\right) \sum_{\substack{K\in\T\\v_{K}>C_m}}{{\rm m}(K)}\,.$$ where $\beta_{\mathbf U} := \inf_{x\in[-\|\mathbf{U}\|_{L^\infty},\|\mathbf{U}\|_{L^\infty}]}B({\rm diam}(\Omega)\,x)$ (because of , $\beta_{\mathbf U}\in(0,1]$).
In order to shorten some expressions hereafter, let us introduce $w_K^m = v_K-C_m$ for all $K\in \T$ and $w_\sigma^{m,D}=v_{\sigma}^D-C_m$ for all $\sigma \in\E^D$. Let us note that we identify ${\mathbf w}^m=(w_K^m)_{K\in\T}$ and the associate piecewise constant function. Therefore, we can write $${\rm m} (\{{\mathbf w}^m>0\}) = \sum_{w_{K}^m>0}{{\rm m}(K)}.$$
First, observe that $E_m$ is the discrete counterpart of $$\int_\Omega \left\vert \nabla \log (1+w^m)\right\vert^2 {\mathbf 1}_{\{w^m>0\}}=\int_\Omega \nabla w^m\cdot\frac{\nabla w^m}{(1+w^m)^2}{\mathbf 1}_{\{w^m>0\}},\
\mbox{ with }w^m=v-C_m\,,$$ [where ${\mathbf 1}_{A}$ is the indicator function of $A$]{}. Let us define $\varphi :s\mapsto s/(1+s){\mathbf 1}_{\{s\geq 0\}}$, which satisfies $\varphi'(s)=1/(1+s)^2{\mathbf 1}_{\{s\geq 0\}}$ and let us introduce $F_m$ another discrete counterpart of the preceding quantity $$F_m=\ds\sum_{\sigma\in\E_{int}\cup\E^D} \tau_\sigma \left((w_{K,\sigma}^m)^+-(w_{K}^m)^+\right)\left(\varphi(w_{K,\sigma}^m)-\varphi(w_{K}^m) \right).$$ It is clear that $E_m\leq F_m$ for all $m\geq 1$, as for all $x,y\in\R$ we have $$\left(\log(1+x^+)-\log(1+y^+)\right)^2\leq (x^+-y^+)\left(\varphi(x)-\varphi(y)\right).$$
Let us now multiply the scheme by $\varphi(w_K^m)$ and sum over $K\in\T$. Due to the non-negativity of $b$ and ${\mathbf v}$, we obtain, after a discrete integration by parts, $$\ds\sum_{\sigma\in\E_{int}\cup\E^D}{\mathcal F}_{K,\sigma}(\varphi(w_K^m)-\varphi(w_{K,\sigma}^m))\leq \sum_{K\in\T} {\rm m}(K) f_K \varphi(w_K^m).$$ Using that $\varphi$ is bounded by 1 and vanishes on $\R_-$, we deduce that $$\label{inegdep}
\ds\sum_{\sigma\in\E_{int}\cup\E^D}{\mathcal F}_{K,\sigma}(\varphi(w_K^m)-\varphi(w_{K,\sigma}^m))\leq { \Vert f\Vert_{L^\infty}\,
{\rm m}(\{{\mathbf w}^m>0\})}.$$
We focus now on the left-hand-side of . Due to and the definition of $w_K^m$, we can rewrite $ {\mathcal F}_{K,\sigma}$ as $${\mathcal F}_{K,\sigma}= \tau_\sigma B(|U_{K,\sigma}|\dsig)(w_K^m-w^m_{K,\sigma})+ {\rm m}(\sigma) \left(U_{K,\sigma}^+ (w_K^m+C_m) - U_{K,\sigma}^-(w_{K,\sigma}^m+C_m)\right) .$$ Observe that since $\varphi$ is a non-decreasing function, one has $$(x-y)\left(\varphi(x)-\varphi(y)\right)\geq (x^+-y^+)(\varphi(x)-\varphi(y)),\quad \forall x,y\in\R.$$ Therefore, using the definition of $\beta_{\mathbf U}$ we obtain that $$\label{ineg1}
\ds\sum_{\sigma\in\E_{int}\cup\E^D}{\mathcal F}_{K,\sigma}(\varphi(w_K^m)-\varphi(w_{K,\sigma}^m))\geq \beta_{\mathbf U} F_m -G_m,$$ with $$G_m=-\sum_{\sigma\in\E_{int}\cup\E^D}{\rm m}(\sigma) \left(U_{K,\sigma}^+ (w_K^m+C_m) - U_{K,\sigma}^-(w_{K,\sigma}^m+C_m)\right) (\varphi(w_K^m)-\varphi(w_{K,\sigma}^m)).$$ For an interior edge, $w_K^m$ and $w_{K,\sigma}^m$ play a symmetric role in the preceding sum. As $w_{\sigma}^{m,D}\leq 0$ for all $\sigma\in\E^D$ and $\varphi$ vanishes on $\R_-$, we can always assume that $w_K^m\geq w_{K,\sigma}^m$ and an edge has a contribution in the sum if at least $w_K^m> 0$. Then, under these assumptions one has $$\begin{gathered}
-{\rm m}(\sigma) \left(U_{K,\sigma}^+ (w_K^m+C_m) - U_{K,\sigma}^-(w_{K,\sigma}^m+C_m)\right) (\varphi(w_K^m)-\varphi(w_{K,\sigma}^m))\\
\leq \Vert {\mathbf U}\Vert_{L^\infty} {\rm m}(\sigma) (w_{K,\sigma}^m+C_m)(\varphi(w_K^m)-\varphi(w_{K,\sigma}^m)).\end{gathered}$$ But, $w_{K,\sigma}^m+C_m\leq 2(1+(w_{K,\sigma}^m)^+)$ and applying the definition of $\varphi$, we get $$\begin{array}{rcl}
(w_{K,\sigma}^m+C_m)(\varphi(w_K^m)-\varphi(w_{K,\sigma}^m))&\leq& 2\ds\frac{(w_K^m)^+-(w_{K,\sigma}^m)^+}{1+(w_{K}^m)^+}\\[1em]
&\leq&2\ds\frac{(w_K^m)^+-(w_{K,\sigma}^m)^+}{\sqrt{1+ (w_K^m)^+}\sqrt{1+ (w_{K,\sigma}^m)^+}}.
\end{array}$$ Therefore, $$G_m\leq 2\Vert {\mathbf U}\Vert_{L^\infty} \sum_{\sigma\in\E_{int}\cup\E^D}{\rm m}(\sigma)\ds\frac{\vert(w_K^m)^+-(w_{K,\sigma}^m)^+\vert}{\sqrt{1+ (w_K^m)^+}\sqrt{1+ (w_{K,\sigma}^m)^+}}.$$ We apply now Cauchy-Schwarz inequality in order to get $$\label{ineg2}
G_m\leq 2\Vert {\mathbf U}\Vert_{L^\infty}(F_m)^{1/2} \left(\sum_{\sigma\in\E^{sp}}{\rm m}(\sigma)\dsig\right)^{1/2},$$ where $\E^{sp}$ is the set of interior and Dirichlet boundary edges on which $(w_K^m)^+-(w_{K,\sigma}^m)^+\neq 0$. It appears that, due to , $$\label{ineg3}
\sum_{\sigma\in\E^{sp}}{\rm m}(\sigma)\dsig\leq \ds\sum_{K\in\T; w_K^m>0}\left(\sum_{\sigma\in \E_{K,int}\cup \E_K^D} {\rm m}(\sigma)\dsig\right)\leq \ds\frac{p}{\xi} {\rm m} (\{{\mathbf w}^m>0\}).$$ We deduce from , , and that $$\beta_{\mathbf U} F_m\leq {2}\Vert {\mathbf U}\Vert_{L^\infty} (F_m)^{1/2}(\frac{p}{\xi}{\rm m} (\{{\mathbf w}^m>0\}))^{1/2}+{\Vert f\Vert_{L^\infty}{\rm m} (\{{\mathbf w}^m>0\})},$$ which yields using Young’s inequality and the bounds $E_m\leq F_m$ and $\beta_{\mathbf U}\leq1$.
Before stating the main result of the section, we need a technical lemma.
\[lem:sequence\] Let $(u_n)_{n\in\N}$ be a sequence of non-negative real numbers and let $K, \rho>0$ and $\alpha >1$. Then if for all $n\in\N$ $$u_{n+1}\,\leq\, K\,\rho^{n}\,u_{n}^\alpha\,,$$ one has $$0\leq u_n\,\leq\, \left(u_0\,\rho^{\frac{1}{(\alpha-1)^2}}\,K^{\frac{1}{\alpha-1}}\right)^{\alpha^n}\,\rho^{-\frac{n(\alpha-1)+1}{(\alpha-1)^2}}\,K^{-\frac{1}{\alpha-1}}$$ for all $n\in\N$ and the bound is optimal. In particular, if $u_0\leq \rho^{-\frac{1}{(\alpha-1)^2}}\,K^{-\frac{1}{\alpha-1}}$, then $\lim u_n=0$.
Just observe that the sequence $v_n = u_n\,\rho^{\frac{n(\alpha-1) + 1}{(\alpha-1)^2}}\,K^{\frac{1}{\alpha-1}}$ satisfies $0\leq v_{n+1}\leq v_{n}^\alpha$ for all $n\geq0$ which directly yields the result.
\[mainprop\] Assume that $f_K\geq 0$ for all $K\in\T$ and $v_\sigma^D \in [0,1]$ for all $\sigma \in \E^D$, so that $v_K\geq 0$ for all $K\in\T$. Then, there exists $\eta>0$ depending only on $\Omega$, $p$ and $\xi$ such that one has the implication $$\label{resprop}
\ds E_1\leq\ \eta\ \frac{\beta_{\mathbf U}^4}{(\|\mathbf{U}\|_{L^\infty}^2+\|f\|_{L^\infty})^2}\quad\Rightarrow\quad (v_K\leq 2,\ \forall K\in\T)\,.$$
The proof consists in establishing an induction property on $E_m$ which guarantees that if $E_1$ is small enough then $\lim E_m =0$. Then, as $\lim C_m=2$ and thanks to the discrete Poincaré inequality, we deduce that $$\ds\sum_{K\in\T} {\rm m}(K) \left( \log (1 +(v_K-2)^+)\right)^2=0,$$ which implies $v_K\leq 2$ for all $K\in\T$.
For establishing the induction, first observe that as $C_m=C_{m-1} +2^{-m+1}$, for any $q>0$ we have: $$\label{eq:nonlinbound}
{\mathbf 1}_{\{{\mathbf w}^m>0\}}\leq \frac{\left(\log (1+({\mathbf w}^{m-1})^+)\right)^q}{(\log (1+2^{-m+1}))^q}{\mathbf 1}_{\{{\mathbf w}^{m-1}>0\}},$$ and thus $${\rm m} (\{{\mathbf w}^m>0\})\leq \frac{1}{(\log (1+2^{-m+1}))^q}\ds\sum_{K\in\T} {\rm m}(K) \left( \log (1 +(w_K^{m-1})^+)\right)^q.$$ We may choose for instance $q=3$ and apply a discrete Poincaré-Sobolev inequality (whose constant $C_{\Omega,p}$ depends only on $\Omega$ and $p$), which leads to $$\label{majmes}
{\rm m} (\{{\mathbf w}^m>0\})\leq\frac{1}{(\log (1+2^{-m+1}))^3}\frac{C(\Omega)}{\xi^{3/2}} E_{m-1}^{3/2}.$$ Noticing that for $x\in[0,1]$, $(\log(1+x))^3\geq (\log 2)^3 x^3$, we deduce from and that $$E_m\leq \frac{4}{\beta_{\mathbf U}^2}\left(\Vert {\mathbf U}\Vert_{L^\infty}^2+\Vert f\Vert_{L^\infty}\right)\frac{{\tilde C}_{\Omega,p}}{\xi^{3/2}}8^{m-1}E_{m-1}^{3/2}.$$ Thus the sequence $(E_m)_{m\geq 0}$ satisfies the hypothesis of Lemma \[lem:sequence\] with $\alpha=3/2$ and $K$ proportional to $(\Vert {\mathbf U}\Vert_{L^\infty}^2+\Vert f\Vert_{L^\infty})/\beta_{\mathbf U}^2$. We deduce the upper bound for $E_1$ under which $\lim E_m =0$.
[*Remark:*]{} The arguments developed in this section still hold, up to minor adaptation, for $f\in L^r(\Omega)$ with $r>p/2$.
Proof of Theorem \[mainthm\] {#sec:proof}
============================
First observe that if one replaces the data $f$ and $v^D$ by either $f^+$ and $(v^D)^+$, or $f^-$ and $(v^D)^-$, in the scheme -, then the corresponding solutions, say respectively $\mathbf{P}=(P_K)_{K\in\T}$ and $\mathbf{N}=(N_K)_{K\in\T}$, are non-negative and such that ${\mathbf v}=\mathbf{P}-\mathbf{N}$ is the solution to - in the original framework.
From there let us show that there is ${\overline M}>V^D_+:=\max(\|(v^D)^+\|_{L^\infty},1)$ such that for all $K\in\T$ one has $0\leq P_K\leq {\overline M}$. The bound for $\mathbf{N}$, which is denoted by ${\underline M}$, can be obtained in the same way.
Let $M>V^D_+$. First observe that $\mathbf{P}^M:=\mathbf{P}/M$ satisfies the scheme - where the source term and boundary data have been replaced by $f^+/M$ and $(v^D)^+/M$ respectively. Moreover, one can apply Proposition \[prop:fund\_ineq\], which yields $$\label{eq:ineqPMagain}
E_1(\mathbf{P}^M)\leq \frac{4p}{\beta_{\mathbf U}^2}\left(\Vert {\mathbf U}\Vert_{L^\infty}^2 + \frac{\Vert f^+\Vert_{L^\infty}}{M}\right) {\rm m} (\{{\mathbf P}^M>1\})\,.$$ Now observe that $\mathbf{P}\,=\,M\,\mathbf{P}^M\,=\,V^D_+\,\mathbf{P}^{V^D_+}$. Therefore, $$\begin{array}{rcl}
\ds E_1(\mathbf{P}^M)&\leq& \ds \frac{4p}{\beta_{\mathbf U}^2}\Big{(}\Vert {\mathbf U}\Vert_{L^\infty}^2\,{\rm m} (\{{\mathbf P}^{ V^D_+}>M / V^D_+\}) + \frac{\Vert f^+\Vert_{L^\infty}}{M}{\rm m}(\Omega)\Big{)} \\[1em]
&\leq&\ds\frac{4p}{\beta_{\mathbf U}^2}\Big{(}\Vert {\mathbf U}\Vert_{L^\infty}^2 \sum_{K\in\T}{\rm m}(K)\,\frac{\log(1+(P_K^{ V^D_+}-1)^+)^2}{\log(M / V^D_+)^2} + \frac{\Vert f^+\Vert_{L^\infty}}{M}{\rm m}(\Omega)\Big{)}\\[1em]
&\leq&\ds\frac{C_{\Omega,p}}{\xi\beta_{\mathbf U}^2}\Vert {\mathbf U}\Vert_{L^\infty}^2\, \frac{E_1(\mathbf{P}^{V^D_+})}{\log(M / V^D_+)^2} + \frac{4p\,{\rm m}(\Omega)}{\beta_{\mathbf U}^2}\frac{\Vert f^+\Vert_{L^\infty}}{M}\,\,,
\end{array}$$ where we used an argument similar to in the second inequality and a discrete Poincaré inequality in the third one. Then, by using again we get $$E_1(\mathbf{P}^{V^D_+})\ \leq\ \frac{4\,p\,{\rm m}(\Omega)}{\beta_{\mathbf U}^2}\left(\Vert {\mathbf U}\Vert_{L^\infty}^2 + \frac{\Vert f^+\Vert_{L^\infty}}{V^D_+}\right)$$ Therefore, the smallness condition of Proposition \[mainprop\] is satisfied by $E_1(\mathbf{P}^M)$ if $$\begin{gathered}
\label{boundM}
\left[\Vert{\mathbf U}\Vert_{L^\infty}^2\left(\Vert {\mathbf U}\Vert_{L^\infty}^2 + \frac{\Vert f^+\Vert_{L^\infty}}{V^D_+}\right)+ \frac{\Vert f^+\Vert_{L^\infty}}{M}\log\left(\frac{M}{V^D_+}\right)^2\right]\,\left(\Vert {\mathbf U}\Vert_{L^\infty}^2 + \frac{\Vert f^+\Vert_{L^\infty}}{M}\right)^2\\\ \leq\ \,C_{\Omega,\xi,p}\,\beta_{\mathbf U}^4\,\log\left(\frac{M}{V^D_+}\right)^2\,.\end{gathered}$$ It is clear that is satisfied for $M$ large enough, which permits to define ${\overline M}$. Observe that if $v^D_+=0$ ($V_+^D=1$) and $\mathbf U=0$, ${\overline M} = \widetilde{C}_{\Omega,\xi,p}\|f^+\|_{L^\infty}$ works as expected.
**Acknowledgements.** The authors thank the Labex CEMPI (ANR-11-LABX-0007-01) and the ANR MOHYCON (ANR-17-CE40-0027-01) for their support. They also want to thank Alexis F. Vasseur for fruitful exchanges on the subject.
|
Fermilab-CONF-13-284-T\
July 16, 2013
[**The Case for a\
Muon Collider Higgs Factory** ]{}
[ Yuri Alexahin$^{(1)}$, Charles M. Ankenbrandt$^{(2)}$, David B. Cline$^{(3)}$, Alexander Conway$^{(4)}$, Mary Anne Cummings$^{(2)}$, Vito Di Benedetto$^{(1)}$, Estia Eichten$^{(1)}$, Jean-Pierre Delahaye$^{(5)}$, Corrado Gatto$^{(6)}$, Benjamin Grinstein$^{(7)}$, Jack Gunion$^{(8)}$, Tao Han$^{(9)}$, Gail Hanson $^{(10)}$, Christopher T. Hill$^{(1)}$, Fedor Ignatov$^{(11)}$, Rolland P. Johnson$^{(2)}$, Valeri Lebedev$^{(1)}$, Leon M. Lederman$^{(1)}$, Ron Lipton$^{(1)}$, Zhen Liu $^{(9)}$, Tom Markiewicz$^{(5)}$, Anna Mazzacane$^{(1)}$, Nikolai Mokhov$^{(1)}$, Sergei Nagaitsev $^{(1)}$, David Neuffer$^{(1)}$, Mark Palmer$^{(1)}$, Milind V. Purohit$^{(12)}$, Rajendran Raja$^{(1)}$, Carlo Rubbia$^{(13)}$ Sergei Striganov$^{(1)}$, Don Summers$^{(14)}$, Nikolai Terentiev$^{(15)}$, Hans Wenzel$^{(1)}$ ]{}\
\[1cm\]
[(1) [*Fermilab, P.O. Box 500, Batavia, Illinois, 60510;* ]{}\
]{} [(2) [*Muons Inc., Batavia, Illinois, 60510* ]{}\
]{} [(3) [*University of California, Los Angeles, California;* ]{}\
]{} [(4) [*University of Chicago, Chicago, Illinois;* ]{}\
]{} [(5) [*SLAC National Accelerator Laboratory, Palo Alto, California;* ]{}\
]{} [(6) [*INFN Naples, Universita Degli Studi di Napoli Federico II, Italia;*]{}\
]{} [(7) [*University of California, San Diego, California;* ]{}\
]{} [(8) [*University of California, Davis, California;* ]{}\
]{} [(9) [*University of Pittsburgh, Pittsburgh, Pennsylvania;*]{}\
]{} [(10) [*University of California, Riverside, California;* ]{}\
]{} [(11) [*Budker Institute of Nuclear Physics, Russia;*]{}\
]{} [(12) [*University of South Carolina, Columbia, South Carolina;* ]{}\
]{} [(13) [*CERN, Geneva, Switzerland;*]{}\
]{} [(14) [*University of Mississippi, Oxford, Miss.;* ]{}\
]{} [(15) [*Carnegie Mellon University, Pittsburgh, Pennsylvania;*]{}\
]{}
Introduction {#introduction .unnumbered}
============
We propose the construction of a compact Muon Collider Higgs Factory. Such a machine can produce up to $\sim 14,000$ at $8\times 10^{31}$ cm$^{-2}$ sec$^{-1}$ clean Higgs events per year, enabling the most precise possible measurement of the mass, width and Higgs-Yukawa coupling constants (, that of the muon).
A Muon Collider Higgs Factory is part of an evolutionary program beginning with aggressive R&D on muon cooling, a possible neutrino factory such as $\nu$STORM, and the construction of Project-X with a rich program of precision physics addressing the $\sim 100$ TeV scale. This is followed by the Muon Collider Higgs factory. This program is upwardly scalable in energy and can lead ultimately to the construction of an energy frontier Muon Collider, reaching energy scales in excess of $\sim 10$ TeV. The Muon Collider Higgs Factory would utilize the intense proton beam from Project-X to produce, collect and cool muons, and by clever staging also use Project-X to accelerate the $\mu^+$ and $\mu^-$ bunches to $\sim 62.5$ GeV. These would be placed in a $\sim 100$ m diameter storage ring (about the size of the Fermilab Booster) and collide at single IP inside of dedicated detector.
The machine and detector issues at a Muon Collider have matured rapidly. Complex timing can be deployed to suppress backgrounds. A complete matrix for the machine now exists. Cooling has reached a conceptual stage at which an R&D program is required to establish proof of principle of this component of the Muon Collider strategy. We are confident that this could be accomplished, with sufficient funding, on a five-year time scale, leading directly to a design and preparation for construction of a Muon Collider Higgs Factory by the middle of the next decade.
For the 2013 Snowmass study we convened a workshop at UCLA. This included a detailed overview of the status of the Muon Collider and its physics justification. We did not focus on 6D cooling or many other machine issues that are the subject of the MAP white paper [@MAPsnowmass]. Key conclusions of the UCLA Workshop were as follows [@key]:
- The study of the Muon Collider Higgs Factory lattice indicates an attainable resolution of a few MeV, adequate to scan and measure the Higgs boson mass with a precision of $\sim 0.1$ MeV.
- Precision measurement of the Higgs boson mass, width, Higgs-Yukawa coupling constants, etc., can reveal the nature of the Higgs boson and discriminate between various theoretical options.
- Preliminary studies to reduce the machine induced backgrounds using e.g. sophisticated timing and requiring tracks to eminate from the IP look very promising but will require further detailed studies including full simulation and reconstruction of physics events on top of background.
- The Muon Collider Higgs Factory is the only proposed machine that can be upgraded to the multi-TeV scale with reasonable luminosity; a lack of significant beamstrahlung and therefore a narrow luminosity spectrum; and power and cost effectiveness.
- The Muon Collider Higgs Factory is part of an evolutionary process, from Project-X to the multi-TeV scale energy frontier, offering a diverse and rich physics program along the way at both the intensity and energy frontiers.
The concept of a Muon Collider was invented in the USSR in 1970’s [@russ]. Neuffer proposed a $90$ GeV Muon Collider “Z-factory” [@Neufferc] in 1979 and the muon storage ring for neutrino experiments [@Neuffern]. Work in the USA ramped up in 1992 with the first dedicated workshop organized by UCLA [@Thiessen], and the Muon Collider Higgs Factory concept emerged from this meeting [@Cline]. During the 1990’s, five workshops were organized by UCLA and two by BNL that served to establish the feasibility of the Muon Collider concept. In 1995 a Muon collider collaboration was formed with FNAL, BNL, LBNL and several universities and regular annual collaboration meetings were held. The muon storage ring neutrino factory concept became increasingly popular with the discovery of Neutrino Oscillations. D. Cline and G. Hanson co-organized the muon collider working groups at the 2001 Snowmass meeting. In 2010 the MAP program was formed and is now coordinating all Muon Collider activity, directed by M. Palmer of Fermilab.
There has been considerable recent progress on the physics studies of the Higgs boson at a Muon Collider and other Higgs factories. The Higgs boson can be located and scanned within a year at luminosities of order $\sim 10^{32}$, as detailed in the 2013 Snowmass Muon Collider Higgs Factory white paper [@Higgs_whitepaper].
Today there are only three realistic options for a lepton based Higgs factory:
1. An ILC type machine to produce the Higgs boson in associated production with a $Z^0$ requiring $10^{34}$ cm$^{-2}$ sec$^{-1}$ luminosity.
2. A compact circular Muon Collider Higgs Factory that produces the Higgs boson directly as an $s$-channel resonance requiring $\sim 10^{32}$ cm$^{-2}$ sec$^{-1}$.
3. A large circular $e^+e^-$ collider, , “TLEP,” of $\sim 80$ km circumference, also producing the Higgs boson in associated production with a $Z^0$ requiring $\sim 10^{34}$ cm$^{-2}$ sec$^{-1}$ luminosity.
In case (2) the collider could be evolved up in energy to a multi-TeV Muon Collider while the core muon production, accumulation and cooling systems would remain almost the same. Likewise, a large circumference circular $e^+e^-$ collider as in case (3) could provide the tunnel for a (VLHC) at $\sim 100$ TeV proton-(anti)proton center-of-mass energy addressing multi-TeV partonic energy scales. Such an evolution to the multi-TeV energy scales is not possible for option (1).
We quote some comments on the Muon Collider and its physics potential from a recent presentation of C. Rubbia [@Rubbia]:
- “In a $\mu^+\mu^-$ collider, when compared to an $e^+ e^-$ collider, the direct Higgs boson production cross section is greatly enhanced since the $s$-channel coupling to a scalar is proportional to the lepton mass.”
- “Therefore the properties of the Higgs boson can be detailed over a larger fraction of model parameter space than at any other proposed accelerator method.”
- “A high energy $\mu^+\mu^-$ collider is the only possible circular high energy lepton collider that can be situated within the CERN or FNAL sites.”
- “The unique feature of the direct production of a Higgs boson in the $s$-state is that the mass, total width and all partial widths can be directly measured with remarkable accuracy.”
Many more details are presented in the 2013 Snowmass Muon Collider and Muon Collider Higgs Factory white papers [@MAPsnowmass; @Higgs_whitepaper].
[99]{}
“Enabling Intensity and Energy Frontier Science with a Muon Accelerator Facility in the U.S.,” A White Paper submitted to the 2013 U.S. Community Summer Study of the Division of Particles and Fields of the American Physical Society, Draft 06/05/2013, U.S. Map Collaboration, ed. J-P Delahaye, ;
For the UCLA Workshop talks see https://hepconf.physics.ucla.edu/higgs2013/;
Proceedings of the First Workshop on the Physics Potential and Development of $\mu+\mu-$ Colliders, Napa, California (1992), Nucl. Instru. and Methods, A350, 24 (1994), D. Cline, Editor;
D. Neuffer, “Colliding Muon Beams at 90-GeV, FERMILAB-FN-0319, (1979); D. Neuffer, “Design Of Muon Storage Rings For Neutrino Oscillations Experiments. (talk),” IEEE Trans. Nucl. Sci. [**28**]{}, 2034 (1981); H. A. Thiessen, P. Channell, C. Hoffman, R. Jameson, S. Schriber, G. Swain, T. -S. Wang and J. Zumbro [*et al.*]{}, “Muon Collider Workshop,” LA-UR-93-866 ;
D. B. Cline, “Physics potential of a few 100-GeV mu+ mu- collider,” Nucl. Instrum. Meth. A [**350**]{}, 24 (1994) ; “Muon Collider Higgs Factory for Smowmass 2013", A White Paper submitted to the 2013 U.S. Community Summer Study of the Division of Particles and Fields of the American Physical Society, Y. Alexahin, , UCLA Physics Preprint-100, FERMILAB-CONF-13-245-T (July, 2013).
Direct quotes from Carlo Rubbia’s talk: "A millimole of muons for a Higgs factory?” talk delivered at GSSI, L’Aquila, Italy, and Institute for Advanced Sustainability Studies, Potsdam, Germany, (March, 2013).
|
---
abstract: |
In this paper, we investigate the accretion on the Reissner-Nordström anti-de-Sitter black hole with global monopole charge. We discuss the general solutions of accretion using the isothermal and polytropic equations of state for steady state, spherically symmetric, non-rotating accretion on the black hole. In the case of isothermal flow, we consider some specific fluids and derive their solutions at the sonic point as well. However, in case of polytropic fluid we calculate the general expressions only, as there exists no global (Bondi) solutions for polytropic test fluids. In addition to this, the effect of fluid on the mass accretion rate are also studied. Moreover, the large monopole parameter $\beta$ greatly suppresses the maximum accretion rate.\
**Keywords**: Accretion; black hole; cosmological constant; global monopole charge.
address:
- '$^1$Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan'
- '$^2$Department of Physics, Faculty of Sciences, Akdeniz University, 07058, Antalya, Turkey'
author:
- 'Ayyesha K. Ahmed$^1$, Ugur Camci$^2$, and Mubasher Jamil$^1$'
title: '**Accretion on Reissner-Nordström-(anti)-de Sitter Black Hole with Global Monopole**'
---
Introduction {#INT}
============
In astrophysics, the accretion of fluids or matter onto compact objects such as neutron stars or black holes, is an interesting physical process as it describes a scenario which is most likely to explain the high energy output from the active galactic nuclei and quasars. Accretion is the process of capturing the matter by a gravitating object towards its center which leads to increase the mass and angular momentum of the accreting body. The stars and planets are formed by the process of accretion in dust clouds. The existence of supermassive black holes at the center of galaxies suggests that such black holes could have been gradually developed through the accretion process. An accretion disk is developed when dust and gases rotate around a compact object and accumulate into a disk. However, accretion does not always increase the mass of the accreting body but it could also decrease the mass such as accretion of phantom energy [@M1; @M2]. A unique feature of black hole is the presence of an event horizon which acts as a boundary through which infalling fluid disappears. This may have various implications. For instance, it provides the inner boundary condition which describes the motion of the fluid and further it helps to avoid the uncertainties regarding the correct boundary conditions.
In $1952$ Bondi investigated the accretion process for a star within the Newtonian framework [@Bondi]. After the evolution of the relativistic theory of gravitation given by Einstein, it became possible for the astrophysicists to study the accretion process within the relativistic framework. So, Michel was the first one, who investigated the accretion process onto the Schwarzschild black hole [@Michel]. Further, Shapiro and Teukolsky [@Shapiro] also contributed to Michel’s idea. Later on, Babichev et al.[@Babichev] discussed the effect of accretion of a dark energy onto Schwarzschild black hole and found that accretion of phantom energy will decrease the mass of the black hole. Debnath [@Debnath] further generalized the Babichev’s idea and presented a framework of a static accretion onto general static and spherically symmetric black holes. Moreover in the series of recent papers [@1t; @t1; @PMach; @Mjamil] accretion of a spherically symmetric spacetime with is investigated. The authors were mainly interested to determine that how the presence of the cosmological constant $\Lambda$ effects the accretion rate.
During the phase transition, many topological defects are produced such as cosmic strings, domain walls, monopoles, etc. Monopoles are the three-dimensional topological defects that are formed when the spherical symmetry is broken during the phase transition. These monopoles have Goldstone fields with high energy density that decrease with the distance only as $r^{-2}$, so that the total energy density is divergent at the large distances [@Bar]. This large density suggests that global monopoles can produce the strong gravitational fields. Another interesting point is that the spacetime around a cosmic string is conical globally but locally the spacetime is flat i.e. just a Minkowski spacetime. So we can think about this spacetime as Minkowski spacetime but, in case of a global monopole the spacetime is not locally flat. However, globally the spacetime at the equatorial plane is conical having deficit angle as $\Delta=8\pi^{2}\eta^{2}$ [@Jusufi]. Hence, it is evident that these monopoles exerts practically no gravitational force on non-relativistic matter but the space around it has a deficit solid angle and all the light rays are deflected by that angle, independent of the impact parameter [@Bar]. It is believed that the charged black holes do not exist in nature, since initial charge will be neutralized quickly [@Punsly]. However, these charged black holes may be produced during the gravitational collapse of massive stars particularly magnetized rotating stars surrounded by a magnetosphere having an equal and opposite charge. The magnetosphere preserves the black hole from a neutralization due to the accretion of charge, so that the black hole remains stable in a typical astrophysical environment having low density.
Barriola and Vilenkin found the approximate solution of the Einstein equations for the static spherically symmetric black hole with a global monopole [@Bar]. Several other authors have also studied the physical properties of the black holes with global monopoles [@Bronnikov; @Hyu; @Pitelli; @Rahaman]. In the absence of the electric charge $Q$ and the cosmological constant $\Lambda$, Letelier [@Letelier] obtained the metric representing the black hole spacetime with a spherical mass $M$ centered at the origin of the system of coordinates, surrounded by a spherical cloud of strings. The accretion process onto the black hole with a string cloud parameter, which can be interpreted as a global monopole, is examined by Ganguly et al.[@ggm]. The circular geodesics and accretion disk of a black hole with global monopole is discussed by Sun et al [@Sun]. Recently, the accretion process of a charged black hole in the anti-de-Sitter spacetime is investigated by Ficek [@Ficek]. The author calculated the analytical solutions for the isothermal and polytropic fluids and found a way to find sonic points as critical points of a Hamiltonian system. He has also proved the existence of closed trajectories and therefore homoclinic solutions in the phase space of subrelativistic isothermal flows. Here, we extend his idea by considering a global monopole charge $\beta$, which is different from the electric charge $Q$ in the black hole. In this paper we investigate the spherically symmetric, steady flow of the most general perfect fluid (gas). The basic motivation comes from the fact that some features of Reissner-Nordström black hole resemble with the Kerr solution as they both share the similar structure of horizons [@Poisson]. The accretion dynamics for Schwarzschild monopole de-Sitter black hole with $Q=0$ has already been discussed in [@Mjamil].
The paper is organized as follows: In sec II we derive the metric (line element) for the Reissner-Nordström anti-de-Sitter (RN-AdS) black hole with global monopole charge. In sec III a general formalism and conservation laws for the accretion process are derived. The speed of sound at the sonic (critical) point is evaluated in sec IV. Next, we have assumed the isothermal equation of state and by choosing different values of state parameter the behaviour of flow for different cases such as ultra-stiff, ultra-relativistic, rotation and sub-relativistic fluids is analysed. In sec VI the general solutions for the polytropic equation of state are obtained as there exists no global (Bondi) solutions for them. In the subsequent section, we have calculated the matter accretion rate of the black hole for the isothermal flow and finally we conclude our discussion. Throughout we use the common relativistic notation while the Greek indices run through $\mu,\nu=0,1,2,3$. The chosen metric signature is $(-,+,+,+)$, and the geometric units as $G=c=1$.
Reissner-Nordström-(anti)-de Sitter spacetime with global monopole
==================================================================
We adopted the metric formalism as given in [@Naresh]. We assume the general static spherically symmetric metric ansatz of the form $$\begin{aligned}
\label{1}
ds^{2}&=&-f(r)dt^2+\frac{1}{f(r)}dr^{2}+r^2(d\theta^2+\sin^2\theta d\phi^2).\end{aligned}$$ The Einstein’s field equations with the cosmological constant $\Lambda$ are $$\begin{aligned}
\label{2}
R_{\mu \nu}-\frac{1}{2} R g_{\mu \nu}+\Lambda g_{\mu \nu}&=&8\pi \left( T^{(EM)}_{\mu \nu} + T^{(GM)}_{\mu \nu} \right),\end{aligned}$$ where $T^{(EM)}_{\mu \nu}$ and $T^{(GM)}_{\mu \nu}$ are the energy-momentum tensors for an electromagnetic field and the energy-momentum tensor for the nonminimally coupled global monopole (GM) field, respectively. The field Eqs. (\[2\]) have non-vacuum solution when the total energy-momentum tensor $T_{\mu \nu} = T^{(EM)}_{\mu \nu} + T^{(GM)}_{\mu \nu}$ does not vanish. The energy momentum tensor for an electromagnetic field is $$\begin{aligned}
\label{3}
T^{(EM)}_{\mu \nu}&=&\frac{1}{4 \pi}(F_{\mu \rho}F_{~\nu}^{\rho}-\frac{1}{4}g_{\mu \nu}F^{\alpha \beta}F_{\alpha \beta}),\end{aligned}$$ where $F^{\alpha \beta}F_{\alpha \beta} \propto 2 E^2$, $E \propto Q / r^2$, $Q$ is the electric charge and the trace of $T^{(EM)}_{\mu \nu}$ is zero. The Lagrangian which describes a global monopole is given by $$\begin{aligned}
\label{n1}
\mathcal{L}&=&\frac{1}{2}(\partial\psi^{a})^{2}-V(\psi^{a}),\end{aligned}$$ with $V(\psi^{a})=\frac{\lambda}{4}(\psi^{a}\psi^{a}-\eta^{2})^{2}$ and $\lambda$ is the self-coupling term and $\eta$ is scale of gauge-symmetry breaking $\eta\sim 10^{6}$ GeV [@Jusufi]. Here $\psi^{a}$ is a triplet scalar field with $a=1,2,3$ given by $$\begin{aligned}
\label{n2}
\psi^{a}&=&\eta h(r) \frac{x^{a}}{r},\end{aligned}$$ with $x^{a}x^{a}=r^{2}$ which yields $\psi^2 = \psi^a \psi^a = \eta^2 h(r)^2$. Hence, the energy-momentum tensor for the global monopole field is $$\begin{aligned}
T^{(GM)}_{\mu \nu}&=&(\partial\psi^{a})^{2} + g_{\mu \nu} \left[\frac{1}{2}(\partial\psi^{a})^{2} -V(\psi^{a}) \right], \label{3-2}\end{aligned}$$ where $\psi = \pm \sqrt{\psi^a \psi^a}$. Outside the monopole core it must be assumed that $h(r) \rightarrow 1$ as $r \rightarrow \infty$ for $V(\psi^{a})$ to vanish asymptotically. Thus far from the core the stress tensor of the system has the components as $$\begin{aligned}
\label{3-4}
& & T_{~0}^{0\,(GM)} = T_{~1}^{1\,(GM)} = \frac{\eta^2 }{r^2}, \quad T_{~2}^{2\,(GM)} = T_{~3}^{3\,(GM)} = 0.\end{aligned}$$ The equation of motion for $\psi^{a}$ is given by $$\begin{aligned}
\label{ra11}
\Box\psi^{a}+\frac{\partial V}{\partial\psi^{a}}=0,\end{aligned}$$ which upon simplification gives $$\begin{aligned}
\label{ra1}
fh_{,rr}+h_{,r}\Big[f_{,r}+\frac{2}{r}f\Big]-\lambda\eta^{2}h(h^{2}-1)&=&0,\end{aligned}$$ where $f$ and $h$ are the function of $r$ due to spherical symmetry and is similar to the equation of motion which we obtain through the energy-momentum conservation i.e. $T^{\mu\nu}_{~~;\mu}=0$ [@Chen]. A global monopole could readily be added to the scalar field by a general prescription due to Dadhich and Patel [@DP] for any spherically symmetric solution.
Using (\[3\]) and (\[3-2\]) together with (\[3-4\]) one can obtain from the Einstein field equations (\[2\]) that $$\begin{aligned}
\label{4}
& & f(r) =1 + 2 \Phi(r) + \frac{Q^{2}}{r^{2}}-\frac{\Lambda r^{2}}{3},\end{aligned}$$ where $\Phi(r)$ is the Newtonian gravitational potential and should satisfy the following constraint equations $$\begin{aligned}
& & \Phi_{,r} + \frac{1}{r} \Phi = \frac{\eta^2}{2 r}, \label{5a} \\
& & \nabla^{2}\Phi = \Phi_{,rr} + \frac{2}{r} \Phi_{,r} = 0. \label{5b}\end{aligned}$$ Clearly, Eq. (\[5b\]) is the Cauchy-Euler equation which gives a well known solution $$\begin{aligned}
\label{6}
\Phi(r)&=&\beta-\frac{M}{r},\end{aligned}$$ where both $\beta = \eta^2 /2$ and $M$ are the constants of integration. These constants are actually the global monopole charge and the mass of the black hole respectively. So using (\[6\]) the metric function $f(r)$ in (\[4\]) gets the form $$\begin{aligned}
\label{7}
f(r)&=&1+2\Big(\beta-\frac{M}{r}\Big)+\frac{Q^{2}}{r^{2}}-\frac{\Lambda r^{2}}{3}.\end{aligned}$$
For $\beta=0$, Eq. (\[7\]) reduces to the RN-AdS black hole while $\beta=\Lambda=0$ gives Schwarzschild black hole. It is known that a horizon is the null-hypersurface. The normal vector to a constant $r$ hypersurface is $t^{\mu}=(0,1,0,0)$. Clearly $t^{\mu}t_{\mu}=f(r)$ therefore, we can say that $r =const$ is the null-hypersurface at $f(r)=0$ and we can determine all the possible horizons from it. For this, we have a polynomial of degree four by using the Eq. (\[7\]). When $\Lambda \neq 0$, we get the algebraic equation of the form $$\label{h-eq-2}
r^4 + a_1 r^2 + a_2 r + a_3 = 0,$$ where $a_1 = -3 (1+ 2 \beta) / \Lambda, \, a_2 = 6 M / \Lambda$ and $a_3 = -3 Q^2 / \Lambda$. This equation can be solved by making it factorizable such that $P^2 - R^2= (P + R) (P - R)$ which gives rise to the resolvent cubic equation. Then the quantities $P$ and $R$ in perfect square have the form given by $$\label{h-eq-3}
P = r^2 + x/2, \qquad R = \sqrt{x-a_1} \left( r - \frac{a_2}{2 (x -a_1)} \right),$$ if the variable $x$ is chosen such that $$\label{h-eq-4}
x^3 - a_1 x^2 - 4 a_3 x + b = 0,$$ i.e. the resolvent cubic with $b = 4 a_1 a_3 - a_2^2 = 36 \left[ (1+ 2\beta) Q^2 - M^2 \right] / \Lambda^2$. Thus, we note that $R$ is linear and $P$ is quadratic in $r$, so each term $P + R$ and $P - R$ is quadratic. Therefore, solving these quadratic formulas one can give all four solutions to the original quartic equation (\[h-eq-2\]). The cubic equation (\[h-eq-4\]) can be simplified by making the substitution $x= y + a_1 /3$. In terms of the new variable $y$, Eq.(\[h-eq-4\]) then becomes $y^3 + 3 L y -2K = 0$, where $K = \left( 2 a_1^3 + 36 a_1 a_3 - 27 b \right) / 54$ and $L = - \left( 12 a_3 + a_1^2 \right) /9$. Now we can solve algebraically the above cubic equation defining the polynomial discriminant $D=K^2 + L^3$. If $D > 0$, one of the roots is real and the other two roots are complex conjugates. If $D < 0$, all roots are real and unequal. In this case, defining $\theta = \arccos \left( K / \sqrt{-L^3}\right)$, then the real valued solutions of (\[h-eq-4\]) are of the form $$\label{h-eq-5}
x_{1,2,3} = \frac{a_1}{3} + 2 \sqrt{-L} \cos\left( \frac{2 \pi \ell}{3} + \frac{\theta}{3} \right),$$ where $\ell \in \{ 0,1,2 \}$ and $L \leq 0$ which yields $Q^2 \leq (1 + 2 \beta)^2 /(4 \Lambda)$ for $\Lambda > 0$ (de-Sitter spacetime) and $Q^2 \geq (1 + 2 \beta)^2 /(4 \Lambda)$ for $\Lambda < 0$ (anti-de-Sitter spacetime). Thus, the value of $\beta$ for $Q >0$ has to satisfy the relation $ |1 + 2 \beta| > 0$, i.e. $\beta > -1/2$ for both $\Lambda > 0$ and $\Lambda < 0$. Let $x_1$ be a real root of (\[h-eq-4\]), the four roots of the original quartic (\[h-eq-2\]) are given by the roots of the quadratic equations $$\label{h-eq-6}
r^2 \pm \sqrt{x_1 -a_1} \, r + \frac{1}{2} \left( x_1 \mp \frac{a_2}{\sqrt{x_1 -a_1}} \right) = 0,$$ which are $$\begin{aligned}
\label{h-eq-7-1}
r_1&=&\frac{1}{2} \left( \sqrt{x_1 -a_1} + \sqrt{\triangle_{-}} \right),\\
r_2&=&\frac{1}{2} \left( \sqrt{x_1 -a_1} - \sqrt{\triangle_{-}} \right),\\
r_3&=&\frac{1}{2} \left( \sqrt{x_1 -a_1} + \sqrt{\triangle_{+}} \right), \\
r_4&=&\frac{1}{2} \left( \sqrt{x_1 -a_1} - \sqrt{\triangle_{+}} \right), \label{h-eq-7-2}\end{aligned}$$ where $\triangle_{\pm} = -(x_1 + a_1) \pm 2 a_2 / \sqrt{x_1 - a_1}$ and $x_1 > a_1 = -3 (1+ 2 \beta) / \Lambda$. The polynomial $f=0$ has at most three real and positive roots, which are representing the Cauchy horizon, event horizon and cosmological horizon respectively. In order to find the generic case where these three horizons exist, the values of the parameters $M, Q, \Lambda$ and $\beta$ must be constraint. When all of those parameters are positive and $0 < \Lambda < 0.17$ then three horizons exist, but when $-1 < \Lambda < 0$ and the remaining parameters positive then there exists only two horizons.
Further, from curvature invariants given by Eq. (\[e1\]), (\[e2\]) and (\[e3\]) in Appendix we see that the curvature is finite everywhere outside the horizon and curvature invariants diverge at $r=0$. Therefore, to remove the singularity at horizon we introduce the Eddington-Finkelstein (EF) coordinate [@Patryk] which is regular at the event horizon, such that $$\begin{aligned}
\label{10}
dt^{\prime}&=&dt-\frac{2\Big(\frac{M}{r}-\beta\Big)-\frac{Q^{2}}{r^{2}}+\frac{\Lambda r^{2}}{3}}{1+2\big{(\beta-\frac{M}{r})}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda r^{2}}{3}}dr.\end{aligned}$$ This leads us to the following form of metric: $$\begin{aligned}
\label{11}
& & \fl ds^{2}=-\left[ 1 + 2\left( \beta-\frac{M}{r} \right) + \frac{Q^{2}}{r^{2}}-\frac{\Lambda r^{2}}{3} \right] dt^{\prime2} -2\left[ 2 \left( \beta-\frac{M}{r}\right) + \frac{Q^{2}}{r^{2}} - \frac{\Lambda r^{2}}{3} \right] dt^{\prime}dr \nonumber \\ && + \left[1-2\left( \beta-\frac{M}{r} \right) - \frac{Q^{2}}{r^{2}}+\frac{\Lambda r^{2}}{3}\right] dr^{2} + r^2(d\theta^2+\sin^2\theta d\phi^2).\end{aligned}$$ The determinant of the metric defined in (\[11\]) is $g=-r^{4}\sin^{2}\theta$ and $\sqrt{\mid g\mid}=r^{2}\sin\theta$.
General equations for spherical accretion
=========================================
Now we derive the governing equations for spherical accretion. Here, we consider the perfect fluid and analyse the accretion rate and flow of a perfect fluid near RN-AdS black hole with a global monopole charge. For this we define the two basic laws of accretion, i.e. particle conservation and energy conservation. Let $n$ be the number of particles and $u^{\mu}$ be the four velocity of the fluid, then the particle flux will be $J^{\mu}=n u^{\mu}$. From the law of particle number conservation there will be no change in the number of particles, their number remain conserved. In other words, we can say that for this system, the divergence of $4$-vector current density vanishes. Mathematically, it means that $$\begin{aligned}
\label{12}
\nabla_{\mu}J^{\mu}&=&0,\end{aligned}$$ where $\nabla_{\mu}$ is the covariant derivative. On the other hand, the energy momentum tensor for a perfect fluid is given by $T^{\mu \nu}=(e+p)u^{\mu}u^{\nu}+pg^{\mu \nu}$ with $e$ as the energy density and $p$ as the pressure and its conservation is given by $$\begin{aligned}
\label{13}
\nabla_{\mu}T^{\mu \nu}&=&0.\end{aligned}$$ The Bondi-type accretion is steady state and spherically symmetric [@1t; @t1], so all the quantities must be function of the radial coordinate only. Furthermore, we are assuming that the fluid is flowing radially in the equatorial plane $(\theta=\frac{\pi}{2})$ therefore, $u^{\theta}=u^{\phi}=0$. By the normalization condition for $4$-velocity $u^{\mu}u_{\mu}=-1$ we obtain, $$\begin{aligned}
\label{14}
u^{t}&=&\frac{\sqrt{ f(r)^2 +(u^{r})^{2}}}{f(r)},\end{aligned}$$ which yields $$\begin{aligned}
\label{15}
u_{t}&=&-\sqrt{f(r)+(u^{r})^{2}}.\end{aligned}$$ At the equatorial plane the continuity equation (\[12\]) becomes $$\begin{aligned}
\label{16}
\nabla_{\mu}(n u^{\mu})&=& \frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}n u^{\mu})\nonumber \\&=&
\frac{1}{r^{2}}\partial_{r}(r^{2}n u^{r})=0,\end{aligned}$$ which upon integration yields $$\begin{aligned}
\label{17}
r^{2}n u^{r}&=&C_{1},\end{aligned}$$ where $C_{1}$ is a constant of integration. For inward flow $u^{r}<0$ and so $C_{1}<0$. Similarly, we know that enthalpy is the ratio between density and the total internal energy of the system at constant pressure. Therefore, by using the first law of thermodynamics, we can define enthalpy as $h(e,p,n)=\frac{e+p}{n}$. As the flow is smooth therefore, Eq. (\[13\]) leads us to $$\begin{aligned}
\label{18}
n u^{\mu}\nabla_{\mu}(hu^{\nu})+g^{\mu\nu}\partial_{\mu}p&=&0.\end{aligned}$$ Further, we assume that the entropy of a fluid moving along a stream line is constant, so the flow must be isentropic [@Pringle]. Hence, Eq. (\[18\]) gives $$\begin{aligned}
\label{20}
u^{\mu}\nabla_{\mu}(hu_{\nu})+\partial_{\nu}h&=&0.\end{aligned}$$ Taking the zeroth component of Eq. (\[20\]) we obtain $$\begin{aligned}
\label{23}
\partial_{r}(hu_{t})=0,\end{aligned}$$ which upon integration gives $$\begin{aligned}
\label{24}
h\sqrt{f(r)+(u^{r})^{2}}&=&C_{2},\end{aligned}$$ where $C_2$ is an another constant of integration. Now, these Eqs. (\[17\]) and (\[24\]) are the main equations which will be used further to analyse the flow of a perfect fluid in the background of RN-AdS with global monopole.
Sonic points
============
Sonic point is the critical point where the four-velocity of the moving fluid becomes equal to the local speed of sound therefore, the flow passing through the sonic point has the maximum accretion rate. If we take constant pressure into account, i.e. $h=h(n)$ then fluid becomes barotropic and the equation of state for barotropic flow can be written as [@Ficek] $$\begin{aligned}
\label{33}
\frac{dh}{h}=a^2 \frac{dn}{n},\end{aligned}$$ where $a$ is the local speed of sound. So Eq. (\[33\]) yields $\ln h = a^2 \ln n$. From Eqs. (\[17\]), (\[24\]) and (\[33\]), we obtain
$$\begin{aligned}
\label{34}
\left[\left(\frac {u^{r}}{u_{t}}\right)^2-a^2\right] \left( \ln u^{r} \right)_{,r}&=&\frac{1}{r(u_{t})^2}\left[2a^{2}(u_{t})^2 - \frac{1}{2} r f_{,r} \right].~~~~~~\end{aligned}$$
In this paper, the quantities referring to the critical point will be denoted with the subscripted letter “$c$". At critical point both sides of Eq. (\[34\]) must be equal to zero. As $(\ln u^{r})_{,r} \neq 0$ so, the local speed of sound at the sonic point becomes $$\begin{aligned}
\label{35}
a_c^{2} = \left(\frac{u_c^{r}} {u_{t_c}}\right)^{2},\end{aligned}$$ where $a_c$ is the value of local speed of sound at sonic point, $r_c$ is the distance of fluid from the black hole at sonic point and $u_c^r$ is the velocity of the fluid at the sonic point. Then, the rhs of Eq. (\[34\]) at the sonic point given by $$\begin{aligned}
\label{36}
2a_c^{2} (u_{t_c})^{2}-\frac{1}{2} r_c f_{c,r_c} = 0,\end{aligned}$$ where $f_c = f(r)|_{r=r_c}$, $f_{c,r_c} = f_{,r} |_{r= r_c}$ and $u_{t_c} = u_t (r_{c},u^r_{c})$. Putting Eq. (\[35\]) into (\[36\]) we obtain the expression for the radial velocity at sonic point as $$\begin{aligned}
\label{37}
(u_c^{r})^2 &=& \frac{1}{4} r_c f_{c,r_c} .\end{aligned}$$ Using Eqs. (\[15\]), (\[36\]) and (\[37\]) we get $$\begin{aligned}
\label{38-1}
r_c f_{c,r_c} = 4 a_c^2 \left[ f_c + (u_c^r)^2 \right], \label{38-2}\end{aligned}$$ which leads to $$\begin{aligned}
\label{38-4}
a_c^2 &=& \frac{r_c f_{c,r_c}}{r_c f_{c,r_c} + 4 f_c}.\end{aligned}$$ This equation allows us to determine $r_c$ once the speed of sound $a^2 = dp / de$ is known. Solving Eqs. (\[37\]) and (\[38-4\]), we may find the values of $r_c$ and $u_c^{r}$ and so we get the critical point as $(r_c,\pm u_c^{r})$.
Isothermal test fluids
======================
Isothermal fluids are those fluids which flow at a constant temperature so that the sound speed throughout the accretion process remains constant. As the fluid is flowing at a very fast speed so it does not take the opportunity to exchange the heat with the surroundings, so it is more likely that our dynamical system is adiabatic. The equation of state for such fluids is of the form $p= k e$, where $k$ is the state parameter such that $0 < k \leq 1$ and $e$ is the energy density [@t1]. In general, the adiabatic sound speed is defined as $a^{2}= {dp}/{de}$. If we compare it to the equation of state, we find $a^{2}=k$. Since there is no change in entropy, so $T dS = 0$ where $S$ denotes the entropy. By the first law of thermodynamics $$\begin{aligned}
\label{39}
\frac{de}{dn}&=&\frac{e+p}{n}=h.\end{aligned}$$ On integrating it from the sonic point to any point inside the fluid, we obtain $$\begin{aligned}
\label{40}
n&=&n_{c}\exp\left(\int_{e_{c}}^{e}\frac{d e^{\prime}}{e^{\prime} + p \left(e^{\prime}\right)} \right).\end{aligned}$$ For the isothermal equation of state $p=ke$, Eq. (\[40\]) becomes $$\begin{aligned}
\label{40a}
n&=&n_{c}\left(\frac{e}{e_{c}}\right)^\frac{1}{k + 1}.\end{aligned}$$ Comparing this to enthalpy, we get $$\begin{aligned}
\label{41}
h &=& \frac{(k+1)e_{c}}{n_{c}} \left( \frac{n}{n_{c}} \right)^{k},\end{aligned}$$ By using Eq. (\[41\]) in Eq. (\[24\]) we have $$\begin{aligned}
\label{42}
n^{k} \sqrt{ f(r) + (u^r)^2 } &=& C_{3},\end{aligned}$$ where $C_3 = C_2 n_c^{1-k} / (k+ 1) e_c$. Comparing Eqs. (\[17\]) and (\[42\]), we get $$\begin{aligned}
\label{43}
\sqrt{ f(r) +(u^{r})^2}= C_{3}r^{2k}(u^{r})^{k}.\end{aligned}$$ On the other hand, we can define the Hamiltonian [@A1; @A2] as $$\begin{aligned}
\label{41b}
\mathcal{H} &=& \frac{f^{1-k}}{(1-v^{2})^{1-k}v^{2k}r^{4k}},\end{aligned}$$ where $v$ is the three-dimensional speed for the radial motion in equatorial plane and is defined as $v\equiv \frac{dr}{fdt}$. Consequently, we have $$\begin{aligned}
\label{41bb}
v^{2}=\Big(\frac{u}{fu^{t}}\Big)^{2}=\frac{u^{2}}{u_{t}^{2}}=\frac{u^{2}}{f+u^{2}},\end{aligned}$$ such that $u^{r}=u=\frac{dr}{d\tau}$, $u^{t}=\frac{dt}{d\tau}$, $u_{t}=-fu^{t}$. As we are mainly concerned in finding the solutions at the sonic point, therefore Eqs. (\[37\]) and (\[38-2\]) lead to $$\begin{aligned}
\label{44a}
(u_c^{r})^2 &=& \frac{1}{4}r_c f_{c,r_c} \\ &=&
k \left( \frac{1}{4} r_c f_{c,r_c} + f_c \right). \label{44b}\end{aligned}$$
It is worthwhile to generalize this discussion from a single point to the continuous flowing fluid. So, we will now analyse the behaviour of fluid by choosing different values of state parameter. For instance, we have $k=1$ (ultra-stiff fluid), $k=1/2$ (ultra-relativistic fluid), $k=1/3$ (radiation fluid) and $k=1/4$ (sub-relativistic fluid).
Solution for $k=1$
------------------
For the ultra-stiff fluids, energy density becomes equal to the pressure so that the equation of state is of the form $p = e$, (i.e. $k=1$). From Eqs. (\[44a\]) and (\[44b\]) one can find $f_c = 0$, i.e. the expression for $r_c$ is identical to the expression for the locations of event horizon, so the sonic point and the event horizon are located at the same place, i.e. $r_h = r_c$. The Hamiltonian (\[41b\]) in this case acquires the form: $$\begin{aligned}
\label{46a}
\mathcal{H}&=&\frac{1}{v^{2}r^{4}}.\end{aligned}$$ The plot in figure \[f1\] shows the two types of fluid motion. First is the supersonic accretion flow in the upper half, i.e. the region where $v>0$ and the other is the subsonic accretion flow in the lower region where $v<0$.
![Plot showing the trajectories of solutions to Eq. (\[41b\]) in phase space and the parameters taken as $k=1$, $M=1$, $Q=0.85$, $\beta=0.075$ and $\Lambda=-0.075$. The black curves show the solution at critical point for which $\mathcal{H}=\mathcal{H}_{c}$, the red curves show the solution at $\mathcal{H}=\mathcal{H}_{c}+15.004$ and the green for $\mathcal{H}=\mathcal{H}_{c}+30.007$.[]{data-label="f1"}](k1.eps){width="8cm"}
Solution for $k=1/2$
--------------------
For ultra-relativistic fluids the equation of state is of the form $p=e / 2$ (i.e. $k=1/2$). In this case pressure is always less than the energy density. Thus, from Eqs. (\[44a\]) and (\[44b\]) we find $r_c f_{c,r_c} - 4 f_c = 0$ which further reduces to the quartic equation $$\begin{aligned}
\label{47}
r_c^{4} + a_1 r_c^{2}+ a_2 r_c + a_3 = 0,\end{aligned}$$ where $a_1 = - 6 (1+ 2 \beta) / \Lambda, a_2 = 15 M / \Lambda$ and $a_3 = -9 Q^2 / \Lambda$. On further simplification, we may find that the values of $r_c$ are same as given in (\[h-eq-7-1\]) and (\[h-eq-7-2\]). Putting this $r_c$ into (\[44a\]) we get the value $u_c^r$ and then we solve these two equations to get the two critical points as $(r_c,\pm u_c^{r})$. For instance, the relation between $r$ and $u^{r}$ is obtained from Eq. (\[43\]) as $$\begin{aligned}
\label{49}
u^{r}=\frac{1}{2}C r^{2}\pm\frac{1}{2} \sqrt{C^{2}r^{4}-4 f(r)},\end{aligned}$$ where $C= C_3^2$. The Hamiltonian (\[41b\]) reduces to $$\begin{aligned}
\label{49b}
\mathcal{H}&=&\frac{\sqrt{f}}{r^{2}v\sqrt{1-v^{2}}}.\end{aligned}$$ The plot in figure \[f2\] shows the different behaviours for the motion of a fluid. The green and red curves are unphysical since they are double valued. The black curves correspond to the transonic behaviour whereas, the blue and magenta curves show the supersonic behaviour of fluid in the region $v > v_{c}$ and subsonic behaviour for $v < v_{c}$, where $v_c$ is the three-dimensional speed for the radial motion in equatorial plane at the sonic point, which is defined as $v_c = \sqrt{u_c^2 / (f_c + u_c^2)}$ by the Eq. (\[41bb\]).
![The plot shows the trajectories of solutions to Eq. (\[41b\]) in phase space where the parameters are $k=1/2$, $M=1$, $Q=0.85$, $\beta=0.075$ and $\Lambda=-0.075$. The black curves show the solution for $\mathcal{H}=\mathcal{H}_{c}$, the red curves show the solution for $\mathcal{H}=\mathcal{H}_{c}-0.04$, the green curves show the solution for $\mathcal{H}=\mathcal{H}_{c}-0.09$, the magenta curve is for $\mathcal{H}=\mathcal{H}_{c}+0.04$ and the blue curve is for $\mathcal{H}=\mathcal{H}_{c}+0.09$.[]{data-label="f2"}](k12.eps){width="8cm"}
Solution for Radiation Fluid $(k=1/3)$
--------------------------------------
The fluid which obeys the equation of state $p=e/3$ (i.e. $k=1/3$) is called radiation fluid. So Eqs. (\[44a\]) and (\[44b\]) lead to $r_c f_{c,r_c} - 2 f_c = 0$, i.e. $$\begin{aligned}
\label{50}
(1+2\beta)r_c^{2}- 3 M r_c + 2Q^{2}&=&0,\end{aligned}$$ which has the solutions $$\begin{aligned}
\label{51}
r_{c\pm} &=& \frac{3M \pm \sqrt{9M^{2}-8Q^{2}(1+2\beta)}}{2},\end{aligned}$$ where $\beta \leq \left( 3 M / 4 Q \right)^2 -1/2$. Using Eq. (\[51\]) in (\[43\]), we obtain $$\begin{aligned}
\label{52}
(u_c^{r})^{2} &=&\frac{[f_c +(u_c^{r})^{2} ]^{3}}{C_3^6 r_c^{4}}.\end{aligned}$$ From here, we can find the value of $u_c^{r}$, and so we get the critical points $(r_c,\pm u_c^{r})$. Thus, we can express $u^{r}$ in terms of $r$ by Eq. (\[44a\]). The Hamiltonian (\[41b\]) in this case will be $$\begin{aligned}
\label{52a}
\mathcal{H}&=&\frac{f^{2/3}}{r^{4/3}v^{2/3}(1-v^{2})^{2/3}}.\end{aligned}$$ In figure \[f3\] we see that fluid’s motion is supersonic in the region $v>v_{c}$ where the curves with blue and magenta color shows the unphysical behaviour of the fluid.
![The plot showing the trajectories of solutions to Eq. (\[41b\]) in phase space with the parameters $k=1/3$, $M=1$, $Q=0.85$, $\beta=0.075$ and $\Lambda=-0.075$. The black curve shows the solution for $\mathcal{H}=\mathcal{H}_{c}$, the red curve shows the solution for $\mathcal{H}=\mathcal{H}_{c}+0.04$, the green curve shows the solution for $\mathcal{H}=\mathcal{H}_{c}+0.09$, the magenta curve is for $\mathcal{H}=\mathcal{H}_{c}-0.04$ and the blue curve is for $\mathcal{H}=\mathcal{H}_{c}-0.09$.[]{data-label="f3"}](k13.eps){width="8cm"}
Solution for $k=1/4$
--------------------
When energy density exceeds the isotropic pressure, we get the sub-relativistic fluids and they obey the equation of state $p= e / 4$ (i.e. $k=1/4$). Eqs. (\[44a\]) and (\[44b\]) yield $4 f_c-3 r_c f_{c,r_c}=0$ which is a quartic equation and has the following form $$\begin{aligned}
\label{53}
r_c^{4}+ a_1 r_c^2 + a_2 r_c + a_3 =0,\end{aligned}$$ where $a_1 = 6 (1 + 2 \beta) / \Lambda, \, a_2 = - 21 M / \Lambda$ and $a_3 = 15 Q^2 / \Lambda$. Now Eq. (\[53\]) has the same form as Eq. (\[h-eq-2\]) so its roots can be found by following the same procedure as given in Eqs. (\[h-eq-7-1\]) to (\[h-eq-7-2\]). Using the value of $r_c$, we obtain $u_c^{r}$ and similarly $C_3$ from (\[43\]). Thus, we can express $u_c^{r}$ in terms of $r_c$ by Eq. (\[43\]) as $$\begin{aligned}
\label{55}
u_c^{r} = \frac{\left[f_c +(u_c^{r})^{2}\right]^{2}}{C_3^{2} r_c^{2}}.\end{aligned}$$
By repeating the same procedure as in previous cases again we can find an explicit form of solution and critical points $(r_c,\pm u_c^{r})$. The Hamiltonian (\[41b\]) takes the form $$\begin{aligned}
\label{55a}
\mathcal{H}&=&\frac{f^{3/4}}{rv^{1/2}(1-v^{2})^{3/4}}.\end{aligned}$$ Figure \[f4\] shows the motion of the fluid in different regions. The motion of the fluid is supersonic in the region where $v>v_{c}$ and subsonic where $v<v_{c}$ whereas, the region of vertical curves show unphysical behaviour of the flow.
![The plot shows the trajectories of solutions to Eq. (\[41b\]) in phase space with parameters as $k=1/4$, $M=1$, $Q=0.85$, $\beta=0.075$ and $\Lambda=-0.075$. The black curve shows the solution for $\mathcal{H}=\mathcal{H}_{c}$, the red curve shows the solution for $\mathcal{H}=\mathcal{H}_{c}-1.04$, the green curve shows the solution for $\mathcal{H}=\mathcal{H}_{c}-1.09$, the magenta curve is for $\mathcal{H}=\mathcal{H}_{c}+1.04$ and the blue curve is for $\mathcal{H}=\mathcal{H}_{c}+1.09$.[]{data-label="f4"}](k14.eps){width="8cm"}
In the above cases we have discussed the non-transonic solutions but we are also concerned with the flows which pass through the transonic point. Transonic flows are often considered in a context of spherical accretion which yields to the maximum accretion rate. So, in figure \[f6\] we have shown the transonic behaviour of the flow for the above mentioned fluids. Since transonic solutions form a closed orbit therefore, the figure demonstrates that it may possible to get homoclinic orbits around a black hole in case of subsonic solutions.
![Transonic solutions obtained for the isothermal equation of state $p=ke$. The other parameters are taken as $M=1$, $\beta=0.075$, $Q=0.85$ and $\Lambda=-0.075$.[]{data-label="f6"}](fig6.eps){width="10cm"}
Polytropic test fluids
=======================
Polytropes are self-gravitating gaseous spheres that are, very useful as crude approximation to relativistic fluid models. The term “polytropic" was originally used to describe a reversible process on any open or closed system of gas or any fluid which involves both heat and work transfer, such that a specified combination of properties were maintained constant throughout the process. In such a process, the expression relating the properties of the system throughout the process is called the polytropic path. There are an infinite number of reversible polytropic paths between two given states; the most commonly used polytropic path is $TdS=C$, where $T$ is temperature, $S$ is entropy, and $C$ is an arbitrary constant which is equal to zero for an adiabatic process. This path is equivalent to the assumption that the same amount of heat is transferred to the system in each equal temperature increment. The equation of state, which the fluid has when it follows this path is called the *Polytropic Equation of State*. Mathematically, it is given as $p=kn^{\Gamma}$, where $k$ and $\Gamma$ are the constants. By keeping this equation of state in mind, we get the following expression for enthalpy [@ggm] $$\begin{aligned}
\label{66}
h=\frac{\Gamma-1}{\Gamma-1-a^{2}}.\end{aligned}$$ Using Eq. (\[66\]), Eq. (\[24\]) has taken the form $$\begin{aligned}
\label{67}
\frac{\sqrt{ f(r)+(u^{r})^{2}}}{\Gamma-1-a^{2}}=C_6,\end{aligned}$$ where $C_{6}$ is the arbitrary constant. If we take our boundary at infinity (i.e $r = r_{\infty}$) then Eq. (\[67\]) will be $$\begin{aligned}
\label{68}
\frac{\sqrt{ f_{\infty} +(u^{r}_{\infty})^{2}}}{\Gamma-1-a^{2}_{\infty}}=C_6,\end{aligned}$$ where $f_{\infty} = f(r_{\infty})$. On comparing Eq. (\[67\]) and Eq. (\[68\]), we obtain $$\begin{aligned}
\label{69}
(\Gamma-1-a^{2}_{\infty})\sqrt{ f(r)+(u^{r})^{2}}=&&(\Gamma-1-a^{2}) \sqrt{ f_{\infty} +(u^{r}_{\infty})^{2}}.\end{aligned}$$ Now at the sonic point, the enthalpy can be given as $$\begin{aligned}
\label{70}
\frac{h}{h_{\infty}}=\left(\frac{n}{n_{\infty}}\frac{a^{2}_{\infty}}{a^{2}}\right)^{\Gamma-1},\end{aligned}$$ which further on using Eq. (\[66\]) gives rise to $$\begin{aligned}
\label{71}
n=n_{\infty}\left(\frac {a^{2}}{a^{2}_{\infty}}\frac{\Gamma-1-a^{2}}{\Gamma-1-a^{2}_{\infty}}\right)^{\frac{1}{\Gamma-1}}.\end{aligned}$$
Since $r^{2}n u^{r}= C_1$, at the spacial infinity it will be $r^{2}_{\infty}n_{\infty}u^{r}_{\infty}= C_1$. On combining Eqs. (\[70\]) and (\[71\]) we obtain $$\begin{aligned}
\label{72}
u^{r}=u^{r}_{\infty} \left(\frac{r_{\infty}}{r} \right)^2 \left[
\left( \frac{a_{\infty}}{a} \right)^2 \left( \frac{\Gamma-1-a^{2}}{\Gamma-1-a^{2}_{\infty}} \right) \right]^{\frac{1}{\Gamma-1}}.\end{aligned}$$ At the critical point $r_c$, together with (\[37\]) and (\[72\]), Eq. (\[69\]) becomes $$\begin{aligned}
\label{73}
(\Gamma-1-a_c^{2})^{2}\left( f_{\infty} + B_3 \right)=&&(\Gamma-1-a^{2}_{\infty})^{2} \left( f_c + \frac{1}{4} r_c f_{c,r_c} \right),\end{aligned}$$ where $$\begin{aligned}
\label{74}
B_3 &\equiv& \left(u_c^{r} \right)^{2}\frac{r_c^{4}}{r_{\infty}^{4}}\left(\frac{a_c^{2}}{a^{2}_{\infty}}
\frac{\Gamma-1-a^{2}_{\infty}}{\Gamma-1-a_c^{2}}\right)^{\frac{2}{\Gamma-1}}.\end{aligned}$$ Note that Eq. (\[73\]) has only one unknown $r_c$. So, if we solve this equation with boundary values of $r_{\infty}$ and $a_{\infty}$, we obtain the position of critical values of $r_c$, $a_c^{2}$ and $u_c^{r}$. Similarly, as in isothermal case we can compute two critical points $(r_c,\pm u_c^{r})$ in the polytropic case also. Furthermore, Eq. (\[72\]) can also be solved numerically to obtain the function $u^{r}$.
Black hole’s accretion rate
===========================
The rate of change in the mass of a black hole is called mass accretion rate and is generally represented by $\dot M$. Basically, it measures the mass of a black hole per unit time. It is defined as the area times flux of a black hole at the event horizon. In this section, we will discuss the effect of radius on the accretion rate. The general expression to calculate it is $\dot M\left|_{r_{h}}=4\pi r^{2} T^{r}_{t}\right|_{r_{h}}$ [@Mjamil], which refers to relativistic statement of the flux of mass-energy density. In non-relativistic treatment, $\dot{M}$ is defined as the flux of rest-mass density *or* $\dot{M} = 4 \pi r^2 \rho u^r$, where $\rho$ is the rest-mass density, $\rho = m_b\, n_b$, $m_b$ is the average mass per baryon and $n_b$ is the baryon density. Then it follows form the perfect fluid energy-momentum tensor (\[3\]) that $T_t^r = (e + p) u_t u^r$ [@M1]. As our dynamical system is conserved so we have $\nabla_{\mu}J^{\mu}=0$ and $\nabla_{\nu} T^{\mu \nu}=0$. From these conservation equations defined in Eqs. (\[17\]) and (\[24\]) we obtain $$\begin{aligned}
\label{56}
& & r^2 u^{r} \, (e + p) \sqrt{f(r) + (u^{r})^{2}} = A_0,\end{aligned}$$ where $A_0$ is an arbitrary constant. Now assuming the equation of state $p=p(e)$, the relativistic energy flux (or continuity) equation gives $$\begin{aligned}
\label{57}
& & \frac{de}{e + p} + \frac{du^{r}}{u^{r}} + \frac{2}{r} dr = 0.\end{aligned}$$ Integration of this equation yields $$\begin{aligned}
\label{58}
& & r^2 u^{r} \, \exp \left[ \int_{e_{\infty}}^{e} \frac{d e'}{e' + p(e')} \right] = - A_1,\end{aligned}$$ where $A_1$ is an integration constant, $e_{\infty}$ is the matter density at infinity and the minus sign is taken because $u^{r}<0$. If we combine the Eq. (\[58\]) with (\[56\]) we obtain $$\begin{aligned}
\label{59}
A_3 = - \frac{A_0}{A_1} =&&(e + p)\sqrt{f(r) + (u^{r})^2} \exp \left[ -\int_{e_{\infty}}^{e} \frac{d e'}{e' + p(e')} \right],\end{aligned}$$ where $A_3$ is an arbitrary constant. If we take the boundary condition at infinity, then the constant $A_3$ becomes $A_3=e_{\infty}+p(e_{\infty})=- A_0/A_1$, where $A_0=(e + p)u_t u^{r} r^2 =- A_1(e_{\infty}+p(e_{\infty}))$. Furthermore, on equatorial plane due to the spherical symmetry, the equation of mass flux $\nabla_{\mu} J^{\mu} = 0$ can also be written as $$\begin{aligned}
\label{60}
& & r^2 u^{r}\, n = A_2,\end{aligned}$$ where $A_2$ is an integration constant. If we divide the Eq. (\[56\]) with (\[60\]), we get an another useful relation $$\begin{aligned}
\label{60-2}
& & \frac{e + p}{n} \sqrt{f(r) + (u^{r})^2} = \frac{A_0}{A_2} \equiv A_4,\end{aligned}$$ where $A_4$ is a constant such that $A_4 = \left( e_{\infty}+p_{\infty}\right) / n_{\infty}$ [@Babichev]. Using (\[56\]), the black hole’s rate of change of mass takes the following form $$\begin{aligned}
\label{61}
\dot{M} &=& - 4 \pi r^2 u^{r} (e + p) \sqrt{f(r) + (u^{r})^2 } = - 4 \pi A_0.\end{aligned}$$ Then, it becomes $$\begin{aligned}
\label{61-1}
\dot{M} &=& 4\pi A_1 (e_{\infty} + p(e_{\infty})),\end{aligned}$$ due to the boundary condition at infinity. This result is valid for any fluid which obeys the equation of state of the form $p = p(e)$. So, the accretion rate for the black hole will be $$\begin{aligned}
\label{61-2}
\dot{M} &=& 4\pi A_1 (e + p) |_{r=r_{h}} ,\end{aligned}$$ at black hole horizon $r_h$. Thus, evaluating Eq.(\[61-2\]) at $r_h$, one can obtain the black hole mass rate for observers at the black hole horizon.
Let us take an isothermal equation of state, i.e. $p=k e$, which implies that $(e+p)=e(1+k)$. Then Eq.(\[58\]) reduces to $r^2 u^{r} e^{\frac{1}{1+k}} = -A_1$, that is $$\begin{aligned}
\label{61-3}
& & e=\left[- \frac{A_1}{ r^2 u^{r} } \right]^{1+k}.\end{aligned}$$ With this expression of $e$, Eq. (\[56\]) yields $$\begin{aligned}
\label{61-4}
& & (u^{r})^2 - \frac{A_0^2 A_1^{-2(1+k)} }{(1+k)^2} r^{4 k} \left( -u^{r} \right)^{2k} + f(r) = 0,\end{aligned}$$ in which the $u^{r}$ can be obtained for the given values of $k$ if the above algebraic equation is solved. Using the obtained $u^{r}$, we find the energy density $e(r)$ from (\[61-3\]). For instance, when $k=1$ (ultra-stiff fluids) it follows from Eq.(\[61-4\]) that $$\begin{aligned}
\label{61-5}
& & u^{r}= \pm 2 A_1^2 \sqrt{ \frac{f(r)}{{A_0^2 r^4 - 4 A_1^4 }} },\end{aligned}$$ which gives $$\begin{aligned}
\label{61-6}
& & e= \frac{\left( A_0^2 r^4 - 4 A_1^4 \right)}{4 A_1^2 r^4 f(r) }.\end{aligned}$$ Putting (\[61-6\]) into (\[61-2\]), we get $$\begin{aligned}
\label{61-7}
& & \dot{M} = \frac{ 6 \pi \left( A_0^2 r^4 - 4 A_1^4 \right)}{ A_1 r^2 \left[ 3 Q^2 - 6 M r + 3(1+ 2 \beta) r^2 - \Lambda r^4 \right]}.~~~~~\end{aligned}$$
![Plot showing the mass accretion rate for different values of monopole parameter for the ultra-stiff fluid $(k=1)$. The other parameters are fixed and are taken as $M=1$, $\Lambda=-0.2$, $Q=0.75$, $A_{0}=2.5$ and $A_{1}=1$ whereas, the vertical line shows the location of horizon corresponding to monopole parameter in each case.[]{data-label="f5"}](1.eps "fig:"){width="5.9cm" height="5.4cm"} \[1\] ![Plot showing the mass accretion rate for different values of monopole parameter for the ultra-stiff fluid $(k=1)$. The other parameters are fixed and are taken as $M=1$, $\Lambda=-0.2$, $Q=0.75$, $A_{0}=2.5$ and $A_{1}=1$ whereas, the vertical line shows the location of horizon corresponding to monopole parameter in each case.[]{data-label="f5"}](2.eps "fig:"){width="5.9cm" height="5.4cm"} \[2\]\
![Plot showing the mass accretion rate for different values of monopole parameter for the ultra-stiff fluid $(k=1)$. The other parameters are fixed and are taken as $M=1$, $\Lambda=-0.2$, $Q=0.75$, $A_{0}=2.5$ and $A_{1}=1$ whereas, the vertical line shows the location of horizon corresponding to monopole parameter in each case.[]{data-label="f5"}](3.eps "fig:"){width="5.9cm" height="5.4cm"} \[1\] ![Plot showing the mass accretion rate for different values of monopole parameter for the ultra-stiff fluid $(k=1)$. The other parameters are fixed and are taken as $M=1$, $\Lambda=-0.2$, $Q=0.75$, $A_{0}=2.5$ and $A_{1}=1$ whereas, the vertical line shows the location of horizon corresponding to monopole parameter in each case.[]{data-label="f5"}](4.eps "fig:"){width="5.9cm" height="5.4cm"} \[1\]
In figure \[f5\] we plot the mass accretion rate for different values of $\beta$. It can be seen that for all values of $\beta$ the accretion rate increases but when it reaches to the maximum limit, it will decrease. Also, we see the large monopole parameter $\beta$ greatly suppresses the maximum accretion rate.
Discussion
==========
In $1989$, the first solution of the Einstein field equations with a global monopole was derived [@Bar]. Here, we have derived the line element (metric) for the Reissner-Nordström-(anti)-de-Sitter spacetime with global monopole. By assuming certain conservation laws for the adiabatic system, we explored the behaviour of fluid. Generally, the equation of state helps us to identify about what kind of fluid is accreting onto black hole. Here, we have focussed mainly on the ultra-stiff, ultra-relativistic, radiation and sub-relativistic fluids. Different kind of fluids with the unique value of state parameter have different kind of evolution onto the black hole. We have not considered the cases for the vacuum energy or cosmological constant since their accretion does not change that much evolution of the black hole. Further, we have assumed that only a single test fluid accretes onto black hole at a certain time and discussed the isothermal as well as the polytropic flows. Though we have derived the general expressions for all type of fluids satisfying the isothermal and polytropic equation of state but for isothermal equation of state we have also find the solutions of the mentioned fluids at the sonic point. However, in case of polytropic fluids there exists no global solutions [@PatrykMach]. Furthermore, we have determined the general analytical expression for the mass accretion rate $\dot{M}$ and found that $\dot{M}$ depends on the mass and charge of the black hole and large monopole parameter $\beta$ suppresses the maximum accretion rate frequently.
A number of extensions to our study is possible. For instance, one may attempt to consider a non-adiabatic system, so that the fluids may have the effects due to viscosity and energy transfer. It is also interesting to extend this result to the region between the sonic point and boundary. A similar analysis can be done for the spherically symmetric but non-static geometry of the fluid where the velocity and energy density of the fluid varies with the time and position.
Acknowledgements {#acknowledgements .unnumbered}
================
UC was supported by The Scientific Research Projects Coordination Unit of Akdeniz University (BAP). We would like to thank the referees for giving insightful comments to improve this work.
Appendix: Curvature Invariants for the RN-AdS Monopole Black Hole {#appendix-curvature-invariants-for-the-rn-ads-monopole-black-hole .unnumbered}
=================================================================
The curvature invariants are given by $$\begin{aligned}
& & \fl I_{1}= g^{\mu\nu}R_{\mu\nu}=4\Big(\frac{\beta}{r^2}-\Lambda\Big),\label{e1} \\& & \fl I_{2}= R^{\mu\nu}R_{\mu\nu}=\frac{4(Q^4-2Q^{2}r^{2}\beta+2r^{4}\beta^{2}-2r^{6}\beta\Lambda+r^{8}\Lambda^{2})}{r^8},\label{e2} \\& & \fl I_{3}= R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma} \nonumber \\& & \fl \,\,\,\, =\frac{8\Big(21Q^{4}+6Q^{2}r(-6M+r\beta)+r^{2}(18M^{2}-12Mr\beta+6r^{2}\beta^{2}-2r^{4}\beta\Lambda+r^{6}\Lambda^{2})\Big)}{3r^{8}}.\nonumber\\
~~~\label{e3}\end{aligned}$$ The above curvature invariants are well defined everywhere except at $r=0$.
References {#references .unnumbered}
==========
[99]{}
Jamil M, Rashid M A and Qadir A 2008 *Eur. Phys. J. C***58** 325 Jamil M 2009 *Eur. Phys. J. C***62** 609 Bondi H 1952 *Mon. Not. R. Astron. Soc.***112** 195 Michel F C 1972 *Astrophys. Space Sci.***15** 153 Shapiro S L and Teukolsky S A 1983 *“Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects"* (Wiley, New York) Babichev E O, Dokuchaev V I and Eroshenko Yu N 2013 *Phys. Usp.***56** 1155 Debnath U 2015 *Eur. Phys. J. C***75** 129 Karkowski J and Malec E 2013 *Phys. Rev. D* **87** 044007 Mach P and Malec E 2013 *Phys. Rev. D***88** 084055 Mach P, Malec E and Karkowski J 2015 *Phys. Rev. D***88** 084056 Bahamonde S and Jamil M 2015 *Eur. Phys. J. C***75** 508 Barriola M and Vilenkin A 1989 *Phys. Rev. Lett.***63** 341 Punsly B 1998 *Ap.J.*[**498**]{} 640
Bronnikov K A, Meierovich B E and Podolyak E R 2002 *J. Exp. and Theor. Phys.***95** 392 Yu H 2002 *Phys. Rev. D***65** 087502 Rahaman F and Bhui B C 2005 *Fizika B***14** 349 Pitelli J P M and Latelier P S 2009 *Phys. Rev. D***80** 104035
Letelier P S 1979 *Phys. Rev. D*[**20**]{} 1294 Ganguly A, Ghosh S G and Maharaj S D 2014 *Phys. Rev. D***90**, 064037 Xu-Dong S, Ju-Hua C and Yon-Jiu W 2014 *Chin. Phys. B***23** 060401 Ficek F 2015 *Class. Quantum Grav.***32** 235008 Poisson E and Israel W 1990 *Phys. Rev. D***41** 1796
Dadhich N, Narayan K and Yajnik U A 1998 *Pramana J. Phys.***50** 307
Jusufi K 2016 *Astrophys. Space Sci.*[**24**]{} 361 Chen S and Jing J 2013 *Class. Quan. Grav.***30** 175012
Dadhich N and Patel L K 1999 *Paramana J. Phys.*[**52**]{} 359
Mach P, Malec E and Karkowski J 2013 *Phys. Rev. D***88** 084056
Pringle J E and King A B 2007 *Astrophysical Flows* (Cambridge University Press) Ahmed A K, Azreg-Ainou M, Faizal M and Jamil M 2016 *Eur. Phys. J. C***76** 280 Ahmed A K, Azreg-Ainou M, Bahamonde S, Capozziello S and Jamil M 2016 *Eur. Phys. J. C***76** 269
Mach P 2015 *Phys. Rev. D***91** 084016
|
---
abstract: 'Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation $[\hat P,\hat M]=1$. In ordinary quantum mechanics $\hat P$ is the derivative and $\hat M$ the coordinate operator. Here we shall realize $\hat P$ as a second order differential operator and $\hat M$ as a first order integral one. We show that this makes it possible to solve large classes of differential and integro-differential equations and to introduce new classes of orthogonal polynomials, related to Laguerre polynomials. These polynomials are particularly well suited for describing so called flatenned beams in laser theory'
address:
- 'ENEA, Dipartimento FIM, CRE Frascati. C. P. 65, 000044 Frascati, Rome, Italy\'
- |
Dipartimento di Ingegneria Elettronica\
Universitá degli Studi Roma Tre and Sezione INFN, Roma Tre\
Via della Vasca Navale 84, 00146 Roma, Italy\
- 'Centre de recherches mathématiques and Department de mathématiques et de statistiques, Université de Montréal, C.P. 6128–Centre Ville, Montréal, QC H3C 3J7, Canada'
author:
- 'G. Dattoli'
- 'D. Levi'
- 'P. Winternitz'
title: 'HEISENBERG ALGEBRA, UMBRAL CALCULUS AND ORTHOGONAL POLYNOMIALS'
---
Introduction
============
The purpose of this article is to construct a realization of the Heisenberg algebra in terms of a second order differential operator $\hat P$ and a first order integral one $\hat M$. We then use this realization to construct families of orthogonal polynomials and study their properties. Finally we discuss applications of these polynomials in mathematics (to solve integro-differential equations), optics (to describe flattened beams in optics) and in astrophysics (to describe distortions of the microwave background radiation).
This article is directly related to several research programs. The most general is that of umbral calculus, also known as finite operator calculus [@1; @2; @3; @4]. This has been described in many different ways. Here we view umbral calculus as an abstract theory of the Heisenberg relation ${\left[ {\hat {P},\hat {M}} \right]} = \hat {1}$ and its mathematical implications.
A related direction is that of [*monomiality*]{}, the essence of which is a systematic investigation of the relations \[eq1\] u\_[n]{} = nu\_[n - 1]{} , u\_[n]{} = u\_[n + 1]{}, leading to the representation of solutions of differential, difference and other equations as [*monomials*]{} $u_{n} = \,\hat {M}^{n}u_{0}$ [@5; @6; @7].
The same algebra is of course obeyed by standard creation-annihilation operators in quantum physics. Monomiality and umbral calculus are closely related to the theory of Fock space ladder operators. For efficient recent applications of these concepts in mathematical physics see e. g. [@5; @6; @7; @8; @9; @10; @11].
A recent article [@12] was devoted to a systematic study of the realization of $\hat P$ as a difference operator and $\hat M$ as an operator the projection of which is the coordinate $x$. This was applied to show that continuous symmetries like Lorentz or Galilei invariance can be implemented in quantum theories on lattices [@12; @13].
In this article we systematically investigate a different realization, namely one in which $\hat P$ is a second order linear differential operator in one real variable and $\hat M$ is a general first order linear integral operator.
The problem is formulated mathematically in Section 2 where we also obtain the general form of the operators $\hat P, \hat M$ in terms of one arbitrary function $Y(x)$ and four constants. We obtain the boundary condition for functions $u(x)$ defining the domain of the operators $\hat P$ and $\hat M$.
Section 3 is devoted to the basis functions $u_{n} = \,\hat {M}^{n}u_{0} $, where $u_{0}$ is the [*seed*]{} or [*vacuum*]{} function.
We show that we can either choose $u_{0} = 1$ and then restrict the form of the function $Y(x)$ or leave $Y(x)$ arbitrary and choose $u_{0} (x) = {Y}'(x)\,(Y(x) +
c_{0})^{\alpha}$ where $c_{0}$ and $\alpha$ are constants. We show that the basis functions $u_{n} (x)$ are eigenfunctions of a second order linear differential operator $L $ with integer eigenvalues $\lambda
_{n} = \,n + 1$. The operator $L$ is [*factorized*]{} in the sense that we have $L = \hat P \hat M$, where $\hat P$, $\hat M$ are the differential and integral operators constructed in Section 2, respectively.
In Section 4 we show that the eigenfunctions $u_{n}$ form an orthogonal set and obtain the corresponding measure and integration limits.
The explicit form of the eigenfunctions is obtained in Section 5 for two different cases, namely $u_{0} = 1$ and $u_{0} (x) = {Y}'(x)\,Y(x)^{\alpha
}$, where $\alpha$ is a constant. In both cases the eigenfunctions are expressed in terms of Laguerre functions of non-trivial arguments.
Section 6 is devoted to three types of applications: the solution of integro-differential equations, the description of flattened beams in optics and the distortion of background radiation in astrophysics.
Conclusions and future outlook are presented in the final Section 7.
LINEAR DIFFERENTIAL AND INTEGRAL OPERATORS SATISFYING THE HEISENBERG ALGEBRA
============================================================================
Let us consider two linear operators that we postulate to have the following form $$\label{eq2}
\begin{array}{l}
\hat {P} = \Phi _{2} (x)\,\hat {D}_{x}^{2} + \Phi _{1} (x)\,\hat {D}_{x} +
\Phi _{0} (x), \\
\hat {M} = g(x)\,\hat {D}_{x}^{ - 1} + k(x) \\
\end{array}$$ where $$\label{eq3}
\begin{array}{l}
\hat {D}_{x} \equiv {\frac{{d}}{{d\,x}}}, \\
\hat {D}_{x}^{ - 1} u(x) \equiv {\int\limits_{x_{0}} ^{x} {u(x)\,dx}} \\
\end{array}$$ and $\Phi _{2,\,1,\,0} (x)$, $g(x)$, $k(x)$ are some sufficiently smooth functions, to be determined below.
We impose the following requirements
1. The products $\hat {P}\,\hat {M}$ and $\hat {M}\,\hat {P}$ are differential operators
2. Their commutator satisfies the Heisenberg relation $$\label{eq4}
{\left[ {\hat {P},\hat {M}} \right]} = \hat {1}.$$
The product $\hat {P}\,\hat {M}$ will a priori contain a term proportional to $\hat {D}_{x}^{ - 1}$. The condition for it to be absent is $$\label{eq5}
\Phi _{2} (x)\,{g}''(x) + \Phi _{1} (x)\,{g}'(x) + \Phi _{0} (x)\,g(x) = 0,$$ where the primes denote derivatives with respect to the argument.
The product $\,\hat {M}\,\hat {P}$ and hence also the commutator ${\left[
{\hat {P},\,\hat {M}} \right]}$ will apriori contain anti-derivatives $\hat
{D}_{x}^{ - k} $of all orders. Indeed, the use of the Leibnitz formula extended to anti-derivatives yields $$\label{eq6}
\hat {D}_{x}^{ - 1} {\left[ {f(x)\,u(x)} \right]} = {\sum\limits_{r =
0}^{\infty} {( - 1)^{r}}} (D^r f(x))\,D^{-(1+r)}u(x),$$ where $f(x)$ can be a function or an $ x$–dependent operator. Calculating the product $\hat {M}\,\hat {P}$we obtain $$\label{eq8}
\begin{array}{l}
\hat {M}\,\hat {P}\,u = k(x)\,[ {\Phi _{2} (x)\,\hat {D}_{x}^{2} +
\Phi _{1} (x)\,\hat {D}_{x} + \Phi _{0} (x)} ]\,u(x)\, +
\,g(x)\,[ \Phi _{2} (x)\,\hat {D}_{x} - \\ {\Phi} '_{2} (x)\,\, + \,\Phi
_{1} (x) ]\,\,
+ \,[{\Phi} ''_{2} (x)\, - \,{\Phi} '_{1} (x)\, + \,\Phi _{0} (x)]\,\hat
{D}_{x}^{ - 1} + ...\vert _{x_{0}} ^{x} \\
\end{array}$$ which will be a differential operators if $$\label{eq9}
{\Phi} ''_{2} (x)\, - \,{\Phi} '_{1} (x)\, + \,\Phi _{0} (x) = 0.$$ The coefficients of $D^{-k}$ with $k \geqslant 2$ are all equal to $0$ (as differential consequences of eq. (\[eq9\])).
The commutation relation (\[eq4\]) then imposes two further equations $$\label{eq10}
\begin{array}{l}
2\,\Phi _{2} (x)\,{g}'(x)\, + \,{\Phi} '_{2} (x)\,g(x) = 1 \\
{k}'(x) = 0, \\
\end{array}$$ and a boundary condition at $x = x_{0} $, namely $$\label{eq11}
[\Phi _{2} (x)\,{u}'(x)\, - \,{\Phi} '_{2} (x)\,u(x)\, + \,\Phi _{1}
(x)\,u(x)] \vert _{x = x_{0}} = 0.$$ Solving the first of eqs. (\[eq10\]) either for $\Phi _{2} (x)$ or $g(x)$, we obtain $$\label{eq12}
\begin{array}{l}
g(x) = {\frac{{1}}{{2\,\sqrt {\Phi _{2} (x)}} }}{\left[
{{\int\limits_{x_{0}} ^{x} {{\frac{{d\xi} }{{\sqrt {\Phi _{2} (\xi )}} }} +
g_{0}} } } \right]}, \\
\mbox{or} \\
\Phi _{2} (x) = {\frac{{1}}{{{\left[ {g(x)} \right]}\,^{2}}}}{\left[
{{\int\limits_{x_{0}} ^{x} {g(\eta )\,d\,\eta + c_{0}} } } \right]}, \\
g_{0} = 2\,\sqrt {c_{0}} , \\
\end{array}$$ where $c_{0}$ is an integration constant. Furthermore from the integration of the second of eqs. (\[eq10\]) we find $$\label{eq13}
k(x) = k_{0}$$ with $k_{0} $ a constant. Eqs. (\[eq5\]) and (\[eq9\]) provide expressions for $\Phi _{1} (x)$ and $\Phi _{0} (x)$ in terms of $\Phi _{2} (x)$ and $g(x)$.
We shall use the second of eqs. (\[eq12\]) and get rid of the integral by introducing a function $Y(x)$, such that we have $$\label{eq14}
g(x) = {Y}'(x),\qquad Y(x) = {\int\limits_{x_{0}} ^{x} {g(}} \xi )\,d\,\xi .$$ The $\Phi $–functions can then be expressed in terms of $Y(x)$ and of some constants as \[eq15\] \_[2]{} (x) &=& ,\
\_[1]{} (x) &=& ,\
\_[0]{} (x) &=& - \[ (Y(x) + c\_[0]{} )([Y]{}”’(x)[Y]{}’(x) - 3[Y]{}”(x)\^[2]{})\
&& + (c\_[1]{} + 3)[Y]{}’(x)\^[2]{}[Y]{}”(x) \]. The boundary condition (\[eq11\]) can be rewritten in terms of the function $Y(x)$ (see below). The operators $\hat P$ and $\hat M$ should be applied only to functions $u(x)$ satisfying this boundary condition.
The results of this section can be summed as a theorem
\[t1\] The operators $$\label{eq16}
\begin{array}{l}
\hat {P} = {\frac{{1}}{{{Y}'(x)^{2}}}}{\left[ {Y(x) + c_{0}}
\right]}\hat {D}_{x}^{2} + {\frac{{1}}{{{Y}'(x)^{3}}}}{\left[ {(c_{1} + 3){Y}'(x)^{2}} - 3(Y(x)
+ c_{0} ){Y}''(x) \right]}\hat {D}_{x} \\
- {\frac{{1}}{{{Y}'(x)^{4}}}}{\left[ {(Y(x) + c_{0} )({Y}'''(x){Y}'(x) -
3\,{Y}''(x)^{2}) + (c_{1} + 3)\,{Y}'(x)^{2}{Y}''(x)} \right]}, \\ \\
\hat {M} = {Y}'\,\hat {D}_{x}^{ - 1} + k_{0}, \\
\end{array}$$ satisfy the Heisenberg relation (\[eq4\]) and both $\hat {P}\,\hat {M}$ and $\,\hat {M}\,\hat {P}$ are second order differential operators. The function $Y(x)$ satisfies the condition $Y(x_{0} ) = 0$ and is otherwise arbitrary. The constants $c_{0} ,\,c_{1} ,\,x_{0} ,\,k_{0} $ are arbitrary. The operators $\hat {P}$ and $\hat {M}$ are defined for functions $u(x)$ satisfying the boundary condition \[eq17\] && \_[x = x\_[0]{}]{} = 0.
The inverse of the theorem is also true. All operators $\hat {P},\,\hat
{M}$ satisfying the above properties are given by (\[eq16\]).
MONOMIALITY AND BASIS FUNCTIONS
===============================
General approach
----------------
The fundamental notion underlying monomiality is the existence of a sequence of basis functions $u_{n} (x)$satisfying $$\label{eq18}
\begin{array}{l}
\hat {P}\,u_{n} (x) = n\,u_{n - 1} (x), \\
\hat {M}\,u_{n} (x) = u_{n + 1} (x), \\
\end{array}$$ these two equations imply $$\label{eq19}
\begin{array}{l}
\hat {P}\,\hat {M}\,u_{n} (x) = (n + 1)\,u_{n} (x), \\
{\left[ {\hat {P},\,\hat {M}} \right]}\,\,u_{n} (x) = \hat {1}\,u_{n} (x)
\\
\end{array}$$ The functions $u_n(x)$ are given explicitly as “monomials” $$\label{eq20}
u_{n} (x) = \hat {M}^{n}\,u_{0} (x),$$ and in terms of the operator $\hat {M}$ and some “seed function” or “vacuum function” $u_{0} (x)$, which is to be defined. The use of the Heisenberg relation along with eq. (\[eq20\]) yields $$\label{eq21}
\hat {P}\,u_{n} (x) = n\,u_{n - 1} (x) + \hat {M}^{n}(\hat {P}\,u_{0} (x)).$$ To satisfy the first of eq. (\[eq18\]) for all $n$ including $n=0$, we must impose $$\label{eq22}
\hat {P}\,u_{0} (x) = 0.$$ The above conditions can be viewed in at least two different ways
a
: As a condition on the operator $\hat {P}$ and thus on the function $Y(x)$
b
: As a condition on the seed function** $u_{0} (x)$.
It is evident that $u_{0} (x)$ plays the role of a physical vacuum.
Conditions on the operator$\hat {P}$ for a constant seed function
-----------------------------------------------------------------
Let us put $u_{0} = 1$ (or any other non-zero constant). Condition (\[eq22\]) implies $\Phi _{0} (x) = 0$ that is, in view of eq. (\[eq16\]) $$\label{eq23}
(Y(x) + c_{0} )\,({Y}'(x)\,{Y}'''(x) - 3\,{Y}''(x)^{2}) + (c_{1} +
3){Y}'(x)^{2}{Y}''(x) = 0.$$ The function $Y(x)$ is therefore no longer arbitrary, but it is subject to eq. (\[eq23\]). This equation has a three-dimensional Lie point symmetry group, generated by the Lie algebra $$\label{eq24}
\begin{array}{l}
\hat {X}_{1} = \partial _{x} , \\
\hat {X}_{2} = x\,\partial _{x} , \\
\hat {X}_{3} = (Y + c_{0} )\,\partial _{Y} \\
\end{array}$$ which can be used to reduce eq. (\[eq23\]) to quadratures. However we obtain implicit solutions of little use in the present context. We therefore use an alternative approach for $u_{0} = 1$, namely we return to the equations solved in Section 2 and impose $\Phi _{0} (x) = 0$ from the beginning. Eqs. (\[eq5\], \[eq9\]) with $\Phi _{0} (x) = 0$ imply $$\label{eq25}
\begin{array}{l}
\Phi _{1} (x) = {\Phi} '_{2} (x) + \alpha , \\
g(x) = {\frac{{1 - 2\,\beta} }{{{\Phi} '_{2} (x) - 2\,\alpha} }},\qquad {\Phi
}'_{2} (x) \ne 2\,\alpha ,\qquad \beta \ne {\frac{{1}}{{2}}} \\
\end{array}$$ where $\alpha$, $\beta$ are integration constants. Eq. (\[eq10\]) implies that $\Phi _{2} (x)$satisfies the condition $$\label{eq26}
(1 - 2\,\beta )\,\Phi _{2} (x)\,{\Phi} ''_{2} (x) + \beta {\left[ {{\Phi
}'_{2} (x)} \right]}^{2} - \alpha \,(1 + 2\,\beta )\,{\Phi} '_{2} (x) +
2\,\alpha ^{2} = 0.$$ By solving the above equation for $\alpha = 0$ we obtain a simple but interesting solution $$\label{eq27}
\begin{array}{l}
\Phi _{2} (x) = {\frac{{(x + \gamma )^{1 - q}}}{{A (1 +
q)}}},\quad \Phi _{1} (x) = {\Phi} '_{2} (x) = {\frac{{1 - q}}{{A (1 +
q)}}}\,\left( {x + \gamma} \right)^{ - q},\quad \Phi _{0} (x)
= 0, \\
q \ne - 1, \qquad \beta = \frac{q}{1+q}. \\
\end{array}$$ Eqs. (\[eq15\]) then imply that $$\label{eq28}
\begin{array}{l}
{Y}'(x) = A\,(x + \gamma )^{q},\qquad c_{0} = {\frac{{A}}{{q + 1}}}(x_{0} + \gamma )^{q + 1}, \\
Y(x) = {\frac{{A}}{{q + 1}}}{\left[
{(x + \gamma )^{q + 1} - (x_{0} + \gamma )^{q + 1}} \right]} \,, \quad c_{1} = -
{\frac{{q + 2}}{{q + 1}}}. \\
\end{array}$$ The operator $\hat {P}$ reduces to $$\label{eq29}
\hat {P} = {\frac{{1}}{{A\,(q + 1)}}}\,\hat {D}_{x} (x + \gamma )^{1 -
q}\hat {D}_{x}$$ and is self-adjoint.
The boundary condition (\[eq17\]) is satisfied identically for $u(x) =
const$ once the constants $c_{0,1}$ are those given in eq. (\[eq28\]).
Condition on the seed function $u(x)$ for arbitrary $Y(x)$
----------------------------------------------------------
Let us now consider the operator $\hat {P}$ of eq. (\[eq16\]) with $Y(x)$ (and hence $g(x) = {Y}'(x)$) arbitrary. The seed function must satisfy eq. (\[eq22\]). From eq. (\[eq5\]) we see that $u_{0} (x) = {Y}'(x)$ is a solution of eq. (\[eq22\]). A second linearly independent solution is easily obtained using the Wronskian. The general solution of eq. (\[eq22\]) can be written as $$\label{eq30}
\begin{array}{l}
u_{0} (x) = {Y}'(x){\left[ {a{}_{1} + a_{2} (Y(x) + c_{0} )^{ - c_{1} - 2}}
\right]}\,,\quad c{}_{1} \ne - 2, \\
\\
\end{array}$$ or $$\label{eq31}
u_{0} (x) = {Y}'(x){\left[ {a{}_{1} + a_{2} \ln (Y(x) + c_{0} )}
\right]}\,,\quad c{}_{1} = - 2.$$ Since we are dealing with linear equations we can consider separately the cases $a_{1} = 1$, $a_{2} = 0$ and $a_{1} = 0$, $a_{2} = 1$ when imposing the boundary condition (\[eq17\]). The boundary condition is satisfied for $ u_{0} (x) = {Y}'(x) (Y(x) + c_{0} )^{ - c_{1} - 2}$ for all values of $c_1$, in particular for the case $c_1 = -2$, i.e. $u_0(x)=Y'(x)$. It is not satisfied for the term with a logarithm, so we discard the solution (\[eq31\]).
The eigenvalue problem for the basis functions $u_{n} (x)$
----------------------------------------------------------
The functions $u_{n} (x)$ defined by the relations (\[eq18\], \[eq19\], \[eq20\]) with $u_{0} (x)$ satisfying eq. (\[eq22\]), will be eigenfunctions of the linear operator $\hat {L} = \hat {P}\,\hat {M}$ and the situation can be summed up according to the following theorem
\[t2\] The functions $u_{n} (x)$ defined by $$\label{eq32}
\begin{array}{l}
u_{n} (x) = \hat {M}^{n}u_{0} (x), \\
\hat {P}\,u_{0} (x) = 0,
\end{array}$$ satisfy the second order *ODE* $$\label{eq33}
\begin{array}{l}
\hat {L}\,u_{n} (x) = (n + 1)\,u_{n} (x), \\
\hat {L} = f_{2} (x)\,\hat {D}_{x}^{2} + f_{1} (x)\,\hat {D}_{x} + f_{0}
(x),
\end{array}$$ with $$\label{eq34}
\begin{array}{l}
f_{2} (x) = {\frac{{k_{0}} }{{{Y}'(x)^{2}}}}{\left[ {Y(x) + c_{0}}
\right]}, \\
f_{1} (x) = {\frac{{1}}{{{Y}'(x)^{3}}}}{\left\{ {\,{\left[ {Y(x) + c_{0} +
k_{0} (c_{1} + 3)} \right]}\,\,{Y}'(x)^{2} - 3\,k_{0} (Y(x) + c_{0}
)\,{Y}''(x)} \right\}}, \\
f_{0} (x) =- {\frac{{1}}{{{Y}'(x)^{4}}}}\,{\left\{ {\begin{array}{l}
{k_{0} (Y(x) + c_{0} )\,({Y}'(x)\,{Y}'''(x) - 3\,{Y}''(x)^{2}) +} , \\
{ + {\left[ {k_{0} (c_{1} + 3) + Y(x) + c_{0}}
\right]}\,{Y}'(x)^{2}{Y}''(x) -} \\
{ - (c_{1} + 3)\,{Y}'(x)^{4}} \\
\end{array}}
\right\}} ,
\end{array}$$ as long as $Y(x)$ satisfies the boundary condition (\[eq17\]). The operators $\hat M$, $\hat P$ are those of Theorem \[t1\]. The seed function $u_{0} (x)$ must be chosen as in eq. (\[eq30\]) for $Y(x)$ arbitrary, or $u_0(x)=1$ for $Y(x)$ as in eq. (\[eq28\]).
The seed function $u_{0} (x)$ satisfies eq. (\[eq17\]), but this does not guarantee the same for all $u_{i} (x),\,i \geqslant 1$.
We will see in Section 5 that this imposes a condition on the constants $c_{0,\,1} $.
Eigenfunctions as functions of two variables.
---------------------------------------------
We have constructed the monomials $u_n(x) = \hat M^n u_o$ as functions of one variable $x$. They also depend in a significant way on the parameter $k_0$. Let us, for the purposes of this paragraph, change notation and write \[3.20\] \_n(x,y) u\_n(x), y k\_0. We then have \[3.21\] \_n(x,y) = M\^n u\_0, M = Y(x) D\_x\^[-1]{} + y.
The functions $\pi_n(x,y)$, as functions of two variables, satisfy a partial differential equation \[3.22\] = P \_n(x,y). Indeed, we have = n M\^[n-1]{} u\_0. We have $\partial \hat M/\partial y = 1$, and eq. (\[eq18\]) leads to eq. (\[3.22\]). Eq. (\[3.22\]) can be written explicitly as a second order partial differential equation, using eq. (\[eq16\]). In particular, for a constant seed function $u_0=1$ we have $\hat P$ as in eq. (\[eq29\]) and eq. (\[3.22\]) reduces to \[3.23\] = D\_x x\^[1-q]{} D\_x \_n, q -1, where we have put $\gamma=0$.
Eq. (\[3.23\]) is a linear heat equation with variable conductivity $x^{1-q}$ and the monomials $u_n(x) = \pi_n(x,y) = \hat M^n u_0$ are solutions of this equation.
Let us put $$y = A (1+q)t, \qquad 1-q = N, \qquad q \ne -1 \,\, (N \ne 2).$$ Eq.(\[3.23\]) now is \[3.9b\] = D\_x x\^N D\_x u(x,t). Its Lie point symmetry algebra can be calculated using standard methods [@13b]. For $N=0$ and $N=\frac{4}{3}$ the algebra is six dimensional and the $N=\frac{4}{3}$ case is isomorphic to the $N=0$ one, i.e. to the symmetry algebra of the constant coefficient heat equation. A basis for the symmetry algebra for $N=0$ and $N=\frac{4}{3}$ is \[40b\] X\_1 &=& \_t,\
X\_2 &=& t\_t + x \_x - u \_u,\
X\_3 &=& t\^2\_t + t x \_x - u \_u,\
X\_4 &=& x\^ \_x - x\^ u \_u,\
X\_5 &=& t x\^ \_x - u \_u,\
X\_6 &=& u \_u.
The fact that the two algebras are isomorphic suggests that the two equations could be transformed into each other by a point transformation (it is a necessary condition for the existence of such a transformation). This is indeed the case here. We put \[41b\] y=t, z=3 x\^[1/3]{}, w(z,y) = x\^[1/3]{} u(x,t). Then, if $u(x,t)$ satisfies \[42b\] = x\^[4/3]{} , $w(z,y)$ will satisfy \[43b\] w\_y = w\_[zz]{} and vice versa.
For $N \ne 0, \frac{4}{3}, 2$ the symmetry algebra of eq, (\[3.9b\]) is four–dimensional with basis given by $X_1$, $X_2$, $X_3$ and $X_6$ of eq. (\[40b\]). The symmetry group in this general case is $GL(2,\mathcal R)$ and the equation (\[3.9b\]) can not be transformed into the usual heat equation (\[43b\]).
ORTHOGONALITY PROPERTIES OF THE EIGENFUNCTIONS
===============================================
Let us first recall some well known results from the theory of linear operators . The adjoint $\hat {L}^{ +}$ of the second linear differential operator $\hat {L}$ of eq. (\[eq33\]) is defined by the relation $$\label{eq35}
{\int\limits_{a}^{b} {U_{2}} } (x)\,\hat {L}\,\,U_{1} (x)\,dx =
{\int\limits_{a}^{b} {U_{1}} } (x)\,\hat {L}^{ +} U_{2} (x)\,dx,$$ where $U_{2,\,1} (x)$ satisfy the boundary condition, $$\label{eq36}
\left( {U{}_{2}(x)\,f_{2} (x)\,{U}'_{1} (x) - U_{1} (x)\,\left( {f_{2}
(x)\,U_{1} (x)} \right)^{\prime} + f_{1} (x)\,U{}_{1}(x)\,U_{2} (x)}
\right)\vert _{a}^{b} = 0,$$ and we have $$\label{eq37}
\hat {L}^{ +} = f_{2} (x)\,\hat {D}_{x}^{2} + (2\,{f}'_{2} (x) - f_{1}
(x))\,\hat {D}_{x} + f_2'' - {f}'_{1} (x) + f_{0} (x).$$ Let us introduce the eigenfunctions of $\hat {L}$ and $ \hat
{L}^{ +} $ $$\label{eq38}
\begin{array}{l}
\hat {L}\,u_{n} (x) = \lambda _{n} \,u_{n} (x), \\
\hat {L}^{ +} v_{m} (x) = \lambda _{m} v_{m} (x) .
\end{array}$$ The functions $u_{n} (x),\,v_{m} (x)$ form mutually orthogonal sets $$\label{eq39}
{\int\limits_{a}^{b} {v_{m}} } (x)\,u_{n} (x)\,dx = 0,\,m \ne n\quad \lambda
_{m} \ne \lambda _{n} ,$$ where $v_{m} (x)$ (in the domain of $\hat {L}^{ +} $) and $u_{n} (x)$ (in the domain of $\hat {L}\,$) must satisfy the same boundary conditions (\[eq36\]) (for some $a$ and $b$ ). Now let us request that the eigenfunctions $v_{n} (x)$ be proportional to $u_{n} (x)$ $$\label{eq40}
v_{n} (x) = w(x)\,u_{n} (x).$$ This implies that $w(x)$ must satisfy the following first order *ODE* $$\label{eq41}
f_{2} (x)\,{w}'(x) + ({f}'_{2} (x) - f_{1} (x))\,w(x) = 0.$$ The boundary condition for $U_{1} (x) = u_{n} (x)$and $U_{2} (x) = v_{m} (x)
= w(x)\,u_{m} (x)$ reduces to $$\label{eq42}
{\left[ {f_{2} (x)\,w(x)(u_{m} (x)\,{u}'_{n} (x) - {u}'_{m} (x)\,u_{n} (x))}
\right]}\,_{a}^{b} = 0.$$ If $w(x)$ satisfies (\[eq41\]), (\[eq42\]) then the eigenfunctions $u_{n} (x)$ of the original operator $\hat {L}$ will be orthogonal with the weight $w(x)$ $$\label{eq43}
{\int\limits_{a}^{b} {w(x)\,u_{m}} } (x)\,u_{n} (x)\,dx = 0,\quad \quad
\lambda _{m} \ne \lambda _{n}.$$ Applying the above general results to the operator $\hat {L}$ of (\[t2\]) we obtain the following theorem
\[t3\] The eigenfunctions $u_{n} (x)$ of eq. (\[eq33\]) will satisfy the orthogonality relation $$\label{eq44}
{\int\limits_{a}^{b} {w(x)\,u_{m}} } (x)\,u_{n} (x)\,dx = N_{n\,m} \delta
_{n\,m} ,$$ with $$\label{eq45}
w(x)\, = {\frac{{(Y(x) + c_{0} )\,^{c_{1} + 2}}}{{{\left| {{Y}'(x)}
\right|}}}}e^{{\frac{{1}}{{k_{0}} }}\,Y(x)},$$ provided they satisfy the boundary conditions $$\label{eq46}
\left\{ \frac{(Y(x) + c_0 )^{c_1 + 3}}{\left| Y'(x)
\right|\,^3} \, e^{\frac{1}{k_0} \,Y(x)} \, \left ( u_m (x)\, u'_n (x) - u'_m (x)\,u_n (x)\right )
\right\}_a^b = 0.$$
We mention that the boundary condition (\[eq46\]) for orthogonality and (\[eq17\]) for (\[t1\], \[t2\]) to hold must be both satisfied and that $x_{0}$, $a$, $b$ are apriori independent constants (to be determined below).
EXPLICIT FORMS OF THE EIGENFUNCTIONS
====================================
Eigenfunctions for $u_{0} (x) = {Y}'(x)\,{\left[ {Y(x) + c_{0}
} \right]}\,^{\alpha} $ with $Y(x)$**arbitrary**
---------------------------------------------------------------
Here we proceed in the spirit of Section (3.3) and put $$\label{eq47}
c_{1} = - 2 - \alpha ,\quad u_{0} (x) = {Y}'(x)\,(Y(x) + c_{0} )\,^{\alpha}$$ i.e. we start from the second solution in eq. (\[eq30\]). The first one in (\[eq30\]) is just the special case of $\alpha = 0$. The condition (\[eq22\]) is satisfied as is the boundary condition (\[eq17\]) for $u_{0} (x)$. The next eigenfunction calculated according to eq. (\[eq20\]) is $$\label{eq48}
u_{1} (x) = {Y}'(x)\,{\left[ {{\frac{{(Y(x) + c_{0} )\,^{\alpha +
1}}}{{\alpha + 1}}} + k_{0} (Y(x) + c_{0} ) - {\frac{{1}}{{\alpha +
1}}}\,c_{0}^{\alpha + 1}} \right]}.$$ Substituting into the boundary condition (\[eq17\]) we obtain $c_{0}^{\alpha +
1} = 0$. It follows that we have $$\label{eq49}
c_{0}^{} = 0,\quad \alpha > - 1.$$ Using eq. (\[eq20\]) we can easily generate $u_{2} (x),\,u_{3} (x),\,...$. Inspired by the form of these functions we make the Ansatz $$\label{eq50}
u_{n} (x) = {Y}'(x)\,Y(x)^{\alpha} g_{n} ( - {\frac{{Y(x)}}{{k_{0}} }}),$$ and substitute it into the eigenvalue equation (\[eq33\]). We find that the function $g_{n} $ satisfies the generalised, or associated [@15; @16] Laguerre equation $$\label{eq51}
( - {\frac{{Y(x)}}{{k_{0}} }})\,{g}''_{n} + {\left[ {\alpha + 1 - ( -
{\frac{{Y(x)}}{{k_{0}} }})} \right]}\,{g}'_{n} (x) + n\,g_{n} (x) = 0$$ Thus eq. (\[eq33\]) is solved in terms of generalised Laguerre polynomials $L_{n}^{(\alpha )} (z)$. Substituting (\[eq50\]) into the boundary condition (\[eq17\]), we see that (\[eq17\]) is satisfied for all values of $n$. The boundary condition (\[eq46\]) for orthogonality is satisfied for all $n$ and $m$ if we choose $$\label{eq52}
a = 0,\,\;b = \infty ,\quad {\left[ {{\frac{{1}}{{k_{0}} }}Y(x)} \right]}_{x\,
\to \infty} \to - \infty$$ (we have put, with no loss of generality, $x_{0} = 0$). Eq. (\[eq32\]) gives us the explicit form of the eigenfunctions for all $n.$
Let us sum up the previous results as the following theorem
\[t4\] The eigenvalue problem $$\label{5.7}
\begin{array}{l}
\frac{k_0}{Y'(x)^2}Y(x) u''_n (x) +
\frac{1}{Y'(x)^3} \left \{ \left[ Y(x) + k_0 (1 - \alpha )
\right] Y'(x)^2 - 3 k_0 Y (x) Y''(x)
\right\} u'_n(x) \\
- \frac{1}{Y'(x)^4} \biggl \{ k_0 Y(x)\,\left(
Y'(x)\,Y'''(x) - 3\, Y''(x)^2 \right) + \left[ k_0 (1 - \alpha
) + Y(x) \right] \,Y'(x)^2 Y''(x) \\ - (1 - \alpha )\, Y'(x)^4
\biggr \} \,u_n =
(n + 1)\,u_n (x),
\end{array}$$ with the boundary condition $$\label{eq53}
{\left\{ {{\frac{{1}}{{{Y}'(x)^{3}}}}\,{\left[ {Y(x)\,{Y}'(x)\,{u}'(x) -
(\alpha \,{Y}'(x)^{2} + Y(x)\,{Y}''(x))\,u(x)} \right]}} \right\}}\vert _{x
= 0} =0,$$ is solved by the monomials $$\label{eq54}
u_{n} (x) = \hat {M}^{n}u_{0} (x), \quad
u_{0} (x) = {Y}'(x)\,Y\left( {x} \right)^{\alpha} , \quad
\hat {M} = {Y}'(x)\,\hat {D}_{x}^{ - 1} + k_{0} .$$ Explicitly the solutions are \[eq55\] u\_[n]{} (x) &=& [Y]{}’(x)Y(x)\^ [\_[j = 0]{}\^[n]{} ]{} (
[\*[20]{}c]{} [n + ]{}\
[n - j]{}\
)Y(x)\^[j]{}k\_[0]{}\^[n - j]{}\
&=& [Y]{}’(x)Y(x)\^ L\_[n]{}\^[()]{} ( [ - ]{} ),> - 1 , where $L_{n}^{(\alpha )} \left( {z} \right)$is a generalized Laguerre polynomial. The function $Y(x)$ and the constants $k_{0}$, $\alpha$ satisfy $$\label{eq56}
\lim _{x \to \infty} {\frac{{Y(x)}}{{k_{0}} }} = - \infty ,\quad \alpha > -
1.$$ The orthogonality relation is $$\label{eq57}
{\int\limits_{0}^{\infty} {{\frac{{Y(x)^{ - \alpha}}}{{{\left| {{Y}'(x)}
\right|}}}}}} e^{{\frac{{1}}{{k_{0}} }}Y(x)}u_{n} (x)\,u_{m} (x)\,dx = ( -
k_{0} )^{\alpha} \delta _{m,n}$$
Eigenfunctions for $u_{0} (x) = 1,\,{Y}'(x) = A\,\left( {x +
\gamma} \right)\,^{q}$
-------------------------------------------------------------
Let us now take ${Y}(x)$, $c_0$ and $c_1$ as in eq. (\[eq28\]). The eigenvalue problem (\[eq33\]) reduces to $$\label{eq58}
\begin{array}{l}
{\frac{{k_{0}} }{{A\,(q + 1)}}}\,(x + \gamma )^{1 - q}\hat {D}_{x}^{2}
u_{n} (x) + {\frac{{1}}{{A\,\left( {q - 1} \right)}}}{\left\{ {A\,\left( {x
+ \gamma} \right) - k_{0} (q - 1) (x + \gamma)^{ - q}} \right\}}\hat {D}_{x} u_{n} (x) =
\\
= n\,u_{n} (x),
\end{array}$$ the boundary condition (\[eq17\]) reduces to $$\label{eq59}
(x + \gamma )^{1 - q}{u}'(x)\vert _{x = x_{0}} = 0.$$ Condition (\[eq59\]) is satisfied for $u_{0} (x) = 1$.
We have $$\label{eq60}
u_{1} (x) = \hat {M}\,u_{0} (x) = A\,(x + \gamma )^{q}(x - x_{0} ) + k_{0}$$ and eq. (\[eq59\]) for $u_{1} (x)$implies $$\label{eq61}
\gamma = - x_{0} ,\quad c_{0} = 0,$$ so that we find $$\label{eq62}
\hat {P} = {\frac{{1}}{{A\,(q + 1)}}}\,\hat {D}_{x} (x - x_{0} )^{1 -
q}\hat {D}_{x} , \quad
\hat {M} = A\left( {x - x_{0}} \right)^{q}\hat {D}_{x}^{ - 1} + k_{0} .$$ We can easily implement eq. (\[eq20\]) and get $$\label{eq63}
u_{n} (x) = {\sum\limits_{j = 0}^{n} {\left( {{\begin{array}{*{20}c}
{n} \hfill \\
{j} \hfill \\
\end{array}} } \right)\,k_{0}^{n - j}} } {\frac{{A^{j}(x - x_{0} )^{j(q +
1)}}}{{(q + 2)\,(2\,q + 3)...((j - 1)\,q + j)}}}$$
Substituting $u_{n}$ into the boundary condition (\[eq59\]) with $\gamma = -
x_{0}$ we obtain the condition $$\label{eq64}
q > - 1.$$ Eq (\[eq63\]) allows us to relate $u_{n} (x)$ to the generalised Laguerre polynomials according to the identity $$\label{eq65}
\begin{array}{l}
u_{n} (x) = {\frac{{\Gamma ({\frac{{1}}{{1 + q}}})\,n!}}{{\Gamma (n +
{\frac{{1}}{{1 + q}}})}}}\,k_{0}^{n} \,L_{n}^{(\alpha )} (z) \\
\alpha = - {\frac{{q}}{{q + 1}}},\quad z = - {\frac{{A}}{{k_{0} \,(q +
1)}}}x^{q + 1} ,
\end{array}$$ where we have set $x_{0} = 0$, with no loss of generality.
The substitution (\[eq65\]) reduces the eigenvalue problem (\[eq58\]) for $\gamma =
- x_{0} = 0$ to the generalized Laguerre equation for $L_{n}^{(\alpha )} (z)$. The orthogonality relation (\[eq44\]) in this case specializes to \[eq66\] && [\_[0]{}\^ [( )]{} ]{}\^e\^u\_[n]{} (x)u\_[m]{} (x)\
&& = ( )\^( - k\_[0]{} )\^\_[n,m]{} . In conclusion we state the following theorem.
\[t5\] The eigenvalue problem $$\label{eq67}
\begin{array}{l}
{u}''_{n} (x) + {\left[ {{\frac{{A}}{{k_{0}} }}x^{q} + (1 - q)\,x^{ - 1}}
\right]}\,{u}'_{n} (x) - {\frac{{A\,(q + 1)}}{{k_{0}} }}n\,x^{q - 1}u_{n}
(x) = 0, \\
x^{1 - q}{u}'_{n} (x)\vert _{x = 0} ,\quad q > - 1,
\end{array}$$ is solved by the *monomials* $$\label{eq68}
\begin{array}{l}
u_{n} (x) = \hat {M}^{n}1, \\
\hat {M} = A\,x^{q}\hat {D}_{x}^{ - 1} + k_{0} .
\end{array}$$ The eigenfunctions $u_{n} (x)$ are expressed in terms of generalised Laguerre polynomials by eq. (\[eq65\]). Their normalization is given in eq. (\[eq66\]).
As a further remark let us note that the polynomials we have derived cannot be framed within the context of the Sheffer type family, the operator $\hat P$ is indeed a function either of $x$ and of the ordinary derivative $\hat {D}_{x} $ (see ref. [@7] for further comments). More generally they belong to the Laguerre family which provides polynomial forms whose generating function cannot be expressed in terms of exponential functions.
APPLICATIONS
============
Applications to the solution of integro-differential equations
--------------------------------------------------------------
An important aspect of umbral calculus is the [*umbral correspondence*]{} [@1; @2; @3; @4]: the correspondence between results obtained in different realizations of the Heisenberg algebra.
Let us first consider a simple example, namely the initial value problem for a first order partial differential equation \[6.1\] = ( + x ) y, y(0,x) = 1. It can be solved by the method of characteristics, or in a more formal way by putting \[6.2\] y(,x) = e\^[(x + \_x)]{} 1. Using the Baker–Campbell–Hausdorff formula[@20a], which in this case reduces to \[6.3\] e\^[(x + \_x)]{} = e\^[x]{} e\^[\_x]{} e\^[- \[ x, \_x \]]{}, we obtain \[6.4\] y(,x) = e\^[\^2/2]{}e\^[x]{} = e\^[\^2/2]{} \_[k=0]{}\^ \^k x\^k. Now let us replace the PDE (\[6.1\]) by an operator equation \[6.5\] y\_(,M) = ( P + M ) y(, M), where $\hat P$ and $\hat M$ are the operators (\[eq2\]) studied in the previous sections. In eq. (\[6.5\]) $y(\tau, \hat M)$ is an operator function. We will apply both sides of eq. (\[6.5\]) to the seed function $u_0$ and this will turn eq. (\[6.5\]) into an integro–differential equation for a function $f(\tau,x)$.
A formal solution of the operator equation (\[6.5\]) is obtained from the expansion (\[6.4\]) simply by inserting the operator $\hat M$ instead of $x$. We apply this operator $y(\tau, \hat M)$ to the seed function $u_0(x)$ and obtain $f(\tau,x) \equiv y(\tau, \hat M) u_0(x)$. i.e. \[6.6\] f(, x) = e\^[\^2/2]{} \_[k=0]{}\^ \^k u\_k(x), where $u_k(x)$ are the basis functions of Section 5, ultimately expressed in terms of generalized Laguerre polynomials.
The function $f(\tau,x)$ is at least a formal solution of the equation \[6.7\] y\_(,M) u\_0(x) = (P + M) y(, M) u\_0(x), in our case the integro–differential equation \[6.8\] f\_(,x) &=& \_2 D\_x\^2 f(,x) + \_1 D\_x f(,x) + (\_0 + k\_0) f(,x)\
&& + Y \_0\^x f(,x) d x. Here $\Phi_2$, $\Phi_1$ and $\Phi_0$ and $Y$ are the functions determined in Section 2. The formal solution (\[6.6\]) is a real solution of eq. (\[6.8\]) if the series converges.
Let us sum up this example. We start from the linear partial diffeential equations (\[6.1\]) for which we know the solution (\[6.4\]) of a given initial value problem. We associate an operator equation (\[6.5\]) and an integro–differential equation (\[6.8\]) to (\[6.1\]) via the umbral correspondence. A solution of an initial value problem for (\[6.8\]) is obtained from (\[6.4\]) by the umbral correspondence: we replace the powers $x^n$ by the basis functions $u_n(x)$ for the appropriate operators $\hat P$ and $\hat M$ of Sections 3, 4 and 5. The solution (\[6.6\]) corresponds to the initial value \[6.9\] f(0,x) = u\_0(x) for eq. (\[6.8\]).
More generally, let us consider an evolution equation of the form \[6.10\] y\_(,x) = F(\_x,x) y(,x), y(0,x) = f(x), where $F(\partial_x,x)$ is a polynomial in $\partial_x$ with coefficients that are power series (or polynomials) in $x$. A solution of the initial value problem (\[6.10\]) is given by \[6.11\] y(,x) = e\^[F(\_x,x)]{} f(x), where (\[6.11\]) can be written (at least in principle) as a power series by applying the Baker–Campbell–Hausdorff formulas in eq. (\[6.11\]): \[6.12\] y(,x) = \_[j=0]{}\^ c\_j() x\^j (or the power series (\[6.12\]) can be obtained in any other way).
To the function (\[6.12\]) we associate another function, namely \[6.12a\] f(,x) = y(, M) u\_0 = \_[j=0]{}\^ c\_j() u\_j(x). The function (\[6.12a\]) will be a solution of the integra–differential equation \[6.13\] y\_(, M) u\_0 = F(P, M) y(, M) u\_0, i.e \[6.14\] f\_(,x) = F(P, M) f(, x), with initial condition \[91\] f(0,x) = \_[j=0]{}\^ c\_j(0) u\_j(x).
In other words, if we know how to solve the PDE (\[6.10\]), we know how to solve the integro–differential equation (\[6.14\]).
Description of flattened optical beams
--------------------------------------
In this subsection we will emphasize the extreme usefulness of the above family of orthogonal polynomials in applications to optics.
We define the orthogonal function \[6.26\] \_n(x,y) = \_n e\^[- x\^[q+1]{}]{} \_n\^[(q)]{}(x,y), with $\alpha _{n}$ a suitable normalization constant such that $$\label{eq93}
{\int\limits_{0}^{\infty} {\Phi {}_{n}}} (x,y)\,\Phi _{m} (x,y)\,dx =
\delta _{m,n} .$$ The functions $\pi_n^{(q)}(x,y)$ are defined similarly as in Section 3.5, i.e. we put \[93\] \_n\^[(q)]{}(x,y)= (Ax\^qD\_x\^[-1]{} + y)\^n 1, y = k\_0. An idea of the shape of the above family of orthogonal functions is given in Fig. 1 where we have plotted the first three “modes” for $q=3$.
![The first three flattened $q=3$ beam modes normalised to unity ($n=0$ continuous, $n=1$ dot, $n=2$ dash). We take $A=1$ and $y=-1$.[]{data-label="fig2"}](fig2.pdf){width="6.78cm" height="4.92cm"}
The shape is that of the so called flattened optical beams. These beams are different from the usual Gaussian beams since they have flat transverse distribution as shown in Fig. 2 and Fig. 3.
![Transverse distribution of a Gaussian and a Flattened beam, the difference in intensities ( denotes the beam waist, with L being the length of the cavity and k the wave vector of the optical field )[]{data-label="fig2a"}](fig2a.pdf){width="6.78cm" height="4.00cm"}
The advantage offered by flattened beams with respect to ordinary Gaussians is that they provide larger suppression of thermal noise on the mirror surface because of a better average on the surface fluctuations, they are therefore particularly useful for high power lasers.
![Transverse distribution of a higher order flattened mode []{data-label="fig2b"}](fig2b.pdf){width="6.78cm" height="4.00cm"}
It is evident that if we take a transverse section of the flat top distribution in Figs. 2, 3 we get the distribution similar to that given in Fig. 1 for $n=0$ and $n=1$ respectively [@20] used in optics to treat a light beam whose cross section has an intensity as uniform as possible.
A typical example of flattened beam is a super–gaussian $$\label{eq94}
f(x) \propto e^{ - x^{p}}$$ whose profile becomes more and more flat as $p$ increases.
It is evident that the function $\Phi _{n} (x,y)$ is a super–gaussian for $n=0$ and that for $n>0$ we have higher order super–gaussian flattened modes.
The distribution of a higher order modes can be obtained from the square moduli of the functions (\[6.26\]) and are shown in Fig. 4 for the cases $n=4$ and $n=30$.
![Flattened beam mode distribution $q=3$ (normalized to unity) and comparison with the fundamental. a) $n=4$ ,b) $n=30$[]{data-label="fig3"}](fig3.pdf "fig:"){width="4.31cm" height="2.92cm"} ![Flattened beam mode distribution $q=3$ (normalized to unity) and comparison with the fundamental. a) $n=4$ ,b) $n=30$[]{data-label="fig3"}](fig4.pdf "fig:"){width="4.33cm" height="2.88cm"}
The advantage offered by this family of orthogonal functions is two-fold
a
: They provide the natural set for the expansion of flattened beams
b
: They offer the possibility of treating the higher order modes and not only the fundamental one.
The study of the propagation of the above family of flattened beams can be performed using the set of functions introduced in this article. We postpone this to a forthcoming investigation.
It is worth pointing out that we have $$\label{eq95}
\Phi _{n} (x,y) = \alpha _{n} ( - y)^{n}e^{ -
{\frac{{z}}{{2}}}}L_{n}^{(\alpha )} (z)$$ and we can hence discuss the relevant evolution using $z$ as the independent transverse coordinate.
It is also worth stressing that the operators $\hat P, \hat M$ can be used to form other Lie algebras, different from the Heisenberg one, similarly as $x$ and $p=-i/dx$ are used to form e.g. su(1,1). In turn these are well suited to describe the optical cavity elements and filters exploited to flatten the beam distribution, as it will be shown in a forthcoming investigation.
Applications in Astrophysics
----------------------------
The Sunyaev–Zeldovich effect [@22; @23; @23b] is the distorsion of the cosmic microwave background radiation spectrum by the inverse Compton scattering of high energy electrons. When describing this effect the authors [@22; @23] obtained the diffusion equation (the Sunyaev–Zeldovich equation): \[95\] = \^4 .
Eq. (\[95\]) has a symmetry algebra with basis: \[96\] X\_1 &=& \_ - w \_w,\
X\_2 &=& \_ + \_ - w \_w,\
X\_3 &=& \^2 \_ + \_ - w \_w,\
X\_4 &=& \_ - w \_w,\
X\_5 &=& \_ - w \_w,\
X\_6 &=& w \_w.
This Lie algebra is isomorphic to the algebra given in eq. (\[40b\]), implying that the Sunyaev–Zeldovich equation (\[95\]) might be equivalent to the heat equation. Indeed it is and the equivalence is realized by the transformation \[97\] t = , x = , u(x,t) = \^[3/2]{} e\^[ ]{} w(,), (see also [@nc]) If $w(\xi,\tau)$ satisfies eq. (\[95\]) then $u(x,t)$ satisfies the heat equation $u_t=u_{xx}$. Hence the Sunyaev–Zeldovich equation can be solved in terms of the functions (\[eq63\]) constructed in this article (for $q=1$ and $k_0= 2 A t$).
CONCLUSIONS
===========
One way of summing up the results of the present article is the following. We have constructed the most general operators $\hat P, \hat M$ of the form (\[eq2\]) that satisfy the Heisenberg relation (\[eq4\]) and used them to construct the monomials $u_{n} (x) = \hat {M}^{n}u_{0} (x)$. They are eigenfunctions of the linear operators $L=\hat P \hat M$ corresponding to positive integer eigenvalues $\lambda
_{n} = n + 1$ .
The second order operator $L $ given in eq. (\[eq33\]) is thus factored into a product of 2 operators $\hat P $ and $\hat M$. The usual factorization of a differential operator is into two lower order differential operators [@17; @18]. Ours is highly non standard: a differential operator $\hat P$ times an integral one $\hat M$.
The functions $u_{n} (x)$ are expressed in terms of generalized Laguerre polynomials $L_{n}^{(\alpha )} (z)$, or equivalently , the confluent hypergeometric function ${}_{2}F_{0} (a,b;z)$. This includes the Hermite polynomials for $\alpha = {\frac{{1}}{{2}}}$, or $\alpha = -
{\frac{{1}}{{2}}}$ but none of the other classical orthogonal polynomials, related to the hypergeometric function (rather then the confluent one).
There is a good reason for this. We have imposed the form (\[eq2\]) on the operators $\hat P, \hat M$ and the monomiality condition then leads to an eigenvalue problem in which the order of the polynomials $n$ enters in the eigenvalues only and enters linearly. For all other classical orthogonal polynomials the dependence on $n$ is more general.
We are looking into possible generalizations in order to obtain monomial realizations of other classes of orthogonal polynomials. This can e. g. be done by imposing all the properties of monomiality, but allowing $\hat P, \hat M$ to depend on $n$, $$\label{eq100}
\hat {P}_{n} u_{n} (x) = n\,u_{n - 1} (x), \qquad
\hat {M}_{n} u_{n} (x) = u_{n + 1} (x).$$ The Heisenberg relation should in this case be modified to $$\label{eq101}
(\hat {P}_{n + 1} \hat {M}_{n} - \hat {M}_{n - 1} \hat P_{n} )\,u_{n} (x) = u_{n}
(x)$$ As also stressed in the previous section the possibility of introducing an extra variable makes it possible to consider our polynomials (or functions) as depending on two variables (and also on two parameters $n$, $\alpha$).
A further topic that is being pursued is that of applications, as outlined in Section 6. A very special case of eq. (\[6.14\]) was studied in Ref. [@8]. We intend to study eq. (\[6.14\]) in general with $\hat P, \hat M$ as specified in this article, for a wide choice of the functions $F(\hat P, \hat M)$.
Regarding applications we have stressed that the form of the introduced orthogonal polynomials is ideally suited for the study of optical flattened beams.
Acknowledgments {#acknowledgments .unnumbered}
===============
D.L thanks the Centro Ricerche ENEA FRASCATI for its hospitality during the time this research was realized. DL was partially supported by PRIN Project ÔMetodi geometrici nella teoria delle onde non lineari ed applicazioni-2006Õ of the Italian Minister for Education and Scientific Research.
The research of P. W. was partially supported by NSERC of Canada. He thanks the Centro Ricerche ENEA FRASCATI and the Università ROMA TRE for their hospitality and financial support during the time this research was realized.
[99]{}
S. Roman and G.C. Rota, The umbral calculus, [*Adv. Math.*]{} [**27**]{}, 95–188 (1978).
S. Roman, [*The Umbral Calculus*]{}, Academic Press, New York, 1984.
G.C. Rota, [*Finite Operator Calculus*]{}, Academic Press, New York, 1975.
A. Di Bucchianico and D. Loeb, Umbral Calculus. [*The Electronic Journal of Combinatorics*]{} DS3 (2000).
G. Dattoli, M. Migliorati and H.M. Srivastava, Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials, [*Math. Comp. Modelling*]{}, [**45**]{}, 1033–1041 (2007).
G. Dattoli, A. Torre and G. Mazzacurati, Monomiality and integrals involving Laguerre polynomials, [*Rend. Mat. Ser. VII*]{}, [**18**]{}, 565–574 (1998).
P. Blasiak, G. Dattoli, A. Horzela and P.A. Penson, Representations of monomiality principle with Sheffer–type polynomials and boson normal ordering, [*Phys. Lett.*]{} [**A352**]{}, 7–12 (2005).
G. Dattoli, M. Migliorati and S. Kahn, Solutions of integro–differential equations and operational methods, [*Appl. Math. Comp*]{} [**186**]{}, 302–308 (2007).
A.V. Turbiner, Quasi–exactly–solvable problems and sl(2) algebra, [*Comm. Math. Phys.*]{} [**118**]{}, 467–474 (1988).
Yu. Smirnov and A.V. Turbiner, Lie algebra discretization of differential equations, [*Mod. Phys. Lett.*]{} [**A10**]{}, 1795–1802 (1995).
C. Chyssomlokos and A.V. Turbiner, Canonical commutation preserving maps, [*J. Phys. A*]{} [**34**]{}, 10475–10485 (2001).
D. Levi, P. Tempesta and P. Winternitz, Umbral calculus, difference equations and the discrete Schrödinger equation, [*J. Math. Phys.* ]{} [**45**]{}, 4077–4105 (2004).
D. Levi, P. Tempesta and P. Winternitz, Lorentz and Galilei invariance on lattices, [*Phys. Rev. D*]{} [**69**]{}, 105011, 1–6 (2004).
P.J. Olver, [*Applications of Lie groups to differential equations*]{}, Springer-Verlag, New York, 1993.
R. Courant and D. Hilbert, [*Methods of Mathematical Physics*]{}, Wiley, New York, 1989.
I.S. Gradshteyn and I.M. Ryzhik, [*Tables of Integrals, Series and Products*]{}, Academic Press, New York, 1965.
M. Abramowitz and I.A. Stegun, [*Handbook of Mathematical Functions*]{}, Dover, New York, 1968.
B.C. Hall, [*Lie Groups, Lie Algebras and Representations: An Elementary Introduction*]{}, Springer & Verlag, New York, 2003; see also arxiv:math-ph/0005032.
A.E. Siegman, Laser beams and resonators: Beyond the 1960s, [*IEEE Journal of Selected Topics in Quantum Electronics*]{} [**6**]{}, 1389Ð-1399 (2000).
F. Gori, Flattened Gaussian beams, [*Opt. Commun.*]{} [**107**]{}, 335–341 (1994).
R.A. Sunyaev and Ya.B. Zeldovich, Small scale fluctuations of relic radiation, [*Astrophys. Space Science*]{} [**7**]{}, 3 (1970).
R.A. Sunyaev and Ya.B. Zeldovich, Microwave background radiation as a probe of the contemporary structure and history of the universe, [*Ann. Rev. Astron. Astrophys.*]{} [**18**]{}, 537–560 (1980).
Y. Rephaeli, S. Sadeh, M. Shimon,The Sunyaev Zeldovich effect, [*Riv. Nuovo Cimento*]{} [**29**]{}(12) 1–18 (2006).
G. Dattoli, M. Migliorati, K. Zhukovsky, An elementary account of relativistic cosmology, [*Riv. Nuovo Cimento*]{} [**29**]{} (10) 1–85 (2006).
L. Infeld and T.E. Hull, The factorization method, [*Rev. Mod. Phys.*]{} [**23**]{}, 21–68 (1951).
B. Mielnik and O. Rosas–Ortiz, Factorization: a little or a great algorithm, [*J. Phys. A*]{} [**37**]{}, 10007–10035 (2004).
|
---
author:
- 'Boris Brimkov[^1]'
- 'Zachary Scherr [^2]'
title: An exact algorithm for the minimum rank of a graph
---
Introduction
============
Let $S_n(\mathbb{R})$ denote the set of real symmetric $n\times n$ matrices. For a matrix $A\in S_n(\mathbb{R})$, $\mathcal{G}(A)$ denotes the graph with vertex set $\{1,\ldots,n\}$ and edge set $\{\{i,j\}:A_{ij}\neq 0,\; 1\leq i<j\leq n\}$. Note that the diagonal of $A$ is not used when constructing $\mathcal{G}(A)$. The set of *symmetric matrices associated with a graph* $G$ is defined as $\mathcal{S}(G)=\{A\in S_n(\mathbb{R}):\mathcal{G}(A)=G\}$. The *minimum rank* of $G$ is defined as $\operatorname{mr}(G)=\min\{\text{rank}(A):A\in\mathcal{S}(G)\}$. The minimum rank problem is a special case of the matrix completion problem which has numerous theoretical and practical applications (such as the million-dollar Netflix challenge [@netflix]); it is also related to the inverse eigenvalue problem [@hogben3], quantum controllability on graphs [@godsil], and various other problems in spectral graph theory and combinatorial matrix theory.
The minimum rank problem was first studied in 1996 by Nylen [@nylen], who gave an algorithm for computing the minimum rank of trees; this algorithm was later improved in [@johnson1; @johnson2; @wei], and generalized to block-cycle graphs in [@barioli1]. The graphs having very large and very small minimum ranks have been characterized in [@barrett2; @barrett; @barrett1; @hogben1; @johnson3]. Decomposition formulas have been derived for computing the minimum ranks of graphs with cut vertices [@barioli2; @Hsieh] and joins of graphs [@barioli3] in terms of the minimum ranks of certain subgraphs. The effects of edge subdivisions [@barrett3; @barrett4], edge deletions [@Edholm], and graph complements [@barioli4; @hogben2] on the minimum rank have also been explored. Upper and lower bounds for the minimum rank of a graph can be obtained using graph theoretic parameters such as the zero forcing number and its variants [@AIM-workshop; @barioli7; @brimkov; @brimkov2; @gentner; @huang], Colin de Verdière type parameters [@barioli7; @barioli5; @hogben1], ordered and induced subgraphs [@mitchell], and other methods. Techniques for computing the minimum rank of small graphs are described in [@DeLoss1], and are combined in [@DeLoss2] with the bounds mentioned above to compute the minimum ranks of all graphs on up to 7 vertices. See [@barioli_zf_mr; @survey] for a survey of recent results on the minimum rank problem.
Despite this extensive research, the literature notably lacks an exact algorithm for computing the minimum rank of an arbitrary graph in finite time; researchers in the field have said it would be “incredibly valuable if such a thing exists" [@hogben_letter]. In this note, we present such an algorithm using two well-known facts from linear algebra and commutative algebra. We demonstrate the capabilities and limitations of our algorithm by computing the minimum ranks of several graphs, some of which could previously not be computed by any automated method. Some possibilities for improvement, extensions to other problems, and directions for future work are also discussed.
Main results {#section_main}
============
Given an $n\times m$ matrix $A$, a *minor* of $A$ with *order* $k$ (or $k$-*minor*, for short) is the determinant of a $k\times k$ submatrix obtained from $A$ by removing some of its rows and columns. A *system of polynomial equations* is a set of simultaneous equations $\{p_1(\vec{x})=\ldots =p_m(\vec{x})=0\}$ where $p_i(\vec{x})$, $1\leq i\leq m$, is a polynomial with rational coefficients in several variables $\vec{x}=[x_1,\ldots,x_n]$. A *solution* of a system of polynomial equations is a set of values for $\vec{x}$ which make all equations true.
The first idea we use in our algorithm is the well-known (yet often forgotten) *determinantal rank* of a matrix, i.e., the order of its largest non-vanishing minor. More precisely, we use the fact that the rank of a matrix $A$ is equal to the largest order of any non-zero minor of $A$:
\[fact1\] For any matrix $A$, $\operatorname{rank}(A)= r$ if and only if all $(r+1)$-minors of $A$ are 0 and not all $r$-minors of $A$ are 0.
The second idea we use concerns the solution of a system of polynomial equations. It follows from the Tarski-Seidenberg theorem [@tarski] that the problem of determining whether a system of polynomial equations has a real solution is decidable:
\[fact2\] It can be determined in finite time whether a system of polynomial equations $\{p_1(\vec{x})=\ldots =p_m(\vec{x})=0\}$ has a real solution.
There are several *tour de force* algorithms for finding the solution of a system of polynomial equations in finite time, if one exists. These include Collins’ algorithm for cylindrical algebraic decomposition [@collins], Buchberger’s algorithm for computing Gröbner bases [@buchberger], and the critical points method of Grigorev and Vorobjov [@grigorev]. See [@basu] for a detailed survey of algorithms and complexity results on solving systems of polynomial equations. Such algorithms are implemented in computer algebra systems like *Mathematica*, *Sage*, *Maple*, and *Magma*, and dedicated solvers like *Bertini* and *PHCpack*.
We are now ready to describe our algorithm for computing the minimum rank of a graph $G$. Let $A^*$ be a matrix that achieves $\operatorname{mr}(G)$. Since $A^*\in \mathcal{S}(G)$, we can represent the diagonal entries of $A^*$ with variables $\vec{x}=[x_1,\ldots,x_n]$, and the off-diagonal non-zero entries in the upper triangle of $A^*$ with variables $\vec{y}=[y_1,\ldots,y_t]$. Since $A^*$ is symmetric, the off-diagonal non-zero entries in the lower triangle of $A^*$ can also be represented by $\vec{y}$. Then, a minor of $A^*$ is simply a polynomial with integer coefficients in the variables $\vec{x}, \vec{y}$. Let $\{f_1(\vec{x},\vec{y}),\ldots,f_p(\vec{x},\vec{y})\}$ be the set of all $k$-minors of $A^*$ for some $k\geq 1$.
By Fact \[fact1\], if the system of polynomials $\{f_1(\vec{x},\vec{y})=\ldots=f_p(\vec{x},\vec{y})=0\}$ has a real solution in which $y_1\neq 0,\ldots,y_t\neq 0$, then $\operatorname{mr}(G)\leq k-1$. Note that for $1\leq i\leq t$, $y_i\neq 0$ if and only if there exists a $\hat{y}_i$ such that $y_i\hat{y}_i=1$. Thus, $\{f_1(\vec{x},\vec{y})=\ldots=f_p(\vec{x},\vec{y})=0\}$ has a real solution in which $y_1\neq 0,\ldots,y_t\neq 0$ if and only if $$\label{reqs}
\{f_1(\vec{x},\vec{y})=\ldots=f_p(\vec{x},\vec{y})=y_1\hat{y}_1-1=\ldots=y_t\hat{y}_t-1=0\}$$ has a real solution. By Fact \[fact2\], it can be determined whether the system has a real solution in finite time. If this procedure is repeated for $k\geq 1$ and terminated as soon as a real solution to the corresponding system of polynomial equations is found, then the iteration at which a real solution is found is equal to $1+\operatorname{rank}(A^*)=1+\operatorname{mr}(G)$. Note that this procedure always terminates, since $\operatorname{mr}(G)\leq n$. We formally summarize this argument in Algorithm \[alg1\].
**Input:** A graph $G$ of order $n$ **Output:** $\operatorname{mr}(G)$ $\vec{x}=[x_1,\ldots,x_n]\leftarrow$ the diagonal entries of a matrix $A^*\in\mathcal{S}(G)$ $\vec{y}=[y_1,\ldots,y_t]\leftarrow$ the non-zero off-diagonal entries in the upper triangle of $A^*$
\[example1\] We illustrate Algorithm \[alg1\] by applying it to a familiar graph. Let $G$ be the path $P_4$ and let $A^*\in \mathcal{S}(G)$ be a matrix such that $\operatorname{rank}(A^*)=\operatorname{mr}(G)$. Then,
$$A^*=\left( \begin{array}{cccc}
a & b & 0 &0 \\
b & c & d &0 \\
0 & d & e &f \\
0 & 0 & f &g\end{array} \right),\qquad \text{where}\; b,d,f\neq 0.$$ Suppose we are in the third iteration of Algorithm \[alg1\], i.e., that we have already examined all 1-minors and 2-minors of $A^*$ and found that there is no assignment of variables that yields a rank 2 matrix. Next we consider the 3-minors of $A^*$, which are $$\left| \begin{array}{ccc}
c & d & 0 \\
d & e & f \\
0 & f & g \end{array} \right|,
\left| \begin{array}{ccc}
b & d & 0 \\
0 & e & f \\
0 & f & g \end{array} \right|,
\left| \begin{array}{ccc}
b & c & 0 \\
0 & d & f \\
0 & 0 & g \end{array} \right|,
\left| \begin{array}{ccc}
b & c & d \\
0 & d & e \\
0 & 0 & f \end{array} \right|,$$ $$\left| \begin{array}{ccc}
b & 0 & 0 \\
d & e & f \\
0 & f & g \end{array} \right|,
\left| \begin{array}{ccc}
a & 0 & 0 \\
0 & e & f \\
0 & f & g \end{array} \right|,
\left| \begin{array}{ccc}
a & b & 0 \\
0 & d & f \\
0 & 0 & g \end{array} \right|,
\left| \begin{array}{ccc}
a & b & 0 \\
0 & d & e \\
0 & 0 & f \end{array} \right|,$$ $$\left| \begin{array}{ccc}
b & 0 & 0 \\
c & d & 0 \\
0 & f & g \end{array} \right|,
\left| \begin{array}{ccc}
a & 0 & 0 \\
b & d & 0 \\
0 & f & g \end{array} \right|,
\left| \begin{array}{ccc}
a & b & 0 \\
b & c & 0 \\
0 & 0 & g \end{array} \right|,
\left| \begin{array}{ccc}
a & b & 0 \\
b & c & d \\
0 & 0 & f \end{array} \right|,$$ $$\left| \begin{array}{ccc}
b & 0 & 0 \\
c & d & 0 \\
d & e & f \end{array} \right|,
\left| \begin{array}{ccc}
a & 0 & 0 \\
b & d & 0 \\
0 & e & f \end{array} \right|,
\left| \begin{array}{ccc}
a & b & 0 \\
b & c & 0 \\
0 & d & f \end{array} \right|,
\left| \begin{array}{ccc}
a & b & 0 \\
b & c & d \\
0 & d & e \end{array} \right|,$$ where the minor in the $i^\text{th}$ row and $j^\text{th}$ column in the array above is obtained by deleting the $i^\text{th}$ row and $j^\text{th}$ column of $A^*$. Evaluating the determinants, we obtain $$\begin{array}{cccc}
ceg-cf^2-d^2g, & beg-bf^2, & bdg, & bdf,\\
beg-bf^2, & aeg-af^2, & adg, & adf,\\
bdg, & adg, & acg-b^2g, & acf-b^2f,\\
bdf, & adf, & acf-b^2f, & ace-ad^2-b^2e.
\end{array}$$ We set each of these determinants equal to zero. Moreover, to ensure that the variables $b$, $d$, and $f$ are nonzero, we introduce three new variables $\hat{b}$, $\hat{d}$, and $\hat{f}$ and the equations $b\hat{b}=1$, $d\hat{d}=1$, and $f\hat{f}=1$. Finally, we check whether the resulting system of polynomial equations has a solution: $$\begin{aligned}
ceg-cf^2-d^2g=beg-bf^2=bdg=bdf=beg-bf^2=aeg-af^2=&\\
adg=adf=bdg=adg=acg-b^2g=acf-b^2f=bdf=adf=&\\
acf-b^2f=ace-ad^2-b^2e=b\hat{b}-1=d\hat{d}-1=f\hat{f}-1=&0.\end{aligned}$$ The Gröbner basis of these polynomials (computed by the computer algebra system *Mathematica*) contains 1, which means the system of polynomial equations has no solution. Hence, there is no assignment of the variables $a,b,c,d,e,f,g$ in $A^*$ which produces a matrix of rank 2. Repeating this procedure with the 4-minor of $A^*$ yields the system of polynomials $$b^2 f^2 - a c f^2 - a d^2 g - b^2 e g + a c e g=b \hat{b}-1 = d \hat{d}-1 =f \hat{f} -1 = 0.$$ The Gröbner basis of these polynomials (again computed by *Mathematica*) does not contain 1, and the two families of solutions are $$\left\{e=\frac{b^2 f^2 - a c f^2 - a d^2 g}{(b^2 - a c) g}, \hat{b}=\frac{1}{b}, \hat{d}= \frac{1}{d}, \hat{f}= \frac{1}{f}\right\},$$ $$\left\{c =\frac{b^2}{a}, g =0, \hat{b}=\frac{1}{b}, \hat{d}= \frac{1}{d}, \hat{f}= \frac{1}{f}\right\}.$$ Then, using the first family of solutions and choosing $a=c=0$, $b=d=e=f=g=1$, we obtain the following rank 3 matrix that achieves the minimum rank of $G$: $$\begin{aligned}
\begin{pmatrix}
0 & 1 & 0 &0 \\
1 & 0 & 1 &0 \\
0 & 1 & 1 &1 \\
0 & 0 & 1 &1
\end{pmatrix}.$$
Computational results
---------------------
We now illustrate the scope of Algorithm \[alg1\] by applying it to some less familiar graphs. As a baseline, we consider the minimum rank program developed by DeLoss et al. in [@DeLoss1; @DeLoss3; @DeLoss2]; this program computes several combinatorial upper and lower bounds for $\operatorname{mr}(G)$ and returns $\operatorname{mr}(G)$ if some lower bound equals some upper bound. The program also attempts connected component and cut vertex decompositions, leveraging the fact that $\operatorname{mr}(G)$ can be expressed in terms of the minimum ranks of the connected and biconnected components of $G$ (or related subgraphs; see [@barioli2] for more details). Finally, if $G$ is a tree, the program uses an exact algorithm for trees to compute $\operatorname{mr}(G)$ (cf. [@AIM-workshop]). If none of the above methods succeed in computing $\operatorname{mr}(G)$, then the program returns the best upper and lower bounds for $\operatorname{mr}(G)$.
We tested Algorithm \[alg1\] on graphs whose minimum ranks could not be found by the program of DeLoss et al. The minimum ranks of these graphs are known, but were obtained by manually finding matrices in $\mathcal{S}(G)$ whose rank matched a lower bound for $\operatorname{mr}(G)$, or by inspecting the structure of $G$ and using combinatorial arguments on a case-by-case basis to deduce $\operatorname{mr}(G)$ (see Proposition 4.1 in [@DeLoss1] and Table 2 in [@DeLoss2]). In contrast, Algorithm \[alg1\] computes the minimum ranks of these graphs without any human intervention. The results and runtimes are reported in Table \[table1\]. The names of the graphs are their Atlas numbers and their adjacencies can be found in *The Atlas of Graphs* [@read]. The computations were performed on a Lenovo Thinkpad with a 2.80GHz Intel i7-7700HQ CPU and 8GB of RAM, running Mathematica 10.0 on Windows 10.
$G$ $|V|$ $|E|$ $\operatorname{mr}(G)$ time (s)
----- ------- ------- ------------------------ ----------
558 7 9 3 37.4
669 7 10 3 198.8
678 7 10 3 267.3
679 7 10 4 40310.4
721 7 10 3 82.6
791 7 11 3 2202.2
801 7 11 3 2831.6
812 7 11 3 155.2
831 7 11 3 392.3
832 7 11 3 644.3
846 7 11 3 957.8
: Minimum ranks of some graphs
$G$ $|V|$ $|E|$ $\operatorname{mr}(G)$ time (s)
----- ------- ------- ------------------------ ----------
863 7 11 3 4682.5
873 7 11 3 4125.9
878 7 11 3 3794.7
913 7 12 3 30178.7
918 7 12 3 2207.3
924 7 12 3 5404.9
932 7 12 3 6958.3
944 7 12 3 11664.0
953 7 12 3 40402.5
956 7 12 3 36576.5
958 7 12 3 32794.7
: Minimum ranks of some graphs
\[table1\]
As can be seen from Table \[table1\], the runtime of Algorithm \[alg1\] typically increases exponentially with $|E(G)|$. For example, graph 558 has the fewest edges among the graphs tested, and took the least amount of time to be solved; graphs 953, 956, and 958 had three more edges than 558 and took roughly three orders of magnitude longer to be solved. This can be explained by the fact that polynomial equation solvers typically require $(pd)^{2^{O(v)}}$ time, where $p$ is the number of polynomials in the system, $d$ is the maximum degree of a polynomial in the system, and $v$ is the number of variables (see [@collins; @wuthrich] and the survey of Ayad [@ayad]; singly exponential methods have been proposed, e.g. in [@grigorev], but have some limitations). Thus, if $G$ has more edges, the system of polynomial equations obtained from $G$ in Algorithm \[alg1\] has more variables and more equations, and thus takes exponentially longer to solve. The runtime of Algorithm \[alg1\] also typically increases significantly with the minimum rank of $G$. For example, graph 679 has the same number of vertices and edges as graphs 669, 678, and 721, but has a larger minimum rank, and took two orders of magnitude longer to be solved. This can be explained by the fact that if $G$ has a larger minimum rank, then Algorithm \[alg1\] has to perform more iterations involving larger $k$-factors, which requires the solution of larger systems of equations with higher degrees. Note that since a symmetric $n\times n$ matrix has $\binom{n}{k}^2$ $k$-minors and at most $\frac{n(n-1)}{2}$ distinct nonzero off-diagonal entries, the $k^{\text{th}}$ iteration of Algorithm \[alg1\] requires solving a system of $\Omega(n^{2k})$ polynomial equations.
Next, we apply Algorithm \[alg1\] to two slightly larger graphs – the paths $P_{11}$ and $P_{12}$. It is well known that $\operatorname{mr}(P_n)=n-1$ for any path $P_n$; nevertheless, $P_{11}$ and $P_{12}$ are illustrative of the algorithm’s runtime and output. The runtimes (in seconds) of the iterations of Algorithm \[alg1\] on $P_{11}$ are as follows: $$0.0, \; 0.0, \; 0.3, \; 2.7, \; 16.3, \; 95.9, \; 294.3, \; 490.4, \; 248.0, \; 23.9, \; 0.2.$$ As expected, at the $11^{\text{th}}$ iteration, a solution to the corresponding system of polynomials was found. One automatically generated instance of the solution is reported below; it can be verified that the rank of this matrix is indeed 10.
$$\label{eq1}
\scriptsize
\left(
\begin{array}{ccccccccccc}
-\frac{22928}{4129} & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
4 & -2 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -3 & -1 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 4 & -3 & -4 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -4 & 1 & -3 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -3 & 0 & -2 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -2 & -4 & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & -2 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & -3 & -2 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 4 & -2 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & -1 \\
\end{array}
\right).$$
The runtimes (in seconds) of the iterations of Algorithm \[alg1\] on $P_{12}$ are as follows: $$0.0, \; 0.0, \; 1.2, \; 4.8, \; 31.6, \; 207.4, \; 1030.5, \; 2518.6, \; 3072.4, \; 2700.5, \; 150.3, \; 0.6.$$ At the $12^{\text{th}}$ iteration, the following rank 11 matrix realizing $\operatorname{mr}(G)$ was returned:
$$\label{eq2}
\scriptsize
\left(
\begin{array}{cccccccccccc}
\frac{114144}{22729} & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-3 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -3 & -4 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -2 & 3 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 2 & -3 & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -2 & 2 & -2 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -2 & 1 & 4 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 4 & 3 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -2 & -2 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & -1 & 4 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & -3 & -4 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 1 \\
\end{array}
\right).$$
Note that the sequences of runtimes of Algorithm \[alg1\] for $P_{11}$ and $P_{12}$ are unimodal, and skewed to the left. This can be explained by the fact that the largest number of polynomial equations have to be solved at iteration $\frac{n}{2}$ (since $\arg\max_k\{{n\choose k}^2\}=\frac{n}{2}$), but the maximum degree of the polynomials is achieved at iteration $n$; as noted above, both the number of equations and the maximum degree increase the runtime of the solver.
Discussion and future work
==========================
In this note, we combined two simple and well-known ideas to obtain a long-sought exact algorithm for the minimum rank of a graph. Overall, the computational limit of our implementation of Algorithm \[alg1\] seems to be at dense graphs with around 7 vertices and sparse graphs with around 9 vertices. For larger graphs, the algorithm runs out of memory or has an impractically long runtime. While the computational scope of Algorithm \[alg1\] is admittedly limited, it settles the decidability of the minimum rank problem and opens the possibility for improved computational methods and new algorithmic paradigms. For example, in addition to $|E(G)|$ and $\operatorname{mr}(G)$, the runtime of Algorithm \[alg1\] depends to a large extent on the efficiency of the subroutine used for solving systems of polynomial equations. While our implementation used a general purpose solver, the systems of polynomial equations that arise from computing determinants of matrices are clearly quite special, and leave much room for specialization and improvement. Thus, an important direction for future work is to develop algorithms that more efficiently determine whether a system of polynomial equations arising from matrix determinants has a solution. Any *a priori* assumptions about the graph whose minimum rank is being computed (e.g., sparsity, symmetry, connectivity) could potentially also be leveraged to obtain a significant speedup. Upper and lower bounds on the minimum rank – such as the zero forcing number – remain relevant, since solving systems of polynomial equations typically takes doubly exponential time, whereas computing the zero forcing number and other graph-based bounds on the minimum rank takes (only) exponential time. Thus, the number of iterations of Algorithm \[alg1\] can be reduced by having good upper and lower bounds for the minimum rank. Binary search can also be used to further decrease the number of iterations.
With slight modifications, Algorithm \[alg1\] can be used to solve several other problems that are related to the minimum rank problem. For example, the *minimum positive semidefinite rank* of $G$, denoted $\operatorname{mr}_+(G)$, is defined as the minimum rank over all positive semidefinite matrices with the same sparsity pattern as $G$. The minimum positive semidefinite rank and similar parameters have been widely studied, mainly involving characterizations for specific graphs and general bounds (see, e.g., [@ekstrand; @osborne; @wang_pos; @yang_pos; @zimmer]); however, until now, the literature did not contain an exact algorithm for computing these parameters. Given an $n \times n$ matrix $A$ and a set $I\subseteq \{1,\ldots,n\}$ with $|I|=k$, a *principal minor* of $A$ is a minor that corresponds to the rows and columns of $A$ indexed by $I$. By Sylvester’s criterion for positive semidefinite matrices [@prussing], a symmetric matrix $A$ is positive semidefinite if and only if all principal minors of $A$ are nonnegative. Thus, augmenting the system of polynomials being solved in line 7 of Algorithm \[alg1\] with a system of polynomial inequalities dictating that all polynomials corresponding to the principal minors of $A^*$ are nonnegative will ensure that any solution to the system corresponds to a positive semidefinite matrix. Since the Tarski-Seidenberg theorem also applies to systems of polynomial inequalities, Algorithm \[alg1\] equipped with a solver for polynomial inequalities can be used to compute $\operatorname{mr}_+(G)$. Other minimum rank parameters restricted by the definiteness of the matrix or by other criteria (e.g. zeros on the diagonal, as in [@zero_diag]) can be handled analogously. As another example, depending on the polynomial equation solver used, Algorithm \[alg1\] could return – in addition to $\operatorname{mr}(G)$ – a matrix whose rank equals $\operatorname{mr}(G)$ (as the matrices in and for $P_{11}$ and $P_{12}$), or a characterization of all matrices whose rank equals $\operatorname{mr}(G)$ (as in Example \[example1\]). This opens possibilities for computationally investigating the set of all matrices that realize the minimum rank of a graph, and finding a matrix that is optimal with respect to certain other criteria.
We also remark that the *maximum nullity* of a graph $G$, defined as $M(G)=\max\{\text{null}(A):A\in\mathcal{S}(G)\}$, and the *maximum multiplicity* of $G$, defined as $mult(G)=\max\{mult_A(\lambda) : A\in \mathcal{S}(G), \lambda\in \mathbb{R}\}$, where $mult_A(\lambda)$ denotes the multiplicity of $\lambda$ as a root of the characteristic polynomial of $A$, are in a sense equivalent to the minimum rank problem and hence can also be solved by Algorithm \[alg1\]. In particular, since $\text{rank}(A)+\text{null}(A)=n$ for any matrix $A$, it follows that $M(G)=n-\text{mr}(G)$; moreover, since $\lambda$ is an eigenvalue of $A$ if and only if $0$ is an eigenvalue of $A-\lambda I$ and since $mult_0(A)=\text{null}(A)$, it follows that $mult(G)=n-\operatorname{mr}(G)$. Note that the analogously defined maximum rank, minimum nullity, and minimum multiplicity problems are not interesting, since a matrix in $\mathcal{S}(G)$ with rank $n$ can be constructed by choosing each diagonal entry to be greater than the sum of the other entries in its row (by the Gershgorin circle theorem, such a matrix is nonsingular).
Finally, we briefly address the minimum rank problem for other fields. Let $S_n(\mathbb{F})$ denote the set of symmetric $n\times n$ matrices whose entries belong to a field $\mathbb{F}$, let $\mathcal{S}_{\mathbb{F}}(G)=\{A\in S_n(\mathbb{F}):\mathcal{G}(A)=G\}$, and let $\operatorname{mr}_{\mathbb{F}}(G)=\min\{\text{rank}(A):A\in\mathcal{S}_{\mathbb{F}}(G)\}$. For any finite field $\mathbb{F}$, $\operatorname{mr}_{\mathbb{F}}(G)$ can clearly be computed in finite time, since there are a finite number of matrices in $\mathcal{S}_{\mathbb{F}}(G)$. Moreover, variants of Algorithm \[alg1\] can be used to compute $\operatorname{mr}_{\mathbb{F}}(G)$ for some other infinite fields, such as $\mathbb{C}$; however, it cannot be used to compute $\operatorname{mr}_{\mathbb{Z}}(G)$, since the problem of determining whether a multivariate polynomial equation has an integer solution (Hilbert’s tenth problem) is undecidable [@matiyasevich]. It would be interesting to determine whether computing $\operatorname{mr}_{\mathbb{Z}}(G)$ is altogether undecidable, or whether there exists a finite time algorithm for it based on a different paradigm. Similarly, it would be interesting to determine whether computing $\operatorname{mr}_{\mathbb{Q}}(G)$ is undecidable; note that the decidability of determining whether a multivariate polynomial equation has a rational solution is still open.
[90]{}
AIM Special Work Group. Zero forcing sets and the minimum rank of graphs. *Linear Algebra and its Applications*, 428(7): 1628–1648, 2008.
A. Ayad. A survey on the complexity of solving algebraic systems. In *International Mathematical Forum*, 5(7): pp.333–353, 2010.
F. Barioli, W. Barrett, S. M. Fallat, H. T. Hall, L. Hogben, B. Shader, P. Van Den Driessche, H. Van Der Holst. Parameters related to tree‐width, zero forcing, and maximum nullity of a graph. *Journal of Graph Theory*, 72(2): 146–177, 2013.
F. Barioli, W. Barrett, S. Fallat, H. T. Hall, L. Hogben, B. Shader, P. Van Den Driessche, H. Van Der Holst. Zero forcing parameters and minimum rank problems. *Linear Algebra and its Applications*, 433(2): 401–411, 2010.
F. Barioli, W. Barrett, S. M. Fallat, H. T. Hall, L. Hogben, H. Van Der Holst. On the graph complement conjecture for minimum rank. *Linear Algebra and its Applications*, 436(12): 4373–4391, 2012.
F. Barioli, S. M. Fallat. On the minimum rank of the join of graphs and decomposable graphs. *Linear Algebra and its Applications*, 421: 252–263, 2007.
F. Barioli, S. M. Fallat, L. Hogben. A variant on the graph parameters of Colin de Verdiere: Implications to the minimum rank of graphs. *Electronic Journal of Linear Algebra*, 13(1): 387–404, 2005.
F. Barioli, S. M. Fallat, L. Hogben. Computation of minimal rank and path cover number for graphs. *Linear Algebra and Its Applications*, 392: 289–303, 2004.
F. Barioli, S. M. Fallat, L. Hogben. On the difference between the maximum multiplicity and path cover number for tree-like graphs. *Linear Algebra and Its Applications*, 409: 13–31, 2005.
W. Barrett, R. Bowcutt, M. Cutler, S. Gibelyou, K. Owens. Minimum rank of edge subdivisions of graphs. *Electronic Journal of Linear Algebra*, 18(1): 530–563, 2009.
W. Barrett, S. Butler, M. Catral, S. M. Fallat, H. T. Hall, L. Hogben, M. Young. The maximum nullity of a complete subdivision graph is equal to its zero forcing number. *Electronic Journal of Linear Algebra*, 27(1): 444–457, 2014.
W. Barrett, J. Grout, R. Loewy. The minimum rank problem over the finite field of order 2: minimum rank 3. *Linear Algebra and its Applications*, 430(4): 890–923, 2009.
W. Barrett, H. Van Der Holst, R. Loewy. Graphs whose minimal rank is two. *Electronic Journal of Linear Algebra*, 11: 258–280, 2004.
W. Barrett, H. Van Der Holst, R. Loewy. Graphs whose minimal rank is two: the finite fields case. *Electronic Journal of Linear Algebra*, 14: 32–42, 2005.
S. Basu. Algorithms in real algebraic geometry: a survey. *arXiv:1409.1534*, 2014.
B. Brimkov, C. C. Fast, I. V. Hicks. Computational approaches for zero forcing and related problems. *European Journal of Operational Research*, 273(3): 889–903, 2019.
B. Brimkov, I. V. Hicks. Complexity and computation of connected zero forcing. *Discrete Applied Mathematics*, 229(1): 31–45, 2017.
B. Buchberger. Theoretical basis for the reduction of polynomials to canonical forms. *ACM SIGSAM Bulletin*, 10(3): 19–29, 1976.
G. E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In *Automata Theory and Formal Languages 2nd GI Conference Kaiserslautern*, pp. 134–183, 1975.
L. DeLoss, J. Grout, L. Hogben, T. McKay, J. Smith, G. Tims. Techniques for determining the minimum rank of a small graph. *Linear Algebra and its Applications*, 432: 2995–3001, 2010.
L. DeLoss, J. Grout, L. Hogben, T. McKay, J. Smith, G. Tims. Program for calculating bounds on the minimum rank of a graph using Sage. *arXiv*:0812.1616, 2008.
L. DeLoss, J. Grout, L. Hogben, T. McKay, J. Smith, G. Tims. Table of minimum ranks of graphs of order at most 7 and selected optimal matrices. *arXiv*:0812.0870, 2008.
C. J. Edholm, L. Hogben, M. Huynh, J. LaGrange, D. D. Row. Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph. *Linear Algebra and its Applications*, 436(12): 4352–4372, 2012.
J. Ekstrand, C. Erickson, H.T. Hall, D. Hay, L. Hogben, R. Johnson, N. Kingsley, S. Osborne, T. Peters, J. Roat, A. Ross. Positive semidefinite zero forcing. *Linear Algebra and its Applications*, 439(7): 1862–1874, 2013.
S. Fallat, L. Hogben. The minimum rank of symmetric matrices described by a graph: A survey. *Linear Algebra and its Applications*, 426: 558–582, 2007.
M. Gentner, L. D. Penso, D. Rautenbach, U. S. Souza. Extremal values and bounds for the zero forcing number. *Discrete Applied Mathematics*, 214: 196–200, 2016.
C. Godsil, S. Severini. Control by quantum dynamics on graphs. *Physical Review A*, 81(5): 052316, 2010.
D. Y. Grigorev, N. N. Vorobjov. Solving systems of polynomial inequalities in subexponential time. *Journal of Symbolic Computation*, 5: 37–64, 1988.
C. Grood, J. Harmse, L. Hogben, T. J. Hunter, B. Jacob, A. Klimas, S. McCathern. Minimum rank with zero diagonal. *Electronic Journal of Linear Algebra*, 27(1): 458–477, 2014.
L. Hogben. Personal communication, Dec 13, 2017.
L. Hogben. Orthogonal representations, minimum rank, and graph complements. *Linear Algebra and its Applications*, 428: 2560–2568, 2008.
L. Hogben. Spectral graph theory and the inverse eigenvalue problem of a graph. *Electronic Journal of Linear Algebra*, 14(1): 12–31, 2005.
L. Hogben, H. Van Der Holst. Forbidden minors for the class of graphs $G$ with $\xi(G) \leq 2$. *Linear Algebra and Its Applications*, 423: 42–52, 2007.
L.-Y. Hsieh. *On Minimum Rank Matrices Having Prescribed Graph*, Ph.D. thesis. University of Wisconsin-Madison, 2001.
L. H. Huang, G. J. Chang, H. G. Yeh. On minimum rank and zero forcing sets of a graph. *Linear Algebra and its Applications*, 432(11): 2961–2973, 2010.
C. R. Johnson, A. L. Duarte. The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree. *Linear and Multilinear Algebra*, 46(1-2): 139–144, 1999.
C. R. Johnson, R. Loewy, P. A. Smith. The graphs for which the maximum multiplicity of an eigenvalue is two. *Linear and Multilinear Algebra*, 57(7): 713–736, 2009.
C. R. Johnson, C. M. Saiago. Estimation of the maximum multiplicity of an eigenvalue in terms of the vertex degrees of the graph of the matrix. *Electronic Journal of Linear Algebra*, 9: 27–31, 2002.
Y. Koren. The BellKor solution to the Netflix grand prize. *Netflix prize documentation*, 81: 1–10, 2009.
Y. Matiyasevich. Enumerable sets are Diophantine (in Russian). *Doklady Akademii Nauk SSSR*, 191(2): 279–282, 1970.
L. H. Mitchell, S. K. Narayan, A. M. Zimmer. Lower bounds in minimum rank problems. *Linear Algebra and its Applications*, 432(1): 430–440, 2010.
P. M. Nylen. Minimum-rank matrices with prescribed graph. *Linear Algebra and its Applications*, 248: 303–316, 1996.
S. Osborne, N. Warnberg. Computing positive semidefinite minimum rank for small graphs. *Involve, a Journal of Mathematics*, 7(5): 595–609, 2014.
J. E. Prussing. The principal minor test for semidefinite matrices. *Journal of Guidance, Control, and Dynamics*, 9(1): 121–122, 1986.
R. C. Read, R. J. Wilson. *An Atlas of Graphs.* Oxford University Press, Oxford, 1998.
A. Tarski. A decision method for elementary algebra and geometry. In *Quantifier Elimination and Cylindrical Algebraic Decomposition*, pp. 24–84, 1998.
L. Wang, B. Yang. Positive semidefinite zero forcing numbers of two classes of graphs. *Theoretical Computer Science*, 786: 44–54, 2019.
P. H. Wei, C. H. Weng. A typical vertex of a tree. *Discrete Mathematics*, 226: 337–345, 2001.
H. R. Wüthrich. Ein Entschiedungsverfahren für die Theorie der reell-abgeschlossenen Körper. *Lecture Notes in Computer Science*, 43: 138–162, 1976.
B. Yang. Positive semidefinite zero forcing: complexity and lower bounds. In *Workshop on Algorithms and Data Structures*, pp. 629–639, 2015.
A. M. Zimmer. A new lower bound for the positive semidefinite minimum rank of a graph. *Linear Algebra and its Applications*, 438(3): 1095–1112, 2013.
[^1]: Department of Mathematics and Statistics, Slippery Rock University, Slippery Rock, PA (boris.brimkov@sru.edu).
[^2]: Department of Mathematical Sciences, Susquehanna University, Selinsgrove, PA (scherr@susqu.edu)
|
---
abstract: 'In complete erasure any arbitrary pure quantum state is transformed to a fixed pure state by irreversible operation. Here we ask if the process of partial erasure of quantum information is possible by general quantum operations, where partial erasure refers to reducing the dimension of the parameter space that specifies the quantum state. Here we prove that quantum information stored in qubits and qudits cannot be partially erased, even by irreversible operations. The no-flipping theorem, which rules out the existence of a universal NOT gate for an arbitrary qubit, emerges as a corollary to this theorem. The ‘no partial erasure’ theorem is shown to apply to spin and bosonic coherent states, with the latter result showing that the ‘no partial erasure’ theorem applies to continuous variable quantum information schemes as well. The no partial erasure theorem suggests an integrity principle that quantum information is indivisible.'
author:
- 'Arun K. Pati'
- 'Barry C. Sanders'
title: No partial erasure of quantum information
---
=10000
\[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Proposition]{} \[theorem\][Conjecture]{}
Introduction
============
Classical information can be stored in distinct macroscopic states of a physical system and processed according to classical laws of physics. That ‘information is physical’ is exemplified by the fact that erasure of classical information is an irreversible operation with a cost of $k T \log 2 $ of energy per bit operating at a temperature $T$ [@Lan61], which is a fundamental source of heat for standard computation [@Ben82]. This is the Landauer erasure principle. In quantum information processing, a qubit cannot be erased by a unitary transformation (see Appendix A) and is subject to Landauer’s principle. In recent years considerable effort has been directed toward investigating possible and impossible operations in quantum information theory.
Impossible operations are stated as no-go theorems, which establish limits to quantum information capabilities and also provide intuition to enable further advances in the field. For example the no-cloning theorem [@Woo82; @dd82; @yuen86] underscored the need for quantum error correction to ensure that quantum information processing is possible in faulty systems despite the impossibility of a quantum FANOUT operation. Other examples of important no-go theorems are the no-deletion theorem [@pb; @Zur00], which proves the impossibility of perfectly deleting one state from two identical states, the no-flipping theorem [@Buz99], which establishes the impossibility of designing a universal NOT gate for arbitrary qubit input states, and the impossibility of universal Hadamard and CNOT gates for arbitrary qubit input states [@Pat02]. The strong no-cloning theorem states that the creation of a copy of a quantum state requires full information about the quantum state [@jozsa]; together with the no-deletion theorem, these establish permanence of quantum information. A profound consequence of the no-cloning and no-deleting theorems suggest a fundamental principle of conservation of quantum information [@Hor03].
Here we establish a new and powerful no-go theorem of quantum information, which suggests both a limitation and protection of quantum information. Our theorem shows that it is impossible to erase quantum information, even partially and even by using irreversible means, where partial erasure corresponds to a reduction of the parameter space dimension for the quantum state that holds the quantum information, namely the qubit or qudit. As a special case, it is impossible to erase azimuthal angle information of a qubit whilst keeping the polar angle information intact, which we show is the no-flipping principle. Our theorem adds new insight into the integrity of quantum information, namely that we can erase complete information but not partial information. This in turn implies that quantum information is indivisible and we have to treat quantum information as a ‘whole entity’. We also introduce the no partial erasure propositions for SU(2) coherent states and for continuous variable quantum information. Since the first e-print release of our work, a no-splitting theorem for quantum information [@Zho05] has been presented, which we show is a straightforward corollary to our Theorem \[theorem:arbitrary\].
An arbitrary qubit is expressed as $$|\Omega\rangle=\cos\tfrac{\theta}{2} |0\rangle
+e^{i\phi}\sin\tfrac{\theta}{2}|1\rangle \in {\cal H}^2$$ with $\Omega\equiv(\theta,\phi)$ and $$\theta\in [0,\pi], \; \phi\in [0,2\pi).$$ Each pure state is uniquely identified with a point on the Poincaré sphere with $\theta$ the polar angle and $\phi$ the azimuthal angle. The states $|0\rangle$ and $|1\rangle$ are the logical zero and one states, respectively. Complete erasure would map all arbitrary qubit states into a fixed qubit state $|\Omega_0\rangle
= |\Sigma \rangle$ regardless of the input state parameters $\theta$ and $\phi$, which is known to be impossible by unitary means. More generally, the $d$-dimensional analogue of the $2$-dimensional qubit is a qudit with quantum state $$\label{eq:qudit}
|\vec{\Omega}\rangle=\sum_{i=1}^d e^{i \phi_i}
\cos\tfrac{\theta_i}{2}|i\rangle \in {\cal H}^d,$$ with $$\vec{\Omega}\equiv(\vec{\theta},\vec{\phi}),$$ and $\cos\tfrac{\theta_i}{2} = |\langle i |\vec{\Omega}\rangle|$ with $\theta_i$ is the Bargmann angle between the $i^\text{th}$ orthonormal vector and the qudit state.
Each vector $\vec{\theta}$ and $\vec{\phi}$ is $d$-dimensional, but normalization of the qudit state and the unphysical nature of the overall phase reduces the number of free parameters for the qudit to $2(d-1)$. A pure qudit can be represented as a point on the projective Hilbert space ${\cal P}$ which is a real $2(d-1)$-dimensional manifold. For the qubit case, $d=2$, and there are two parameters, so we see the reduction of the formula for two qubits is correct.
The organization of our paper is as follows. In the section II, we present our no-partial erasure theorem for non-orthogonal qubits and qudits. Also we show how the no-flipping theorem for arbitrary qubits emerges as a corollary to our theorem. Then, we prove the no-partial erasure theorem for an arbitrary qudit using linearity of quantum theory. Furthermore, we show that the no-splitting theorem for quantum information also follows from our theorem. In the section III, we present the no-partial erasure result for spin coherent state. In the section IV, we generalize the no-partial erasure theorem for continuous variable quantum information. Lastly, in the section V, we conclude our paper.
No-Partial Erasure of Qubit and Qudit
=====================================
In this section, we prove powerful theorems that establish the impossibility of partial erasure of arbitrary qudit states but first begin with a definition of partial erasure.
Partial erasure is a completely positive (CP), trace preserving mapping of all [*pure*]{} states $$|\{\zeta_i;i=1,\ldots,n\}\rangle,$$ with real parameters $\zeta_i$, in an $n$-dimensional Hilbert space to [*pure*]{} states in an $m$-dimensional Hilbert subspace via a constraint $$\kappa(\{\zeta_i;i=1,\ldots,n\}$$ such that $m<n$.
The process of partial erasure reduces the dimension of the parameter domain and [*does not leave the state entangled with any other system*]{}. One may wonder why we emphasize that the process does not entangle with other system; it is because we want to analyze this process in parallel with the complete erasure process.
Recall that, in complete erasure, an arbitrary pure state of a qubit is mapped to a fixed pure state, i.e., $|\Omega \rangle \mapsto |\Sigma \rangle = |0\rangle$. If we allow the original system to be entangled with ancilla, then we would trivially be able to erase partial information. For example, if we commence with a qubit in the state $$|\Omega \rangle = \alpha |0 \rangle + \beta |1 \rangle$$ and enjoin an ancilla in the state $|0 \rangle$, then a controlled-NOT gate would entangle these two qubits together. Then the resulting state of the original qubit has no phase information about the input state. So we have a process that maps $$|\Omega \rangle \langle\Omega | \mapsto \rho(\theta).$$
Therefore, we do not want that final state of the quantum system is in a mixed state. We would like to see if the partial information can be erased and yet we retain purity of a quantum state in question. One example of partial erasure would be reducing the parameter space for the qubit from $\Omega$ to $\theta$ by fixing $\phi$ as a constant (say $\phi=0)$, i.e., $$|\theta,\phi\rangle \mapsto |\theta \rangle$$ corresponds to partial erasure of a qubit where the phase information or azimuthal angle information about a qubit is lost.
We now prove that there cannot exist a physical operation capable of erasing any pair of non-orthogonal qudit states.
\[theorem:nonorthogonal\] In general, there is no physical operation that can partially erase any pair of non-orthogonal qudits.
The partial erasure quantum operation is a CP, trace-preserving mapping that transforms an arbitrary qudit state $|\vec{\Omega}\rangle$ into the constrained qudit state $|\vec{\Omega}\rangle_\kappa$ for $\kappa(\vec{\Omega})=0$ a constraining equation that effectively reduces the parameter space by at least one dimension. Arbitrary qudit states can be represented as points on the projective Hilbert space parametrized by $\Omega$, and $\kappa$ constrains these points to a one-dimensional subset of the projective Hilbert space.
We can introduce the parametrization $\vec{\tau}$ so that the constraint $\kappa$ allows us to write the parameters as $\vec{\Omega}(\vec{\tau})$. Consider the mapping of two distinct qudit states $|\vec{\Omega} \rangle$ and $|\vec{\Omega}' \rangle$ to $|\vec{\Omega(\tau}) \rangle$ and $|\vec{\Omega}'(\vec{\tau}')
\rangle$, respectively, for some values $\vec{\tau}$ and $\vec{\tau}'$.
By attaching an ancilla, the quantum operation ${\cal E}$ that maps $$|\vec{\Omega}\rangle \langle \vec{\Omega}| \mapsto
\mathcal{E}\left(|\vec{\Omega}\rangle \langle \vec{\Omega}|\right) =
|\vec{\Omega}(\vec{\tau}) \rangle \langle
\vec{\Omega}(\vec{\tau})|$$ can be represented as a unitary evolution on the enlarged Hilbert space, so partial erasure of qudits can be expressed as $$\begin{aligned}
|\vec{\Omega} \rangle |A\rangle &\mapsto & |\vec{\Omega}(\vec{\tau})
\rangle |A_{\Omega}\rangle,\; \nonumber\\
|\vec{\Omega'} \rangle |A\rangle & \mapsto &
|\vec{\Omega}'(\vec{\tau}')
\rangle |A_{\Omega'}\rangle,\end{aligned}$$ where $|A \rangle$ is the initial state, $|A_{\Omega}\rangle$, and $|A_{\Omega'}\rangle$ are the final states of the ancilla. Now, by unitarity, taking the inner product we have $$\langle \vec{\Omega}|\vec{\Omega'}\rangle
=\langle \vec{\Omega}(\vec{\tau})| \vec{\Omega'}(\vec{\tau'})\rangle
\langle A_{\Omega}|A_{\Omega'} \rangle.$$ However, the inner product of the resultant two qudit states is not same as the inner product of the original qudit states. Hence we cannot partially erase a pair of non-orthogonal qudits by any physical operation.
If $d=2$, we readily obtain the no partial erasure theorem for any pair of non-orthogonal qubits.
As a special, and instructive, case, let us consider the impossibility of erasing azimuthal angle information for qubits. Consider the partial erasure of two non-orthogonal qubit states $|\Omega \rangle =
|\theta, \phi \rangle $ and $|\Omega' \rangle = |\theta', \phi' \rangle$ by removing the azimuthal angle information. In the enlarged Hilbert space the unitary transformations for these two states are given by $$\begin{aligned}
|\theta, \phi \rangle | A \rangle & \mapsto &
|\theta \rangle |A_{\Omega}\rangle, \; \nonumber\\
|\theta', \phi' \rangle | A \rangle
& \mapsto & |\theta' \rangle |A_{\Omega'}\rangle.\end{aligned}$$ As unitary evolution must preserve the inner product, we have $$\langle \theta, \phi|\theta', \phi' \rangle
= \langle \theta|\theta' \rangle \langle A_{\Omega}|A_{\Omega'} \rangle.$$
More explicitly, in terms of these real parameters we have $$\begin{aligned}
\cos \frac{\theta}{2} \cos \frac{\theta'}{2} +
\sin \frac{\theta}{2} \cos \frac{\theta'}{2} e^{i (\phi' -\phi)} =
\cos \frac{\theta - \theta'}{2} \langle A_{\Omega}|A_{\Omega'} \rangle.\end{aligned}$$ However, for arbitrary values of $\phi$ and $\phi'$ the above equation cannot hold. Therefore, it is impossible to erase azimuthal angle information of a qubit by physical operations.
The above equation suggests that there may be special classes of qubit states that can be partially erased. The general condition is that if $$\phi = \phi' + 2n\pi,$$ $n$ being an integer, then any qubit that differs in phase by $2n\pi$ can be partially erased. This implies if we restrict our qubits to be chosen from any great circle passing through north and south poles of the Poincaré sphere, then those qubits can be partially erased by a physical operation. Similarly, we can show that it is impossible to erase the information about the polar angle $\theta$ of an arbitrary qubit, i.e., the transformation $$| \theta, \phi \rangle | A \rangle \mapsto | \phi \rangle |A_{\Omega}\rangle$$ is not allowed.
Although there does not exist a completely positive, trace-preserving mapping that partially erases a qubit, there exists a proper subset of qudit or qubit states that are erased by a given mapping. For example the set of qubit states whose parameters satisfy the constraint $\kappa$ can have partial erasure according to the already-imposed constraint $\kappa$. Partial erasure can also be effected on an arbitrary qubit by a unitary mapping if the state is known simply because there always exists a unitary map between any two states in a Hilbert space; hence there exists a unitary mapping from every qubit state to constrained qubit states. Also a qubit or a qudit in known orthogonal states can be partially erased.
Now we show that for $d=2$ and $$\kappa(\Omega=(\theta,\phi))= \kappa(\theta,\phi_0)$$ for all $\Omega$, with the azimuthal phase $\phi_0$ fixed, we obtain the no flipping principle for an arbitrary qubit. We know that a classical bit like $0$ or $1$ can be flipped, so also a qubit in an orthogonal state like $|0\rangle$ or $|1\rangle$. However, an unknown qubit $|\Omega \rangle$ cannot be flipped. That is there is no exact universal NOT gate for an arbitrary qubit. This is because the flipping operation is an anti-unitary operation which is not a CP map and thus cannot be implemented physically. The no-flipping principle for an unknown qubit is another important limitation in quantum information.
Erasure of the azimuthal phase from the parameter domain of a qubit, whilst leaving the polar phase parameter unchanged by the mapping, implies the existence of a universal NOT gate. \[lemma:polarerasure\]
Suppose we can erase phase information of an arbitrary qubit. For an orthogonal qubit state $|\theta, \phi\rangle^\perp$ partial erasure effects the mapping $$|\theta, \phi \rangle^\perp|A\rangle\mapsto|\theta \rangle^\perp |A_{\Omega}\rangle^\perp.$$ If this holds, then after the partial erasure one can apply a local unitary NOT gate to $|\theta\rangle^\perp$ and convert it to $|\theta\rangle $ (in this case by applying $i \sigma_y$).
Next an application of the inverse of the partial erasure transformation yields the state $|\theta,\phi \rangle$. This means by applying a sequence of unitary transformations one can flip an unknown qubit state, that is, map an arbitrary qubit to its complement. Hence erasure of azimuthal phase but not polar phase implies the existence of a universal NOT gate.
Now we can apply the above to prove easily the non-existence of a universal NOT gate [@Buz99].
A universal NOT gate is impossible.
To prove the no flipping principle, we show that a universal NOT gate requires partial erasure. Suppose there is a universal NOT gate for an arbitrary qubit that takes $$|\theta, \phi\rangle \mapsto |\theta, \phi \rangle^\perp.$$ However, it is known that such an operation exists [@ghosh; @arun] if and only if the qubit belongs to a great circle, that is, the qubit parameter domain is constrained by $\kappa$ to a great circle on the sphere defined by $\theta$ and $\phi$. This means that the arbitrary qubit must have been mapped to a qubit on the great circle (this mapping is a partial erasure machine) before passing through the universal NOT.
After the universal NOT it must have passed through a reverse partial erasure machine. Thus to be able to design a universal NOT gate for an arbitrary qubit we need a partial erasure operation from $$|\theta,\phi \rangle \mapsto |\theta \rangle.$$ However, we know that this is impossible. Hence no partial erasure of phase information implies the non-existence of a universal NOT gate for a qubit.
Theorem \[theorem:nonorthogonal\] establishes that there is no physical operation that can partially erase any pair of nonorthogonal qudit states, from which the ‘no flipping principle’ emerges as a simple corollary. However, Theorem 1 applies to a set of quantum states which are not arbitrary. One can ask a more general question: Can we partially erase an arbitrary quantum state by a linear transformation? Now we show that linearity of quantum theory establishes that there cannot exist a physical operation that can partially erase a qudit, which is a stronger result.
\[theorem:arbitrary\] An arbitrary qudit cannot be partially erased by an irreversible operation.
We know that partial erasure operation for known orthogonal states is possible. Let $|\vec{\Omega}_n \rangle$ be a known orthonormal basis in $\mathcal{H}^{d}$. Then a partial erasure operation for these states yields $$|\vec{\Omega}_n \rangle | A \rangle \mapsto
|\vec{\Omega}_n(\vec{\tau}) \rangle |A_{\Omega_n}\rangle.$$ Consider an arbitrary qudit $|\vec{\Omega} \rangle$ of Eq. (\[eq:qudit\]) which is a linear superposition of the basis states $\{ |\Omega_n \rangle \}$. Suppose partial erasure of $|\vec{\Omega} \rangle$ is possible. Then linearity of the partial erasure transformation requires that $$\begin{aligned}
|\vec{\Omega} \rangle | A \rangle &= \sum_{n=1}^d e^{i \phi_n}
\cos\tfrac{\theta_n}{2}|\vec{\Omega}_n \rangle |A \rangle \mapsto
\nonumber\\
& \sum_{n=1}^d e^{i \phi_n}
\cos\tfrac{\theta_n}{2}|\vec{\Omega}_n(\vec{\tau}) \rangle
|A_{\Omega_n}\rangle
=| \bf{\Omega} \rangle.\end{aligned}$$ Ideally the partial erasure of an arbitrary qudit should have yielded a [*pure*]{} output state that takes constrained values for $\theta_n$ and $\phi_n$. However, the resultant state is not a pure qudit state but rather is entangled with the ancilla. By definition partial erasure maps a pure state to a pure state, hence a contradiction. Thus, linearity (including irreversible operations) prohibits partial erasure of arbitrary quantum information.
For $d=2$ we obtain the impossibility of partial erasure of an arbitrary qubit. For example, we cannot omit either polar or azimuthal angle information of a qubit by irreversible operation.
Here we give another proof for no-partial erasure of an arbitrary qubit solely using linearity and without using ancilla states. Suppose we have the partial erasure operation for two known orthogonal states such as $| \Psi(\theta_0, \phi_0) \rangle$ and $| {\bar \Psi}(\theta_0, \phi_0) \rangle$. Then the partial erasure operation can be represented by $$\begin{aligned}
&| \Psi(\theta_0, \phi_0) \rangle \mapsto | \psi(\theta_0) \rangle,
\nonumber\\
&| {\bar \Psi}(\theta_0, \phi_0) \rangle
\mapsto | {\bar \psi}(\theta_0) \rangle.\end{aligned}$$
Let an arbitrary qubit $| \Phi(\theta, \phi) \rangle$ be in a linear superposition of these two basis states: $$| \Phi(\theta, \phi) \rangle =
\cos \frac{\theta}{2} | \Psi(\theta_0, \phi_0) \rangle +
\sin \frac{\theta}{2} \exp(i \phi) | {\bar \Psi}(\theta_0, \phi_0) \rangle .$$ If we want to have partial erasure of $| \Phi(\theta, \phi) \rangle$ then, by linearity of the erasure transformation we have $$\begin{aligned}
| \Phi(\theta, \phi) \rangle = &
\cos \frac{\theta}{2} | \Psi(\theta_0, \phi_0) \rangle +
\sin \frac{\theta}{2} e^{i \phi} | {\bar \Psi}(\theta_0, \phi_0) \rangle
\nonumber\\
&\mapsto
\cos \frac{\theta}{2} | \psi(\theta_0) \rangle +
\sin \frac{\theta}{2} e^{i \phi} | {\bar \psi}(\theta_0) \rangle \nonumber\\
=&
| {\tilde \Phi}(\theta, \phi) \rangle\end{aligned}$$ Again, ideally the partial erasure of an arbitrary qubit should have yielded an output state that is completely independent of $\phi$, i.e., $| \Phi(\theta, \phi) \rangle \mapsto | \Phi(\theta) \rangle$. However, we have a state $| {\tilde \Phi}(\theta, \phi) \rangle$ that has complete information about both $\theta$ and $\phi$. Hence, this shows that linearity (which includes also irreversible operations) of quantum theory does not allow partial erasure of quantum information. If we include ancilla, then the original qubit will be entangled with the ancilla and by throwing out ancilla, we will be left with a qubit state that is no more pure. Note that if we allow irreversible operation (unitary evolution of combined system and tracing out of the ancilla), we can eliminate complete information of an arbitrary qubit (albeit the fact that the original information still remains in the ancilla) as in the complete erasure. Thus, one can erase the complete information of a qubit but not the partial information by an irreversible operation and yet retain its purity.
The implication of being able to completely erase, but not partially erase quantum information implies that quantum information is indivisible. There is no classical analogue for this result: no partial erasure is a strictly quantum phenomenon. We introduce the term *integrity principle* to refer to this inability to partially erase quantum information.
Since the release of our e-print proving ‘no partial erasure’ theorem a ‘no-splitting theorem’ for quantum information has been presented [@Zho05], where ‘no splitting’ refers to the impossibility of splitting a qubit $|\theta,\phi\rangle$ into a product state $|\theta\rangle|\phi\rangle$ with one qubit representing the $\theta$ information and the other representing the $\phi$ information. Here we show that the no-splitting follows from Theorem \[theorem:arbitrary\].
\[corollary:nosplitting\] No-partial erasure theorem implies a no-splitting of quantum information.
Suppose quantum information can be split. Then there exists an operation that transforms $$|\theta, \phi \rangle \mapsto |\theta \rangle |\phi \rangle.$$ We can append an ancillary qubit in a specific state and swap with the second qubit, then trace to eliminate all information about $\phi$. Thus splitting implies partial erasure, which contradicts Theorem \[theorem:arbitrary\]. Hence, it is impossible to split quantum information.
No-partial erasure of spin coherent state
=========================================
Our theorem that no partial erasure of qudits is possible is important because quantum information is clearly not only conserved but also indivisible. However, the ‘no partial erasure’ theorem yields another important result for erasure of spin coherent states, also known as SU(2) coherent states [@Are72; @gilm72; @perel72].
The SU(2) coherent states are a generalization of qubits, which can be thought of as spin-$\tfrac{1}{2}$ states, to states of higher spin $j$. The SU(2) raising and lowering operators are $\hat{J}_+$ and $\hat{J}_-$, respectively, and their commutator $$[\hat{J}_+,\hat{J}_-]=2\hat{J}_z$$ yields the weight operator $\hat{J}_z$ with spectrum $$\{m;-j \leq m \leq j\}$$ and integer spacing between successive values of $m$. The weight basis is $|j\,m\rangle$ with $j(j+1)$ the eigenvalue for states in the $j^\text{th}$ irrep of the Casimir invariant $\hat{J^2}$.
The SU(2) coherent states are obtained by ‘rotations’ of the highest-weight state $|j\,j\rangle$. Here we use the stereographic parameter $$\gamma = e^{i\phi} \tan(\theta/2)$$ that corresponds to the coordinates of the state on the complex plane obtained by a stereographic projection of the point on the Poincaré sphere for the given state, with parameters $\theta$ and $\phi$ are the polar and azimuthal angular coordinates of the Poincaré sphere defined earlier; here the sphere represents states of a $(2j+1)$-dimensional system, not just the two-dimensional qubit. For $|j\,j\rangle$ the highest-weight state, the SU(2) coherent state is [@San89] $$|j,\gamma\rangle=R_j(\gamma)|j\,j\rangle = \sum_{m=0}^{2j}
\begin{pmatrix} 2j \\ m \end{pmatrix}^{1/2}
\frac{\gamma^m}{(1+|\gamma|^2)^j} |j\,j-m\rangle$$ for $$\begin{aligned}
R_j(\gamma)&=\exp\left[\tfrac{1}{2}\theta\left(\hat{J}_-e^{i\phi}-\hat{J}_
+e^{-i\phi}\right)\right] \nonumber \\ &= \exp(\gamma \hat{J}_-)
\exp[-\hat{J}_z \ln(1 + |\gamma|^2 )]
\exp(- \gamma^*\hat{J}_+ ).\end{aligned}$$
We can now prove the following no go result using our theorem.
Partial erasure of SU(2) coherent states is impossible.
The SU(2) coherent state is a qudit with the constraint that $$e^{i\phi_m}\cos\tfrac{\theta_m}{2}
=\begin{pmatrix}2j\\m\end{pmatrix}^{1/2}\frac{\gamma^m}{(1+|\gamma|^2)}$$ for each $m$. Partial erasure of the SU(2) coherent states corresponds to partial erasure over a subspace of qudits, which we have shown is impossible.
No-Partial erasure of Continuous Variable state
===============================================
Next, we prove the ‘no partial erasure’ theorem for continuous variable quantum information. Ideally continuous variable (CV) quantum information encodes quantum information as superpositions of eigenstates of the position operator $\hat{x}$, namely $$\hat{x}|x\rangle= x|x\rangle; \{ x\in\mathbb{R}\}$$ with complex amplitude $\Psi(x)$ [@qicv]. We can represent a CV state as $$|\Psi\rangle=\int_\mathbb{R} dx \Psi(x)|x\rangle,\;\;\Psi(x)=\langle
x|\Psi\rangle.$$ Note that $\Psi(x)$ can be any complex-valued function, subject to the requirement of square-integrability and normalization.
Now let us reduce $\Psi(x)$ to a real-valued function, so we have effectively reduced the parameter space even in infinity dimensional Hilbert space. Does there exist a completely positive, trace preserving mapping from the set of states with $\Psi(x)$ a general complex-valued function to the new $\psi(x)$ a general real-valued function?
Partial erasure of continuous variable quantum information is a completely positive map of all arbitrary pure states with complex wavefunctions to pure states with real wavefunctions.
There is no physical operation that can partially erase any pair of complex wavefunctions.
We prove this theorem for a system with one degree of freedom, namely canonical position $x$; this proof is readily extended to more than one degrees of freedom. Suppose there is a CP map that can partially erase a wavefunction $\Psi(x)$ via $$|\Psi \rangle |A\rangle \mapsto |\psi \rangle |A_{\Psi}\rangle,\;$$ where $$||\Psi||^2 = \int_\mathbb{R} dx \; |\Psi(x)|^2,\;\; || \psi||^2 =
\int_\mathbb{R} dx \; \psi(x)^2.$$ If this holds for another arbitrary wavefunction $\Phi(x)$, then we have $$|\Phi \rangle |A\rangle \mapsto |\phi \rangle |A_{\Phi}\rangle,\;$$ where $$||\Phi||^2 = \int_\mathbb{R} dx \; |\Phi(x)|^2$$ and $$||\phi||^2 = \int_\mathbb{R} dx \; \phi(x)^2.$$ However, the inner product preserving condition $$\int_\mathbb{R} dx\, \Psi(x)^*\Phi(x) = \int_\mathbb{R} dx\,\psi(x) \phi(x)
\int_\mathbb{R} dx\, A_{\Psi}^*(x) A_{\Phi}(x)$$ cannot hold for general complex-valued wavefunctions. Hence, we cannot partially erase a pair of complex wavefunction.
This result is analogous to partial erasure of qudits. Furthermore, the restriction should apply for any erasure of the complex domain by one dimension (such as a circle where amplitude is fixed and phase varies). Similarly, one can give a general proof of no partial erasure of continuous variable quantum information, not just complex to real but complex to any one-dimensional subset of the complex space.
For example, let the partial erasure process transforms the wavefunction such that one of the complex amplitudes becomes real (which is one way to reduce the parameter space by one dimension). Then we can prove that it is also impossible. Under partial erasure the continuous variable state $$|\Psi\rangle= \int_\mathbb{R} dx \Psi(x)|x\rangle,$$ with $$\Psi(x) = \sum_{n=0}^{\infty} c_n \Psi_n(x),$$ $c_n = |c_n|\exp(i\theta_n)$ transforms as $$\sum_{n=0}^{\infty} c_n \Psi_n(x) \mapsto \sum_{n=0}^{\infty} d_n \Psi_n(x)$$ where the constraint is that all $c_n = d_n$ are complex except for one $d_k$, which is a real number. Consider a pair of wavefunctions $(\Psi(x),\Phi(x))$ with $$\Phi(x) = \sum_{n=0}^{\infty} c_n' \Phi_n(x),$$ $c_n' = |c_n'| \exp(i\theta_n')$ and partial erasure of $\Phi(x)$ is given by $$\sum_{n=0}^{\infty} c_n' \Phi_n(x) \mapsto \sum_{n=0}^{\infty} d_n' \Phi_n(x)$$ with similar constraints.
For clarity, let us not include an ancilla. Unitarity implies that $$\exp(i[ \theta_k' -\theta_k]) = 1,$$ which is impossible for arbitrary values of $\theta_k$ and $\theta_k'$. Hence, we cannot forget even one parameter of the complex wavefunction.
Conclusions
===========
In summary we have introduced a new process called partial erasure of quantum information and asked if quantum information can undergo partial erasure. We have shown that partial erasure of qubits, qudits, SU(2) coherent states, and continuous variable quantum information is impossible. These results point to the integrity principle for quantum information, namely that it is indivisible and robust even against partial erasure. This principle gives a new meaning to quantum information and nicely complements the recent profound observation of the principle of conservation of quantum information [@Hor03].
Furthermore, the impossibility theorems presented here underscore essential differences between classical information (which could be stored in orthogonal quantum states) and general quantum information, analogous to related but distinct impossibility results such as the no-cloning, no-deleting and no-flipping principles. Our principle of quantum information integrity may have implications for investigations into quantum mechanics over real, complex, and quaternionic number fields [@Adl95; @Stu60]: a unitary equivalence between complex and real quantum theories would appear to contradict the no partial erasure theorem. Interesting problems that warrants further investigation is approximate deterministic partial erasure and exact probabilistic partial erasure over restricted classes of states.
*Acknowledgments: —* AKP thanks C. H. Bennett for useful discussions. BCS has been supported by Alberta’s Informatics Circle of Research Excellence (iCORE), the Canadian Institute for Advanced Research, and the Australian Research Council.
No complete erasure by reversible operations
============================================
In quantum theory a reversible operation can be represented by a unitary operator. Erasure of a qubit state $|\Psi\rangle$ transforms this to a fixed state $|\Sigma \rangle$, which contains no information about the input state.
Consider erasure of a pair of qubits $|\Omega \rangle$ and $|\Omega' \rangle$ such that $$|\Omega \rangle \mapsto |\Sigma \rangle$$ and $$|\Omega' \rangle \mapsto |\Sigma \rangle.$$ As unitary evolution preserves the inner product, we will have $$\langle \Omega |\Omega' \rangle = 1,$$ which cannot be true. Furthermore, even for two orthogonal states such as $|0\rangle$ and $|1\rangle$, this evolution implies a contradiction.
This paradox demonstrates, in a simplest and yet profound way, that neither classical information nor quantum information can be erased by any reversible operation.
[99]{}
R. Landauer, IBM J. Res. Develop. **5**, 183 (1961). C. H. Bennett, Int. J. Theor. Phys. **21**, 905 (1982). W. K. Wootters and W. H. Zurek, Nature **299**, 802 (1982).
D. Dieks, Phys. Lett. A **92**, 271 (1982).
H. P. Yuen, *ibid* **113**, 405 (1986).
A. K. Pati and S. L. Braunstein, Nature (Lond,) **404**, 164 (2000).
W. H. Zurek, Nature (Lond.) **404**, 40 (2000);
V. Bužek, M. Hillery and R. F. Werner, **60**, R2626 (1999).
A. K. Pati, **66**, 062319 (2002).
R. Jozsa, IBM J. Res. & Dev **48**, 79 (2004).
M. Horodecki, R. Horodecki, A. Sen De, U. Sen, quant-ph/0306044 (2003).
D. Zhou, B. Zeng and L. You, quant-ph/0503168 (2005).
S. Ghosh, A. Roy, and U. Sen, Phys. Rev. A 63, 014301 (2001).
A. K. Pati, Phys. Rev. A 63, 014302 (2001).
F. T. Arecchi, E. Courtens, R. Gilmore and H. Thomas, **6**, 2211 (1972).
R. Gilmore, Ann. Phys. (N.Y.) **74**, 391 (1972).
A. M. Perelomov, Commun. Math. Phys. **26**, 222 (1972).
B. C. Sanders, **40**, 2417 (1989).
S. L. Braunstein and A. K. Pati, *Quantum Information with Continuous Variables* (Kluwer, Dordrecht, 2003).
S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, New York, 1995). E. C. G. Stueckelberg, Helv. Phys. Acta 33, 727 (1960).
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---
abstract: |
An elegant procedure which characterizes a decomposition of some class of binomial configurations into two other, resembling a definition of Pascal’s Triangle, was given in [@gevay]. In essence, this construction was already presented in [@perspect]. We show that such a procedure is a result of fixing in configurations in some class $\mathcal K$ suitable hyperplanes which both: are in this class, and deleting such a hyperplane results in a configuration in this class. By a way of example we show two more (added to that of [@gevay]) natural classes of such configurations, discuss some other, and propose some open questions that seem also natural in this context.
Mathematics Subject Classification: 05B30, 51E30 (51E20)
Keywords: Pascal Triangle (of binomials), binomial, configuration, hyperplane, combinatorial Grassmannian, combinatorial Veronesian, Pascal Triangle of Configurations
author:
- 'Krzysztof Pra[ż]{}mowski'
title: 'Hyperplanes in Configurations, decompositions, and Pascal Triangle of Configurations'
---
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Introduction {#sec:intro .unnumbered}
============
On one hand, “Pascal Triangle” is a term which is known to all mathematicians: it characterizes an arrangement of binomial coefficients in a form of a ‘pyramid’ such that each item is the sum of items placed immediately above it. In another view: the sum of each neighbour items in a row equals to the item which is their common neighbour (in the row below). Clearly, binomial coefficients are simply values of a two-argument function $b(n,k)$ defined on nonnegative integers ($n=0,1,\ldots$, $k=0,\ldots,n$) and nothing ‘magic’ is in the pyramid defined above. It is a visual presentation of recursive equation which these coefficients satisfy. Clearly, the sequences of boundary values $b(n,0)$ and $b(n,n)$ uniquely determine then the function $b$. Nevertheless the recurrence in question is extremely simple...
Quite recently, Gabor G[é]{}vay in [@gevay] noted that there is family of point-line configurations which can be arranged in such a pyramid, with a suitably defined “sum” of the configurations in question. Or: each (nontrivial, non-boundary) configuration in this family can be decomposed into two other members of this family. In essence, this decomposition (even in a more general form) was presented also earlier in [@perspect Representation 2.12]; the class in question consists of configurations which generalize Desargues configuration considered as schemes of mutual perspectives between several simplexes. On other hand, such systems of (geometrical) perspectives can be found even in the classical book of Veblen and Young [@class:proj] (G[é]{}vay quotes also explicitly Danzer and Cayley) and its combinatorial schemes are special instances of so called binomial graphs, investigated in the context of association schemes (cf. e.g. [@klin]), and associated incidence structures. Combinatorial schemes characterizing these configurations can be found already in [@levi] and [@coxet]. So:
> the subject was known, but its regular nature was not known – was not stated explicitly until [@gevay].
But then it appeared that the “sum” of two configurations is not a well defined operation that depends solely on the summands, and the associated decomposition is, in fact, associated with a choice of a hyperplane in the decomposed configuration. After that become clear (we present these observation in Section \[sec:binconfy\], Theorem \[thm:decompo0\] and equation ) there appeared that there are other natural known classes of configurations that can be arranged into respective triangles. These are, in particular, so called combinatorial Veronesians (defined originally in [@combver], without any connections with studying hyperplanes in configurations). In Section \[sec:exm\] we discuss some of the classes which appear within this theory.
Notations, standard constructions {#sec:nota}
=================================
Elementary combinatorics {#ssec:intro:combin}
------------------------
There are well known formulas concerning binomial coefficients, frequently referred to as “Pascal Triangle of Binomials". To be more precise, these formulas correspond to the arrangement of the binomial coefficients in a pyramid with consecutive rows:
$\Big( \left(\binom{n}{k}\colon k=0,\ldots,n \right)\colon n = 0,1,2,\ldots \Big)$.
Then the corresponding recursive formula is the following $$\begin{aligned}
\label{pyramid:0}
\binom{n}{k} & = & \binom{n-1}{k-1} + \binom{n-1}{k};\end{aligned}$$ equation yields immediately next two: $$\begin{aligned}
\label{pyramid:1}
\textstyle{\binom{n}{k} - \binom{n-1}{k}} & = & \textstyle{\binom{n-1}{k-1}}, \text{ and}
\\ \label{pyramid:2}
\textstyle{\binom{n}{k} - \binom{n-1}{k-1}} & = & \textstyle{\binom{n-1}{k}}.\end{aligned}$$
For purposes of our next investigations it will be more convenient to arrange binomial coefficients into a (infinite) matrix:
$\big[ \ginomx(m,k)\colon m,k = 0,1,\ldots \big]$,
where $$\ginomx(m,k) = \ginom(m,k);$$ clearly, $\ginomx(m,k) = \ginomx(k,m)$; the fundamental recursive formula for the binomial coefficients takes the form $$\label{rec:ginom}
\ginomx(m,k) = \ginomx(m,k-1) + \ginomx(m-1,k).$$
Rudiments of geometry of configurations {#ssec:intro:config}
---------------------------------------
We say that a structure ${\goth K} = {{\ensuremath{\langle U,\lines,\inc \rangle}}}$ with $\inc\; \subset U\times\lines$ is a [*$\konftyp(\nu,\rho,\beta,\kappa)$-configuration*]{} if $\goth K$ is a partial linear space (i.e. $a,b \inc A,B$ yields $a =b$ or $A = B$) such that $|U| = \nu$, $|\lines| = \beta$, exactly $\rho$ elements of $\lines$ are in the relation $\inc$ with $a\in U$, for each $a\in U$, and exactly $\kappa$ elements of $U$ are in the relation $\inc$ with $A \in \lines$, for each $A\in\lines$.
Let $\goth K$ be a configuration as above, then the following equation (a specialized form of the so called fundament equation of partial linear spaces) holds $$\label{equ:pls}
\nu \cdot \rho = \beta \cdot \kappa.$$ The elements of $U$ are called [*points*]{} of $\goth K$, the elements of $\lines$ are called [*lines*]{} of $\goth K$, and the relation $\inc$ is [*the incidence*]{}. The numbers $\rho$ and $\kappa$ are referred to as [*point rank*]{} and [*line size/rank*]{} resp. It is a folklore, that every configuration as above with $\kappa\geq 2$ is isomorphic to a configuration, whose lines are sets of points, and the incidence is the standard membership relation $\in$. If this will not cause a confusion (as it may happen in particular examples) we shall frequently assume that the incidence of $\goth K$ is the membership relation.
A subset $\cal H$ of the set of points of $\goth K$ is called [*a hyperplane*]{} of $\goth K$ when
- $\cal H$ is [*a subspace*]{} of $\goth K$, i.e. if the conditions $a,b \inc A\in\lines$ and $a,b\in{\cal H}$, $a\neq b$ yield $x \in {\cal H}$ for every $x$ such that $x \inc A$,
- and
- each line of $\goth K$ crosses $\cal H$, i.e. for each $A\in\lines$ there is $x\in {\cal H}$ such that $x\inc A$.
Let $\cal H$ be a hyperplane of $\goth K$. Then, for each line $A$ of $\goth K$ either there is a unique $x\in{\cal H}$ with $x \inc A$ (we write $x = A^\infty$ in that case) or every point incident with $A$ belongs to $\cal H$: the set of such lines will be denoted by $\lines[{\cal H}]$. Clearly,
${\goth K}\restriction{\cal H} := {{\ensuremath{\langle {\cal H},\lines[{\cal H}], \inc \cap \big({\cal H}\times\lines[{\cal H}]\big) \rangle}}}$
is a partial linear space; quite frequently in the sequel we shall make no distinction between $\cal H$ and ${\goth K}\restriction{\cal H}$. Clearly, the set $U$ of all the points of $\goth K$ is a hyperplane of $\goth K$. In what follows we shall assume that a hyperplane means a [*proper*]{} (i.e. ${\cal H}\neq U$) subspace that satisfies suitable conditions.
Given a hyperplane $\cal H$ of $\goth H$ we define [*the reduct*]{}
${\goth K} \setminus {\cal H} :=
{{\ensuremath{\langle U\setminus {\cal H},\, \lines\setminus \lines[{\cal H}],\, \inc \cap \big((U\setminus {\cal H})\times(\lines\setminus \lines[{\cal H}])\big) \rangle}}}$;
if $\kappa \geq 3$ then ${\goth K}\setminus {\cal H}$ is a partial linear space with all the lines of size (rank) $\kappa -1$. Let us write, for symmetry, ${\goth K}_1 = {\goth K}\setminus {\cal H}$ and ${\goth K}_2 = {\goth K}\restriction{\cal H}$. Recall, that we have a function $\infty$ from the lines of ${\goth K}_1$ into the points of ${\goth K}_2$. Let us try to “reverse” this decomposition:
\[def:zlepka\] Let ${\goth K}_i = {{\ensuremath{\langle U_i,\lines_i,\inc_i \rangle}}}$ be a partial linear space for $i=1,2$. Assume that $U_1 \cap U_2 = \emptyset = \lines_1 \cap \lines_2$. Let $\infty\colon\lines_1\longrightarrow U_2$ be a map such that the following holds
if $U_1\ni x \inc A,B\in\lines_1$ and $A^\infty = B^\infty$ then $A = B$.
We define
$U:$
: $= U_1 \cup U_2$,
$\lines:$
: $= \lines_1\cup\lines_2$,
$\inc:$
: $= \inc_1 \cup \inc_2 \cup \{(x,A) \colon U_2\ni x=A^\infty, A\in\lines_1\}$.
Finally, we set $$\label{def:zlepka0}
{\goth K}_1 \rtimes_\infty {\goth K}_2 := {{\ensuremath{\langle U,\lines,\inc \rangle}}}.$$ It is evident that [*${\goth K}_1 \rtimes_\infty {\goth K}_2$ is a partial linear space*]{}.
Let ${\goth K} = {\goth K}_1 \rtimes_\infty {\goth K}_2$ with ${\goth K}_i$ as in \[def:zlepka\]. Then $U_2$ is a hyperplane in $\goth K$ and $\lines_2 = \lines[U_2]$.
It suffices to state directly that if $A\in\lines$ then either $A\in\lines_2$ and then $x\inc A$ gives $x\in U_2$, or $A\in\lines_1$ and then $x\inc A$ yields $x\in U_1$ or $U_2 \ni x = \infty(A)$.
The construction of the type \[def:zlepka\] is quite frequent in geometry. One particular case let us mention below:
Let ${\goth K}_1 = {{\ensuremath{\langle U_1,\lines_1,\inc_1,\parallel_1 \rangle}}}$ be a partial linear space with parallelism of lines; we write $[A]_{\parallel_1}$ for the equivalence class of $A\in\lines_1$ w.r.t. the relation $\parallel_1$ (i.e. simply for the direction of $A$). Suppose that there is a formula $\Phi$ in the language of ${\goth K}_1$ such that the relation
$\{ ([A_1]_{\parallel_1},[A_2]_{\parallel_1},[A_3]_{\parallel_1})\colon (A_1,A_2,A_3)\in\lines_1^3,\ \Phi(A_1,A_2,A_3) \}$
is a ternary equivalence relation on the set $\big( \lines_1\diagup\parallel_1 \big)^3$ (cf. [@ternequiv]); let $\lines_2$ be the set of its equivalence classes, and ${\goth K}_2 = {{\ensuremath{\langle {\lines_1\diagup\parallel_1},\lines_2,\in \rangle}}}$. With $A^\infty = [A]_{\parallel_1}$ for $A\in\lines_1$ we obtain the structure ${\goth K} = {\goth K}_1 \rtimes_\infty {\goth K}_2$ which is called, in that context, the [*closure of an affine structure*]{} ${\goth K}_1$.
In particular cases of this construction, practically, the structures ${\goth K}_1$ and $\goth K$ are given, and [*we search for an appropriate formula $\Phi$*]{} (see [@afclos]: affine completion, [@polarclos], [@segreclos]).
Other examples of this construction will appear in the next Section.
Dualization {#ssec:duale}
-----------
Let ${\goth K} = {{\ensuremath{\langle U,\lines,\inc \rangle}}}$ be an incidence structure; we call the structure
$\dual{{\goth K}} = {{\ensuremath{\langle \lines,U,\inc^{-1} \rangle}}}$
the dual of $\goth K$. It is evident that $\dual{{\goth K}}$ is a partial linear space whenever $\goth K$ is so. In particular
if $\goth K$ is a $\konftyp(\nu,\rho,\beta,\kappa)$-configuration then $\dual{\goth K}$ is a $\konftyp(\beta,\kappa,\nu,\rho)$-configuration.
\[prop:dual-hypy\] Let $\cal H$ be a hyperplane of a partial linear space ${\goth K} = {{\ensuremath{\langle U,\lines \rangle}}}$ such that the induced correspondence $\infty$ is bijective. Then $\lines\setminus\lines[{\cal H}]$ is a hyperplane of $\dual{\goth K}$.
Let $L_1,L_2 \in \lines\setminus\lines[{\cal H}]$. Assume that $L_1,L_2\inc^{-1} x\in U$ and $\lines\ni L \inc^{-1} x$. Suppose that $L\notin\lines\setminus\lines[{\cal H}]$, then $L\in\lines[{\cal H}]$ and, consequently, $x\in{\cal H}$. This gives $x = \infty(L_1) = \infty(L_2)$; we have $L_1 = L_2$ then. This proves that $\lines[{\cal H}]$ is a subspace of $\dual{\goth K}$.
Let $L$ be an arbitrary line of $\dual{\goth K}$, then $L \in U$. If $L\notin{\cal H}$ then each line of $\goth K$ (each point of $\dual{\goth K}$) that passes through $L$ is in $\lines\setminus\lines[{\cal H}]$. If $L\in{\cal H}$ then $\infty^{-1}(L) \inc^{-1} L$. This suffices for the proof.
Standard examples show that the condition [*$\infty$ is bijective*]{} assumed in \[prop:dual-hypy\] cannot be removed. Indeed, the plane in a projective 3-space $\goth P$ is a hyperplane, but the family of lines of the resulting affine 3-space is not even a subspace of $\dual{\goth P}$. However, \[prop:dual-hypy\] appears useful when we deal with (binomial) configurations. Proposition \[prop:dual-hypy\] can be easily (re)formulated in a more ‘constructive’ fashion:
\[cor:dual-hipy\] Let ${\goth K}_i$ be configurations as in \[def:zlepka\] with a suitable map $\infty$ defined. Assume that $\infty$ is a bijection and ${\goth K} = {\goth K}_1\rtimes_\infty {\goth K}_2$. Then $$\label{wzor:dual-hipy}
\dual{\goth K}\quad = \quad \dual{{\goth K}_2} \rtimes_{\infty^{-1}} \dual{{\goth K}_1}$$
Binomial configurations {#sec:binconfy}
=======================
Generalities
------------
The main subject of this section consists in investigations on the family of [*binomial configurations*]{} i.e. of configurations of the type $\ginconf(k,m)$ for some positive integers $k,m$. It is easily seen that each parameters of this form satisfy . Let us write
$\ginkonfx(k,m)$ for the class of all $\ginconf(k,m)$-configurations.
\[thm:decompo0\] Let ${\goth K}\in\ginkonfx(k,m)$ and let $\cal H$ be a hyperplane of $\goth K$. Assume that
-2pt
\[war1\] $\cal H$ is a configuration (in this case this means simply that ${\goth K}\restriction{\cal H}$ has constant point rank), and
\[war2\] ${\goth K}\setminus{\cal H}$ is a binomial configuration.
Then
-2pt
\[war3\] $\cal H$ is a binomial configuration, more precisely: ${\goth K}_2 = {\goth K}\restriction{\cal H}\in\ginkonfx(k-1,m)$;
\[war4\] ${\goth K}_1 = {\goth K}\setminus{\cal H}\in\ginkonfx(k,m-1)$;
\[war5\] there is a 1-1 correspondence $\infty\colon\text{lines of }{\goth K}_1 \longrightarrow\text{ points of }{\goth K}_2$ such that ${\goth K} = {\goth K}_1 \rtimes_\infty {\goth K}_2$.
Recall that, right from the definition, the points of $\goth K$ have rank $k$, and the lines of $\goth K$ have size $m$. Set $n=m+k-1$.
Then, from the definition we get immediately that the points of ${\goth K}_1$ are all of the same rank $k$ and the lines are all of the size $m-1$, so, in accordance with , ${\goth K}_1$ is a $\ginconf(k,m-1)$-configuration, which justifies . The number of points in ${\goth K}_1$ is $\binom{n-1}{k}$ and the number of points of $\goth K$ is $\binom{n}{k}$; from the Pascalian equations the number of points of ${\goth K}_2$ is $\binom{n}{k} - \binom{n-1}{k} = \binom{n-1}{k-1}$. Similarly we compute the number of lines of ${\goth K}_2$: it equals to $\binom{n}{m} - \binom{n-1}{m-1} = \binom{n-1}{m}$. The size of the lines in $\cal H$ is $m$; from assumption and applied to ${\goth K}_2$ we get that the point rank in ${\goth K}_2$ equals to $k-1$. So, ${\goth K}_2$ is a $\ginconf(k-1,m)$-configuration. This justifies . Finally, since each point in ${\goth K}_2$ has its rank on one less than in $\goth K$ we get that through each one of these points there passes exactly one line of ${\goth K}_1$, so $\infty$ is a bijection, as required in .
Informally speaking, \[thm:decompo0\] gives a decomposition $$\label{eq:decomp0}
\ginconfx(k,m) = \ginconfx(k,m-1) \rtimes_\infty \ginconfx(k-1,m),$$ which resembles reverent Pascalian equation . But note, that the “operation” $\rtimes_\infty$ is not commutative, and it depends essentially on the parameter $\infty$.
Indeed, it suffices to have a look on hyperplanes in binomial partial Steiner triple systems, either in a more general approach of [@hypinbin:psts] or in a more particular case of [@hypingendes] and note that in the Desargues configuration a line accomplished with a point not joinable with any point on this line is a hyperplane, it contains three points of rank $3$ and one point of rank $0$ so, it is not a configuration.
Let us consider the smallest sensible and possible case: $\ginconfx(2,3) \rtimes \ginconfx(3,2) = \ginconfx(3,3)$. If ${\goth K}\in\ginconfx(3,3)$ then $\goth K$ is a $\ginconf(3,3)= \konftyp(10,3,10,3)$-configuration: one of ten possible. If ${\goth K}_1$ is a $\ginconf(3,2)=\konftyp(4,3,6,2)$-configuration then it is the complete graph $K_4$. If ${\goth K}_2$ is a $\ginconf(2,3)=\konftyp(6,2,4,3)$-configuration then it is simply the Pasch-Veblen configuration $\goth V$. It was shown in [@klik:VC] that there are exactly six maps $\infty$ which yield pair wise non isomorphic configurations $K_4 \rtimes_\infty {\goth V}$. So, [*there are binomial configurations ${\goth K}_1,{\goth K}_2$ and bijections $\infty',\infty''\colon\text{lines of }{\goth K}_1 \longrightarrow\text{ points of }{\goth K}_2$ such that ${\goth K}_1\rtimes_{\infty'}{\goth K}_2 \not\cong {\goth K}_1\rtimes_{\infty''}{\goth K}_2$.*]{} Consequently, the symbol $\rtimes$ [*is not a well defined operation, without the argument $\infty$ defined explicitly*]{}.
Let ${\goth K}_1,{\goth K}_2$ be binomial configurations, let a map $\infty\colon\text{lines of }{\goth K}_1 \longrightarrow\text{ points of }{\goth K}_2$ be a bijection.
From assumption, ${\goth K}_i\in \ginconfx(k_i,m_i)$ for some integers $k_i,m_i$, $i=1,2$. Moreover, the two numbers: of lines of ${\goth K}_1$ and of points of ${\goth K}_2$ coincide. This means than $\binom{k_1+m_1-1}{m_1} = \binom{k_2+m_2-1}{k_2}$. Then $k_1+m_1-1 = k_2+m_2-1$ and one of the following holds:
either $m_1 = m_2-1$ – in this case $k_2 = k_1-1$ and [*${\goth K} = {\goth K}_1 \rtimes_\infty{\goth K}_2$ is a binomial configuration*]{},
or $m_1 = k_2$ and then $k_1 = m_2$. Consider e.g. the case $k_1=m_1=k_2=m_2=3$, then ${\goth K}_i$ are $\konftyp(10,3,10,3)$-configurations. But then ${\goth K} = {\goth K}_1 \rtimes_\infty{\goth K}_2$ has $20$ points and $20$ lines. Ten lines have size $3$, and ten have size $4$. So, in this case [*$\goth K$ is not even a configuration.*]{}
This shows that a ‘sum’ of two binomial configurations, even determined by constructing ‘improper points’, may be not a binomial configuration.
In the next Section we present two remarkable families of binomial configurations which yield families indexed by positive integers and which yield “a Pascal Triangle".
Examples {#sec:exm}
========
Example: the family of combinatorial Grassmannians {#exm:grasy}
--------------------------------------------------
For an integer $k$ and a set $X$ we write ${\raise.5ex\hbox{\ensuremath{\wp}}}_k(X)$ for the family of $k$-subsets of $X$. Nowadays the notation $\binom{X}{k}$ instead of ${\raise.5ex\hbox{\ensuremath{\wp}}}_k(X)$ becomes widely used. We prefer, however, not to mix integers and sets.
Let ${\goth K}\in\ginconfx(k,m)$; then the points of $\goth K$ can be identified with the $k$-subsets of a fixed $n$-element set $X$, where $n = m+k-1$. Let us identify the lines of $\goth K$ with the elements of ${\raise.5ex\hbox{\ensuremath{\wp}}}_m(X)$ and define $$\label{def:inc:gras0}
a \inc A :\iff a \in {\raise.5ex\hbox{\ensuremath{\wp}}}_k(X) \land A \in {\raise.5ex\hbox{\ensuremath{\wp}}}_m(X) \land |a \cap A|=1.$$ Suppose that $a\neq b$ and $a,b \inc A$ with $\inc$ defined by . Then $a\cap b = X\setminus A$ and therefore $A$ is uniquely determined by its two points $a$ and $b$. So, the structure
${\goth G}(k,m) := {{\ensuremath{\langle {\raise.5ex\hbox{\ensuremath{\wp}}}_k(X),{\raise.5ex\hbox{\ensuremath{\wp}}}_m(X),\inc \rangle}}}$
is a partial linear space. It is not too hard to verify that it is a configuration with the lines of size $m$ and the points of rank $k$, so ${\goth G}(k,m)\in\ginconfx(k,m)$.
In practice, the above presentation is not so easy to handle with and not too intuitive.
There is a one-to-one correspondence between the elements of ${\raise.5ex\hbox{\ensuremath{\wp}}}_m(X)$ and the elements of ${\raise.5ex\hbox{\ensuremath{\wp}}}_{k-1}(X)$: indeed, $n = m + (k-1)$ so, the boolean complementation $\varkappa$ is a bijection in question. Then we see that the pair of maps $(\id,\varkappa)$ maps ${\goth G}(k,m)$ onto the structure ${{\ensuremath{\langle {\raise.5ex\hbox{\ensuremath{\wp}}}_k(X),{\raise.5ex\hbox{\ensuremath{\wp}}}_{k-1}(X),\supset \rangle}}}$, which coincides with the $DCD(n,k)$ introduced in [@gevay].
Analogously, there is a one-to-one correspondence between the elements of ${\raise.5ex\hbox{\ensuremath{\wp}}}_{m-1}(X)$ and the elements of ${\raise.5ex\hbox{\ensuremath{\wp}}}_k(X)$; set $k_0 = m-1$, then $(\varkappa,\id)$ maps ${\goth G}(m,k)$ onto the structure ${{\ensuremath{\langle {\raise.5ex\hbox{\ensuremath{\wp}}}_{k_0}(X),{\raise.5ex\hbox{\ensuremath{\wp}}}_{k_0+1}(X),\subset \rangle}}}$, which coincides with the [*combinatorial Grassmannian*]{} $\GrasSpace(X,k_0)$ defined in [@perspect].
Let us concentrate upon the presentation given in [@perspect], let us drop out the superfluous index $0$ and let ${\goth K} = \GrasSpace(X,k)$, $|X| = n$; remember that $\GrasSpace(X,k) \in \ginconfx(n-k,k+1)$. We write $\GrasSpace(n,k)$ for the type of $\GrasSpace(X,k)$ where $|X| = n$.
Let us fix an element $i\in X$, then ${\raise.5ex\hbox{\ensuremath{\wp}}}_k(X)$ is the disjoint union ${\raise.5ex\hbox{\ensuremath{\wp}}}_k(X) = {\cal X}_1 \cup {\cal X}_2$, where ${\cal X}_1 = \{ a\in {\raise.5ex\hbox{\ensuremath{\wp}}}_k(X)\colon i\in a \}$ and ${\cal X}_2 = \{ a\in{\raise.5ex\hbox{\ensuremath{\wp}}}_k(x)\colon i \notin a\} = {\raise.5ex\hbox{\ensuremath{\wp}}}_k(X\setminus\{i\})$. The following is easily seen:
${\cal X}_2$ is a hyperplane of $\goth K$, ${\goth K}_2 := {\goth K}\restriction{{\cal X}_2} = \GrasSpace(X\setminus\{i\},k)$
${\goth K}_1 = {\goth K}\setminus{\cal X}_2$, with the point-set ${\cal X}_1$, is isomorphic under the map ${\cal X}_1\ni a \longmapsto a\setminus\{ i \}\in{\raise.5ex\hbox{\ensuremath{\wp}}}_{k-1}(X\setminus\{ i \})$ to the structure $\GrasSpace(X\setminus\{i\},k-1)$.
\[jawne1\] Let $A$ be a line of ${\goth K}_1$, so $A\in{\raise.5ex\hbox{\ensuremath{\wp}}}_{k+1}(X)$ where $i\in A$. Then $A \setminus \{ i \}\in {\raise.5ex\hbox{\ensuremath{\wp}}}_k(A)\cap {\cal X}_2$, so $A^\infty = A\setminus\{i\}$.
In view of the above and \[thm:decompo0\] we get that
If $i \in X$ is arbitrary then $$\GrasSpace(X,k) \cong \GrasSpace(X\setminus\{i\},k-1)\rtimes_\infty\GrasSpace(X\setminus\{i\},k)$$ with $\infty$ defined by above.
In numerical symbols we can write: $$\GrasSpace(n,k) = \GrasSpace(n-1,k-1)\rtimes_\infty\GrasSpace(n-1,k).$$ This decomposition was studied in many details in [@gevay], it was also noticed in [@perspect Representation 2.12]. While expressed in terms of ${\goth G}(k,m)$ it assumes the form $${\goth G}(k_0,m_0) = {\goth G}(k_0,m_0-1)\rtimes_\infty{\goth G}(k_0-1,m_0),$$ where $k_0 = n-k$, $m_0 = k+1$.
Example: the family of combinatorial Veronesians {#exm:very}
------------------------------------------------
Let $X$ be an $m$-element set; we write ${\mbox{\large$\goth y$}}_k(X)$ for the $k$-element multisets with the elements in $X$. In naive words, a multiset is a ‘set’ whose elements belong to $X$, and each one of them can occur several times. Formally, it is a function $f$ defined on $X$ with values in the set of natural numbers (with zero); this function ‘counts’ how many times given item from $X$ occurs in $f$. It is a convenient way to symbolize such a function $f$ in the form $f = \prod_{x\in X} x^{f(x)}$ (with the natural relations like $x^ix^j = x^{i+j}$, $x^i y^j = y^j x^i$, $x^0 = 1$, $1 x = x$, etc...). Then the cardinality of $f$ is $|f| = \sum_{x\in X}f(x)$. We write $\suport(f) = \{ x\in X\colon f(x) > 0 \}$; clearly, $|f| = \sum_{x\in\suport(f)} f(x)$
Let us write $\bigcup_{i=0}^{i=k-1} {\mbox{\large$\goth y$}}_i(X) =: {\mbox{\large$\goth y$}}_{<k}(X)$. On the set ${\mbox{\large$\goth y$}}_k(X) \times {\mbox{\large$\goth y$}}_{<k}(X)$ we define the incidence relation $\inc$ by the formula: $$\label{def:inc:ver0}
e \inc f :\iff f = e\, x^{k-|e|} \text{ for some } x\in X.$$ The structure
$\VerSpace(m,k) = {{\ensuremath{\langle {\mbox{\large$\goth y$}}_k(X),{\mbox{\large$\goth y$}}_{<k}(X),\inc \rangle}}}$
is called a [*combinatorial Veronesian*]{}; the class of combinatorial Veronesians was introduced in [@combver]. It was proved that $\VerSpace(m,k)$ is a partial linear space with the points of rank $k$ and the lines of size $m$; the formulas counting the cardinality of ${\mbox{\large$\goth y$}}_k(X)$ and of ${\mbox{\large$\goth y$}}_{<k}(X)$ are known in the elementary combinatorics; summing up we get that $\VerSpace(m,k)\in\ginconfx(k,m)$.
Let us fix $a\in X$ and define ${\cal X}_2 = \{f\in{\mbox{\large$\goth y$}}_k(X)\colon a\in\suport(f)\}$ and ${\cal X}_1 = \{f\in{\mbox{\large$\goth y$}}_k(X)\colon a\notin\suport(f)\}$; then ${\mbox{\large$\goth y$}}_k(X)$ is the disjoint union ${\cal X}_1 \cup {\cal X}_2$.
It is seen that the map ${\mbox{\large$\goth y$}}_{k-1}(X)\ni f\longmapsto f\,a^1\in {\cal X}_2$ is a bijection. Suppose that $f' a^1, f'' a^1 \inc e$ where $e\in{\mbox{\large$\goth y$}}_{<k}(X)$. Then $a\in\suport(e)$ and $f',f''\inc \frac{e}{a}\in{\mbox{\large$\goth y$}}_{<k-1}(X)$. Finally, $a\in\suport(f)$ for every $f$ with $f\inc e$, which yields that ${\cal X}_2$ is a subspace of $\VerSpace(X,k)$; as we noted, it is isomorphic to $\VerSpace(X,k-1)$.
Let $e\in{\mbox{\large$\goth y$}}_{<k}(X)$ be a line of $\VerSpace(m,k)$. If $a \in\suport(e)$ then $f\in{\cal X}_2$ for every $f$ with $f\inc e$. If $a\notin\suport(e)$ then $e^\infty = e\,a^{k-|e|}$ is the unique element incident with $e$ which belongs to ${\cal X}_2$.
\[jawne2\] Evidently, the points in ${\cal X}_1$ can be considered as the points of $\VerSpace(X\setminus\{a\},k)$. Let $e\in{\mbox{\large$\goth y$}}_{<k}(X\setminus\{ a \})$ be a line of $\VerSpace(X\setminus\{a\},k)$; then $e^\infty = e\,a^{k-|e|}\inc e$ is well defined.
In particular, the above yields that ${\cal X}_2$ is a hyperplane of $\VerSpace(m,k)$.
Summing up, we obtain
Let $a\in X$ be arbitrary. $$\VerSpace(X,k) = \VerSpace(X\setminus\{ a\},k) \rtimes_\infty \VerSpace(X,k-1),$$ where $\infty$ is defined by above.
In (numerical) symbols we can express this fact by $$\VerSpace(m,k) = \VerSpace(m-1,k)\rtimes_\infty \VerSpace(m,k-1).$$
As a consequence of [@combver Cor. 4.8, Thm. 4.5], $\VerSpace(m,k)$ is a combinatorial Grassmannian only for $k=2$ or $m=2$ so, Grassmannians and Veronesians are essentially distinct families.
Example: the family of dual combinatorial Veronesians {#exm:duvery}
-----------------------------------------------------
In Subsections \[exm:grasy\] and \[exm:very\], we have found decompositions of the scheme $\ginconfx(k,m) = \ginconfx(k,m-1)\rtimes\ginconfx(k-1,m)$. Clearly, $\ginconfx(m,k)$ are dual to $\ginconfx(k,m)$; therefore, in view of \[cor:dual-hipy\] one can expect that each of these decompositions determines a decomposition of the scheme
$\ginconfx(m,k) =\dual{\ginconfx(k,m)} =
\dual{\ginconfx(k-1,m)} \rtimes \dual{\ginconfx(k,m-1)} =
\ginconfx(m,k-1)\rtimes \ginconfx(m-1,k)$
In case of combinatorial Grassmannians the dualization procedure does not yield any new family of configurations:
Let $n = |X|$ for a set $X$. Then $\dual{\GrasSpace(X,k)} \cong \GrasSpace(X,n-k)$.
However, the dual Veronesians yield another, third family: if $\dual{\VerSpace(m,k)}$ is (isomorphic to) a combinatorial Grassmannian then either $k=2$ or $m=2$; if it is isomorphic to a combinatorial Veronesian then $k=2$, or $m=2$, or $k=3=m$. Even $\dual{\VerSpace(k,k)} \cong \VerSpace(k,k)$ is not valid for $k > 3$ (see [@combver Thm.’s 4.14, 4.15])!
Let us adopt notation of Subsection \[exm:very\] and let ${\goth K} = {{\ensuremath{\langle U,\lines \rangle}}} = \VerSpace(X,k)$; let us remind that ${\cal X}_2 = \{ f\in{\mbox{\large$\goth y$}}_k(X)\colon a \in \suport(f) \}$ is a hyperplane of $\goth K$ and then $\lines[{{\cal X}_2}] = \{ e\in {\mbox{\large$\goth y$}}_{<k}(X)\colon a \in\suport(e) \} =:\lines_2$. Consequently, $\lines_1 := \lines\setminus\lines_2 = {\mbox{\large$\goth y$}}_{<k}(X\setminus\{ a\})$ is a hyperplane of $\dual{\goth K}$; set ${\cal X}_1 := U\setminus{\cal X}_2 = {\mbox{\large$\goth y$}}_k(X\setminus\{ a \})$. Consider a line $f\in{\mbox{\large$\goth y$}}_{k}(X)$ of ${{\ensuremath{\langle \lines_2,{\cal X}_2 \rangle}}}$; then $a \in \suport(f)$: let $dg(a,f)$ be the greatest integer $s$ such that $f=a^s g$ for a multiset $g$. We associate with such an $f$ the point $f^\infty = \frac{f}{a^{dg(a,f)}}\in\lines_1$, it is seen that we obtain
$\dual{\VerSpace(m,k)} =
{{\ensuremath{\langle \lines_2,{\cal X}_2,\inc^{-1} \rangle}}} \rtimes_{\infty}{{\ensuremath{\langle \lines_1,{\cal X}_1,\inc^{-1} \rangle}}}
\cong \dual{\VerSpace(m,k-1)} \rtimes_{\infty} \dual{\VerSpace(m-1,k)}$.
With the symbols $\VerSpacex(m,k) = \dual{\VerSpace(m,k)}\in\ginconfx(m,k)$ we arrive to $$\VerSpacex(m,k) = \VerSpacex(m,k-1) \rtimes_\infty \VerSpacex(m-1,k)$$ Consequently, following \[cor:dual-hipy\] we can explicitly characterize the Pascal Triangle of Configurations consisting of dual combinatorial Veronesians.
Comments and problems
=====================
We have shown three families $\mathscr K$ of configurations ${\goth K}(m,k)\colon m,k=1, 2\ldots$ such that the formula ${\goth K}(m,k) = {\goth K}(m,k-1)\rtimes_{\infty_{m,k}}{\goth K}(m-1,k)$ is valid for all $m,k$ and suitable maps $\infty_{m,k}$. One can expect that there are more such families: the point is to find a suitable family
$\big[\infty_{m,k}\colon{\raise.5ex\hbox{\ensuremath{\wp}}}_{m-1}(m+k-2)\longrightarrow{\raise.5ex\hbox{\ensuremath{\wp}}}_{k-1}(m+k-2)\colon m,k=1,2,\ldots \big]$
It is seen how huge variety of binomial partial triple systems can be obtained via ‘completing’ complete graphs (see [@hypinbin:psts]): one can expect that our procedure produces much more required configurations (cf. Problem \[prob:rozmnoz\]).
However, one essential question appears: which of them can be realized in a Desarguesian projective space: we call them [*projective*]{} then. It is known that all the combinatorial Grassmannians are projective. It is also known that (practically all) combinatorial Veronesians are not projective (only $\VerSpace(3,3)$ and $\VerSpace(2,k)$, $\VerSpace(m,2)$ are realizable). Similarly, dual of combinatorial Veronesians are also not projective (besides the exceptions indicated before), [@combver Thm.’s 6.9, 6.10].
The statement like [*if ${\goth K}_1$ and ${\goth K}_2$ are realizable then ${\goth K}_1\rtimes{\goth K}_2$, if it is a (binomial) configuration then is realizable as well*]{} is false, in general. It suffices to present $\VerSpace(4,3)$ as the “sum” of projectively realizable structures $\VerSpace(3,3)$ and $\VerSpace(4,2)$. So, a natural question arises
\[prob:rozmnoz\] Assume that ${\goth K}_1$ and ${\goth K}_2$ are projective (binomial) configurations which satisfy corresponding ‘recursive equation’ $${\goth K}_1\in\ginconfx(k,m-1) \text{ and } {\goth K}_2\in\ginconfx(k-1,m)
\text{ for some } k,m\geq 2.$$ Then there is a bijection $\infty\colon\text{ lines of }{\goth K}_1\longrightarrow \text{ points of }{\goth K}_2$ so as ${\goth K}_1 \rtimes_\infty{\goth K}_2 \in\ginconfx(k,m)$. This observation enables us to construct ‘Pascal Triangle of Configurations’ from, practically, arbitrary boundary sequences of configurations, considering arbitrary $\infty$’s.
For which maps $\infty$ (is there necessarily at least one) the structure ${\goth K}_1\rtimes_\infty{\goth K}_2$ is projective?
Note that “boundary” sequences $\ginconfx(2,k)$ and $\ginconfx(k,2)$ are known: $\ginconfx(2,k) = \{ \dual{K_{k+1}} \}$ and $\ginconfx(k,2) = \{ K_{k+1} \}$, and these two sequences consist of projective configurations.
So, considering configurations decomposed with the following schemes
$\ginconfx(3,k) = \ginconfx(3,k-1)\rtimes \ginconfx(2,k) =
\ginconfx(3,k-1)\rtimes \dual{K_{k+1}}$, $\ginconfx(k,3) = \ginconfx(k,2)\rtimes \ginconfx(k-1,3) =
K_{k+1} \rtimes \ginconfx(k-1,3)$.
the real problem lies in the classification/choice of bijections $\infty$!
In particular, there are known binomial partial Steiner triple systems not in the families $\VerSpace(?,?)$ nor among $\GrasSpace(?,?)$, and nor among $\dual{\VerSpace(?,?)}$ which are projective, for example, so called quasi-Grassmannians of [@skewgras]. Each such structure $\vergras_n$ has parameters as the corresponding $\GrasSpace(n,2)$. So, there arises a very particular, but intriguing
Is there a map $\infty$ such that the structure $\vergras_{n-1}\rtimes_\infty\GrasSpace(n-1,3)$ (which has the parameters of $\GrasSpace(n,3)$) is realizable in a Desarguesian projective space.
Addendum {#addendum .unnumbered}
========
The paper is a result of discussions during Combinatorics 2018 in Arco.
[9]{}-2pt (1987) [*Affine-projective relationship: applications*]{}, \[In:\] Berger M. (eds) Geometry I, Universitext. Springer, Berlin, Heidelberg (1987), Chapter 5, 111–121. , [Geom. Dedicata 35]{} (1990), 43–76. , [*Self-dual configurations and regular graphs*]{}, Bull. Amer. Math. Soc. [**56**]{}(1950), 413–455. , [*Pascal’s triangle of configurations*]{}, \[in:\] [*Marston D. E. Conder, Antoine Deza, Asia Ivi[ć]{} Weiss*]{}(ed’s), [*Discrete Geometry and Symmetry*]{}, GSC 2015. Springer Proceedings in Mathematics & Statistics, vol 234. Springer, Cham (2018), 181–199, DOI:10.1007/978-3-319-78434-2\_10 , [*Angewandte Algebra f[ü]{}r Mathematiker und Informatiker*]{}, VEB Deutcher Verlag der Wissenschaften, Berlin 1988. , [*Geometrische Konfigurationen*]{}, Leipzig, S. Hirzel (1929). , [*Affinization of Segre products of partial linear spaces*]{}, Bull. Iranian Math. Soc. 43(2017), no. 5, 1101–1126. , [*Binomial partial Steiner triple systems with complete graphs: structural problems*]{}, 2015, arXiv:1508.05974 , [*${10}_{3}$-configurations and projective realizability of multiplied configurations*]{}, Des. Codes Cryptogr. [**51**]{}, no. 1 (2009), 45–54. , [*Multiple perspectives and generalizations of the Desargues configuration*]{}, Demonstratio Math. [**39**]{} (2006), no. 4, [887–906]{}. , [*Combinatorial Veronese structures, their geometry, and problems of embeddability*]{}, Results Math. [**51**]{} (2008), 275–308. , [*On some regular multi-veblen configurations, the geometry of combinatorial quasi Grassmannians*]{}, Demonstratio Math. [**42**]{} (2009), no.2 387–402. , [*From Cayley-Dickson Algebras to Combinatorial Grassmannians*]{}, Mathematics 2015 3(4), 1192–1221, DOI:10.3390/math3041192 , [*On $n$-ary equivalence relations in algebra and their applications to geometry*]{}, Warsaw, Dissertationes PAS (1981) , [*Projective Geometry*]{}, University of Michigan Library (January 1, 1910)
Authors’ address:\
Krzysztof Pra[ż]{}mowski,\
Institute of Mathematics, University of Bia[ł]{}ystok\
K. Cio[ł]{}kowskiego 1M, 15-245 Bia[ł]{}ystok, Poland\
e-mail: `krzypraz@math.uwb.edu.pl`,
|
---
abstract: 'We present SUSY\_LATTICE - a C++ program that can be used to simulate certain classes of supersymmetric Yang–Mills (SYM) theories, including the well known ${\cal N} = 4$ SYM in four dimensions, on a flat Euclidean space-time lattice. Discretization of SYM theories is an old problem in lattice field theory. It has resisted solution until recently when new ideas drawn from orbifold constructions and topological field theories have been brought to bear on the question. The result has been the creation of a new class of lattice gauge theories in which the lattice action is invariant under one or more supersymmetries. The resultant theories are local, free of doublers and also possess exact gauge-invariance. In principle they form the basis for a truly non-perturbative definition of the continuum SYM theories. In the continuum limit they reproduce versions of the SYM theories formulated in terms of [*twisted*]{} fields, which on a flat space-time is just a change of the field variables. In this paper, we briefly review these ideas and then go on to provide the details of the C++ code. We sketch the design of the code, with particular emphasis being placed on SYM theories with ${\cal N} = (2, 2)$ in two dimensions and ${\cal N} = 4$ in three and four dimensions, making one-to-one comparisons between the essential components of the SYM theories and their corresponding counterparts appearing in the simulation code. The code may be used to compute several quantities associated with the SYM theories such as the Polyakov loop, mean energy, and the width of the scalar eigenvalue distributions.'
address:
- 'Department of Physics, Syracuse University, Syracuse, NY 13244, USA'
- |
Department of Physics, Syracuse University, Syracuse, NY 13244, USA\
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA[^1]
author:
- Simon Catterall
- Anosh Joseph
title: 'An object oriented code for simulating supersymmetric Yang–Mills theories'
---
Lattice Gauge Theory ,Supersymmetric Yang–Mills ,Rational Hybrid Monte Carlo ,Object Oriented Programming
11.15.Ha ,12.60.Jv ,12.10.-g ,12.15.-y ,87.55.kd ,87.55.kh
Introduction {#sec:intro}
============
The problem of formulating supersymmetric theories on lattices has a long history going back to the earliest days of lattice gauge theory. However, after initial efforts failed to produce useful supersymmetric lattice actions the topic languished for many years. Indeed a folklore developed that supersymmetry and the lattice were mutually incompatible. However, recently, the problem has been re-examined using new tools and ideas such as topological twisting [@Sugino:2003yb; @Sugino:2004qd; @Sugino:2004uv; @Catterall:2004np; @Catterall:2005fd; @D'Adda:2005zk; @Catterall:2005eh; @Sugino:2006uf; @Catterall:2006jw; @Catterall:2006is; @D'Adda:2007ax; @Catterall:2007kn; @Catterall:2007hk; @Catterall:2008dv; @Catterall:2009it], orbifold projection and deconstruction [@Kaplan:2002wv; @Nishimura:2003tf; @Cohen:2003xe; @Cohen:2003qw; @Kaplan:2005ta; @Unsal:2006qp; @Damgaard:2007be; @Damgaard:2007xi; @Matsuura:2007ec; @Damgaard:2008pa], and a class of lattice models have been constructed which maintain one or more supersymmetries exactly at non-zero lattice spacing[^2].
The availability of a supersymmetric lattice construction is clearly very exciting. For example, having a lattice construction of the well known ${{\cal N}}=4$ SYM in four-dimensions is very advantageous from the point of view of exploring the connection between gauge theories and string/gravitational theories. But even without this connection to string theory, it is clearly of great importance to be able to give a non-perturbative formulation of a supersymmetric theory via a lattice path integral, in the same way that one can formally define QCD as a limit of lattice QCD as the lattice spacing goes to zero and the box size to infinity. From a practical point of view, one can also hope that some of the technology of lattice field theory such as strong coupling expansions and Monte Carlo simulation can be brought to bear on such supersymmetric theories.
In this paper, we will briefly describe the key ingredients that go into the lattice constructions of a variety of SYM theories and then provide the details of C++ code that can be used to simulate these theories. In Section \[sec:method-twist-SYM\] we provide the general method of twisting the supersymmetries of certain classes of SYM theories to provide twisted SYM theories that are compatible with discretization on the lattice. We start with the twisted ${{\cal N}}=2$ SYM in two dimensions as a warm up example and after writing down the discretization of this theory we go on to describe the twisted versions of ${{\cal N}}=4$ SYM in three dimensions and ${{\cal N}}=4$ SYM in four dimensions. In Section \[sec:simulating-sym-alg\] we describe the algorithms used to simulate these resultant lattice theories: rational hybrid Monte Carlo (RHMC) algorithm to compute the fermion determinant with fractional power, leapfrog algorithm to evolve the system of equations and then a Metropolis test to accept or reject the configurations. In Section \[sec:overall-structure\] we provide the overall structure of the C++ code and describe how the code advances by generating new configurations using RHMC algorithm, saves the field configurations after some number of Monte Carlo sweeps and measures the observables in the theory. We provide some simulation results in Section \[sec:sim-results\], specific to the ${{\cal N}}=2$ SYM in two-dimensions. We compute the eigenvalues of the scalars of the theory, study the Pfaffian phases and the presence of fermionic sign problem in the theory and also investigate the restoration of supersymmetry by checking if the scalar supersymmetry has indeed been implemented correctly in our C++ code. We provide some conclusions and outlook in Section \[sec:conclu\]. We also provide three appendices: \[sec:install\] details the installation of the program, \[sec:file-list\] lists the files included in SUSY\_LATTICE library with a brief description of their purpose and in \[sec:input-param\] we provide a sample file with input parameters.
We hope that this work will motivate elementary particle physicists as well as high energy computational physicists to pursue numerical studies of supersymmetric lattice theories in particular, the ${{\cal N}}=4$ Yang–Mills in four dimensions.
The method of topological twisting in SYM theories {#sec:method-twist-SYM}
==================================================
First, let us explain why discretization of supersymmetric theories resisted solution for so long. The central problem is that naive discretizations of continuum supersymmetric theories break supersymmetry completely and radiative effects lead to a profusion of relevant supersymmetry breaking counterterms in the renormalized lattice action. The coefficients to these counterterms must then be carefully fine tuned as the lattice spacing is sent to zero in order to arrive at a supersymmetric theory in the continuum limit. In most cases this is both unnatural and practically impossible – particularly if the theory contains scalar fields. Of course, one might have expected problems – the supersymmetry algebra is an extension of the Poincaré algebra, which is explicitly broken on the lattice. Specifically, there are no infinitesimal translation generators on a discrete space-time so that the algebra $\{Q,\overline{Q}\}=\gamma_a p_a$, where $a$ is the space-time index, is already broken at the classical level. Equivalently, it is a straightforward exercise to show that a naive supersymmetry variation of a naively discretized supersymmetric theory fails to yield zero as a consequence of the failure of the Leibniz rule when applied to lattice difference operators. In the last five years or so this problem has been revisited using new theoretical tools and ideas and a set of lattice models have been constructed which retain exactly some of the continuum supersymmetry at non-zero lattice spacing. The basic idea is to maintain a particular sub-algebra of the full supersymmetry algebra in the lattice theory. The hope is that this exact symmetry will constrain the effective lattice action and protect the theory from dangerous supersymmetry violating counterterms.
Two approaches have been pursued to produce such supersymmetric actions: one based on ideas drawn from the field of topological field theory [@Sugino:2003yb; @Catterall:2004np; @Catterall:2005fd] and another pioneered by David B. Kaplan. Mithat Ünsal and collaborators using ideas of orbifolding and deconstruction [@Cohen:2003xe; @Cohen:2003qw; @Kaplan:2005ta]. Remarkably, these two seemingly independent approaches lead to the same lattice theories – see [@Catterall:2007kn; @Unsal:2006qp; @Damgaard:2007be; @Damgaard:2007eh] and the recent reviews [@Catterall:2009it; @Giedt:2009yd; @Joseph:2011xy]. This convergence of two seemingly completely different approaches to the problem leads one to suspect that the final lattice theories may represent essentially unique solutions to the simultaneous requirements of locality, gauge-invariance and at least one exact supersymmetry. In this paper, we will use the language of topological twisting to discuss these supersymmetric lattice constructions, but the reader should remember that the orbifold methods lead to the same lattice theories.
Twisting the supersymmetries in $d$ dimensions
----------------------------------------------
The basic idea of twisting goes back to Witten in his seminal paper on topological field theory [@Witten:1988ze] but actually had been anticipated in earlier work on staggered fermions [@Elitzur:1982vh]. In our context the idea is decompose the fields of the theory in terms of representations not of the original (Euclidean) rotational symmetry $SO_{\rm rot}(d)$ but a twisted rotational symmetry $SO(d)^\prime$, which is the diagonal subgroup of this symmetry and an $SO_{\rm R}(d)$ subgroup of the R-symmetry of the theory, SO(d)\^=[diag]{}(SO\_[Lorentz]{}(d)SO\_[R]{}(d)) . To be explicit, consider the case where the total number of supersymmetries is $Q=2^d$. In this case we can treat the supercharges of the twisted theory as a $2^{d/2}\times 2^{d/2}$ matrix $q$. This matrix can be expanded on products of gamma matrices: q = [[Q]{}]{}I + [[Q]{}]{}\_a \_a + [[Q]{}]{}\_[ab]{}\_a\_b + …The $2^d$ antisymmetric tensor components that arise in this basis are the twisted supercharges and satisfy a corresponding supersymmetry algebra following from the original algebra [[Q]{}]{}\^2 &=& 0 ,\
{[[Q]{}]{},[[Q]{}]{}\_a} &=& p\_a ,\
&& The presence of the nilpotent scalar supercharge ${{\cal Q}}$ is most important: it is the algebra of this charge that we can hope to translate to the lattice. The second piece of the algebra expresses the fact that the momentum is the ${{\cal Q}}$-variation of something, which makes plausible the statement that the energy-momentum tensor and hence the entire action can be written in ${{\cal Q}}$-exact form[^3]. Notice that an action written in such a ${{\cal Q}}$-exact form is trivially invariant under the scalar supersymmetry, provided the latter remains nilpotent under discretization.
The rewriting of the supercharges in terms of twisted variables can be repeated for the fermions of the theory and yields a set of antisymmetric tensors $(\eta, \psi_a, \chi_{ab}, \ldots)$, which for the case of $Q=2^d$ matches the number of components of a real [[Kähler-Dirac ]{}]{}field. This repackaging of the fermions of the theory into a [[Kähler-Dirac ]{}]{}field is at the heart of how the discrete theory avoids fermion doubling as was shown by Becher, Joos and Rabin in the early days of lattice gauge theory [@Rabin:1981qj; @Becher:1982ud].
It is important to recognize that the transformation to twisted variables corresponds to a simple change of variables in flat space – one more suitable to discretization. A true topological field theory only results when the scalar charge is treated as a true BRST charge and attention is restricted to states annihilated by this charge. In the language of the supersymmetric parent theory such a restriction corresponds to a projection to the vacua of the theory. It is [*not*]{} employed in the lattice constructions we discuss in this paper.
A warm up example: Twisted ${{\cal N}}=2$ SYM in two dimensions
---------------------------------------------------------------
We look at the twisted ${{\cal N}}=2$ SYM in two dimensions as a warm up example. This theory satisfies our requirements for supersymmetric latticization: its R-symmetry possesses an $SO(2)$ subgroup corresponding to rotations of the two degenerate Majorana fermions into each other. Explicitly the theory can be written in twisted form as S = [[Q]{}]{}d\^2x (\_[ab]{} [[F]{}]{}\_[ab]{} + - d ) . \[2daction\] The degrees of freedom are just the twisted fermions $(\eta, \psi_a, \chi_{ab})$ previously described and a complex gauge field ${{\cal A}}_a$. The latter is built from the usual gauge field and the two scalars present in the untwisted theory ${{\cal A}}_a = A_a + i B_a$ with corresponding complexified field strength ${{\cal F}}_{ab}$.
The complex covariant derivatives appearing in these expressions are defined by [[D]{}]{}\_a &=& \_a + [[A]{}]{}\_a = \_a + A\_a + iB\_a ,\
\_a &=& \_a + \_a = \_a + A\_a- i B\_a . All fields take values in the adjoint representation of $U(N)$[^4]. It should be noted that despite the appearance of a complexified connection and field strength, the theory possesses only the usual $U(N)$ gauge-invariance corresponding to the real part of the gauge field.
Notice that the original scalar fields transform as vectors under the original R-symmetry and hence become vectors under the twisted rotation group while the gauge fields are singlets under the R-symmetry and so remain vectors under twisted rotations. This structure makes possible the appearance of a complex gauge field in the twisted theory.
The nilpotent transformations associated with ${{\cal Q}}$ are given explicitly by [[Q]{}]{} [[A]{}]{}\_a &=& \_a\
[[Q]{}]{} \_a &=& 0\
[[Q]{}]{} \_a &=& 0\
[[Q]{}]{} \_[ab]{} &=& -\_[ab]{}\
[[Q]{}]{} &=& d\
[[Q]{}]{} d&=&0
Performing the ${{\cal Q}}$-variation and integrating out the auxiliary field $d$ yields S = d\^2x (-\_[ab]{}[[F]{}]{}\_[ab]{} + \[ \_a, [[D]{}]{}\_a\]\^2 - \_[ab]{}[[D]{}]{}\_ - \_a\_a) . \[twist\_action\]
To untwist the theory and verify that indeed in flat space it just corresponds to the usual theory one can do a further integration by parts to produce S = d\^2x (-F\^2\_[ab]{} + 2B\_a D\_b D\_b B\_a - \[B\_a,B\_b\]\^2 + L\_F ) , where $F_{ab}$ is the usual Yang–Mills term. It is now clear that the imaginary parts of the gauge fields $B_a$ can now be given an interpretation as the scalar fields of the usual formulation. Similarly one can build spinors out of the twisted fermions and write the action in the manifestly Dirac form L\_F = (
[cc]{}\_[12]{}&
) (
[cc]{}-D\_2-iB\_2&D\_1+iB\_1\
D\_1-iB\_1&D\_2-iB\_2
) (
[c]{}\_1\
\_2
) .
Discretization of the twisted ${{\cal N}}=2$, $d=2$ theory
----------------------------------------------------------
{width="50.00000%"} \[fig:2dlatt\]
The twisted theory described in the previous section may be discretized using the techniques developed in [@Catterall:2007kn; @Damgaard:2007be; @Damgaard:2008pa]. (Complex) gauge fields are represented as complexified Wilson gauge links ${{\cal U}}_a({ {\bf n} })=e^{{{\cal A}}_a({ {\bf n} })}$ living on links $({ {\bf n} }, { {\bf n} }+{\widehat{\boldsymbol {\mu}}}_a)$ of a lattice, which for the moment we can think of as hypercubic. These transform in the usual way under $U(N)$ lattice gauge transformations [[U]{}]{}\_a([ [**n**]{} ]{})G([ [**n**]{} ]{})[[U]{}]{}\_a([ [**n**]{} ]{})G\^([ [**n**]{} ]{}+ \_a) . Supersymmetric invariance then implies that $\psi_a({ {\bf n} })$ live on the same links as ${{\cal U}}_a({ {\bf n} })$ and transform identically. The scalar fermion $\eta({ {\bf n} })$ is clearly most naturally associated with a site and transforms accordingly ([ [**n**]{} ]{})G([ [**n**]{} ]{})([ [**n**]{} ]{})G\^([ [**n**]{} ]{}) . The field $\chi_{ab}({ {\bf n} })$ is slightly more difficult. Naturally as a 2-form it should be associated with a plaquette. In practice we introduce diagonal links running through the center of the plaquette and choose $\chi_{ab}({ {\bf n} })$ to lie [*with opposite orientation*]{} along those diagonal links. This choice of orientation will be necessary to ensure gauge-invariance. Fig. 1 shows the unit cell of the resultant lattice theory.
To complete the discretization we need to describe how continuum derivatives are to be replaced by difference operators. A natural technology for accomplishing this in the case of adjoint fields was developed many years ago and yields expressions for the derivative operator applied to arbitrary lattice p-forms [@Aratyn:1984bd]. In the case discussed here we need just two derivatives given by the expressions [[D]{}]{}\^[(+)]{}\_a f\_b([ [**n**]{} ]{}) &=& [[U]{}]{}\_a([ [**n**]{} ]{})f\_b([ [**n**]{} ]{}+\_a)-f\_b([ [**n**]{} ]{})[[U]{}]{}\_a([ [**n**]{} ]{}+\_b) ,\
\^[(-)]{}\_a f\_a([ [**n**]{} ]{}) &=& f\_a([ [**n**]{} ]{})\_a([ [**n**]{} ]{})-\_a([ [**n**]{} ]{}-\_a)f\_a([ [**n**]{} ]{}- \_a) . The lattice field strength is then given by the gauged forward difference ${{\cal F}}_{ab}({ {\bf n} }) = {{\cal D}}^{(+)}_a {{\cal U}}_b({ {\bf n} })$ and is automatically antisymmetric in its indices. Furthermore, it transforms like a lattice 2-form and yields a gauge-invariant loop on the lattice when contracted with $\chi_{ab}({ {\bf n} })$. Similarly the covariant backward difference appearing in ${{\overline{\cal D}}}^{(-)}_a {{\cal U}}_a({ {\bf n} })$ transforms as a 0-form or site field and hence can be contracted with the site field $\eta({ {\bf n} })$ to yield a gauge-invariant expression.
This use of forward and backward difference operators guarantees that the solutions of the theory map one-to-one with the solutions of the continuum theory and hence fermion doubling problems are evaded [@Rabin:1981qj]. Indeed, by introducing a lattice with half the lattice spacing one can map this [[Kähler-Dirac ]{}]{}fermion action into the action for staggered fermions [@Banks:1982iq]. Notice that, unlike the case of QCD, there is no rooting problem in this supersymmetric construction since the additional fermion degeneracy is already required by the continuum theory. The number of fermions of the twisted theory exactly fills out the components of a [[Kähler-Dirac ]{}]{}field and corresponds to the taste degeneracy of (reduced) staggered fermions.
Twisted ${{\cal N}}=4$ SYM in three dimensions
----------------------------------------------
The twist of ${{\cal N}}=4$ SYM in three dimensions[^5] can be most succinctly written in the form where S &=&Q d\^3x (\_[ab]{}[[F]{}]{}\_[ab]{} + + d + B\_[abc]{}\_c \_[ab]{}) . \[action1\] The fermions comprise a multiplet of p-form fields $(\eta, \psi_a, \chi_{ab}, \theta_{abc})$[^6] where in three dimensions $p=0, \cdots, 3$. This multiplet of twisted fermions corresponds to a single [[Kähler-Dirac ]{}]{}field and here possesses eight single component fields as expected for a theory with ${{\cal N}}=4$ supersymmetry in three dimensions.
The imaginary parts of the complex gauge field ${{\cal A}}_a$ with $a=1, 2, 3$ appearing in this construction yield the three scalar fields of the conventional (untwisted) theory. Fields $d$ and $B_{abc}$ are auxiliaries introduced to render the scalar nilpotent supersymmetry $Q$ nilpotent off-shell. The latter acts on the twisted fields as follows Q [[A]{}]{}\_a &=& \_a\
Q \_a &=& 0\
Q \_a &=& 0\
Q \_[ab]{} &=& \_[ab]{}\
Q &=& d\
Q d &=& 0\
Q B\_[abc]{} &=& \_[abc]{}\
Q \_[abc]{} &=& 0\[susy\] Notice that this construction differs slightly from the one discussed in [@Catterall:2010ng]. The fermion term involving a 3-form is here trivially rewritten as a $Q$-exact rather than $Q$-closed form.
{width="50.00000%"}
\[fig1\]
Doing the $Q$-variation, integrating out the field $d$ and using the Bianchi identity \_[abc]{}\_c \_[ab]{} = 0 , yields \[action\] S &=& d\^3 x (-\_[ab]{}[[F]{}]{}\_[ab]{} + \[ \_a, [[D]{}]{}\_a\]\^2 -\_[ab]{}[[D]{}]{}\_\
&&- \_a\_a- \_[abc]{}\_) . The terms appearing in the bosonic part of the action can then be written in the following form exposing the $B_a$ dependence explicitly \_[ab]{}[[F]{}]{}\_[ab]{} &=& (F\_[ab]{} - \[B\_a, B\_b\])(F\_[ab]{} - \[B\_a, B\_b\]) + (D\_) (D\_) ,\
\^2 &=& -2(D\_a B\_a)\^2 , where $F_{ab}$ and $D_a$ denote the usual field strength and covariant derivative depending on the real part of the connection ${{\cal A}}_a$.
Discretization of the three-dimensional ${{\cal N}}=4$ SYM theory
-----------------------------------------------------------------
The transition to the lattice from the continuum theory is similar to the case of the two-dimensional ${{\cal N}}=2$ SYM theory. We replace the continuum complex gauge field ${{\cal A}}_a(x)$ at every point by an appropriate complexified Wilson link ${{\cal U}}_a({ {\bf n} })=e^{{{\cal A}}_a({ {\bf n} })},~a = 1, 2, 3$. These lattice fields are taken to be associated with unit length vectors in the coordinate directions ${{\boldsymbol a}}$ in an three-dimensional hypercubic lattice. By supersymmetry the fermion fields $\psi_a({ {\bf n} }),~a = 1, 2, 3$ lie on the same oriented link as their bosonic superpartners running from ${ {\bf n} }\to { {\bf n} }+ {\widehat{\boldsymbol {\mu}}}_a$. In contrast the scalar fermion $\eta({ {\bf n} })$ is associated with the site ${ {\bf n} }$ of the lattice and the tensor fermions $\chi_{ab}({ {\bf n} }),~a < b = 1, 2, 3$ with a set of diagonal face links running from ${ {\bf n} }+ {\widehat{\boldsymbol {\mu}}}_a + {\widehat{\boldsymbol {\mu}}}_b \to { {\bf n} }$. The final 3-form field $\theta_{abc}({ {\bf n} })$ is then naturally placed on the body diagonal running from ${ {\bf n} }\to { {\bf n} }+ {\widehat{\boldsymbol {\mu}}}_a + {\widehat{\boldsymbol {\mu}}}_b + {\widehat{\boldsymbol {\mu}}}_c$. The unit cell and fermionic field orientations of the three-dimensional theory is given in Fig. 2. The construction then posits that all link fields transform as bi-fundamental fields under gauge transformations ([ [**n**]{} ]{}) && G([ [**n**]{} ]{}) ([ [**n**]{} ]{}) G\^([ [**n**]{} ]{})\
\_m([ [**n**]{} ]{}) && G([ [**n**]{} ]{}) \_m([ [**n**]{} ]{}) G\^([ [**n**]{} ]{}+ \_m)\
\_[mn]{}([ [**n**]{} ]{}) && G([ [**n**]{} ]{}+ \_m + \_n) \_[mn]{}([ [**n**]{} ]{}) G\^([ [**n**]{} ]{})\
\_[mnq]{}([ [**n**]{} ]{}) && G([ [**n**]{} ]{}) \_[mnq]{}([ [**n**]{} ]{}) G\^([ [**n**]{} ]{}+ \_m + \_n + \_q)\
[[U]{}]{}\_m([ [**n**]{} ]{}) && G([ [**n**]{} ]{}) [[U]{}]{}\_m([ [**n**]{} ]{}) G\^([ [**n**]{} ]{}+ \_m)\
\_m([ [**n**]{} ]{}) && G([ [**n**]{} ]{}+ \_m) \_m([ [**n**]{} ]{}) G\^([ [**n**]{} ]{}) \[eq:gaugetrans-3d\]
The action of the lattice theory resembles to its continuum cousin with the one modification that the continuum field ${{\cal A}}_a(x)$ is replaced with the Wilson link ${{\cal U}}_m({ {\bf n} })$ and the lattice field strength being defined as ${{\cal F}}_{mn}({ {\bf n} }) = {{\cal D}}_m^{(+)}{{\cal U}}_n({ {\bf n} })$. Thus the supersymmetric and gauge-invariant lattice action is S &=& [[Q]{}]{}\_[[ [**n**]{} ]{},m,n,q]{} (\_[mn]{}([ [**n**]{} ]{})[[F]{}]{}\_[mn]{}([ [**n**]{} ]{}) + ([ [**n**]{} ]{}) \_m\^[(-)]{}[[U]{}]{}\_m([ [**n**]{} ]{})\
&&+ ([ [**n**]{} ]{}) d([ [**n**]{} ]{}) + B\_[mnq]{}([ [**n**]{} ]{})\_q\^[(+)]{}\_[mn]{}([ [**n**]{} ]{})) .
The covariant difference operators appearing in these expressions are defined by [@Damgaard:2008pa] [[D]{}]{}\_m\^[(-)]{}f\_m([ [**n**]{} ]{}) &=& [[U]{}]{}\_m([ [**n**]{} ]{})f\_m([ [**n**]{} ]{}) - f\_m([ [**n**]{} ]{}- \_m)[[U]{}]{}\_m([ [**n**]{} ]{}- \_m)\
[[D]{}]{}\_m\^[(+)]{}f\_n([ [**n**]{} ]{}) &=& [[U]{}]{}\_m([ [**n**]{} ]{})f\_n([ [**n**]{} ]{}+ \_m) - f\_n([ [**n**]{} ]{})[[U]{}]{}\_m([ [**n**]{} ]{}+ \_n)\
\_m\^[(-)]{}f\_m([ [**n**]{} ]{}) &=& f\_m([ [**n**]{} ]{})\_m([ [**n**]{} ]{}) - \_m([ [**n**]{} ]{}-\_m)f\_m([ [**n**]{} ]{}-\_m)\
\_m\^[(+)]{}f\_[nq]{}([ [**n**]{} ]{}) &=& f\_[nq]{}([ [**n**]{} ]{}+\_m)\_m([ [**n**]{} ]{}) - \_m([ [**n**]{} ]{}+\_n+\_q)f\_[nq]{}([ [**n**]{} ]{}) These expressions are determined by the twin requirements that they reduce to the corresponding continuum results for the adjoint covariant derivative in the naive continuum limit ${{\cal U}}_m \rightarrow {\mathbb I}_N + {{\cal A}}_m$ and that they transform under gauge transformations like the corresponding lattice link field carrying the same indices. This allows the terms in the action to correspond to gauge-invariant closed loops on the lattice.
Upon following the prescription [@Damgaard:2008pa] for lattice covariant derivatives, we write down the lattice action in terms of the link fields ${{\cal U}}_m({ {\bf n} })$ and ${{\overline{\cal U}}}_m({ {\bf n} })$ \[eq:4action\] S &=& \_[[ [**n**]{} ]{},m,n,q]{} (-\_[mn]{}([ [**n**]{} ]{})[[F]{}]{}\_[mn]{}([ [**n**]{} ]{}) + (\_m\^[(-)]{}[[U]{}]{}\_m([ [**n**]{} ]{}))\^2\
&&-\_[mn]{}([ [**n**]{} ]{})[[D]{}]{}\_[\[m]{}\^[(+)]{}\_[n\]]{}([ [**n**]{} ]{}) - ([ [**n**]{} ]{}) \_m\^[(-)]{}\_m([ [**n**]{} ]{}) - \_[mnq]{}([ [**n**]{} ]{})\_[\[q]{}\^[(+)]{}\_[mn\]]{}([ [**n**]{} ]{})) .
The bosonic part of the action is S\_B &=& \_[[ [**n**]{} ]{},m,n]{}\
&=& \_[[ [**n**]{} ]{},m,n]{} , and the fermionic part S\_F &=& -\_[[ [**n**]{} ]{},m,n,q,r,e,f]{} {(\_[mq]{}\_[nr]{} - \_[mr]{}\_[nq]{})\
&&\
&&+ ([ [**n**]{} ]{})(\_m([ [**n**]{} ]{})\_m([ [**n**]{} ]{}) - \_m([ [**n**]{} ]{}- \_m)\_m([ [**n**]{} ]{}- \_m))\
&&+(\_[mr]{}\_[ne]{}\_[qf]{} + \_[qr]{}\_[me]{}\_[nf]{} + \_[nr]{}\_[qe]{}\_[mf]{})\
&&\_[ref]{}([ [**n**]{} ]{})(\_[re]{}([ [**n**]{} ]{}+\_f)\_f([ [**n**]{} ]{}) - \_f([ [**n**]{} ]{}+\_r+\_e)\_[re]{}([ [**n**]{} ]{}))} .
It is easy to see that each term in the lattice action forms a gauge-invariant loop on the lattice.
Twisted ${{\cal N}}=4$ SYM in four dimensions
---------------------------------------------
In four dimensions the constraint that the target theory possess 16 supercharges singles out a single theory for which this construction can be undertaken – the ${{\cal N}}=4$ SYM.
The continuum twist of ${{\cal N}}=4$ that is the starting point of the twisted lattice construction was first written down by Marcus in 1995 [@Marcus:1995mq] although it now plays an important role in the Geometric-Langlands program and is, hence, sometimes called the GL-twist [@Kapustin:2006pk]. This four-dimensional twisted theory is most compactly expressed as the dimensional reduction of a five-dimensional theory in which the ten (one gauge field and six scalars) bosonic fields are realized as the components of a complexified five-dimensional gauge field while the 16 twisted fermions naturally span one of the two [[Kähler-Dirac ]{}]{}fields needed in five dimensions. Remarkably, the action of this theory contains a ${{\cal Q}}$-exact piece of precisely the same form as the two dimensional theory given in (\[2daction\]) provided one extends the field labels to run now from one to five. In addition, the Marcus twist requires a new ${{\cal Q}}$-closed term, which was not possible in the two-dimensional theory. S\_[closed]{} = -\_[mnpqr]{}\_[qr]{}\_p\_[mn]{}\[closed\] . The supersymmetric invariance of this term then relies on the Bianchi identity \_[mnpqr]{}\_p\_[qr]{}=0 .
Discretization of the four-dimensional ${{\cal N}}=4$ SYM theory
----------------------------------------------------------------
In two and three dimensions we were able to accommodate the bosonic fields of the theory in a natural way by assigning them to the links of a hypercubic lattice. For the ${{\cal Q}}=16$ theory this is not possible; the theory can be parametrized in terms of five complex gauge fields in the continuum. We are thus motivated to search for a four-dimensional lattice with five basis vectors ${\widehat{\boldsymbol {\mu}}}_a$, $a=1,\cdots, 5$. One simple solution is to use a hypercubic lattice with an additional body diagonal \[eq:mu-vectores\] \_1 &=& (1, 0, 0, 0)\
\_2 &=& (0, 1, 0, 0)\
\_3 &=& (0, 0, 1, 0)\
\_4 &=& (0, 0, 0, 1)\
\_5 &=& (-1, -1, -1, -1)The field ${{\cal U}}_5$ is then placed on the body diagonal link. Actually, we will indeed utilize such a hypercubic lattice when building the C++ data structure needed to code the resulting theory. Notice that the basis vectors sum to zero, consistent with the use of such a linearly dependent basis.
However, it should also be clear that a more symmetrical choice is possible in which the five basis vectors are entirely equivalent and the lattice theory possesses a large point group symmetry $S^5$ corresponding to permutations of the set of basis vectors. Such a discrete structure exists in four dimensions: it is called the $A_4^*$ lattice. It is constructed from the set of five basis vectors ${\widehat{\boldsymbol e}}_a$ pointing from the center of a four-dimensional equilateral simplex out to its vertices together with their inverses $-{\widehat{\boldsymbol e}}_a$. It is the four-dimensional analog of the two-dimensional triangular lattice. A specific basis for the $A_4^*$ lattice is given in the form of five lattice vectors \_1 &=& (, , , )\
\_2 &=& (-, , , )\
\_3 &=& (0, -, , )\
\_4 &=& (0, 0, -, )\
\_5 &=& (0, 0, 0, -) The basis vectors satisfy the relations \_[m=1]{}\^[5]{} \_m = 0; \_m \_n = (\_[mn]{} - ); \_[m=1]{}\^[5]{}(\_m)\_(\_m)\_ = \_; , = 1,,4. Notice that $S^5$ is a subgroup of the twisted rotation symmetry group $SO(4)^\prime$. Furthermore, the lattice fields transform in reducible representations of this discrete group - for example, the vector ${{\cal U}}_a$ decomposes into a four component vector ${{\cal U}}_\mu$ and a scalar field $\phi=\sum_a {{\cal U}}_a$ under $S^5$ and hence also under $SO(4)^\prime$ in the continuum limit. Invariance of the lattice theory with respect to $S^5$ then guarantees that the lattice theory will inherit full invariance under twisted rotations as the lattice spacing is sent to zero.
Complexified Wilson gauge link variables ${{\cal U}}_a$ are then placed on these links together with their ${{\cal Q}}$-superpartners $\psi_a$. The ten twisted fermions $\chi_{ab}$ are associated with additional diagonal links ${\widehat{\boldsymbol e}}_a+{\widehat{\boldsymbol e}}_b$ with $a>b$ while a single fermion $\eta$ is placed at each lattice site.
We can connect the basis vectors of the hypercubic lattice and the $A_4^*$ lattice through a set of linear transformations - see [@Unsal:2006qp; @Catterall:2011pd]. The integer-valued hypercubic lattice site vector ${ {\bf n} }$ can be related to the physical location in space-time using the $A_4^*$ basis vectors ${\widehat{\boldsymbol e}}_a$ [[R]{}]{}= a \_[=1]{}\^4 (\_ [ [**n**]{} ]{})\_ = a \_[=1]{}\^[4]{}n\_\_ , where $a$ is the lattice spacing. On using the fact that $\sum_{m}{\widehat{\boldsymbol e}}_m = 0$, we can show that a small lattice displacement of the form $d{ {\bf n} }= {\widehat{\boldsymbol {\mu}}}_m$ corresponds to a space-time translation by $(a{\widehat{\boldsymbol e}}_m)$: d[[R]{}]{}= a \_[=1]{}\^[4]{}(\_ d [ [**n**]{} ]{})\_ = a \_[=1]{}\^[4]{} (\_ \_m)\_ = a \_m . The lattice action corresponds to a discretization of the Marcus twist on this $A_4^*$ lattice and can be represented as a set of traced closed bosonic and fermionic loops. It is invariant under the exact ${{\cal Q}}$ scalar supersymmetry, lattice gauge transformations and a global permutation (point group) symmetry $S^5$, and can be proven free of fermion doubling problems as discussed before. The ${{\cal Q}}$-exact part of the lattice action is again given by (\[twist\_action\]) with the indices $\mu,\nu$ now labeling the five basis vectors of $A_4^*$ or equivalently its hypercubic cousin.
Finally, it is important to note that while the true lattice in space-time is this rather complicated looking $A_4^*$ structure, we can represent all of the lattice fields in our theory by giving only their coordinates on the abstract hypercubic lattice. Indeed, since the lattice action only depends on the structure of the hypercubic lattice we will not need the explicit coordinates of the $A_4^*$ lattice to generate Monte Carlo configurations during the simulation. The explicit mapping of hypercubic coordinates to space-time coordinates in the $A_4^*$ lattice is only needed when, for example, we want to compute spatially dependent objects such as correlation functions of fields. In this case we should compute distances relative to the underlying $A_4^*$ lattice [*not*]{} its hypercubic partner.
While the supersymmetric invariance of the ${{\cal Q}}$-exact term is manifest in the lattice theory it is not immediately clear how to discretize the continuum ${{\cal Q}}$-closed term. Remarkably, it is possible to discretize (\[closed\]) in such a way that it is indeed exactly invariant under the twisted supersymmetry: S\_[closed]{} = - \_[[ [**n**]{} ]{},m,n,p,q,r]{} \_[mnpqr]{}\_[qr]{}([ [**n**]{} ]{}+\_m+\_n+\_p) \^[(-)]{}\_p \_[mn]{}([ [**n**]{} ]{}+\_p) , which can be seen to be supersymmetric since the lattice field strength satisfies an exact Bianchi identity [@Aratyn:1984bd] \_[mnpqr]{}\^[(+)]{}\_p\_[qr]{}=0 .
Simulating the SYM theories: Algorithms {#sec:simulating-sym-alg}
=======================================
Although the fields entering into these twisted descriptions appear somewhat different to the usual fields used in QCD the basic algorithms we use to simulate them are borrowed directly from lattice QCD; namely we integrate out the fermions to produce a Pfaffian which is in turn represented by the square root of a determinant[^7] and can be simulated using the usual RHMC algorithm [@Clark:2006wq].
If we denote the set of twisted fermions by the field $\Psi=(\eta, \psi_\mu, \chi_{\mu\nu})$ we first introduce a corresponding pseudo-fermion field $\Phi$ with action S\_[PF]{}=\^(M\^M)\^[-]{} , \[pseudo\] where $M=M({{\cal U}},{{\cal U}}^\dagger)$ is the antisymmetric twisted lattice fermion operator given, for example, in (\[eq:4action\])[^8].
Integrating over the fields $\Phi$ will then yield (up to a possible phase) the Pfaffian of the operator $M({{\cal U}}, {{\cal U}}^\dagger)$ as required. The fractional power is approximated by the partial fraction expansion = \_0 + \_[i=1]{}\^P , \[partial\] where the coefficients $\{\alpha_i, \beta_i\}$ are evaluated offline using the Remez algorithm to minimize the error in some interval $(\epsilon, A)$. Typically we have used $P=15$ which yields a fractional error of $0.00001$ for the interval $0.0000001 \to 1000.0$, which conservatively covers the range we are interested in.
Following the standard procedure, we introduce momenta $(p_{{\cal U}}, p_F)$ conjugate to the coordinates $({{\cal U}}, \Phi)$ and evolve the coupled system using a discrete time leapfrog algorithm according to the classical Hamiltonian \[eq:class-Hamilt\] H = S\_B + S\_[PF]{} + p\_[[U]{}]{}|[p]{}\_[[U]{}]{}+ p\_|[p]{}\_ . Notice that the bosonic action[^9] S\_B=\_[[ [**x**]{} ]{},m,n]{} , is real, positive semi-definite in all these theories.
One step of the discrete time update is given by p\_[[U]{}]{}&=&|[f]{}\_[[U]{}]{}\
p\_&=&|[f]{}\_\
[[U]{}]{}&=&(e\^[t p\_[[U]{}]{}]{}-I)[[U]{}]{}\
&=&t p\_\
p\_[[U]{}]{}&=&|[f]{}\_[[U]{}]{}\
p\_&=&|[f]{}\_where the forces $f_{{\cal U}}$ and $f_\Phi$ are given by f\_[[U]{}]{}&=&-\
f\_&=&- and the bar denotes complex conjugation. Using the partial fraction expansion given in (\[partial\]) the fermionic contributions to these forces take the form f\^[fermionic]{}\_[[U]{}]{}&=& \_[i=1]{}\^P\_i\
f\^[fermionic]{}\_&=& -\_0|-\_[i=1]{}\^P\_i|[s]{}\_i\
where (M\^M+\_i)s\_i &=&\
t\_i &=& Ms\_i The latter set of sparse linear equations is solved using a multi-mass conjugate gradient (MCG) solver [@Jegerlehner:1996pm], which allows for the simultaneous solution of all $P$ systems in a single CG solve.
At the end of one such classical trajectory the final configuration is subjected to a standard Metropolis test based on the Hamiltonian $H$. The symplectic and reversible nature of the discrete time update is then sufficient to allow for detailed balance to be satisfied and hence expectation values are independent of $\delta t$. After each such trajectory the momenta are refreshed from the appropriate Gaussian distribution as determined by $H$, which renders the simulation ergodic.
The fermionic contribution to the forces are shown below f\^[fermionic]{}\_[[[U]{}]{}\_m]{} &=& = \_[i=1]{}\^[P]{} \_i F\^ (M\^M) F\
&=& -\_[i=1]{}\^[P]{} \_i ()\^ (M\^M) ()\
&=& -\_[i=1]{}\^[P]{} \_i ()\^ (M\^ + M) ()\
&=& -\_[i=1]{}\^[P]{} \_i\
&=& -\_[i=1]{}\^[P]{} \_i . f\^[fermionic]{}\_[F]{} &=&\
&=& \_0 (F\^F) + \_[i=1]{}\^[P]{} \_i (F\^)\
&=&\_0 F\^ + \_[i=1]{}\^[P]{} \_i s\_i\^ .
Overall structure of the C++ code {#sec:overall-structure}
=================================
Typically the bosons lie on the usual nearest neighbor links of a hyeprcubic lattice while the fermions occupy both these links and additional site, face and body diagonal links. In the case of ${{\cal N}}=4$ in four dimensions we have to augment the set of boson links with one additional gauge field associated with the body diagonal link of the hypercube. We introduce the [Lattice\_Vector]{} class to store the coordinates of the lattice sites and also the vector between sites. Such lattice vectors can be added or subtracted by overloading the ‘$+$’ or ‘$-$’ operators. These operations also respect the lattice boundary conditions. Associated with this class is a general function [loop\_over\_lattice(x sites)]{} that implements a loop over all lattice sites indexed by their coordinate vector; thus a simple loop looks like
The bosonic and pseudo fermionic fields are stored in various objects which are indexed via their lattice site vector and whose type corresponds directly to the tensor structure of the associated continuum field so that one finds C++ classes labeled [Site\_Field]{}, [Link\_Field]{}, [Plaq\_Field]{}, [Body\_Field]{} etc. in the header file [utilities.h]{}. (We provide the list of C++ files that goes into the code in \[sec:file-list\].) The full [[Kähler-Dirac ]{}]{}field is contained in the class [Twist\_Fermion]{} while the [Gauge\_Field]{} class contains the complexified Wilson gauge link. All these objects are in turn built from objects of type [Umatrix]{} corresponding to complex [NCOLOR x NCOLOR]{} matrices. Simple arithmetric operations which overload the usual arithmetic operations are defined for manipulating these objects.
{width="80.00000%"}
\[fig:structure\]
Let us briefly describe how the code works. The general organizational structure of the code is given in Fig. 3. We begin with [sym.cpp]{}. It reads the input parameters such as number of sweeps ([SWEEPS]{}), number of thermalization steps ([THERM]{}), gap in measurements ([GAP]{}), the ‘t Hooft coupling ([LAMBDA]{}), etc., using functions contained in the file [read\_param.cpp]{}. It can also read in previously generated field configurations using [read\_in.cpp]{}. [ ]{}\
The code [sym.cpp]{} performs three major tasks:
- Generates new configurations using a rational hybrid Monte Carlo (RHMC) algorithm. This is accomplished by calling the function [update(U,F)]{} contained in [update.cpp]{}.
- Saves the current field configuration after some number of Monte Carlo sweeps (using the functions in [write\_out.cpp]{}).
- Measures the observables in the theory. This is done by function calls within [measure.cpp]{}.
Let us focus on the task of updating field configurations first. After reading the initial parameters and field configurations [update()]{} is called. Here we refresh the momenta ${\tt p\_U}$ and ${\tt p\_F}$ (using a Gaussian distribution) and then go to [kinetic\_energy.cpp]{} to compute the kinetic energy: . Compare this with the first two terms in the classical Hamiltonian (\[eq:class-Hamilt\]): $\overline{p}_{{{\cal U}}}p_{{{\cal U}}} + \overline{p}_{\Phi}p_{\Phi}$ . After computing kinetic energy the boson and pseudo-fermion actions (\[eq:class-Hamilt\]) are computed with a call to the function [action()]{}.
The computation of the bosonic action $S_B$ is straightforward. In the code it is accomplished with the line . Here [KAPPA]{} is the dimensionless lattice coupling. It is defined in [read\_param.cpp]{} and depends on the number of dimensions ([D]{}), size of the lattice ([LX]{}, [LY]{}, [LZ]{}, [T]{}) and number of colors ([NCOLOR]{}).
The code associated with spefcific terms in the bosonic action can easily be identified with its analytic expression. We have $\rightarrow$ [Umu(x)\*Udagmu(x)-Udagmu(x-e\_mu)\*Umu(x-e\_mu)]{} ,\
[Fmunu(x)]{} $\rightarrow$ [Umu(x)\*Unu(x+e\_mu)-Unu(x)\*Umu(x+e\_nu)]{} .
The code used to compute the fermionic part of the action is given by , where [n]{} runs from ${\tt 0}$ to [DEGREE]{} (which is equal to number of terms in the Remez approximation $P$), [ampdeg]{} corresponds to $\alpha_0$, [F]{} the twisted pseudo-fermion $F$, [Cjg(F)]{} is $F^{\dagger}$, [amp\[n\]]{} is $\alpha_i$ and [sol\[n\]]{} corresponds to $s_i \equiv (M^{\dagger}M + \beta_i)^{-1}F$.
Again one should compare this code with the form of the pseudo-fermion action $S_{pf} = \alpha_0 F^{\dagger}F + \sum_{i=1}^{P} \alpha_i F^{\dagger}\Big[(M^{\dagger}M + \beta_i)^{-1}F\Big]$ .
We invoke a multi-mass conjugate gradient solver [MCG\_solver()]{} given in [MCG\_solver.cpp]{} to help compute the terms needed in the fermionic action. The MCG solver can return the solutions to $(M^{\dagger}M + \beta_i) s_i = F$ for all shifts $\beta_i$.
Once the Hamiltonian is computed we evolve the fields along a classical trajectory. This is handled by the function [evolve\_fields]{}. The evolution of the fields and momenta is achieved through a leapfrog algorithm. In the first half step we have
------------ --------------- ------------------------------
[p\_Umu]{} $\rightarrow$ [p\_Umu + 0.5\*DT\*f\_Umu]{}
[p\_F]{} $\rightarrow$ [p\_F + 0.5\*DT\*f\_F]{}
[Umu]{} $\rightarrow$ [Umu + exp(DT\*p\_Umu)]{}
[F]{} $\rightarrow$ [F + DT\*p\_F]{}
------------ --------------- ------------------------------
Immediately after computing the change in fields ([Umu]{} and [F]{}) and momenta ([p\_Umu]{} and [p\_F]{}), we update the forces by calling [force()]{}. The bosonic force contribution to [f\_Umu]{} is given by
--------------- --------------- -----------------------------------------------------
[f\_Umu(x)]{} $\rightarrow$ [f\_Umu(x)+Umu(x)\*Udagmu(x)\*DmuUmu(x)]{}
[-Umu(x)\*DmuUmu(x+e\_mu)\*Udagmu(x)]{}
[+2.0\*Umu(x)\*Unu(x+e\_mu)\*Adj(Fmunu(x))]{}
[-2.0\*Umu(x)\*Adj(Fmunu(x-e\_nu))\*Unu(x-e\_nu)]{}
--------------- --------------- -----------------------------------------------------
The computation of the fermionic force [f\_F]{} requires first a call to the MCG solver ${\tt MCG\_solver()}$. We find . Once we have this solution an additional contribution to the gauge force coming from the pseudo-fermions is gotten by a call to the function [fermion\_forces()]{}. Each fermionic term in the action yields a contribution. We provide a part of this code in Fig. 4.
In the second half step of the leapfrog algorithm the momenta [p\_U]{} and [p\_F]{} are again updated with the new forces. These final forces are then saved for the next iteration.
In practice, it is important to use a multi-time step integrator for this evolution [@Sexton:1992nu]. In this case while the fermions are evolved with a time step of [DT]{}, the bosons are integrated with the time step [DT/MSTEP]{}. Provided the boson force is substantially larger than the fermionic contribution this can result in fewer costly fermion inversions for a fixed acceptance rate. In practice the parameter [MSTEPS]{} can be tuned to optimize the update - typically [MSTEPS=10]{}.
Finally, control returns to [update()]{} and the updated Hamiltonian [H\_new]{} is computed. A simple Metropolis test is used to accept or reject the field configuration at the end of the trajectory.
Site, Link and Plaquette type operators
---------------------------------------
The bosonic and fermionic fields, and the covariant difference operators living on the hypercubic lattice are associated with various geometric structures such as sites, links and plaquettes. They are implemented in the code using various user defined C++ classes: [Site\_Field]{}, [Link\_Field]{}, [Plaq\_Field]{}, [Body\_Field]{}, etc. They are constructed such that they can take values in $U(N)$ or $SU(N)$. They make appearances in the code in many ways and we summarize their general structure in the table below:
[Site\_Field]{} $S({ {\bf x} })$ ${{\overline{\cal D}}}^{(-)}_\mu L_\mu({ {\bf x} })$
---------------------- --------------------------------- ------------------------------------------------------------- ------------------------------------------------------------------
[Link\_Field]{} $L_\mu({ {\bf x} })$ ${{\overline{\cal D}}}^{(+)}_\mu S({ {\bf x} })$ ${{\cal D}}^{(-)}_\nu P_{\mu \nu}({ {\bf x} })$
[Plaquette\_Field]{} $P_{\mu \nu}({ {\bf x} })$ ${{\cal D}}^{(+)}_\mu L_\nu({ {\bf x} })$ ${{\overline{\cal D}}}^{(-)}_\rho B_{\mu \nu \rho}({ {\bf x} })$
[Body\_Field]{} $B_{\rho \mu \nu}({ {\bf x} })$ ${{\overline{\cal D}}}^{(+)}_\rho P_{\mu \nu}({ {\bf x} })$
As an instructive example let us look at the coding details of the [Link\_Field]{} class. In Fig. 5 we show how the [Link\_Field]{} class is defined along with overloading of basic operators such as ‘+’ and ‘-’.
We look at the structure of the fermionic term $\eta {{\overline{\cal D}}}_\mu \psi_\mu$ on the lattice and the structure of the corresponding fermionic operator in the code. On the lattice this fermionic term takes the form \_\_&&\
&=& , where $T^a$ are the generators of the gauge group.
On expanding the lattice covariant difference operators we have \_\_&&\
&=& . In the code we compute the combination ${\rm Tr}(T^a{{\cal U}}_\mu({ {\bf x} })T^b)$ as $V_\mu({ {\bf x} })^{ab}$ and store it as the object [Adjoint\_Link\_Field]{}. It is this field that is passed into the functions that require the action of the twisted fermion operator in the inverter. Explicitly, the contribution to the operator coming from the term ${\rm Tr}~(\eta {{\overline{\cal D}}}_\mu \psi_\mu)$ in the action takes the following form in the code:
------------------------------------------------------------ --
[+0.5\*conjug(V.get(x,mu).get(a,b))]{}
[-0.5\*conjug(V.get(x-e\_mu,mu).get(b,a))\*BC(x,-e\_mu)]{}
[+0.5\*conjug(V.get(x,mu).get(a,b))\*BC(x,e\_mu)]{}
[-0.5\*conjug(V.get(x,mu).get(b,a))]{}
------------------------------------------------------------ --
Simulation results {#sec:sim-results}
==================
In this section we provide some numerical results obtained through the recent simulations of the two-dimensional ${{\cal N}}=2$ lattice SYM theory [@Catterall:2011aa; @Mehta:2011ud].
The results we show in this section were obtained using the orbifold prescription for the parametrization of the complexified gauge fields ${{\cal A}}_\mu(x)$ on the lattice. The continuum fields ${{\cal A}}_a(x)$ are mapped to link fields ${{\cal U}}_a({ {\bf n} })$ living on the link between ${ {\bf n} }$ and in ${ {\bf n} }+{\widehat{\boldsymbol {\mu}}}_a$ through the mapping: [[U]{}]{}\_a([ [**n**]{} ]{}) = [[[A]{}]{}\_a([ [**n**]{} ]{})]{} , where ${{\cal A}}_a({ {\bf n} })=\sum_{i=1}^{N_G} {{\cal A}}_a^i T^i$ where $T^i=1 \ldots N_G$ are the anti-hermitian generators of a $U(N)$ group. Notice though that in spite of the appearance of a complex connection the theory only possesses the usual $U(N)$ gauge symmetry. [^10] Simulations with linear gauge links of this type have been investigated in [@Catterall:2011aa].
Eigenvalues of scalars
----------------------
The requirement that the lattice theory target the continuum theory as the lattice spacing is sent to zero demands vanishing of the fluctuations of all lattice fields and [*in particular the fluctuations of the trace part of the scalar field $B^0_a$*]{}. It is also important that the trace mode develops a nonzero expectation value of unity in order that the lattice action yield the appropriate kinetic terms in the naive continuum limit. Given the absence of any classical potential guaranteeing these features, we find that it is necessary to add a suitable gauge-invariant potential to the lattice theory to ensure these conditions hold[^11]. In principle, once this mode is regulated one can examine whether this potential can be sent to zero in the continuum limit.
We have added a simple potential term of the following form to regulate the trace mode in the simulations[^12]: \[eq:u1-mass-term\] S\_M = \^2 \_[ [**x**]{} ]{}(([[U]{}]{}\_a\^([ [**x**]{} ]{}) [[U]{}]{}\_a([ [**x**]{} ]{}))-1)\^2 . This term fixes the vev of the scalar trace mode $B^0_a$ to unity and constrains the fluctuations of the trace mode $\delta B^0_a$ with a quadratic mass term at leading order in the lattice spacing. The remaining traceless fluctuations feel only a soft quartic potential. S\_M \^2\_[ [**x**]{} ]{}\[B\^0\_a \]\^2 + …Since this $U(1)$ scalar sector decouples in the naive continuum limit this should not break the supersymmetry of the remaining $SU(N)$ sector for small enough lattice spacing (indeed all susy breaking terms should vanish as $\mu\to 0$)
In the C++ code the mass term (\[eq:u1-mass-term\]) is implemented using in [action.cpp]{}. The $U(1)$ mass coefficient $\mu$ is denoted by the parameter [BMASS]{} and should be held [*fixed*]{} as we take the continuum limit. This implies that the physical mass is taken to infinity in this limit for any non-zero $\mu$ and hence that the expectation value of $B^0_a$ is frozen at unity in this limit.
We rescale all lattice fields by powers of the lattice spacing to make them dimensionless. This leads to an overall dimensionless coupling parameter of the form $N/(2\lambda a^2)$, where $a=\beta/T$ is the lattice spacing, $\beta$ is the physical extent of the lattice in the Euclidean time direction and $T$ is the number of lattice sites in the time-direction. The coupling $\lambda = g^2N$ is the usual ’t Hooft parameter. Thus, the lattice coupling \[eq:lattice-coupling\] = , for the symmetric two-dimensional lattice where the spatial length $L=T$[^13] Note that $\lambda\beta^2$ is the dimensionless physical ‘t Hooft coupling measured in units of the area. In these two dimensional simulations, the continuum limit can be approached by fixing $t=\lambda\beta^{2}$ and $N$, and increasing the number of lattice points $L \rightarrow \infty$. We have taken three different values for this coupling $t=0.5, 1.0, 2.0$ and lattice sizes ranging from $L=2, \cdots, 12$. In Fig. 6 we show the average scalar eigenvalue given by ${{\cal U}}_a^\dagger {{\cal U}}_a - I$ for the ${{\cal Q}}=4$ model as a function of the lattice size $L$. This figure confirms that as $L \rightarrow \infty$ we are indeed approaching a continuum limit since the scalar eigenvalues (which contain a factor of $a$ to render them dimensionless) are driven to zero.
{width="10cm"} \[fig:q4u2scalar\]
Pfaffian phase/sign problems
----------------------------
The models we have discussed may encounter an additional difficulty in the context of simulation - the fermionic sign problem. After integration over the fermions the effective bosonic action picks up a contribution from the logarithm of the fermionic Pfaffian ${\rm Pf}(M)$ which is not necessarily real. Indeed for the supersymmetric lattice constructions we described above, $M$ at non zero lattice spacing is a complex operator and one might worry that the resulting Pfaffian could exhibit a fluctuating phase $e^{i\alpha}$. Since Monte Carlo simulations must necessarily be performed with a positive definite measure the only way to incorporate this phase is through a re-weighting procedure which folds the phase in with the observables of the theory. Expectation values of observables derived from such simulations can then suffer huge statistical errors which swamp the signal rendering the Monte Carlo techniques effectively useless.
In Fig. 7 we show results for $\langle|\sin(\alpha)|\rangle$ as a function of $L$ for the ${{\cal Q}}=4$ model with gauge group $U(2)$ (edit [utilities.h]{} to change number of supercharges). Three values of $t=\lambda\beta^2$ are shown in each plot but the behavior is qualitatively similar for all $t$. We have used the mass parameter controlling the $U(1)$ mode as [BMASS]{} = 1. These numerical results show that while this model appears to suffer from a sign problem for coarse lattices these effects disappear as the lattice is refined and the phase fluctuations are driven to zero as the continuum limit is taken. This is consistent with the work reported in [@Hanada:2010qg]
{width="10cm"}
Restoration of supersymmetry
----------------------------
The topological nature of the twisted theory formulated on a torus with periodic boundary conditions can be used to show that the partition function of the lattice model is actually independent of the coupling constant. Thus derivatives of the partition function with respect to the coupling constant such as the expectation value of the action must vanish. Since the fermions enter only quadratically, their contribution can be evaluated simply using a scaling argument and thence a simple expression derived for the expectation value of the bosonic action. Thus measurements of $\langle S_B({{\cal U}}, {{\overline{\cal U}}})\rangle$ provide us with a check that the scalar supersymmetry has indeed been implemented correctly in our codes. Actually, since in practice we use supersymmetry breaking (thermal) boundary conditions (and also employ a supersymmetry breaking potential for the scalar $U(1)$ mode) to do simulations, measuring this quantity provides some insight into the magnitude of supersymmetry breaking effects in the theory.
In the case of two-dimensional ${{\cal Q}}=4$ theory, we have the expression for the mean action S = - = [[Q]{}]{}= 0 , where $\kappa$ is the coupling constant of the twisted action and the last equality follows from the ${{\cal Q}}$-exact nature of the twisted theory and shows that the vanishing mean action can be thought of as arising as a consequence of a simple ${{\cal Q}}$-Ward identity.
If we integrate out the twisted fermions and the auxiliary field $d$ we find the following expression for the partition function of the two-dimensional ${{\cal Q}}=4$ theory Z = \^[4N\_G V/2]{} \^[-N\_GV/2]{} D[[U]{}]{}De\^[-S\_B([[U]{}]{}, )]{}[ ]{}\^ (M([[U]{}]{}, )) , where $N_G$ is the number of generators of the gauge group and $V$ is the number of lattice points. The first pre-factor arises from the fermion integration and the second derives from the Gaussian integration over the auxiliary field. From this we find the following condition on the mean bosonic action as a consequence of the scalar supersymmetry ${{\cal Q}}$: \[eq:exact-baction\] S\_B = N\_G V . In Fig. 8 we show the mean bosonic action on the lattice against the lattice size $L$. The thick solid line represents the exact value of the bosonic action given in \[eq:exact-baction\].
{width="10cm"}
Clearly the lattice measurements approach the exact result for sufficiently small lattice spacing. The deviations that are visible are presumably related to the fact that we have a sign problem (these measurements do [*not*]{} incorporate re-weighting) for small $L$ and the simulations are also conducted at non zero temperature. We have shown that the sign problem disappears in the continuum limit which is consistent with the much better agreement at large $L$. To recover the true zero temperature result requires in principle that we extrapolate our measurements to $t\to\infty$ after taking the thermodynamic limit.
Conclusions and outlook {#sec:conclu}
=======================
In this paper we have described in some detail the construction of an object oriented code suitable for the simulation of a recently discovered class of lattice field theories possessing exact supersymmetry. The continuum construction and lattice discretization of ${{\cal Q}}=4,8,16$ supercharge SYM theories in two, three and four dimensions are all covered in detail. The structure of the problem requires the construction of unusual data structures for representing the fermions, which is the primary difference between the code described here and more conventional codes suitable for simulating QCD. Nevertheless the basic algorithms employed (RHMC and multi-mass CG solvers) are borrowed directly from lattice QCD and adapted to the problem at hand. We verify the correctness of the resultant code by showing results from simulations of the two-dimensional SYM model. Acceleration of this code can be achieved by off-loading the linear solver calculation to a GPU card - we refer the reader to [@Galvez:2011cd] for details. It is also possible to parallelize the code with suitable distributed libraries layered over MPI [@DiPierro:2005jd] and work in both these directions is ongoing.
Acknowledgments
===============
This work is supported in part by DOE under grant number DE-FG02-85ER40237. Simulations were performed using USQCD resources at Fermilab. We would like to thank useful discussions with Richard Galvez, Joel Giedt, Dhagash Mehta and Greg van Anders.
Installation of the program {#sec:install}
===========================
It is very easy to perform the installation and execution of SUSY\_LATTICE. Below we provide the necessary steps on Unix or Linux systems.
- Download the code from CPC Program Library and unpack it.
- Change the directory to SUSY\_LATTICE.
- Compile the code (g++ -O \*.cpp -o SUSY\_LATTICE -llapack -lblas).
- Modify the input parameters located in file [*parameters*]{}
- Type ./SUSY\_LATTICE $>$& log & to run the code.
The authors have tested the code on Linux machines. After slight modifications of above steps the code may be installed on other machines.
The output of the code produces the following files in the running directory:
- [cgs:]{} Average number of conjugate gradient (CG) iterations. (See [MCG\_solver.cpp]{}).
- [config:]{} File to read in containing the site, link and plaquette field configurations from a previous run. (See [read\_in.cpp]{}.)
- [corrlines:]{} Correlation function between temporal Polyakov lines as function of spatial separation (See [corrlines.cpp]{}.)
- [data:]{} Boson (1st column) and fermion (2nd column) contributions to the total action. (See [measure.cpp]{}.)
- [dump:]{} Site, link and plaquette field configurations stored as ASCII (See [write\_out.cpp]{}.)
- [eigenvalues:]{} Eigenvalues of ${{\cal U}}^dagger_a(x){{\cal U}}_a(x)$ $N$ real numbers for each lattice point $x$ and direction $a$ (See [measure.cpp]{}.)
- [hmc\_test:]{} $e^{-Delta H}$ from HMC test (See [update.cpp]{}.)
- [lines\_s:]{} Spatial Polyakov line. (See [measure.cpp]{}.)
- [lines\_t:]{} Temporal Polyakov line. (See [measure.cpp]{}.)
- [log:]{} Log file.
- [loops:]{} Wilson loops. (See [loop.cpp]{}.)
- [scalars:]{} ${\rm Tr} (U^\dagger U)$ (See [measure.cpp]{}.)
- [ulines\_s:]{} The spatial Polyakov line computed using the unitary part of the link (See [measure.cpp]{}.)
- [ulines\_t:]{} The temporal Polyakov line computed using the unitary part of the link (See [measure.cpp]{}.)
The list of files in SUSY\_LATTICE library {#sec:file-list}
==========================================
We list the files included in SUSY\_LATTICE library with a brief description of their purpose.
- [action.cpp:]{} Compute the total action - fermionic and bosonic.
- [corrlines.cpp:]{} Finds the traced product of the link matrices at various lattice sites.
- [evolve\_fields.cpp:]{} Leapfrog evolution algorithm. Also stores the fermion and boson forces for the next iteration.
- [fermion\_forces.cpp:]{} Computes the fermion kick to gauge link force.
- [force.cpp:]{} Bosonic and pseudo-fermionic contribution to the force.
- [kinetic\_energy.cpp:]{} Computes the kinetic energy term in the Hamiltonian.
- [line.cpp:]{} Computes the Polyakov lines.
- [loop.cpp:]{} Computes the Wilson loops.
- [matrix.cpp:]{} Builds the fermion matrix (sparse and full forms) and also computes the Pfaffian of the fermion operator.
- [MCG\_solver.cpp:]{} multi-mass CG solver needed for RHMC alg.
- [measure.cpp:]{} Performs measurements on field configurations. Writes out scalar eigenvalues, Polyakov/Wilson loops and the action.
- [my\_gen.cpp:]{} Computes $SU(N)$ generator matrices.
- [obs.cpp:]{} Computes fermion and gauge actions. Also returns the unitary piece of the complex link field.
- [read\_in.cpp:]{} Reads in the previously generated field configurations - file [config]{}
- [read\_param.cpp:]{} Reads in the simulation parameters from a data file called [parameters]{}
- [setup.cpp:]{} Contains the partial fraction coefficients necessary to represent fractional power of fermion operator - used by Remez algorithm.
- [sym.cpp:]{} The main program - performs warm up on field configurations and commences measurement sweeps once the configurations are warmed up.
- [unit.cpp:]{} Extracts the unitary piece of the complex gauge links.
- [update.cpp:]{} Updates the field configurations based on HMC test.
- [utilities.cpp:]{} Utility functions. Contains constructors for site, link, plaquette fields, gauge fields, twist fermions etc. Edit to change number of supercharges and size of lattice dimensions.
- [write\_out.cpp:]{} Writes out the values of gauge and twist fermion fields on to a file called [dump]{}.
A sample input parameter file for SUSY\_LATTICE {#sec:input-param}
===============================================
This is a sample input parameter file called [parameters]{} located in the SUSY\_LATTICE folder.
--------------- ------------ ------------ ------------- ------------- -------------- ------------- ------------
${\tt 10000}$ ${\tt 50}$ ${\tt 10}$ ${\tt 0.5}$ ${\tt 1.0}$ ${\tt 0.02}$ ${\tt 0.0}$ ${\tt 0}$
[SWEEPS]{} [THERM]{} [GAP]{} [LAMBDA]{} [BETA]{} [DT]{} [ALPHA]{} [READIN]{}
--------------- ------------ ------------ ------------- ------------- -------------- ------------- ------------
There are the definitions of the parameters:
- [SWEEPS:]{} Total number of Monte Carlo time steps intended for taking measurement steps.
- [THERM:]{} Total number of Monte Carlo time steps intended for thermalizing the field configurations.
- [GAP:]{} The gap between measurement steps.
- [LAMBDA:]{} The ‘t Hooft coupling.
- [BETA:]{} Inverse temperature.
- [DT:]{} The time step put in the integrator for leapfrog evolution.
- [ALPHA:]{} A supersymmetric mass (deformation) parameter.
- [READIN:]{} Determines whether to read in the previously generated field configurations or not. The program will read in the previous configurations if [READIN]{} is set to ${\tt 1}$.
[9]{}
F. Sugino, “A Lattice formulation of super Yang–Mills theories with exact supersymmetry,” JHEP [**0401**]{}, 015 (2004). \[hep-lat/0311021\].
F. Sugino, “Super Yang–Mills theories on the two-dimensional lattice with exact supersymmetry,” JHEP [**0403**]{}, 067 (2004). \[hep-lat/0401017\].
F. Sugino, “Various super Yang–Mills theories with exact supersymmetry on the lattice,” JHEP [**0501**]{}, 016 (2005). \[hep-lat/0410035\].
S. Catterall, “A Geometrical approach to N=2 super Yang–Mills theory on the two dimensional lattice,” JHEP [**0411**]{}, 006 (2004). \[hep-lat/0410052\].
S. Catterall, “Lattice formulation of N=4 super Yang–Mills theory,” JHEP [**0506**]{}, 027 (2005). \[hep-lat/0503036\].
A. D’Adda, I. Kanamori, N. Kawamoto, K. Nagata, “Exact extended supersymmetry on a lattice: Twisted N=2 super Yang–Mills in two dimensions,” Phys. Lett. [**B633**]{}, 645-652 (2006). \[hep-lat/0507029\].
S. Catterall, “Dirac-Kahler fermions and exact lattice supersymmetry,” PoS [**LAT2005**]{}, 006 (2006). \[hep-lat/0509136\].
F. Sugino, “Two-dimensional compact N=(2,2) lattice super Yang–Mills theory with exact supersymmetry,” Phys. Lett. [**B635**]{}, 218-224 (2006). \[hep-lat/0601024\].
S. Catterall, “Simulations of N=2 super Yang–Mills theory in two dimensions,” JHEP [**0603**]{}, 032 (2006). \[hep-lat/0602004\].
S. Catterall, “On the restoration of supersymmetry in twisted two-dimensional lattice Yang–Mills theory,” JHEP [**0704**]{}, 015 (2007). \[hep-lat/0612008\].
A. D’Adda, I. Kanamori, N. Kawamoto, K. Nagata, “Exact Extended Supersymmetry on a Lattice: Twisted N=4 Super Yang–Mills in Three Dimensions,” Nucl. Phys. [**B798**]{}, 168-183 (2008). \[arXiv:0707.3533 \[hep-lat\]\].
S. Catterall, “From Twisted Supersymmetry to Orbifold Lattices,” JHEP [**0801**]{}, 048 (2008). \[arXiv:0712.2532 \[hep-th\]\].
S. Catterall, A. Joseph, “Lattice actions for Yang–Mills quantum mechanics with exact supersymmetry,” Phys. Rev. [**D77**]{}, 094504 (2008). \[arXiv:0712.3074 \[hep-lat\]\].
S. Catterall, “First results from simulations of supersymmetric lattices,” JHEP [**0901**]{}, 040 (2009). \[arXiv:0811.1203 \[hep-lat\]\].
S. Catterall, D. B. Kaplan, M. Unsal, “Exact lattice supersymmetry,” Phys. Rept. [**484**]{}, 71-130 (2009). \[arXiv:0903.4881 \[hep-lat\]\].
D. B. Kaplan, E. Katz, M. Unsal, “Supersymmetry on a spatial lattice,” JHEP [**0305**]{}, 037 (2003). \[hep-lat/0206019\].
J. Nishimura, S. -J. Rey, F. Sugino, “Supersymmetry on the noncommutative lattice,” JHEP [**0302**]{}, 032 (2003). \[hep-lat/0301025\].
A. G. Cohen, D. B. Kaplan, E. Katz, M. Unsal, “Supersymmetry on a Euclidean space-time lattice. 1. A Target theory with four supercharges,” JHEP [**0308**]{}, 024 (2003). \[hep-lat/0302017\].
A. G. Cohen, D. B. Kaplan, E. Katz, M. Unsal, “Supersymmetry on a Euclidean space-time lattice. 2. Target theories with eight supercharges,” JHEP [**0312**]{}, 031 (2003). \[hep-lat/0307012\]
D. B. Kaplan, M. Unsal, “A Euclidean lattice construction of supersymmetric Yang–Mills theories with sixteen supercharges,” JHEP [**0509**]{}, 042 (2005). \[hep-lat/0503039\].
M. Unsal, “Twisted supersymmetric gauge theories and orbifold lattices,” JHEP [**0610**]{}, 089 (2006). \[hep-th/0603046\].
P. H. Damgaard, S. Matsuura, “Classification of supersymmetric lattice gauge theories by orbifolding,” JHEP [**0707**]{}, 051 (2007). \[arXiv:0704.2696 \[hep-lat\]\].
P. H. Damgaard, S. Matsuura, “Relations among Supersymmetric Lattice Gauge Theories via Orbifolding,” JHEP [**0708**]{}, 087 (2007). \[arXiv:0706.3007 \[hep-lat\]\].
S. Matsuura, “Exact vacuum energy of orbifold lattice theories,” JHEP [**0712**]{}, 048 (2007). \[arXiv:0709.4193 \[hep-lat\]\]
P. H. Damgaard, S. Matsuura, “Geometry of Orbifolded Supersymmetric Lattice Gauge Theories,” Phys. Lett. [**B661**]{}, 52-56 (2008). \[arXiv:0801.2936 \[hep-th\]\].
M. Hanada, J. Nishimura, S. Takeuchi, “Non-lattice simulation for supersymmetric gauge theories in one dimension,” Phys. Rev. Lett. [**99**]{}, 161602 (2007). \[arXiv:0706.1647 \[hep-lat\]\].
K. N. Anagnostopoulos, M. Hanada, J. Nishimura, S. Takeuchi, “Monte Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature,” Phys. Rev. Lett. [**100**]{}, 021601 (2008). \[arXiv:0707.4454 \[hep-th\]\].
T. Azeyanagi, M. Hanada, T. Hirata, “On Matrix Model Formulations of Noncommutative Yang–Mills Theories,” Phys. Rev. [**D78**]{}, 105017 (2008). \[arXiv:0806.3252 \[hep-th\]\].
M. Hanada, L. Mannelli, Y. Matsuo, “Four-dimensional N=1 super Yang–Mills from matrix model,” Phys. Rev. [**D80**]{}, 125001 (2009). \[arXiv:0905.2995 \[hep-th\]\].
A. D’Adda, N. Kawamoto, J. Saito, “Formulation of Supersymmetry on a Lattice as a Representation of a Deformed Superalgebra,” Phys. Rev. [**D81**]{}, 065001 (2010). \[arXiv:0907.4137 \[hep-th\]\].
M. Hanada, I. Kanamori, “Lattice study of two-dimensional N=(2,2) super Yang–Mills at large-N,” Phys. Rev. [**D80**]{}, 065014 (2009). \[arXiv:0907.4966 \[hep-lat\]\].
M. Hanada, S. Matsuura, F. Sugino, “Two-dimensional lattice for four-dimensional N=4 supersymmetric Yang–Mills,”
\[arXiv:1004.5513 \[hep-lat\]\].
M. Hanada, “A proposal of a fine tuning free formulation of 4d N = 4 super Yang–Mills,” JHEP [**1011**]{}, 112 (2010). \[arXiv:1009.0901 \[hep-lat\]\].
P. H. Damgaard, S. Matsuura, “Lattice Supersymmetry: Equivalence between the Link Approach and Orbifolding,” JHEP [**0709**]{}, 097 (2007). \[arXiv:0708.4129 \[hep-lat\]\].
J. Giedt, “Progress in four-dimensional lattice supersymmetry,” Int. J. Mod. Phys. [**A24**]{}, 4045-4095 (2009). \[arXiv:0903.2443 \[hep-lat\]\].
A. Joseph, “Supersymmetric Yang–Mills theories with exact supersymmetry on the lattice,” Int. J. Mod. Phys. [**A26**]{}, 5057-5132 (2011) arXiv:1110.5983 \[hep-lat\].
E. Witten, “Topological Quantum Field Theory,” Commun. Math. Phys. [**117**]{}, 353 (1988).
S. Elitzur, E. Rabinovici, A. Schwimmer, “Supersymmetric Models On The Lattice,” Phys. Lett. [**B119**]{}, 165 (1982).
J. M. Rabin, “Homology Theory Of Lattice Fermion Doubling,” Nucl. Phys. [**B201**]{}, 315 (1982).
P. Becher, H. Joos, “The Dirac-Kahler Equation and Fermions on the Lattice,” Z. Phys. [**C15**]{}, 343 (1982).
H. Aratyn, M. Goto, A. H. Zimerman, “A Lattice Gauge Theory For Fields In The Adjoint Representation,” Nuovo Cim. [**A84**]{}, 255 (1984).
T. Banks, Y. Dothan, D. Horn, “Geometric Fermions,” Phys. Lett. [**B117**]{}, 413 (1982).
M. Blau, G. Thompson, “Aspects of $N_{T}\geq 2$ topological gauge theories and D-Branes,” Nucl. Phys. [**B492**]{}, 545-590 (1997). \[hep-th/9612143\].
S. Catterall, “Topological gravity on the lattice,” JHEP [**1007**]{}, 066 (2010). \[arXiv:1003.5202 \[hep-lat\]\].
N. Marcus, “The Other topological twisting of N=4 Yang–Mills,” Nucl. Phys. [**B452**]{}, 331-345 (1995). \[hep-th/9506002\].
A. Kapustin, E. Witten, “Electric-Magnetic Duality And The Geometric Langlands Program,”
\[hep-th/0604151\].
S. Catterall, E. Dzienkowski, J. Giedt, A. Joseph, R. Wells, “Perturbative renormalization of lattice N=4 super Yang–Mills theory,” JHEP [**1104**]{}, 074 (2011). \[arXiv:1102.1725 \[hep-th\]\].
M. A. Clark, “The Rational Hybrid Monte Carlo Algorithm,” PoS [**LAT2006**]{}, 004 (2006). \[hep-lat/0610048\].
B. Jegerlehner, “Krylov space solvers for shifted linear systems,”
\[hep-lat/9612014\].
J. C. Sexton, D. H. Weingarten, “Hamiltonian evolution for the hybrid Monte Carlo algorithm,” Nucl. Phys. [**B380**]{}, 665-678 (1992).
S. Catterall, R. Galvez, A. Joseph and D. Mehta, “On the sign problem in 2D lattice super Yang–Mills,” arXiv:1112.3588 \[hep-lat\].
D. Mehta, S. Catterall, R. Galvez and A. Joseph, “Supersymmetric gauge theories on the lattice: Pfaffian phases and the Neuberger 0/0 problem,” arXiv:1112.5413 \[hep-lat\].
M. Hanada, I. Kanamori, “Absence of sign problem in two-dimensional N = (2,2) super Yang–Mills on lattice,” JHEP [**1101**]{}, 058 (2011). \[arXiv:1010.2948 \[hep-lat\]\].
R. Galvez and G. van Anders, “Accelerating the solution of families of shifted linear systems with CUDA,” arXiv:1102.2143 \[hep-lat\].
M. Di Pierro, “Parallel programming with matrix distributed processing,” arXiv:hep-lat/0505005.
[**PROGRAM SUMMARY**]{}\
[*Manuscript Title:*]{} An object oriented code for simulating supersymmetric Yang–Mills theories\
[*Authors:*]{} Simon Catterall and Anosh Joseph\
[*Program Title:*]{} SUSY\_LATTICE\
[*Journal Reference:*]{}\
[*Catalogue identifier:*]{}\
[*Licensing provisions:*]{} None\
[*Programming language:*]{} C++\
[*Operating system:*]{} Any, tested on Linux machines\
[*Keywords:*]{} Lattice Gauge Theory ,Supersymmetric Yang–Mills ,Rational Hybrid Monte Carlo ,Object Oriented Programming\
[*PACS:*]{} 11.15.Ha, 12.60.Jv, 12.10.-g, 12.15.-y, 87.55.kd, 87.55.kh\
[*Classification:*]{} 11.6 Phenomenological and Empirical Models and Theories\
[*Nature of problem:*]{}\
To compute some of the observables of supersymmetric Yang–Mills theories such as supersymmetric action, Polyakov/Wilson loops, scalar eigenvalues and Pfaffian phases.\
[*Solution method:*]{}\
We use the Rational Hybrid Monte Carlo algorithm followed by a Leapfrog evolution and a Metroplois test. The input parameters of the model are read in from a parameter file.\
[*Restrictions:*]{}\
This code applies only to supersymmetric gauge theories with extended supersymmetry, which undergo the process of maximal twisting. (See Section \[sec:method-twist-SYM\] of the manuscript for details.)\
[*Unusual features:*]{}\
[*Running time:*]{}\
From a few minutes to several hours depending on the amount of statistics needed.\
[*References:*]{}
[^1]: $^\dagger$ Present address.
[^2]: There exist other attempts to study various supersymmetric models on the lattice. See [@Hanada:2007ti; @Anagnostopoulos:2007fw; @Azeyanagi:2008bk; @Hanada:2009kz; @D'Adda:2009kj; @Hanada:2009hq; @Hanada:2010kt; @Hanada:2010gs].
[^3]: Actually in the case of the four-dimensional ${{\cal N}}=4$ there is an additional ${{\cal Q}}$-closed term needed.
[^4]: The generators are taken to be [*anti-hermitian*]{} matrices satisfying $\operatorname{Tr}(T^aT^b)=-\delta^{ab}$.
[^5]: This twist of ${{\cal N}}=4$, $d=3$ SYM is known as the Blau-Thompson twist [@Blau:1996bx].
[^6]: It is common in the continuum literature to replace the 2- and 3-form fields in these expressions by their Hodge duals; a second vector $\hat{\psi}_a$ and scalar $\hat{\eta}$ see, for example [@Blau:1996bx].
[^7]: Of course this ignores a possible sign ambiguity. We return to this issue later when we discuss whether the phase quenched simulations we use suffer from a sign problem.
[^8]: The antisymmetry is guaranteed if the fermion action is rewritten as the sum of the original terms plus their lattice transposes.
[^9]: From now on we interchangeably use ${{\bf x}}$ and ${ {\bf n} }$ to denote the lattice site.
[^10]: Notice that our lattice gauge fields are dimensionless and hence contain an implicit factor of the lattice spacing $a$.
[^11]: It was precisely this requirement that led to a truncation of the $U(N)$ symmetry to $SU(N)$ in the original simulations of these theories corresponding to a delta function potential for the $U(1)$ part of the field [@Catterall:2008dv].
[^12]: A potential term of this type was first introduced and tested in [@Hanada:2010qg].
[^13]: To obtain this dimensionally reduced model from the ${{\cal N}}=4$ theory one merely sets the parameters ${\tt LX}=1$, ${\tt LY}=1$ in [utilities.h]{}.
|
---
abstract: 'We consider the chiral Lagrangian for baryon fields with $J^P =\frac{1}{2}^+$ or $J^P =\frac{3}{2}^+$ quantum numbers as constructed from QCD with up, down and strange quarks. The specific class of counter terms that are of chiral order $Q^3$ and contribute to meson-baryon interactions at the two-body level is constructed. Altogether we find 24 terms. In order to pave the way for realistic applications we establish a set of 22 sum rules for the low-energy constants as they are implied by QCD in the large-$N_c$ limit. Given such a constraint there remain only 2 independent unknown parameters that need to be determined by either Lattice QCD simulations or directly from experimental cross section measurements. At subleading order we arrive at 5 parameters.'
author:
- 'Yonggoo Heo$^1$, C. Kobdaj$^1$[^1] and Matthias F.M. Lutz$^{2,3 \,\rm a}$'
bibliography:
- 'literature.bib'
title: |
Constraints from a large-$N_c$ analysis on\
meson-baryon interactions at chiral order $Q^3$
---
Introduction
============
Still after many decades of vigorous studies the outstanding challenge of modern physics is to establish a rigorous link of QCD to low-energy hadron physics as it is observed in the many experimental cross section measurements. After all it is the only fundamental field theory there is that leads to the emergence of structure as a consequence of truly non-perturbative interactions in a quantum field theory. On the one-hand the data set is extended recently by LHCb, BES, COMPASS, Belle with more and more exciting new phenomena, on the other hand there is a huge data set on pion and photon induced reactions in the resonance region which still up today is not understood in terms of QCD dynamics [@Lutz:2015ejy; @Pennington:2016dpj]. Such reactions constitute the doorway of understanding non-perturbative QCD, like studies of the hydrogen atom paved the way of understanding QED.
While simulations of QCD on finite lattices made considerable progress the last decade it is still not feasible to derive cross sections systematically as measured in the laboratory in the resonance region of QCD. Thus at present it may be of advantage to resort to a well established method of modern physics. Derive the implications of the fundamental theory by matching it to effective field theory approaches that are formulated in terms of the relevant degrees of freedom.
With the great advances of lattice QCD simulations such an approach is going through a revolution at present since the effective field theory can now be scrutinized systematically by QCD lattice data. In turn, the typically quite large set of low-energy constants can be derived from QCD prior to confronting the effective field theory to scattering data taken in the laboratory. This has been emphasized and illustrated in the recent work [@Guo:2018kno]. Some results for sets of low-energy constants have already been obtained from the masses of baryons and mesons in their ground states with $J^P = \frac{1}{2}^+, \frac{3}{2}^+$ and $J^P = 0^-, 1^-$ quantum numbers [@Lutz:2014oxa; @Lutz:2018cqo; @Guo:2018kno; @Bavontaweepanya:2018yds; @Heo:2018qnk].
Since the majority of available lattice data was taken at unphysical quark masses it is mandatory to establish reliable tools to translate such data back to the physical case. The fact that lattice data are typically for unphysical hadrons so far we see as a fortunate circumstance since this way information on QCD is provided that cannot be inferred from the PDG or any experimental cross section so easily. Moreover, the determination of large sets of low-energy constants from lattice data on the hadron ground state masses at various unphysical quark masses appears to be much easier and better controlled as compared to their extraction from the first few available phase shifts as computed on QCD lattices at unphysical quark masses.
Here we wish to emphasize that our strategy how to pave the way towards the understanding of non-perturnative QCD relies heavily on our recent claim that the chiral Lagrangian properly formulated for the physics of up, down and strange quarks, can be successfully applied to low-energy QCD once it is set up in terms of on-shell meson and baryon masses. It was demonstrated that then the size of the physical strange quark mass does not prohibit the application of the chiral Lagrangian. This is contrasted by the conventional $\chi$PT approach, in which bare masses are to be used inside any loop expression. Here any low-orders application to the flavor SU(3) case should be avoided, being of no physical significance.
The purpose of the current study is to further prepare the quantitative application of the chiral Lagrangian with three light flavors to meson-baryon scattering data. Our target is the set of counter terms that carry chiral order $Q^3$ and contribute to meson-baryon scattering at the two-body level. Such $Q^3$ counter terms play a decisive role in the chiral dynamics of the meson-baryon systems. As was pointed out already in [@Lutz2002a] only in the presence of such terms it may be feasible to establish a universal set of $Q^2$ counter terms that describe pion, kaon and antikaon nucleon scattering data. Though there is a plethora of works [@Kaiser:1995eg; @Lutz:1997wt; @Oset:1997it; @Jamin:2000wn; @Jido:2003cb; @Mai:2009ce; @Bruns:2010sv; @Bruns:2012eh; @Ikeda:2012au; @Oller:2013dxa; @Feijoo:2015yja; @Ramos:2016odk] that fit the $Q^2$ counter terms to pion-nucleon, kaon-nucleon or antikaon-nucleon scattering, the only so far a univeral approach is documented in [@Lutz2002a]. In turn there are various mutually non-compatible sets of the $Q^2$ counter terms available.
We would argue that there are also still some residual deficiencies in [@Lutz2002a] which may hamper the direct use of the most comprehensive set low-energy constants as extracted from the published lattice data set on the baryon octet and decuplet masses in [@Lutz:2018cqo]. Most severe, we would argue, are the particularities of the unitarization schemes. Within the flavor SU(3) framework so far all published works rely on neglect or improper treatment of left-hand branch points. Though we do not expect this to lead to huge qualitative issues, a quantitative and controlled study of in particular p-wave phase shifts should consider it in a reliable manner. We feel this to be an achievable request owing to the fact that such a scheme exists by now with [@Gasparyan:2010xz; @Danilkin:2010xd; @Gasparyan:2011yw]. So far it was applied only to the flavor SU(2) case with the $\pi N$ and $\gamma N$ channels.
Within a flavor SU(3) context such $Q^3$ terms were first used in [@Lutz2002a]. Later the complete order $Q^3$ Lagrangian was constructed in [@Frink:2006hx; @Oller:2007qd] for the baryon octet fields. To the best knowlege of the authors such counter terms have not been constructed so far involving the baryon decuplet fields. We are aware of the recent work [@Jiang:2018mzd], which, however, provides partial results only. Since we wish to derive sum rules for the $Q^3$ low-energy constants from large-$N_c$ QCD [@tHooft74; @Witten1979] a reliable construction of the latter terms is the target of the first part of our work in section II. It follows the second part with section III in which we apply large-$N_c$ QCD in order to derive sum rules for the set of $Q^3$ low-energy constants. Here we follow the framework previously established in [@Luty1994; @Dashen1995; @Lutz:2010se]. In our case we compute the contributions of the $Q^3$ counter terms to the correlation function with two axial-vector and one vector current in the baryon ground states. From a study of the latter the desired sum rules will be derived.
Chiral Lagrangian with baryon octet and decuplet fields {#section:chiral-lagrangian}
=======================================================
We recall the conventions for the chiral Lagrangian as used in the current work [@Krause1990; @Lutz2002a; @LutzSemke2010; @Lutz:2018cqo]. The hadronic fields as decomposed into their isospin multiplets are $$\begin{aligned}
&& \Phi = \tau \cdot \pi (140)
+ \alpha^\dagger \cdot K (494) + K^\dagger(494) \cdot \alpha
+ \eta(547)\,\lambda_8\,,
\nonumber\\
&& \sqrt{2}\,B
= \alpha^\dag\cdot{N}(939) + \lambda_8\,\Lambda(1115)
+ \vec\tau\cdot\vec\Sigma(1195) + {\Xi}^T(1315) \,i\sigma_2 \cdot\alpha
\,,\nonumber\\
&& \alpha^\dagger = {\textstyle{1\over \sqrt{2}}}\,(\lambda_4+i\,\lambda_5 ,\lambda_6+i\,\lambda_7 )\,,\qquad \qquad \qquad
\vec\tau = (\lambda_1,\lambda_2, \lambda_3)\,,\end{aligned}$$ where the matrices $\lambda_i$ are the Gell-Mann generators of the SU(3) algebra. The numbers in the brackets recall the approximate masses of the particles in units of MeV. Of central importance is the covariant derivative $$\begin{aligned}
( D_\mu B)^i_j = \partial_\mu B^i_j + (\Gamma_\mu)^i_h\,B^h_j - B^i_h\,(\Gamma_\mu)^h_j\,,
\label{example-Dmu}\end{aligned}$$ as introduced in terms of the chiral connection $\Gamma_\mu$. The chiral connection with $\Gamma_\mu=-\Gamma_\mu^\dagger$ and other convenient chiral building blocks are constructed in terms of the chiral fields $\Phi$ in a non-linear fashion such that the all chiral Ward identities of QCD are recovered in systematic applications of the chiral Lagrangian [@Gasser1983a; @Gasser1983b; @Krause1990]. We write $$\begin{aligned}
&&\Gamma_\mu ={\textstyle{1\over 2}}\,u^\dagger\,\Big[\partial_\mu -i\,(v_\mu + a_\mu) \Big]\,u
+{\textstyle{1\over 2}}\, u\,\Big[\partial_\mu -i\,(v_\mu - a_\mu)\Big]\,u^\dagger \,,
\nonumber\\
&& \textcolor{black}{ U_\mu = {\textstyle{1\over 2}}\,u^\dagger \, \big(
\partial_\mu \,e^{i\,\frac{\Phi}{f}} \big)\, u^\dagger
-{\textstyle{i\over 2}}\,u^\dagger \,(v_\mu+ a_\mu)\, u
+{\textstyle{i\over 2}}\,u \,(v_\mu-a_\mu)\, u^\dagger\;, \qquad \qquad
u = e^{i\,\frac{\Phi}{2\,f}} } \,,
\nonumber\\
&& H_{\mu\nu} =
{D}_\mu\,i\,U_\nu + {D}_\nu \,i\,U_\mu
\, ,\qquad \qquad \qquad \qquad
{D}_{\mu\nu} = {D}_\mu{D}_\nu + {D}_\nu{D}_\mu \,,\end{aligned}$$ where we emphasize the presence of the classical vector and axial-vector source fields, $v_\mu$ and $a_\mu$ of QCD [@Gasser1983a; @Gasser1983b]. The important merit of all building blocks $B, U_\mu, H_{\mu\nu}$ and $D_{\mu \nu}$ lies in their identical chiral transformation properties. Thus, the action of the covariant derivatives is implied by the example case (\[example-Dmu\]).
As derived first in [@Lutz2002a] there are 10 independent symmetry conserving $Q^3$ terms that are needed in the baryon octet sector. Such terms were studied in momentum space properly projected onto the kinematics required in meson-baryon scattering process. Initially there were 20 terms considered. It was shown in [@Lutz2002a] that only 10 terms are independent. This result was established by an evaluation of the s- and p-wave projections of their contributions to the scattering amplitudes. Explicit expressions how such terms contribute to the meson-baryon interaction kernel were provided in Appendix B of that work.
This result was confirmed later in [@Frink:2006hx; @Oller:2007qd] based on a complementary strategy. In fact, initially the authors of [@Oller:2007qd] claimed the relevance of 11 terms in [@Oller2006]. A result inconsistent with the original finding in [@Lutz2002a]. This error was corrected first in [@Frink:2006hx]. In the current work we use the 10 terms in their following representation $$\begin{aligned}
&& {\mathcal L}^{(3)}_{[8]\,[8]} =- u^{}_{1}\,\tr\,\bar{B}\,\gamma^\mu\,{B}\,[{U}^\nu,\,{H}_{\mu\nu}]_-
- u^{}_{2}\,\tr\,\bar{B}\,[{U}^\nu,\,{H}_{\mu\nu}]_-\,\gamma^\mu\,{B}
\nonumber\\
&& \qquad \quad - \,\tfrac{1}{2}\,u^{}_{3}\,\big(
\tr\,\bar{B}\,{U}^\nu\,\gamma^\mu\,\tr\,{H}_{\mu\nu}\,{B} + \text{h.c.}
\big)
\nonumber\\
&& \qquad \quad - \,\tfrac{1}{2}\,u^{}_{4}\,\big(
\tr\,\bar{B}\,\gamma^\lambda\,({D}^{\mu\nu}{B})\,[{U}_\lambda,\,{H}_{\mu\nu}]_-
+ \tr\,({D}^{\mu\nu}\bar{B})\,\gamma^\lambda\,{B}\,[{U}_\lambda,\,{H}_{\mu\nu}]_-
\big)
\nonumber\\
&& \qquad \quad - \,\tfrac{1}{2}\,u^{}_{5}\,\big(
\tr\,\bar{B}\,[{U}_\lambda,\,{H}_{\mu\nu}]_-\,\gamma^\lambda\,({D}^{\mu\nu}{B})
+ \tr\,({D}^{\mu\nu}\bar{B})\,[{U}_\lambda,\,{H}_{\mu\nu}]_-\,\gamma^\lambda\,{B}
\big)
\nonumber\\
&& \qquad \quad
-\, \tfrac{1}{4}\,u^{}_{6}\,\big(
\tr\,\bar{B}\,{U}_\lambda\,\gamma^\lambda\,\tr\,{H}_{\mu\nu}\,({D}^{\mu\nu}{B})
+\tr\ ({D}^{\mu\nu}\bar{B})\,{U}_\lambda\,\gamma^\lambda\,\tr\,{H}_{\mu\nu}\,{B}
+ \text{h.c.}
\big)
\nonumber\\
&& \qquad \quad
-\, \tfrac{1}{2}\,u^{}_{7}\,\big(
\tr\,\bar{B}\,\sigma^{\lambda \mu}\,({D}^{\nu}{B})\,[{U}_\lambda,\,{H}_{\mu \nu}]_+
- \tr\,({D}^{\nu} \bar{B})\,\sigma^{\lambda \mu}\,{B}\,[{U}_\lambda,\,{H}_{\mu \nu}]_+
\big)
\nonumber\\
&& \qquad \quad
-\, \tfrac{1}{2}\,u^{}_{8}\,\big(
\tr\,\bar{B}\,[{U}_\lambda,\,{H}_{\mu \nu}]_+\,\sigma^{\lambda \mu}\,({D}^{\nu}{B})
- \tr\,({D}^{\nu}\bar{B})\,[{U}_\lambda,\,{H}_{\mu \nu}]_+\,\sigma^{\lambda \mu}\,{B}
\big)
\nonumber\\
&& \qquad \quad
- \,\tfrac{1}{4}\,u^{}_{9}\,\big(
\tr\,\bar{B}\,{U}_\lambda\,\sigma^{\lambda \mu }\,({D}^{\nu}{B})\,{H}_{\mu \nu}
- \tr\,({D}^{\nu}\bar{B})\,{U}_\lambda\,\sigma^{\lambda \mu }\,{B}\,{H}_{\mu \nu}
+ \text{h.c.}
\big)
\nonumber\\
&& \qquad \quad - \,\tfrac{1}{2}\,u^{}_{10}\,\big(
\tr\,\bar{B}\,\sigma^{\lambda \mu}\,({D}^{\nu}{B})\,\tr\,{U}_\lambda\,{H}_{\mu \nu}
- \tr\,({D}^{\nu}\bar{B})\,\sigma^{\lambda \mu}\,{B}\,\tr\,{U}_\lambda\,{H}_{\mu \nu}
\big)
\,.\end{aligned}$$
We turn to the decuplet sector. The construction of the chiral Lagrangian is straightforward following the rules established by Krause in [@Krause1990]. We use here the conventional Rarita-Schwinger fields to interpolate to the decuplet of the spin-three-half states. The baryon decuplet field $B^{ijk}_\mu$ comes with three fully symmetric flavor indices, $i,j,k = 1,2,3$ as $$\begin{aligned}
\label{def:field-representation-decuplet}
\begin{array}{llll}
B_{\mu}^{111} = \Delta^{++}_\mu
\,, \qquad \;\;\;&
B_{\mu}^{112} = \Delta^{+}_\mu/\sqrt3
\,, \qquad &
B_{\mu}^{122} = \Delta^{0}_\mu/\sqrt3
\,, \qquad &
B_{\mu}^{222} = \Delta^{-}_\mu
\,,\\
B_{\mu}^{113} = \Sigma^{+}_\mu/\sqrt3
\,,&
B_{\mu}^{123} = \Sigma^{0}_\mu/\sqrt6
\,,&
B_{\mu}^{223} = \Sigma^{-}_\mu/\sqrt3
\,,& \\
B_{\mu}^{133} = \Xi^{0}_\mu/\sqrt3
\,,&
B_{\mu}^{233} = \Xi^{-}_\mu/\sqrt3
\,,& & \\
B_{\mu}^{333} = \Omega^{-}_\mu
\,,& & &
\end{array}\end{aligned}$$ where the components are identified with the states in the particle basis for convenience. The covariant derivative takes the form $$\begin{aligned}
(D_\mu {B}_\nu)^{ijh}
= \,
\partial_\mu {B}_\nu^{ijh}
+ \Gamma^{i}_{\mu,l}\,{B}_\nu^{ljh}
+ \Gamma^{j}_{\mu,l}\,{B}_\nu^{ilh}
+ \Gamma^{h}_{\mu,l}\,{B}_\nu^{ijl}
\,,\end{aligned}$$ where again the chiral connection $\Gamma_\mu$ is needed. In order to keep track of the various flavor index contraction in the many terms of the chiral Lagrangian we use here a powerful notation already introduced by one of the authors in [@Lutz:2001yb]. The idea behind the notation is to introduce a few auxiliary objects in terms of which any interaction term can be written down in terms of simple $3 \times 3$ matrix products like it is the case in the baryon octet sector. Indeed this is achieved by the consideration of suitable ’dot’ products of the decuplet fields. We need to discriminate the following three cases only $$\begin{aligned}
(\bar{B}^{\mu} \cdot {B}_\nu)^i_j
= \bar{B}^{\mu}_{jkl}\,{B}_\nu^{ikl}
\,,\qquad
(\bar{B}^{\mu} \cdot \Phi)^i_j = \epsilon^{kli}\,\bar{B}^{\mu}_{kmj}\,\Phi^m_l
\,,\qquad
(\Phi \cdot {B}_\nu)^i_j = \epsilon_{klj}\,\Phi^l_m\,{B}_\nu^{kmi}
\,,\end{aligned}$$ where any of such product yields a two-index object that transforms as a flavor octet field again. Note that it takes a bit of group theory that indeed all our terms in the chiral Lagrangian can be written down in such a notation. Given this fact, it is however, rather convenient to apply such a notation, since the painful write-down of flavor redundant terms can be avoided to a large extent. In [@Lutz:2001yb; @LutzSemke2010] all terms at order $Q^2$ that are relevant for meson-baryon scattering were written down for the first time. Such terms were recently rediscovered in [@Jiang:2018mzd; @Holmberg:2018dtv] using a less transparent notation. The first partial list of $Q^2$ terms involving the baryon decuplet field was published in [@Tiburzi:2004rh].
We now turn to the symmetry preserving $Q^3$ terms that involve a decuplet field. A complete list of 14 = 8+6 terms is readily worked out with $$\begin{aligned}
&& {\mathcal L}^{(3)}_{[10]\,[10]} = \tfrac{1}{2}\,v^{}_{1}\,\big(
\tr\,(\bar{B}_{\tau}\cdot\,{U}^\nu)\,\gamma^\mu\,({H}_{\mu\nu}\cdot\,{B}^{\tau})
+ \text{h.c.}
\big)
\nonumber\\ && \quad \,
+ \tfrac{1}{2}\,v^{}_{2}\,\big(
\tr\,(\bar{B}_{\lambda}\cdot\,{U}^{\lambda})\,\gamma^\mu\,({H}_{\mu\nu}\cdot\,{B}^{\nu})
+\text{h.c.}
\big)
+ \,\tfrac{1}{2}\,v^{}_{3}\,\big(
\tr\,(\bar{B}^{\nu}\cdot\,{U}^{\lambda})\,\gamma^\mu\,({H}_{\mu\nu}\cdot\,{B}_{\lambda})
+ \text{h.c.}
\big)
\nonumber\\ && \quad \,
+ \,\tfrac{1}{4}\,v^{}_{4}\,\big(
\tr\,(\bar{B}_{\tau}\cdot\,{U}_\lambda)\,\gamma^\lambda\,({H}_{\mu\nu}\cdot\,({D}^{\mu\nu}{B}^{\tau}))
+ \tr\,(({D}^{\mu\nu}\bar{B}_{\tau})\cdot\,{U}_\lambda)\,\gamma^\lambda\,({H}_{\mu\nu}\cdot\,{B}^{\tau})
+ \text{h.c.}
\big)
\nonumber\\ && \quad \,
+ \,\tfrac{1}{2}\,v^{}_{5}\,\big(
\tr\,(\bar{B}_{\tau}\cdot\,\sigma^{\lambda \mu}\,({D}^{\nu}{B}^{\tau}))\,[{U}_\lambda,\,{H}_{\mu \nu }]_+
- \tr\,(({D}^{\nu}\bar{B}_{\tau})\cdot\,\sigma^{\lambda \mu}\,{B}^{\tau})\,[{U}_\lambda,\,{H}_{\mu \nu }]_+
\big)
\nonumber\\ && \quad \,
+\, \tfrac{1}{4}\,v^{}_{6}\,\big(
\tr\,(\bar{B}_{\tau}\cdot\,{U}_\lambda)\,\sigma^{\lambda\mu}\,({H}_{\mu\nu}\cdot\,({D}^{\nu}{B}^{\tau}))
- \tr\,(({D}^{\nu}\bar{B}_{\tau})\cdot\,{U}_\lambda)\,\sigma^{\lambda\mu}\,({H}_{\mu\nu}\cdot\,{B}^{\tau})
+ \text{h.c.} \big)
\nonumber\\ && \quad \,
+ \,\tfrac{1}{2}\,v^{}_{7}\,\big(
\tr\,(\bar{B}_{\tau}\cdot\,\sigma^{\lambda \mu}\,({D}^{\nu}{B}^{\tau}))\,\tr\,{U}_\lambda\,{H}_{\mu \nu}
- \tr\,(({D}^{\nu}\bar{B}_{\tau})\cdot\,\sigma^{\lambda \mu}\,{B}^{\tau})\,\tr\,{U}_\lambda\,{H}_{\mu \nu}
\big)
\nonumber\\ && \quad \,
+\, \tfrac{1}{4}\,v^{}_{8}\,\big(
\tr\,(\bar{B}^{\mu}\cdot\,{U}_\lambda)\,({H}_{\mu\nu}\cdot\,({D}^{\lambda}{B}^{\nu}))
- \tr\,(({D}^{\lambda}\bar{B}^{\mu})\cdot\,{U}_\lambda)\,({H}_{\mu\nu}\cdot\,{B}^{\nu})
+ \text{h.c.} \big)
\,,\end{aligned}$$ and $$\begin{aligned}
&& {\mathcal L}^{(3)}_{[8]\,[10]} =
\tfrac{1}{2}\,w^{}_{1}\,\big(
\tr\,\big( \bar{B}^\nu \cdot [{U}_\lambda,\,{H}_{\mu\nu}]_+ \big)\,i\,\sigma^{\lambda\mu}\gamma_5\,{B}
+ \text{h.c.}
\big)
\nonumber\\ && \,
+ \tfrac{1}{4}\,w^{}_{2}\,\big(
\tr\,\big( \bar{B}^\lambda \cdot[{U}_\lambda,\,{H}_{\mu\nu}]_+ \big)\,i\,\gamma^\mu\gamma_5\,({D}^\nu{B})
- \tr\,\big( ({D}^\nu\bar{B}^\lambda) \cdot[{U}_\lambda,\,{H}_{\mu\nu}]_+ \big)\,i\,\gamma^\mu\gamma_5\,{B}
+ \text{h.c.}
\big)
\nonumber\\ && \,
+ \tfrac{1}{4}\,w^{}_{3}\,\big(
\tr\,\big( \bar{B}^\lambda \cdot[{U}_\mu,\,{H}_{\lambda\nu}]_+ \big)\,i\,\gamma^\mu\gamma_5\,({D}^\nu{B})
- \tr\,\big( ({D}^\nu\bar{B}^\lambda) \cdot[{U}_\mu,\,{H}_{\lambda\nu}]_+ \big)\,i\,\gamma^\mu\gamma_5\,{B}
+ \text{h.c.}
\big)
\nonumber\\
&& \, +\,
\tfrac{1}{2}\,w^{}_{4}\,\big(
\tr\,\big( \bar{B}^\nu \cdot [{U}_\lambda,\,{H}_{\mu\nu}]_- \big)\,i\,\sigma^{\lambda\mu}\gamma_5\,{B}
+ \text{h.c.}
\big)
\nonumber\\ && \,
+ \tfrac{1}{4}\,w^{}_{5}\,\big(
\tr\,\big( \bar{B}^\lambda \cdot[{U}_\lambda,\,{H}_{\mu\nu}]_- \big)\,i\,\gamma^\mu\gamma_5\,({D}^\nu{B})
- \tr\,\big( ({D}^\nu\bar{B}^\lambda) \cdot[{U}_\lambda,\,{H}_{\mu\nu}]_- \big)\,i\,\gamma^\mu\gamma_5\,{B}
+ \text{h.c.}
\big)
\nonumber\\ && \,
+ \tfrac{1}{4}\,w^{}_{6}\,\big(
\tr\,\big( \bar{B}^\lambda \cdot[{U}_\mu,\,{H}_{\lambda\nu}]_- \big)\,i\,\gamma^\mu\gamma_5\,({D}^\nu{B})
- \tr\,\big( ({D}^\nu\bar{B}^\lambda) \cdot[{U}_\mu,\,{H}_{\lambda\nu}]_- \big)\,i\,\gamma^\mu\gamma_5\,{B}
+ \text{h.c.}
\big) \,.\end{aligned}$$ We observe a significant mismatch with the number of seven terms claimed in [@Jiang:2018mzd].
Correlation function from the chiral Lagrangian
===============================================
We consider QCD’s axial-vector and vector currents, $$\begin{aligned}
&& A_\mu^{(a)}(x) = \bar \Psi (x)\,\gamma_\mu \,\gamma_5\,\frac{\lambda_a}{2}\,\Psi(x) \,, \qquad \qquad
V^{(a)}_\mu(x) = \bar \Psi (x)\,\gamma_\mu\,\frac{\lambda_a}{2}\,\Psi(x) \,,
\label{def-amu}\end{aligned}$$ where we recall their definitions in terms of the Heisenberg quark-field operators $\Psi(x)$. With $\lambda_a$ we denote the Gell-Mann flavor matrices. Our target is an evaluation of the following matrix elements $$\begin{aligned}
C^{(abe)}_{\mu\nu\lambda}(q,q')
= \int d^4{x}\,d^4{y}\,e^{+i\,q\cdot({x}-{y})} \,e^{+i\,q'\cdot y}\,
{\langle \,\bar p,\,\bar \chi|} \,
{\cal T}\,A_\mu^{(a)}(x)\,A_\nu^{(b)}(0)\,V_\lambda^{(e)}(y) \,{|p,\,\chi\rangle} \,,
\label{def-C}
\end{aligned}$$ in the baryon ground states. Here the spin projections of the initial and final baryon states we denote by $\chi$ and $\bar \chi$. Similarly the initial and final three momenta of the states are $p$ and $\bar p$. The flavor structure in (\[def-C\]) is incomplete since also the initial and final baryon states come in different flavor copies. We return to this issue below in more detail. Given the chiral Lagrangian, it is well defined how to derive the contributions to such matrix elements in application of the classical matrices of source functions, $a_\mu$ and $v_\mu$.
The particular correlation function is chosen as to selectively probe our $Q^3$ terms. This is so since any such term in the chiral Lagrangian is linear in the $U_\lambda$ field but also in the $H_{\mu \nu}$ field. Upon an expansion of those building blocks in powers of the meson fields one finds $$\begin{aligned}
i\,U_\mu = a_\mu + \cdots \,,\qquad \qquad \qquad
i\,H_{\mu \nu} = \big[ v_\mu, \, a_\nu \big]_- + \big[ v_\nu, \, a_\mu \big]_- + \cdots \,.
\end{aligned}$$ From here we conclude that the tree-level evaluation of the chiral Lagrangian is charcterized by the symmetry conserving $Q^3$ terms, as anticipated above.
The motivation for our study of this correlation function is twofold. First, it serves as a convenient tool as to verify whether we use only independent sets of the symmetry conserving $Q^3$ terms. We checked for the flavor octet case, that any additional term leads to a contribution that can be linear combined in terms of the 10 terms originally used in [@Lutz2002a] and confirmed later in [@Frink:2006hx; @Oller:2007qd]. An analogous computation consolidates our claim about the smallest set of independent terms in the decuplet sector. Second, such a correlation function can be scrutinized also in large-$N_c$ QCD. This will lead to sum rules amongst the set of low-energy constants introduced in this work. We will turn to this issue in the next section.
We close this section with explicit results for the correlation function. It suffices to evaluate the matrix elements in the strict flavor SU(3) limit. In this case a baryon octet or a decuplet state $$\begin{aligned}
{|p, \,\chi,\, c \rangle}\,, \qquad \qquad \qquad {|p, \,\chi, \,klm \rangle} \,,
\label{def-states}\end{aligned}$$ is specified by its three-momentum $p$ and the flavor indices $c=1,\cdots ,8$ or $k,l,m=1,2,3$. The spin-polarization label is $\chi = 1,2$ for the octet and $\chi =1,\cdots ,4$ for the decuplet states. In order to discriminate flavor structures from the currents versus those from the baryon states we introduce the operator $$\begin{aligned}
{\mathcal O}^{(abe)}_{ijh}(q,q')
= \int d^4{x}\,d^4{y}\,e^{+i\,q\cdot({x}-{y})} \,e^{+i\,q'\cdot y}\,
{\cal T}\,A_i^{(a)}(x)\,A_j^{(b)}(0)\,V_h^{(e)}(y) \,,
\label{def-O}
\end{aligned}$$ which matrix elements in the baryon states (\[def-states\]) are considered in the following. Note that in (\[def-O\]) we already focus on the space components of the three currents. From the study of such components the anticipated large-$N_c$ sum rules for the low-energy constants, the main target of our work, can be derived.
Since we will encounter many flavor indices in our work, which either run from one to three or from one to eight, we found it useful to split the alphabet into two parts. We use the Roman small letters from $a$ to $g$ for flavor indices with $a=1,..,8$ and letters from $h$ to $z$ for indices with $h=1,..,3$. With this convention it is easily confirmed over which range a given flavor index goes.
We are now prepared to present results for the matrix elements introduced with (\[def-C\]). A somewhat tedious but straightforward evaluation leads to the explicit results $$\begin{aligned}
&& \qquad \qquad \qquad \qquad \qquad \qquad
{\langle \bar{p}, \bar\chi, d|}\, {\mathcal O}^{(abe)}_{ijh}(q,q') \,{| p, \chi, c\rangle}
\nonumber\\ \nonumber\\
&& = \,
\bar{u}(\bar{p},\bar\chi)\frac{
2\,g^{{i}{j}}\,\gamma^{h}
+ g^{{i}{h}}\,\gamma^{j} + g^{{j}{h}}\,\gamma^{i}
}{4}\,{u}({p},\chi)
\,\Big\{
- (u^{}_{1}+u^{}_{2})\,\delta_{ab}\,d_{{d}{c}e}
- 3\,(u^{}_{1}+u^{}_{2})\,d_{abg}\,d_{efg}\,d_{{d}{c}f}
\nonumber\\
&& \qquad -\, (u^{}_{1}-u^{}_{2})\,\delta_{ab}\,if_{{d}{c}e}
- 3\,(u^{}_{1}-u^{}_{2})\,d_{abg}\,d_{efg}\,if_{{d}{c}f}
+ (u^{}_{1}+u^{}_{2})\,(\delta_{ae}\,d_{b{d}{c}} + \delta_{be}\,d_{a{d}{c}})
\nonumber\\
&& \qquad
+ \,(u^{}_{1}-u^{}_{2})\,(\delta_{ae}\,if_{b{d}{c}} + \delta_{be}\,if_{a{d}{c}})
- \tfrac{1}{2}\,u^{}_{3}\,(
\delta_{a{d}}\,if_{be{c}} + \delta_{b{d}}\,if_{ae{c}}
- \delta_{a{c}}\,if_{be{d}} - \delta_{b{c}}\,if_{ae{d}}
)
\Big\}
\nonumber\\ &&
+\, \bar{u}(\bar{p},\bar\chi)\,\frac{
g^{{i}{h}}\,\gamma^{j} - g^{{j}{h}}\,\gamma^{i}
}{4}\,{u}({p},\chi)
\,\Big\{
(u^{}_{1}+u^{}_{2})\,f_{abg}\,f_{efg}\,d_{{d}{c}f}
+ (u^{}_{1}-u^{}_{2})\,f_{abg}\,f_{efg}\,if_{{d}{c}f}
\nonumber\\
&& \qquad
-\, \tfrac{1}{2}\,u^{}_{3}\,(
\delta_{a{d}}\,if_{be{c}} - \delta_{b{d}}\,if_{ae{c}}
- \delta_{a{c}}\,if_{be{d}} + \delta_{b{c}}\,if_{ae{d}}
)
\Big\}
\nonumber\\ &&
+\, \bar{u}(\bar{p},\bar\chi)\,\frac{
i\,\sigma^{{i}{h}}\,\delta_{jk}
+ i\,\sigma^{{j}{h}}\,\delta_{ik}
}{8}\,{u}({p},\chi)
\,(\bar{p}+{p})^{k}
\,\Big\{
- (u^{}_{7} + u^{}_{8} - u^{}_{9})\,d_{abg}\,i\,f_{efg}\,d_{{d}{c}f}
\nonumber\\
&& \qquad
- \,(u^{}_{7} - u^{}_{8})\,d_{abg}\,i\,f_{efg}\,i\,f_{{d}{c}f}
- \tfrac{1}{2}\,u^{}_{9}\,(
\delta_{a{d}}\,i\,f_{be{c}} + \delta_{b{d}}\,i\,f_{ae{c}}
+ \delta_{a{c}}\,i\,f_{be{d}} + \delta_{b{c}}\,i\,f_{ae{d}} )
\Big\}
\nonumber\\ &&
+\, \bar{u}(\bar{p},\bar\chi)\, \frac{2\, i\,\sigma^{{i}{j}}\,\delta_{hk}
+ i\,\sigma^{{i}{h}}\,\delta_{jk}
- i\,\sigma^{{j}{h}}\,\delta_{ik}
}{8}
\,{u}({p},\chi)
\,\Big\{ - (
\tfrac{4}{3}\,u^{}_{7} + \tfrac{4}{3}\,u^{}_{8}
- \tfrac{1}{3}\,u^{}_{9}
+ 2\,u^{}_{10}
)\,if_{abe}\,\delta_{{d}c}
\nonumber \\
&& \qquad
+\, (u^{}_{7} + u^{}_{8} - \,u^{}_{9})\,(if_{aeg}\,d_{bgf} - if_{beg}\,d_{agf})\,d_{{d}{c}f}
\nonumber \\
&& \qquad +\, (u^{}_{7} - u^{}_{8})\,(if_{aeg}\,d_{bgf} - if_{beg}\,d_{agf})\,i\,f_{{d}{c}f}
\nonumber \\
&& \qquad
-\, \tfrac{1}{2}\,u^{}_{9}\,(
\delta_{a{d}}\,if_{be{c}} -\delta_{b{d}}\,if_{ae{c}}
+ \delta_{a{c}}\,if_{be{d}} - \delta_{b{c}}\,if_{ae{d}}
)
\Big\} \,(\bar{p}+{p})^{k}
\nonumber\\ &&
+\, \bar{u}(\bar{p},\bar\chi)\,\frac{
\bar{p}^{i}\,\gamma^{j}
+ \bar{p}^{j}\,\gamma^{i}
}{2}\,{u}({p},\chi)\,(\bar{p}+p)^{h}
\,\Big\{
- (u^{}_{4} + u^{}_{5})\,\delta_{ab}\,d_{{d}{c}e}
- 3\,(u^{}_{4} + u^{}_{5})\,d_{abg}\,d_{efg}\,d_{{d}{c}f}
\nonumber\\
&& \qquad
-\, (u^{}_{4} - u^{}_{5})\,\delta_{ab}\,if_{{d}{c}e}
- 3\,(u^{}_{4} - u^{}_{5})\,d_{abg}\,d_{efg}\,if_{{d}{c}f}
+ (u^{}_{4} + u^{}_{5})\,(\delta_{ae}\,d_{b{d}{c}} + \delta_{be}\,d_{a{d}{c}})
\nonumber\\
&& \qquad
+\, (u^{}_{4} - u^{}_{5})\,(\delta_{ae}\,i\,f_{b{d}{c}} + \delta_{be}\,if_{a{d}{c}})
\nonumber\\
&& \qquad
-\, \tfrac{1}{2}\,u^{}_{6}\,\big(
\delta_{a{d}}\,if_{be{c}} + \delta_{b{d}}\,if_{ae{c}}
- \delta_{a{c}}\,if_{be{d}} - \delta_{b{c}}\,\,if_{ae{d}}
\big)
\Big\}
\nonumber\\ &&
- \,
\bar{u}(\bar{p},\bar\chi)\,\frac{
\bar{p}^{i}\,\gamma^{j}
- \bar{p}^{j}\,\gamma^{i}
}{2}\,{u}({p},\chi)
\,\Big\{
(u^{}_{4} + u^{}_{5})\,f_{abg}\,f_{efg}\,d_{{d}{c}f}
+ (u^{}_{4} - u^{}_{5})\,f_{abg}\,f_{efg}\,if_{{d}{c}f}
\nonumber\\
&& \qquad
- \,\tfrac{1}{2}\,u^{}_{6}\,(
\delta_{a{d}}\,if_{be{c}} - \delta_{b{d}}\,if_{ae{c}}
- \delta_{a{c}}\,if_{be{d}} + \delta_{b{c}}\,if_{ae{d}}
)
\Big\}\,(\bar{p}-p)^{h}
\,.
\label{res-15}\end{aligned}$$ Corresponding expressions for matrix elements in the baryon decuplet states are collected in Appendix A. We wish to emphasize that the computation of such matrix elements serves as a powerful consistency check whether the terms of the chiral Lagrangian were constructed properly. Our results (\[res-15\]) show that all terms shown are independent, i.e. it is not possible to eliminate any term.
Current correlation function in large-$N_c$ QCD
===============================================
Consider ${\mathcal O}_{QCD}$ to be the time ordered product of any combination of local currents in large-$N_c$ QCD, where (\[def-O\]) may serve as a specific example for $N_c = 3$. The generic form of the large-$N_c$ operator expansion can be taken as $$\begin{aligned}
{\langle \,\bar p,\bar\chi\,|}\,{\mathcal O}_{QCD}\,
{|p,\,\chi\,\rangle} = \sum_{n=0}^\infty\, c_n( \bar p, p)\,
{(\bar\chi \,|} \,{\mathcal O}^{(n)}_{\rm static} \,
{|\chi\,)} \,,
\label{def-largeN-expansion}\end{aligned}$$ where it is important to note that unlike the physical baryon states, ${|p,\,\chi\,\rangle}$, the effective baryon states, ${|\chi\,)}$, do not depend on the three-momentum $p$. All dynamical information in (\[def-largeN-expansion\]) is moved into appropriate coefficient functions $c_n(\bar p, p)$. Moreover, in the decomposition (\[def-largeN-expansion\]) the coefficients $ c_n(\bar p, p) $ depend on neither the flavor nor the spin quantum number of the initial or the final baryon state. The merit of (\[def-largeN-expansion\]) lies in the fact that the contributions on its right-hand-side can be sorted according to their relevance at large values of $N_c$.
The effective baryon states ${|c, \chi)}$ and ${|klm, \chi)}$ have a mean-field structure that can be generated in terms of effective quark operators. They correspond to the baryon states already introduced with (\[def-states\]) for the particular choice $N_c =3$. A complete set of color-neutral one-body operators may be constructed in terms of the very same static quark operators $$\begin{aligned}
&& \quarknumberoperator = q^\dagger ( \one \otimes \one \otimes \one )\,q \,, \qquad \qquad \;\;\,
J_i = q^\dagger \Big(\frac{\sigma_i }{2} \otimes \one \otimes \one \Big)\, q \,,
\nonumber\\
&& T^a = q^\dagger \Big(\one \otimes \frac{\lambda_a}{2} \otimes \one \Big)\, q\, ,\qquad \quad \;\;
G^a_i = q^\dagger \Big( \frac{\sigma_i}{2} \otimes \frac{\lambda_a}{2} \otimes \one \Big)\, q\,,
\label{def:one-body-operators}\end{aligned}$$ with operators $q=(u,d,s)^T$ introduced for the up, down and strange quarks. With $\lambda_a$ we denote the Gell-Mann matrices. While the action of any of the spin-flavor operators introduced in (\[def:one-body-operators\]) on the tower of large-$N_c$ states is quite involved at large $N_c \neq 3$ matters turn quite simple and straightforward at the physical value $N_c = 3$. For this physical case where there is a flavor octet with spin-one-half or a flavor decuplet with spin-three-half only, we recall the well established results of [@Lutz2002a; @LutzSemke2010] with $$\begin{aligned}
&& \quarknumberoperator \,{|c,\chi)} = 3\,{|c,\chi)}\,,
\qquad \qquad
\nonumber \\
&& J_i \,{|c,\chi)}=\frac{1}{2}\, \sigma^{(i)}_{{\bar \chi} \chi}\, {|c,{\bar \chi})}\,,
\qquad \qquad
T^a\, {|c,\chi)} = i\,f_{cda}\, {|d,\chi)}\,,
\nonumber \\
&& G^{a}_i\, {|c,\chi)} = \sigma^{(i)}_{{\bar \chi} \chi}\, \Big(\frac12\,d_{cda} + \frac{i}{3}\, f_{cda}\Big)\,
{|d,{\bar \chi})} + \frac{1}{2\sqrt{2}}\, S^{(i)}_{{\bar \chi} \chi}\, \Lambda_{ac}^{klm}
\, {|klm,{\bar \chi})}\,,
\nonumber \\
\nonumber \\
&& \quarknumberoperator \,{|klm, \chi)}=3\,{|klm, \chi)}\,,
\qquad \qquad
\nonumber\\
&& J_i \,{|klm, \chi)}= \frac{3}{2}\,\Big(\vec S \,\sigma_i\, \vec S^\dagger \Big)_{{\bar \chi} \chi}\, {|klm, {\bar \chi})},
\qquad \qquad
T^a \,{|klm, \chi)}= \frac{3}{2}\,\Lambda^{a,nop}_{klm}\, {|nop,\chi)},
\nonumber \\
&& G^a_i \,{|klm, \chi)}= \frac34 \, \Big(\vec S\,\sigma_i\, \vec S^\dagger \Big)_{{\bar \chi} \chi}\,
\Lambda^{a,nop}_{klm}\, {|nop, {\bar \chi})} +
\frac{1}{2\sqrt{2}} \, \Big(S^{\dagger}_i \,\Big)_{{\bar \chi} \chi}\, \Lambda^{ac}_{klm}\, {|c,{\bar \chi})}\,,
\label{result:one-body-operators}\end{aligned}$$ with the Pauli matrices $\sigma_i$ and the spin-transition matrices $S_i$ characterized by $$\begin{aligned}
&& S^\dagger_i\, S_j= \delta_{ij} - \frac{1}{3}\sigma_i \sigma_j \,, \qquad S_i\,\sigma_j - S_j\,\sigma_i = -i\,\varepsilon_{ijk} \,S_k\,,
\qquad \vec S\cdot \vec S^\dagger= \one_{(4\times 4)}\,,
\nonumber\\
&& \vec S^\dagger \cdot \vec S =2\, \one_{(2\times 2)}\,, \qquad \vec S \cdot \vec \sigma = 0 \,,\qquad
\epsilon_{ijk}\,S_i\,S^\dagger_j = i\,\vec S \,\sigma_k\,\vec S^\dagger\,.
\label{def:spin-transition-matrices}\end{aligned}$$ We recall some instrumental flavor structures $$\begin{aligned}
&& \Lambda_{ab}^{klm} = \Big[\varepsilon_{ijk}\, \lambda^{(a)}_{li}\,
\lambda^{(b)}_{mj} \,\Big]_{\mathrm{sym}(klm)}\,,\qquad \quad
\delta^{\,klm}_{\,nop} \;\;= \Big[\delta_{kn}\,\delta_{lo}\,\delta_{mp} \,\Big]_{\mathrm{sym}(nop)}\,,
\nonumber\\
&& \Lambda^{ab}_{klm} = \Big[\varepsilon_{ijk}\, \lambda^{(a)}_{il}\,
\lambda^{(b)}_{jm} \,\Big]_{\mathrm{sym}(klm)}\,,\qquad \quad
\Lambda^{a,klm}_{nop} = \Big[\lambda^{(a)}_{kn} \delta_{lo} \,\delta_{mp}\Big]_{\mathrm{sym}(nop)}\,,
\label{def:flavor-transition-matrices}\end{aligned}$$ that occur frequently in our previous and current works [@Lutz:2010se; @Lutz:2014jja; @Lutz:2018cqo].
In the sum of (\[def-largeN-expansion\]) there are infinitely many terms one may write down. The static operators ${\mathcal O}_{\rm static}^{(n)}$ are finite products of the one-body operators $J_i\,, T^a$ and $G^a_i$. In contrast the counting of $N_c$ factors is intricate since there is a subtle balance of suppression and enhancement effects. An $r$-body operator consisting of the $r$ products of any of the spin and flavor operators receives the suppression factor $N_c^{-r}$. This is counteracted by enhancement factors for the flavor and spin-flavor operators $T^a$ and $G^a_i$ that are produced by taking baryon matrix elements at $N_c \neq 3$. Altogether this leads to the effective scaling laws [@Dashen1994] $$\begin{aligned}
J_i \sim \frac{1}{N_c} \,, \qquad \quad T^a \sim N^0_c \,, \qquad \quad G^a_i \sim N^0_c \,.
\label{effective-counting}\end{aligned}$$ According to (\[effective-counting\]) there are an infinite number of terms contributing at a given order in the the $1/N_c$ expansion. Taking higher products of flavor and spin-flavor operators does not reduce the $N_c$ scaling power. A systematic $1/N_c$ expansion is made possible by a set of operator identities [@Dashen1994; @Lutz:2010se], that allows a systematic summation of the infinite number of relevant terms. As a consequence of the SU(6) Lie algebra any commutator of one-body operators can be expressed in terms of one-body operators again. Therefore it suffices to consider anti commutators of the one-body operators [@Dashen1994; @Lutz:2010se]. For instance, consider the following two identities that hold in matrix elements of the baryon states $$\begin{aligned}
d_{gab}\,[T_a,\,T_b]_+
&& \, = \,
- 2\,T_g
+ 2\,[J^i,\,G^i_g]_+
\,,\nonumber\\
d_{gab}\,[G^i_a,\,G^j_b]_+
&& \, = \,
\tfrac{1}{3}\,\delta^{ij}\,\big(
\tfrac{9}{2}
\,T_g - \tfrac{3}{2}\,[J^k,\,G^k_g]_+\big)
+ \tfrac{1}{6}\,\big([J^i,\,G^j_g]_++[J^j,\,G^i_g]_+\big)\,.
$$ Altogether the expansion scheme is implied by two reduction rules:
- All operator products in which two flavor indices are contracted using $\delta_{ab}$, $f_{abc}$ or $d_{abc}$ or two spin indices on $G$’s are contracted using $\delta_{ij}$ or $\varepsilon_{ijk}$ can be eliminated.
- All operator products in which two flavor indices are contracted using symmetric or antisymmetric combinations of two different $d$ and/or $f$ symbols can be eliminated. The only exception to this rule is the antisymmetric combination $f_{acg}\,d_{bch}-f_{bcg}\,d_{ach}$.
As a consequence the infinite tower of spin-flavor operators truncates at any given order in the $1/N_c$ expansion. We can now turn to the $1/N_c$ expansion of the baryon matrix elements of our specific product of QCD’s axial-vector and vector currents. In application of the operator reduction rules, the baryon matrix elements of time-ordered products of the current operators are expanded in powers of the effective one-body operators according to the counting rule (\[effective-counting\]) supplemented by the reduction rules.
Sum rules for the low-energy constants
======================================
As compared to previous works [@Luty1994; @Dashen1995; @Lutz:2010se; @Lutz:2018cqo] that dealt with correlation functions of one or two currents only, it turned out that the systematic construction of the large-$N_c$ operator hierachy for the correlation function of three currents is considerably more involved. While it is straightforward to write down a set of operators to a given order almost any single term cannot be matched to the matrix elements as implied by the chiral Lagrangian. This is so since the role of charge conjugation and parity invariances is not so transparent in the given frame work. We derived our operators by considering all possible combinations and then performed the matching in application of a suitable computer algebra code. This then generated the following leading order decomposition $$\begin{aligned}
&& {\cal O}^{ijh}_{abe}
= \delta^{(ij)_+}_{h}\,\Big\{
\hat{g}_{1}\,\big(
\delta_{ab}\,T_e
- (\delta_{ae}\,T_b + \delta_{be}\,T_a)
+ 3\,d_{abg}\,d_{efg}\,T_f
\big)
\nonumber\\ && \, \qquad\qquad
-\, \tfrac{1}{2}\,\hat{g}_{4}\,\big\{
d_{aeg}\,[J^l,\,([T_g,\,G^l_b]_+ - [T_b,\,G^l_g]_+)]_+
+ d_{beg}\,[J^l,\,([T_g,\,G^l_a]_+ - [T_a,\,G^l_g]_+)]_+
\nonumber\\ && \, \qquad\qquad\qquad
- 2\,d_{abg}\,[J^l,\,([T_g,\,G^l_e]_+ - [T_e,\,G^l_g]_+)]_+
\big\} \Big\}
\nonumber\\
&& \qquad +\, (\bar{p}+{p})^q\,\Big\{\big(i\,\epsilon^{iju}\,\delta_{hq} + \delta^{(ij)_-}_{(vq)_-}\,i\,\epsilon^{huv}\big)\,\Big[
\hat{g}_{2}\,(i\,f_{aeg}\,d_{gbf} - i\,f_{beg}\,d_{gaf})\,G^u_f
+ \hat{g}_{5}\,i\,f_{abe}\,J^u
\nonumber\\ && \, \qquad\qquad \qquad
+ \,\hat{g}_{6}\,(i\,f_{aeg}\,d_{bfg} - i\,f_{beg}\,d_{afg})\,[J^u,\,T_f]_+
\Big]
\nonumber\\ && \, \qquad \qquad
+\, \delta^{(ij)_+}_{(vq)_+}\,i\,\epsilon^{huv}\,\Big[
\hat{g}_{3}\,d_{abg}\,i\,f_{feg}\,G^u_f
+\hat{g}_{7}\,d_{abg}\,if_{efg}\,[J^u,\,T_f]_+
\Big]
\Big\}
\,,
\label{QCD-identity-AVA}\end{aligned}$$ where the parameters $\hat g_{1-3}$ and $\hat g_{4-7}$ are relevant at leading and subleading orders respectively. In (\[QCD-identity-AVA\]) we use the notation $$\begin{aligned}
\delta_{(mn)_\pm}^{(ij)_\pm} =
\tfrac{1}{2}\,(\delta_{mi}\,\delta_{nj} \pm \delta_{mj}\,\delta_{ni})\,,\quad
\delta^{(ij)_\pm}_h = \tfrac{1}{2\,(\bar{M}+{M})}\,\big(\bar p^i \,\bar p^h + p^i \, p^h\big) \,\big(\bar p+ p\big)^j \pm (i \leftrightarrow j) \,.\end{aligned}$$
------------------------------------------------------------------------------------ ------------------------------------------------------------------------------- --------------------------------
$ u^{}_{1} = 0 $ $ v^{}_{1} = 0 $ $ w^{}_{1} = 0 $
$ u^{}_{2} = 0 $ $ v^{}_{2} = 0 $ $ w^{}_{2} = - 4\,\hat{g}_{2}$
$ u^{}_{3} = 0 $ $ v^{}_{3} = 0 $ $ w^{}_{3} = 4\,\hat{g}_{2} $
$ u^{}_{4} = \tfrac{1}{2}\,\hat{g}_{1} + \tfrac{1}{2}\,\hat{g}_{4} $ $ v^{}_{4} = - 3\,\hat{g}_{1} $ $ w^{}_{4} = 0 $
$ u^{}_{5} = -\tfrac{1}{2}\,\hat{g}_{1} - \tfrac{1}{2}\,\hat{g}_{4} $ $ v^{}_{5} = -3\,\hat{g}_{2} - 18\,\hat{g}_{6} $ $ w^{}_{5} = 0 $
$ u^{}_{6} = 3\,\hat{g}_{4} $ $ v^{}_{6} = 0 $ $ w^{}_{6} = 0 $
$ u^{}_{7} = \tfrac{1}{3}\,\hat{g}_{2} - 2\,\hat{g}_{6} $ $ v^{}_{7} = 2\,\hat{g}_{2} + 3\,\hat{g}_{6} + 12\,\hat{g}_{6} \qquad \qquad$
$ u^{}_{8} = \tfrac{5}{3}\,\hat{g}_{2} + 2\,\hat{g}_{6} $ $ v^{}_{8} = 0$
$ u^{}_{9} = 0 $
$ u^{}_{10}= -\tfrac{4}{3}\,\hat{g}_{2} - \hat{g}_{5} \qquad \qquad $
------------------------------------------------------------------------------------ ------------------------------------------------------------------------------- --------------------------------
: Matching of the large-$N_c$ operators to the LEC. []{data-label="tab:algebraic-sol"}
Owing to the matching condition $$\begin{aligned}
\hat g_2 + \hat g_3 = 0 \,,
\label{matching-condition-1}\end{aligned}$$ there are two leading order operators only. This is a non-trivial result in view of the fact that one may write down many more leading order operators. For instance consider the particular term $$\begin{aligned}
\delta^{(ij)_+}_{h}\,\Big( 2\,[[T_a,\,T_b]_+,\,T_e]_+
- ([[T_a,\,T_e]_+,\,T_b]_+ + [[T_b,\,T_e]_+,\,T_a]_+) \Big) \,,
\label{eliminate}\end{aligned}$$ which matrix elements can be shown to be proportional to the matrix elements of the operator associated with $\hat g_1$. At subleading order we find three additional operators only. Here the matching condition $$\begin{aligned}
\hat g_6 + \hat g_7 = 0 \,,\end{aligned}$$ eliminates one term.
The number of independent coupling constants in the chiral Lagrangian is 24. At leading order in the $1/N_c$ expansion all of them can be expressed in terms of $\hat g_1$ and $\hat g_2$ as detailed in Tab. \[tab:algebraic-sol\]. At subleading order the additional three parameters $\hat g_4, \hat g_5 $ and $\hat g_6$ enter. The desired sum rules follow upon eliminating the parameters $\hat g_n$. There are 15 common sum rules applicable at LO and NLO $$\begin{aligned}
&& u^{}_{1,2,3} = 0 = u^{}_{9} \,,\qquad \quad
v^{}_{1,2,3} = 0 = v^{}_{6,8}\,, \qquad \quad w^{}_{4,5,6} = 0= w^{}_1\,,
\nonumber\\
&& u^{}_{5} = -u^{}_{4}\,,\qquad \qquad \quad \;
w^{}_{3}
= -w^{}_{2}
\,.\end{aligned}$$ They are supplemented by 4 and 7 additional sum rules at NLO and LO respectively as $$\begin{aligned}
&& v^{}_{5} = 6\,u^{}_{7}-3\,u^{}_{8}
\,,\qquad \qquad \qquad
v^{}_{4} = 6\,u^{}_{5}+u^{}_{6}
\,,
\nonumber\\
&& v^{}_{7} = -6\,u^{}_{7}-3\,u^{}_{10}
\,,\qquad \qquad \quad \!
w^{}_{3}
= 2\,(u^{}_{7}+u^{}_{8}) \,,
$$ and $$\begin{aligned}
&& v^{}_{5}
= -9\,u^{}_{7}
= \tfrac{9}{4}\,u^{}_{10}
\,,\qquad \qquad
v^{}_{4} = 6\,u^{}_{5}
\,,\qquad \qquad
u^{}_{6}=0
\,,
\nonumber\\ &&
v^{}_{7}
= 6\,u^{}_{7}
\,,\qquad \qquad \qquad
w^{}_{3}
= 12\,u^{}_{7}
= \tfrac{12}{5}\,u^{}_{8}
\,.\end{aligned}$$
Summary
=======
In this work we further prepared the ground for realistic applications of the chiral Lagrangian with the baryon octet and the baryon decuplet fields. For the first time all symmetry preserving $Q^3$ counter terms were constructed as they are relevant for any two-body meson-baryon reaction process. Altogether we find 24 terms. In order to pave the way towards applications of this set of low-energy parameters we derived a set of sum rules. We considered matrix elements of a correlation function with two axial-vector and one vector currents in the baryon ground states as they arise in QCD at a large number of colors ($N_c$). From a systematic operator expansion thereof we deduced our set of 22 sum rules valid at leading order in the $1/N_c$ expansion. At subleading order there remain 19 relations. With our result we now deem it feasible to perform significant coupled-channel studies of meson-baryon scattering processes considering channels with the baryon octet and decuplet fields on an equal footing as it is requested by large-$N_c$ QCD.
0.3cm [**[Acknowledgments]{}**]{} 0.3cm Y. Heo and C. Kobdaj acknowledge partial support from Suranaree University of Technology, the Office of the Higher Education Commission under NRU project of Thailand (SUT-COE: High Energy Physics and Astrophysics) and SUT-CHE-NRU (Grant No. FtR.11/2561).
Appendix A
==========
$$\begin{aligned}
&& \qquad \qquad \qquad \qquad \qquad \qquad
{\langle \bar{p},\,\bar \chi,\,nop\, |}\, {\mathcal O}^{(abe)}_{ijh}(q,q') \,{|p, \,\chi,\, klm\, \rangle}
\nonumber\\ \nonumber\\
&& = \,
-\,\tfrac{1}{4}\,
\delta^{{nop}}_{xyz}\,\Lambda^{d,xyz}_{klm}\,
\big( 3\,d_{abg}\,d_{edg}
+ \delta_{ab}\,\delta_{ed}
- \delta_{ae}\,\delta_{bd} - \delta_{ad}\,\delta_{be} \big)
\,\Big\{
\nonumber\\
&& \qquad
+\, \tfrac{1}{2}\,v^{}_{1} \bar{u}_{\tau}(\bar{p},\bar\chi)\,\big(
2\,\gamma^{{h}}\,g^{{i}{j}}
+ \gamma^{{i}}\,g^{{h}{j}}
+ \gamma^{{j}}\,g^{{h}{i}}
\big)\,{u}^\tau({p},\chi)\,
\nonumber\\
&& \qquad + \, \tfrac{1}{2}\,v^{}_{2}\left(
\bar{u}^{{i}}(\bar{p},\bar\chi)\,\gamma^{{j}}\,{u}^{h}({p},\chi)
+ \bar{u}^{{j}}(\bar{p},\bar\chi)\,\gamma^{{i}}\,{u}^{h}({p},\chi)
+ \bar{u}^{{i}}(\bar{p},\bar\chi)\,\gamma^{{h}}\,{u}^{j}({p},\chi)
+ \bar{u}^{{j}}(\bar{p},\bar\chi)\,\gamma^{{h}}\,{u}^{i}({p},\chi)
\right)\,
\nonumber\\
&& \qquad
+ \, \tfrac{1}{2}\,v^{}_{3}\left(
\bar{u}^{{i}}(\bar{p},\bar\chi)\,\gamma^{{h}}\,{u}^{j}({p},\chi)
+ \bar{u}^{{j}}(\bar{p},\bar\chi)\,\gamma^{{h}}\,{u}^{i}({p},\chi)
+ \bar{u}^{{h}}(\bar{p},\bar\chi)\,\gamma^{{i}}\,{u}^{j}({p},\chi)
+ \bar{u}^{{h}}(\bar{p},\bar\chi)\,\gamma^{{j}}\,{u}^{i}({p},\chi)
\right)\,
\Big\}\,
\nonumber\\ && \,
-\,\tfrac{1}{4}\,
\delta^{{nop}}_{xyz}\,\Lambda^{d,xyz}_{klm}\, f_{abg}\,f_{edg}
\,\Big\{
\tfrac{1}{2}\,v^{}_{1}\,\bar{u}_{\tau}(\bar{p},\bar\chi)\,\big(
\gamma^{{i}}\,g^{{h}{j}}
- \gamma^{{j}}\,g^{{h}{i}}
\big)\,{u}^\tau({p},\chi)\,
\nonumber\\
&& \qquad
- \,\tfrac{1}{2}\,v^{}_{2} \left(
\bar{u}^{{i}}(\bar{p},\bar\chi)\,\gamma^{{j}}\,{u}^{h}({p},\chi)
- \bar{u}^{{j}}(\bar{p},\bar\chi)\,\gamma^{{i}}\,{u}^{h}({p},\chi)
+ \bar{u}^{{i}}(\bar{p},\bar\chi)\,\gamma^{{h}}\,{u}^{j}({p},\chi)
- \bar{u}^{{j}}(\bar{p},\bar\chi)\,\gamma^{{h}}\,{u}^{i}({p},\chi)
\right)\,
\nonumber\\
&& \qquad
+\, \tfrac{1}{2}\,v^{}_{3} \left(
\bar{u}^{{i}}(\bar{p},\bar\chi)\,\gamma^{{h}}\,{u}^{j}({p},\chi)
- \bar{u}^{{j}}(\bar{p},\bar\chi)\,\gamma^{{h}}\,{u}^{i}({p},\chi)
+ \bar{u}^{{h}}(\bar{p},\bar\chi)\,\gamma^{{i}}\,{u}^{j}({p},\chi)
- \bar{u}^{{h}}(\bar{p},\bar\chi)\,\gamma^{{j}}\,{u}^{i}({p},\chi)
\right)\,
\Big\}\,
\nonumber\\ && \,
-\,\tfrac{1}{8}\,\bar{u}_{\tau}(\bar{p},\bar\chi)\,\left(
i\sigma^{{i}{h}}\,(\bar{p}+{p})^{j} + i\sigma^{{j}{h}}\,(\bar{p}+{p})^{i}
\right)\,
{u}^\tau({p},\chi)\,
\,\Big\{
(v^{}_{5} + \tfrac{3}{4}\,v^{}_{6})\,d_{abg}\,if_{efg}\,
\delta^{nop}_{xyz}\,\Lambda^{f,xyz}_{klm}
\nonumber\\
&& \qquad
- \,\tfrac{3}{8}\,v^{}_{6}\,
\delta^{nop}_{rst}\,\big(
(\Lambda^{a,rst}_{xyz}\,if_{beg} + \Lambda^{b,rst}_{xyz}\,if_{aeg})\,\Lambda^{g,xyz}_{klm}
+ \Lambda^{g,rst}_{xyz}\,(\Lambda^{a,xyz}_{klm}\,if_{beg} + \Lambda^{b,xyz}_{klm}\,if_{aeg})
\big)
\Big\}
\nonumber\\ && \,
-\, \tfrac{1}{8}\,\bar{u}_{\tau}(\bar{p},\bar\chi)\,\left(
2\,i\sigma^{{i}{j}}\,(\bar{p}+{p})^{h}
+ i\sigma^{{i}{h}}\,(\bar{p}+{p})^{j} - i\sigma^{{j}{h}}\,(\bar{p}+{p})^{i}
\right)\,
{u}^\tau({p},\chi)\,\Big\{
\nonumber\\
&& \qquad +\,
2\,(
\tfrac{2}{3}\,v^{}_{5} + \tfrac{1}{2}\,v^{}_{6} + v^{}_{7}
)\,\delta^{nop}_{klm}\,if_{abe}
+ (v^{}_{5} + \tfrac{3}{4}\,v^{}_{6})\,(if_{beg}\,d_{agf} - if_{aeg}\,d_{bgf})\,
\delta^{nop}_{xyz}\,\Lambda^{f,xyz}_{klm}
\nonumber\\
&& \qquad
- \tfrac{3}{8}\,v^{}_{6}\,
\delta^{nop}_{rst}\,\big(
(\Lambda^{a,rst}_{xyz}\,if_{beg} - \Lambda^{b,rst}_{xyz}\,if_{aeg})\,\Lambda^{g,xyz}_{klm}
+ \Lambda^{g,rst}_{xyz}\,(\Lambda^{a,xyz}_{klm}\,if_{beg} - \Lambda^{b,xyz}_{klm}\,if_{aeg})
\big)
\Big\}
\nonumber\\ && \,
- \, \tfrac{1}{8}\,\delta^{nop}_{xyz}\,\Lambda^{f,xyz}_{klm}\,\big(
3\,d_{abg}\,d_{feg}
+ \delta_{ab}\,\delta_{fe}
- \delta_{af}\,\delta_{be} - \delta_{ae}\,\delta_{bf}
\big)\,
\Big\{
\nonumber\\
&& \qquad +\,
\tfrac{1}{2}\,v_{8}\, \big(
(
\bar{u}^{h}(\bar{p},\bar\chi)\,{u}^{j}({p},\chi)
+ \bar{u}^{j}(\bar{p},\bar\chi)\,{u}^{h}({p},\chi)
)\,(\bar{p}+{p})^{i}
\nonumber\\
&& \qquad\qquad +\,
(
\bar{u}^{h}(\bar{p},\bar\chi)\,{u}^{i}({p},\chi)
+ \bar{u}^{i}(\bar{p},\bar\chi)\,{u}^{h}({p},\chi)
)\,(\bar{p}+{p})^{j}
\big)\,
\Big\}
\nonumber\\ &&\,
- \, \tfrac{1}{8}\,\delta^{nop}_{xyz}\,\Lambda^{f,xyz}_{klm}\,
f_{abg}\,f_{feg}\,
\Big\{
\nonumber\\
&& \qquad +\,
\tfrac{1}{2}\,v_{8}\, \big(
(
\bar{u}^{h}(\bar{p},\bar\chi)\,{u}^{j}({p},\chi)
+ \bar{u}^{j}(\bar{p},\bar\chi)\,{u}^{h}({p},\chi)
)\,(\bar{p}+{p})^{i}
\nonumber\\
&& \qquad\qquad -\,
(
\bar{u}^{h}(\bar{p},\bar\chi)\,{u}^{i}({p},\chi)
+ \bar{u}^{i}(\bar{p},\bar\chi)\,{u}^{h}({p},\chi)
)\,(\bar{p}+{p})^{j}
\big)\,
\Big\}
\nonumber\\ && \,
+\, \tfrac{1}{4}\, \delta^{nop}_{xyz}\,\Lambda^{d,xyz}_{klm}\,
\big( 3\,d_{abg}\,d_{edg}
+ \delta_{ab}\,\delta_{ed}
- \delta_{ae}\,\delta_{bd} - \delta_{ad}\,\delta_{be} \big)
\,\Big\{
\nonumber\\
&& \qquad
+ v^{}_{4}\,\bar{u}_\tau(\bar{p},\bar\chi)\,\big(
\bar{p}^{i}\,\gamma^{j}
+ \bar{p}^{j}\,\gamma^{i}\,
\big)\,{u}^\tau({p},\chi)\,(\bar{p}+p)^{h}
\Big\}
\nonumber\\ && \,
- \, \tfrac{1}{4}\, \bar{u}_\tau(\bar{p},\bar\chi)\,\big(
\, \bar{p}^{i}\,\gamma^{j}\,
- \bar{p}^{j}\,\gamma^{i}
\big)\,{u}^\tau({p},\chi)\,(\bar{p}-p)^{h}
\,\delta^{nop}_{xyz}\,\Lambda^{d,xyz}_{klm}\,
v^{}_{4}\, f_{abg}\,f_{edg} \,,\end{aligned}$$
and $$\begin{aligned}
&& \qquad \qquad \qquad \qquad \qquad \qquad
{\langle \bar{p},\,\bar \chi,\,nop\, |}\, {\mathcal O}^{(abe)}_{ijh}(q,q') \,{| p, \chi, c\rangle}
\nonumber\\ \nonumber\\
&& = \,
\tfrac{1}{8\,\sqrt{2}} \,\big(
\bar{u}^{i}(\bar{p},\bar\chi)\,i\,\sigma^{{j}{h}} + \bar{u}^{j}(\bar{p},\bar\chi)\,i\,\sigma^{{i}{h}}
\big)\,\gamma_5\,{u}({p},\chi)\,\Big\{
- w^{}_{1}\,d_{abf}\,i\,f_{egf}
\nonumber\\
&& \qquad
-\, w^{}_{4}\,\big(3\,d_{abf}\,d_{gef}
+ \delta_{ab}\,\delta_{ge} - (\delta_{ag}\,\delta_{be} + \delta_{bg}\,\delta_{ae})\big)
\Big\}\,\Lambda^{nop}_{gc}
\nonumber\\ && \,
+ \,\tfrac{1}{16\,\sqrt{2}} \,\big(
( \bar{u}^{i}(\bar{p},\bar\chi)\,\gamma^{j} + \bar{u}^{j}(\bar{p},\bar\chi)\,\gamma^{i}
)\,(\bar{p}+{p})^{h}
\nonumber\\
&& \qquad \qquad \qquad
+(
\bar{u}^{i}(\bar{p},\bar\chi)\,(\bar{p}+{p})^{j} + \bar{u}^{j}(\bar{p},\bar\chi)\,(\bar{p}+{p})^{i}
)\,\gamma^{h}
\big)\,\gamma_5\,{u}({p},\chi)\,\Big\{
- w^{}_{2}\,d_{abf}\,i\,f_{egf}
\nonumber\\
&& \qquad
- w^{}_{5}\,\big(3\,d_{abf}\,d_{gef}
+ \delta_{ab}\,\delta_{ge} - (\delta_{ag}\,\delta_{be} + \delta_{bg}\,\delta_{ae})\big)
\Big\}\,\Lambda^{nop}_{gc}
\nonumber\\ && \,
+\,\tfrac{1}{16\,\sqrt{2}} \,\big(
(
\bar{u}^{i}(\bar{p},\bar\chi)\,\gamma^{j} + \bar{u}^{j}(\bar{p},\bar\chi)\,\gamma^{i}
)\,(\bar{p}+{p})^{h}
\nonumber\\
&& \qquad \qquad \qquad
+\,\bar{u}^{h}(\bar{p},\bar\chi)\,(
\gamma^{i}\,(\bar{p}+{p})^{j} + \gamma^{j}\,(\bar{p}+{p})^{i}
)
\big)\,\gamma_5\,{u}({p},\chi)\,\Big\{
- w^{}_{3}\,d_{abf}\,if_{egf}
\nonumber\\
&& \qquad
- w^{}_{6}\,\big(3\,d_{abf}\,d_{gef}
+ \delta_{ab}\,\delta_{ge} - (\delta_{ag}\,\delta_{be} + \delta_{bg}\,\delta_{ae})\big)
\Big\}\,\Lambda^{nop}_{gc}
\nonumber\\ && \,
+\tfrac{1}{8\,\sqrt{2}}\, \big(
2\, \bar{u}^{h}(\bar{p},\bar\chi)\,i\,\sigma^{{i}{j}}
- (\bar{u}^{i}(\bar{p},\bar\chi)\,i\,\sigma^{{j}{h}} - \bar{u}^{j}(\bar{p},\bar\chi)\,i\,\sigma^{{i}{h}})
\big)\,\gamma_5\,{u}({p},\chi)
\,\Big\{
w^{}_{1}\,\big(if_{aef}\,d_{bgf} - if_{bef}\,d_{agf}\big)
\nonumber\\
&& \qquad
- w^{}_{4}\,f_{abf}\,f_{gef}
\Big\}\,\Lambda^{nop}_{gc}
\nonumber\\ && \,
+\,\tfrac{1}{16\,\sqrt{2}} \,\big(
(
\bar{u}^{i}(\bar{p},\bar\chi)\,\gamma^{j} - \bar{u}^{j}(\bar{p},\bar\chi)\,\gamma^{i}
)\,(\bar{p}+{p})^{h}
\nonumber\\
&& \qquad \qquad \qquad
+\,(
\bar{u}^{i}(\bar{p},\bar\chi)\,(\bar{p}+{p})^{j} - \bar{u}^{j}(\bar{p},\bar\chi)\,(\bar{p}+{p})^{i}
)\,\gamma^{h}
\big)\,\gamma_5\,{u}({p},\chi)\,\Big\{
w^{}_{2}\,\big(i\,f_{aef}\,d_{bgf} - i\,f_{bef}\,d_{agf}\big)
\nonumber\\
&& \qquad
- w^{}_{5}\,f_{abf}\,f_{gef}
\Big\}\,\Lambda^{nop}_{gc}
\nonumber\\ && \,
+\,\tfrac{1}{16\,\sqrt{2}} \,\big(
-\,(
\bar{u}^{i}(\bar{p},\bar\chi)\,\gamma^{j} - \bar{u}^{j}(\bar{p},\bar\chi)\,\gamma^{i}
)\,(\bar{p}+{p})^{h}
\nonumber\\
&& \qquad \qquad \qquad
+\,\bar{u}^{h}(\bar{p},\bar\chi)\,(
\gamma^{i}\,(\bar{p}+{p})^{j} - \gamma^{j}\,(\bar{p}+{p})^{i}
)
\big)\,\gamma_5\,{u}({p},\chi)\,\Big\{
w^{}_{3}\,\big(if_{aef}\,d_{bgf} - if_{bef}\,d_{agf}\big)
\nonumber\\
&& \qquad
- w^{}_{6}\,f_{abf}\,f_{gef}
\Big\}\,\Lambda^{nop}_{gc}
\,.\end{aligned}$$
Appendix B
==========
$$\begin{aligned}
{({d}, {\bar\chi}|}\,
[[T_a,\,T_b]_+,\,T_e]_+
\,{|{c}, \chi)}
& = &
\delta_{\bar\chi\chi}\,\big[
2\,\delta_{ab}\,if_{e{c}{d}}
+ 3\,d_{abg}\,(
d_{cgf}\,if_{{d}ef}
- d_{{d}gf}\,if_{{c}ef}
)
\nonumber\\ && \qquad
- \delta_{b{d}}\,if_{e{c}a}
- \delta_{a{d}}\,if_{e{c}b}
- \delta_{ac}\,if_{eb{d}}
- \delta_{bc}\,if_{ea{d}}
\big]
\,,\nonumber\\
{({d}, {\bar\chi}|}\,
[[G^i_a,\,G^j_b]_+,\,T_e]_+
\,{|{c}, \chi)}
& = &
\tfrac{1}{4}\,\delta_{\bar\chi\chi}\,\delta^{ij}\,\big[
\tfrac{10}{3}\,\delta_{ab}\,if_{e{c}{d}}
\nonumber\\ && \qquad
- d_{abg}\,(
d_{g{d}f}\,if_{e{c}f}
- d_{gcf}\,if_{e{d}f}
)
+ \tfrac{8}{3}\,d_{abg}\,(
f_{g{d}f}\,f_{e{c}f}
+ f_{gcf}\,f_{e{d}f}
)
\nonumber\\ && \qquad
+ \tfrac{1}{3}\,(
\delta_{a{d}}\,if_{eb{c}}
+ \delta_{b{d}}\,if_{ea{c}}
- \delta_{ac}\,if_{eb{d}}
- \delta_{bc}\,if_{ea{d}}
)
\big]
\nonumber\\ &+&
\tfrac{1}{4}\,i\,\epsilon^{ijk}\,\sigma^k_{\bar\chi\chi}\,\big[
\delta_{a{d}}\,if_{eb{c}}
- \delta_{b{d}}\,if_{ea{c}}
+ \delta_{ac}\,if_{eb{d}}
- \delta_{bc}\,if_{ea{d}}
\nonumber\\ && \qquad
+ 2\,if_{abg}\,(
d_{g{d}f}\,if_{e{c}f}
- d_{gcf}\,if_{e{d}f}
)
\nonumber\\ && \qquad
+ \tfrac{5}{3}\,if_{abg}\,(
f_{g{d}f}\,f_{e{c}f}
+ f_{gcf}\,f_{e{d}f}
)
\big]
\,,\nonumber\\
{({d}, {\bar\chi}|}\,
[[T_a,\,T_b]_+,\,G^i_e]_+
\,{|{c}, \chi)}
& = &
\tfrac{1}{2}\,\sigma^i_{\bar\chi\chi}\,\big[
2\,\delta_{ab}\,d_{ec{d}}
+ \tfrac{4}{3}\,\delta_{ab}\,if_{ec{d}}
\nonumber\\ && \qquad
+ 3\,d_{abg}\,d_{fg{d}}\,d_{ecf}
+ 2\,d_{abg}\,d_{fg{d}}\,if_{ecf}
\nonumber\\ && \qquad
+ 3\,d_{abg}\,d_{cgf}\,d_{ef{d}}
+ 2\,d_{abg}\,d_{cgf}\,if_{ef{d}}
\nonumber\\ && \qquad
- \delta_{ac}\,d_{eb{d}}
- \delta_{bc}\,d_{ea{d}}
- \delta_{b{d}}\,d_{eca}
- \delta_{a{d}}\,d_{ecb}
\nonumber\\ && \qquad
- \tfrac{2}{3}\,\delta_{ac}\,if_{eb{d}}
- \tfrac{2}{3}\,\delta_{bc}\,if_{ea{d}}
- \tfrac{2}{3}\,\delta_{b{d}}\,if_{eca}
- \tfrac{2}{3}\,\delta_{a{d}}\,if_{ecb}
\big]
\,,\nonumber\\
{({d}, {\bar\chi}|}\,
[[T_a,\,T_b]_+,\,J^i]_+
\,{|{c}, \chi)}
& = &
\sigma^i_{\bar\chi\chi}\,\big[
\delta_{ab}\,\delta_{{c}{d}} - \delta_{a{c}}\,\delta_{b{d}} - \delta_{a{d}}\,\delta_{b{c}}
+ 3\,d_{abg}\,d_{{c}g{d}}
\big]
\,,\nonumber\\
{({d}, {\bar\chi}|}\,
[[G^i_a,\,G^l_b]_+,\,J^l]_+
\,{|{c}, \chi)}
& = &
\tfrac{1}{4}\,\sigma^i_{\bar\chi\chi}\,\big[
\tfrac{1}{3}\,(
5\,\delta_{ab}\,\delta_{c{d}} - \delta_{ac}\,\delta_{b{d}} - \delta_{a{d}}\,\delta_{bc}
)
\nonumber\\ && \qquad
- d_{{d}cg}\,d_{abg}
+ \tfrac{8}{3}\,i\,f_{c{d}g}\,d_{abg}
\big]
\,,\nonumber\\ \nonumber\\\nonumber\\
{({nop}, {\bar\chi}|}\,
[[T_a,\,T_b]_+,\,T_e]_+
\,{|{klm}, \chi)}
& = &
\tfrac{27}{8}\,\delta_{\bar\chi\chi}\,\delta^{{nop}}_{rst}\,\Big\{
\Lambda^{a,rst}_{xyz}\,\Lambda^{b,xyz}_{uvw}\,\Lambda^{e,uvw}_{klm}
+ \Lambda^{e,rst}_{uvw}\,\Lambda^{a,uvw}_{xyz}\,\Lambda^{b,xyz}_{klm}
+\,(a \leftrightarrow b)
\Big\}
\,,\nonumber\\
{({nop}, {\bar\chi}|}\,
[[G^i_a,\,G^j_b]_+,\,T_e]_+
\,{|{klm}, \chi)}
& = &
\tfrac{3}{2}\,\delta^{{nop}}_{rst}\,\Big\{
\nonumber\\ && \,
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
- \tfrac{3}{8}\,(S^{i}\,S^{j\dagger} + S^{j}\,S^{i\dagger} - \tfrac{3}{2}\,\delta^{ij}\,\one_{(4\times 4)}
)_{\bar\chi\chi}\,\big[
\Lambda^{a,rst}_{xyz}\,\Lambda^{b,xyz}_{uvw}\,\Lambda^{e,uvw}_{klm}
+ \Lambda^{e,rst}_{uvw}\,\Lambda^{a,uvw}_{xyz}\,\Lambda^{b,xyz}_{klm}
+\,(a \leftrightarrow b)
\big]
\nonumber\\ && \,
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+ \tfrac{3}{16}\,i\,\epsilon^{ijk'}\,(\vec{S}\,\sigma^{k\prime}\,\vec{S}^\dagger
)_{\bar\chi\chi}\,\big[
\Lambda^{a,rst}_{xyz}\,\Lambda^{b,xyz}_{uvw}\,\Lambda^{e,uvw}_{klm}
+ \Lambda^{e,rst}_{uvw}\,\Lambda^{a,uvw}_{xyz}\,\Lambda^{b,xyz}_{klm}
-\,(a \leftrightarrow b)
\big]
\nonumber\\ && \,
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+ \tfrac{1}{16}\,(S^{i}\,S^{j\dagger} + S^{j}\,S^{i\dagger}
)_{\bar\chi\chi}\,\big[
\Lambda^{rst}_{ag}\,\Lambda^{bg}_{uvw}\,\Lambda^{e,uvw}_{klm}
+ \Lambda^{e,rst}_{uvw}\,\Lambda^{uvw}_{ag}\,\Lambda^{bg}_{klm}
+\,(a \leftrightarrow b)
\big]
\nonumber\\ && \,
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+ \tfrac{1}{16}\,i\,\epsilon^{ijk'}\,(\vec{S}\,\sigma^{k\prime}\,\vec{S}^\dagger
)_{\bar\chi\chi}\,\big[
\Lambda^{rst}_{ag}\,\Lambda^{bg}_{uvw}\,\Lambda^{e,uvw}_{klm}
+ \Lambda^{e,rst}_{uvw}\,\Lambda^{uvw}_{ag}\,\Lambda^{bg}_{klm}
-\,(a \leftrightarrow b)
\big]
\Big\}
\,,\nonumber\\
{({nop}, {\bar\chi}|}\,
[[T_a,\,T_b]_+,\,G^i_e]_+
\,{|{klm}, \chi)}
& = &
\tfrac{27}{16}\,(\vec{S}\,\sigma^i\,\vec{S}^\dagger)_{\bar\chi\chi}\,\delta^{{nop}}_{rst}\,
\Big\{
\nonumber\\ && \qquad
\Lambda^{a,rst}_{xyz}\,\Lambda^{b,xyz}_{uvw}\,\Lambda^{e,uvw}_{klm}
+ \Lambda^{e,rst}_{xyz}\,\Lambda^{a,xyz}_{uvw}\,\Lambda^{b,uvw}_{klm}
+\,(a \leftrightarrow b)
\Big\}
\,,\nonumber\\
{({nop}, {\bar\chi}|}\,
[[T_a,\,T_b]_+,\,J^i]_+
\,{|{klm}, \chi)}
& = &
\tfrac{27}{4}\,(\vec{S}\,\sigma^i\,\vec{S}^\dagger)_{\bar\chi\chi}\,
\delta^{{nop}}_{rst}\,\Big\{
\Lambda^{a,rst}_{xyz}\,\Lambda^{b,xyz}_{klm}
+\,(a \leftrightarrow b)
\Big\}
\,,\nonumber\\
{({nop}, {\bar\chi}|}\,
[[G^i_a,\,G^l_b]_+,\,J^l]_+
\,{|{klm}, \chi)}
& = &
\tfrac{3}{2}\,(\vec{S}\,\sigma^i\,\vec{S}^\dagger)_{\bar\chi\chi}\,
\delta^{{nop}}_{rst}\,\Big\{
\nonumber\\ && \qquad
\tfrac{13}{8}\,\,\Lambda^{a,rst}_{xyz}\,\Lambda^{b,xyz}_{klm}
- \tfrac{1}{12}\,\Lambda^{rst}_{ag}\,\Lambda^{bg}_{klm}
+\,(a \leftrightarrow b)
\Big\}
\Big\}
\,,\nonumber\\ \nonumber\\\nonumber\\
{({nop}, {\bar\chi}|}\,
[[T_a,\,T_b]_+,\,T_e]_+
\,{|{c}, \chi)}
& = &
0
\,,\nonumber\\
{({nop}, {\bar\chi}|}\,
[[G^i_a,\,G^j_b]_+,\,T_e]_+
\,{|{c}, \chi)}
& = &
\tfrac{1}{8\,\sqrt{2}}\,\delta^{{nop}}_{rst}\,\Big\{
\nonumber\\ && \,
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
(S^i\,\sigma^j+S^j\,\sigma^i)_{\bar\chi\chi}\,\big[
if_{e{c}f}\,(d_{afg}+i\,f_{afg})\,\Lambda^{rst}_{bg}
+ \tfrac{3}{2}\,\Lambda^{e,rst}_{uvw}\,(d_{acg}+i\,f_{acg})\,\Lambda^{uvw}_{bg}
+\,(a \leftrightarrow b)
\big]
\nonumber\\ && \,
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+ i\,\epsilon^{ijk}\,S^k_{\bar\chi\chi}\,\big[
if_{e{c}f}\,\big(
(d_{afg}+\tfrac{2}{3}\,i\,f_{afg})\,\Lambda^{rst}_{bg}
+ \tfrac{5}{3}\,(
i\,f_{afg}\,\Lambda^{rst}_{bg}
- i\,f_{abg}\,\Lambda^{rst}_{fg}
)
\big)
\nonumber\\ &&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!
+ \tfrac{3}{2}\,\Lambda^{e,rst}_{uvw}\,\big(
(d_{acg}+\tfrac{2}{3}\,i\,f_{acg})\,\Lambda^{uvw}_{bg}
+ \tfrac{5}{3}\,(
i\,f_{acg}\,\Lambda^{uvw}_{bg}
- i\,f_{abg}\,\Lambda^{uvw}_{cg}
)
\big)
-\,(a \leftrightarrow b)
\big]
\Big\}
\,,\nonumber\\
{({nop}, {\bar\chi}|}\,
[[T_a,\,T_b]_+,\,G^i_e]_+
\,{|{c}, \chi)}
& = &
\tfrac{1}{2\sqrt{2}}\,S^i_{\bar\chi\chi}\,\Big\{
\tfrac{9}{4}\,\delta^{{nop}}_{rst}\,(
\Lambda^{a,rst}_{xyz}\,\Lambda^{b,xyz}_{uvw}
+ \Lambda^{b,rst}_{xyz}\,\Lambda^{a,xyz}_{uvw}
)\,\Lambda^{{uvw}}_{ec}
\nonumber\\ && \qquad
+ \Lambda^{{nop}}_{ef}\big(
\delta_{ab}\,\delta_{cf} - \delta_{ac}\,\delta_{bf} - \delta_{af}\,\delta_{bc}
+ 3\,d_{abg}\,d_{cgf}
\big)
\Big\}
\,,\nonumber\\
{({nop}, {\bar\chi}|}\,
[[T_a,\,T_b]_+,\,J^i]_+
\,{|{c}, \chi)}
& = &
0
\,,\nonumber\\
{({nop}, {\bar\chi}|}\,
[[G^i_a,\,G^l_b]_+,\,J^l]_+
\,{|{c}, \chi)}
& = &
\tfrac{1}{8\,\sqrt{2}}\,S^i_{\bar\chi\chi}\,\Big\{
5\,(d_{a{c}g}+i\,f_{a{c}g})\,\Lambda^{{nop}}_{bg}
+ 5\,(d_{b{c}g}+i\,f_{b{c}g})\,\Lambda^{{nop}}_{ag}
\nonumber\\ && \qquad
- 3\,(d_{a{c}g}+\tfrac{2}{3}\,i\,f_{a{c}g})\,\Lambda^{{nop}}_{bg}
+ 3\,(d_{b{c}g}+\tfrac{2}{3}\,i\,f_{b{c}g})\,\Lambda^{{nop}}_{ag}
\nonumber\\ && \qquad
- 5\,(
i\,f_{a{c}g}\,\Lambda^{{nop}}_{bg}
- i\,f_{b{c}g}\,\Lambda^{{nop}}_{ag}
- 2\,i\,f_{abg}\,\Lambda^{{nop}}_{{c}g}
)
\Big\}
\label{res-three-body}
\,,\end{aligned}$$
where Eq. (A.2) in [@Lutz:2010se] was used. We correct a typo with $$\begin{aligned}
{({nop}, {\bar\chi}|}\,[G^i_a,G^j_b]_+\,{|{c}, \chi)}
& = &
\tfrac{1}{8\,\sqrt{2}}\,i\,\epsilon^{ijk}\,S^k_{\bar\chi\chi}\,\big[
(d_{ace}+\tfrac{2}{3}\,i\,f_{ace})\,\Lambda^{{nop}}_{be}
+ \tfrac{5}{3}\,(
i\,f_{ace}\,\Lambda^{{nop}}_{be}
- i\,f_{abe}\,\Lambda^{{nop}}_{ce}
)
-\,(a \leftrightarrow b)
\big]
\nonumber\\ &+&
\tfrac{1}{8\,\sqrt{2}}\,(S^i\,\sigma^j+S^j\,\sigma^i)_{\bar\chi\chi}\,\big[
(d_{ace}+i\,f_{ace})\,\Lambda^{{nop}}_{be}
+\,(a \leftrightarrow b)
\big]
\,.\end{aligned}$$
[^1]:
|
---
abstract: 'Belief revision is the process in which an agent incorporates a new piece of information together with a pre-existing set of beliefs. When the new information comes in the form of a report from another agent, then it is clear that we must first determine whether or not that agent should be trusted. In this paper, we provide a formal approach to modeling trust as a pre-processing step before belief revision. We emphasize that trust is not simply a relation between agents; the trust that one agent has in another is often restricted to a particular domain of expertise. We demonstrate that this form of trust can be captured by associating a state-partition with each agent, then relativizing all reports to this state partition before performing belief revision. In this manner, we incorporate only the part of a report that falls under the perceived domain of expertise of the reporting agent. Unfortunately, state partitions based on expertise do not allow us to compare the relative strength of trust held with respect to different agents. To address this problem, we introduce pseudometrics over states to represent differing degrees of trust. This allows us to incorporate simultaneous reports from multiple agents in a way that ensures the most trusted reports will be believed.'
author:
- |
Aaron Hunter\
British Columbia Institute of Technology\
Burnaby, Canada\
`aaron_hunter@bcit.ca`
bibliography:
- 'action.bib'
title: Belief Revision and Trust
---
Introduction {#sec:introduction}
============
The notion of trust must be addressed in many agent communication systems. In this paper, we consider one isoloated aspect of trust: the manner in which trust impacts the process of belief revision. Some of the most influential approaches to belief revision have used the simplifying assumption that all new information must be incorporated; however, this is clearly untrue in cases where information comes from an untrusted source. In this paper, we are concerned with the manner in which an agent uses an external notion of trust in order to determine how new information should be integrated with some pre-existing set of beliefs.
Our basic approach is the following. We introduce a simple model of trust that allows an agent to determine if a source can be trusted to distinguish between different pairs of states. We use this notion of trust as a precursor to belief revision. Hence, before revising by a new formula, an agent first determines to what extent the source of the information can be trusted. In many cases, the agent will only incorporate “part” of the formula into their beliefs. We then extend our model of trust to a more general setting, by introducing quantitative measures of trust that allow us to compare the degree to which different agents are trusted. Fundamental properties are introduced and established, and applications are considered.
Preliminaries
=============
Intuition
---------
It is important to note that an agent typically does not trust another agent universally. As such, we will not apply the label “trusted” to another agent; instead, we will say that an agent is trusted with respect to a certain domain of knowledge. This is further complicated by the fact that there are different reasons that an agent may not be trusted. For example, an agent might not be trusted due to their perceived knowledge of a domain. In other cases, an agent might not be trusted due to their perceived dishonesty, or bias. In this paper, our primary focus is on trust as a function of the perceived expertise of other agents. Towards the end, we briefly address the different formal mechanisms that would be required to deal with deceit.
Motivating Example
------------------
We introduce a motivating example in commonsense reasoning where an agent must rely on an informal notion of trust in order to inform rational belief change; we will return to this example periodically as we introduce our formal model.
Consider an agent that visits a doctor, having difficulty breathing. Incidentally, the agent is wearing a necklace that prominently features a jewel on a pendant. During the examination, the doctor checks the patient’s throat for swelling or obstruction; at the same time, the doctor happens to look at the necklace. Following the examination, the doctor tells the patient “you have a viral infection in your throat - and by the way, you should know that the jewel in your necklace is not a diamond.”
The important part about this example is the fact that the doctor provides information about two distinct domains: human health and jewelry. In practice, a patient is very likely to trust the doctor’s diagnosis about the viral infection. On the other hand, the patient really has very little reason to trust the doctor’s evaluation of the necklace. We suggest that a rational agent should actually incorporate the doctor’s statement about the infection into their own beliefs, while essentially ignoring the comment on the necklace. This approach is dictated by the kind of trust that the patient has in the doctor. Our aim in this paper is to formalize this kind of “localized” domain-specific trust, and then demonstrate how this form of trust is used in practice to inform belief revision.
Trust
-----
Trust consists of two related components. First, we can think of trust in terms of how likely an agent is to believe what another agent says. Alternatively, we can think of trust in terms of the degree to which an agent is likely to allow another to perform actions on their behalf. In this paper, we will be concerned only with the former.
A great deal of existing work on trust focuses on the manner in which an agent develops a [*reputation*]{} based on past behaviour. A brief survey of reputation systems is given in [@Huynh06]. Reputation systems can be used to inform the allocation of tasks [@Ramchurn09], or to avoid deception [@Salehi09]. The model of trust presented in this paper is not intended to be an alternative to existing reputation systems; we are not concerned with the manner in which an agent learns to trust another. Instead, our focus is simply on developing a suitable model of trust that is expressive enough to inform the process of [*belief revision*]{}. The manner in which this model of trust is developed over time is beyond the scope of this paper.
Belief Revision
---------------
[*Belief revision*]{} refers to the process in which an agent must integrate new information with some pre-existing beliefs about the state of the world. One of the most influential approaches to belief revision is the AGM approach, in which an agent incorporates the new information while keeping as much of the intial belief state as consistently possible [@AlchourronGardenforsMakinson85].
This approach was originally defined with respect to a finite set $P$ of propositional variables representing properties of the world. A [*state*]{} is a propositional interpretation over $P$, representing a possible state of the world. A [*belief set*]{} is a deductively closed set of formulas, representing the beliefs of an agent. Since $P$ is finite, it follows that every belief set defines a corresponding [*belief state*]{}, which is the set of states that an agent considers to be possible. A revision operator is a function that takes a belief set and a formula as input, and returns a new belief set. An AGM revision operator is a revision operator that satisfies the AGM postulates, as specified in [@AlchourronGardenforsMakinson85].
It turns out that every AGM revision operator is characterized by a total pre-order over possible worlds. To be more precise, a [*faithful assignment*]{} is a function that maps each belief set to a total pre-order over states in which the models of the belief set are the minimal states. When an agent is presented with a new formula $\phi$ for revision, the revised belief state is the set of all minimal models of $\phi$ in the total pre-order given by the faithful assignment. We refer the reader to [@KatsunoMendelzon92] for a proof of this result, as well as a complete description of the implications. For our purposes, we simply need to know that each AGM revision operator necessarily defines a faithful assignment.
A Model of Trust
================
Domain-Specific Trust
---------------------
Assume we have a fixed propositional signature $\mathbf{F}$ as well as a set of agents $\mathbf{A}$. For each $A\in\mathbf{A}$, let $Bel_A$ denote a deductively closed set of formulas over $\mathbf{F}$ called the [*belief set*]{} of $A$. For each $A$, let $*_A$ denote an AGM revision operator that intuitively captures the way that the agent $A$ revises their beliefs when presented with new information. This revision operator represents sort of an “ideal” revision situation, in which $A$ has complete trust in the new information. We want to modify the way this operator is used, by adding a representation of the extent to which $A$ trusts each other agent $B\in\mathbf{A}$ over ${\mathbf F}$.
We assume that all new information is reported by an agent, so each formula for revision can be labelled with the name of the reporting agent.[^1] At this point, we are not concerned with degrees of trust or with resolving conflicts between different sources of information. Instead, we start with a binary notion of trust, where $A$ either trusts $B$ or does not trust $B$ with respect to a particular domain of expertise.
We encode trust by allowing each agent $A$ to associate a partition $\Pi^B_A$ over possible states with each agent $B$.
A [*state partition*]{} $\Pi$ is a collection of subsets of $2^{\mathbf F}$ that is collectively exhaustive and mutually exclusive. For any state $s\in 2^{\mathbf{F}}$, let $\Pi(s)$ denote the element of $\Pi$ that contains $s$.
If $\Pi=\{2^{\mathbf F}\}$ then we call $\Pi$ the [*trivial partition*]{} with respect to ${\mathbf F}$. If $\Pi=\{\{s\} \mid s\in2^{\mathbf F}\}$, then we call $\Pi$ the [*unit partition*]{}.
For each $A\in\mathbf{A}$ the [*trust function*]{} $T_A$ is a function that maps each $B\in{\mathbf A}$ to a state partition $\Pi^B_A$.
The partition $\Pi^B_A$ represents the trust that $A$ has in $B$ over different aspects of knowledge. Informally, the partition encodes states that $A$ will trust $B$ to distinguish. If $\Pi^B_A(s_1)\ne\Pi^B_A(s_2)$, then $A$ will trust that $B$ can distinguish between states $s_1$ and $s_2$. Conversely, if $\Pi^B_A(s_1)=\Pi^B_A(s_2)$, then $A$ does not see $B$ as an authority capable of distinguishing between $s_1$ and $s_2$. We clarify by returning to our motivating example.
Let $\mathbf{A}=\{A,D,J\}$ and let $\mathbf{F}=\{sick, diam\}$. Informally, the fluent $sick$ is true if $A$ has an illness and the fluent $diam$ is true if a certain piece of jewelry that $A$ is wearing contains a real diamond. If we imagine that $D$ represents a doctor and $J$ represents a jeweler, then we can use state partitions to represent the trust that $A$ has in $D$ and $J$ with respect to different domains. Following standard shorthand notation, we represent a state $s$ by the set of fluent symbols that are [*true*]{} in $s$. In order to make the descriptions of a partition more readable, we use a $\mid$ symbol to visually separate different cells. The following partitions are then intuitively plausible in this example: $$\begin{aligned}
\Pi^D_A &:=& \{sick,diam\},\{sick\} \mathbf{\mid} \{diam\},\emptyset\\
\Pi^J_A &:=& \{sick,diam\},\{diamond\} \mathbf{\mid} \{sick\},\emptyset \end{aligned}$$ Hence, $A$ trusts the doctor $D$ to distinguish between states where $A$ is sick as opposed to states where $A$ is not sick. However, $A$ does not trust $D$ to distinguish between worlds that are differentiated by the authenticity of a diamond. The formula $sick\wedge\neg diamond$ encodes the doctor’s statement that the agent is sick, and the necklace they are wearing has a fake diamond.
Although the preceding example is simple, it illustrates how a partition can be used to encode the perceived expertise of agents. In the doctor-jeweler example, we could equivalently have defined trust with respect to the set of fluents. In other words, we could have simply said that $D$ is trusted over the fluent $sick$. However, there are many practical cases where this is not sufficient; we do not want to rely on the fluent vocabulary to determine what is a valid feature with respect to trust. For example, a doctor may have specific expertise over lung infections for those working in factories, but not for lung infections for those working in a space shuttle. By using state partitions to encode trust, we are able to capture a very flexible class of distinct areas of trust.
Incorporating Trust in Belief Revision
--------------------------------------
As indicated previously, we assume each agent $A$ has an AGM belief revision operator $*_A$ for incorporating new information. In this section, we describe how the revision operator $*_A$ is combined with the trust function $T_A$ to define a new, trust-incorporating revision operator $*^B_A$. In many cases, the operator $*^B_A$ will not be an AGM operator because it will fail to satisfy the AGM postulates. In particular, $A$ will not necessarily believe a new formula when it is reported by an untrusted source. This is a desirable feature.
Our approach is to define revision as a two-step process. First, the agent considers the source and the relevant state partition to determine how much of the new information to incorporate. Second, the agent performs standard AGM revision using the faithful assignment corresponding to the belief revision operator.
Let $\phi$ be a formula and let $T_A(B)=\Pi_A^B$. Define: $$\Pi_A^B[\phi] = \bigcup\{\Pi_A^B(s) \mid s\models\phi\}.$$
Hence $\Pi_A^B[\phi]$ is the union of all cells that contain a model of $\phi$.
If $A$ does not trust $B$ to distinguish between states $s$ and $t$, then any report from $B$ that provides evidence that $s$ is the actual state is also evidence that $t$ is the actual state. When $A$ performs belief revision, it should be with respect to the distinctions that $B$ can be trusted to make. It follows that $A$ need not believe $\phi$ after revision; instead $A$ should interpret $\phi$ to be evidence of any state $s$ that is $B$-indistinguishable from a model of $\phi$. Formally, this means that the formula $\phi$ is construed to be evidence for each state in $\Pi_A^B[\phi]$.
\[def:trrev\] Let $A,B\in\mathbf{A}$ with $T_A(B)=\Pi^B_A$, and let $*_A$ be an AGM revision operator for $A$. For any belief set $K$ with corresponding ordering $\prec_K$ given by the underlying faithful assignment, the trust-sensitive revision $K*^B_A\phi$ is the set of formulas true in $$\min_{\prec_K}(\{s \mid s\in\Pi_A^B[\phi]\}).$$
So rather than taking the minimal models of $\phi$, we take all minimal states that $B$ can not be trusted to distinguish from the minimal models of $\phi$.
It is worth remarking that this notion can be formulated synactically as well. Since ${\mathbf F}$ is finite, each state $s$ is defined by a unique, maximal conjunction over literals in ${\mathbf F}$; we simply take the conjunction of all the atomic formulas that are true in $s$ together with the negation of all the atomic formulas that are false in $s$.
For any state $s$, let $prop(s)$ denote the unique, maximal conjunction of literals true in $s$.
This definition can be extended for a cell in a state partition.
Let $\Pi$ be a state partition. For any state $s$, $$prop(\Pi(s)) = \bigvee \{prop(s^{\prime}) \mid s^{\prime}\in\Pi(s)\}.$$
Note that $prop(\Pi(s))$ is a well-defined formula in disjunctive normal form, due to the finiteness of ${\mathbf F}$. Intuitively, $prop(\Pi(s))$ is the formula that defines the partition $\Pi(s)$. In the case of a trust partition $\Pi^B_A$, we can use this idea to define the [*trust expansion*]{} of a formula.
\[def:trustexpansion\] Let $A,B\in\mathbf{A}$ with the corresponding state partition $\Pi^B_A$, and let $\phi$ be a formula. The [*trust expansion*]{} of $\phi$ for $A$ with respect to $B$ is the formula $$\phi^B_A := \bigvee \{prop(\Pi^B_A(s)) \mid s\models\phi\}.$$
Note that this is a finite disjunction of disjunctions, which is again a well defined formula. We refer to $\phi^B_A$ as the trust expansion of $\phi$ because it is true in all states that are consistent with $\phi$ with respect to distinctions that $A$ trusts $B$ to be able to make. It is an expansion because the set of models of $\phi^B_A$ is normally larger than the set of models of $\phi$. The trust sensitive revision operator could equivalently be defined as the normal revision, following translation of $\phi$ to the corresponding trust expansion.
Returning to our example, we consider a few different formulas for revision:
1. $\phi_1 = sick$
2. $\phi_2 = \neg diam$
3. $\phi_3 = sick \wedge \neg diam$.
Suppose that the agent initially believes that they are not sick, and that the diamond they have is real, so $K=\neg sick \wedge diam$. For simplicity, we will assume that the underlying pre-order $\prec_K$ has only two levels: those states where $K$ is true are minimal, and those where $K$ is false are not. We have the following results for revision
1. $K *_A^D \phi_1 = sick\wedge diam$
2. $K *_A^D \phi_2 = \neg sick\wedge diam$
3. $K *_A^D \phi_3 = sick\wedge diam$.
The first result indicates that $A$ believes the doctor when the doctor reports that they are sick. The second result indicates that $A$ essentially ignores a report from the doctor on the subject of jewelry. The third result is perhaps the most interesting. It demonstrates that our approach allows an agent to just incorporate a part of a formula. Hence, even though $\phi_3$ is given as a single piece of information, the agent $A$ only incorporates the part of the formula over which the doctor is trusted.
Formal Properties
=================
Basic Results
-------------
We first consider extreme cases for trust-sensitive revision operators. Intuitively, if $T_A(B)$ is the trivial partition, then $A$ does not trust $B$ to be able to distinguish between any states. Therefore, $A$ should not incorporate any new information obtained from $B$. The following proposition makes this observation explicit.
\[prop1\] If $T_A(B)$ is the trivial partition, then $K*^B_A\phi = K$ for all $K$ and $\phi$.
The other extreme situation occurs when $T_A(B)$ is the unit partition, which consists of all singleton sets. In this case, $A$ trusts $B$ to be able to distinguish between every possible pair of states. It follows from this result that trust sensitive revision operators are not AGM revision operators.
\[prop2\] If $T_A(B)$ is the unit partition, then $*^B_A=*_A$.
Hence, if $B$ is universally trusted, then the corresponding trust sensitive revision operator is just the a priori revision operator for $A$.
Refinements
-----------
There is a partial ordering on partitions based on the notion of [*refinement*]{}. We say that $\Pi_1$ is a refinement of $\Pi_2$ just in case, for each $S_1\in\Pi_1$, there exists $S_2\in\Pi_2$ such that $S_1\subseteq S_2$. We also say that $\Pi_1$ is [*finer*]{} than $\Pi_2$. In terms of trust-partitions, refinement has a natural interpretation in terms of “breadth of trust.” If the partition corresponding to $B$ is finer than that corresponding to $C$, it means that $B$ is trusted more broadly than $C$. To be more precise, it means that $B$ is trusted to distinguish between all of the states that $C$ can distinguish, and possibly more. If $B$ is trusted more broadly that $C$, it follows that a report from $B$ should give give $A$ more information. This idea is formalized in the following proposition.
For any formula $\phi$, if $\Pi^B_A$ is a refinement of $\Pi^C_A$, then $|K*^B_A\phi| \subseteq |K*^C_A\phi|$.
This is a desirable property; if $B$ is trusted over a greater range of states, then fewer states are possible after a report from $B$.
Multiple Reports
----------------
One natural question that arises is how to deal with multiple reports of information from different agents, with different trust partitions. In our example, for instance, we might get a conflicting report from a jeweler with respect to the status of the necklace. In order to facilitate the discussion, we introduce a precise notion of a [*report*]{}.
A [*report*]{} is a pair $(B,\phi)$, where $B\in\mathbf{A}$ and $\phi$ is a formula.
We can now extend the definition of trust senstive revision to reports in the obvious manner. In fact, if the revising agent $A$ is clear from the context, we can use the short hand notation: $$K * (\phi,B) = K *^B_A \phi.$$ The following definition extends the notion of revision to incorporate multiple reports.
\[def:multi\] Let $\{A\}\cup\mathbf{B}\subseteq \mathbf{A}$, and let $\Phi=\{(\phi_i,B_i) \mid i<n\}$ be a finite set of reports. Given $K$, $*$ and $\prec_K$, the trust-sensitive revision $K*_A \Phi$ is the set of formulas true in $$\min_{\prec_K}(\{s \mid s\in\Pi_A^{Bi}[\phi_i]\}).$$
So the trust sensitive revision for a finite set of reports from different agents is essentially the normal, single-shot revision by the conjunction of formulas. The only difference is that we expand each formula with respect to the trust partition for a particular reporting agent.
In the doctor and jeweler domain, we can consider how how an agent might incorporate a set of reports from $D$ and $J$. We start with the same initial belief set as before: $K=\neg sick \wedge diam$. Consider the following reports:
1. $\Phi_1 = \{(sick, D), (\neg diam, D)\}$
2. $\Phi_2 = \{(sick,J),(\neg diam, J)\}$
3. $\Phi_3 = \{(sick, D), (\neg diam, J)\}$
4. $\Phi_4 = \{(sick,J),(\neg diam, D)\}$
We have the following results following revision:
1. $K *_A \Phi_1 = sick\wedge diam$
2. $K *_A \Phi_2 = \neg sick\wedge \neg diam$
3. $K *_A \Phi_3 = sick\wedge \neg diam$
4. $K *_A \Phi_4 = \neg sick \wedge diam$.
These results demonstrate how the agent $A$ essentially incorporates information from $D$ and $J$ in domains where they are trusted, and ignores information when they are not trusted. Note that, in this case, $D$ and $J$ are trusted over disjoint sets of states. As a result, it is not possible to have contradictory reports that are equally trusted.
The problem with Definition \[def:multi\] is that the set of states in the minimization may be empty. This occurs when multiple agents give conflicting reports, and we trust each agent on the domain. In order to resolve this kind of conflict, we need a more expressive form of trust that allows some agents to be trusted more than others. We introduce such a representation in the next section.
Trust Pseudometrics
===================
Measuring Trust
---------------
In the previous section, we were concerned with a binary notion of trust that did not include any measure of the strength of trust held in a particular agent or domain. Such an approach is appropriate in cases where we only receive new information from a single source, or from a set of sources that are equally reliable. However, it is not sufficient if we consider cases where several different sources may provide conflicting information. In such cases, we need to determine which information source is the most trust worthy with respect to the domain currently under consideration.
In the binary approach, we associated a partition of the state space with each agent. In order to capture different levels of trust, we would like to introduce a measure of the distance between two states from the perspective of a particular agent. In other words, an agent $A$ would like to associate a distance function $d_B$ over states with each other agent $B$. If $d_B(s,t)=0$, then $B$ can not be trusted to distinguish between the states $s$ and $t$. On the other hand, if $d_B(s,t)$ is very large, then $A$ has a high level of trust in $B$’s ability to distinguish between $s$ and $t$. The notion of distance that we introduce will be a [*psuedometric*]{} on the state space. A pseudometric is a function $d$ that satisfies the following properties for all $x,y,z$ in the domain $X$:
1. $d(x,x)=0$
2. $d(x,y) = d(y,x)$
3. $d(x,z) \le d(x,y) + d(y,z)$
The difference between a [*metric*]{} and a pseudometric is that we do not require that $d(x,y)=0$ implies $x=y$ (the so-called law of indiscernables). This would be undesirable in our setting, because we want to use the distance 0 to represent states that are indistinguishable rather than identical. The first two properties are clearly desirable for a measure of our trust in another agent’s ability to discern states. The third property is the triangle inequality, and it is required to guarantee that our trust in other agents is transititive across different domains.
For each $A\in\mathbf{A}$, a [*pseudometric trust function*]{} ${\cal T}_A$ is a function that maps each $B\in{\mathbf A}$ to a pseudometric $d_B$ over $2^{\mathbf F}$.
The pair $(2^{\mathbf F}, {\cal T}_A)$ is called a pseudometric trust space. We would like to model the situation where a sequence of formulas $\Phi = \phi_1,\dots,\phi_n$ is received from the agents $\mathbf{B}=B_1,\dots,B_n$, respectively. Note that the order does not matter, we think of the formulas as arriving at the same instant with no preference between them other than the preference induced by the pseudometric trust space.
We associate a sequence of state partitions with each pseudometric trust space.
Let $(2^{\mathbf F}, {\cal T}_A)$ be a pseudometric trust space, let $B\in\mathbf{A}-A$, and let $i$ be a natural number. For each state $s$, define the set $\Phi_B^A(i)(s)$ as follows: $$\Pi_B^A(i)(s) = \{t\mid d_B(s,t)\le i\}.$$ The collection of sets $\{\Pi_B^A(i)(s) \mid s\in 2^{\mathbf F}\}$ is a state partition.
We let $\Pi_B^A(i)$ denote the state partition obtained from this proposition. The cells of the partition $\Pi_B^A(i)$ consist of all states are separated by a distance of no more than $i$. The following proposition is immediate.
$\Pi_B^A(i)$ is a refinement of $\Pi_B^A(j)$, for any $i<j$.
Hence, a pseudometric trust space defines a sequence of partitions for each agent. This sequence of partitions gets coarser as we increase the index; increasing the index corresponds to requiring a higher level of trust that an agent can distinguish between states. Since we can use Definition \[def:trrev\] to define a trust sensitive revision operator from a state partition, we can now define a trust sensitive revision operator for any fixed distance $i$ between states. Informally, as $i$ increases, we require $B$ to have a greater degree of certainty in order to trust them to distinguish between states. However, it is not clear in advance exactly which $i$ is the right threshold. Our approach will be to find the lowest possible threshold that yields a consistent result.
Note that $\Pi_B^A(i)$ will be a trivial partition for any $i$ that is less than the minimum distance assigned by the underlying pseudometric trust function.
Let $(2^{\mathbf F}, {\cal T}_A)$ be a pseudometric trust space, and let $m$ be the least natural number such that $\Pi_B^A(m)$ is non-trival. The [*trust sensitive revision operator*]{} for $A$ with respect to $B$ is the trust sensitive revision operator given by $\Pi_B^A(m)$.
This is a simple extension of our approach based on state partitions. In the next section, we take advantage of the added expressive power of pseudometrics.
We modify the doctor example. In order to consider different levels of trust, it is more interesting to consider a domain involving two doctors: a general practitioner $D$ and a specialist $S$. We also assume that the vocabulary includes two fluents: $ear$ and $skin$. Informally, $ear$ is understood to be true if the patient has an ear infection, whereas $skin$ is true if the patient has skin cancer. The important point is that an ear infection is something that can easily be diagnosed by any doctor, whereas skin cancer is typically diagnosed by a specialist. In order to capture these facts, we define two pseudometrics $d_D$ and $d_S$. For simplicity, we label the possible states as follows: $$\begin{aligned}
s_1 &=& \{ear,skin\}\\
s_2 &=& \{ear\}\\
s_3 &=& \{skin\}\\
s_4 &=& \emptyset\end{aligned}$$ We define the pseudometrics as follows:
$s_1,s_2$ $s_1,s_3$ $s_1,s_4$ $s_2,s_3$ $s_2,s_4$ $s_3,s_4$
------- ----------- ----------- ----------- ----------- ----------- ----------- --
$d_D$ 1 2 2 2 2 1
$d_S$ 2 2 2 2 2 2
With these pseudometrics, it is easy to see that both $D$ and $S$ can distinguish all of the states. However, $S$ is more trusted to distinguish between states related to a skin cancer diagnosis. In our framework, we would like to ensure that this implies $S$ will be trusted in the case of conflicting reports from $D$ and $S$ with respect to skin cancer.
Multiple Reports
----------------
We view the distances in a pseudometric trust space as absolute measurements. As such, if $d_B(s,t)>d_C(s,t)$, then we have greater trust in $B$ as opposed to $C$ as far as the ability to discern the states $s$ and $t$ is concerned. We would like to use this intuition to resolve conflicting reports between agents.
Let $\{A\}\cup\mathbf{B}\subseteq \mathbf{A}$, and let $\Phi=\{(\phi_i,B_i) \mid i<n\}$ be a finite set of reports. There exists a natural number $m$ such that $$\bigcap_{i<n}(\Pi_A^{Bi}[\phi_i](m)) \ne \emptyset.$$
Hence, for any set of reports, we can get a non-intersecting intersection if we take a sufficiently coarse state partition. In many cases this partition will be non-trival. Using this proposition, we define multiple report revision as follows.
\[def:multitr\] Let $(2^{\mathbf F}, {\cal T}_A)$ be a pseudometric trust space, let $\Phi=\{(\phi_i,B_i) \mid i<n\}$ be a finite set of reports, and let $m$ be the least natural number such that $\bigcap_{i<n}(\Pi_A^{Bi}[\phi_i](m)) \ne \emptyset.$ Given $K$, $*$ and $\prec_K$, the trust-sensitive revision $K*^{\mathbf B}_A \Phi$ is the set of formulas true in $$\min_{\prec_K}(\{s \mid s\in\Pi_A^{Bi}[\phi_i](m)\}).$$
Hence, trust-sensitive revision in this context involves finding the finest possible partition that provides a meaningful combination of the reports, and then revising with the corresponding state partition.
Trust and Deceit
================
To this point, we have only been concerned with modeling the trust that one agent holds in another due to perceived knowledge or expertise. Of course, the issue of trust also arises in cases where one agent suspects that another may be dishonest. However, the manner in which trust must be handled differs greatly in this context. If $A$ does not trust $B$, then there is little reason for $A$ to believe any part of a message sent directly from $B$.
Discussion
==========
Related Work
------------
We are not aware of any other work on trust that explicitly deals with the interaction between trust and formal belief revision operators. There is, however, a great deal of work on frameworks for modelling trust. As noted previously, the focus of such work is often on building reputations. One notable approach to this problem with an emphasis on knowledge representation is [@Wang07], in which trust is built based on evidence. This kind of approach could be used as a precursor step to build a trust metric, although one would need to account for domain expertise.
Different levels of trust are treated in [@Krukow07], where a lattice structure is used to represent various levels of trust strength. This is similar to our notion of a trust pseudometric, but it permits incomparable elements. There are certainly situations where this is a reasonable advantage. However, the emphasis is still on the representation of trust in [*an agent*]{} as opposed to trust in an agent with respect to a domain.
One notable approach that is similar to ours is the semantics of trust presented in [@Krukow07], which is a domain-based approach to differential trust in an agent. The emphasis there is on [*trust management*]{}, however. That is, the authors are concerned with how agents maintain some record of trust in the other agents; they are not concerned with a differential approach to belief revision.
Conclusion
----------
In this paper, we have developed an approach to trust sensitive belief revision in which an agent is trusted only with respect to a particular domain. This has been formally accomplished first by using state partitions to indicate which states an agent can be trusted to distinguish, and then by using distance functions to quantify the strength of trust. In both cases, the model of trust is used as sort of a precursor to belief revision. Each agent is able to perform belief revision based on a pre-order over states, but the actual formula for revision is parametrized and expanded based on the level of trust held in the reporting agent.
There are many directions for future work, in terms of both theory and applications. As noted previously, one of the subtle distinctions that must be addressed is the difference between trusted [*expertise*]{} and trusted [*honesty*]{}. The present framework does not explicitly deal with the problem of deception or [*belief manipulation*]{} [@Hunter13]; it would be useful to explore how models of trust must differ in this context. In terms of applications, our approach could be used in any domain where agents must make decisions based on beliefs formulated from multiple reports. This is the case, for example, in many networked communication systems.
[^1]: This is not a significant restriction. In domains involving sensing or other forms of discovery, we could simply allow an agent $A$ to self-report information with complete trust.
|
---
abstract: 'We show that deep convolutional neural networks (CNN) can massively outperform traditional densely-connected neural networks (both deep or shallow) in predicting eigenvalue problems in mechanics. In this sense, we strike out in a new direction in mechanics computations with strongly predictive NNs whose success depends not only on architectures being deep, but also being fundamentally different from the widely-used to date. We consider a model problem: predicting the eigenvalues of 1-D and 2-D phononic crystals. For the 1-D case, the optimal CNN architecture reaches $98\%$ accuracy level on unseen data when trained with just 20,000 samples, compared to $85\%$ accuracy even with $100,000$ samples for the typical network of choice in mechanics research. We show that, with relatively high data-efficiency, CNNs have the capability to generalize well and automatically learn deep symmetry operations, easily extending to higher dimensions and our 2D case. Most importantly, we show how CNNs can naturally represent mechanical material tensors, with its convolution kernels serving as local receptive fields, which is a natural representation of mechanical response. Strategies proposed are applicable to other mechanics’ problems and may, in the future, be used to sidestep cumbersome algorithms with purely data-driven approaches based upon modern deep architectures.'
address:
- 'Department of Mechanical, Materials, and Aerospace Engineering, Illinois Institute of Technology, Chicago, IL, 60616 USA'
- Amazon AWS AI Group
author:
- David Finol
- Yan Lu
- Vijay Mahadevan
- Ankit Srivastava
bibliography:
- 'ReferencesBib.bib'
title: Deep Convolutional Neural Networks for Eigenvalue Problems in Mechanics
---
Convolutional Neural Networks ,Phononic crystal ,Deep learning in Mechanics
Introduction
============
There has been a recent surge of interest in using deep learning using CNNs for machine learning problems in the areas of speech recognition, image and natural language processing, where advanced pattern recognition is required in data which is generally arranged in grid-like topologies [@rawat2017deep]. Beginning with image classification tasks, deep CNNs have consistently outperformed baseline mathematical models with prediction accuracies, in some tasks exceeding 98% [@lecun1998mnist]. In several areas, deep CNNs have achieved and even surpassed human level performance. For instance, in speech recognition in particular the baseline approaches [@bahl1990maximum; @bahl1986maximum; @levinson1983introduction] have been completely outperformed by statistical learning techniques, including CNN model architectures [@hinton2012deep; @deng2013recent]. The latter techniques have recently reached human-level accuracy [@Xiong2017; @Stolcke2017]. This rise of the deep CNNs over the last half a decade has been aided by a similar improvement in the computational capabilities which are required by such convolutional neural networks. Graphical Processing Units have proved particularly adept at training very large CNNs using increasingly large amounts of available data[@schmidhuber2015deep].
All this raises an important question: can deep architectures, such as CNNs, also be used to replace or aid certain mathematical computations central to the area of mechanics? To our knowledge, this question has not been considered before by other researchers in the field. There has admittedly been a recent push towards purely data driven mechanics computing [@kirchdoerfer2016data; @kirchdoerfer2017data; @ibanez2018manifold; @mosavi2017reviewing; @bessa2017framework]. However, these studies did not employ CNNs, but other predictive tools, such as regression and principal component analyses. There has also been a very recent push towards physics informed neural networks [@karpatne2017physics] as a way to make deep networks more data efficient, but they also seem to limit to traditional NNs or fully-connected Multilayer Perceptrons (MLPs) as their architecture. Similarly, authors have recently used deep networks for numerical quadrature calculations in FEM but have, again, limited the architecture to MLPs [@oishi2017computational]. More traditionally, the mechanics community has been using MLPs for various pattern recognition tasks and computations for several decades now [@le2015computational; @Lefik2009; @su2004intelligent; @challis1996ultrasonic; @legendre2001neural]. Furthermore, the neural networks which have been used have tended to be shallow in model depth characterized by a small number of layers, a small number of nodes per layer and a single set of computational elements. In this respect, the neural network approaches of the mechanics community resemble the pre-CNN era approaches of the machine learning community. In the machine learning community, it has been well established that deep CNN models significantly overpower both shallow and deep MLPs due to the following[@bengio2009learning; @delalleau2011shallow; @pascanu2013number; @montufar2014number]:
- Shallow networks show poorer generalization capabilities for highly nonlinear input-output relationships.
- Deep networks provide a more compact distributed representation of input output relationships.
A principal contribution of this paper is to develop some basic aspects of the framework which would allow deep CNNs to be employed for mechanics computations. Since no previous study exists in this domain, here we choose to focus on a simple eigenvalue problem in mechanics and train networks of various architectures to predict its eigenvalues. We show that deep CNNs massively outperform MLPs in this task. Another important contribution is to move from shallow networks in mechanics research to deep networks which are then efficiently trained over Graphical Processing Units, thereby progressing towards achieving research parity with the more traditional areas of machine learning such as vision and speech recognition.
There are some key ideas that make CNNs particularly suitable for solving problems encountered in the mechanics field. First, CNNs employ the concept of receptive fields in their core architecture exploiting spatially local correlation in grid-like topologies [@goodfellow2016deep]. This implies that CNNs may represent a powerful tool for pattern recognition in various computational mechanics and materials problems which are characterized by local interactions.
Second, there is a well-established consensus in the machine learning community that CNNs are efficiently able to learn representations which certain exhibit underlying symmetries [@gens2014deep; @jaderberg2015spatial]. CNN architectures tend to be equivariant to general affine input transformations of the nature of translations, rotations and small distortions [@lecun2012learning; @lenc2015understanding]. Instinctively, therefore, we expect that CNNs would be naturally suited to mechanics problems which exhibit such symmetries. These include, but is not limited to, areas in mechanics which are built on symmetries such as the mechanics and dynamics of periodic structures.
Finally, unlike MLPs, CNNs allow for multiple channels to be associated with each input node a feature that has been successfully exploited to represent the RGB channels of individual pixels for image recognition purposes. For problems in mechanics, the concept can be naturally extended where the components of the material tensors of individual discretized elements can be identified as multiple channels to individual nodes in a CNN. CNNs require minimal preprocessing to be able to classify high-dimensional patterns from a complex decision surface [@lecun2015lenet]. Our aim with this paper is, then, to demonstrate the potential of CNNs in the field of computational mechanics by deploying deep CNN architectures for a narrowly focused eigenvalue problem. We compare our results with results from MLPs showing the massive performance boost that is possible with the use of CNNs.
Problem Description
===================
In this section, we explain the physical model for the eigenvalue problem which we wish to emulate using CNNs. There is nothing particularly special about the chosen problem except for the fact that, being an eigenvalue problem, it represents a highly nonlinear input-output relationship and it serves well as a test bed for evaluating the capabilities of various neural networks. The frequency domain dynamics of a linear elastic medium with a spatially varying constitutive tensor $\mathbf{C}$ and density $\rho$ is given by: $$\boldsymbol{\Lambda}(\mathbf{u})+\mathbf{f}=\lambda\mathbf{u};\quad \boldsymbol{\Lambda}(\mathbf{u})\equiv \frac{1}{\rho}\left[C_{ijkl}u_{k,l}\right]_{,j}$$ where $\lambda=-\omega^2$, $\mathbf{u}$ is the displacement field, $\mathbf{f}$ is the body force, and $\boldsymbol{\Lambda}$ is a linear differential operator. For phononic problems the problem domain is periodic and is defined by a repeating unit cell. For the general 3-dimensional case, the unit cell ($\Omega$) is characterized by 3 base vectors $\mathbf{h}^i$, $i=1,2,3$. Any point within the unit cell can be uniquely specified by the vector $\mathbf{x}=H_i\mathbf{h}^i=x_i\mathbf{e}^i$ where $\mathbf{e}^i$ are the orthogonal unit vectors and $0\leq H_i\leq 1,\forall i$. The unit cell is associated with a set of reciprocal base vectors $\mathbf{q}^i$ such that $\mathbf{q}^i\cdot\mathbf{h}^j=2\pi\delta_{ij}$. Reciprocal lattice vectors are represented as a linear combination of the reciprocal base vectors, $\mathbf{G}^\mathbf{n}=n_i\mathbf{q}^i$, where $n_i$ are integers. Fig. (\[fVectors\]) shows the schematic of a 2-D unit cell, clearly indicating the unit cell basis vectors, the reciprocal basis vectors and the orthogonal basis vectors.
![Schematic of a 2-dimensional periodic composite. The unit cell vectors ($\mathbf{h}^1,\mathbf{h}^2$), reciprocal basis vectors ($\mathbf{q}^1,\mathbf{q}^2$), and the orthogonal vectors ($\mathbf{e}^1,\mathbf{e}^2$) are shown.[]{data-label="fVectors"}](Fig1)
The material properties have the following periodicity: $$C_{jkmn}(\mathbf{x}+n_i\mathbf{h}^i)=C_{jkmn}(\mathbf{x});\quad \rho(\mathbf{x}+n_i\mathbf{h}^i)=\rho(\mathbf{x})$$ where $n_i(i=1,2,3)$ are integers. Due to the periodic nature of the problem, it accepts solutions of the form $\mathbf{u}(\mathbf{x})=\mathbf{u}^p(\mathbf{x})e^{\mathrm{i}\mathbf{k}.\mathbf{x}}$ where $\mathbf{k}$ is the Bloch wavevector with components $\mathbf{k}=Q_i\mathbf{q}^i$ where $0\leq Q_i\leq 1,\forall i$ and $\mathbf{u}^p$ is a periodic function. Under the substitution, the harmonic elastodynamic problem can be formally written as (neglecting the body force): $$\boldsymbol{\Lambda}^{(\mathbf{k})}(\mathbf{u}^p)=\lambda\mathbf{u}^p
\label{eq:Eigenvalue}$$ where the superscript $(\mathbf{k})$ is now included to emphasize that the operator depends upon the Bloch wavevector. Explicitly we have: $$\boldsymbol{\Lambda}^{(\mathbf{k})}(\mathbf{u}^p)\equiv \frac{1}{\rho}\left[C_{ijkl}u_{k,l}^p\right]_{,j}+\frac{\mathrm{i}q_jC_{ijkl}}{\rho}u_{k,l}^p+\frac{\mathrm{i}q_l}{\rho}\left[C_{ijkl}u_k^p\right]_{,j}-\frac{q_lq_jC_{ijkl}}{\rho}u_k^p$$ For a suitable span of the wavevector, the sets of eigenvalues $\lambda_n$ (and the corresponding frequencies $\omega_n$) constitute the phononic dispersion relation of the composite. There are several numerical techniques for calculating the eigenvalues but a common method is to expand the field variable $\mathbf{u}^p$ in an appropriate basis and then use the basis to convert the differential equation into a set of linear equations. Both the Plane Wave Expansion[@kushwaha1993acoustic] method and Rayleigh quotient[@lu2016variational] follow this strategy.
The 1-D Problem
---------------
There is only one possible Bravais lattice in 1-dimension with a unit cell vector whose length equals the length of the unit cell itself (Fig. \[1-DResult\]a). Without any loss of generality we take the direction of this vector to be the same as $\mathbf{e}^1$. If the length of the unit cell is $a$, then we have $\mathbf{h}^1=a\mathbf{e}^1$. The reciprocal vector is given by $\mathbf{q}^1=(2\pi/a)\mathbf{e}^1$. The wave-vector of a Bloch wave traveling in this composite is specified as $\mathbf{k}=Q_1\mathbf{q}^1$. To completely characterize the band-structure of the unit cell it is sufficient to evaluate the dispersion relation in the irreducible Brillouin zone ($0\leq Q_1\leq .5$).
![a. Schematic of a 1-dimensional 2-phased periodic composite. The unit cell vector ($\mathbf{h}^1)$, reciprocal basis vector ($\mathbf{q}^1$), and the orthogonal vector ($\mathbf{e}^1$) are shown, b. Bandstructure of a 1-D, 2-phase phononic crystal showing the frequency eigenvalues constituting the first two passbands. Unit cell details in [@srivastava2016metamaterial].[]{data-label="1-DResult"}](1-DResult)
For plane longitudinal wave propagating in the $\mathbf{e}^1$ direction the only displacement component of interest is $u_1$ and the only relevant stress component is $\sigma_{11}$. The equation of motion and the constitutive law are: $$\label{equationofmotion1D}
\sigma_{11,1}=-\lambda\rho(x_1) u_1; \quad \sigma_{11}=E(x_1)u_{1,1}$$ where $E(x_1)$ is the spatially varying Young’s modulus. The exact dispersion relation for 1-D longitudinal wave propagation in a periodic layered composite can be solved using the transfer matrix method [@srivastava2016metamaterial]. For 2-phase composites, the relation is particulary simple [@rytov1956acoustical], $$\begin{aligned}
\label{Rytov}
\nonumber \cos(\mathbf{k} a)=\cos(\omega h_1/c_1)\cos(\omega h_2/c_2)-\Gamma \sin(\omega h_1/c_1)\sin(\omega h_2/c_2),\\
\Gamma=(1+\kappa^2)/(2\kappa), \; \kappa=\rho_1c_1/(\rho_2c_2), \end{aligned}$$ where $h_i$ is the thickness, $\rho_i$ is the density, and $c_i$ is the longitudinal wave velocity of the $i$th layer $(i = 1,2)$ in a unit cell. We can solve for the corresponding wave number $\mathbf{k}$ (or, equivalently, $Q_1$) by providing a frequency, $\omega$ (or, equivalently, $f=\omega/2\pi$), using (\[Rytov\]). These $f-Q_1$ (or $\omega-\mathbf{k}$) pairs constitute the eigenvalue band-structure of the composite when the wavevector is made to span the first Brillouin zone ($Q_1\in[0,0.5]$). A representative example of the bandstructure, calculated for a representative 1-D, 2-phase composite (Fig. \[1-DResult\]a), is shown in Fig. (\[1-DResult\]b). The frequency values in Fig. (\[1-DResult\]b) are, thus, related to the eigenvalues of the phononic crystal when a certain wavenumber $Q_1$ is specified. Results in Fig. (\[1-DResult\]) are calculated by using the exact physical solution of the system given by the Rytov equation (\[Rytov\]). Our aim in this paper is to train neural networks of varying architectures to sidestep the physical model as represented by Eqs. (\[equationofmotion1D\],\[Rytov\]).
Input-output relationship framework
-----------------------------------
The phononic eigenvalue problem described above can be represented as an input-output relationship. Formally, we can write: $$\mathbf{e}=\mathbf{f}_c(E(x_1),\rho(x_1),Q_1)
\label{eq:1DIOC}$$ where $\mathbf{e}$ is a vector of eigenvalues which is obtained through a vector of nonlinear functions $\mathbf{f}_c$ which operates on the space dependent material properties ($E,\rho$) and a choice of the wavenumber $Q_1$. An approximation to the eigenvalues can be generated by considering the material properties as discretely defined over the unit cell. We first normalize the unit cell with its length and discretize the range of $x_1$ ($x_1\in [0,1]$) into $N$ segments. We identify the material properties over these segments as the input variables. The material properties now become vectors themselves, individually defined over each segment, and the input-output relationship becomes: $$\begin{aligned}
\mathbf{e}=\mathbf{f}_d(\mathbf{E},\boldsymbol{\rho},Q_1),\; \mathbf{E}=\{E_i\},\boldsymbol{\rho}=\{\rho_i\};\;i=1,2...N
\label{eq:1DIOD}\end{aligned}$$ where $\mathbf{f}_d$ are the sought approximations to the continuous input-output relationships $\mathbf{f}_c$ in (\[eq:1DIOC\]). In this paper we have taken $N=100$ and we have sought to train neural networks to learn and predict $\mathbf{f}_d$ from a set of training data. The training data consists of input and output datasets created from the solution of the exact problem (\[equationofmotion1D\],\[Rytov\],\[eq:1DIOC\]). Specifically, the input data consists of randomly generated instantiations of vectors $\mathbf{E},\boldsymbol{\rho}$ and the output data consists of the corresponding first two eigenvalues, appropriately normalized, at chosen values of $Q_1$. For the set of problems under consideration, we seek to estimate the first two eigenvalues at 10 different $Q_1$ points within the range of $Q_1$ (\[0,0.5\]). This essentially translates into an output vector size of 20 and a flat input vector size of 200. 100 elements of the input vector correspond to $\mathbf{E}$ and the rest correspond to $\boldsymbol{\rho}$. In training neural networks to learn this input-output relationships, there are some questions of primary concern. Some of these pertain to the architecture of the network, the number of training data needed to achieve a certain desired error, the method of training the network, the method by which the training data is generated, the effect of the training data on the ability of the network to generalize for unseen examples, and in unseen regions of the space. These are discussed in detail in the next sections.
Neural Networks and the Representation Learning Approach
========================================================
A central problem in mechanics, as in other areas, is the approximation of a function which relates an input to an output in a system of interest (\[eq:1DIOC\]). The traditional method of attempting this problem is to manually convert the inputs to a set of representative features, which could then be related to the outputs. Representation learning, on the other hand, is a set of techniques that allows a system to automatically discover the optimal representations needed for feature detection, classification or real-value mapping from raw data. This replaces manual feature engineering and allows a machine to both learn the features and use them to perform a specific task (Fig. \[fig:replearning\]). Neural networks are a set of algorithms which perform this task by creating automatic representations of input data in their hidden layers. The first precursors to the modern neural networks were proposed by Rosenblatt [@rosenblatt1958perceptron]. Since then, significant advances in the area of machine learning has led to modern neural networks with complex function approximation capabilities [@schmidhuber2015deep]. Artificial Neural Networks (ANN) are comprised of multiple interconnected simple computational elements called *neurons*. Both the fully-connected multi-layer perceptrons and convolutional neural networks fit within the class of ANNs and they differ primarily in their neuronal architecture and interconnectivity.
![Manual feature crafting vs. the representation learning approach[]{data-label="fig:replearning"}](replearning)
Multi-Layer Perceptrons
-----------------------
The Multi-layer Perceptron (MLP) concept, more generally referred as feed-forward neural network, was initially proposed as an universal function approximator [@hornik1991approximation]. The objective of an MLP is to approximate a function $f$ between inputs $\mathbf{x}$ and outputs $\mathbf{y}$: $${}
\mathbf{y} = f(\mathbf{x})$$ This process can be represented in NN terms as a function mapping of inputs to the outputs through a set of optimizable parameters. For the single hidden layer MLP shown in Fig. (\[fig:FCMLP\]), these optimization parameters are the weight matrices $\mathbf{V,W}$ and the mapping is approximated as: $${}
y \approx f(x,\theta)= \mathbf{W}g(\mathbf{V}x)$$ where appropriate matrix multiplications are assumed. $g$ represents the activation function which is generally a non-linear transformation. The weights of the network are typically stochastically initialized and are subsequently tuned within the training phase of the network. One method of training the network is to relate its known input-output data and minimize its prediction deviation from the output by appropriately changing the weights, through an optimization process.
![Graph of a simple Multi-layer Perceptron with fully-connected layers[]{data-label="fig:FCMLP"}](2layerFCMLP)
Convolutional Neural Networks
-----------------------------
Three key ideas differentiate convolutional networks from conventional networks, which have made them highly successful in various engineering and science fields [@Ji2013; @Sermanet2013; @bojarski2016end; @aurisano2016convolutional; @wallach2015atomnet; @tajbakhsh2016convolutional; @goh2017deep] : the use of convolution operation, the implementation of the Rectified Linear (ReLU) activation function, and a representation-invariance imposing operation termed Pooling.
### Convolution
An important difference between a fully-connected MLP and a CNN is the use of the convolution operation instead of the standard matrix multiplication operation. While in the case of the MLP in Fig. (\[fig:FCMLP\]) the inputs to the neurons in the middle layer are obtained by a simple matrix multiplication of the weight $\mathbf{V}$ and the input $\mathbf{x}$, for a CNN, the input will instead be transformed using a convolution operation through a kernel $\mathbf{w}$ into a feature map $\mathbf{h}$. Assuming that the input is a 2-D vector of length $N$ and depth $d$, this can be represented as a 3-D matrix of size $N\times 1\times d$. The kernel is similarly assumed to be a matrix of size $m\times 1\times d$ where $m<N$. In this case, one element of the feature map will be calculated by computing the Einstein sum $h_l=x_{j+l,1,k}w_{j,1,k};j=1,...m,k=1,...d$ where $l$ is the current location of the filter. The filter is then advanced from its current location by a predetermined step size termed stride and the next element of the feature map is calculated. In our examples presented below, a good value of the stride is determined to be 1. This process is repeated until the kernel spans $\mathbf{x}$ in the length dimension and completes the calculation of $\mathbf{h}$. This process is symbolically represented as: $${}
\mathbf{h} = \mathbf{x}*\mathbf{w}$$ where $*$ represents the convolution operation. For instance, in our 1D phononic eigenvalue problem, the inputs are the spatially ordered sequence of material property values. These are represented by an input vector $\mathbf{X}$ whose depth is 2. In the depthwise direction, the first element corresponds to the modulus of a given finite element and the second element corresponds to the density value. The convolutional layer, for our case, convolves each input matrix with $k$ kernels ${\mathbf{w}_i}$ resulting in a total of $k$ feature maps. Each feature map $\mathbf{h}_i$ is computed as follows: $${}
\mathbf{h}_i = \mathbf{X}*\mathbf{w}_i + b_{i},i = 1,...,k.$$ where ${b_{i}}$ are bias parameters. CNNs work by tuning the parameters of the kernels $\mathbf{w}_i$ and the bias parameters $b_i$ in order to learn the desired input-output relationships. There are some interesting points to note here. First, the feature maps corresponding to the modulus and density channels are not completely independent of each other as they are generated from the dot product of the same kernel $\mathbf{w}_i$. This allows the neural network to associate the different material properties at a point as belonging to the same material. Second, the concepts of kernels allows for the feature maps to represent local interactions. This sparsity allows CNNs to learn an input-output relationship dependent upon spatial and temporal structure more efficiently than MLPs.
### ReLU Activation
Having calculated the pre-activation feature maps $\mathbf{h}_i$, they are passed through nonlinear activation functions commonly referred to as ReLU. The Rectifier Linear Unit Rectifier Linear Unit (ReLU) [@Glorot2011] has proven to provide high computational efficiency and is a piece-wise linear activation function that outputs zero if the input is negative and outputs the input if it is greater than zero. Mathematically, given a feature map $\mathbf{h}_i$, a ReLU function is defined as follows: $${}
\hat{\mathbf{h}}_ {i} = \mathrm{max}(0, \mathbf{h}_{i})$$ in which $\mathbf{h}_i$ and $\hat{\mathbf{h}}_i$ represent the ReLU input and output respectively.
### Pooling Operation
Once the convolution operations have been applied along with ReLU activations, the outputs pass through a parameter-reducing layer commonly referred to as a max-pooling layer. The output of this layer is the maximum unit from $p$ adjacent input units. In our 1-D problem, pooling is performed along both the transformed modulus and density data axes (Fig. \[fig:convMax\].) Following the findings of optimal CNN achitectures in computer vision and speech recognition [@LeCun1995; @krizhevsky2012imagenet] and our own optimization work, the pooling operation is performed only once in our case, after two consecutive convolution layers. The general intuition is that as more pooling layers are applied, units in higher layers would be less discriminative with respect to the variations in input features [@Zhang2017].
![Convolution and max-pooling transformations of modulus and density axes[]{data-label="fig:convMax"}](maxpool2)
Optimization Algorithm
----------------------
For both MLPs and CNNs in this paper, we employed the Stochastic Gradient Descent with Momemtum (SGDM) through backpropagation for network training. The training is performed on small groups of datasets at a time, termed mini-batches. The algorithm updates the model parameter by taking small steps in the direction of negative gradient: $${}
\theta_{l+1} = \theta_l - \alpha \nabla E(\theta_l)+\gamma(\theta_{l}-\theta_{l-1})$$ where $\theta_l$ represents the optimizable model parameters at iteration $l$, $\alpha$ is the learning rate, $E(\theta_l)$ is the current minibatch cost function and the $\gamma(\theta_{l}-\theta_{l-1})$ is the contribution of the previous gradient step to the current iteration [@bishop2006pattern]. The specific algorithm employed is termed as Adam and it involves adaptive learning rates for different parameters from estimates of first and second moments of the gradients [@Kingma2014].
Objective Function
------------------
Hung et al and Lippmann [@Hung1996; @Richard1991] have demonstrated that neural networks based on squared-error functions are able to accurately estimate posterior probabilities for finite sample sizes. The mean squared error objective function is defined as, $${}
MSE = \frac{1}{m} \sum {(\hat{y} -y)}^2$$ where $\hat{y}$ is the prediction and $y$ is the actual target output (eigenvalue in our case.) The summation is over all outputs and over all the data points in a mini-batch and $m$ is the total number of terms in the summation.
Data Mapping and Normalization
------------------------------
As is standard practice in modern neural network implementations, all input material property data is normalized through mean normalization: $${}
x_{norm} = \frac{x-\bar{x}}{x_\mathrm{max}-x_\mathrm{min}}$$ where $\bar{x}$ is the mean of the input vector. Furthermore, the eigenvalues are also normalized by a reference maximum, as is standard practice in the machine learning field.
Numerical Experiments
=====================
Data Generation
---------------
The input data for the CNN described above was generated following standard physical principles that guide the phononic eigenvalue problem as previously described. The training, validation, and test data sets can be divided into two broad categories. In the first category, the material property vectors $\mathbf{E}$ and $\boldsymbol{\rho}$ were generated using a uniform probability distribution on the modulus and density values. The probability distribution was created with lower and upper bounds for both the modulus (lower bound: 100MPa, upper bound: 300GPa) and density (lower bound: 800kg/m$^3$, upper bound: 8000kg/m$^3$). From these distributions, we created 100-element unit cells $\Omega^i$ randomly generated , which are characterized by modulus vectors $\mathbf{E}^i$ and density vectors $\boldsymbol{\rho}^i$. Corresponding to these unit cells, we calculated the 20 eigenvalues of interest which constitute the output data vectors $\mathbf{e}^i$. This dataset category is termed Dataset-A for future reference and has a total of 300,000 data samples. In the respective results, a small fraction of this dataset was employed purely for model training, validation and, in an initial phase, testing of the networks. We note that since the modulus and density values in Dataset-A are independently sampled from two different probability distributions, the material properties assigned to the individual elements in any given $\Omega^i$ very likely do not correspond to any real material.
The second broad category of data used in this paper is aimed at testing specific capabilities of the model in making predictions for those unit cells $\Omega$ which might be of more practical interest but which are highly unlikely to be represented in Dataset-A. One such example is the case where $\Omega$ is composed of only 2 different material phases - a configuration which appears frequently in the phononics and metamaterial literature [@hussein2014dynamics; @srivastava2016metamaterial; @srivastava2017evanescent]. To create a dataset corresponding to this configuration, we divided the 100-element unit cell into two zones of 50 elements each. All elements in these individual zones were then assigned randomly generated but same modulus and density values. The input-output data for this dataset is referred to as Dataset-2. We similarly created datasets for 3 and 10 material phases and refer to them as Dataset-3 and Dataset-10 respectively which, in addition to Dataset-2, is collectively referred to as Dataset-B. At this point it should be noted that it is highly unlikely that any individual case appearing in Dataset-B also appears in Dataset-A.
Model Architectures
-------------------
After an iterative process of line search in the hyperparameters of the CNNs employed, it was determined that a typical optimal model for our phononic eigenvalue problem has the following architecture: There are 100 input nodes corresponding to the 100-element unit cell for the 1-D case (Fig. \[fig:CNN\]). Each input node has two channels with individual channels corresponding to the density and modulus values of the element. This is a crucial piece of information in our CNN implementation in that we choose to identify the various material properties of an element with individual channels in a CNN. Not only does this create a direct correspondence with the practice of identifying the RGB information at a pixel with individual channels in computer vision applications, it also guides how CNNs could be employed for problems in mechanics in higher dimensions. In higher dimensions, more material properties (components of elasticity tensor) are required to describe the behavior of materials. Given the success of CNNs in our current 1-D problem, a promising strategy going forward would be to identify these material properties as different channels in a CNN input node layer.
\[fig:cnn2new\] ![CNN architecture used for the 1D study[]{data-label="fig:CNN"}](cnn3new "fig:")
Following the input node is a set of two convolution layers with ReLU activation functions. The convolution filters has a size of 3 in the dimension of the input vector. A max-pooling layer follows the set of convolution layers with a filter dimension of 2$\times$1. These layers are followed by two fully connected dense layers with ReLU activation functions. Subsequently, a fully connected layer with linear or Gaussian connections leads to the outputs of the network which are the eigenvalues of the problem. For the purposes of most of the results shown below, the network was trained on 28,000 samples which took a little less than 13 minutes on a GPU (NVIDIA GTX980). Once the network is trained, prediction obviously takes a much smaller amount of time (fraction of a second.) The same input-output mapping was also performed using a multi-layer perceptron with only fully connected layers, which was used to generate some of the key results presented in this paper. The architecture that yielded the best prediction accuracy was found to have 6 hidden layers with 1024 computational units in each layer (Fig. \[fig:MLP\].) The final output layer consisted of gaussian or linear connections to the output units.
![MLP architecture used for the 1D study[]{data-label="fig:MLP"}](MLPnew)
Results
-------
### Approximation Capabilities of CNNs
Our initial focus was on obtaining average eigenvalue prediction accuracies higher than 95$\%$ on unseen examples and comparing the performance of CNNs vs regular MLPs in achieving this metric. We define mean absolute accuracy of our predictions: $${}
e = 1-\frac{1}{p} \sum \frac{|(\hat{y} -y)|}{\hat{y}}$$ where the summation is being carried out over all eigenvalues and all test data and $p$ is the total number of terms in the summation. The error is then expressed as a percentage. For the purpose of comparison, we considered Dataset-A for training, validation, and testing purposes. As mentioned earlier, Dataset-A has 300,000 data samples. A randomly selected portion of these is used for training the networks and the rest of the unseen examples are used for validation and testing purposes.
![Comparison of prediction accuracy as a function of training data size (CNN vs MLP)[]{data-label="fig:MLPvsCNN"}](MLPvsCNNnew)
The striking point about this comparison is that the CNN easily outperforms the MLP in terms of eigenvalue prediction accuracy as can be seen in Fig. (\[fig:MLPvsCNN\]). Both networks improve in their prediction accuracies as they are trained on larger fractions of Dataset-A. However, the CNN already has a higher than 95$\%$ prediction accuracy when it has been trained only on 20,000 samples. At this level of training, the MLP has an accuracy of slighly greater than 70$\%$. In fact, the MLP only reaches about 85$\%$ accuracy at 100,000 training samples at which point the CNN is already above 98$\%$ prediction accuracy. The 30,000-40,000 samples range seems to be a breaking point where both networks show the largest percental decrease of mean absolute error. In summary, these comparisons show that CNNs have the potential to achieve high prediction accuracies in problems of mechanics with a fraction of the training data required by MLPs.
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The CNN architecture employed was eventually able to achieve 1.13 $\%$ of mean absolute error in its eigenvalue predictions with a normalized 1-$\sigma$ error of 2.11e-05. The small value of the standard deviation shows that most of the distribution of prediction errors is heavily centered around its mean value. In other words, a large fraction of the predictions have absolute errors close to the reported mean of the distribution, whereas only a small fraction of the predictions has larger prediction errors. Fig. (\[fig:performance\]b) shows all the predicted eigenvalues along with the associated analytically computed eigenvalues and clarifies the high correlation which exists between the two sets. This correlation can be measured in terms of the Pearson correlation coefficient which, for the present case, is at the level of 0.999 indicating that the regression between the two sets is strongly linear. This can also be seen from the distribution of prediction errors in Fig. (\[fig:performance\]a) which shows that a vast majority of the predictions have errors in the $\pm 2\%$ range. Fig. (\[fig:performance\]c) shows an example prediction of the CNN compared with actual eigenvalue calculations for a specific unseen phononic crystal configuration, which represents a sample from Dataset-2. Specifically, this sample represents a 2-phase material with two equal phases. One phase has a density of 5.80 g/cm$^3$ and a modulus of 76.27 GPa and the other phase has a density of 7.23 g/cm$^3$ and a modulus of 23.61 GPa. It can then be seen that all the 20 eigenvalues have been predicted with a reasonable accuracy by the CNN.
### Generalization Capabilities of CNNs
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From the previous section, it is evident that CNNs can massively outperform MLPs with fully connected layers in terms of their prediction accuracies for comparable training datasets. Another important consideration in evaluating the efficacy of any approximating method is its capability to generalize to unseen examples. To a limited extent, we have already shown that the trained CNN achieves very high prediction accuracies on unseen test data when the test data is extracted from the same distribution as the training and validation datasets. Here we consider the ability of the CNN to generalize to examples which are derived from significantly different distributions. For this we consider Dataset-A, Dataset-2, Dataset-3, and Dataset-10 as described earlier. While each of the elements in the samples in Dataset-A have materials properties that come from a random uniform distribution, the rest of the datasets used in this section contains contrasting phases that naturally yields a broader range of eigenvalue distributions. Representative 3D histograms for Dataset-A and Dataset-2 can be seen in Fig. (\[fig:GeneralizationInput\]). The colormap projection in Fig. (\[fig:GeneralizationInput\]a) shows that there is largely an even distribution of samples spanning the property ranges for Dataset-A. For dataset-2 (\[fig:GeneralizationInput\]b), the distribution is less even and it misses some density ranges that are present in Dataset-A. Most importantly, however, since Dataset-B corresponds to only 2-phase cases, its corresponding eigenvalue ranges are very different from those of the samples in Dataset-A. This is clarified in Fig. (\[fig:GeneralizationOutput\]) which shows histogram plots of all the eigenvalues for Dataset-A, Dataset-2, and Dataset-10. It shows that the eigenvalue range spanned by Dataset-2 is about twice as large as that spanned by Dataset-A.
![Uniform sample from the N100 and the 2 real materials datasets, regarding the 1st upper eigenvalue from the band structure[]{data-label="fig:GeneralizationOutput"}](eigvaloutputnew)
Fig. (\[fig:GeneralizationOutput\]) shows some results which underline the generalization capabilities of the employed CNNs. First, when the network is only trained on Dataset-A, it is able to achieve $<16\%$ prediction errors on Dataset-2, $<15\%$ prediction errors on Dataset-3, and $<13\%$ prediction errors on Dataset-10 (blue bars.) This is interesting because not only does any input configuration in Dataset-2, Dataset-3, and Dataset-10 most likely does not exist in Dataset-A, there exist large eigenvalue ranges in these datasets which the trained network was never trained on and has never seen. These prediction errors come down substantially when Dataset-A is augmented with some samples from Dataset-2 and Dataset-3 for training purposes. The results are shown by orange bars in Fig. (\[fig:GeneralizationOutput\]). In this case, the prediction errors for Dataset-2, Dataset-3, and Dataset-10 are all below $6\%$. Note that in this case, although the network was never trained on Dataset-10, it was able to generalize well when tested on this dataset. From these results, there is strong indication that deep CNNs applied as we present seem to be able to successfully generalize on completely new regions of the input-output space, improving upon the typical issues brought by implementations of shallow and less diverse networks in mechanics. These deep networks also seem to require much less data to achieve and surpass the performance of traditional MLPs.
![Model Generalization[]{data-label="fig:GeneralizationError"}](1DGenPlot)
### Automatic learning of translational invariance
An important and interesting feature of our CNN implementation is that it seems to have learned translational invariance of our problems automatically. Since phononic crystals are periodic composites, their eigenvalues (Eq. \[eq:Eigenvalue\]) are invariant under unit cell translations. In effect, it means that a 2-phase unit cell with a 50-50 distribution of the two material phases ($P_1-P_2$) will have the same eigenvalues as a 3 phase unit cell made by periodically translating the two materials phases. For the latter, let’s consider as an example, a unit cell made of phases $P_1-P_2-P_1$ with a 25-50-25 distribution respectively. Our CNN accurately predicts the eigenvalues for the two cases (both unseen), showing that it has learned the translation invariance of the problem. A deeper understanding of this can be done by reconstructing the activations of the deeper layers of the network and understanding the filter activations when a prediction is being made. For the most part, it is often not a trivial task to try to reconstruct the deeper layers of the network into the input space. However, a simple activation analysis on the maxpooling layer is performed in this study. The activations of the maxpooling filters feed directly to the regression layers. Given those activations are similar in the two cases, then the network prediction will also be the same. In theory and if the network’s optimization is done successfully, filters should specialize such that their unique and sparse activation leads to the best output prediction possible. This means that input samples with similar input-output mappings would activate the same filters, while keeping the rest inactive.
![Mean post-pooling filter activations versus filter number for two symmetric samples from Dataset-2 (which yields the same eigenvalue output) and a Dataset-10 sample[]{data-label="fig:activations1"}](activations2)
Fig. (\[fig:activations1\]) shows the maxpooling filter activation results when the network is presented with three different input unit cells. Two of these (from Dataset-2) are physically the same, differing only with a translation, with the third one being significantly different as it is taken from Dataset-10. We notice from this figure that for the cases of the translationally equivalent unit cells, not only are the same filters activated, but their activation values are also practically the same. On the other hand, for the sample from Dataset-10, the filter activations differ not only in their values, but also in which filters are activated.
Convolutional Neural Networks for 2-D phononic eigenvalues
----------------------------------------------------------
![a. Schematic of the 2-D periodic composite made from steel cylinders distributed in hexagonal packing in epoxy matrix, b. Discretization of the unit cell, c. Irreducible Brouillon Zone in the reciprocal lattice, d. Band-structure calculation results using the mixed variational formulation.[]{data-label="fig:fCompHexagonal"}](fCompHexagonal)
The formalism of input-output relationships as described in Eqs. (\[eq:1DIOC\], \[eq:1DIOD\]) immediately applies to higher dimensional eigenvalue problems. Here, as a demonstration of the convolutional neural networks in higher dimensions, we consider the phononic eigenvalue problem emerging in a hexagonal phononic crystal. A representative example of 2-D phononic eigenvalues is shown in the bandstructure calculations given in Fig. (\[fig:fCompHexagonal\]). More details about the calculations can be found in Ref. [@srivastava2014mixed]. In order to apply a CNN framework to the 2-D problem, we divide the hexagonal unit cell into a hexagonal grid of size ($128\times 128$). Now the eigenvalue input-output formalism is given by: $$\begin{aligned}
\mathbf{e}=\mathbf{f}_d(\mathbf{E},\boldsymbol{\nu},\boldsymbol{\rho},Q_1,Q_2),\; \mathbf{E}=\{E_{ij}\},\boldsymbol{\nu}=\{\nu_{ij}\},\boldsymbol{\rho}=\{\rho_{ij}\};\;i,j=1,2...128
\label{eq:2DIOD}\end{aligned}$$ where $E_{ij},\nu_{ij}$ are the value of the Young’s modulus and Poisson’s ratio respectively for the $i,j$ element in the hexagonal grid and $Q_1,Q_2$ are the normalized wavenumbers in the directions $\mathbf{q}^1,\mathbf{q}^2$ respectively. The training-testing-validation datasets are created by randomly generating material property matrices $\mathbf{E},\boldsymbol{\nu},\boldsymbol{\rho}$ by considering the Young’s modulus as a random variable distributed between 100 MPa and 300 GPa, Poisson’s ratio as a random variable betweeen 0.2 and 0.45, and density as a random variable between 800 and 8000 kg/m$^3$. As inputs to the neural networks, two additional matrices of size $128\times 128$ corresponding to $Q_1, Q_2$ are also considered. All the elements of one of these matrices are equal to $Q_1$ and all the elements of the other matrix are equal to $Q_2$. The matrices corresponding to $\mathbf{E},\boldsymbol{\nu},\boldsymbol{\rho},Q_1,Q_2$ are now concatenated into a 3-D matrix $\mathbf{I}$ of size $128\times 128\times 5$. The output $\mathbf{e}$ is a 1-D vector which consists of the lowest 50 eigenvalues of the phononic problem corresponding to $\mathbf{I}$. Four different datasets were created to demonstrate the training and testing of the neural network. Dataset-A consists of 100,000 samples created by considering the material properties in each element as independent random variables. Dataset-2, Dataset-3, and Dataset-5 each consist of 20,000 samples where, in each data-sample, the material properties are only allowed to take 2, 3, and 5 different randomly generated values respectively.
The optimized network architecture used for this problem is a natural extension of the one used in the 1-D problem. The input space $\mathbf{I}$, with depth 5 in this case, is processed by 2D convolution and pooling filters, with size of 2x2 instead. These then lead to fully-connected units, which eventually connect to the output vector $\mathbf{e}$. Fig. (\[fig:2DResults\]) shows some results which underline the generalization capabilities of the employed CNNs for the 2-D case. First, when the network is only trained on Dataset-A, it is able to achieve prediction errors below $50\%$ prediction errors on Dataset-2, Dataset-3, and Dataset-5 (blue bars). This difference with respect to the 1D case is likely tied to the significant increase on the input dimensionality. A significant decrease on prediction error is observed when the optimization dataset includes Dataset-A, Dataset-2 and Dataset-3 samples. The results are shown by orange bars in Fig. (\[fig:2DResults\]a). As seen in the orange bars in Fig. (\[fig:2DResults\]a), the prediction errors for Dataset-2, Dataset-3, and Dataset-5 are all below $5\%$. Just as in the 1D case, the network was never trained on Dataset-5, but it was able to also generate predictions with errors below the target threshold in this new space. Finally, Figure (\[fig:2DResults\]b) shows how the prediction error on unseen Dataset-5 samples changes as the number of total samples in the optimized mixed dataset increases. This demonstrates the data efficiency and accuracy of the CNN implementation stands for this higher dimensional case as well.
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Conclusion
==========
We have shown that key mathematical properties that have made modern deep neural network architectures, such as CNNs, and that have proven to be very successful in typical machine learning implementation fields, can translate to complex input-output mapping tasks in mechanics. Through our results, not only were we able to demonstrate that CNNs can successfully approximate the complex function mapping of material tensors of a phononic crystal to its eigenvalues, but also that it can be done with much higher data efficiency and performance than traditional neural networks (densely connected MLPs). The CNN model architectures used in this study present a much more diverse set of computational elements and deeper graph depth that aid both the feature abstraction process and the function approximation capabilities, as is well established in the machine learning literature. The basic filter activation study in this paper also provided a hint at how the feature abstraction and function mapping occurs in our implementation, leaving the door open for more a comprehensive study in future applications. We found that when using an optimized network to predict a case with a characteristic input translation that leads to the same output, exactly the same post-pooling filters are consistently activated with quite similar activation values. This fact provides clues that these filters are specializing in characteristic input patterns that aids the prediction of the output eigenvalues. This process has been noted to occur in computer vision applications of deep CNNs and have proven useful in our application in mechanics as well.
There are many interesting questions worth pursuing after this study. The primary one is the extension of this process to dimensions higher than 2 for the prediction of eigenvalues in more complex mechanics problems. In higher dimensions more material properties would need to be taken into account. For instance, even if we consider an isotropic material, its mechanical description would generally require 3 independent constants (shear modulus, bulk modulus, and density). For anisotropic materials this number further goes up. However, within the CNN framework presented in this paper, taking these into account merely requires the addition of more channels or depth to the input vectors. With this framework, it would be interesting to see if deep CNNs can be efficiently trained to act as substitutes for eigenvalue algorithms in the case of 2-, and 3-D mechanics problems. Such networks would have no fundamental computational complexity limitations with which all eigenvalue algorithms suffer and, therefore, would provide a way to explore design spaces which have not been probed yet. The idea of representing the elements of material property tensors as different channels in a CNN, as proposed in this paper, is clearly not limited to eigenvalue problems. Therefore, it is conceivable that CNNs can similarly lead to significant improvements in regression and classification tasks in non-eigenvalue mechanics problems including time domain problems.
Acknowledgements {#acknowledgements .unnumbered}
================
A.S. acknowledges support from the NSF CAREER grant \# 1554033 to the Illinois Institute of Technology.
References {#references .unnumbered}
==========
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---
abstract: 'We study the phenomenology of a supersymmetric bulk in the scenario of large extra dimensions. The virtual exchange of gravitino KK states in selectron pair production in polarized collisions is examined. The leading order operator for this exchange is dimension six, in contrast to that of graviton KK exchange which induces a dimension eight operator at lowest order. Some kinematic distributions for selectron production are presented. These processes yield an enormous sensitivity to the fundamental higher dimensional Planck scale.'
author:
- 'JoAnne L. Hewett and Darius Sadri'
title: Phenomenology of Supersymmetric Large Extra Dimensions
---
One might wonder whether supersymmetry plays a role in the recently proposed ADD scenario [@Arkani-Hamed:1998rs] of large extra dimensions. Clearly, bulk supersymmetry is not in conflict with the basic assumptions of the model. In fact, various reasons exist for believing in a supersymmetric bulk, not least of which is the motivation of string theory. D-branes of string theory provide a natural mechanism for the confinement of SM fields. If string theory is the ultimate theory of nature then the proposal of ADD might be embedded within it with a bulk supporting a supersymmetric gravitational action, with supersymmetry serving as a mechanism for stabilizing the bulk radii. In [@sadri], we investigated the consequences of a supersymmetric bulk in the ADD scenario. If bulk supersymmetry remains unbroken away from the brane, then it is natural to ask what happens to the superpartners of the bulk gravitons, the gravitinos. The bulk gravitinos must also expand into a Kaluza-Klein (KK) tower of states and induce experimental signatures.
We focus on the effects of the virtual exchange of the bulk gravitino and graviton KK tower states in the process $e^+e^-\to\tilde e^+\tilde e^-$ at a high energy Linear Collider (LC). This process is well-known as a benchmark for collider supersymmetry studies [@Baer:1988kx], as the use of incoming polarized beams enables one to disentangle the neutralino sector and determine the degree of mixing between the various pure gaugino states. The effects of the virtual exchange of a graviton KK tower in selectron pair production has been examined in [@tgr] for the case of non-supersymmetric large extra dimensions. The introduction of gravitino KK exchange greatly alters the phenomenology of this process by modifying the angular distributions and by substantially increasing the magnitude of the cross section. We find that the leading order behavior for this process is given by a dimension-6 operator, in contrast to the dimension-8 operator corresponding to graviton KK exchange. This yields a tremendous sensitivity to the existence of a supersymmetric bulk, resulting in a search reach for the ultraviolet cut-off of the theory of order $20-25\times\sqrt s$.
We assume that the Standard Model fields are confined to a 3-brane. In string theory, D-branes are extended objects on which open strings terminate [@Polchinski:1996na]. Only closed strings can propagate far away from the D-brane, which on a ten dimensional background is described locally by a type II string theory whose spectrum contains two gravitinos (their vertex operators carry one vector and one spinor index). The D-brane introduces open string boundary conditions which are invariant under just one supersymmetry, so only a linear combination of the two original supersymmetries survives for the open strings. Since open and closed strings couple to each other, the D-brane breaks the original $N=2$ supersymmetry down to $N=1$. The low energy effective theory will then be a $D=10, N=1$ supergravity theory with a single Majorana-Weyl gravitino which couples to a conserved space-time supercurrent.
We take the vacuum of space-time to be of the form $M^{4} \times T^{6}$, where $M^{4}$ is four dimensional Minkowski space-time and $T^{6} = S^{1} \times \ldots \times S^{1}$, the direct product of six dimensions each compactified on a circle, insuring four dimensional Poincaré invariance. For simplicity, we take a common radius of compactification $R_c$ for all extra dimensions. The gravitino kinetic term of the Lagrangian is $$\label{RS:action}
E^{-1} \mathcal{L} \: = \:
\frac{i}{2} \: \bar\Psi_{\hat{\mu}} \:
\Gamma^{\hat{\mu}\hat{\nu}\hat{\rho}}
\: \nabla_{\hat{\nu}}
\Psi_{\hat{\rho}} \,,$$ with $E$ being the determinant of the vielbein in ten dimensions, $\Gamma^{\hat{\mu}\hat{\nu}\hat{\rho}}$ the antisymmetric product of three $\Gamma$ matrices, and $\Psi_{\hat{\mu}}$ a Majorana-Weyl vector-spinor. Here the hatted indices range over all dimensions of the space-time.
The explicit derivation of an effective four dimensional action is presented in [@sadri]. Here we present a summary of the results. The 4-d effective Lagrangian can be written as $$\label{effective:Lagrangian:KK:diag}
e^{-1} \: \mathcal{L}_{eff}^{\vec{s}} ( x ) \: = \:
\sum_{j=1}^{4}
\left\{
\frac{i}{2} \:
\bar{\omega}_{m}^{\vec{s},j}(x)
\gamma^{mnp} ( \partial_{n} \omega_{p}^{\vec{s},j} (x) )
\: + \:
i \: \bar{\omega}_{m}^{\vec{s},j}(x)
\sigma^{mp} m_{\vec{s}}^j
\omega_{p}^{\vec{s},j}(x)
\right\}\,,$$ with the mass given by $m_{\vec{s}}^j = (-1)^j \frac{\sqrt{\vec{s} \cdot \vec{s}}}{R_{c}}$. The fields associated with the negative mass eigenvalues can be redefined to remove this sign, however, care must be taken with the Feynman rule for the coupling of the gravitino to matter. The sum in Eq. (\[effective:Lagrangian:KK:diag\]) runs over the four Majorana vector-spinors. We have applied the Majorana-Weyl condition in ten dimensions, which yields four Majorana spinors after the decomposition into four dimensions. Generally, the masses of the four gravitinos at each Kaluza-Klein level can be shifted by supersymmetry breaking effects on the brane. When studying the phenomenology of such models, we will assume that the $N=4$ supersymmetry is broken at scales near the fundamental scale $M_D$, with only $N=1$ supersymmetry surviving down to the electroweak scale. The phenomenological contributions from the heavy gravitinos associated with the breaking of the extended supersymmetry near the fundamental scale will be highly suppressed, due to the large mass of these individual excitations.
In [@sadri], we derive the coupling of fermions and scalars to the gravitinos. The term coupling scalars, spinors and gravitinos, and minimally coupled to gravity, is $$\mathcal{L}_I \: = \:
- \frac{\kappa}{\sqrt{2}} | e |
\left\{
\left( \partial_\mu \Phi_L \right) \bar{\Omega}_\nu
\gamma^\mu \gamma^\nu \psi_L \: + \:
\left( \partial_\mu \Phi_R \right) \bar{\Omega}_\nu
\gamma^\mu \gamma^\nu \psi_R
\right\}\, \: + \: h.c. \,,$$ with $\bar{\Omega}_\nu$ a Majorana vector-spinor. Expanding $|e|$ to leading order in the vierbein yields the appropriate Feynman rules.
There are numerous tree level processes contributing to selectron pair production in the presence of a supersymmetric bulk. In addition to the standard $\gamma,Z$ s-channel exchange and $\tilde B^0,\tilde W^0$ t-channel contributions present in the MSSM, we now have contributions arising from the s-channel exchange of the bulk graviton KK tower and the t-channel exchange of the bulk gravitino KK tower. There are no u-channel contributions due to the non-identical final states. The contributions from neutral higgsino states are negligible due to the smallness of the Yukawa coupling. The diagrammatic contributions to the individual scattering processes for left- and right-handed selectron production with initial polarized electron beams are summarized in Table \[diag\]. Note that the $\tilde W^0$ exchange only contributes to the process $e^-_Le^+\to\sl^-\sl^+$, and that the t-channel gravitino and the $\tilde B^0$ contributions are isolated in the reaction $e^-_{L,R}e^+\to\tilde e^-_{L,R}\tilde e^+_{R,L}$.
---------------------------------------------------------------------------------------------------------------------------------------------------------
$\tilde e^-_L\tilde e^+_L$ $\tilde e^-_R\tilde e^+_L$ $\tilde e^-_L\tilde e^+_R $ $\tilde e^-_R\tilde e^+_R$
------------ -------------------------------------------- -------------------------------- ----------------------------- --------------------------------
$e^-_Le^+$ s-channel $\gamma\,, Z\,, G_n$ s-channel $\gamma\,, Z\,,
G_n$
t-channel $\tilde W\,, \tilde B\,, \Psi_n$ t-channel $\tilde B\,,
\Psi_n$
$e^-_Re^+$ s-channel $\gamma\,, Z\,, G_n$ s-channel $\gamma\,, Z\,,
G_n$
t-channel $\tilde B\,, \Psi_n$ t-channel $\tilde B\,, \Psi_n$
---------------------------------------------------------------------------------------------------------------------------------------------------------
: The diagrammatic contributions to individual scattering processes for polarized electron beams. A blank box indicates that there are no contributions for that polarization configuration.[]{data-label="diag"}
The unpolarized matrix element for the case of massive gravitino KK exchanges is $${\cal M}={\frac{\kappa^2}{2}}\sum_{\vec n} {\frac{k_1^\nu k_2^\rho}
{t-m^2_{\vec n}}}\bar e(p_1)\gamma_\mu\gamma_\nu P^{\vec n,\mu\tau}
\gamma_\rho\gamma_\tau e(p_2)\,,$$ where the sum extends over the gravitino KK modes and $\kappa=\sqrt{8\pi G_N} =\bar M_P^{-1}$ is the reduced Planck scale. $P^{\vec n,\mu\tau}$ represents the numerator of the propagator for a Rarita-Schwinger field of mass $m_{\vec n}$ and is given by $$\label{propagator}
P^{\vec{n}, \mu \nu} \: = \:
i \: \left( \slash{k} + m_{\vec{n}} \right)
\left( \frac{k^{\mu} k^{\nu}}{m_{\vec{n}}^2} - \eta^{\mu \nu} \right)
- \frac{i}{3}
\left( \gamma^{\mu} + \frac{k^{\mu}}{m_{\vec{n}}} \right)
\left( \slash{k} - m_{\vec{n}} \right)
\left( \gamma^{\nu} + \frac{k^{\nu}}{m_{\vec{n}}} \right)\,.$$ The mass splitting between the evenly spaced bulk gravitino KK excitations is given by $1/R_c$, which lies in the range $10^{-4}$ eV to few MeV for $\delta=2$ to 6 assuming $M_D\sim 1$ TeV; their number density is thus large at collider energies. The sum over the KK states can then be approximated by an integral which is log divergent for $\delta=2$ and power divergent for $\delta>2$. We employ a cut-off to regulate these ultraviolet divergences, with the cut-off being set to $\Lambda_c$, which in general is different from $M_D$, to account for the uncertainties from the unknown ultraviolet physics. This approach is the most model independent and is that generally used in the case of virtual graviton exchange [@Hewett:1998sn]. In practice, the integral over the gravitino KK states is more complicated than that in the case with spin-2 gravitons due to the dependence of the gravitino propagator on $m_{\vec n}$. We find that the leading order term for $\sqrt{|t|}\ll\Lambda_c$ results in the replacement (in the case of $\delta=6$) $${\frac{\kappa^2}{2}}\sum_{\vec n} {\frac{P^{\vec n,\mu\tau}}{t-m^2_{\vec n}}}
\to {\frac{-i8\pi}{5\Lambda_c^3}}\left( \eta^{\mu\nu}-{\frac{1}{3}}
\gamma^\mu\gamma^\nu\right) \,$$ in the matrix element; the structure of the summed gravitino propagator is thus altered from that of a single massive state. Hence the leading order behavior for gravitino KK exchange results in a dimension-6 operator! This is in stark contrast to graviton KK exchange which yields a dimension-8 operator at leading order. We thus expect an increased sensitivity to the scale $\Lambda_c$ in the case of a supersymmetric bulk.
In order to perform a numerical analysis of this process, we need to specify a concrete supersymmetric model. We choose that of Gauge Mediated Supersymmetry Breaking (GSMB) as it naturally contains a light zero-mode gravitino. We specify a sample set of input parameters at the messenger scale, where the supersymmetry breaking is mediated via the messenger sector, and use the Renormalization Group Equations (RGE) to obtain the low-energy sparticle spectrum. We choose two sets of sample input parameters describing the messenger sector which are consistent with our model. The RGE evolution of these parameter sets results in the sparticle spectrum $$\begin{aligned}
{\rm Set~I} : & m_{\tilde e_L} & =217.0\, {\rm GeV},\quad
m_{\tilde e_R}=108.0\, {\rm GeV}, \quad
\chi^0_i=(76.5,\, 141.5,\, 337.0,\, 367.0)\, {\rm GeV} \,,\nonumber \\
{\rm Set~II}: & m_{\tilde e_L} & =210.5\, {\rm GeV},\quad
m_{\tilde e_R}=104.5\, {\rm GeV}, \quad
\chi^0_i=(110.5,\, 209.6,\, 322.5,\, 324.0)\, {\rm GeV} \,,
\nonumber\end{aligned}$$ where $\chi^0_i$ with $i=1,4$ corresponds to the four mixed neutralino states. The first set of parameters yields a bino-like state for the lightest neutralino, whereas the second set results in a Higgsino-like state for $\chi^0_1$. These input parameters were selected in order to obtain a sparticle spectrum which is kinematically accessible to the Linear Collider; our results are essentially insensitive to the exact details of the spectrum.
Figure \[8782\_fig5\] shows the angular distributions with $100\%$ electron beam polarization for each helicity configuration listed in Table \[diag\] for the two sets of parameters discussed above, with and without the contributions from supersymmetric extra dimensions. In each case, the solid curve corresponds to the bino-like case and the dashed curve represents the Higgsino-like scenario. The top set of curves are those for a supersymmetric bulk with $\Lambda_c=1.5$ TeV, while the bottom set corresponds to our two $D=4$ supersymmetric models, , without the graviton and gravitino KK contributions. We note that the $D=4$ results (i.e. MSSM) agree with those in the literature [@Baer:1988kx]. We see from the figure that in the process where the gravitino contributions are dominant, $e^-_{L,R}e^+\to\sl^\pm\sr^\mp$, there is little difference in the shape or magnitude between the two $\chi^0_1$ compositions. The use of selectron pair production in polarized collisions as a means of determining the composition of the lightest neutralino is thus made more difficult in the scenario with supersymmetric large extra dimensions. In what follows, we present results only for the bino-like $\chi^0_1$ as a sample case; our conclusions will not be dependent on the assumptions of the composition of the lightest neutralino.
![Angular distributions for each helicity configuration with supersymmetric bulk contributions for $\Lambda_c=1.5$ TeV (top curves), and for the $D=4$ supersymmetric models (bottom curves). The solid (dashed) curves correspond to a bino-like (Higgsino-like) composition of the lightest neutralino.[]{data-label="8782_fig5"}](slac8782_fig5a.ps "fig:"){width="5.1cm"} ![Angular distributions for each helicity configuration with supersymmetric bulk contributions for $\Lambda_c=1.5$ TeV (top curves), and for the $D=4$ supersymmetric models (bottom curves). The solid (dashed) curves correspond to a bino-like (Higgsino-like) composition of the lightest neutralino.[]{data-label="8782_fig5"}](slac8782_fig5b.ps "fig:"){width="5.1cm"}
![Angular distributions for each helicity configuration with supersymmetric bulk contributions for $\Lambda_c=1.5$ TeV (top curves), and for the $D=4$ supersymmetric models (bottom curves). The solid (dashed) curves correspond to a bino-like (Higgsino-like) composition of the lightest neutralino.[]{data-label="8782_fig5"}](slac8782_fig5c.ps "fig:"){width="5.1cm"} ![Angular distributions for each helicity configuration with supersymmetric bulk contributions for $\Lambda_c=1.5$ TeV (top curves), and for the $D=4$ supersymmetric models (bottom curves). The solid (dashed) curves correspond to a bino-like (Higgsino-like) composition of the lightest neutralino.[]{data-label="8782_fig5"}](slac8782_fig5d.ps "fig:"){width="5.1cm"}
![Angular distributions for each helicity configuration with supersymmetric bulk contributions for $\Lambda_c=1.5$ TeV (top curves), and for the $D=4$ supersymmetric models (bottom curves). The solid (dashed) curves correspond to a bino-like (Higgsino-like) composition of the lightest neutralino.[]{data-label="8782_fig5"}](slac8782_fig5e.ps){width="5.1cm"}
Figure \[8782\_fig3\] shows the angular distribution for the process $e^-_Re^+\to\sr^+\sr^-$ with $\sqrt s=500$ GeV assuming 100% polarization of the electron beam, detailing the effects of each class of contributions to selectron pair production. The bottom curve represents the full contributions (s- and t-channel) from the 4-dimensional standard gauge-mediated supersymmetric model discussed above in the case where the $\chi^0_1$ is bino-like, corresponding to parameter set I. The middle curve displays the effects of adding only the s-channel contributions of the bulk graviton KK tower in the scenario of a non-supersymmetric bulk with $\Lambda_c=1.5$ TeV. We see that there is little difference in the distribution between the $D=4$ supersymmetric case and with the addition of the graviton KK tower, in either shape or magnitude. It would hence be difficult to disentangle the effects of graviton exchange from an accurate measurement of the underlying supersymmetric parameters using this process alone. The top curve corresponds to the full set of contributions from a supersymmetric bulk, , our standard supersymmetric model plus KK graviton and KK gravitino tower exchange for the case of six extra dimensions with $\Lambda_c=1.5$ TeV. Here we see that the exchange of bulk gravitino KK states yields a large enhancement in the cross section and a substantial shift in the shape of the angular distribution, particularly at forward angles, even for $\Lambda_c=3\sqrt s$. This provides a dramatic signal for a supersymmetric bulk!
We now compute the potential sensitivity to the cut-off scale from selectron pair production using our sample case with a bino-like lightest neutralino state. We employ the usual $\chi^2$ procedure, including statistical errors only. We sum over both initial left- and right-handed electron polarization states, assuming $P_{e^-}=80\%$. The resulting 95% C.L. search for $\Lambda_c$ from each final state, $\sl^+\sl^-\,, \sr^+\sr^-$, and $\sl^\pm\sr^\mp$, is given as a function of integrated luminosity in Fig. \[8782\_fig9\] for $\sqrt s = 0.5$ and 1.0 TeV. We see that for 500 of integrated luminosity, corresponding to design values, the search reach in the left- and right-handed selectron pair production channels is given roughly by $\Lambda_c\simeq 6-10
\times\sqrt s$, which is essentially what is achievable for bulk graviton KK exchange in the reaction $e^+e^-\to f\bar f$ [@Hewett:1998sn]. However, the $\sl^\pm\sr^\mp$ production channel yields an enormous search capability with a 95% C.L. sensitivity to $\Lambda_c$ of order $25\times\sqrt s$ for design luminosity. This process thus has the potential to either discover a supersymmetric bulk, or eliminate the possibility of supersymmetric large extra dimensions as being relevant to the hierarchy problem. We stress that there is nothing special about our choice of supersymmetric parameters; our results will hold as long as selectrons are kinematically accessible to high energy colliders. We conclude that selectron pair production provides a very powerful tool in searching for a supersymmetric bulk.
![The angular distribution for $e^-_Re^+\to\sr^+\sr^-$ from the $D=4$ supersymmetric model I, plus the addition of bulk graviton KK tower exchange, and with bulk gravitino KK tower exchange, corresponding to the bottom, middle, and top curves, respectively.[]{data-label="8782_fig3"}](slac8782_fig3.ps){width="6cm"}
![95% C.L. search reach for $\Lambda_c$ in each production channel as a function of integrated luminosity for $\sqrt s= 0.5$ and 1.0 TeV.[]{data-label="8782_fig9"}](slac8782_fig9a.ps "fig:"){width="5.1cm"} ![95% C.L. search reach for $\Lambda_c$ in each production channel as a function of integrated luminosity for $\sqrt s= 0.5$ and 1.0 TeV.[]{data-label="8782_fig9"}](slac8782_fig9b.ps "fig:"){width="5.1cm"}
In summary, we have examined the phenomenological consequences of a supersymmetric bulk in the scenario of large extra dimensions. We assumed that supersymmetry is unbroken in the bulk, with gravitons and gravitinos being free to propagate throughout the higher dimensional space, and that the SM and MSSM gauge and matter fields are confined to a 3-brane. Motivated by string theory, we worked in the framework of $D=10$ supergravity, and found that the KK reduction of the bulk gravitinos yields four Majorana spinors in four dimensions. We then assumed that the residual $N=4$ supersymmetry is broken near the fundamental scale $M_D$, with only $N=1$ supersymmetry surviving at the electroweak scale.
Starting with the $D=10$ action for this scenario, we expanded the bulk gravitino into a KK tower of states, and determined the 4-d action for the spin-3/2 KK excitations. We then presented the coupling of the bulk gravitino KK states to fermions and their scalar partners on the brane. We applied these results to a phenomenological analysis by examining the effects of virtual exchange of the gravitino KK tower in superparticle pair production. We focused on the reaction $\epem\to\tilde e^+\tilde e^-$ as this process is a benchmark for collider supersymmetry studies. Our numerical analysis was performed in the framework of gauge mediated supersymmetry breaking as it naturally affords a light zero-mode gravitino. However, our results do not depend on the specifics of this particular model, with the exception of the existence of a light zero-mode gravitino state.
Performing the sum over the KK propagators, we found that the leading order contribution to this process arises from a dimension-6 operator, and is independent of the zero-mode mass. This is in stark contrast to the virtual exchange of spin-2 graviton KK states, which yields a dimension-8 operator at leading order. We thus found that the gravitino KK contributions substantially alter the production rates and angular distributions for selectron pair production, and may essentially be isolated in the $\sl^\pm\sr^\mp$ channel. The resulting sensitivity to the cut-off scale is tremendous, being of order $20-25\times\sqrt s$.
We expect that the virtual exchange of gravitino KK states in hadronic collisions will have somewhat less of an effect in squark and gluino pair production than what we have found here. The reason is that these processes are initiated by both quark annihilation and gluon fusion sub-processes, only one of which will be sensitive to tree-level gravitino exchange for a given production channel. The sensitivity to the cut-off scale will then depend on the relative weighting of the quark and gluon initial states. In addition, t-channel gravitino contributions will only be numerically relevant for up- and down-squark production due to flavor conservation; hence their effect will be diluted by the production of the other degenerate squark flavors and the relative weighting of the parton densities.
We note also that virtual exchange of gravitino KK states may also have a large effect on selectron pair production in $e^-e^-$ collisions, which are tailor-made for t-channel Majorana exchanges. High energy Linear Colliders thus provide an excellent probe for the existence of supersymmetric large extra dimensions, and have the capability of discovering this possibility or eliminating it as being relevant to the hierarchy problem.
The authors are supported by the US Department of Energy, Contract DE-AC03-76SF00515.
\#1 \#2 \#3 [Mod. Phys. Lett. [**\#1**]{}, \#2 (\#3)]{} \#1 \#2 \#3 [Nucl. Phys. [**\#1**]{}, \#2 (\#3)]{} \#1 \#2 \#3 [Phys. Lett. [**\#1**]{}, \#2 (\#3)]{} \#1 \#2 \#3 [Phys. Rep. [**\#1**]{}, \#2 (\#3)]{} \#1 \#2 \#3 [Phys. Rev. [**\#1**]{}, \#2 (\#3)]{} \#1 \#2 \#3 [Phys. Rev. Lett. [**\#1**]{}, \#2 (\#3)]{} \#1 \#2 \#3 [Rev. Mod. Phys. [**\#1**]{}, \#2 (\#3)]{} \#1 \#2 \#3 [Nuc. Inst. Meth. [**\#1**]{}, \#2 (\#3)]{} \#1 \#2 \#3 [Z. Phys. [**\#1**]{}, \#2 (\#3)]{} \#1 \#2 \#3 [E. Phys. J. [**\#1**]{}, \#2 (\#3)]{} \#1 \#2 \#3 [Int. J. Mod. Phys. [**\#1**]{}, \#2 (\#3)]{} \#1 \#2 \#3 [J. High En. Phys. [**\#1**]{}, \#2 (\#3)]{}
[99]{}
N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B [**429**]{}, 263 (1998) \[arXiv:hep-ph/9803315\]; N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Rev. D [**59**]{}, 086004 (1999) \[arXiv:hep-ph/9807344\]; N. Arkani-Hamed, S. Dimopoulos and J. March-Russell, Phys. Rev. D [**63**]{}, 064020 (2001) \[arXiv:hep-th/9809124\]; N. Arkani-Hamed, S. Dimopoulos, N. Kaloper and R. Sundrum, Phys. Lett. B [**480**]{}, 193 (2000) \[arXiv:hep-th/0001197\]. I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B [**436**]{}, 257 (1998) \[arXiv:hep-ph/9804398\]. N. Arkani-Hamed, S. Dimopoulos and J. March-Russell, Phys. Rev. D [**63**]{}, 064020 (2001) \[arXiv:hep-th/9809124\]. The details of this work are presented in: J. L. Hewett and D. Sadri, \[arXiv:hep-ph/0204063\], where more extensive references can be found.
A. Bartl, H. Fraas and W. Majerotto, Z. Phys. C [**34**]{}, 411 (1987); H. Baer, A. Bartl, D. Karatas, W. Majerotto and X. Tata, Int. J. Mod. Phys. A [**4**]{}, 4111 (1989); T. Tsukamoto, K. Fujii, H. Murayama, M. Yamaguchi and Y. Okada, Phys. Rev. D [**51**]{}, 3153 (1995). T. G. Rizzo, Phys. Rev. D [**60**]{}, 075001 (1999) \[arXiv:hep-ph/9903475\]. J. Polchinski, arXiv:hep-th/9611050. J. L. Hewett, Phys. Rev. Lett. [**82**]{}, 4765 (1999) \[arXiv:hep-ph/9811356\].
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---
abstract: 'There are some basic differences between the observed properties of galaxies and clusters and the predictions from current hydrodynamical simulations. These are particularly pronounced in the central regions of galaxies and clusters. The popular NFW (Navarro, Frenk, & White) profile, for example, predicts a density cusp at the center, a behavior that (unsurprisingly) has not been observed. While it is not fully clear what are the reasons for this discrepancy, it perhaps reflects (at least partly) insufficient spatial resolution of the simulations. In this paper we explore a purely phenomenological approach to determine dark matter density profiles that are more consistent with observational results. Specifically, we deduce the gas density distribution from measured X-ray brightness profiles, and substitute it in the hydrostatic equilibrium equation in order to derive the form of dark matter profiles. Given some basic theoretical requirements from a dark matter profile, we then consider a number of simple profiles that have the desired asymptotic form. We conclude that a dark matter profile of the form $\rho=\rho_0\left(1+r/r_a \right)^{-3}$ is most consistent with current observational results.'
address:
- 'School of Physics and Astronomy, Tel Aviv University, Tel Aviv, 69978, Israel'
- |
School of Physics and Astronomy, Tel Aviv University, Tel Aviv, 69978, Israel,\
Center for Astrophysics and Space Sciences, University of California, San Diego, La Jolla, CA92093-0424
author:
- Yinon Arieli
- Yoel Rephaeli
title: 'Dark Matter Profiles in Clusters of Galaxies: a Phenomenological Approach'
---
dark matter profiles - Galaxies: clusters: general - X-rays: galaxies: clusters 98.65.Cw ; 95.35.$+d$ ; 95.85Nv
Introduction
============
Mass density profiles of galaxies and clusters of galaxies play a central role in the study of the intrinsic properties of these systems as well as in models of their formation, evolution, and their use as probes of the mass density of the universe. The formation of cold dark matter (CDM) halos has been studied extensively over the years. Considerable theoretical work has been done to describe the shape of DM profiles, mostly by numerical simulations (, Suto 2002). Early attempts were severely limited in predicting the profile in the central region due to insufficient spatial resolution. Recent improvements in numerical techniques led to the attainment of high central resolution ($\sim 10$ kpc) in the dynamical simulation of DM profiles. However, a realistic description of the distribution of DM and gas in the central cluster regions necessitates a coupled dynamical and hydrodynamical simulation that can follow the evolution of DM and gas under their mutual interactions. In particular, heating and cooling processes in the gas can occur on timescales which are shorter than the Hubble time with possible ramifications also for the DM profile.
Based on large, N-body simulations of the evolution of DM configurations, Navarro, Frenk & White (hereafter NFW, 1995, 1997) showed that CDM density profiles are independent of the halo mass, and can be accurately fit over a large range of sizes by a simple algebraic form which is said to be universal (but see Jing & Suto 2000). The proposed NFW profile has a cusp-like $r^{-1}$ behavior close to the center, and an asymptotic $r^{-3}$ falloff at large $r$. It has been argued that DM profiles may even exhibit a steeper inner cusp; for example, Moore et al. (1999) claimed that high resolution simulations indicate that the central profile is $\propto r^{-1.5}$.
An uninterrupted steep rise of the density towards the center is clearly unphysical, and such a characterization could be due to insufficient level of spatial resolution in the numerical simulations, other important factors in the simulations (such as the particular choice of initial conditions, e.g., Bartschiger & Labini 2001), or a result of physical limitations in the description of the cluster density over small (O\[10 kpc\]), typically [ *galactic*]{} scales. That the latter are perhaps the more likely reasons is indicated by even a steeper inner cusp that is deduced in recent (Governato et al. 2001) higher resolution simulations. Even the improved hydrodynamical simulations do not include all the relevant physical processes that could affect the nature of the deduced mass profiles. For example, it is clear that the properties of intracluster (IC) gas, whose fractional contribution to the total mass is $\sim
10\%$, must play some role in the determination of the density profile.
Recent [*Chandra*]{} observations of the central regions of a few nearby clusters have not (yet) provided unequivocal evidence on the shape of the central DM profile. Strong evidence was found for a flat profile in the central region of Abell 1795 (Ettori 2002), a behavior similar to the trend seen in observations in some galaxies (mainly low surface brightness and dwarf galaxies). But there is also evidence for a central cusp in the clusters Abell 2029 (Lewis et al 2002), Hydra A (David 2001), and EMSS 1358 (Arabadjis 2002). However, evidence for the latter is weak, given the low quality of the fits (generally large $\chi^2/dof$), central CD galaxy (in Abell 2029), low central resolution (in EMSS 1358), and relatively high value of the concentration parameter with respect to expectations from numerical simulations (in Hydra A). Clearly, many more [*Chandra*]{} and XMM observations of the central regions of clusters are needed in order to discern a clear trend in the shape of DM profiles there.
Over the last few years the NFW profile has been adopted in calculations of the structure and evolution of CDM halos, this in spite of its unappealing central cusp. There is a clear need for further exploration of cluster mass profiles with the aim of either modifying the NFW profile, or finding an alternative profile that is more consistent with observations. The inconsistency between observational results and predictions from simulations provides considerable motivation for a more [*phenomenological*]{} approach that is based on dynamical deductions from the observed properties of IC gas, an approach that is adopted in this paper. The gas density and temperature profiles can be determined from current high quality X-ray measurements; these profiles can then be used to probe the total density distribution.
We begin with a short review of IC gas density profiles,and their use to probe the DM profile based on the hydrostatic equilibrium (HE) equation. In section 3 we describe the limitations of the NFW profile for the IC DM density. Alternative DM profiles are discussed in section 4; we consider the requirements from a DM profile, and attempt a solution to the divergence problem of the NFW profile in section 4.1. This consists of a slight but physically important modification of the NFW profile which results in a finite density at the center. A theoretical discussion of new phenomenological DM profiles is given in section 4.2. Next (section 5), we confront several different DM profiles with observational data, primarily a sample of ROSAT measurements of 24 nearby and moderately distant clusters at redshifts $z\leq 0.2$, and draw some conclusions on the form of the most viable profile. In Section 6 we summarize and briefly discuss a few other aspects of the subject matter.
Spatial Distribution of IC Gas
==============================
Spectral measurements of thermal bremsstrahlung X-ray emission from IC gas provide an integrated measure of the emissivity-weighted temperature, while the gas density profile is deduced from measurements of the surface brightness (SB) distribution across the cluster. A starting point in a theoretical description of the gas in a relaxed cluster is the attainment of HE, and although this includes aspehrical configurations, the assumption of spherical symmetry is still reasonable for at least a subset of rich and regular clusters. The expected availability of uniform datasets of measurements of many clusters with the XMM and [*Chandra*]{} satellites motivates a more realistic modeling of the gas thermal and spatial distributions than afforded by an isothermal $\beta$ model. An example for a more general description is a polytropic equation of state $P \propto \rho_g^{\gamma}$ relating the (thermal, assumed dominant) pressure and ([*gas*]{}) density, with the index $\gamma$ as a free parameter. The HE equation is then $$\frac{k T_{0}\gamma}{\mu m_{p}
\rho_{g0}^{\gamma-1}}\rho_g^{\gamma-1}\frac{d ln\rho_g}{dr}=-\frac{G
M(r)}{r^2}\,,$$ where k, $\mu$ and $m_{p}$ are the Boltzmann constant, the mean molecular weight, and the proton mass, respectively; $M(r)$ is the total cluster mass interior to $r$ – a sum of the masses of DM, gas and galaxies.
The gas density is usually represented by an analytic (King) $\beta$ profile $$\rho_g(r)=\frac{\rho_{g0}}{\left[1+y^2\right]^{3 \beta/2}}\,,$$ with $y=r/r_c$; $r_c$ is the gas core radius. In the case of isothermal gas, $\gamma = 1$, the (sky) projected X-ray SB profile that corresponds to this density is of the form $$S_x^{\beta}(R)=S_0\left(1+\frac{R^2}{r_c^2}\right)^{-3\beta+1/2}\,,$$ where $S_0$ is the central SB, $R$ denotes the projected distance from the cluster center, and $\beta$ is a fit parameter. The X-ray deduced gas parameters are used in the HE equation to determine $M(r)$. In the simplest treatment the gas contribution to the gravitational field can be ignored, to first approximation, since the gas mass fraction is $\leq 10\%$.
The NFW Profile
===============
The NFW DM profile is $$\rho^{NFW}(x)=\frac{\rho_{0}}{x \left( 1+x
\right)^2}\,,$$ where $x = r/r_{s}$; $r_{s}$ is a scaling radial parameter. Both $r_{s}$ and the central density $\rho_0$ are related to the cosmological parameters (NFW 1997). While it is generally deduced from N-body simulations that the DM density is asymptotically $\propto r^{-3}$, the behavior of the NFW profile in the inner core is problematic: There is no clear observational evidence for such density cusps in clusters. Specifically, the deduced mass distribution in clusters disagrees with that deduced from the NFW profile, and the X-ray SB calculated from this profile has a much smaller core radius than deduced from observations (Suto et al. 1998). Of course, the rise of density can be truncated below a certain inner radius and replaced by a constant value; this, however, is too arbitrary. The discrepancy is even larger in cD clusters.
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DM profile from polytropic gas
------------------------------
We first show that there are appreciable differences between the DM profile deduced from an isothermal $\beta$ gas and the NFW profile. The spherically symmetric HE equation – in the limit when the gas contribution to the gravitational field can be ignored – yields in this case $$\rho^{poly}(y)=-\tilde{A} \frac{\left[\left( 1+y^2\right)^{-3
\beta/2}\right]^{\gamma-1}\left\{\left[ -3
\beta\left(\gamma-1\right)+1\right] y^2 +3\right\}}{\left(1+y^2
\right) ^2}\,,$$ where $y = r/r_c$ and $$\tilde{A} = \frac{3 k \beta T_{0}}{4 \pi G \mu m_p r_c^2}\,.$$
In Figure 1 we show the resulting DM profiles and the NFW profile for typical values of the parameters; the left panel is for isothermal gas, while the right panel shows the results for $\gamma=$ 1.2, 4/3 and 5/3. There is a noticeable difference between these profiles and the NFW profile, especially at $r\leq
r_c$. The calculated profiles converge to a constant central density, while the NFW profile diverges in this region. In the outer region the falloff of the NFW profile is more moderate than those of the other profiles.
Next we calculate the gas profile deduced from the NFW model; substituting the NFW profile (4) into the isothermal HE equation we have (as was obtained already by Suto, Sasaki & Makino 1998) $$\rho_{g}(r)=\rho_{g0} exp[-Bf(r/r_{s})]\,,$$ where $$\begin{aligned}
f(x)=1-\frac{1}{x}ln(1+x)\,,\end{aligned}$$ for the NFW profile, and B is the dimensionless parameter $$\begin{aligned}
B\equiv\frac{4\pi G \mu m_{p} \rho_{0} r_{s}^2}{k T_{g0}}\,.\end{aligned}$$
In Figure 2 we plot the gas profiles deduced from the NFW profile for three values of B that correspond to a wide temperature range. Also shown are best fits of each of the three curves to a $\beta$ profile. The right panel shows the ratio between each profile and its best-fit $\beta$ profile.
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As can be seen in these figures the gas distribution deduced from the NFW profile is quite different from the $\beta$ profile. Only in the extreme and highly unlikely case of small B $(B\approx 1)$ the two profiles are similar, with differences smaller than 10%.
Differences between the gas distribution deduced from the NFW profile and the $\beta$ profile can also be demonstrated by their related X-ray SB. Fitting this predicted SB profile to the observed profile, which is known to have a $\beta$ model shape, results in the dependence displayed in Figure 3.
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Clearly, the SB predicted from the NFW profile does not provide a reasonably good fit to the $\beta$-model SB function. Rather, we find that the former function can be well fit by a generalized $\beta$ distribution of the form $$S_x^{new}(\theta/\theta_c)=\frac{S^{new}(0)}{\left[1+\left(\theta/\theta_c^{new}\right)^\xi
\right]^{\tilde{\beta}}}\,,$$ where $\xi \neq 2$, and $\tilde{\beta}\neq 3\beta-1/2$, i.e. values different than those of the $\beta$ model (see dotted line in Figure 3). These results are consistent with those of Suto et al. (1998). Thus, it is clear that the general SB function $S_x^{new}$ gives a better fit to $S_x^{NFW}$ than does $S_x^{\beta}$.
We conclude that the NFW profile seems to be inconsistent with X-ray observations of clusters. The main physical reason for the above differences is the excessively high DM concentration that is predicted by the NFW model in the central cluster region.
The NFW profile and polytropic gas
----------------------------------
When the NFW profile is substituted in the more general polytropic HE equation (1), the resulting gas density is $$\rho_g(r)=\rho_{g0}\left[1-\frac{B(\gamma-1)}{\gamma}f(r/r_s)
\right]^{\frac{1}{\gamma-1}}\,.$$ This function assumes negative values beyond a critical radius and is therefore physically unacceptable at larger radii.
For a cluster with a maximal radius $r_{max}$, the requirement that the profile is non-negative constrains the value of the polytropic index to be $$\gamma \leq \frac{Bf(x_{max})}{Bf(x_{max})-1}\,,$$ where we took $x=x_{max}$ since $f(x)$ is a monotonically increasing function for the NFW profile. In table 1 we list the upper limit on the polytropic index for various values of $r_{max}$ and $B$, with a typical value of the scale radius, $r_s=0.2\,Mpc$. The results weakly depend on the value of the scale radius; for larger values of $r_s$, , $0.5$ $Mpc$, the value of $\gamma_{max}$ increases only by $\sim
5\%$. It would seem from Table 1 that in this model the polytropic index is limited to a relatively narrow range of values.
[|c|c|c|c|c|c|c|c|c|c|]{} $\gamma_{max}$ & 1.07& 1.16& 1.38& 1.07& 1.15& 1.35& 1.06& 1.13& 1.32\
$r_{max}$ (Mpc) & 1.5& 1.5& 1.5& 2& 2& 2& 3& 3& 3\
B& 20& 10& 5& 20& 10& 5& 20& 10& 5\
Alternative DM Profiles
=======================
Having discussed the possibly problematic features of the NFW model, we now want to find alternative DM profiles that are more physically viable and are consistent with X-ray SB measurements. To do so we first specify the properties desired of a DM profile, and then consider a slightly modified NFW profile, and – more generally – the simplest functional forms that satisfy the requirements from a DM profile. We require that a DM profile has the following properties:
$1.$ Finite, positive-definite at all $r$.
$2.$ Asymptotic $r^{-3}$ behavior at large $r$.
$3.$ An associated gas profile (from HE equation) that has the form of a $\beta$-profile with a value of $\beta$ which is consistent with X-ray SB measurements.
The first property is an obvious physical requirement; the second is based on results of many N-body simulations, and the third is based on fits to observed SB profiles of many clusters. In assessing the viability of a DM profile we will also consider whether the central mass density is high enough for observable effects of gravitational lensing. In some clusters the measurements of either giant arcs, strong, or weak gravitational lensing imply that the DM central density needs to be sufficiently high to produce these lensing effects. Specifically, the central surface density has to be typically higher than the critical value of $\Sigma_0 \sim 0.5\, gr/cm^2$ (with $H_0=50\; km\; s^{-1}\;
Mpc^{-1}$ and $\Omega=1$) in order that multiple lensed images are produced (Subramanian & Cowling 1986).
A Modified NFW profile
----------------------
An operational approach to the divergence problem of the NFW profile is to replace it in the inner region, $r \leq r_b$, with a non-divergent form. From figures 1 & 2, it is clear that in the central region of the cluster the NFW profile is much steeper than either $\rho^{iso}$ or $\rho^{poly}$, intersecting these curves inside the cluster core. Therefore, in order to remove the divergence of the NFW profile, in this inner region we replace the NFW profile with the functional form obtained as a solution – eq. (5) – to the HE equation when taking a $\beta$-profile for the gas. This form can then be tailored to the NFW profile outside the inner region, namely $$\begin{aligned}
\rho^{new-poly}(r)=\left\{ \begin{array}{ll}
\tilde{A} \frac{\left\{\left[ 1+(r/r_c)^2\right]^{-3
\beta/2}\right\}^{\gamma-1}\left\{\left[-3
\beta\left(\gamma-1\right)+1\right] (r/r_c)^2 +3\right\}}{\left[
1+(r/r_c)^2 \right] ^2}
& \mbox{$ r\leq r_{b}$} \\ \\
\frac{\rho_{0}}{(r/r_s) \left( 1+r/r_s
\right)^2} & \mbox{$r > r_{b}$}
\end{array}
\right.\,, \end{aligned}$$ for the (general) polytropic case.
The new modified profile is not smooth at $r = r_b$, but it is continuous at this point, and at larger radii it falls off asymptotically as $r^{-3}$. If we were to fit it by the functions of the kind of $\rho^{iso}$ and $\rho^{poly}$, the fit would be excellent in the central region, but the quality of the fit would deteriorate outside this region. Thus, the overall discrepancy between this modified DM profile and the profile deduced from the measured gas density would still remain. (Note that the DM profile can be similarly changed in clusters with a giant cD galaxy whose DM density distribution is generally different from the NFW profile.)
New DM profiles
---------------
Adopting a purely phenomenological approach we can readily write down a family of DM profiles that satisfy the first two requisite properties, resembling an isothermal profile in the inner region and falloff asymptotically as $r^{-3}$ at large radii. These profiles can be characterized by a scale radius $r_a$ (which is generally different from the NFW scale radius $r_s$), and the set of three parameters $(\eta,\nu$,$\lambda)$: $$\rho=\frac{\rho_0^{\star}}{\left[1+x^\eta
\right]\left[1+x^\nu\right]^\lambda}\,,$$ for which $\eta + \nu + \lambda = 3$, and $x=r/r_a$. Here we consider the four simplest profiles characterized by ($\eta,\nu$,$\lambda$)= (0,1,3), (1,2,1), (0,2,3/2) and (3,$\nu$,0), with $\rho_0^{\star} = 2\rho_0$ for $\eta = 0$, and $\rho_0^{\star} = \rho_0$ for $\eta = 1, 3$. The second profile was previously deduced to provide a good fit to the distribution of DM in dwarf galaxies (Burkert, 1995).
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In order to be able to meaningfully compare the different models, we take a nominal value of $10^{15}$ M$_{\odot}$ for the total mass of the cluster (mostly that of the DM and gas) at the virial radius, $r_{vir}\approx1.5\;Mpc$. Doing so relates the central density and scale radius, so the values of these quantities cannot be arbitrarily selected. The requirements of the equality of the total masses at the virial radius and the $r^{-3}$ falloff at large radius completely specify the profile parameters. The four new DM profiles are shown in figure 4 together with the NFW profile. Clearly, values of the central densities span a wide range, and the convergence of the first profile ($\rho^I$) to the asymptotic $r^{-3}$ law is the fastest.
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To check whether the third condition is also satisfied, we substitute the new profiles into the HE isothermal equation, neglecting the gas and galaxy contributions to the total mass, and obtain the following gas profiles, $$\rho_g^i(x)=\rho^i_{g0}\exp[-B f^i(x)]$$ where $$\begin{aligned}
f^I(x)&=&\frac{2+x}{2+2x}-\frac{\ln(1+x)}{x}\\
f^{II}(x)&=&\frac{1}{4x}\left\{2(1+x)\left[\arctan(x)-\ln(1+x)\right]+(x-1)\ln(1+x^2)\right\}\\
f^{III}(x)&=&1-\frac{arcsinh(x)}{x}\\
f^{IV}(x)&=&\frac{1}{6x}\left[ 3 x^3 \;_2F_1\left(
\frac{2}{3},1,\frac{5}{3},-x^3\right)-2 \ln(1+x^3)\right]\end{aligned}$$ where $_2F_1$ is the hypergeometric function. The parameter B was, defined in eq. (9), depends on the DM parameters $r_a$ and $\rho_0$, and the gas parameters $T_0$ and $\beta$. Due to the stipulated constancy of the mass at the virial radius, $B$ essentially depends only on the gas temperature, $T_0$. In Table 2 we list values of B for temperatures in the range 5-15 keV.
[|c|c|c|c|c|]{} $B_{NFW}$ & $B_I$ & $B_{II}$ & $B_{III}$ & $B_{IV}$\
20 & 35.13 & 18.94 & 13.64 & 11.39\
10 & 17.56 & 9.47 & 6.82 & 5.69\
5 & 8.78 & 4.73 & 3.41 & 2.84\
Fits of the new gas profiles to a $\beta$ model are shown in figure 5. Although all the fits provide better approximation to the $\beta$ model than the corresponding fit from the NFW model, values of the fit parameters for the third and fourth profiles are somewhat unlikely. The second profile yields good results only on small scales ($r\leq 0.7-0.8\;Mpc$); the fit is poor at larger radii. The best fit is obtained with the first profile.
We checked whether each of the above DM profiles can be closely approximated by fitting a solution of the HE equation obtained with an isothermal $\beta$ profile for the gas. The fit parameters were $\tilde{A}, r_c, \rho_{g0}$ and $\beta$. Not surprisingly, we find that all of the above four profiles can be very well fit by a solution of the HE equation. Here again the fit to the NFW profile is very poor. We have also compared the gas profiles deduced from the HE equation for each of the DM models directly to the observed quantity by numerically evaluating the SB. (In these computations the gas density was truncated at $x=20$, corresponding to a limiting radius which is larger than the virial radius.)
Results of the fits of the deduced SB profiles to a $\beta$ model are shown in Figure 6. The first two profiles yield good fits to a SB that has a form of a $\beta$ profile, with reasonable parameter values. The third and fourth profiles are better fit by a function of the form of $S_x^{new}$ (Eq. 10), with $\xi\neq 2$ and $\tilde{\beta}\neq 3 \beta-1/2$, as does the modified NFW model.
Next we derived the gas density distributions from the HE equation for polytropic gas and with the DM mass corresponding to the above profiles. The deduced gas density profiles are represented in terms of the function $f(r)$ in eq. (11). Since this function is monotonically increasing with $r$, there is an upper limit on the value of $\gamma$ below which the density is positive definite. We have determined the limiting values of $\gamma$ for typical values of B and $r_{max}=1.5 \,Mpc$. Only for the first DM profile these limits are acceptable. For the other three profiles the deduced values of $\gamma$ are either unrealistic or negative, and thus unacceptable for the observed range of gas temperatures.
As discussed previously, the mass obtained from the HE equation for isothermal gas closely approximates the cluster total mass since the gas is approximately isothermal and the fractional mass contribution of the gas is small. We have compared the DM masses obtained from direct integrations of the DM density distributions and those obtained from the HE equation for isothermal gas. For isothermal gas, the deduced cluster mass is $$M^{iso}(r)=\frac{3 k T_0 \beta}{\mu m_p G r_c^2}\frac{r^3}{\left(1
+\frac{r^2}{r_c^2}\right)}\,.$$
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The DM mass profiles for the new profiles can written as $$M(r)=4 \pi \rho_0 r_a^3 m(x)\,,$$ where $m(x)$ are obtained by integrating the four profiles: $$\begin{aligned}
m^I(x)&=&\ln\left(1+x\right)-\frac{x\left(2+3x\right)}{2\left(
1+x\right)^2}\\
m^{II}(x)&=&\frac{1}{4}\left[2\ln(1+x)+\ln(1+x^2)-2\arctan(x)
\right]\\
m^{III}(x)&=&arcsinh(x)-\frac{x}{\sqrt{1+x^2}}\\
m^{IV}(x)&=&\frac{1}{3}\ln(1+x^3) .\end{aligned}$$ The new profiles can be well described by such a function, while only a poor fit is obtained to the NFW mass profile.
Finally, the presence of gravitational arcs in some clusters implies sufficiently high central mass densities. We have evaluated the central surface densities for the new profiles using the Abel integral; these are $$\begin{aligned}
\Sigma^I(0)&=&\rho_0^I r_a^I \,\\
\Sigma^{II}(0)&=&\frac{\pi}{2} \rho_0^{II} r_a^{II} \,\\
\Sigma^{III}(0)&=&2\rho_0^{III} r_a^{III}\,\\
\Sigma^{IV}(0)&=&\frac{4 \pi}{3 \sqrt{3}} \rho_0^{IV} r_a^{IV}\end{aligned}$$ for the first to the fourth profile. Actual values are $\sim 1$ g/cm$^{2}$ for the first DM profile, to about a factor $\sim 4$ lower for the fourth profile. Since these values are quite comparable to the critical density $\Sigma_c \approx 0.5$ g/cm$^{2}$ needed to produce arcs, no additional constraint is imposed on these models by this consideration.
[ l@ c@ c@ c@ c@ c@ c@ c@ c@ ]{} cluster & z & &\
& & $r_a, \, B_I$ & $r_a, \, B_{II}$ & $r_s, \, B_{N}$ & $\chi_{I}^2$ & $\chi_{II}^2$ & $\chi_{N}^2$ & $\sigma_{\chi}$\
A401 & 0.0748 & 0.29 , 14.88 & 0.32 , 8.51 & 0.87 , 8.92 & 1.69 & 2.33 & 1.92 & 0.20\
A478 & 0.0881 & 0.17 , 16.28 & 0.19 , 9.35 & 0.50 , 9.61 & 5.71 & 18.75 & 2.98 & 0.35\
A520 & 0.203 & 0.59 , 19.31 & 0.56 , 10.08 & 2.73 , 15.01 & 1.19 & 1.03 & 1.70 & 0.33\
A545 & 0.153 & 0.45 , 19.31 & 0.44 , 10.39 & 1.66 , 13.1 & 1.49 & 0.97 & 2.89 & 0.45\
A586 & 0.171 & 0.22 , 16.43 & 0.24 , 9.27 & 0.68 , 9.88 & 1.38 & 1.10 & 1.98 & 0.33\
A644 & 0.0704 & 0.24 , 16.07 & 0.25 , 9.10 & 0.68 , 9.50 & 2.87 & 2.45 & 4.30 & 0.28\
A1413 & 0.1427 & 0.22 , 16.24 & 0.24 , 9.21 & 0.69 , 9.86 & 1.97 & 2.94 & 2.10 & 0.32\
A1651 & 0.0825 & 0.24 , 15.67 & 0.25 , 8.82 & 0.63 , 8.96 & 1.82 & 1.68 & 1.72 & 0.28\
A1656 & 0.0232 & 0.54 , 18.76 & 0.54 , 10.29 & 2.20 , 13.09 & 1.16 & 0.98 & 3.39 & 0.32\
A1689 & 0.181 & 0.22 , 17.55 & 0.24 , 9.94 & 0.70 , 10.71 & 2.69 & 5.40 & 2.08 & 0.3\
A1763 & 0.187 & 0.36 , 15.79 & 0.38 , 8.90 & 1.15 , 9.76 & 4.03 & 6.32 & 2.36 & 0.39\
A2029 & 0.0765 & 0.14 , 15.06 & 0.16 , 8.71 & 0.40 , 8.80 & 3.55 & 11.15 & 4.19 & 0.39\
A2163 & 0.203 & 0.41 , 15.86 & 0.42 , 8.75 & 1.40 , 10.21 & 1.37 & 1.35 & 2.52 & 0.34\
A2204 & 0.1523 & 0.15 , 15.96 & 0.17 , 9.22 & 0.43 , 9.29 & 0.83 & 1.29 & 0.93 & 0.38\
A2218 & 0.175 & 0.34 , 17.21 & 0.35 , 9.49 & 1.19 , 11.17 & 1.00 & 1.30 & 2.76 & 0.31\
A2244 & 0.097 & 0.18 , 15.68 & 0.20 , 9.04 & 0.54 , 9.30 & 1.06 & 1.23 & 1.44 & 0.24\
A2255 & 0.0809 & 0.86 , 21.53 & 0.76 , 10.62 & 6.91 , 26.09 & 0.99 & 1.14 & 0.95 & 0.21\
A2319 & 0.0559 & 0.31 , 14.12 & 0.36 , 8.15 & 0.94 , 8.49 & 3.02 & 5.21 & 1.46 & 0.22\
A2507 & 0.196 & 0.58 , 16.04 & 0.56 , 8.52 & 2.97 , 13.39 & 1.02 & 0.95 & 1.08 & 0.21\
A3112 & 0.0746 & 0.11 , 15.23 & 0.15 , 9.02 & 0.29 , 8.85 & 1.44 & 2.15 & 1.61 & 0.27\
A3667 & 0.0542 & 0.32 , 13.29 & 0.34 , 7.49 & 1.02 , 8.22 & 4.99 & 7.81 & 5.33 & 0.35\
A3888 & 0.168 & 0.47 , 22.09 & 0.47 , 12.24 & 2.38 , 18.15 & 1.54 & 1.65 & 1.60 & 0.16\
PKS0745 & 0.1028 & 0.13 , 16.19 & 0.15 , 9.42 & 0.38 , 9.52 & 2.85 & 3.47 & 4.65 & 0.25\
Triang & 0.051 & 0.38 , 15.87 & 0.41 , 8.91 & 1.85 , 9.84 & 3.86 & 6.29 & 2.06 & 0.23\
Parameters of the DM profiles from a ROSAT Sample
=================================================
In an attempt to further distinguish between the four DM profiles described in the previous section and possibly select the most realistic profile based on available observational data, we have used results from a sample of SB profiles of clusters measured with the ROSAT PSPC. The sample – a subset of a dataset which was compiled and investigated by Ettori & Fabian (1999; the data were kindly provided by Ettori) – consists of 24 clusters with X-ray luminosities $\geq 10^{45}$ erg $s^{-1}$ (taking $H_0=50\;
km\;s^{-1}\;Mpc^{-1}$), at redshifts in the range $0.051-0.203$. We have selected nearby and moderately distant clusters for which the ROSAT PSPC provides some – albeit not optimal – spatial resolution (for more details, see Ettori & Fabian 1999). We performed fits of the measured SB profiles to those predicted from three different models for the isothermal gas density. The assumption of gas isothermality is a reasonable approximation to the temperature profile at radii larger than $~0.1r_{vir} \; (\approx 0.2Mpc)$. All three models were deduced from the HE equation adopting these DM profiles: NFW, and our first ($\rho^I$) and second ($\rho^{II}$) models. The corresponding gas density distributions are $$\rho_g^{N}(x)=\rho_{g0}^{N}\exp^{-B_{N}}(1+x)^{B_{N}/x}
\,,$$ where $x=r/r_s$, $$\rho_g^I(x)=\rho_{g0}^I\exp^{-\frac{B_I(2+x)}{2+2x}}(1+x)^{B_I/x}
\,,$$ and $$\rho_g^{II}(x)=\rho_{g0}^{II}\exp^{-\frac{B_{II}}{4x}\left\{2(1+x)\left[\arctan(x)-\ln(1+x)\right]+(x-1)\ln(1+x^2)\right\}} \,,$$ where $x=r/r_a$. The fits were performed by $\chi^2$ minimization (using the ’Minuit’ CERN program).
Results of the fits are summarized in Table 3, where in addition to listing the best-fit values of the scale radii and $B$, values of the reduced $\chi^2$ ($\chi^2/dof$) and the standard deviation for the reduced $\chi^2$ are also specified. Of the three models, the first DM profile provides the best fit to the data of 11 out of the 24 clusters, with the second profile providing the best fit in 8 clusters. Furthermore, for most of the latter 8 clusters the differences between the quality of the fits based on the first and second DM models are not statistically significant. Based on the results from this ROSAT dataset it is apparent that the first DM profile is most consistent with the data while the NFW profile is the least favored. (We note that all three fits to the data on three clusters – A478, A1763 & A3667 – are very poor, raising doubts on the validity of the assumption of hydrostatic equilibrium of the gas in these clusters, perhaps due to ongoing merger activity?).
In order to assess the impact of a more precise but similar database, we have repeated the fits by artificially reducing the observational errors in the measurements of the SB. Doing so does not affect appreciably the quality of the fit to the SB from the first DM model, but reduces the consistency with the second DM model and significantly worsening the viability of the NFW profile. Clearly, since this test is based on the current database it does not add independent confirmation of the results, but rather just a simulation of what might be feasible to do when higher quality data become available.
Discussion
==========
The aim of this work has been to find alternative DM profiles that are finite at the cluster center and are consistent with observed X-ray SB profiles. Our approach is purely phenomenological and is based on the selection of simple non-cusped profiles that falloff asymptotically as $\propto r^{-3}$ at large $r$. We first constructed two modified NFW profiles ($\rho^{iso}$ and $\rho^{poly}$) by truncating the NFW below some inner radius ($r_b$) merely to remedy the central divergence of the NFW model. Since $r_b \sim 50$ kpc, the new profiles quickly converge to the NFW profile. But the requirement that the related gas density profile has an associated thermal bremsstrahlung SB with the typical $\beta$ model shape led us to abandon these modified profiles as realistic alternatives to the NFW model.
We then considered four new profiles characterized by the three parameters ($\mu,\nu,\lambda$) with the requisite features, and tested their viability by contrasting their associated SB profiles with ROSAT data on a sample of 24 clusters. Comparison of the gas profiles resulting from the new DM models with $\beta$ gas profiles clearly shows that all of these give better results than the NFW profile. Below a radius of about 1 Mpc, the behavior of all four profiles is acceptable, but progressively degrades at larger radii. Examining the SB profile calculated from the different DM profiles we saw that the results are not unequivocal, namely that the general shape of the SB function can be fitted quite well to a $\beta$ SB profile. This is so for a fit done over a large range of radii; the fit is particularly good over the radial range $r\leq 1.5-2\;Mpc$. The reason for this is the fact that a SB that has the shape of a $\beta$ profile necessarily has a flat slope in the central region. Upon detailed comparison of the results of the fits, as well as consistency with polytropic gas distributions, we concluded that the first of these four profiles, for which $(\eta,\nu,\lambda)=(0,1,3)$: $$\rho(r)=\frac{\rho_0}{\left(1+r/r_a\right)^3}\,,$$ is most consistent with the ROSAT sample. We emphasize, however, that due to the large observational uncertainties the preference of the first profile over the second is not large. It is quite likely that the availability of more precise X-ray SB and temperature measurements will enable a more definite distinction to be made between these viable alternatives to the NFW profile.
Independent recent work (El-Zant, Shlosman & Hoffman 2001) also leads to a resolution of the ’core catastrophe’ in galaxies – the discrepancy between the diverging inner density profile of DM from CDM N-body simulations and the finite core deduced from observations. These authors suggest the gas in galaxies is not initially smoothly distributed in the DM halo, but rather is concentrated in small clumps containing $\sim 0.01\%$ of the total mass. The orbital energy of the clumps dissipates by dynamical friction as they move in background of DM particles, thereby transferring energy to the DM and heating it. This process is said to be sufficiently effective to turn the primordial cusp of the DM profile into a non-diverging core, resulting in a profile of the form $$\rho=\frac{C}{\left(r+r_c\right)\left(A+r\right)^2}\,,$$ where $C$ is a fixed parameter, $A$ and $r_c$ are a scale parameter and core radius, respectively. Furthermore, it was found that best fits require that $A=r_c$, which then yields essentially the same profile that we have deduced for clusters. This similarity between the behavior of the DM profile in galaxies and in clusters is indeed expected in theories of formation and evolution of the large scale structure. Gravitational drag could also be important in clusters (e.g., Rephaeli & Salpeter 1980), so a similar process of transfer of kinetic energy of the galaxies (initially with their DM halos) to the IC DM could have flattened also cluster profiles.
Flattening of DM profiles in the centers of galaxies and clusters is suggested by other theoretical considerations. For example, D’Onghia, Firmani & Chincarini (2002) have recently argued that the flattening can occur in galactic and cluster centers if the DM in these systems consists of weakly self-interacting particles. Collisions between the particles during system collapse and the associated inward transfer of heat lead to expansion of the core. They propose that this process is implemented in N-body simulations by modifying the initial conditions and taking a self-interaction cross section that is inversely proportional to the particle velocity.
Finally, the physical motivation to find a more acceptable form for DM density profiles in galaxies and clusters, and the already available observational data, provide a viable basis for selecting between simple, well-behaved profiles. We have identified what seems to be the most consistent form of the DM distribution in clusters. Our work has been based on a simplified theoretical description of clusters – such as the sphericity of the cluster and isothermality of IC gas – assumptions that can be relaxed when higher quality spectral and spatial XMM and [*Chandra*]{} measurements of clusters will be available.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
We are grateful to Dr. Stefano Ettori for providing the ROSAT dataset discussed in this paper.
Arabadjis J.S., Bautz M.W. & Garmire G.P., 2002, ApJ 572, 66-78. Baertschiger, T. & Labini F.S., astro-ph/0109199. Burkert, A., astro-ph/9504041. David, L.P. et al, 2001, ApJ 557, 546-559. D’Onghia, E., Firmani C. & Chicarini G., astro-ph/0203255. El-Zant, A., Shlosman I. & Hoffman Y., astro-ph/0103386. Ettori, S. & Fabian A.C., astro-ph/9901304. Ettori, S. & Fabian A.C., Allen S.W & Johnstone R.M., 2002, MNRS 331, 635-648. Governato F., Ghigna S. & Moore B., astro-ph/0105443. Jing, Y.P. & Suto Y., 2000, ApJ 529, L000. Lewis, A.D., Boute D.A. & Stocke J.T., asrto-ph/0209205. Moore, B., Quinn T., Governato F., Stadel J. & Lake G., astro-ph/9903164. Navarro, J., Frenk C.S.& White S.D.M., 1995, MNRAS 275, 720. Navarro, J., Frenk C.S.& White S.D.M., 1997, ApJ 490, 493. Rephaeli, Y. & Salpeter E.E., 1980, ApJ, 240, 20. Subramanian, K. & Cowling S.A., 1986, MNRS., 219,333-346. Suto, Y. Sasaki, S. & Makino N., astro-ph/9807112. Suto, Y. Sasaki, S. & Makino N., 1998A, ApJ 497,555. Suto, Y., astro-ph/0207202.
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abstract: 'The electronic structure of semi-metallic transition-metal dichalcogenides, such as WTe$_2$ and orthorhombic $\gamma-$MoTe$_2$, are claimed to contain pairs of Weyl points or linearly touching electron and hole pockets associated with a non-trivial Chern number. For this reason, these compounds were recently claimed to conform to a new class, deemed type-II, of Weyl semi-metallic systems. A series of angle resolved photoemission experiments (ARPES) claim a broad agreement with these predictions detecting, for example, topological Fermi arcs at the surface of these crystals. We synthesized single-crystals of semi-metallic MoTe$_2$ through a Te flux method to validate these predictions through measurements of its bulk Fermi surface (FS) *via* quantum oscillatory phenomena. We find that the superconducting transition temperature of $\gamma-$MoTe$_2$ depends on disorder as quantified by the ratio between the room- and low-temperature resistivities, suggesting the possibility of an unconventional superconducting pairing symmetry. Similarly to WTe$_2$, the magnetoresistivity of $\gamma-$MoTe$_2$ does not saturate at high magnetic fields and can easily surpass $10^{6}$ %. Remarkably, the analysis of the de Haas-van Alphen (dHvA) signal superimposed onto the magnetic torque, indicates that the geometry of its FS is markedly distinct from the calculated one. The dHvA signal also reveals that the FS is affected by the Zeeman-effect precluding the extraction of the Berry-phase. A direct comparison between the previous ARPES studies and density-functional-theory (DFT) calculations reveals a disagreement in the position of the valence bands relative to the Fermi level $\varepsilon_F$. Here, we show that a shift of the DFT valence bands relative to $\varepsilon_F$, in order to match the ARPES observations, and of the DFT electron bands to explain some of the observed dHvA frequencies, leads to a good agreement between the calculations and the angular dependence of the FS cross-sectional areas observed experimentally. However, this relative displacement between electron- and hole-bands eliminates their crossings and, therefore, the Weyl type-II points predicted for $\gamma-$MoTe$_2$.'
author:
- 'D. Rhodes'
- 'R. Schönemann'
- 'N. Aryal'
- 'Q. Zhou'
- 'Q. R. Zhang'
- 'E. Kampert'
- 'Y.-C. Chiu'
- 'Y. Lai'
- 'Y. Shimura'
- 'G. T. McCandless'
- 'J. Y. Chan'
- 'D. W. Paley'
- 'J. Lee'
- 'A. D. Finke'
- 'J. P. C. Ruff'
- 'S. Das'
- 'E. Manousakis'
- 'L. Balicas'
title: 'Bulk Fermi-surface of the Weyl type-II semi-metal candidate $\gamma$-MoTe$_2$'
---
Introduction
============
The electronic structure of the transition-metal dichalcogenides (TMDs) belonging to the orthorhombic and non-centrosymmetric $Pmn2_1$ space group, e.g. WTe$_2$, were recently recognized as candidates for possible topologically non-trivial electronic states. For instance, their monolayer electronic bands were proposed to be characterized by a non-trivial $Z_2 = 1$ topological invariant based on the parity of their valence bands, making their monolayers good candidates for a quantum spin Hall insulating ground-state [@TP_transition]. This state is characterized by helical edge states that are protected by time-reversal symmetry from both localization and elastic backscattering. Hence, these compounds could provide a platform for realizing low dissipation quantum electronics and spintronics [@TP_transition; @MacDonald].
However, the majority of gapped TMDs, such as semiconducting MoS$_2$ or WSe$_2$, crystallize either in a trigonal prismatic coordination or in a triclinic structure with octahedral coordination [@review1; @review2] as is the case of ReS$_2$. Those crystallizing in the aforementioned orthorhombic phase, e.g. WTe$_2$, are semi-metals albeit displaying remarkable transport properties such as an enormous, non-saturating magnetoresistivity [@cava]. Strain is predicted to open a band gap [@TP_transition] in WTe$_2$, which might make it suitable for device development. In fact, simple exfoliation of its isostructural $\gamma-$MoTe$_2$ compound (where $\gamma$ refers to the orthorhombic semi-metallic phase) into thin atomic layers was claimed to induce a band gap [@MoTe2_MI_transition] in the absence of strain. Such a transition would contrast with band structure calculations finding that WTe$_2$ should remain semi-metallic when exfoliated down to a single atomic layer [@Lv]. The insulating behavior reported for a few atomic layers of WTe$_2$ was ascribed to an increase in disorder due to its chemical instability in the presence of humidity which would induce Anderson localization [@Morpurgo], although more recently it was claimed to be intrinsic from transport measurements on encapsulated few-layered samples [@Cobden].
Orthorhombic $\gamma-$MoTe$_2$ and its isostructural compound WTe$_2$ were also claimed, based on density functional theory calculations, to belong to a new class of Weyl semi-metals, called type-II, which is characterized by a linear touching between hole and electron Fermi surface pockets [@bernevig; @felser; @bernevig2; @Hasan]. As for conventional Weyl points [@Weyl1; @Weyl2], these Weyl type-II points would also act as topological charges associated with singularities, i.e., sources and sinks, of Berry-phase pseudospin [@bernevig; @felser; @bernevig2; @Hasan] which could lead to anomalous transport properties. A series of recent angle-resolved photoemission spectroscopy (ARPES) measurements [@ARPES_Huang; @ARPES_Deng; @ARPES_Jiang; @ARPES_Liang; @ARPES_Xu; @ARPES_Tamai; @ARPES_Belopolski; @thirupathaiah] claim to observe a good overall agreement with these predictions. These studies observe the band crossings predicted to produce the Weyl type-II points, which would be located slightly above the Fermi-level, as well as the Fermi arcs projected on the surface of this compound [@ARPES_Huang; @ARPES_Deng; @ARPES_Jiang; @ARPES_Liang; @ARPES_Xu; @ARPES_Tamai; @ARPES_Belopolski; @thirupathaiah].
Here, motivated by the scientific relevance and the possible technological implications of the aforementioned theoretical predictions [@bernevig; @felser; @bernevig2; @Hasan; @TP_transition], we evaluate, through electrical transport and torque magnetometry in bulk single-crystals, the electronic structure at the Fermi level and the topological character of orthorhombic $\gamma-$MoTe$_2$. Our goal is to contrast our experimental observations with the theoretical predictions and the reported ARPES results in order to validate their findings. This information could, for example, help us predict the electronic properties of heterostructures fabricated from single- or a few atomic layers of this compound. An agreement between the calculated geometry of the FS of $\gamma-$MoTe$_2$ with the one extracted from quantum oscillatory phenomena, would unambiguously support the existence of Weyl nodes in the bulk [@Weyl1; @Weyl2] and, therefore, the existence of related non-trivial topological surface states or Fermi arcs [@Weyl2; @Hasan; @bernevig; @felser; @bernevig2; @Hasan]. However, quantum oscillatory phenomena from $\gamma-$MoTe$_2$ single-crystals reveals a Fermi surface whose geometry is quite distinct from the one predicted by the DFT calculations based on its low temperature crystallographic structure. The extracted Berry-phase is found to be field-dependent. Still one does not obtain evidence for the topological character predicted for this compound when the Berry-phase is evaluated at low fields. Here, we show that shifts in the relative position of the electron and hole bands, implied by previous ARPES studies [@ARPES_Huang; @ARPES_Deng; @ARPES_Jiang; @ARPES_Liang; @ARPES_Xu; @ARPES_Tamai; @ARPES_Belopolski; @thirupathaiah], can replicate the angular dependence of the observed Fermi surface cross-sectional areas. However, these band shifts imply that the valence and electron bands would no longer cross and, therefore, that $\gamma-$MoTe$_2$ would not display the predicted Weyl type-II semi-metallic state.
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{width="14cm"}
Methods and Experimental Results
================================
Very high quality single crystals of monoclinic $\beta-$MoTe$_2$ were synthesized through a Te flux method: Mo, 99.9999%, and Te 99.9999 % powders were placed in a quartz ampoule in a ratio of 1:25 heated up to 1050 $^{\circ}$C and held for 1 day. Then, the ampoule was slowly cooled down to 900 $^{\circ}$C and centrifuged. The “as harvested" single-crystals were subsequently annealed for a few days at a temperature gradient to remove the excess Te. Magneto-transport measurements as a function of temperature were performed in a Physical Property Measurement System using a standard four-terminal configuration. Measurements of the Shubnikov-de Haas (SdH) and the de Haas-van Alphen (dHvA) effects were performed in dilution refrigerator coupled to a resistive Bitter magnet, with the samples immersed in the $^3$He-$^4$He mixture. Measurements of the dHvA-effect were performed *via* a torque magnetometry technique, i.e. by measuring the deflection of a Cu-Be cantilever capacitively. Electrical transport measurements in pulsed magnetic fields were performed at the Dresden High Magnetic Field Laboratory using a 62 T magnet with a pulse duration of 150 ms. The sample temperature was controlled using a $^4$He bath cryostat (sample in He atmosphere) with an additional local heater for temperatures above 4.2 K. Synchrotron based X-ray measurements were performed in three single-crystals at the CHESS-A2 beam line using a combination of photon energies and cryogenic set-ups. The crystallographic data was reduced with XDS [@XDS]. The structures were solved with direct methods using SHELXS [@shelx]. Outlier rejection and absorption correction was done with SADABS. Least squares refinement on the intensities were performed with SHELXL [@shelx]. For additional detailed information on the experimental set-ups used, see Supplemental Information file [@supplemental].
As illustrated by Fig. 1(a), the as synthesized single-crystals display resistivity ratios *RRR* = $\rho(T = 300 \text{ K})/\rho(T = 2 \text{ K})$ ranging from $380$ to $> 2000$ which is one to two orders of magnitude higher than the $RRR$ values currently in the literature (see, for example, Ref. ). Although not clearly visible in Fig. 1(a) due to its logarithmic scale, a hysteretic anomaly is observed in the resistivity around $ 240 $ K corresponding to the monoclinic to orthorhombic structural transition which stabilizes what we denominate as the orthorhombic $\gamma-$MoTe$_2$ phase. For a clearer exposure of this transition and related hysteresis, see Ref. . These single-crystals were subsequently measured at much lower temperatures allowing us to determine their superconducting transition temperature $T_c$. Remarkably, and as seen in Fig. 1(b), we find that $T_c$ depends on sample quality, increasing considerably as the $RRR$ increases, suggesting that structural disorder suppresses $T_c$. For these measurements, particular care was taken to suppress the remnant field of the superconducting magnet since the upper critical fields are rather small (see Supplemental Fig. S1 [@supplemental]). The sample displaying the highest $RRR$ and concomitant $T_c$ was measured in absence of a remnant field. To verify that these differences in $T_c$ are not due to a poor thermal coupling between the sample and the thermometers, $T_c$ was measured twice by increasing and decreasing $T$ very slowly. The observed hysteresis is small relative to $T_c$ indicating that the measured $T_c$s are not an artifact. The values of the residual resistivities $\rho_0$ depend on a careful determination of the geometrical factors such as the size of the electrical contacts. Therefore, the $RRR$ provides a more accurate determination of the single-crystalline quality. In the past, the suppression of $T_c$ by impurities and structural defects was systematically taken as evidence for unconventional superconductivity [@andy; @satoru1; @satoru2], e.g. triplet superconductivity [@Maeno] in Sr$_2$RuO$_4$. Nevertheless, the fittings of the upper-critical fields $H_{c2}$ to a conventional Ginzburg-Landau expression, shown in Fig. S1 [@supplemental], points towards singlet pairing.
We have also evaluated the quality of our single crystals through Hall-effect [@qiong] and heat capacity measurements (see, Supplemental Fig. S2 [@supplemental]). Hall-effect reveals a sudden increase in the density of holes below $T = 40$ K, suggesting a possible temperature-induced Lifshitz-transition. While the heat capacity reveals a broad anomaly around $T = 66$ K, well-below its Debye temperature ($\Theta_D \simeq 120$ K), that would suggest that the structural degrees of freedom continue to evolve upon cooling below $T = 100$ K. Given that such structural evolution could affect the electronic band structure predicted for $\gamma-$MoTe$_2$ [@felser; @bernevig2; @Hasan], we performed synchrotron X-ray scattering down to $ \sim 12$ K (see, Supplemental Fig. S3 [@supplemental]). We observe some variability in the lattice constants extracted among several single-crystals and a sizeable hysteresis in the range $125 \text{K} \leq T \leq 250 \text{K}$ associated with the structural transition observed at $T \simeq 250$ K, but no significant evolution in the crystallographic structure below 100 K. As we discuss below, there are negligible differences between the electronic bands calculated with the crystal structures collected at 100 K and at 12 K, respectively.
Figures 1(c) and 1(d) display the change in magnetoresistivity $\Delta \rho (\mu_0 H)/\rho_0 = (\rho(\mu_0 H)-\rho_0)/\rho_0$ as a function of the field $\mu_0 H$ for a crystal characterized by $RRR \sim 450$ when the electrical current flows along the crystalline *a*-axis and the field is applied either along the *c*- or the *b*-axes, respectively. Similarly to WTe$_2$, for both orientations $\Delta \rho /\rho_0$ shows no sign of saturation under fields all the way up to 60 T while surpassing $1 \times 10^{6}$ % for $\mu_0 H \| c$-axis [@cava]. For WTe$_2$ such anomalous magnetoresistivity was attributed to compensation between the density of electrons and holes [@cava; @pletikosic; @pippard]. Nevertheless, there are a number of subsequent observations [@Daniel] contradicting this simple scenario, such as i) a non-linear Hall response [@Joe], ii) the suppression of the magnetoresistivity at a pressure where the Hall response vanishes [@WTe2_SC_1] (i.e. at perfect compensation), and iii) the observation of a pronounced magnetoresistivity in electrolyte gated samples with a considerably higher density of electrons with respect to that of holes [@Fuhrer]. It remains unclear if the proposed unconventional electronic structure [@bernevig; @felser; @bernevig2] would play a role on the giant magnetoresistivity of WTe$_2$, while its measured FS differs from the calculated one [@Daniel; @behnia]. In contrast, we have previously shown that $\gamma$-MoTe$_2$ indeed is a well compensated semi-metal [@qiong]. The slightly smaller magnetoresistivity of $\gamma-$MoTe$_2$ relative to WTe$_2$ is attributable to heavier effective effective masses, according to de Haas-van Alphen-effect discussed below, or concomitantly lower mobilities.
The best $\gamma-$MoTe$_2$ samples, i.e. those with $RRR \geq 2000$, display even more pronounced $\Delta \rho/\rho_0$ under just $\mu_0 H \simeq 10$ T. The oscillatory component superimposed on the magnetoresistivity corresponds to the Shubnikov-de Haas (SdH) effect resulting from the Landau quantization of the electronic orbits. Figure 1(e) shows the oscillatory, or the SdH signal as a function of inverse field $(\mu_0H)^{-1}$ for three temperatures. The SdH signal was obtained by fitting the background signal to a polynomial and subtracting it. Notice how for this sample and for $\mu_0 H \|$ *c*-axis, the SdH signal is dominated by a single frequency. However for all subsequent measurements performed under continuous fields (discussed below) one observes the presence of two main frequencies very close in value, each associated to an extremal cross-sectional area $A$ of the FS through the Onsager relation $F = A(\hbar / 2\pi e)$ where $\hbar$ is the Planck constant and $e$ is the electron charge. To illustrate this point, we show in Fig. 1(f) the oscillatory signal extracted from the magnetic torque, i.e. $\mathbf{\tau} = \mathbf{M} \times \mu_0 \mathbf{H}$, or the de Haas-van Alphen effect (dHvA) collected from a $\gamma-$MoTe$_2$ single-crystal for fields aligned nearly along its *c*-axis. Here $M = \chi \mu_0 H$ is the magnetization and $\chi(\mu_0 H, T)$ is its magnetic susceptibility. Figure 1(f) shows the oscillatory component of the magnetic susceptibility $\Delta \chi = \partial (\tau/\mu_0H)/ \partial(\mu_0H)$. The envelope of the oscillatory signal displays the characteristic “beating" pattern between two close frequencies. This becomes clearer in the fast Fourier transform of the oscillatory signal shown below. According to the Lifshitz-Onsager quantization condition [@kim; @nagaosa], the oscillatory component superimposed onto the susceptibility is given by: $$\begin{aligned}
\Delta \chi[(B)^{-1}] \propto \frac{T}{B^{5/2}}\sum_{l=1}^{\infty} \frac{\exp^{-l \alpha \mu T_D/ B}\cos(l g \mu \pi/2)}{l^{3/2}\sinh(\alpha \mu T/B)}\end{aligned}$$ $$\begin{aligned}
\times \cos\left\{ 2\pi \left[ \left( \frac{F}{B}-\frac{1}{2}+\phi_B \right)l + \delta \right] \right\}\end{aligned}$$ where $F$ is the dHvA frequency, $l$ is the harmonic index, $\omega_c$ the cyclotron frequency, $g$ the Landé *g*-factor, $\mu$ the effective mass in units of the free electron mass $m_0$, and $\alpha$ is a constant. $\delta$ is a phase shift determined by the dimensionality of the FS which acquires a value of either $\delta = 0$ or $ \pm 1/8$ for two- and three-dimensional FSs [@kim; @nagaosa; @kopelevich], respectively. $\phi_B$ is the Berry phase which, for Dirac and Weyl systems, is predicted to acquire a value $\phi_B = \pi$ [@kim; @nagaosa; @kopelevich]. Finally, $T_D= \hbar /(2 \pi k_B \tau $) is the so-called Dingle temperature from which one extracts $\tau$ or the characteristic quasiparticle scattering time.
In Supplementary Figs. S4 and S5 [@supplemental], we discuss the extraction of the Berry-phase of $\gamma-$MoTe$_2$ via fits to Eq. (1) of the oscillatory signal shown in Fig. 1(f). However, the geometry of the FS of $\gamma-$MoTe$_2$ evolves slightly as the field increases due to the Zeeman-effect, which precludes the extraction of its Berry phase. More importantly, one cannot consistently extract a value $\phi_B \simeq \pi$ when one limits the range in magnetic fields to smaller values in order to minimize the role of the Zeeman-effect. In other words, the dHvA-effect does not provide evidence for the topological character of $\gamma-$MoTe$_2$. Nevertheless, it does indicate that the Dingle temperature decreases as the field increases implying a field-induced increase in the quasiparticle lifetime. This effect should contribute to its large and non-saturating magnetoresistivity. We reported a similar effect for WTe$_2$ [@Daniel].
Since the Berry-phase extracted from the dHvA-effect does not provide support for a topological semi-metallic state in $\gamma-$MoTe$_2$, it is pertinent to ask if the DFT calculations predict the correct electronic band-structure and related FS geometry for this compound, since both are the departing point for the predictions of Refs. . To address this issue, we studied the dHvA-effect as a function of the orientation of the field with respect to the main crystallographic axes. Here, our goal is to compare the angular dependence of the cross-sectional areas determined experimentally, with those predicted by DFT.
Figures 2(a) and 2(b) display both the dHvA (red traces) and the SdH signals (black traces) measured in two distinct single crystals and for two field orientations, respectively along the $c-$ and the $a-$axes. As previously indicated, the dHvA and SdH signals were obtained after fitting a polynomial and subtracting it from the background magnetic torque and magnetoresistivity traces, respectively. The SdH signal was collected from a crystal displaying a $RRR \gtrsim 1000 $ at $T \simeq 25$ mK under fields up to 18 T, while the dHvA one was obtained from a crystal displaying $RRR \gtrsim 2000$ at $T \simeq 35$ mK under fields up to 35 T. Both panels also display the Fast Fourier transform (FFT) of the oscillatory signal. For fields along the $c-$axis, one observes two main peaks at $F_{\alpha} = 231$ T and at $F_{\beta} = 242$ T, as well as their first- and second harmonics and perhaps some rather small frequencies which could result from imperfect background subtraction. We obtain the same two dominant frequencies regardless of the interval in $H^{-1}$ used to extract the FFTs. Supplemental Fig. S6 [@supplemental] displays the dHvA signal for $H$ aligned nearly along the $b-$axis along with the corresponding FFT spectra which are again dominated by two prominent peaks. The observation of just two main frequencies for $\mu_0H \| c-$axis is rather surprising since, as we show below, DFT calculations, including the effect of the spin-orbit interaction, predict several pairs of electron-like corrugated cylindrical FSs along with pairs of smaller three-dimensional electron-like sheets in the First-Brillouin zone. Around the $\Gamma-$point, DFT predicts at least a pair of four-fold symmetric helix-like large hole sheets. This complex FS should lead to a rich oscillatory signal, contrary to what is observed. One might argue that the non-observation of all of the predicted FS sheets would be attributable to an experimental lack of sensitivity or to poor sample quality which would lead to low carrier mobility. Nevertheless, our analysis of the Hall-effect within a two-carrier model [@qiong], yields electron- and hole-mobilities ranging between $10^4$ and $10^5$ cm$^2$/Vs at low $T$s which is consistent with both the small residual resistivities and the large resistivity ratios of our measured crystals. Given that the magnetic torque is particularly sensitive to the anisotropy of the FS, such high mobilities should have allowed us to detect most of the predicted FSs, particularly at the very low $T$s and very high fields used for our measurements. Hence, we conclude that the geometry of the FS ought to differ considerably from the one predicted by DFT.
In Figs. 2(c) and 2(d) we plot the amplitude of the main peaks observed in the FFT spectra for fields along the $c-$axis as a function of the temperature. Red lines are fits to the Lifshitz-Kosevich (LK) temperature damping factor, i.e. $x/\sinh x$ with $x = 14.69 \mu T/H$ and with $\mu$ being the effective mass in units of the free electron mass, from which we extract the masses associated with each frequency. As seen, for $H \| c-$axis one obtains $\mu_{\alpha} = 0.85$ $m_0$ and $\mu_{\beta}= 0.8$ $m_0$, which contrasts with the respective values obtained for $H \| a-$axis, namely $\mu_{\alpha, \beta} \simeq 1.5$ $m_0$, see Figs. 2(e) and 2(f). As previously mentioned for $\mu_0H \|b-$axis, we observe two main frequencies, but by reducing the $H^{-1}$ window to focus on the higher field region, we detect additional frequencies (See, Fig. S6 [@supplemental]) which are characterized by heavier effective masses, i.e. in the order of $2.5-2.9$ $m_0$. This indicates that $\gamma-$MoTe$_2$ displays a higher anisotropy in effective masses when compared to WTe$_2$ [@Daniel], although these masses are consistent with its sizeable $\gamma_e$ coefficient. Supplemental Fig. S7 [@supplemental] displays several traces of the dHvA signal as functions of the inverse field for several angles between all three main crystallographic axes. These traces are used to plot the angular dependence of the FS cross-sectional areas in order to compare these with the DFT calculated ones.
Comparison between experiments and the DFT calculations
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Several recent angle-resolved photoemission spectroscopy (ARPES) studies [@ARPES_Huang; @ARPES_Deng; @ARPES_Jiang; @ARPES_Liang; @ARPES_Xu; @ARPES_Tamai; @ARPES_Belopolski; @thirupathaiah] claim to find a broad agreement between the band structure calculations, the predicted geometry of the Fermi surface, the concomitant existence of Weyl type-II points [@bernevig; @felser; @bernevig2], and the related Fermi arcs on the surface states of $\gamma-$MoTe$_2$. Several of these experimental and theoretical studies claim that the electronic structure of this compound is particularly sensitive to its precise crystallographic structure. Inter-growth of the $2H-$phase or the temperature used to collect to X-ray diffraction data, typically around 100 to 230 K, are claimed to have a considerable effect on the calculations [@bernevig2; @ARPES_Tamai]. Given the few frequencies observed by us, it is pertinent to ask if the mild evolution of the crystallographic structure as a function of the temperature shown in Fig. S3 [@supplemental] would affect the geometry of the FS of $\gamma-$MoTe$_2$. To address this question we performed a detailed angular-dependent study of the frequencies extracted from both the SdH and the dHvA effects in $\gamma-$MoTe$_2$ in order to compare these with the angular dependence of the FS cross-sectional areas predicted by the calculations.
In the subsequent discussion we compare the angular dependence of our dHvA frequencies with calculations performed with the Quantum Espresso [@QE] implementation of the density functional theory in the GGA framework including spin-orbit coupling (SOC). The Perdew-Burke-Ernzerhof (PBE) exchange correlation functional [@PBE] was used with fully relativistic norm-conserving pseudopotentials generated using the optimized norm-conserving Vanderbilt pseudopotentials as described in Ref. . The 4*s*, 4*p*, 4*d* and 5*s* electrons of Mo and the 4*d*, 5*s* and 5*p* electrons of Te were treated as valence electrons. After careful convergence tests, the plane-wave energy cutoff was taken to be 50 Ry and a $k-$point mesh of $20\times 12\times 6$ was used to sample the reducible Brillouin Zone (BZ) used for the self-consistent calculation. The Fermi surfaces were generated using a more refined $k-$point mesh of $45\times 25\times 14$. FS sheets were visualized using the XCrysden software [@xcrysden]. The related angular dependence of the quantum oscillation frequencies was calculated using the skeaf code [@skeaf]. As shown in Fig. 3 the results are very close to those obtained by using the VASP and the Wien2K implementations of DFT (see, Fig. S8 in SI [@supplemental]), and also to those reported by Refs. .
Figure 3(a) displays the electronic band structure of $\gamma-$MoTe$_2$, based on its structure determined at $T = 100$ K, with and without the inclusion of SOC. As previously reported [@felser; @bernevig2], electron- and hole-bands intersect along the $\Gamma-X$ direction at energies slightly above $\varepsilon_F$ creating a pair of Weyl type-II points. Figure 3(b) shows a comparison between band structures based on the crystallographic lattices determined at 12 K and at 100 K, respectively. Both sets of electronic bands are nearly identical and display the aforementioned crossings between hole- and electron-bands thus indicating that the electronic structure remains nearly constant below 100 K. Figures 3(c) and 3(d) provide a side perspective and a top view of the overall resulting FS within BZ, respectively. The main features of the DFT calculations are the presence of two-dimensional electron pockets, labeled $e_1$ and $e_2$ in Figs. 3(e) and 3(f) and of large “star-shaped” hole-pockets near the $\Gamma-$point, labeled as the $h_2$ and the $h_3$ sheets in Figs. 3(k) and 3(l). These electron and hole pockets nearly “touch”. Due to the broken inversion symmetry, these bands are not Kramer’s degenerate, and hence the spin-orbit split partners of the corresponding electron and hole pockets are located inside the corresponding bigger sheets. The $h_1$ hole pocket and the $e_3$ and $e_4$ electron pockets are very sensitive to the position of $\varepsilon_F$ disappearing when $\varepsilon_F$ is moved by only $\pm 15$ meV.
Figures 4(a) and 4(b) present the angular dependence of the calculated and of the measured FFT spectra of the oscillatory signal (raw data in Fig. S7 [@supplemental]), respectively. In this plot the Onsager relation was used to convert the theoretical FS cross-sectional areas into oscillatory frequencies. In Fig. 4(b) $\theta$ refers to angles between the $c-$ and the $a-$axis, where $\theta=0^{\circ}$ corresponds to $H\parallel c-$axis, while $\phi$ corresponds to angles between the $c-$ and the $b-$axis, again relative to the $c-$axis. As seen, there are striking differences between both data sets with the calculations predicting far more frequencies than the measured ones. More importantly, for fields oriented from the *c*-axis towards either the $a-$ or the $b-$axis, one observes the complete absence of experimental frequencies around $\sim 1$ kT which, according to the calculations, would correspond to the cross-sectional areas of the hole-pockets $h_2$ and $h_3$. In addition, while many of the predicted electron orbits show a marked two-dimensional character, diverging as the field is oriented towards the $a-$ or the $b-$axis, the experimentally observed frequencies show finite values for fields along either axis. This indicates that these orbits are three-dimensional in character, despite displaying frequencies close to those predicted for the $e_1$ and the $e_2$ pockets for fields along the $c-$axis. These observations, coupled to the non-detection of all of the predicted orbits, in particular the large hole $h_2$ and $h_3$ Fermi surfaces, indicate unambiguously that the actual geometry of the FS of $\gamma-$MoTe$_2$ is different from the calculated one. Notice that frequencies inferior to $F = 100$ T, which correspond to the smaller electron- and hole-pockets and which are particularly sensitive to the position of $\varepsilon_F$ as previously mentioned, were not included in Fig. 4(a) for the sake of clarity.
The calculation shows a significant difference between the SOC-split theoretical bands, which is highlighted by the absence of a frequency around 0.5 kT associated with $e_2$ pocket, along with its presence in association with the $e_1$ pocket. This contrast between both orbits is due to the presence of a “handle-like” structure (see Fig. 3(e)) in $e_1$ which gives a maximum cross-section close to the BZ edge. However, at this position there is no maximum cross-section within the BZ for the $e_2$ pocket since its “handle” is missing (see Fig. 3(f)). This marked difference in topology between the FSs of both spin-orbit split partners indicates that the strength of the SOC provided by the DFT calculations tends to be considerably larger than the one implied by our experiments. In fact, from the twin peaks observed in the experimental FFT spectra having frequencies around 250 T for fields along the *c*-axis, which are likely to correspond to SOC-split bands due to their similar angular dependence, we can infer that the actual SO-splitting is far less significant than the value predicted by the calculations. We have investigated the possibility of an overestimation of the strength of the SOC within our calculations which is the mechanism driving the DFT prediction of a large number of dHvA frequencies displaying remarkably different angular dependencies. For instance, we calculated the angular dependence of the FSs without the inclusion of SOC. This leads to just one, instead of a pair of distinct SOC-split bands, which in fact display angular dependencies very similar to those of orbits $h_3$, $e_1$ and $e_3$ in Fig. 4(a). Notice that part of the discrepancy is attributable to the inter-planar coupling which is not well captured by the DFT calculations [@Son]. DFT suggests that this compound is van der Waals like by predicting several two-dimensional (i.e. cylindrical like) FS sheets, when the experiments indicate that the overall FS displays a marked three-dimensional character. This indicates that the inter-planar coupling is stronger than implied by DFT. In any case, from Figs. 4(a) and 4(b) and the above discussion, it is clear that there are significant discrepancies between the calculated and the measured FSs. Given that the proposed Weyl type-II scenario [@bernevig; @felser; @bernevig2] hinges on a possible touching between electron- and hole-pockets, it is critical to understand their exact geometry, or the reason for the disagreement between predictions and experiments, before one can make any assertion on the existence of the Weyl type-II points in $\gamma-$MoTe$_2$.
To understand the source of the disagreement between calculations and our measurements, we now focus on a detailed comparison between our DFT calculations and a selection of ARPES studies. Figure 5(a) corresponds to data from Ref. depicting an ARPES energy distribution map (EDM) along $k_y$ while keeping $k_x =0$. Figure 5(b) plots its derivative. In both figures the $\Gamma-$point corresponds to $k_y=0$. According to the calculations, this EDM should reveal two valence bands intersecting $\varepsilon_F$ around the $\Gamma-$point; the first leading to two small hole-pockets, or the $h_1$ sheets at either side of $\Gamma$, with the second SOC-split band producing the larger $h_2$ and $h_3$ sheets. Instead, ARPES observes just one band intersecting $\varepsilon_F$ which leads to a single FS sheet of cross-sectional area $S_{\text{FS}} \sim \pi (0.1 \text{ \AA}^{-1})^2$ as indicated by the vertical blue lines in Fig. 5(b). This observation by ARPES questions the existence of the band (or of its intersection with $\varepsilon_F$) responsible for the large $h_2$ and $h_3$ hole-pockets with this band being the one previously reported to touch the electron band that produces the $e_1$ pocket and creating in this way the Weyl type-II points [@ARPES_Huang; @ARPES_Deng; @ARPES_Jiang; @ARPES_Tamai; @ARPES_Belopolski]. Notice that our dHvA measurements do not reveal any evidence for the original $h_2$ and $h_3$ sheets, thus, being in agreement with this ARPES observation. Furthermore, the Onsager relation $F= S(\hbar/2\pi e)$ yields a frequency of $\sim 330$ T for $S_{\text{FS}}$ which, in contrast, is close to the frequencies observed by us for $\mu_0H \| c-$axis. Figure 5(c) displays the band structure calculated “ribbons” obtained by projecting the $k_z$ dependence of the bands onto the $k_x-k_y$ plane. This representation of the band-structure provides a better comparison with the ARPES EDMs. As seen, there is a good overall agreement between the ARPES and the DFT bands, as previously claimed [@ARPES_Huang; @ARPES_Deng; @ARPES_Jiang; @ARPES_Tamai; @ARPES_Belopolski], except for the exact position of $\varepsilon_F$. The purple line depicts the position of $\varepsilon_F$ according to the DFT calculations while the white line depicts the position of $\varepsilon_F$ according to ARPES. A nearly perfect agreement between DFT and ARPES is achievable by shifting the DFT valence bands by $\sim - 50$ meV, which is what ends suppressing the $h_2$ and $h_3$ FS hole sheets from the measured ARPES EDMs. As shown through Figs. 5(d) and 5(e), this disagreement between the ARPES and the DFT bands is observed in the different ARPES studies[@ARPES_Tamai]. Figure 5(d) corresponds to an EDM along the $k_x$ $(k_y=0)$ direction of the BZ. As shown in Fig. 5(e), DFT reproduces this EDM quite well. Nevertheless, as indicated by the yellow lines in both figures, which are positioned at the top of the deepest valence band observed by ARPES, the ARPES bands are displaced by $\sim -45$ meV with respect to the DFT ones. Red dotted lines in Fig. 5(d) indicate the cross-sections of the observed electron pockets or $\sim \pi (0.1 \text{ \AA}^{-1})^2$. Therefore, to match our main dHvA frequencies, the electron bands would have to be independently and slightly displaced towards higher energies to decrease their cross-sectional area. To summarize, DFT and ARPES agree well on the overall dispersion of the bands of $\gamma-$MoTe$_2$, but not on their relative position with respect to $\varepsilon_F$.
Therefore, guided by ARPES, we shifted the overall valence bands of $\gamma-$MoTe$_2$, shown in Figs. 3(a) and 3(b), by -50 meV and the electron ones by +35 meV to recalculate the FS cross-sectional areas as a function of field orientation relative to the main crystallographic axes. The comparison between the measured dHvA cross-sectional areas and those resulting from the shifted DFT bands are shown in Fig. 6. Figure 6(a) displays the Fourier spectra, previously shown in Fig. 4(b), with superimposed colored lines identifying shifted electron (magenta) and hole (blue) orbits according to Fig. 6(b) which displays these frequencies as a function of field orientation for shifted non-SOC-split DFT bands. As seen, the qualitative and quantitative agreement is good, but not perfect. In contrast, Fig. 6(c) displays these orbits/frequencies as a function of field orientation for SOC-split DFT bands. Clearly, and as previously discussed, the approach used to evaluate the effect of the SOC in $\gamma-$MoTe$_2$ seems to overestimate it for reasons that remain to be clarified. Concerning the Weyl physics in $\gamma-$MoTe$_2$, the displacement of the bands, introduced here to explain our observations based on the guidance provided by previous ARPES studies, would eliminate the crossings between the electron- and the hole-bands as shown in Fig. 7(a). Finally, Figs. 7(b) to 7(g) display the geometry of the Fermi surface resulting from the shifted bands. Overall, the FS displays a distinctly more marked three-dimensional character, with the electrons and the-hole pockets remaining well-separated in *k*-space. This three-dimensionality is consistent with the observations of Ref. which finds that the electronic bands do disperse along the $k_z-$direction implying that $\gamma-$MoTe$_2$ cannot be considered a van der Waals coupled solid. Notice that DFT tends to underestimate the inter-planar coupling in weakly coupled compounds which is at the heart of the disagreement between the calculations and our observations.
Conclusions
===========
In conclusion, quantum oscillatory phenomena reveal that the geometry of the Fermi surface of $\gamma-$MoTe$_2$ is quite distinct from the one predicted by previous electronic band-structure calculations. Our low-temperature structural analysis *via* synchrotron X-ray diffraction measurements indicates the absence of an additional structural transition below the monoclinic to orthorhombic one that would explain this disagreement, while heat-capacity measurements provide no evidence for an electronic phase-transition upon cooling. In contrast, a direct comparison between DFT calculations and the band-structure reported by angle resolved photoemission spectroscopy reveals a disagreement on the position of the valence bands relative to the Fermi-level, with the experimental valence bands shifted by $\sim -50$ meV relative to the DFT ones. Therefore, one should be careful concerning the claims of a broad agreement between the calculations and the electronic bands revealed by ARPES measurements [@ARPES_Huang; @ARPES_Deng; @ARPES_Jiang; @ARPES_Liang; @ARPES_Xu; @ARPES_Tamai].
Here, we show that it is possible to describe the angular-dependence of the observed de Haas-van Alphen Fermi surface cross-sectional areas by shifting the position of the DFT bands relative to the Fermi level as indicated by ARPES. However, with this adjustment, the Weyl points, which result from band-crossings that are particularly sensitive to small changes in the lattice constants, are no longer present in the band-structure of $\gamma-$MoTe$_2$. Although our approach of modifying the band structure in order to obtain an agreement with both ARPES and de Haas-van Alphen experiments has only a phenomenological basis, our findings do shed a significant doubt on the existence of the Weyl points in the electronic band structure of $\gamma-$MoTe$_2$.
Finally, this study combined with the ARPES results in Ref. , indicate that there ought to be a Lifshitz-transition [@wu] upon W doping in the $\gamma-$Mo$_{1-x}$W$_x$Te$_2$ series, leading to the disappearance of the central hole pockets in $\gamma-$MoTe$_2$ in favor of the emergence of hole-pockets at either side of the $\Gamma-$point in $\gamma-$Mo$_{1-x}$W$_x$Te$_2$.
We acknowledge helpful discussions with R. M. Osgood and A. N. Pasupathy. J.Y.C. is supported by NSF-DMR-1360863. L. B. is supported by DOE-BES through award DE-SC0002613 for experiments under high magnetic fields and at very low temperatures, and by the U.S. Army Research Office MURI Grant W911NF-11-1-0362 for the synthesis and physical characterization of two-dimensional materials and their heterostructures. We acknowledge the support of the HLD-HZDR, member of the European Magnetic Field Laboratory (EMFL). Research conducted at the Cornell High Energy Synchrotron Source (CHESS) is supported by the NSF & NIH/NIGMS via NSF award DMR-1332208. The NHMFL is supported by NSF through NSF-DMR-1157490 and the State of Florida.
[0]{} X. F. Qian, J. W. Liu , L. Fu, and J. Li, *Quantum spin Hall effect in two-dimensional transition metal dichalcogenides*, Science **346**, 1344 (2014). D. Pesin, and A. H. MacDonald, *Spintronics and pseudospintronics in graphene and topological insulators*, Nat. Mater. **11**, 409-416 (2012). M. Chhowalla, H. S. Shin, G. Eda, L. -J. Li, K. P. Loh, and H. Zhang, *The chemistry of two-dimensional layered transition metal dichalcogenide nanosheets*, Nat. Chem. **5**, 263-275 (2013). S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson, R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl and J. E. Goldberger, *Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene*, ACS Nano **7**, 2898-2926 (2013). M. N. Ali, J. Xiong, S. Flynn, J. Tao, Q. D. Gibson, L. M. Schoop, T. Liang, N. Haldolaarachchige, M. Hirschberger, N. P. Ong and R. J. Cava, *Large, non-saturating magnetoresistance in WTe$_2$*, Nature **514**, 205-208 (2014). D. H. Keum, S. Cho, J. H. Kim, D. -H. Choe, H. -J. Sung, M. Kan, H. Kang, J. -Y. Hwang, S. W. Kim, H. Yang, K. J. Chang and Y. H. Lee, *Bandgap opening in few-layered monoclinic MoTe$_2$*, Nat. Phys. **11**, 482-486 (2015). H. Y. Lv, W. J. Lu, D. F. Shao, Y. Liu, S. G. Tan, and Y. P. Sun, *Perfect charge compensation in WTe$_2$ for the extraordinary magnetoresistance: From bulk to monolayer*, EPL **110**, 37004 (2015). L. Wang, I. Gutiérrez-Lezama, C. Barreteau, N. Ubrig, E. Giannini, and A. F. Morpurgo, *Tuning magnetotransport in a compensated semimetal at the atomic scale*, Nat. Commun. **6**, 8892 (2015). Z. Fei, T. Palomaki, S. Wu, W. Zhao, X. Cai, B. Sun, P. Nguyen, J. Finney, X. Xu, and D. H. Cobden, *Edge conduction in monolayer WTe$_2$*, Nat. Phys. **13**, 677 (2017). A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai and B. A. Bernevig, *A New Type of Weyl Semimetals*, Nature **527**, 495-498 (2015). Y. Sun, S.C. Wu, M. N. Ali, C. Felser, and B. Yan, *Prediction of the Weyl semimetal in the orthorhombic MoTe$_2$*, Phys. Rev. B **92**, 161107 (2015). Z. Wang, D. Gresch, A. A. Soluyanov, W. Xie, S. Kushwaha, X. Dai, M. Troyer, R. J. Cava, and B. A. Bernevig, *MoTe$_2$: A Type-II Weyl Topological Metal*, Phys. Rev. Lett. **117**, 056805 (2016). T. R. Chang, S. -Y. Xu, G. Chang, C. -C. Lee, S. -M. Huang, B. Wang, G. Bian, H. Zheng, D. S. Sanchez, I. Belopolski, N. Alidoust, M. Neupane, A. Bansil, H. -T. Jeng, H. Lin, and M. Z. Hasan, *Prediction of an arc-tunable Weyl Fermion metallic state in Mo$_x$W$_{1-x}$Te$_2$*, Nat. Commun. **7**, 10639 (2016). H. M. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, *Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides*, *Phys. Rev. X* **5**, 011029 (2015). S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. -C. Lee, S. -M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, M. Z. Hasan, *Discovery of a Weyl fermion semimetal and topological Fermi arcs*, Science **349**, 613-617 (2015). L. Huang, T. M. McCormick, M. Ochi, Z. Zhao, M. -T. Suzuki, R. Arita, Y. Wu, D. Mou, H. Cao, J. Yan, N. Trivedi, and A. Kaminski, *Spectroscopic evidence for type II Weyl semimetal state in MoTe$_2$*, Nat. Mater. **15**, 1155-1160 (2016). K. Deng, G. Wan, P. Deng, K. Zhang, S. Ding, E. Wang, M. Yan, H. Huang, H. Zhang, Z. Xu, J. Denlinger, A. Fedorov, H. Yang, W. Duan, H. Yao, Y. Wu, S. Fan, H. Zhang, X. Chen and S. Zhou, *Experimental observation of topological Fermi arcs in type-II Weyl semimetal MoTe$_2$*, Nat. Phys. **12**, 1105 (2016). J. Jiang, Z. K. Liu, Y. Sun, H. F. Yang, R. Rajamathi, Y. P. Qi, L. X. Yang, C. Chen, H. Peng, C. -C. Hwang, S. Z. Sun, S. -K. Mo, I. Vobornik, J. Fujii, S. S. P. Parkin, C. Felser, B. H. Yan, Y. L. Chen, *Observation of the Type-II Weyl Semimetal Phase in MoTe$_2$*, Nat. Commun. **8**, 13973 (2017). A. Liang, J. Huang, S. Nie, Y. Ding, Q. Gao, C. Hu, S. He, Y. Zhang, C. Wang, B. Shen, J. Liu, P. Ai, L. Yu, X. Sun, W. Zhao, S. Lv, D. Liu, C. Li, Y. Zhang, Y. Hu, Y. Xu, L. Zhao, G. Liu, Z. Mao, X. Jia, F. Zhang, S. Zhang, F. Yang, Z. Wang, Q. Peng, H. Weng, X. Dai, Z. Fang, Z. Xu, C. Chen, X. J. Zhou, *Electronic Evidence for Type II Weyl Semimetal State in MoTe$_2$*, arXiv:1604.01706 (2016). N. Xu, Z. J. Wang, A. P. Weber, A. Magrez, P. Bugnon, H. Berger, C. E. Matt, J. Z. Ma, B. B. Fu, B. Q. Lv, N. C. Plumb, M. Radovic, E. Pomjakushina, K. Conder, T. Qian, J. H. Dil, J. Mesot, H. Ding, M. Shi, *Discovery of Weyl semimetal state violating Lorentz invariance in MoTe$_2$*, arXiv:1604.02116 (2016). A. Tamai, Q. S. Wu, I. Cucchi, F. Y. Bruno, S. Ricco, T. K. Kim, M. Hoesch, C. Barreteau, E. Giannini, C. Bernard, A. A. Soluyanov, F. Baumberger, *Fermi arcs and their topological character in the candidate type-II Weyl semimetal MoTe$_2$*, Phys. Rev. X **6**, 031021 (2016), and references therein. I. Belopolski, D. S. Sanchez, Y. Ishida, X. C. Pan, P. Yu, S. Y. Xu, G. Q. Chang, T. R. Chang, H. Zheng, N. Alidoust, G. Bian, M. Neupane, S. M. Huang, C. C. Lee, Y. Song, H. Bu, G. Wang, S. Li, G. Eda, H.-T. Jeng, T. Kondo, H. Lin, Z. Liu, F. Song, S. Shin and M. Z. Hasan, *Discovery of a new type of topological Weyl fermion semimetal state in Mo$_x$W$_{1-x}$Te$_2$*, Nat. Commun. **7**, 13643 (2016). S. Thirupathaiah, R. Jha, B. Pal, J. S. Matias, P. K. Das, P. K. Sivakumar, I. Vobornik, N. C. Plumb, M. Shi, R. A. Ribeiro, and D. D. Sarma, *MoTe$_2$: An uncompensated semimetal with extremely large magnetoresistance*, Phys. Rev. B **95**, 241105(R) (2017). W. Kabsch, *XDS*, Acta Cryst. **D66**, 125 (2010). G. M. Sheldrick, *A short history of SHELX*, Acta Cryst. A **64**, 112 (2008). See supplemental material at http://link.aps.org/ Y. Qi, P. G. Naumov, M. N. Ali, C. R. Rajamathi, W. Schnelle, O. Barkalov, M. Hanfland, S. -C. Wu, C. Shekhar, Y. Sun, V. Sü[ß]{}, M. Schmidt, U. Schwarz, E. Pippel, P. Werner, R. Hillebrand, T. Förster, E. Kampert, S. Parkin, R. J. Cava, C. Felser, B. Yan and S. A. Medvedev, *Superconductivity in Weyl semimetal candidate MoTe$_2$*, Nat. Commun. **7**, 11038 (2016). Q. Zhou, D. Rhodes, Q. R. Zhang, S. Tang, R. Schönemann, and L. Balicas, *Hall effect within the colossal magnetoresistive semimetallic state of MoTe$_2$*, Phys. Rev. B **94**, 121101(R) (2016). A. P. Mackenzie, R. K. W. Haselwimmer, A. W. Tyler, G. G. Lonzarich, Y. Mori, S. Nishizaki, and Y. Maeno, *Extremely strong dependence of superconductivity on disorder in Sr$_2$RuO$_4$*, Phys. Rev. Lett. **80**, 161-164 (1998). S. Nakatsuji, K. Kuga, Y. Machida, T. Tayama, T. Sakakibara, Y. Karaki, H. Ishimoto, S. Yonezawa, Y. Maeno, E. Pearson, G. G. Lonzarich, L. Balicas, H. Lee and Z. Fisk, *Superconductivity and quantum criticality in the heavy-fermion system beta-YbAlB$_4$*, Nat. Phys. **4**, 603-607 (2008). M. Tsujimoto, Y. Matsumoto, T. Tomita, A. Sakai, S. Nakatsuji, *Heavy-Fermion Superconductivity in the Quadrupole Ordered State of PrV$_2$Al$_{20}$*, Phys. Rev. Lett. **113**, 267001 (2014). A. P. Mackenzie, and Y. Maeno, *The superconductivity of Sr$_2$RuO$_4$ and the physics of spin-triplet pairing*, Rev. Mod. Phys. **75**, 657-712 (2003).I. Pletikosić, M. N. Ali, A. V. Fedorov, R. J. Cava, and T. Valla, *Electronic Structure Basis for the Extraordinary Magnetoresistance in WTe$_2$*, Phys. Rev. Lett. **113**, 216601 (2014). A. B. Pippard, *Magnetoresistance in Metals* (Cambridge University, Cambridge, 1989). D. Rhodes, S. Das, Q. R. Zhang, B. Zeng, N. R. Pradhan, N. Kikugawa, E. Manousakis, and L. Balicas, *Role of spin-orbit coupling and evolution of the electronic structure of WTe$_2$ under an external magnetic field*, Phys. Rev. B **92**, 125152 (2015). Y. Luo, H. Li, Y. M. Dai, H. Miao, Y. G. Shi, H. Ding, A. J. Taylor, D. A. Yarotski, R. P. Prasankumar, and J. D. Thompson, *Hall effect in the extremely large magnetoresistance semimetal WTe$_2$*, Appl. Phys. Lett. **107**, 182411 (2015). D. Kang, Y. Zhou, W. Yi, C. Yang, J. Guo, Y. Shi, S. Zhang, Z. Wang, C. Zhang, S. Jiang, A. Li, K. Yang, Q. Wu, G. Zhang, L. Sun and Z. Zhao *Superconductivity emerging from a suppressed large magnetoresistant state in tungsten ditelluride*, Nat. Commun. **6**, 7804 (2015). Y. L. Wang, K. F. Wang, J. Reutt-Robey, J. Paglione, and M. S. Fuhrer, *Breakdown of compensation and persistence of nonsaturating magnetoresistance in gated WTe$_2$ thin flakes*, Phys. Rev. B **93**, 121108 (2016). Z. Zhu, X. Lin, J. Liu, B. Fauqué, Q. Tao, C. Yang, Y. Shi, and K. Behnia, *Quantum Oscillations, Thermoelectric Coefficients, and the Fermi Surface of Semimetallic WTe$_2$*, Phys. Rev. Lett. **114**, 176601 (2015). Y. B. Zhang, Y. W. Tan, H. L. Stormer, P. Kim, *Experimental observation of the quantum Hall effect and Berry’s phase in graphene*, Nature **438**, 201-204 (2005). H. Murakawa, M. S. Bahramy, M. Tokunaga, Y. Kohama, C. Bell, Y. Kaneko, N. Nagaosa, H. Y. Hwang, Y. Tokura, *Detection of Berry’s Phase in a Bulk Rashba Semiconductor*, Science **342**, 1490-1493 (2013). I. A. Luk’yanchuk, and Y. Kopelevich, *Phase Analysis of Quantum Oscillations in Graphite*, Phys. Rev. Lett. **93**, 166402 (2004). P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari and R. M. Wentzcovitch, *QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials*, J. Phys.: Condens. Matter *21*, 395502 (2009). J. P. Perdew, K. Burke, M. Ernzerhof, *Generalized Gradient Approximation Made Simple*, Phys. Rev. Lett. **77**, 3865-3868 (1996). D. R. Hamann, *Optimized norm-conserving Vanderbilt pseudopotentials*, Phys. Rev. B **88**, 085117 (2013). A. Kokalj, *Computer graphics and graphical user interfaces as tools in simulations of matter at the atomic scale*, Comp. Mater. Sci. **28**, 155-168 (2003). Code available at http://www.xcrysden.org/. P. M. C. Rourke, and S. R. Julian, *Numerical extraction of de Haas-van Alphen frequencies from calculated band energies*, Comp. Phys. Commun. **183**, 324 (2012). H. -J. Kim, S. -H. Kang, I. Hamada, and Y. -W. Son, *Origins of the structural phase transitions in MoTe$_2$ and WTe$_2$*, Phys. Rev. B **95**, 180101(R) (2017). D. Rhodes, D. A. Chenet, B. E. Janicek, C. Nyby, Y. Lin, W. Jin, D. Edelberg, E. Mannebach, N. Finney, A. Antony, T. Schiros, T. Klarr, A. Mazzoni, M. Chin, Y. -c. Chiu, W. Zheng, Q. R. Zhang, F. Ernst, J. I. Dadap, X. Tong, J. Ma, R. Lou, S. Wang, T. Qian, H. Ding, R. M. Osgood, Jr, D. W. Paley, A. M. Lindenberg, P. Y. Huang, A. N. Pasupathy, M. Dubey, J. Hone, and L. Balicas, *Engineering the structural and electronic phases of MoTe$_2$ through W substitution*, Nano Lett. **17**, 1616 (2017). Y. Wu, N. H. Jo, M. Ochi, L. Huang, D. Mou, S. L. Bud’ko, P. C. Canfield, N. Trivedi, R. Arita, and A. Kaminski, *Temperature induced Lifshitz transition in WTe$_2$*, Phys. Rev. Lett. **115**, 166602 (2015).
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abstract: 'Extraordinary progress has been made towards developing neural network architectures for classification tasks. However, commonly used loss functions such as the multi-category cross entropy loss are inadequate for ranking and ordinal regression problems. Hence, approaches that utilize neural networks for ordinal regression tasks transform ordinal target variables into a series of binary classification tasks but suffer from inconsistencies among the different binary classifiers. Thus, we propose a new framework (Consistent Rank Logits, CORAL) with theoretical guarantees for rank-monotonicity and consistent confidence scores. Through parameter sharing, our framework also benefits from lower training complexity and can easily be implemented to extend conventional convolutional neural network classifiers for ordinal regression tasks. Furthermore, the empirical evaluation of our method on a range of face image datasets for age prediction shows a substantial improvement compared to the current state-of-the-art ordinal regression method.'
author:
- '\'
- '\'
- '\'
bibliography:
- 'bibliography.bib'
title: 'Rank-consistent Ordinal Regression for Neural Networks'
---
Deep Learning, Ordinal Regression, Convolutional Neural Networks, Age Prediction, Machine Learning, Biometrics.
Introduction {#sec:introduction}
============
Ordinal regression (sometimes also referred to as *ordinal classification*), describes the task of predicting labels on an ordinal scale. Here, a ranking rule or classifier $h$ maps each object $\mathbf{x}_i \in \mathcal{X}$ into an ordered set ${h: \mathcal{X} \rightarrow \mathcal{Y}}$, where ${\mathcal{Y}=\{r_1 \prec ... \prec r_K\}}$. In contrast to classification, the ranks include ordering information. In comparison with metric regression, which assumes that $\mathcal{Y}$ is a continuous random variable, ordinal regression regards $\mathcal{Y}$ as a finite sequence where the metric distance between ranks is not defined.
Along with age estimation [@niu2016ordinal], popular applications for ordinal regression include predicting the progression of various diseases, such as Alzheimer’s [@doyle2014predicting], Crohn’s [@weersma2009molecular], artery [@streifler1995lack], and kidney disease [@sigrist2007progressive]. Also, ordinal regression models are common choices for text message advertising [@rettie2005text] and various recommender systems [@parra2011implicit].
While the field of machine learning developed many powerful algorithms for predictive modeling, most algorithms were designed for classification tasks. In 2007, Li and Lin proposed a general framework for ordinal regression via extended binary classification [@li2007ordinal], which has become the standard choice for extending machine learning algorithms for ordinal regression tasks. However, implementations of this approach commonly suffer from classifier inconsistencies among the binary rankings [@niu2016ordinal], which we address in this paper with a new method and theorem for guaranteed classifier consistency that can easily be implemented in various machine learning algorithms. Furthermore, we present an empirical study of our approach on challenging real-world datasets for predicting the age of individuals from face images using our method with convolutional neural networks (CNN).
Aging can be regarded as a non-stationary process, since age progression in early childhood is primarily associated with changes in the shape of the face whereas aging during adulthood is largely defined by changes in skin texture. Thus, many traditional approaches for age estimation employed ordinal regression [@yang2010ranking; @chang2011ordinal; @cao2012human; @li2012learning]. In addition to traditional machine learning algorithms that require manual feature extraction, CNNs were proposed that conduct both feature learning and ordinal regression to estimate the apparent age [@niu2016ordinal]. However, existing CNNs for ordinal regression approaches still suffer from classifier inconsistencies, which we aim to address in this work.
The main contributions of our paper are as follows:
1. the Consistent Rank Logits (CORAL) framework for ordinal regression with theoretical guarantees for classifier consistency and well-defined generalization bounds with and without dataset- and task-specific importance weighting;
2. CNN architectures with CORAL formulation for ordinal regression tasks that come with the added side benefit of reducing the number of parameters to be trained compared to CNNs for classification;
3. experiments showing a substantial improvement of the CORAL method with guaranteed classifier consistency over the state-of-the-art CNN for ordinal regression applied to age estimation from face images.
The remainder of this paper is organized as follows: Section \[sec:related-work\] provides a concise overview of the related work. We explain the theory behind the CORAL framework in Section \[sec:proposed\], followed by the CNN architecture implementation. Section \[sec:experiments\] describes the experimental setup of the empirical validation. In Section \[sec:results\], we provide a description and summary of the experimental results. Finally, Section \[sec:conclusions\] concludes the paper with a summary and outlook.
Related Work {#sec:related-work}
============
Ordinal Regression and Ranking
------------------------------
Several multivariate extensions of generalized linear models have been developed in the past for ordinal regression, including the popular proportional odds and the proportional hazards models [@mccullagh1980regression]. Moreover, ordinal regression has become a popular topic of study in the field of machine learning to extend classification algorithms by reformulating the problem to utilize multiple binary classification tasks. Early work in this regard includes the use of perceptrons [@crammer2002pranking; @shen2005ranking] and support vector machines [@herbrich1999support; @shashua2003ranking; @rajaram2003classification; @chu2005new]. A general reduction framework that unified the view of a number of these existing algorithms for ordinal regression was later proposed by Li and Lin [@li2007ordinal].
While earlier works on using CNNs for ordinal targets have employed conventional classification approaches [@levi2015age; @rothe2015dex], the general reduction framework from ordinal regression to binary classification by Li and Lin [@li2007ordinal] was recently adopted by Niu et al. [@niu2016ordinal]. In [@niu2016ordinal], an ordinal regression problem with $K$ ranks was transformed into $K-1$ binary classification problems, with the $k$th task predicting whether the age label of a face image exceeds rank $r_k$, ${k=1,...,K-1}$. All $K-1$ tasks share the same intermediate layers but are assigned distinct weight parameters in the output layer. However, this approach does not guarantee that the predictions are consistent such that predictions for individual binary tasks may disagree. For example, in an age estimation setting, it would be contradictory if the $k$th binary task predicted that the age of a person was larger than 30, but a previous task predicted the person’s age was smaller than 20, which is suboptimal when the $K-1$ task predictions are combined to obtain the estimated age.
While the ordinal regression CNN yielded state-of-the-art results on several age estimation datasets, the authors acknowledged the classifier inconsistency as not being ideal but also noted that ensuring that the $K-1$ binary classifiers are consistent would increase the training complexity substantially [@niu2016ordinal]. Our proposed method addresses both of these issues with a theoretical guarantee for classifier consistency without increasing the training complexity.
{width="0.98\linewidth"}
CNN Architectures for Age Estimation
------------------------------------
Due to its broad utility in social networking, video surveillance, and biometric verification, age estimation from human faces is an active area of research. Likely owed to the rapid advancements in computer vision based on deep learning, most state-of-the-art age estimation methods are now utilizing CNN architectures [@rothe2015dex; @chen2016cascaded; @niu2016ordinal; @ranjan2017all; @chen2017using].
Related to the idea of training binary classifiers separately and combining the independent predictions for ranking [@frank2001simple], a modification of the ordinal regression CNN [@niu2016ordinal] was recently proposed for age estimation, called Ranking-CNN, that trains an ensemble of CNNs for binary classifications and aggregates the predictions to predict the age label of a given face image [@chen2017using]. The researchers showed that training a series of CNNs improves the predictive performance over a single CNN with multiple binary outputs. However, ensembles of CNNs come with a substantial increase in training complexity and do not guarantee classifier consistency, which means that the individual binary classifiers used for ranking can produce contradictory results. Another approach for utilizing binary classifiers for ordinal regression is the siamese CNN architecture by Polania et al. [@polania2018ordinal]. Since this siamese CNN has only a single output neuron, comparisons between the input image and multiple, carefully selected anchor images are required to compute the rank.
Recent research has also shown that training a multi-task CNN for various face analysis tasks, including face detection, gender prediction, age estimation, etc., can improve the overall performance across different tasks compared to a single-task CNN [@ranjan2017all] by sharing lower-layer parameters. In [@chen2016cascaded], a cascaded convolutional neural network was designed to classify face images into age groups followed by regression modules for more accurate age estimation. In both studies, the authors used metric regression for the age estimation subtasks. While our paper focuses on the comparison of different ordinal regression approaches, we hypothesize that such all-in-one and cascaded CNNs can be further improved by our method, since, as shown in [@niu2016ordinal], ordinal regression CNNs outperform metric regression CNNs in age estimation tasks.
Proposed Method {#sec:proposed}
===============
This section describes the proposed CORAL framework that addresses the problem of classifier inconsistency in ordinal regression CNNs based on multiple binary classification tasks for ranking.
Preliminaries
-------------
Let ${D=\{\mathbf{x}_i,y_i\}_{i=1}^N}$ be the training dataset consisting of $N$ examples. Here, $\mathbf{x}_i\in \mathcal{X}$ denotes the $i$th image and $y_i$ denotes the corresponding rank, where $y_i\in \mathcal{Y}=\{r_1,r_2,...r_K\}$ with ordered rank $r_K\succ r_{K-1}\succ \ldots\succ r_1$. The symbol $\succ$ denotes the ordering between the ranks. The ordinal regression task is to find a ranking rule $h: \mathcal{X}\rightarrow \mathcal{Y}$ such that some loss function $L(h)$ is minimized.
Let $\mathcal{C}$ be a $K\times K$ *cost matrix* [@li2007ordinal], where $\mathcal{C}_{y,r_k}$ is the cost of predicting an example $(\mathbf{x},y)$ as rank $r_k$. Typically, $\mathcal{C}_{y,y}=0$ and $\mathcal{C}_{y,r_k}>0$ for $y\neq r_k$. In ordinal regression, we generally prefer each row of the cost matrix to be *V-shaped*. That is ${\mathcal{C}_{y,r_{k-1}}\geq \mathcal{C}_{y,r_k}}$ if $r_{k}\leq y$ and ${\mathcal{C}_{y,r_k}\leq \mathcal{C}_{y,r_{k+1}}}$ if $r_{k}\geq y$. The *classification cost matrix* has entries ${\mathcal{C}_{y,r_k}=\mathbbm{1}\{y\neq r_k\}}$, which does not consider ordering information. In ordinal regression, where the ranks are treated as numerical values, the *absolute cost matrix* is commonly defined by ${\mathcal{C}_{y,r_k}=|y-r_k|}$.
In [@li2007ordinal], the researchers proposed a general reduction framework for extending an ordinal regression problem into several binary classification problems. This framework requires the use of a cost matrix that is convex in each row (${\mathcal{C}_{y,r_{k+1}}-\mathcal{C}_{y,r_k}\geq \mathcal{C}_{y,r_{k}}-\mathcal{C}_{y,r_{k-1}}}$ for each $y$) to obtain a rank-monotonic threshold model. Since the cost-related weighting of each binary task is specific for each training example, this approach was described as infeasible in practice due to its high training complexity [@niu2016ordinal]. Our proposed CORAL framework does neither require a cost matrix with convex-row conditions nor explicit weighting terms that depend on each training example to obtain a rank-monotonic threshold model and to produce consistent predictions for each binary task. Moreover, CORAL allows for an optional task importance weighting. The optional assignment of non-uniform task importance weights, for example, may be used to address label imbalances (Section \[sec:task-importance\]), which makes the CORAL framework more applicable to real-world datasets.
Ordinal Regression with a Consistent Rank Logits Model
------------------------------------------------------
We propose the Consistent Rank Logits (CORAL) model for multi-label CNNs with ordinal responses. Within this framework, the binary tasks produce consistently ranked predictions. The following two subsections describe the label extension into binary tasks performed during training as well as the loss function for parameterizing the neural network to predict ordinal labels.
#### Label Extension and Rank Prediction.
Given the training dataset $D=\{\mathbf{x}_i,y_i\}_{i=1}^N$, we first extend a rank label $y_i$ into $K-1$ binary labels $y_i^{(1)},\ldots,y_i^{(K-1)}$ such that $y_i^{(k)} \in \{0,1\}$ indicates whether $y_i$ exceeds rank $r_k$, i.e., $y_i^{(k)}=\mathbbm{1}\{y_i>r_k\}$. The indicator function $\mathbbm{1}\{\cdot\}$ is $1$ if the inner condition is true, and $0$ otherwise. Providing the extended binary labels as model inputs, we train a single CNN with $K-1$ binary classifiers in the output layer. Here, the $K-1$ binary tasks share the same weight parameter but have independent bias units, which solves the inconsistency problem among the predicted binary responses and reduces the model complexity (Figure \[fig:resnet\]).
Based on the binary task responses, the predicted rank for an input $\mathbf{x}_i$ is then obtained via $$\begin{aligned}
\label{eq:predicted-label}
& h(\mathbf{x}_i)=r_q, \\[7pt]
\vspace{0.5cm}
& {q = 1 + \sum_{k=1}^{K-1} f_k(\mathbf{x}_i)},\end{aligned}$$ where $f_k(\mathbf{x}_i)\in \{0,1\}$ is the prediction of the $k$th binary classifier in the output layer. We require that $\{f_k\}_{k=1}^{K-1}$ reflect the ordinal information and are *rank-monotonic*,
$$f_1(\mathbf{x}_i)\geq f_2(\mathbf{x}_i) \geq \ldots, f_{K-1}(\mathbf{x}_i),$$
which guarantees that the predictions are consistent.
#### Loss Function.
Let $\mathbf{W}$ denote the weight parameters of the neural network excluding the bias units of the final layer. The penultimate layer, whose output is denoted as $g(\mathbf{x}_i, \mathbf{W})$, shares a single weight with all nodes in the final output layer. $K-1$ independent bias units are then added to $g(\mathbf{x}_i,\mathbf{W})$ such that ${\{g(\mathbf{x}_i,\mathbf{W})+b_k\}_{k=1}^{K-1}}$ are the inputs to the corresponding binary classifiers in the final layer. Let $s(z)=1/(1+\exp(-z))$ be the logistic sigmoid function. The predicted empirical probability for task $k$ is defined as
(y\_i\^[(k)]{}=1)=s(g(\_i,)+b\_k).
For model training, we minimize the loss function
\[eq:loss\_fun\] L(,)=\
- \_[i=1]{}\^N\_[k=1]{}\^[K-1]{} \^[(k)]{} ,
which is the weighted cross-entropy of $K-1$ binary classifiers. For rank prediction (Eq. \[eq:predicted-label\]), the binary labels are obtained via
\[eq:proba-to-binary\]
f\_k(\_i) ={(y\_i\^[(k)]{}=1)>0.5}.
In Eq. , $\lambda^{(k)}$ denotes the weight of the loss associated with the $k$th classifier (assuming $\lambda^{(k)}>0$). In the remainder of the paper, we refer to $\lambda^{(k)}$ as the importance parameter for task $k$. Some tasks may be less robust or harder to optimize, which can be taken into consideration by choosing a non-uniform task weighting scheme. The choice of task importance parameters is covered in more detail in Section \[sec:task-importance\]. Next, we provide a theoretical guarantee for classifier consistency under uniform and non-uniform task importance weighting given that the task importance weights are positive numbers.
Theoretical Guarantees for Classifier Consistency {#sec:theoretical-guarantees}
-------------------------------------------------
In the following theorem, we show that by minimizing the loss $L$ (Eq. \[eq:loss\_fun\]), the learned bias units of the output layer are non-increasing such that ${b_1\geq b_2\geq \ldots \geq b_{K-1}}$. Consequently, the predicted confidence scores or probability estimates of the $K-1$ tasks are decreasing, i.e.,
$$\widehat{P}\left(y_i^{(1)}=1\right)\geq \widehat{P}\left(y_i^{(2)}=1\right) \geq \ldots \geq \widehat{P}\left(y_i^{(K-1)}=1\right)$$
for all $i$, ensuring classifier consistency. Consequently, $\{f_k\}_{k=1}^{K-1}$ given by Eq. \[eq:proba-to-binary\] are also rank-monotonic.\
\[th:ordered\_thres\] By minimizing loss function defined in Eq. , the optimal solution $(\mathbf{W}^*,\mathbf{b}^*)$ satisfies $b_1^*\geq b_2^* \geq \ldots \geq b_{K-1}^*$.
Suppose $(\mathbf{W},b)$ is an optimal solution and [$b_k < b_{k+1}$]{} for some $k$. Claim: by either replacing $b_k$ with $b_{k+1}$ or replacing $b_{k+1}$ with $b_k$, we can decrease the objective value $L$. Let $$\begin{aligned}
& A_1 =\{n: y_n^{(k)}=y_n^{(k+1)}=1\},
\\
& A_2=\{n: y_n^{(k)} = y_n^{(k+1)}=0\}, \\
& A_3 =\{n: y_n^{(k)}=1, \, y_n^{(k+1)}=0\}.\end{aligned}$$ By the ordering relationship we have $$A_1\cup A_2 \cup A_3= \{1,2,\ldots,N\}.$$ Denote $p_n(b_k)= s(g(\mathbf{x}_n,\mathbf{W})+b_k)$ and $$\begin{aligned}
& \delta_{n} = \log(p_n(b_{k+1}))-\log(p_n(b_k)), \\
& \delta\,'_{n} = \log(1-p_n(b_{k}))-\log(1-p_n(b_{k+1})).\end{aligned}$$ Since $p_n(b_k)$ is increasing in $b_k$, we have $\delta_{n}>0$ and $\delta\,'_n>0$.
If we replace $b_k$ with $b_{k+1}$, the loss terms related to $k$th task are updated. The change of loss $L$ (Eq. \[eq:loss\_fun\]) is given as $$\Delta_1 L = \lambda^{(k)}\Big[- \sum_{n\in A_1} \delta_{n} + \sum_{n\in A_2} \delta\,'_n - \sum_{n\in A_3} \delta_{n}\Big].$$
Accordingly, if we replace $b_{k+1}$ with $b_k$, the change of $L$ is given as $$\Delta_2 L = \lambda^{(k+1)}\Big[\sum_{n\in A_1}\delta_{n} - \sum_{n\in A_2} \delta\,'_n - \sum_{n\in A_3} \delta\,'_n \Big].$$
By adding $\frac{1}{\lambda^{(k)}}\Delta_1 L$ and $\frac{1}{\lambda^{(k+1)}}\Delta_2 L$, we have $$\frac{1}{\lambda^{(k)}}\Delta_1 L+ \frac{1}{\lambda^{(k+1)}}\Delta_2 L
= -\sum_{n\in A_3} (\delta_n + \delta\,'_n)<0,$$ and know that either $\Delta_1 L<0$ or $\Delta_2 L <0$. Thus, our claim is justified, and we conclude that any optimal solution $(\mathbf{W}^*,b^*)$ that minimizes $L$ satisfies $$b_1^*\geq b_2^* \geq \ldots \geq b_{K-1}^*.$$
Note that the theorem for rank-monotonicity in [@li2007ordinal], in contrast to Theorem \[th:ordered\_thres\], requires the use of a cost matrix $\mathcal{C}$ with each row $y_n$ being convex. Under this convexity condition, let $\lambda_{y_n}^{(k)}=|\mathcal{C}_{y_n,r_k}-\mathcal{C}_{y_n,r_{k+1}}|$ be the weight of the loss associated with the $k$th task on the $n$th example, which depends on the label $y_n$. In [@li2007ordinal], the researchers proved that by using example-specific task weights $\lambda_{y_n}^{(k)}$, the optimal thresholds are ordered. This assumption requires that $\lambda_{y_n}^{(k)}\geq \lambda_{y_n}^{(k+1)}$ when $r_{k+1}< y_n$, and $\lambda_{y_n}^{(k)}\leq \lambda_{y_n}^{(k+1)}$ when $r_{k+1}>y_n$. Theorem \[th:ordered\_thres\] is free from this requirement and allows us to choose a fixed weight for each task that does not depend on the individual training examples, which greatly reduces the training complexity. Moreover, Theorem \[th:ordered\_thres\] allows for choosing either a simple uniform task weighting or taking dataset imbalances into account (Section \[sec:task-importance\]) while still guaranteeing that the predicted probabilities are non-decreasing and the task predictions are consistent.
Generalization Bounds
---------------------
Based on well-known generalization bounds for binary classification, we can derive new generalization bounds for our ordinal regression approach that apply to a wide range of practical scenarios as we only require $C_{y,r_k} = 0 \text{ if } r_k=y$ and $C_{y,r_k} > 0 \text{ if } r_k \neq y$. Moreover, Theorem \[th:gener-error\] shows that if each binary classification task in our model generalizes well in terms of the standard 0/1-loss, the final rank prediction via $h$ (Eq. \[eq:predicted-label\]) also generalizes well.
\[th:gener-error\] Suppose $\mathcal{C}$ is the cost matrix of the original ordinal label prediction problem, with $\mathcal{C}_{y,y}=0$ and $\mathcal{C}_{y,r_k}>0$ for $k\neq y$. $P$ is the underlying distribution of $(\mathbf{x},y)$, i.e., $(\mathbf{x},y)\sim P$. If the binary classification rules $\{f_k\}_{k=1}^{K-1}$ obtained by optimizing Eq. \[eq:loss\_fun\] are rank-monotonic, then
$$\label{eq:gen-bound1}
\resizebox{.99\hsize}{!}{$\underset{(\mathbf{x},y)\sim P}{\mathbb{E}}\mathcal{C}_{y,h(\mathbf{x})}
\leq
\sum_{k=1}^{K-1}\big|\mathcal{C}_{y,r_k}-\mathcal{C}_{y,r_{k+1}}\big| \underset{(\mathbf{x},y)\sim P}{\mathbb{E}}\mathbbm{1}\{f_k(\mathbf{x})\neq y^{(k)}\}$}.$$
For any $\mathbf{x}\in \mathcal{X}$, we have $$f_1(\mathbf{x})\geq f_2(\mathbf{x}) \geq \ldots \geq f_{K-1}(\mathbf{x}).$$
If $h(\mathbf{x})=y$, then $\mathcal{C}_{y,h(\mathbf{x})}=0$.\
If $h(\mathbf{x})=r_q\prec y=r_s$, then $q<s$. We have $$f_1(\mathbf{x})=f_2(\mathbf{x})=\ldots=f_{q-1}(x)=1$$ and $$f_q(\mathbf{x})=f_{q+1}(\mathbf{x})=\ldots=f_{K-1}(\mathbf{x})=0.$$ Also, $$y^{(1)}=y^{(2)}=\ldots=y^{(s-1)}=1$$ and $$y^{(s)}=y^{(s+1)}=\ldots=y^{(K-1)}=0.$$ Thus, $\mathbbm{1}\{f_k(\mathbf{x})\neq y^{(k)}\}=1$ if and only if $q\leq k\leq s-1$. Since $\mathcal{C}_{y,y}=0,$
$$\begin{aligned}
\mathcal{C}_{y,h(\mathbf{x})} & = \sum_{k=q}^{s-1}(\mathcal{C}_{y,r_k}-\mathcal{C}_{y,r_{k+1}})\cdot \mathbbm{1}\{f_k(\mathbf{x})\neq y^{(k)}\} \\
& \leq \sum_{k=q}^{s-1}\big|\mathcal{C}_{y,r_k}-\mathcal{C}_{y,r_{k+1}}\big|\cdot \mathbbm{1}\{f_k(\mathbf{x})\neq y^{(k)}\} \\
& \leq \sum_{k=1}^{K-1}\big|\mathcal{C}_{y,r_k}-\mathcal{C}_{y,r_{k+1}}\big|\cdot \mathbbm{1}\{f_k(\mathbf{x})\neq y^{(k)}\}.\end{aligned}$$
Similarly, if $h(x)=r_q\succ y=r_s$, then $q>s$ and $$\begin{aligned}
\mathcal{C}_{y,h(\mathbf{x})} &= \sum_{k=s}^{q-1}(\mathcal{C}_{y,r_{k+1}}-\mathcal{C}_{y,r_{k}})\cdot \mathbbm{1}\{f_k(\mathbf{x})\neq y^{(k)}\} \\
& \leq \sum_{k=1}^{K-1}\big|\mathcal{C}_{y,r_{k+1}}-\mathcal{C}_{y,r_{k}}\big|\cdot \mathbbm{1}\{f_k(\mathbf{x})\neq y^{(k)}\}.\end{aligned}$$ In any case, we have $$\mathcal{C}_{y,h(\mathbf{x})}\leq \sum_{k=1}^{K-1}\big|\mathcal{C}_{y,r_k}-\mathcal{C}_{y,r_{k+1}}\big|\cdot \mathbbm{1}\{f_k(\mathbf{x})=y^{(k)}\}.$$ By taking the expectation on both sides with $(\mathbf{x},y)\sim P$, we arrive at Eq. .
In [@li2007ordinal], by assuming the cost matrix to have V-shaped rows, the researchers define generalization bounds by constructing a discrete distribution on $\{1,2,\ldots,K-1\}$ conditional on each $y$, given that the binary classifications are rank-monotonic or every row of $\mathcal{C}$ is convex. However, the only case they provided for the existence of rank-monotonic binary classifiers was the ordered threshold model, which requires a cost matrix with convex rows and example-specific task weights. In other words, when the cost matrix is only V-shaped but does not meet the convex row condition, i.e., $\mathcal{C}_{y,r_k}-\mathcal{C}_{y,r_{k-1}}>\mathcal{C}_{y,r_{k+1}}-\mathcal{C}_{y,r_k}>0$ for some $r_k>y$, the method proposed in [@li2007ordinal] did not provide a practical way to bound the generalization error. Consequently, our result does not rely on cost matrices with V-shaped or convex rows and can be applied to a broader variety of real-world use cases.
Task Importance Weighting {#sec:task-importance}
-------------------------
![Example of the task importance weighting according to Eq. shown for the AFAD dataset (Section \[sec:datasets\]).[]{data-label="fig:afad-importance"}](figures/afad-taskimp){width="\columnwidth"}
According to Theorem \[th:ordered\_thres\], minimizing the loss of the CORAL model guarantees that the bias units are non-increasing and thus the binary classifiers are consistent as long as the task importance parameters are positive: ${\forall k \in \{1, ..., K-1\}: \lambda^{(k)} > 0}$.
In many real-world applications, features between certain adjacent ranks may have more subtle distinctions. For example, facial aging is commonly regarded as a non-stationary process [@ramanathan2009age] such that face feature transformations could be more detectable during certain age intervals. Moreover, the relative predictive performance of the binary tasks may also be affected by the degree of binary data imbalance for a given task that occurs as a side-effect of extending a rank label into $K-1$ binary labels. Hence, we hypothesize that choosing non-uniform task weighting schemes improves the predictive performance of the overall model.
We first experimented with a weighting scheme proposed in [@niu2016ordinal] that aims to address the class imbalance in the face image datasets. However, compared to using a uniform scheme ($\forall k \in \{1, ..., K-1\}: \lambda^{(k)} = 1$), we found that it had a negative effect on the predictive performance of all models evaluated in this study.
Hence, we propose a weighting scheme that takes the rank distribution of the training examples into account but also considers the label imbalance for each classification task after extending the original ranks into binary labels. Note that CORAL, according to Theorem \[th:ordered\_thres\], guarantees that the predicted probabilities are non-decreasing and the task predictions are consistent as long as the task importance weights are non-negative as described in Section \[sec:theoretical-guarantees\].
Specifically, our task weighting scheme is defined as follows. Let $S_k = \sum_{i=1}^N \mathbbm{1}\{y_i^{(k)}=1\}$ be the number of examples whose ranks exceed $r_k$. By the rank ordering we have $S_1\geq S_2\geq \ldots \geq S_{K-1}$. Let $M_k = \max(S_k, N-S_k)$ be the number of majority binary label for each task. We define the importance of the $k$th task as the scaled $\sqrt{M_k}$:
\[eq:our-task-importance\] \^[(k)]{} = .
Under this weighting scheme, the general class imbalance of a dataset is taken into account. Moreover, in our examples, classification tasks corresponding to the edges of the distribution of unique rank labels receive a higher weight than the classification tasks that see more balanced rank label vectors during training, which may help improve the predictive performance of the model. The lowest weight may not always be assigned to the center-rank: if $S_{K-1}>0.5$, the last task has the lowest weight, and if $S_1<0.5$, the first task has the lowest weight. An example of an importance weight distribution is shown in Figure \[fig:afad-importance\].
It shall be noted that the task importance weighting is only used for model parameter optimization; when computing the predicted rank by adding the binary results (Eq. \[eq:predicted-label\]), each task has the same influence on the final rank prediction. Since $\lambda^{(k)}>\frac{\sqrt{M_k}}{\sqrt{N}}>\frac{M_k}{N}\geq 0.5$, it prevents tasks from having negligible weights as in [@niu2016ordinal] when a dataset contains only a small number of examples for certain ranks. We provide an empirical comparison between a uniform task weighting and task weighting according to Eq. in Section \[sec:weighting\].
Experiments {#sec:experiments}
===========
Datasets and Preprocessing {#sec:datasets}
--------------------------
The MORPH-2 dataset [@ricanek2006morph] (55,608 face images; <https://www.faceaginggroup.com/morph/>) was preprocessed by locating the average eye-position in the respective dataset using facial landmark detection [@sagonas2016300] via MLxtend v0.14 [@raschka2018mlxtend] and then aligning each image in the dataset to the average eye position. The faces were then re-aligned such that the tip of the nose was located in the center of each image. The age labels used in this study ranged between 16-70 years. The CACD database [@chen2014cross] was preprocessed similar to MORPH-2 such that the faces spanned the whole image with the nose tip being in the center. The total number of images is 159,449 in the age range 14-62 years (<http://bcsiriuschen.github.io/CARC/>). For both the Asian Face Database [@niu2016ordinal] (AFAD; 165,501 faces, age labels 15-40 years; <https://github.com/afad-dataset/tarball>) and the UTKFace database [@zhifei2017cvpr] (16,434 images, age labels 21-60 years; <https://susanqq.github.io/UTKFace/>) centered images were already provided.
Each image database was randomly divided into 80% training data and 20% test data. All images were resized to 128$\times$128$\times$3 pixels and then randomly cropped to 120$\times$120$\times$3 pixels to augment the model training. During model evaluation, the 128$\times$128$\times$3 face images were center-cropped to a model input size of 120$\times$120$\times$3. The training and test partitions for all datasets, along with all preprocessing code used in this paper, are available at <https://github.com/Raschka-research-group/coral-cnn/tree/master/datasets>.
Convolutional Neural Network Architectures {#sec:architecture}
------------------------------------------
To evaluate the performance of CORAL for age estimation from face images, we chose the ResNet-34 architecture [@he2016deep], which is a modern CNN architecture that achieves good performance on a variety of image classification tasks. For the remainder of this paper, we refer to the original ResNet-34 CNN with cross entropy loss as CE-CNN. To implement CORAL, we replaced the last output layer with the corresponding binary tasks (Figure \[fig:resnet\]) and refer to this CNN as [CORAL-CNN]{}. Similar to [CORAL-CNN]{}, we replaced the cross-entropy layer of the ResNet-34 with the binary tasks for ordinal regression described in [@niu2016ordinal] and refer to this architecture as [OR-CNN]{}.
Note that next to guaranteed rank-consistency, another advantage of CORAL-CNN method over OR-CNN is the reduction of parameters in the output layer. Suppose that there are $m$ output nodes in the last fully connected layer, which is connected to the output layer. In [@niu2016ordinal], the output layer consists of $(K-1)\times 2$ output nodes: there are $K-1$ binary classification tasks with 2 neurons each. Thus, the number of parameters in the final layer is $(m+1) \times (K-1) \times 2$. The output layer of the CORAL-network, however, uses one neuron for each task and the weights are shared among all $(K-1)$ tasks. Hence, the number of parameters is $m+K-1$, such that the CORAL-CNN has a substantially lower training complexity as the ORDINAL-CNN. For example, CORAL-CNN has 219,190 fewer parameters than OR-CNN in the case of ResNet-34 (Figure \[fig:resnet\]) and MORPH-2, where $K=55$ and $m=2048$.
Training and Evaluation
-----------------------
{width="85.00000%"}
For model evaluation and comparison, we computed the mean absolute error (MAE) and root mean squared error (RMSE), which are standard metrics used for age prediction, on the test set after the last training epoch:
$$\begin{aligned}
\text{MAE} &= \frac{1}{N}\sum_{i=1}^{N} \big|y_i - h(\mathbf{x}_i)\big|,
\\ \text{RMSE} &= \sqrt{\frac{1}{N}\sum_{i=1}^{N} \big(y_i - h(\mathbf{x}_i)\big)^2},\end{aligned}$$
where $y_i$ is the ground truth rank of the $i$th test example and $h(\mathbf{x}_i)$ is the predicted rank, respectively.
In addition, we computed the Cumulative Score (CS) as the proportion of images for which the absolute differences between the predicted rank labels and the ground truth are below a threshold $T$:
$$\begin{aligned}
\text{CS}(T) = \frac{1}{N}\sum_{i=1}^N 1\big{\{}|y_i - h(\mathbf{x}_i)| \leq T \big{\}}.\end{aligned}$$
By varying the threshold $T$, CS curves were plotted to compare the predictive performances of the different age prediction models (the larger the area under the curve, the better).
The model training was repeated three times with different random seeds for model weight initialization while the random seeds were consistent between the different methods to allow for fair comparisons. All CNNs were trained for 200 epochs with stochastic gradient descent via adaptive moment estimation using exponential decay rates [@kingma2015adam] $\beta_0=0.90$ and $\beta_2=0.99$ (default settings). To avoid introducing empirical bias by designing our own CNN architecture for comparing the ordinal regression approaches, we adopted a standard architecture for this comparison, namely, ResNet-34 [@he2016deep].
Exploring different learning rates for the different losses (cross-entropy, ordinal regression CNN [@niu2016ordinal], and the CORAL approach), we found that a learning rate of $\alpha=5 \times 10^{-5}$ performed best across all models, which is likely due to the similar base architecture (ResNet-34). Also, for all models, the loss converged after 200 epochs.
Comparisons with additional neural network architectures, Inception-v3 [@szegedy2016rethinking] and VGG-16 [@simonyan2014very], are included in the Supplementary Materials.
Hardware and Software
---------------------
All loss functions and neural network models were imple- mented in PyTorch 1.1.0 [@paszke2017automatic] and trained on NVIDIA GeForce 1080Ti and Titan V graphics cards. The source code is available at <https://github.com/Raschka-research-group/coral-cnn>.
Results and Discussion {#sec:results}
======================
We conducted a series of experiments on four independent face image datasets for age estimation (Section \[sec:datasets\]) to compare our CORAL approach (CORAL-CNN) with the ordinal regression approach described in [@niu2016ordinal], denoted as OR-CNN. All implementations were based on the ResNet-34 architecture as described in Section \[sec:architecture\], including the standard ResNet-34 with cross-entropy loss (CE-CNN) as performance baseline.
-- --- ------ ------ ------ ------ ------ ------ ------ ------
MAE MAE MAE
0 3.40 4.88 3.98 5.55 6.57 9.16 6.18 8.86
1 3.39 4.87 4.00 5.57 6.24 8.69 6.10 8.79
2 3.37 4.87 3.96 5.50 6.29 8.78 6.13 8.87
0 2.98 4.26 3.66 5.10 5.71 8.11 5.53 7.91
1 2.98 4.26 3.69 5.13 5.80 8.12 5.53 7.98
2 2.96 4.20 3.68 5.14 5.71 8.11 5.49 7.89
0 2.68 3.75 3.49 4.82 5.46 7.61 5.56 7.80
1 2.63 3.66 3.46 4.83 5.46 7.63 5.37 7.64
2 2.61 3.64 3.52 4.91 5.48 7.63 5.25 7.53
-- --- ------ ------ ------ ------ ------ ------ ------ ------
\[tab:all-results\]
Estimating the Apparent Age from Face Images
--------------------------------------------
Across all datasets (Table \[tab:all-results\]), we found that both OR-CNN and CORAL-CNN outperform the standard cross-entropy loss (CE-CNN) on these ordinal regression tasks as expected. Similarly, as summarized in Table \[tab:all-results\] and Figure \[fig:cs-curves\], our CORAL method shows a substantial improvement over OR-CNN [@niu2016ordinal], which does not guarantee classifier consistency. Moreover, we repeated each experiment three times using different random seeds for model weight initialization and dataset shuffling, to ensure that the observed performance improvement of CORAL-CNN over OR-CNN is reproducible and not coincidental. We may conclude that guaranteed classifier consistency via CORAL has a substantial, positive effect on the predictive performance of an ordinal regression CNN (a more detailed analysis regarding the rank inconsistency by Niu et al’s OR-CNN is provided in Section \[sec:inconsistencies\]).
Furthermore, it can be observed that for all methods, the overall predictive performance on the different datasets appears in the following order: MORPH-2 $>$ AFAD $>$ CACD $>$ UTKFace (Table \[tab:all-results\] and Figure \[fig:cs-curves\]). A possible explanation is that MORPH-2 has the best overall image quality and the photos were taking under relatively consistent lighting conditions and viewing angles. For instance, we found that AFAD includes some images of particularly low resolution (e.g., 20x20). While UTKFace and CACD also contain some lower-quality images, a possible reason why the methods perform worse on UTKFace compared to AFAD is that UTKFace is about ten times smaller than AFAD. Because CACD has approximately the same size as AFAD, the lower performance may be explained by the wider age range that needs to be considered (14-62 in CACD compared to 15-40 in AFAD).
Inconsistencies Inccurred by OR-CNN {#sec:inconsistencies}
-----------------------------------
This section analyzes the rank inconsistency issue of Niu et al.’s method [@niu2016ordinal] in more detail. Figure \[fig:inconsistency-plot\] shows an example of an inconsistent rank prediction for OR-CNN on a single image in the MORPH-2 test dataset.
{width="95.00000%"}
------------- ----------------- ------------------------------- ------------------------------- -------------------------------
CORAL-CNN Ordinal-CNN [@niu2016ordinal] Ordinal-CNN [@niu2016ordinal] Ordinal-CNN [@niu2016ordinal]
All predictions All predictions Only correct predictions Only incorrect predictions
**Morph**
Seed 0 0 2.74 2.02 2.89
Seed 1 0 2.74 2.08 2.88
Seed 2 0 3.00 2.20 3.16
**AFAD**
Seed 0 0 2.32 1.78 2.40
Seed 1 0 2.35 1.83 2.43
Seed 2 0 2.55 1.97 2.63
**UTKFace**
Seed 0 0 4.79 3.64 4.92
Seed 1 0 5.73 4.05 5.95
Seed 2 0 5.07 3.84 5.21
**CACD**
Seed 0 0 5.06 4.55 5.10
Seed 1 0 5.40 4.76 5.44
Seed 2 0 5.56 4.87 5.61
------------- ----------------- ------------------------------- ------------------------------- -------------------------------
[|l|c|c|c|c|c|]{} Method &
-------------------
Importance Weight
-------------------
& & & &\
------------------------------------------------------------------------
--------------------------
OR-CNN [@niu2016ordinal]
--------------------------
& NO & 2.97 $\pm$ 0.01 & 3.68 $\pm$ 0.02 & 5.74 $\pm$ 0.05 & 5.52 $\pm$ 0.02\
------------------------------------------------------------------------
--------------------------
OR-CNN [@niu2016ordinal]
--------------------------
& YES & 2.91 $\pm$ 0.02 & 3.65 $\pm$ 0.03 & 5.76 $\pm$ 0.19 & 5.49 $\pm$ 0.02\
------------------------------------------------------------------------
------------------
CORAL-CNN (ours)
------------------
& NO & 2.64 $\pm$ 0.04 & 3.49 $\pm$ 0.03 & 5.47 $\pm$ 0.01 & 5.39 $\pm$ 0.16\
------------------------------------------------------------------------
------------------
CORAL-CNN (ours)
------------------
& YES & **2.59** $\pm$ **0.03** **** & **3.48 $\pm$ 0.03** & **5.39 $\pm$ 0.07** & **5.35 $\pm$ 0.09**\
\[tab:task-imp\]
Table \[tab:inconsistency\] lists the average numbers of inconsistencies that were observed for the different test datasets predictions, where an inconsistency occurs if the predictions of the binary classification tasks are not rank-monotonic (as shown in the example in Figure \[fig:inconsistency-plot\]). As expected, due to the theoretical guarantees, no rank inconsistencies were observed for CORAL-CNN. The average number of rank inconsistencies inccurred by Ordinal-CNN is between 2.32 and 5.56, depending on the dataset.
When comparing the average number of rank inconsistencies that occur among the test predictions that predict the age labels correctly (Table \[tab:inconsistency\], penultimate column) to the number of inconsistencies among the incorrect predictions (Table \[tab:inconsistency\], last column), it can be seen that the ordinal regression method by Niu et al [@niu2016ordinal] has a smaller number of rank inconsistencies if it predicts the age label correctly. This observation suggests that the rank inconsistencies are detrimental to the predictive performance, and the better performance of CORAL-CNN compared to Ordinal-CNN (Table and Figure) may be achieved by eliminating the rank inconsistency issue.
Task Importance Weighting {#sec:weighting}
-------------------------
While all results described in the previous section are based on experiments without task importance weighting (i.e., $\forall k: \lambda^{(k)}=1$), we repeated all experiments using our weighting scheme proposed in Section \[sec:task-importance\], which takes label imbalances into account. Note that according to Theorem \[th:ordered\_thres\], CORAL still guarantees classifier consistency under any chosen task weighting scheme as long as weights are assigned positive values. From the results provided in Table \[tab:task-imp\], we find that by using a task weighting scheme that also takes label imbalances into account (Eq. \[eq:our-task-importance\]), we can further improve the performance of the CORAL-CNN models across all four datasets.
To test the hypothesis that the improvements in predictive performance via the task importance weighting are due to addressing the label imbalance, we conducted additional experiments using balanced datasets (Figure \[fig:balanced-tasks\]B). To balance the MORPH-2 and AFAD datasets (Figure \[fig:balanced-tasks\]A), we selected a smaller age range (18-39 years for AFAD and 18-45 years for MORPH-2) and randomly removed samples from ages larger than the age with the smallest number of examples. The task importance weight distributions for the imbalanced and balanced datasets are shown in Figure \[fig:balanced-tasks\].
![Age label distributions and task importance weights for the original, imbalanced datasets (A) and balanced datasets (B). []{data-label="fig:balanced-tasks"}](figures/balanced-data.pdf){width="50.00000%"}
As shown in Table \[tab:all-balanced-results\], both Niu et al.’s ordinal regression method [@niu2016ordinal] and CORAL perform equally well with and without optional task importance weighting if the datasets are balanced. Considering this observation in the context of the predictive performance improvements measured on imbalanced datasets (Table \[tab:task-imp\]), we conclude that the task importance weighting is an effective measure for working with imbalanced datasets.
[|c|c|c|c|c|c|c|]{} & & & &\
Methods &
------------
Importance
Weight
------------
& & & & &\
OR-CNN [@niu2016ordinal] & No & 0 & 2.95 & 4.12 & 3.82 & 5.20\
& No & 1 & 3.33 & 4.66 & 3.91 & 5.31\
& No & 2 & 2.98 & 4.17 & 3.83 & 5.19\
& No & 3 & 2.91 & 4.08 & 3.80 & 5.18\
& No & 4 & 3.09 & 4.40 & 3.83 & 5.24\
& No & 5 & 3.01 & 4.22 & 3.75 & 5.10\
& No & 6 & 2.87 & 4.08 & 3.78 & 5.13\
& No & 7 & 2.85 & 4.01 & 3.83 & 5.22\
& No & 8 & 2.93 & 4.12 & 3.76 & 5.08\
& No & 9 & 2.85 & 4.03 & 3.79 & 5.19\
& No & AVG $\pm$ SD & 2.98 $\pm$ 0.15 & 4.19 $\pm$ 0.20 & 3.81 $\pm$ 0.05 & 5.18 $\pm$ 0.07\
& & & & & &\
OR-CNN [@niu2016ordinal] & Yes & 0 & 2.86 & 4.02 & 3.77 & 5.13\
& Yes & 1 & 2.88 & 4.07 & 3.90 & 5.29\
& Yes & 2 & 2.94 & 4.14 & 3.77 & 5.08\
& Yes & 3 & 2.96 & 4.19 & 3.83 & 5.20\
& Yes & 4 & 3.06 & 4.28 & 3.83 & 5.19\
& Yes & 5 & 2.79 & 3.95 & 3.80 & 5.15\
& Yes & 6 & 2.81 & 3.96 & 3.90 & 5.27\
& Yes & 7 & 2.84 & 3.97 & 3.75 & 5.10\
& Yes & 8 & 3.46 & 4.75 & 3.74 & 5.07\
& Yes & 9 & 2.95 & 4.16 & 3.86 & 5.27\
& Yes & AVG $\pm$ SD & 2.96 $\pm$ 0.20 & 4.15 $\pm$ 0.24 & 3.82 $\pm$ 0.06 & 5.18 $\pm$ 0.08\
& & & & & &\
CORAL-CNN (ours) & No & 0 & 2.61 & 3.64 & 3.62 & 4.91\
& No & 1 & 2.58 & 3.57 & 3.60 & 4.98\
& No & 2 & 2.64 & 3.64 & 3.57 & 4.83\
& No & 3 & 2.68 & 3.70 & 3.63 & 4.90\
& No & 4 & 2.81 & 3.81 & 3.59 & 4.87\
& No & 5 & 2.58 & 3.57 & 3.59 & 4.86\
& No & 6 & 2.56 & 3.52 & 3.68 & 5.00\
& No & 7 & 2.67 & 3.70 & 3.61 & 4.88\
& No & 8 & 2.61 & 3.61 & 3.61 & 4.84\
& No & 9 & 2.64 & 3.67 & 3.56 & 4.83\
& No & AVG $\pm$ SD & **2.64** $\pm$ **0.08** & **3.64** $\pm$ **0.09** & 3.61 $\pm$ 0.04 & 4.89 $\pm$ 0.06\
& & & & & &\
CORAL-CNN (ours) & Yes & 0 & 2.62 & 3.61 & 3.59 & 4.86\
& Yes & 1 & 2.69 & 3.73 & 3.60 & 4.88\
& Yes & 2 & 2.62 & 3.58 & 3.54 & 4.80\
& Yes & 3 & 2.63 & 3.61 & 3.68 & 4.99\
& Yes & 4 & 2.62 & 3.62 & 3.60 & 4.87\
& Yes & 5 & 3.11 & 4.26 & 3.57 & 4.86\
& Yes & 6 & 2.60 & 3.61 & 3.60 & 4.90\
& Yes & 7 & 2.62 & 3.60 & 3.56 & 4.84\
& Yes & 8 & 2.66 & 3.68 & 3.63 & 4.92\
& Yes & 9 & 2.63 & 3.63 & 3.57 & 4.83\
& Yes & AVG $\pm$ SD & 2.68 $\pm$ 0.15 & 3.69 $\pm$ 0.20 & **3.59** $\pm$ **0.04** & **4.88** $\pm$ **0.05**\
& & & & & &\
Conclusions {#sec:conclusions}
===========
In this paper, we developed the CORAL framework for ordinal regression via extended binary classification with theoretical guarantees for classifier consistency. Moreover, we proved classifier consistency without requiring rank- or training label-dependent weighting schemes, which permits straightforward implementations and efficient model training. CORAL can be readily implemented to extend common CNN architectures for ordinal regression tasks. Applied to four independent age estimation datasets, the results unequivocally showed that the CORAL framework substantially improved the predictive performance of CNNs for age estimation. Our method can be readily generalized to other ordinal regression problems and different types of neural network architectures, including multilayer perceptrons and recurrent neural networks.
Acknowledgements
================
Support for this research was provided by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation. Also, we thank the NVIDIA Corporation for a generous donation via an NVIDIA GPU grant to support this study.
|
---
abstract: 'We apply the Velocity Distribution Function (VDF) to a sample of Sunyaev-Zel’dovich (SZ)-selected clusters, and we report preliminary cosmological constraints in the $\sigma_8$-$\Omega_m$ cosmological parameter space. The VDF is a forward-modeled test statistic that can be used to constrain cosmological models directly from galaxy cluster dynamical observations. The method was introduced in [@2017ApJ...835..106N] and employs line-of-sight velocity measurements to directly constrain cosmological parameters; it is less sensitive to measurement error than a standard halo mass function approach. The method is applied to the Hectospec Survey of Sunyaev-Zeldovich-Selected Clusters (<span style="font-variant:small-caps;">HeCS-SZ</span>) sample, which is a spectroscopic follow up of a [*Planck*]{}-selected sample of 83 galaxy clusters. Credible regions are calculated by comparing the VDF of the observed cluster sample to that of mock observations, yielding . These constraints are in tension with the [*Planck*]{}Cosmic Microwave Background (CMB) TT fiducial value, which lies outside of our 95% credible region, but are in agreement with some recent analyses of large scale structure that observe fewer massive clusters than are predicted by the [*Planck*]{}fiducial cosmological parameters.'
author:
- Michelle Ntampaka
- Ken Rines
- Hy Trac
bibliography:
- '../../references.bib'
title: 'Cluster Cosmology with the Velocity Distribution Function of the HeCS-SZ Sample '
---
Introduction {#sec:intro}
============
Galaxy clusters are massive, gravitationally bound collections of hundreds to thousands of galaxies. The standard cosmological model predicts that the abundance of clusters as a function of their mass and redshift depends sensitively on the underlying cosmological parameters. Because cluster counts depend on the cosmological model, cluster abundance measurements can be used to put constraints on cosmological parameters such as the amplitude of matter fluctuations, [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}, and the matter density parameter, [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}[e.g. @1998ApJ...504....1B]. More recently, forward-modeling approaches have been used to describe cluster abundance, not by the cluster counts as a function of mass and redshift, but by the distribution of direct observables . Forward-modeling approaches can minimize biases that would otherwise be introduced by mass measurement error and can provide a complementary and mass-independent way to evaluate the abundance of clusters.
Cluster abundance and other large scale structure (LSS) measurements can be used to constrain cosmological models, but some current LSS data are in tension with [*Planck*]{}CMB constraints of cosmological parameters. Notably, SZ surveys find fewer massive clusters than are predicted by the [*Planck*]{}CMB fiducial cosmology ; one interpretation of this is that the [*Planck*]{}SZ survey prefers a [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}lower than the CMB fiducial cosmology. While some LSS probes are consistent with [*Planck*]{}constraints [e.g. @2015MNRAS.446.2205M], others prefer a low [<span style="font-variant:small-caps;">$\sigma_8$</span>]{} [e.g. @2013MNRAS.432.2433H; @2017MNRAS.465.1454H]. In a rigorous analysis of WiggleZ, SDSS RSD, CFHTLenS, CMB lensing and SZ cluster count by [@2017arXiv170809813L], these LSS probes are found to be inconsistent with [*Planck*]{}CMB cosmological constraints. This tension may be due to unaccounted systematic errors or it may be indicative of more interesting physics that needs to be included in the model.
A number of explanations have been suggested to resolve this tension. X-ray cluster mass measurements under the assumption of hydrostatic equilibrium can incorporate nonthermal pressure support in the form of a bias parameter, $b$, that relates the SZ mass to the true cluster mass, $M_\mathrm{SZ}=(1-b)M_\mathrm{true}$. SZ cluster masses calibrated on X-ray observations must also take this bias factor into account. When this bias factor is allowed to be large, the tension between CMB and SZ parameters is significantly reduced; the mass bias required to bring cluster observations into agreement with the CMB anisotropy is $b\approx0.42$ . Adding a non-zero neutrino mass can also reduce the tension, but at the same time increases tension in other parameters predicted by [*Planck*]{}CMB and SZ. Notably, it lowers the [*Planck*]{}constraints for the Hubble constant . Though a number of explanations have been suggested to resolve this tension, the source of the disagreement is not yet agreed-upon within the community.
[Dynamical cluster mass estimates provide a complementary way to explore the LSS-CMB tension]{}. Dynamical mass estimates utilize the virial theorem, relating the LOS velocity dispersion of cluster members, $\sigma_v$, to cluster mass, $M$, as a power law. This approach famously led to [@zwicky1933rotverschiebung’s [-@zwicky1933rotverschiebung]]{} inference of dark matter in the Coma cluster. Hydrodynamical simulations find that the velocity dispersion of dark matter particles are similar to those of galaxies [e.g. @2010ApJ...708.1419L], though selecting only the brightest galaxies in a cluster tends to bias the galaxy velocity dispersion low [@2013ApJ...772...47S; @2013MNRAS.436..460W]. Because the dynamical mass method provides a relatively unbiased probe of cluster masses that is complementary to other cluster mass estimates, virial-theorem-based approaches are used in modern cluster mass estimates [e.g. @2010ApJ...721...90B; @2010ApJ...715L.180R; @2013ApJ...772...25S; @2014ApJ...792...45R; @2015ApJ...799..214B]. [Cluster counts as a function of velocity dispersion have also been proposed as a method for constraining cosmological parameters [@2016MNRAS.462.4117C].]{}
This work utilizes the Hectospec Survey of Sunyaev-Zeldovich-Selected Clusters [[<span style="font-variant:small-caps;">HeCS-SZ</span>]{}, @2016ApJ...819...63R]. The [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample is an MMT/Hectospec spectroscopic follow up of a sample of 83 clusters selected from [*Planck*]{}observations. [In their analysis of the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample, [@2016ApJ...819...63R] find that these clusters are dynamically colder than expected, having smaller velocity dispersions, $\sigma_v$, than is predicted by a [*Planck*]{}-selected sample of clusters for the [*Planck*]{}fiducial cosmology.]{} A mass bias together with a velocity bias of galaxies could explain the discrepancy. However, the biases required to remove the tension between CMB and cluster cosmological parameters would need to be large, in some cases, outside of the range presented in recent literature. [@2016ApJ...819...63R] conclude that another explanation may be necessary to resolve the tension.
[To explore the LSS-CMB tension, we utilize cluster line-of-sight velocity observations of the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample in conjunction with a forward modeling dynamical approach.]{} The cluster sample is analyzed using the Velocity Distribution Function (VDF), which was first introduced in [@2017ApJ...835..106N]. This method utilizes a forward-modeled test statistic that quantifies the abundance of clusters by measurements of their dynamics, through comparing summed [probability density functions (PDFs) of clusters’ measured line-of-sight (LOS) velocities to that of a mock catalog. ]{}
We present a summary of the cluster observations, the mock observations, and the VDF methodology in Section \[sec:methods\], results and constraints on [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}and [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}in Section \[sec:results\], and a discussion of the results in Section \[sec:discussion\].
Methods {#sec:methods}
=======
---------------------------------------- ----------------------------------------
{width="50.00000%"} {width="50.00000%"}
---------------------------------------- ----------------------------------------
HeCS-SZ {#sec:hecs}
-------
The Hectospec Survey of Sunyaev-Zeldovich-Selected Clusters [[<span style="font-variant:small-caps;">HeCS-SZ</span>]{}, @2016ApJ...819...63R] is an SZ-selected sample of clusters . [The full [*Planck*]{}-selected sample contains 87 clusters selected above the 80% completeness limit of the [*Planck*]{}medium-deep sky or the 50% completeness limit of the [*Planck*]{}shallow sky ;]{} [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}comprises 83 clusters selected randomly from these 87 clusters. The sample ranges from redshift [0.05]{} up to [0.3]{} and utilizes SDSS DR6 Legacy and SDSS DR10 imaging as well as MMT/Hectospec spectroscopic follow up of potential cluster members. [When two high-error galaxies are excluded, the remaining 25,112 galaxies in the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample have mean line of sight velocity error of [$\approx 30\,{\mathrm{km \, s^{-1}}}$]{}.]{}
The [cluster]{} selection function of the [[<span style="font-variant:small-caps;">HeCS-SZ</span>]{}]{} sample is shown in Figure \[fig:catcomp\]. As redshift increases, so does the minimum mass a cluster must have to be detected by [*Planck*]{}; this is due to the [*Planck*]{}beam size. At high $z$ and low $M$, the SZ signal is diluted by the large beam size, causing these clusters to have low signal to noise. [The selection function is discontinuous at $z=0.2$ because the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample utilizes observations from two SDSS data releases. The sample for $z<0.2$ utilizes observations from SDSS DR10 and has an effective sky area of $11589 \, {\mathrm{deg}^2\xspace}$ and a selection function that corresponds to [*Planck*]{}’s 80% completeness limit for the medium-deep sky. The sample for $0.2 < z < 0.3$ utilizes imaging observations from SDSS DR6 Legacy and spectroscopic observations from the Hectospec Cluster Survey [@2013ApJ...767...15R]. It has an effective sky area of $8417 \, {\mathrm{deg}^2\xspace}$ and the completeness limit jumps by 20% compared to the $z<0.2$ sample.]{}
Figure \[fig:catcomp\] shows the selection function of the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}observation, as well as the redshift and SZ-determined $M_{500}$ values of the 83 clusters in the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample. [The Planck medium-deep survey zone comprises 41.3% of the full sky, while the shallow survey zone comprises 56% (including the region around the Galactic Plane). Along the selection function shown in Figure \[fig:catcomp\], the likelihood of a cluster being detected by [*Planck*]{}is 80% for the medium-deep survey zone and 50% for the shallow survey zone. This likelihood increases with increased cluster SZ mass, $M_{500,\,\mathrm{SZ}}$, therefore clusters lying above the selection function are more likely to be detected (See for complete details of the cluster selection function, survey zones, and completeness). To account for the varying completeness in this range, we define an integral completeness, $$\mathcal{C}=\frac{N_\mathrm{observed}}{N_\mathrm{true}},
\label{eq:completeness}$$ the ratio of the number of [*Planck*]{}-detected clusters above the selection function to the true number of clusters in the sky and redshift regions of interest. We adopt a conservative flat prior of $\mathcal{C} \in [0.6, 1.0]$ on the integral completeness of the cluster sample.]{}
The [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}$M_{500}$ shown in this figure are calculated from the $Y_\mathrm{SZ}$ signal and reported by . Figure \[fig:catcomp\] shows results for two sample values of the mass bias $b$, which parameterizes the scaling between a cluster’s true mass and the observed $M_\mathrm{SZ}$ mass as $M_\mathrm{SZ} = (1-b) M_\mathrm{true}$. A bias parameter of $b=0.42$ is needed to bring SZ cluster masses into agreement with [*Planck*]{}CMB , and employing this bias effectively increases the $M_{500}$ values of the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}clusters reported on the vertical axis of Figure \[fig:catcomp\]. See [@2016ApJ...819...63R] for further details of the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}observations.
A simple cylindrical cut is used to select apparent cluster members, including both true cluster members and interloping field galaxies, from the full [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}observation. This cylindrical cut uses angular extent and velocity cuts to correspond with the size and velocity dispersion of a cluster with [$M_\mathrm{200c} \geq 1\times10^{15} {h^{-1} \, \mathrm{M_{\odot}}}$]{} at the [*Planck*]{}fiducial cosmology: a comoving [$1.6{h^{-1}\,\mathrm{Mpc}}$ outer aperture, initial velocity cut of $2500 \,{\mathrm{km \, s^{-1}}}$ about the cluster center and no $\sigma_v$ velocity paring]{}, according to the method detailed in [@2017ApJ...835..106N].
The location of the cluster center in the plane of the sky and the location of the cluster velocity center are both chosen iteratively. A cylinder with [$R_\mathrm{aperture} = 1.6{h^{-1}\,\mathrm{Mpc}}$ and $|v|\leq2500 \,{\mathrm{km \, s^{-1}}}$]{} is centered initially on the SZ center and redshift of the cluster reported by . [The plane of sky center of mass and the mean velocity are calculated and the cylinder is recentered on this new location. The cylinder center iteratively moves in this manner until convergence (defined as [$\Delta R < 0.02 {h^{-1}\,\mathrm{Mpc}}$ and $\Delta v < 50 \,{\mathrm{km \, s^{-1}}}$]{}); most clusters converge in one or two iterations.]{}
The central region of the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}observation is then removed; the resulting cylindrical selection has a [$0.25{h^{-1}\,\mathrm{Mpc}}$ axial hole]{} through the center. [The velocity bias between substructure and dark matter particles is small in the outer regions of clusters but increases toward the center, making the outer region a better probe of the cluster’s dynamical state (Aung et al., *in prep.*).]{} Furthermore, there may be systematic differences between the mock catalogs and [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}clusters due to the high concentration of galaxies in this region stemming from fiber collisions, observational selection effects, or distance resolution in the $N$-body simulation used to build the mock observation that destroys substructure near the center of simulated cluster.
Determining physical comoving distances to apply the fixed aperture requires some knowledge of the underlying cosmological parameters, and the measurement is particularly sensitive to [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}. We note that varying [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}by $\approx 10\%$ changes the effective aperture by $\lesssim 2\%$. Because this effect is relatively small, we opt to adopt the straightforward approach of assuming the [*Planck*]{}fiducial cosmology and a [$1.6 {h^{-1}\,\mathrm{Mpc}}$]{} comoving aperture for the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}clusters.
Mock Observations
-----------------
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The mock observations are created from UniverseMachine [@2018arXiv180607893B] galaxies incorporated into the publicly available Multidark MDPL2 $N$-body simulation[^1] [@2016MNRAS.457.4340K]. The MDPL2 simulation uses a $\Lambda$CDM cosmology, with cosmological parameters consistent with the [*Planck*]{}CMB fiducial cosmology : $\Omega_{\Lambda} = 0.693$, $\Omega_m = 0.307$, $\Omega_b = 0.048$, $h = 0.678$, $n_s=0.96$, and $\sigma_8 = 0.823$.
The MDPL2 simulation contains $3840^3$ particles in a box with comoving length $1.0h^{-1}\rm{Gpc}$ with mass resolution of $1.51\times10^9{h^{-1} \, \mathrm{M_{\odot}}}$. Halo and subhalo properties are reported from the publicly available Rockstar halo catalog [@2012ascl.soft10008B], and halo masses reported here are with respect to the critical density $\rho_\mathrm{crit}$.
UniverseMachine is an empirical model of the merger history and star formation history of galaxies. The model paints galaxies onto halos and subhalos in the Multidark simulation using halo merger trees; halo mass, halo assembly history, and redshift all inform galaxy star formation rate. UniverseMachine tracks orphan galaxies, which are galaxies for which the hosting substructure has been numerically disrupted in the simulation, allowing for richer cluster mock observations than would be possible with the $N$-body simulation alone. All UniverseMachine galaxies with stellar mass $\geq 10^{9.5}{h^{-1} \, \mathrm{M_{\odot}}}$ are used to construct the mock observations. [Because the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample has an average measurement error of [$\approx 30\,{\mathrm{km \, s^{-1}}}$]{}, each galaxy in the mock catalog is given a velocity error selected from a Gaussian with width [$30\,{\mathrm{km \, s^{-1}}}$]{}]{}. For further details about the UniverseMachine galaxy catalog, see [@2018arXiv180607893B] and references therein.
The UniverseMachine galaxies in comoving $1.0h^{-1}\rm{Gpc}$ snapshots for twelve redshifts ranging from $z=0.0$ to $z=0.304$ are used to construct [eight]{} full-sky light cones. The light cones differ only in the location at which the observer is placed. Each full-sky light cone is used to construct [ten]{} mock observations. These mock observations are modeled to follow the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}observation closely: they adopt the redshift range [($0.05<z<0.3$)]{} and sky area ($11589 \, {\mathrm{deg}^2\xspace}$ for $z<0.2$ and $8417 \, {\mathrm{deg}^2\xspace}$ for $0.2 < z < 0.3$) of [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}.
[$Y_\mathrm{SZ}$]{}signals are assigned to each cluster with [$M_{200} \geq 10^{14.2} {h^{-1} \, \mathrm{M_{\odot}}}$]{} according to the power law relation from , which assumes the universal pressure profile of . The power law relation is employed with bias parameter [$b=0$]{} and lognormal scatter $\sigma_{\ln{Y}}$ included. The $Y_\mathrm{SZ}(M_\mathrm{SZ})$ relationship is inverted, and $M_\mathrm{SZ}$ values assigned to each cluster based on the cluster’s $Y_\mathrm{SZ}$ signal. Because this process includes scatter $\sigma_{\ln{Y}}$, it properly forward-models the sample of clusters which would be realistically observed and, therefore, accounts for Eddington bias.
![A PDF of bias, $b$, needed to scale the selection function (shown in Figure \[fig:catcomp\]) to match the expected cluster counts for $\mathcal{C}= 0.8$. A significant bias is necessary to bring the observed cluster masses into agreement with that of a simulation at the [*Planck*]{}CMB fiducial cosmology. The VDF provides a way to explore this tension by utilizing cluster dynamical measurements. []{data-label="fig:biashist"}](fig3.pdf "fig:"){width="45.00000%"}\
Apparent cluster members, including both true cluster members and interloping field galaxies, are selected with the same method described in Section \[sec:hecs\]: a simple cylindrical cut with a [$1.6{h^{-1}\,\mathrm{Mpc}}$]{} aperture, initial velocity cut of [$2500 \,{\mathrm{km \, s^{-1}}}$]{} about the cluster center, and with [no $\sigma_v$ clipping]{}. The cluster centers on the plane of the sky, as well as in velocity space, are determined by the same [iterative scheme]{} described in \[sec:hecs\], and the central [$0.25{h^{-1}\,\mathrm{Mpc}}$]{} is removed from the sample.
Figure \[fig:masshist\] shows a PDF of mock catalog and [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample SZ masses and redshift. For $b=0$, the distribution of masses disagrees significantly, but invoking a mass bias of $b=0.42$ can bring the mass distributions into agreement. Alternately, a mock catalog with a low value of [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}can also bring the mass distributions of the observed and simulated clusters into agreement. The redshift distribution of of the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample agrees with that of the mock catalog.
[Mock observations for both the fiducial cosmology as well as for the nonfiducial cosmologies]{} are calculated as in [@2017ApJ...835..106N]. Briefly, each cluster is given a weight according to the cluster’s redshift, $z$, and mass, $M_{200}$[, both of which are known quantities that are extracted from the simulation]{}. This weight, $w$ is given by $$w=\frac{\mathrm{HMF}(M, z, {\textsc{$\sigma_8$}\xspace}{}, {\textsc{$\Omega_m$}\xspace})}{\mathrm{HMF}(M, z, {\textsc{$\sigma_8$}\xspace}{}_\mathrm{, sim}, {\textsc{$\Omega_m$}\xspace}_\mathrm{, sim})}$$ [the ratio of the nonfiducial analytic halo mass function (HMF) to the simulation HMF for the full $1.0 {h^{-1}\,\mathrm{Gpc}}$ box at the most appropriate redshift snapshot. For the analytic HMF we use that of [@2008ApJ...688..709T]]{} with an average halo density of [$200\rho_\mathrm{crit}$. Further details about the HMF of the Multidark suite of simulations can be found in [@2017MNRAS.469.4157C].]{}
[The selection function shown in Figure \[fig:catcomp\] is scaled by a multiplicative factor until the clusters above the selection have $\sum w = 87 \mathcal{C}^{-1}$, where 87 is the number of clusters in the [*Planck*]{}-selected sample from which the 83 [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}clusters were randomly selected and $\mathcal{C}$ is the integral completeness above the selection function. ]{}
[The multiplicative scaling of the selection function of the mock observations relates to the bias factor that may be applied to the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}observations, $b$, [as $(1-b)^{-1}$]{}. Therefore, at the fiducial cosmology, a multiplicative scaling of $1.72$ applied to the scaling relation shown in Figure \[fig:catcomp\] is equivalent to the bias $b=0.42$]{} that is needed to bring [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}cluster observations into agreement with the [*Planck*]{}CMB TT fiducial cosmology. Figure \[fig:biashist\] shows a histogram of biases needed to scale the simulation at the fiducial cosmology to match expected cluster counts for $\mathcal{C}=0.8$. Biases that are slightly larger than the [*Planck*]{}-reported value are needed to bring the mock observations into full agreement with the observed cluster counts. This is due to the fact that the simulation uses the cosmological model at the central point of the [*Planck*]{}fiducial cosmology; simulations at lower [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}cosmologies, but still within the CMB TT constraint contours, would prefer lower biases.
In the case where cluster mass is defined by an average halo density with respect to the critical density, e.g. the $200\rho_\mathrm{crit}$ definition used in this work, it is unlikely that the $M(\sigma_v)$ relation changes dramatically with changing [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}[e.g. @Evrard:2008aa; @2016MNRAS.456.3068O]. Therefore, for nonfiducial, non-simulated cosmologies, cluster velocity distributions for individual clusters [[are]{}]{} assumed to remain unchanged.
Mock Observations with Systematics {#sec:altcat}
----------------------------------
In addition to the standard method outlined in Section \[sec:methods\], we explore how systematic differences between the mock catalog and the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample might affect the resulting [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}constraints. We construct [eleven]{} mock catalogs with systematics to explore this. [The first four catalogs explore changes in parameters applied to both the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}and mock catalog, while the remaining seven catalogs explore systematic changes to the mock catalogs only.]{}
1. **Large Aperture**: To explore how the choice of cylinder size may bias the results, a large cylinder with $R_\mathrm{ap}=2.3{h^{-1}\,\mathrm{Mpc}}$, $v_\mathrm{cut}=3785{\mathrm{km \, s^{-1}}}$, and $R_\mathrm{hole}=0.25{h^{-1}\,\mathrm{Mpc}}$ is used to select galaxies in both the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample and also in the mock catalog. The $R_\mathrm{ap}$ and $v_\mathrm{cut}$ values are selected to correspond to the typical radius and $2\sigma_v$ of a $3\times10^{15}{h^{-1} \, \mathrm{M_{\odot}}}$ cluster.
2. **No Axial Hole**: To explore how the choice of axial hole may bias the constraints, no axial hole is removed from the cylinder center. All other parameters of the standard catalog remain unchanged. This change is applied to both the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample and also to the mock catalog.
3. **Sigma Clipping**: To reduce the effects of interlopers, 2-sigma iterative velocity clipping is applied both to the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample and also to the mock catalogs. Additional details about this iterative clipping can be found in [@2017ApJ...835..106N].
4. **Low Redshift**: To explore whether the constraints are driven by the high-mass sample at high redshift, these clusters are removed from the sample. Both the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample and also the mock catalogs are limited to samples with $z\leq0.2$.
5. **Y-M Scatter**: To explore whether an underestimation of the scatter in the Y-M scaling relation may bias the results, the level of scatter in the Y-M scaling relation is increased by 50%. This change is applied only to the mock catalog to assess for a potential systematic difference between the mock and [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}samples. The increase in scatter effectively makes the average mass of the mock sample smaller by scattering low-mass clusters above the selection function shown in Figure \[fig:catcomp\].
6. **Radial Selection**: To explore whether small differences between the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample and the mock observation in the the radial distribution function (RDF, discussed in detail in Section \[sec:rdf\]) bias the results, the mock clusters are subsampled to match the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}RDF.
7. **Rich Clusters**: To explore how the choice of richness may bias the constraints, only rich clusters, defined as having at least $40$ galaxies, are included in the mock catalog.
8. **Reduced Interlopers**: To explore whether an overabundance of interlopers may bias the results, the threshold mass for interlopers is increased to $M_{200}\geq10^{12}{h^{-1} \, \mathrm{M_{\odot}}}$. This change is applied only to the mock catalog to assess for a potential systematic difference between the mock and [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}samples. The increase in interloper mass cut effectively decreases the number of interloping galaxies in the mock catalog.
9. **Red Fraction**: [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}galaxies are preferentially selected to lie on the red sequence and therefore are more likely to be cluster members (and less likely to be interlopers) than a stellar-mass-selected sample. To test for a bias from preselecting likely cluster members, we select galaxies from the mock catalog with a similar bias towards cluster members. This is done by subsampling true cluster members to 80% of the original galaxy population and interloping field galaxies to 20% of the original galaxy population. This change is applied to the mock sample only.
10. [**Quiescent Galaxies**: [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}galaxies are preferentially selected to lie on the red sequence, while the sample in the standard catalog is mass-selected. Differences are found between the velocity dispersions of red galaxies versus the velocity dispersions of all galaxies in a cluster sample, with velocity dispersions of a color-selected sample mildly biased compared to than that of a stellar-mass-selected sample . To test for a bias stemming from differences in galaxy selection, we select galaxies from the mock catalog with specific star formation rate (sSFR) $\leq\,10^{-2}\,\mathrm{Gyr}^{-1}$ [as in, e.g., @2015MNRAS.446..521S]. This change is applied to the mock sample only.]{}
11. [**Biased Velocities**: To further explore how velocity bias may affect the resulting constraints, [[we artificially impose a velocity bias]{}]{} on the galaxies in the standard mock catalog, reducing the velocity of every galaxy in the mock catalog to $0.95$ times its true value. It should be noted that this check likely overemphasizes the true velocity bias in two ways: first, by imposing the bias on all galaxies in the sample, including interlopers, and second, by failing to disentangle the fact that this bias tends toward zero for well-sampled clusters [@2013ApJ...772...47S]. This change is applied to the mock sample only.]{}\[item:biasedv\]
Unless otherwise noted, all parameters are identical to the standard catalog and the method is applied to both the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample as well as to the mock catalog. Table \[table:summary\] summarizes the key details of these mock catalogs with systematics.
The Velocity Distribution Function
----------------------------------
The velocity distribution function [VDF, @2017ApJ...835..106N] is a forward-modeled test statistic that can be used to compare distributions of observed cluster member velocities to those predicted with simulations. The VDF can be used directly to explore constraints on cosmological parameters and is less affected by measurement errors associated with dynamical masses or the resulting Eddington bias [@1913MNRAS..73..359E]. The VDF, $dn(v)/dv$, is the sum of [probability distribution functions (PDFs) of galaxy LOS velocities]{} and is given by $$\frac{dn}{dv} (v) = \left[\frac{1}{N}\sum_{i=1}^{N} \left[ \mathrm{PDF}(|v|) \right]_i \right]_{A, z_\mathrm{min}, z_\mathrm{max}},
\label{eq:VDF}$$ where $\mathrm{PDF}(|v|)$ denotes a probability distribution function of the absolute value of galaxy LOS velocities, the index $i$ denotes a sum over $N$ clusters (83 for the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample and [$87 \times \mathcal{C}^{-1}$ for the mock observations]{}), $A$ indicates that the VDF is calculated for a given sky area, and $z_\mathrm{min}, z_\mathrm{max}$ indicates that the VDF is calculated for a given redshift range[^2]. [We use [six]{} velocity bins of width [$\Delta v=250\,{\mathrm{km \, s^{-1}}}$, [with edges]{} from $0$ to $1500\,{\mathrm{km \, s^{-1}}}$]{}.]{}
![[Top panel: The VDF of the mock observations at two different cosmologies (colored curves with band indicating the middle 68% of mock observations), for [the standard catalog with]{} integral completeness $\mathcal{C}=0.8$. There is a dearth of high-velocity members in the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}catalog (blue points with Poisson error bars) compared to the fiducial cosmology (${\textsc{$\mathcal{S}_8$}\xspace}=0.82$, red solid), but a lower [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}cosmology (e.g. ${\textsc{$\mathcal{S}_8$}\xspace}=0.75$, green dash) is more consistent with the data. This agrees with the results of the top left panel of Figure \[fig:masshist\], which shows that unless one invokes a significant mass bias, there are fewer high-mass clusters in the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}observation than in the mock observations. Bottom panel: The fractional difference between models, $\delta$, is the largest at very high and very low velocities, where the technique has the most resolving power. Near $\approx 800 {\mathrm{km \, s^{-1}}}$, $\delta$ crosses 0, indicating that bins near these central velocities are not as useful in differentiating models. ]{} []{data-label="fig:vdf"}](fig4.pdf){width="50.00000%"}
The resulting VDF is shown in Figure \[fig:vdf\]. As in [@2017ApJ...835..106N], the VDF has the most resolving power at high and low velocities. The crossover point where velocity bins have no resolving power has shifted downward from $\approx 1200 \, {\mathrm{km \, s^{-1}}}$ in [@2017ApJ...835..106N] to [$\approx 800\, {\mathrm{km \, s^{-1}}}$]{}; this is due to the changes in the cluster selection method and effective volume compared to the previous tests on mock catalogs.
The VDF in Figure \[fig:vdf\] shows that the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}clusters are dynamically colder than those in the simulation at the fiducial cosmology, since the VDF has a dearth of high-velocity members compared to the mock observation at the fiducial cosmology. [As was noted in [@2016ApJ...819...63R], the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}clusters have a smaller $\sigma_v$ than expected from a [*Planck*]{}-selected cluster sample at the [*Planck*]{}CMB fiducial cosmology. This could be caused by a number of factors, including a true dearth of high-mass clusters or a bias between velocity dispersions in the simulations and those of observed clusters.]{}
[Figure \[fig:vdf\_alternate\] shows the VDF for mock catalogs with systematics, which explore how parameter choices or possible biases between the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample and mock observations may introduce bias to the results. For complete details on the parameter choices for each of the mock catalogs with systematics, see Section \[sec:altcat\].]{}
[Four of the mock catalogs with systematics (Large Aperture, No Axial Hole, Sigma Clipping, and Low Redshift) assess parameter choices. For these catalogs, we change one of the standard parameter choices, and apply this change to both the mock catalog as well as the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample. As can be seen in Figure \[fig:vdf\_alternate\], both the mock catalog VDF as well as the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}VDF change compared to the standard version in Figure \[fig:vdf\]. This change is most pronounced in the sigma clipping example, with a decrease in signal at the high velocity end of the VDF. In all cases, the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}VDF signal lies below (above) the fiducial mock VDF signal at high (low) velocities, and is in much closer agreement with a low [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}cosmology. ]{}
The remaining [seven]{} mock catalogs with systematics [(Y-M Scatter, Radial Selection, Rich Clusters, Reduced Interlopers, Red Fraction, Quiescent Galaxies, and Biased Velocities)]{} involve changes to the mock VDF only; the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}VDF signal is the same as is shown for the standard catalog in Figure \[fig:vdf\]. The changes impart small changes on the mock VDF, but none of these changes are sufficient to pull the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample into agreement with the ${\textsc{$\mathcal{S}_8$}\xspace}=0.82$ fiducial cosmology VDF. [(There is one exception, the Biased Velocities catalog, which is within 1$\sigma$ of ${\textsc{$\mathcal{S}_8$}\xspace}=0.82$, but as discussed previously, this catalog likely overestimates a realistic velocity bias, with the Quiescent Galaxies catalog being a more proper modeling of this effect.)]{} The result is robust to these explorations of possible systematics: the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}clusters are dynamically colder than those in the simulation at the fiducial cosmology.
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Standard Large Aperture No Axial Hole
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\[5ex\] Sigma Clipping Low Redshift Y-M Scatter
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\[5ex\] Radial Selection Rich Clusters Reduced Interlopers
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\[5ex\] Red Fraction Quiescent Galaxies Biased Velocities
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\[5ex\]
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Radial Distribution Function {#sec:rdf}
----------------------------
The radial distribution function (RDF) is used as a self-consistency check to evaluate the relative radial distributions of galaxies. It is calculated as a sum of comoving radial distance PDFs and is given by $$\frac{dn}{dR_\mathrm{sep}} (R_\mathrm{sep}) = \left[\frac{1}{N}\sum_{i=1}^{N} \left[ \mathrm{PDF}(R_\mathrm{sep}) \right]_i \right]_{A, z_\mathrm{min}, z_\mathrm{max}},
\label{eq:RDF}$$ where $R_\mathrm{sep}$ is the [projected]{} radial distance from the galaxy to the cluster center, $\mathrm{PDF}(R_\mathrm{sep})$ denotes a probability distribution function of the projected comoving distance from the cluster center under the assumption of the fiducial cosmology, the index $i$ denotes a sum over $N$ clusters, and $A, z_\mathrm{min}, z_\mathrm{max}$ indicates that the RDF is calculated for a given sky area and redshift.
Figure \[fig:rdf\] shows the RDF for the mock clusters, which are based on the fiducial cosmology, and the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}clusters. [The error bars on the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}RDF is created by allowing [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}and $h$ to vary within an ellipse defined by the TT+lowP 2-$\sigma$ constraints reported by : [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}varies from [$0.29$ to $0.34$]{} and $h$ varies from [$0.65$ to $0.69$]{}.]{} The error band on the RDF of the mock observations shows the middle 68% and 95% of the individual mock observation RDFs at the fiducial cosmology.
When evaluated at the fiducial cosmology, the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}RDF is somewhat more centrally peaked than the mock observations, though the difference is subtle. One explanation for this is that the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample may contain fewer massive clusters. Alternately, galaxy selection effects, cluster selection effects, or baryonic effects may be the source of the disagreement, which is not fully explained by allowing [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}and $h$ to vary to extreme values. The effect of galaxy selection is explored by subsampling the mock catalog to match the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample RDF in the Radial Selection catalog described in Section \[sec:results\] and Table \[table:summary\].
![The radial distribution function (RDF) of the mock observations and [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}clusters. Parameters [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}and $h$ affect the inferred $R_\mathrm{sep}$, and allowing these parameters to vary within the 2-$\sigma$ values reported by produces a band of $RDF$ curves for the mock observations (dark and light red bands showing the middle 68% and 95%, respectively). The RDF of the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample (blue with Poisson error bars) is subtly more centrally peaked than the RDF of the mock observation. [Subsampling the mock observation to match the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}RDF to account for possible galaxy selection effects is explored in the Radial Selection catalog (see Sections \[sec:altcat\] and Table \[table:summary\])]{}. Alternately, the difference may be due to the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample containing less massive clusters with a smaller physical extent than the mock observation clusters.[]{data-label="fig:rdf"}](fig6.pdf){width="50.00000%"}
[Results]{} {#sec:results}
===========
Credible Regions
----------------
The posterior probability, $P({\sigma_8, \, \Omega_m}{}|y)$, of a model given the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}observation is calculated as in [@2017ApJ...835..106N]. Briefly, the estimated covariance matrix, , is given by $$\hat{C}= \frac{1}{n_\mathrm{mock}-1} \sum_{i=1}^{n_\mathrm{mock}} \left[ (y_i-\bar{y})(y_i-\bar{y})^T\right],
\label{eq:cov}$$ where $i$ denotes a sum over the $n_\mathrm{mock}=80$ fiducial mock observations, $n_\mathrm{bin}$ is the number of velocity bins, $y_i$ is a $n_\mathrm{bin}\times1$ column vector of the $i^{th}$ mock observation’s $n_\mathrm{bin}$ bin values, and $\bar{y}$ is a $n_\mathrm{bin}\times1$ column vector of the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}observation.
As detailed in, e.g., , an unbiased estimator, $\hat{\Psi}^{-1}$, of the inverse covariance matrix is given by $$\hat{\Psi}^{-1} = \frac{n_\mathrm{mock}-n_\mathrm{bin}-2}{n_\mathrm{mock}-1} \hat{C}^{-1},
\label{eq:invcov}$$ where $\hat{C}^{-1}$ denotes a standard matrix inversion of the covariance matrix. The $\chi^2$ values are calculated by $$\chi^2(y|\sigma_8, \Omega_m) = (\bar{y}-y^\star)^T \, \hat{\Psi}^{-1} \, (\bar{y}-y^\star),
\label{eq:chi2}$$ [where $y^\star$ is a $n_\mathrm{bin}\times1$ column vector of average mock observation VDF signal at a single cosmology and $\bar{y}$ is the measured [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}VDF signal. The unbiased estimator of the inverse covariance matrix, $\hat{\Psi}^{-1}$, is calculated at the fiducial cosmology and assumed to be constant across the [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}-[<span style="font-variant:small-caps;">$\Omega_m$</span>]{}plane.]{} This assumption breaks down as models farther from the fiducial model are considered. To properly analyze the VDF and the constraints in the [$\sigma_8$-$\Omega_m$]{} plane, one would need a suite of simulations across multiple [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}and [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}models.
![VDF cosmological constraints on [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}and [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}(black, 68% and 95% contours, all plots). The [*Planck*]{}CMB TT fiducial cosmology (green, 1- and 2-$\sigma$ contours, all plots) prefers a high [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}. constraints from [*Planck*]{}SZ cluster counts (red, top plot) prefers a low [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}if mass bias is not invoked, while more recent [cluster-based]{} results invoking mass bias from CCCP + BAO + BBN \[Baseline\] (blue, top plot) lie below, though not in tension with, the [*Planck*]{}CMB fiducial cosmology. [The KiDS+VIKING-450 cosmic shear analysis [orange, center plot, @2018arXiv181206076H]]{} also lies below the [*Planck*]{}CMB fiducial cosmology. [@2017arXiv170801530D] Year 1 Results for $\Lambda CDM$ (purple, middle plot), Weighing the Giants [peach, bottom plot, @2015MNRAS.446.2205M], and South Pole Telescope [pink, bottom plot, @2016ApJ...832...95D] lie below, though not in tension with, the [*Planck*]{}fiducial cosmology. []{data-label="fig:likelihood"}](fig7a.pdf "fig:"){width="40.00000%"}\
![VDF cosmological constraints on [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}and [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}(black, 68% and 95% contours, all plots). The [*Planck*]{}CMB TT fiducial cosmology (green, 1- and 2-$\sigma$ contours, all plots) prefers a high [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}. constraints from [*Planck*]{}SZ cluster counts (red, top plot) prefers a low [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}if mass bias is not invoked, while more recent [cluster-based]{} results invoking mass bias from CCCP + BAO + BBN \[Baseline\] (blue, top plot) lie below, though not in tension with, the [*Planck*]{}CMB fiducial cosmology. [The KiDS+VIKING-450 cosmic shear analysis [orange, center plot, @2018arXiv181206076H]]{} also lies below the [*Planck*]{}CMB fiducial cosmology. [@2017arXiv170801530D] Year 1 Results for $\Lambda CDM$ (purple, middle plot), Weighing the Giants [peach, bottom plot, @2015MNRAS.446.2205M], and South Pole Telescope [pink, bottom plot, @2016ApJ...832...95D] lie below, though not in tension with, the [*Planck*]{}fiducial cosmology. []{data-label="fig:likelihood"}](fig7b.pdf "fig:"){width="40.00000%"}\
![VDF cosmological constraints on [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}and [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}(black, 68% and 95% contours, all plots). The [*Planck*]{}CMB TT fiducial cosmology (green, 1- and 2-$\sigma$ contours, all plots) prefers a high [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}. constraints from [*Planck*]{}SZ cluster counts (red, top plot) prefers a low [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}if mass bias is not invoked, while more recent [cluster-based]{} results invoking mass bias from CCCP + BAO + BBN \[Baseline\] (blue, top plot) lie below, though not in tension with, the [*Planck*]{}CMB fiducial cosmology. [The KiDS+VIKING-450 cosmic shear analysis [orange, center plot, @2018arXiv181206076H]]{} also lies below the [*Planck*]{}CMB fiducial cosmology. [@2017arXiv170801530D] Year 1 Results for $\Lambda CDM$ (purple, middle plot), Weighing the Giants [peach, bottom plot, @2015MNRAS.446.2205M], and South Pole Telescope [pink, bottom plot, @2016ApJ...832...95D] lie below, though not in tension with, the [*Planck*]{}fiducial cosmology. []{data-label="fig:likelihood"}](fig7c.pdf "fig:"){width="40.00000%"}\
We find that, for a given integral completeness $\mathcal{C}$, the likelihood $\mathcal{L}({\textsc{$\mathcal{S}_8$}\xspace}|\mathcal{C})$ is well modeled by a Gaussian in [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}with constant variance $\sigma^2$ and mean $\mu(\mathcal{C})$ that varies linearly with integral completeness, $\mathcal{C}\equiv N_\mathrm{observed}/N_\mathrm{true}$.
[To account for uncertainties in integral completeness, which was not an unknown quantity in [@2017ApJ...835..106N], we add an additional step to the statistical inference by including priors on [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}and $\mathcal{C}$.]{} The posterior $\pi$ takes the form $$\pi({\textsc{$\mathcal{S}_8$}\xspace}\mid \bar{y}, \mathcal{C}, \sigma) \propto \mathcal{L}\left( {\textsc{$\mathcal{S}_8$}\xspace}\mid \mathcal{C}, \sigma\right) \pi_1({\textsc{$\mathcal{S}_8$}\xspace}) \pi_2(\mathcal{C})$$ where the likelihood is a Gaussian $$\mathcal{L}\left( {\textsc{$\mathcal{S}_8$}\xspace}\mid \mathcal{C}, \sigma\right) \propto \exp{\left(\frac{-\left[{\textsc{$\mathcal{S}_8$}\xspace}- \mu(\mathcal{C})\right]^2}{(2 \sigma)^2}\right)},$$ $\pi_1(S_8)$ is a flat prior on [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}, [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}$\in[0.0, 1.0]$, and $\pi_2(\mathcal{C})$ is a flat prior on completeness, $\mathcal{C}\in[0.6, 1.0]$.
[The [credible regions]{} are calculated between ${\textsc{$\Omega_m$}\xspace}= 0.28$ and ${\textsc{$\Omega_m$}\xspace}=0.33$;]{} this function takes the form $$\mathcal{S}_8 = \sigma_8 \left( \frac{\Omega_m}{0.3}\right)^\gamma$$ where the power law parameter $\gamma$ describes the direction of the degeneracy in the [$\sigma_8$-$\Omega_m$]{} plane and the parameter [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}can be interpreted as the preferred value of [<span style="font-variant:small-caps;">$\sigma_8$</span>]{}at $\Omega_m=0.3$.
Figure \[fig:likelihood\] shows the 68% and 95% credible regions. We find constraints in the [$\sigma_8$-$\Omega_m$]{} plane given by , as shown in Figure \[fig:likelihood\]. The [*Planck*]{}CMB TT fiducial cosmology lies outside of our 95% credible region, but constraints from [*Planck*]{}SZ cluster counts , [the KiDS+VIKING-450 cosmic shear analysis [@2018arXiv181206076H]]{} and DES Year 1 Results for $\Lambda CDM$ [@2017arXiv170801530D] lie within 1 $\sigma$ of our reported constraints. More recent [*Planck*]{}cluster count results invoking mass bias , Weighing the Giants [@2015MNRAS.446.2205M], and South Pole Telescope [@2016ApJ...832...95D] have maximum likelihood functions that are within 2$\sigma$ of our reported constraints near ${\textsc{$\Omega_m$}\xspace}=0.3$.
Note that while some other definitions of [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}use a $\gamma$ parameter of either $\gamma=0.5$ [e.g. @2017MNRAS.465.1454H] or $\gamma=0.3$ . We allow $\gamma$ to be informed by the data rather than adopting a more standard definition, but for values of [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}near the [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}normalization value of $0.3$, these disparate definitions of [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}can be compared. It is only far from the [<span style="font-variant:small-caps;">$\Omega_m$</span>]{}normalization value of $0.3$ that the degeneracy direction $\gamma$ separates these different definitions of [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}.
{width="80.00000%"}
[<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}Constraints for Mock Catalogs with Systematics
---------------------------------------------------------------------------------------------------------------
[l l r ]{}
Standard & $R_\mathrm{ap}=1.6{h^{-1}\,\mathrm{Mpc}}$, [$v_{\mathrm{cut}}$]{}$= 2500\,{\mathrm{km \, s^{-1}}}$, $R_\mathrm{hole}=0.25{h^{-1}\,\mathrm{Mpc}}$, &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.751\pm0.037$]{}\
& $N_\mathrm{min}=20$ &\
Large Aperture & $R_\mathrm{ap}=2.3{h^{-1}\,\mathrm{Mpc}}$, [$v_{\mathrm{cut}}$]{}$= 3785\, {\mathrm{km \, s^{-1}}}$ &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.735\pm0.052$]{}\
No Axial Hole & $R_\mathrm{hole}=0.0{h^{-1}\,\mathrm{Mpc}}$ &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.735\pm0.041$]{}\
Sigma Clipping & $\sigma_\mathrm{clip}=2.0$ &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.731\pm0.035$]{}\
Low Redshift &Limit catalogs to clusters with $z\leq0.2$. &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.743\pm0.040$]{}\
Y-M Scatter & 50% additional scatter in Y-M relation. &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.751 \pm0.037$]{}\
Radial Selection & Force simulation RDF to match observed RDF by downsampling &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.747 \pm0.044$]{}\
& mock observation. &\
Rich Clusters & $N_\mathrm{min}=40$ &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.749\pm0.038$]{}\
Reduced Interlopers & Higher threshold for interlopers; interlopers must live in a halo &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.751\pm0.037$]{}\
& with $M_\mathrm{200}\geq10^{12}{h^{-1} \, \mathrm{M_{\odot}}}$. &\
[Red Fraction]{} & Randomly select 80% of true members and 20% of interlopers &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.741\pm0.034$]{}\
& to form the mock observation. &\
[Quiescent Galaxies]{} &Select only galaxies with sSFR $\leq10^{-1}\,\mathrm{Gyr}^{-1}$. &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.761\pm0.036$]{}\
[Biased Velocities]{} &[[Artificially bias the velocities of all mock galaxies.]{}]{} &[${\textsc{$\mathcal{S}_8$}\xspace}= \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.793\pm0.037$]{}\
\[table:summary\]
\[sec:altconstraints\]
Figure \[fig:s8summary\] shows a summary of [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}constraints evaluated at ${\textsc{$\Omega_m$}\xspace}=0.3$. This figure shows the sample of recent probes of [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}that were highlighted in Figure \[fig:likelihood\], constraints of [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}from the VDF standard catalog, and also constraints of [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}for the mock catalogs with systematics. Our constraints on [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}are remarkably robust to the changes in catalog parameters that are explored by the [[eleven]{}]{} additional mock catalogs.
[The Biased Velocities and Quiescent Galaxies catalogs constraints prefer the largest [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}. The Biased Velocities catalog ([$0.793\pm0.037$]{}, with a center that is $\approx 1.15 \sigma$ from the standard catalog), as discussed in Section \[sec:altcat\], overemphasizes the true velocity bias in two ways: first, by imposing the bias on all galaxies in the sample, including interlopers, and second, by failing to disentangle the fact that this bias tends toward zero for well-sampled clusters [@2013ApJ...772...47S]. Supporting the relevance of the second of these caveats, four clusters from the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}sample are studied in detail in [@2013ApJ...767...15R], with the authors finding no significant bias in the estimates of velocity dispersions of red-sequence galaxies compared to the full galaxy sample. Though the value of $0.95$ may be an overestimate of the bias found in the [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}cluster sample, it nonetheless provides an important cross-check for understanding how velocity bias might affect the resulting cosmological constraints. ]{}
[In the case of the Quiescent Velocities catalog ([${\textsc{$\mathcal{S}_8$}\xspace}= 0.761\pm0.036$]{}), in selecting only the quiescent galaxies in the mock catalog, the number of high-velocity galaxies is somewhat reduced. This results in constraints preferring a slightly larger [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}, though still within $\frac{1}{2}\sigma$ of the standard method. This can be understood in the context of velocity segregation , with red or quiescent galaxies having a velocity dispersion that may be $\approx5\%$ smaller than that of the full cluster sample , and can be attributed to the fact that blue galaxies are typically infalling and, therefore, have higher velocities. However, it should be noted that the effect becomes smaller with well-sampled clusters [@2013ApJ...772...47S] and recent simulations find a stellar-mass-selected sample to be relatively unbiased ($<5\%$ in [@2018MNRAS.474.3746A]), and both of these may contribute to the fact that the Quiescent Velocities constraints are not as extreme as the Biased Velocities constraints.]{}
[Table \[table:summary\] briefly describes the mock catalogs with systematics and also gives the parameter constraints on [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}for each. Regardless of the choice of parameters, each approach yields a preference for a low [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}and [all catalogs (except for the Biased Velocities catalog) have a central value within $\frac{1}{2}\sigma$ of the standard method. While properly assessing the systematic error due to the choice of modeling parameters would require more thoroughly sampling the high-dimensional space of these parameters, this result suggests that the systematic error is of the order one-half the statistical error.]{} ]{}
Discussion & Conclusion {#sec:discussion}
=======================
The abundance of clusters as a function of mass and redshift is a valuable tool for constraining cosmological models. Notably, the cosmological parameters based on [*Planck*]{}CMB observations predict more high-mass clusters than are observed. Biases in cluster mass estimates remain at the forefront of the discussion in interpreting cluster mass observations and resolving the tension between [*Planck*]{}CMB cosmological parameters and cluster counts.
Eddington bias, caused by the steeply declining mass function coupled with errors in mass estimates, produces an observed halo mass function with an upscatter of high-mass clusters. Accounting for this bias must be done correctly, and assumptions about the distribution of scatter handled carefully. The velocity distribution function (VDF) is a forward-modeled test statistic that can be used to quantify the abundance of galaxy clusters in a way that is less sensitive to biases introduced by measurement error than more standard HMF approach.
We have used the VDF to compare the summed velocity PDFs of observed [<span style="font-variant:small-caps;">HeCS-SZ</span>]{}clusters to mock observations produced by $N$-body simulations. In agreement with [@2016ApJ...819...63R], we find that this collection of clusters are dynamically colder than expected, having smaller velocities than one would predict for a [*Planck*]{}-selected sample of clusters for the [*Planck*]{}fiducial cosmology[[. This suggests]{}]{} that the observed clusters may be less massive than an ${\textsc{$\mathcal{S}_8$}\xspace}=0.82$ cosmology would predict. [We have explored several possible sources of systematic error, including parameter choices, interloper fraction, [velocity bias]{}, and galaxy selection effects. Our results are remarkably robust to reasonable changes to the standard mock catalog. ]{}While the precise fit of our preliminary constraints has a small dependence on the details of the model, all approaches show a preference for a low [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}, and our standard approach gives credible regions in the [$\sigma_8$-$\Omega_m$]{} plane given by [${\textsc{$\mathcal{S}_8$}\xspace}\equiv \sigma_8 \left(\Omega_m/0.3\right)^{0.25} = 0.751\pm0.037$]{}.
The constraints presented here should not be overinterpreted. The missing high-velocity members may be caused by a true dearth of high-mass clusters, or may alternately be caused by a bias between simulated cluster substructure velocity and cluster member velocities. Also, assumptions of a smoothly-varying covariance matrix break down far from the fiducial model. [Similarly, the fraction and characteristics of interlopers are a function of cosmological parameters; these will also change far from the simulated fiducial cosmology.]{} Properly evaluating the covariance matrices [and interloper population]{} of non-fiducial (or non-simulated) cosmologies would require a suite of large volume, high resolution simulations to capture the nuanced correlations. [Furthermore, properly evaluating these with realistic galaxy selection effects may require a large suite of hydrodynamical simulations that properly model galaxy properties.]{} Such a suite of simulations could also eliminate the need to assume cosmological parameters in the calculation of comoving distances, which could instead be replaced with invariant parameters such as angular extent.
Other LSS and cluster-based analyses have found similar results, seeing a tension between observations and the [*Planck*]{}CMB cosmology, and preferring smaller [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}values (or needing to invoke new physical models) to explain the discrepancy [e.g. @2013MNRAS.432.2433H; @2017MNRAS.465.1454H; @2017MNRAS.467.3024L; @2017arXiv170809813L]. The VDF provides a complementary test to use clusters as a cosmological probe. As observations probe larger areas of the sky to deeper magnitudes become available, these data sets will provide opportunities to understand and resolve the tension in [<span style="font-variant:small-caps;">$\mathcal{S}_8$</span>]{}constraints.
[^1]: http://www.cosmosim.org/
[^2]: Note that the VDF takes a slightly different form than the one presented in [@2017ApJ...835..106N]. Here, we have removed the volume element, which is sensitive to the underlying cosmological parameters, and instead defined the test statistic by limits on parameters that are invariant under changes in cosmology: sky area, redshift range, and the number of clusters observed.
|
---
abstract: 'We prove that a curve of degree $dk$ on a very general surface of degree $d \geq 5$ in ${\mathbb{P}}^3$ has geometric genus at least $\frac{dk(d-5)+k}{2} + 1$. This improves bounds given by G. Xu. As a corollary, we conclude that the very general quintic surface in ${\mathbb{P}}^3$ is algebraically hyperbolic.'
address: |
Department of Mathematics, Statistics and CS\
University of Illinois at Chicago, Chicago, IL 60607
author:
- Izzet Coskun
- Eric Riedl
title: 'Algebraic hyperbolicity of the very general quintic surface in ${\mathbb{P}}^3$'
---
[^1]
Introduction
============
A complex projective variety $X$ is [*algebraically hyperbolic*]{} if there exists an $\epsilon > 0$ such that for any curve $C \subset X$ of geometric genus $g(C)$ we have $$2g(C)-2 \geq \epsilon \deg(C).$$ In particular, algebraically hyperbolic varieties contain no rational or elliptic curves. Algebraic hyperbolicity is intimately related to metric notions of hyperbolicity. Recall that a variety $X$ is [*Brody hyperbolic*]{} if it admits no holomorphic maps from ${\mathbb{C}}$. By Brody’s Theorem [@Brody], Brody hyperbolicity is equivalent to the nondegeneracy of the Kobayashi metric on compact manifolds. Brody hyperbolicity is conjectured to control many geometric and arithmetic properties of $X$. For example, Demailly [@Demailly] proves that Brody hyperbolic varieties are algebraically hyperbolic and conjectures the following.
A smooth complex projective variety $X$ is Brody hyperbolic if and only if it is algebraically hyperbolic.
Hypersurfaces provide a natural testing ground for deep conjectures on hyperbolicity. Hence, we are led to the following question.
\[ques-algHyp\] For which $n$ and $d$ is a very general hypersurface of degree $d$ in ${\mathbb{P}}^n$ algebraically hyperbolic?
More generally, one can ask the following.
\[ques-whichCurves\] For a very general hypersurface of degree $d$ in ${\mathbb{P}}^n$, for which pairs $(e,g)$ does there exist a curve of degree $e$ and geometric genus $g$?
The study of the hyperbolicity of very general hypersurfaces in ${\mathbb{P}}^n$ has a long history (see [@CoskunHyperbolicity; @Demailly; @DemaillyElGoul; @Mcquillan; @Siu]). For example, Question \[ques-algHyp\] has been resolved in many cases. Results of Voisin [@Voisin; @Voisincorrection] prove that if $n \geq 4$, then $$2g-2 \geq e(d-2n+2).$$ Therefore, a very general hypersurface of degree $d \geq 2n-1$ in ${\mathbb{P}}^n$ is algebraically hyperbolic for $n \geq 4$. In the case $n=3$, any curve on $X$ is a complete intersection of type $(d,k)$. Geng Xu [@Xu] imporved results of Ein [@Ein] and proved that $$g \geq \frac{dk(d-5)}{2} + 2,$$ showing that surfaces in ${\mathbb{P}}^3$ of degree at least $6$ are algebraically hyperbolic. However, despite all this interest, the case of quintics in ${\mathbb{P}}^3$ has remained open for the past 20 years (see [@Demailly; @Demaillynew]). Our goal in this paper is to prove the following.
\[thm-algHyp\] Let $X \subset {\mathbb{P}}^3$ be a very general surface of degree $d \geq 5$. Then the geometric genus of any curve in $X$ of degree $dk$ is at least $\frac{dk(d-5)+k}{2} + 1$.
In particular, when $d=5$, the bound on the genus of a curve of degree $5k$ specializes to $$2 g(C) - 2 \geq k = \frac{1}{5} \deg(C).$$ We obtain the following corollary.
A very general quintic surface in ${\mathbb{P}}^3$ is algebraically hyperbolic.
Organization of the paper {#organization-of-the-paper .unnumbered}
-------------------------
In §\[sec-Prelim\], we recall the basic setup developed by Ein, Voisin, Pacienza, Clemens and Ran. In §\[sec-Proof\], we prove Theorem \[thm-algHyp\].
Acknowledgments {#acknowledgments .unnumbered}
---------------
We would like to thank Lawrence Ein, Mihai Păun and Matthew Woolf for useful discussions.
preliminaries {#sec-Prelim}
=============
In this section, we review the basic setup due to [@ClemensRan; @Ein; @Ein2; @Pacienza; @Pacienza2; @Voisin; @Voisincorrection] and collect facts relevant to the rest of this paper. The main goal of this section is to prove Corollary \[cor-mapToNormalBundle\]. We always work over the complex numbers ${\mathbb{C}}$.
Let $S_d = H^0({\mathbb{P}}^n, {\mathcal{O}}_{{\mathbb{P}}^n}(d))$. Suppose that a general hypersurface of degree $d$ in ${\mathbb{P}}^n$ admits a generically injective map from a smooth projective variety of deformation class $Y$. After an étale base change $U \to S_d$, we can find a generically injective map $$h:{\mathcal{Y}}\to {\mathcal{X}}$$ over $U$, where ${\mathcal{X}}$ is the universal hypersurface of degree $d$ in ${\mathbb{P}}^n$ over $U$ and ${\mathcal{Y}}$ is a smooth family of pointed varieties over $U$ with members in the deformation class of $Y$. We choose ${\mathcal{Y}}$ so that the codimension of $h({\mathcal{Y}})$ in ${\mathcal{X}}$ is $n-1-\dim Y$. Let $$\pi_1: {\mathcal{X}}\to U \quad \mbox{and} \quad \pi_2: {\mathcal{X}}\to {\mathbb{P}}^n$$ denote the two natural projections of the universal hypersurface over $U$. Let the [*vertical tangent sheaf*]{} $T_{{\mathcal{X}}}^{{\operatorname{vert}}}$ be defined by the natural sequence $$0 \rightarrow T_{{\mathcal{X}}}^{{\operatorname{vert}}} \rightarrow T_{{\mathcal{X}}} \rightarrow \pi_2^* T_{{\mathbb{P}}^n} \rightarrow 0.$$
Since every hypersurface contains a variety of deformation class $Y$, the family ${\mathcal{Y}}$ dominates $U$ under $\pi_1 \circ h$. Moreover, without loss of generality, we may assume that ${\mathcal{Y}}$ is stable under the $GL_{n+1}$ action on ${\mathbb{P}}^n$, so $\pi_2 \circ h$ dominates ${\mathbb{P}}^n$ [@Pacienza; @Voisin]. Furthermore, the invariance under $GL_{n+1}$ also implies that the map $T_{{\mathcal{Y}}} \rightarrow h^* (\pi_2^* T_{{\mathbb{P}}^n})$ is surjective. Let $T_{{\mathcal{Y}}}^{{\operatorname{vert}}}$ denote the kernel $$0 \rightarrow T_{{\mathcal{Y}}}^{{\operatorname{vert}}} \rightarrow T_{{\mathcal{Y}}} \rightarrow h^* (\pi_2^* T_{{\mathbb{P}}^n}) \rightarrow 0.$$
Let $t \in U$ be a general closed point. Let $X_t$ denote the corresponding fiber of ${\mathcal{X}}$. Let $$h_t: Y_t \to X_t$$ denote the corresponding map from the fiber $Y_t$ of ${\mathcal{Y}}$ over $t$ to $X_t$. Let $$i_t: X_t \rightarrow {\mathcal{X}}\quad \mbox{and} \quad j_t: Y_t \rightarrow {\mathcal{Y}}$$ denote the inclusion of $X_t$ in ${\mathcal{X}}$ and $Y_t$ in ${\mathcal{Y}}$, respectively. Observe that $i_t \circ h_t = h \circ j_t$. Let $N_{h/{\mathcal{X}}}$ denote the normal sheaf to the map $h: {\mathcal{Y}}\to {\mathcal{X}}$ and let $N_{h_t/X_t}$ denote the normal sheaf to $h_t$ defined as the cokernels of the following natural sequences $$0 \to T_{{\mathcal{Y}}} \to h^*T_{{\mathcal{X}}} \to N_{h/{\mathcal{X}}} \to 0$$ $$0 \rightarrow T_{Y_t} \rightarrow h_t^* T_{X_t} \rightarrow N_{h_t/X_t} \rightarrow 0.$$
Following Ein, Voisin, Pacienza, Clemens and Ran, it is standard to relate $N_{h_t/X_t}$ to sheaves arising from the family ${\mathcal{Y}}\to {\mathcal{X}}$. Unfortunately, the prior work does not explicitly state the theorem we will use. For the reader’s convenience, we reprove the main statements, emphasizing the key points from our perspective. We will make repeated use of the following lemma.
\[lem-staysExact\] Let $\phi: Z_0 \rightarrow Z$ be a morphism of projective varieties. Let $$\label{seq-lemstaysexact}
0 \to E \to F \to G \to 0$$ be a short exact sequence of sheaves on $Z$, where $E$ and $F$ are vector bundles. If $\phi^* E \to \phi^* F$ is generically injective, then the sequence $$0 \to \phi^*E \to \phi^* F \to \phi^* G \to 0$$ on $Z_0$ is exact.
Apply the derived pullback functor to the sequence (\[seq-lemstaysexact\]) to obtain the sequence $$0 \to L^1 \phi^* G \to \phi^*E \to \phi^* F \to \phi^* G \to 0 ,$$ where $L^1 \phi^* F =0$ since $F$ is a vector bundle. Since $\phi^* E \to \phi^* F$ is generically injective, $L^1 \phi^* G$ must be torsion. Since $E$ is a vector bundle, $\phi^*E$ is torsion-free and we conclude that $L^1 \phi^* G = 0$. This proves the lemma.
We first relate the normal bundles $N_{h_t/X_t}$ and $N_{h/{\mathcal{X}}}$.
\[lem-firstDiagram\] We have $$N_{h_t/X_t} \cong j_t^* N_{h/{\mathcal{X}}}.$$
Consider the following diagram. ‘=10 =3em
cdstrut[[by 2em]{}]{}
\#1\#2[ d\#1 .cdstrut u\#1 .cdstrut r\#1 \^[\#2]{} l\#1 \^[\#2]{}]{} ‘=10
=3em $$\begin{matrix}
& & 0 & & 0 \cr
& & \arrow{u}{} & & \arrow{u}{} \cr
& & {\mathcal{O}}_{Y_t}^{N} & \arrow{r}{=} & {\mathcal{O}}_{Y_t}^N \cr
& & \arrow{u}{} & & \arrow{u}{} \cr
0 & \arrow{r}{} & j_t^* T_{{\mathcal{Y}}} & \arrow{r}{} & h_t^* i_t^* T_{{\mathcal{X}}} & \arrow{r}{} & j_t^* N_{h/{\mathcal{X}}}& \arrow{r}{} & 0 \cr
& & \arrow{u}{} & & \arrow{u}{} & & \arrow{u}{\cong} \cr
0 & \arrow{r}{} & T_{Y_t} & \arrow{r}{} & h_t^{*}T_{X_t} & \arrow{r}{} & N_{h_t/X_t} & \arrow{r}{} & 0 \cr
& & \arrow{u}{} & & \arrow{u}{} \cr
& & 0 & & 0 \cr
\end{matrix}$$ Since $Y_t$ in ${\mathcal{Y}}$ and $X_t$ in ${\mathcal{X}}$ are fibers of fibrations, their normal bundles are trivial of rank $N= \dim S_d$. The first column in the diagram is the definition of the normal bundle $N_{Y_t/{\mathcal{Y}}}$. The second column in the diagram is the pullback under $h_t$ of the sequence $$0 \rightarrow T _{X_t} \rightarrow T_{{\mathcal{X}}}|_{X_t} \rightarrow N_{X_t/{\mathcal{X}}} \rightarrow 0$$ defining $N_{X_t/{\mathcal{X}}}$. Since $h_t$ is generically injective and $T_{X_t} \to T_{{\mathcal{X}}}|_{X_t}$ is everywhere injective, by Lemma \[lem-staysExact\] the pullback of the sequence by $h_t$ remains exact. The top row is an isomorphism of sheaves in the natural way. The bottom row is the sequence defining $N_{h_t/X_t}$. The middle row is the pullback of the sequence defining $N_{h/{\mathcal{X}}}$ under $j_t$ using the identification $h \circ j_t = i_t \circ h_t$. It is exact by Lemma \[lem-staysExact\] and the fact that $Y_t$ passes through a general point of ${\mathcal{Y}}$. The required isomorphism $N_{h_t/X_t} \cong j_t^* N_{h/{\mathcal{X}}}$ follows from the diagram by the Nine Lemma.
We can also express the normal bundle $N_{h_t/X_t}$ in terms of vertical tangent bundles. Let $K$ denote the cokernel of the map $T_{{\mathcal{Y}}}^{{\operatorname{vert}}} \to h^* T_{{\mathcal{X}}}^{{\operatorname{vert}}}$.
We have $$N_{h_t/X_t} \cong j_t^* K.$$
This is another diagram chase. Consider the following diagram.
‘=11 =3em
cdstrut[[by 2em]{}]{}
\#1\#2[ d\#1 .cdstrut u\#1 .cdstrut r\#1 \^[\#2]{} l\#1 \^[\#2]{}]{} ‘=12
=3em $$\begin{matrix}
& & 0 & & 0 \cr
& & \arrow{u}{} & & \arrow{u}{} \cr
& & j_t^* h^* \pi_2^* T_{{\mathbb{P}}^n} & \arrow{r}{=} & h_t^* i_t^* \pi_2^*T_{{\mathbb{P}}^n} \cr
& & \arrow{u}{} & & \arrow{u}{} \cr
0 & \arrow{r}{} & j_t^* T_{{\mathcal{Y}}} & \arrow{r}{} & h_t^* i_t^*T_{{\mathcal{X}}} & \arrow{r}{} & j_t^* N_{h/{\mathcal{X}}} & \arrow{r}{} & 0 \cr
& & \arrow{u}{} & & \arrow{u}{} & & \arrow{u}{\cong} \cr
0 & \arrow{r}{} & j_t^* T_{{\mathcal{Y}}}^{{\operatorname{vert}}} & \arrow{r}{} & h_t^{*} i_t^*T_{{\mathcal{X}}}^{{\operatorname{vert}}} & \arrow{r}{} & j_t^* K & \arrow{r}{} & 0 \cr
& & \arrow{u}{} & & \arrow{u}{} \cr
& & 0 & & 0 \cr
\end{matrix}$$
The first two columns come from the definitions of $T_{{\mathcal{Y}}}^{{\operatorname{vert}}}$ and $T_{{\mathcal{X}}}^{{\operatorname{vert}}}$, respectively. The restrictions of each to $Y_t$ remain exact by Lemma \[lem-staysExact\]. The top isomorphism is also the natural one, since $j_t^* h^* = h_t^* i_t^*$.
The bottom row is exact by Lemma \[lem-staysExact\], and the middle row is exact just as in the proof of Lemma \[lem-firstDiagram\].
Thus, in order to study positivity of $N_{h_t/X_t}$, we may study positivity of $K$.
Let $M_d^{{\mathbb{P}}^n}$ be the bundle defined by the sequence $$0 \to M_d^{{\mathbb{P}}^n} \to {\mathcal{O}}_{{\mathbb{P}}^n} \otimes S_d \to {\mathcal{O}}_{{\mathbb{P}}^n}(d) \to 0 .$$ Then $T_{{\mathcal{X}}}^{{\operatorname{vert}}} \cong \pi_2^*M^{{\mathbb{P}}^n}_d$.
This is yet another diagram chase. Consider the following diagram.
‘=11 =3em
cdstrut[[by 2em]{}]{}
\#1\#2[ d\#1 .cdstrut u\#1 .cdstrut r\#1 \^[\#2]{} l\#1 \^[\#2]{}]{} ‘=12
=3em $$\begin{matrix}
& & 0 & & 0 \cr
& & \arrow{u}{} & & \arrow{u}{} \cr
& & \pi_2^* {\mathcal{O}}(d) & \arrow{r}{=} & \pi_2^* {\mathcal{O}}(d) \cr
& & \arrow{u}{} & & \arrow{u}{} \cr
0 & \arrow{r}{} & {\mathcal{O}}\otimes S_d & \arrow{r}{} & \pi_2^*T_{{\mathbb{P}}^n} \oplus {\mathcal{O}}\otimes S_d & \arrow{r}{} & \pi_2^*T_{{\mathbb{P}}^n} & \arrow{r}{} & 0 \cr
& & \arrow{u}{} & & \arrow{u}{} & & \arrow{u}{\cong} \cr
0 & \arrow{r}{} & T_{{\mathcal{X}}}^{{\operatorname{vert}}} & \arrow{r}{} & T_{{\mathcal{X}}} & \arrow{r}{} & \pi_2^*T_{{\mathbb{P}}^n} & \arrow{r}{} & 0 \cr
& & \arrow{u}{} & & \arrow{u}{} \cr
& & 0 & & 0 \cr
\end{matrix}$$
The bottom row is the defining sequence for $T_{{\mathcal{X}}}^{{\operatorname{vert}}}$. The middle row comes from the natural splitting of $\pi_2^*T_{{\mathbb{P}}^n} \oplus {\mathcal{O}}\otimes S_d$.
The second column is the normal bundle sequence for ${\mathcal{X}}\subset {\mathbb{P}}^n \times U$. The maps in the first column are naturally induced by the maps in the second and prove the desired result.
\[lem-sCalc\] There exists an integer $s$ so that there is a surjective map $$\bigoplus_{i=1}^s h^* \pi_2^* M_1^{{\mathbb{P}}^n} \to K.$$
Given a degree $d-1$ polynomial $P$, we get a natural map $M_1^{{\mathbb{P}}^n} \to M_d^{{\mathbb{P}}^n}$, given by multiplication by $P$. Since polynomials of the form $x_i P$ generate the fiber of $M_d^{{\mathbb{P}}^n}$ at a point, we see that we can inductively keep adding copies of $M_1^{{\mathbb{P}}^n}$ to decrease the cokernel of the combined map.
In Voisin, Pacienza, and Clemens-Ran [@Voisincorrection; @Pacienza; @Pacienza2; @ClemensRan], the strategy is to show that if $Y_t$ is not contained in the locus on $X_t$ swept out by lines, then $s$ in Lemma \[lem-sCalc\] is not too large relative to $n$. Since the first Chern class of $M_1^{{\mathbb{P}}^n}$ is negative the hyperplane class on ${\mathbb{P}}^n$, this bounds the negativity of the normal bundle of any subvariety $Y_t \subset X_t$ that is not contained in the locus swept out by lines.
If $n=3$, we get a map with torsion cokernel if we merely let $s=1$, so bounding $s$ is not useful in studying curves on surfaces in ${\mathbb{P}}^3$. Instead, we combine the fact that $M_1^{{\mathbb{P}}^n} \cong \Omega_{{\mathbb{P}}^n}(1)$ with our understanding of $\Omega_{{\mathbb{P}}^n}(1)$ pulled back to $Y_t$ to get better bounds on the degree of $N_{h_t/X_t}$. The statement we use for our purposes is the following.
\[cor-mapToNormalBundle\] If $n=3$ and ${\mathcal{Y}}$ is a family of curves, there is a map $h_t^* i_t^* \pi_2^* \Omega_{{\mathbb{P}}^n}(1) \to N_{h_t/X_t}$ with torsion cokernel.
Pull back the surjective map $\bigoplus_{i=1}^s h^* \pi_2^* M_1^{{\mathbb{P}}^n} \to K$ from Lemma \[lem-sCalc\] under $j_t$. Since $K$ has rank $1$ and $N_{h_t/X_t} \cong j_t^* K$, the result follows.
Scrolls and degree considerations {#sec-Proof}
=================================
In this section, we prove Theorem \[thm-algHyp\]. We specialize the discussion in the previous section to the case of curves.
\[prop-Scrolls\] Let $h: C \to {\mathbb{P}}^n$ be a generically injective map of degree $e$ from a smooth curve $C$ to ${\mathbb{P}}^n$. Then line bundle quotients of $h^* \Omega_{{\mathbb{P}}^n}(1)$ of degree $-m$ give rise to surface scrolls in ${\mathbb{P}}^n$ of degree at most $e-m$ that contain $h(C)$.
The pullback of the Euler sequence on ${\mathbb{P}}^n$ to $C$ gives rise to the sequence $$0 \to h^*\Omega_{{\mathbb{P}}^n}(1) \to {\mathcal{O}}_{C}^{n+1} \to {\mathcal{O}}_C(1) \to 0 .$$ Suppose that $Q$ is a degree $-m$ line bundle quotient of $h^*\Omega_{{\mathbb{P}}^n}(1)$ $$0 \rightarrow S \rightarrow h^*\Omega_{{\mathbb{P}}^n}(1) \rightarrow Q \rightarrow 0$$ with kernel $S$. Then $S$ is a vector sub-bundle of $h^*\Omega_{{\mathbb{P}}^n}(1)$ of degree $m-e$. Composing $S$ with the inclusion $h^*\Omega_{{\mathbb{P}}^n}(1) \rightarrow {\mathcal{O}}_C^{n+1}$ realizes $S$ as a vector sub-bundle of ${\mathcal{O}}_C^{n+1}$, with quotient $Q'$ of rank $2$ and degree $e-m$. By the universal property of projective space, this gives a map from $\psi: {\mathbb{P}}(Q') \to {\mathbb{P}}^n$ whose image contains $h(C)$. If the map $\psi$ is generically injective, the scroll has degree $e-m$. If $\psi$ is generically $d_0$ to one, then the scroll has degree $\frac{e-m}{d_0}$.
\[cor-degreeOfrk1Quot\] Let $h: C \to {\mathbb{P}}^n$ be a generically injective map of degree $e$ from a smooth curve $C$ to ${\mathbb{P}}^n$. Assume that $h(C)$ does not lie on a surface scroll of degree less than $k$. Then any rank $1$ quotient of $h^* \Omega_{{\mathbb{P}}^n}(1)$ (not necessarily a line bundle) has degree at least $k-e$.
Let $Q$ be a rank $1$ quotient, and let $L$ be the line bundle given by $Q$ mod torsion. Then $\deg Q \geq \deg L \geq e-k$ by Proposition \[prop-Scrolls\].
Let $X \subset {\mathbb{P}}^3$ be a very general degree $d$ surface. Let $h: C \to X$ be the normalization of a complete intersection curve of type $(d, k)$ in $X$. Assume that $C$ has genus $g$. Then by Corollary \[cor-mapToNormalBundle\], there is a map $\alpha: h^* \Omega_{{\mathbb{P}}^n}(1) \to N_{h/X}$ with torsion cokernel. Let $Q$ be the image of $\alpha$. Then since $Q$ injects into $N_{h/X}$, $Q$ is a rank 1 quotient of $h^* \Omega_{{\mathbb{P}}^n}(1)$ with $\deg Q \leq \deg N_{h/X}$. In order to use Corollary \[cor-degreeOfrk1Quot\], we need to understand the smallest possible degree of a scroll containing $h(C)$.
If $k \leq d$, then the degree $k$ hypersurface containing $h(C)$ is the smallest degree surface containing $h(C)$, so any scroll containing $h(C)$ will have degree at least $k$. If $d > k$, then the only irreducible surface of degree less than $k$ containing $h(C)$ is $X$. Since $d \geq 5$ and $X$ is very general, $X$ is irreducible and of general type. In particular, $X$ is not a scroll. Thus, the complete intersection curve $h(C)$ lies on no surface scrolls of degree less than $k$. Hence, by Corollary \[cor-degreeOfrk1Quot\], $$\deg N_{h/X} \geq \deg Q \geq k-dk.$$
On the other hand, we know that $$\deg N_{h/X} = dk(4-d)+2g-2.$$ Rearranging the inequality $$dk(4-d)+2g-2 \geq k-dk ,$$ we obtain $$2g-2 \geq dk(d-5)+k$$ or $$g \geq \frac{dk(d-5)+k}{2}+1$$ as desired.
If one had a better bound for the minimal degree of the universal line of a scroll containing $h(C)$, one could get better genus bounds. In particular, if the degree were at least $k+s$, then we would get a genus bound of $$g \geq \frac{dk(d-5)+k+s}{2} + 1 .$$ One can ask if the cone over $h(C)$ with vertex at the most singular point of $h(C)$ is the scroll containing $h(C)$ with minimal degree of the universal line. If this were true, then the corresponding genus bound would be $$g \geq \frac{dk(d-4)-\lfloor \sqrt{dk^2+1} \rfloor +1}{2} .$$
[ABCH]{} R. Brody. Compact manifolds in hyperbolicity. Trans. Amer. Math. Soc. 235(1978), 213–219.
C. Ciliberto and M. Zaidenberg. Scrolls and hyperbolicity. Internat. J. Math. 24(2013), no. 3
H. Clemens and Z. Ran. Twisted genus bounds for subvarieties of generic hypersurfaces. Amer. J. Math. 126(2004), no. 1, 89–120.
I. Coskun. The arithmetic and the geometry of Kobayashi hyperbolicity. Snowbird lectures in algebraic geometry. Contemp. Math., 388(2005), 77–88.
J.P. Demailly. Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. In Algebraic geometry Santa Cruz 1995, volume 62 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, (1997), 285–360.
J.P. Demailly. Recent results on the Kobayashi and Green-Griffiths-Lang conjectures. arXiv:1801.04765
J.P. Demailly and J. El Goul. Hyperbolicity of generic surfaces of high degree in projective 3-space. Amer. J. Math. 122(2000), 515–546.
L. Ein. Subvarieties of generic complete intersections. Invent. Math. 94(1988), 163–169.
L. Ein. Subvarieties of generic complete intersections II. Math. Ann. 289(1991), 465–471.
M. McQuillan. Holomorphic curves on hyperplane sections of 3-folds. Geom. Funct. Anal. (1999), no. 9, p. 370–392.
G. Pacienza. Rational curves on general projective hypersurfaces. J. Algebraic Geom. 12(2003), no. 2, 245–267.
G. Pacienza. Subvarieties of general type on a general projective hypersurface. Trans. Amer. Math. Soc. 356(2004), no. 7, 2649–2661.
Y.T. Siu. Hyperbolicity in complex geometry. The legacy of Niels Henrik Abel. Springer, Berlin, 2004, p. 543–566.
C. Voisin. On a conjecture of Clemens on rational curves on hypersurfaces. J. Differential Geom. 44(1996), 200–214.
C. Voisin. A correction: “On a conjecture of Clemens on rational curves on hypersurfaces". J. Differential Geom. 49(1998), 601–611.
G. Xu. Subvarieties of general hypersurfaces in projective space. J. Differential Geom. 39(1994), 139–172.
G. Xu. Divisors in generic complete intersections in projective space. Trans. Amer. Math. Soc. 348(1996), 2725–2736.
[^1]: During the preparation of this article the first author was partially supported by the NSF grant DMS-1500031 and NSF FRG grant DMS 1664296 and the second author was partially supported by the NSF RTG grant DMS-1246844.
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---
abstract: 'We investigate the existence of ground states with prescribed mass for the focusing nonlinear Schrödinger equation with $L^2$-critical power nonlinearity on noncompact quantum graphs. We prove that, unlike the case of the real line, for certain classes of graphs there exist ground states with negative energy for a whole interval of masses. A key role is played by a thorough analysis of Gagliardo-Nirenberg inequalities and on estimates of the optimal constants. Most of the techniques are new and suited to the investigation of variational problems on metric graphs.'
author:
- |
Riccardo Adami[^1], Enrico Serra, Paolo Tilli\
\
[Dipartimento di Scienze Matematiche “G.L. Lagrange”, Politecnico di Torino ]{}\
[Corso Duca degli Abruzzi, 24, 10129 Torino, Italy]{}
title: |
Negative energy ground states\
for the $L^2$-critical NLSE on metric graphs
---
[AMS Subject Classification: 35R02, 35Q55, 81Q35, 49J40.]{}
[Keywords: Minimization, metric graphs, critical growth, nonlinear Schrödinger\
1.65cm Equation.]{}
Introduction
============
In this paper we investigate the existence of ground states for the *critical* NLS energy functional $$\label{NLSe} E (u,{\mathcal G})
= \frac 1 2 \| u' \|^2_{L^2 ({\mathcal G})}
- \frac 1 6 \| u \|^6_{L^6 ({\mathcal G})} =\frac 1 2 \int_{\mathcal G}|u'|^2dx
-\frac 1 6 \int_{\mathcal G}|u|^6 dx$$ on a noncompact metric graph ${\mathcal G}$, under the [*mass constraint*]{} $$\label{mass} \| u \|^2_{L^2 ({\mathcal G})} \ = \ \mu.$$ The subcritical case, where the $L^6$ norm is replaced by an $L^p$ norm with $p\in (2,6)$, has been investigated in [@ast; @ast2]. The energy in is *critical* in the sense that, under the mass-preserving transformations $$u(x)\quad\mapsto\quad u_\lambda(x):=\lambda^{1/2} \,u(\lambda\, x)\qquad
(\lambda>0),$$ the kinetic and the potential terms in scale in the same way, namely $$\label{scalE}
E(u_\lambda,\lambda^{-1}{\mathcal G}) \quad=\quad
\lambda^2\, E(u,{\mathcal G}),$$ which is typical of critical problems with a strong loss of compactness.
Throughout the paper, ${\mathcal G}$ denotes a *noncompact metric graph*, i.e. a connected metric space obtained by gluing together, by the identification of some of their endpoints, a finite number of closed line intervals (not necessarily bounded), according to the topology of a graph, self-loops and multiple edges being allowed. Any bounded edge $e$ is identified with an interval $[0,\ell_e]$, while unbounded edges are referred to as “half-lines”, and are identified with (copies of) the positive half-line ${{\mathbb R}}^+=[0,+\infty)$; at least one edge is assumed to be unbounded, so that ${\mathcal G}$ is noncompact (two very special cases are when ${\mathcal G}={{\mathbb R}}^+$ and when ${\mathcal G}={{\mathbb R}}$, the latter being obtained by gluing together two copies of ${{\mathbb R}}^+$). We refer to Section \[sec2\] (see also [@berkolaiko; @exner; @ast]) for more details.
In this framework, by a “ground state of mass $\mu$” we mean a solution to the minimization problem $$\label{minprob} \min_{u\in H^1_\mu({\mathcal G})} E(u,{\mathcal G}),\qquad
H^1_\mu({\mathcal G}):=\left\{ u\in H^1({\mathcal G}) :\,\Vert
u\Vert_{L^2({\mathcal G})}^2=\mu\right\},$$ for which it is clearly sufficient to work with real valued, nonnegative functions. Obviously ground states solve, for some $\omega \in {{\mathbb R}}$, the stationary quintic NLS equation $$u'' + |u|^4u = \omega u$$ on each edge of ${\mathcal G}$, with Kirchhoff boundary conditions at the vertices (see Prop. 3.3 in [@ast]).
The existence of ground states for a given $\mu$ is strictly related to the behavior of the *ground-state energy level* function $$\label{defelevel}
{{\mathcal E}}_{\mathcal G}(\mu)=\inf_{u\in H^1_\mu({\mathcal G})} E(u,{\mathcal G}),\quad\mu\geq 0,$$ which will play a central role throughout this paper.
As is wellknown (see Sec. 2), when ${\mathcal G}={{\mathbb R}}$ there exists a *critical mass* ${\mu_{{\mathbb R}}}$ such that the minimization problem has a solution if and only if $\mu={\mu_{{\mathbb R}}}$, and the same occurs when ${\mathcal G}={{\mathbb R}}^+$ (with a *smaller* critical mass ${\mu_{{{\mathbb R}}^+}}={\mu_{{\mathbb R}}}/2$). This severe restriction is due to the scaling rule and the dilation-invariance of ${{\mathbb R}}$ and ${{\mathbb R}}^+$. Thus, when ${\mathcal G}={{\mathbb R}}$ or ${\mathcal G}={{\mathbb R}}^+$, the minimization process is extremely unstable and, in a sense, of little interest.
When ${\mathcal G}$ is a generic (noncompact) metric graph, however, the problem can be highly nontrivial and, depending on the topology of ${\mathcal G}$, entirely new phenomena may arise, such as problem having solutions if, and only if, $\mu$ belongs to some *whole interval* of masses.
For each graph ${\mathcal G}$ we can define, in a natural way, a *critical mass* ${\mu_{\mathcal G}}$, that depends on ${\mathcal G}$ via the best constant $K_{\mathcal G}$ in the Gagliardo-Nirenberg inequality , and it turns out that ${\mu_{{{\mathbb R}}^+}}\leq {\mu_{\mathcal G}}\leq{\mu_{{\mathbb R}}}$, so that ${{\mathbb R}}^+$ and ${{\mathbb R}}$ are extremal graphs, as concerns the critical mass (see Proposition \[intermediate\]). The mass ${\mu_{\mathcal G}}$ is the precise threshold such that ${{\mathcal E}}_{\mathcal G}(\mu)<0$ (possibly $-\infty$) as soon as $\mu>{\mu_{\mathcal G}}$ and, on a general ground, a *necessary condition* for the existence of ground states in is that $\mu\in [{\mu_{\mathcal G}},{\mu_{{\mathbb R}}}]$ (see Proposition \[banali\]).
This condition, however, is far from being sufficient: the true nature of problem strongly depends on the topology of ${\mathcal G}$, and the following (mutually exclusive) cases are possible:
- ${\mathcal G}$ has a *terminal point* (a tip, Fig. \[figbaffo\]). Then ${\mu_{\mathcal G}}={\mu_{{{\mathbb R}}^+}}$, and problem has no solution unless $\mu={\mu_{{{\mathbb R}}^+}}$ and ${\mathcal G}$ is isometric to ${{\mathbb R}}^+$.
- ${\mathcal G}$ admits a *cycle covering* (Fig. \[figH\]). Then ${\mu_{\mathcal G}}={\mu_{{\mathbb R}}}$, and problem has no solution unless $\mu={\mu_{{\mathbb R}}}$ and ${\mathcal G}$ is isometric to ${{\mathbb R}}$ (or to one of a few very special, and completely classified, other structures, see Theorem 2.5 in [@ast]).
- ${\mathcal G}$ has exactly one half-line and no terminal point (Fig. \[figonehalf\]). Then ${\mu_{\mathcal G}}={\mu_{{{\mathbb R}}^+}}$, and problem has a solution if and only if $\mu\in({\mu_{{{\mathbb R}}^+}},{\mu_{{\mathbb R}}}]$.
- In all other cases (Fig. \[figD\]): if ${\mu_{\mathcal G}}<{\mu_{{\mathbb R}}}$, then problem has a solution if and only if $\mu\in[{\mu_{\mathcal G}},{\mu_{{\mathbb R}}}]$.
at (-.5,2) \[nodo\] (02) ; at (2,2) \[nodo\] (22) ; at (2,4) \[nodo\] (24) ; at (3.6,1.6) \[nodo\] (42) ; at (3,3) \[nodo\] (33) ; at (5,2) \[nodo\] (52) ; at (3,3) \[nodo\] (32) ; at (4,3) \[nodo\] (43) ; at (-1,4) \[nodo\] (04) ; at (2,4) \[nodo\] (24) ; at (.5,1) \[nodo\] (11) ; at (2,0) \[nodo\] (20) ; at (4,0) \[nodo\] (40) ; at (-4,2) \[minimum size=0pt\] (meno) ; at (-4,4) \[minimum size=0pt\] (menoalt) ; at (7.9,2) \[minimum size=0pt\] (piu) ; at (-4.1,2) \[infinito\] (infmeno) [$\scriptstyle\infty$]{}; at (-4.1,4) \[infinito\] (infmenoalt) [$\scriptstyle\infty$]{}; at (8,2) \[infinito\] (infpiu) [$\scriptstyle\infty$]{};
at (6,4) \[nodo\] (term); (43)–(term);
(02)–(04); (04)–(24); (04)–(22); (24)–(22); (02)–(11); (11)–(22); (11)–(20); (20)–(22); (22)–(33); (24)–(33); (24)–(43); (33)–(43); (43)–(52); (33)–(42); (20)–(42); (20)–(40); (40)–(42); (42)–(52); (40)–(52); (02)–(meno); (52)–(piu); (04)–(menoalt);
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Some remarks are in order, to better clarify the scope of this scenario (the precise statements, which are the main results of the paper, are given in Theorems \[teobaffo\], \[teoH\], \[teounasemi\] and \[quartocaso\]).
In the first two cases ground states, as a rule, do not exist. In case [(a)]{}, the presence of a tip –hence of a terminal edge– allows the construction of “monotone” functions of mass ${\mu_{{{\mathbb R}}^+}}$, that decrease away from the tip and mimic a half-soliton on ${{\mathbb R}}^+$, with an energy level arbitrarily close to zero (albeit strictly positive, unless ${\mathcal G}$ is exactly ${{\mathbb R}}^+$): thus, in a sense, graphs with a tip behave much like a half-line. In case [(b)]{}, by contrast, the covering assumption is not compatible with “monotone” functions and, due to a rearrangement argument from ${\mathcal G}$ to ${{\mathbb R}}$, no function of mass ${\mu_{{\mathbb R}}}$ can have a negative energy on ${\mathcal G}$. This rigidity rules out ground states, unless ${\mathcal G}$ supports a soliton, and this, in turn, occurs only when ${\mathcal G}={{\mathbb R}}$ (possibly with the identification of some pairs of points, in a way compatible with the even symmetry of a soliton). Thus, dually, a graph as in [(b)]{} behaves much like ${{\mathbb R}}$.
The last two cases are, on the contrary, extremely nontrivial. In [(c)]{}, ${\mathcal G}$ consists of a compact core ${\mathcal K}$ (with no terminal edge) attached to a half-line, the simplest example being the “tadpole” graph in Fig. \[figtadpole\].
at (-2.5,0) \[infinito\] (1) [$\scriptstyle\infty$]{}; at (1.8,0) \[nodo\] (2) ; (1) – (2) ; (2.2,0) circle (0.4);
If $\mu\in ({\mu_{{{\mathbb R}}^+}},{\mu_{{\mathbb R}}}]$, a ground state of mass $\mu$ always exists –with a *strictly negative* energy– and this is a completely new phenomenon. Over ${{\mathbb R}}^+$, due to , one has ${{\mathcal E}}_{{{\mathbb R}}^+}(\mu)=-\infty$ (and no ground state) as soon as $\mu>{\mu_{{{\mathbb R}}^+}}$: here, on the contrary, the compact core ${\mathcal K}$ attached to ${{\mathbb R}}^+$ has the effect of a *stabilizer* as regards ground states: due to ${\mathcal K}$, ${\mathcal G}$ loses dilation invariance, and high concentration is no longer energetically convenient, which accounts for *strictly negative* (yet finite!) ground-state energy levels.
Finally, in [(d)]{}, ${\mathcal G}$ has no tip, no cycle covering and (being noncompact) at least two half-lines. This case becomes very interesting if one *further* assumes that ${\mu_{\mathcal G}}\!<\!{\mu_{{\mathbb R}}}$, which guarantees the existence of ground states for every mass $\mu\in [{\mu_{\mathcal G}},{\mu_{{\mathbb R}}}]$: here, contrary to [(c)]{}, a ground state exists also when $\mu={\mu_{\mathcal G}}$, with a zero energy level. A particularly interesting, and specific, feature of the case $\mu={\mu_{\mathcal G}}$ is the coexistence of compact and noncompact minimizing sequences. Thus an additional difficulty in this case is the [*choice*]{} of a proper sequence to work with.
Explicit examples of graphs can be constructed (e.g. the “signpost" graph in Fig. \[figD\], as explained in Sec. 3), where ${\mu_{\mathcal G}}\!<\!{\mu_{{\mathbb R}}}$: this extra assumption, however, is crucial to prove the existence of ground states, and we believe that it cannot be dropped in general.
More precisely we believe that, within case [(d)]{}, the *sole* topology of ${\mathcal G}$ is not enough, in general, to establish whether ${\mu_{\mathcal G}}<{\mu_{{\mathbb R}}}$ or ${\mu_{\mathcal G}}={\mu_{{\mathbb R}}}$ and, in the latter case, whether a ground states exists, of mass ${\mu_{\mathcal G}}$. It is an open problem, at present, to fully understand problem in case [(d)]{}, when ${\mu_{\mathcal G}}={\mu_{{\mathbb R}}}$.
Thus, summing up, the four cases [(a)]{}–[(d)]{} cover all the possible topologies of a (noncompact) metric graph ${\mathcal G}$. The first two cases are extremely rigid, with ground states being the exception rather than the rule. Case [(c)]{} is, on the contrary, very interesting, with ground states in a universal range of prescribed masses, and one may consider such graphs as “intermediate” between ${{\mathbb R}}^+$ and ${{\mathbb R}}$. Finally, case [(d)]{} is also nontrivial (with ground states in a whole, closed interval of prescribed masses), but this is subordinated to the condition that ${\mu_{\mathcal G}}<{\mu_{{\mathbb R}}}$: in this case ${\mathcal G}$ shows, again, an intermediate behavior between ${{\mathbb R}}^+$ and ${{\mathbb R}}$. To conclude this discussion we wish to emphasize that all the ground states of cases [(c)]{} and [(d)]{} (except those of mass ${\mu_{\mathcal G}}$) have [*negative energy*]{}. This is in sharp contrast with the behavior of the NLS equation on ${{\mathbb R}}$ (or ${{\mathbb R}}^n$), where every solution with negative energy blows up in finite time ([@cazenave]), and will be the object of a forthcoming paper.
The problem of the minimization of under the constraint can be interpreted according to the Gross-Pitaevskii theory for the ground state of the Bose-Einstein condensates. Indeed, following a series of results obtained in the last decade by several authors (see e.g. [@ls; @lsy; @ly; @agt; @esy1; @esy2; @esy3; @pickl; @bos; @holmer]), under some physical conditions the dynamics of a gas of interacting identical bosons can be described through a one-body nonlinear equation, called [*Gross-Pitaevskii equation*]{}. More precisely, if the particles in the gas interact in pairs, then the resulting equation displays a [*cubic*]{} nonlinearity. On the other hand, in [@chen] a system of identical bosons interacting through a [*three*]{}-body potential was considered, and the resulting equation was shown to be a [*quintic*]{} NLS.
In actual dilute Bose-Einstein condensates, two- and three-body interactions may coexist, so that in the resulting one-body equation both cubic and quintic terms would arise, even though normally the effect of the cubic term overwhelms the effect of the quintic, that is therefore neglected. Concerning the sign of the nonlinear terms, it turns out to depend on the character attractive or repulsive of the interaction among the particles, so that it is possible to realize experimentally condensates that display either a focusing or a defocusing behaviour ([@wieman]).
For these reasons, and since both the functional and the $L^2$-norm are conserved by the evolution driven by the quintic NLS equation $$\label{quintic}
i \partial_t u (t,x) \ = \ - u'' (t,x) - |u(t,x)|^4 u (t,x),$$ the issue of the existence of a minimizer of the constrained energy can be interpreted as the search for the ground state of a particular Bose-Einstein condensate where two-body interactions are absent.
With respect to the problems currently studied as regards ground states of a Bose-Einstein condensate and to the rigorous derivation of equation obtained in [@chen], let us stress two remarkable differences: first, we consider a [*focusing*]{} nonlinearity, second, we set the problem on a graph. Both features have nowadays an established experimental counterpart: on the one hand, self-concentrating condensates are currently realised [@wieman], on the other hand, condensation on graph-like structures has been recently observed in [@lorenzo].
While linear dynamics on quantum graphs is nowadays a well-known branch of mathematical physics (see e.g. [@berkolaiko; @exner; @post]), its nonlinear counterpart has gained an increasing interest only quite recently. A seminal study of nonlinear evolution equation on ramified structures appeared in [@ali] and then was extended to several physical domains, involving various mathematical issues [@bona; @vonbelow]. The study of the evolution of solitary waves on star graphs was performed in [@acfn1], while the search for ground states states was carried out in [@matrasulov; @sobirov; @acfn2; @acfn3; @acfn4] for the case of star graphs, and then extended to more general graphs (dealing with the same class considered in this paper) in [@ast; @ast2; @sven]. Stationary states and related bifurcation were recently investigated in [@noja; @marzuola; @pelinovsky]. In particular, [@schneider] treats the case of periodic graphs, which is not covered in this paper, and shows the occurrence of a bifurcation phenomenon. The integrability of the cubic NLS on star graphs was proved in [@caudrelier]. All cited papers deal with the subcritical case or even specialize to the cubic case. To our knowledge, the present work is the first contribution to the understanding of the role of the $L^2$-criticality in the framework of graphs. The fact that this role appears to be different from what happens on standard domains like ${{\mathbb R}}^n$ is in our opinion worth being stressed.
The paper is organised as follows: in Section 2 we give some preliminary results and introduce the notion of critical mass; in Section 3 we state the four main theorems. The core of the proofs is a result stated and proved in Section 4. Finally, in Section 5 we conclude the proofs of the main existence results stated in Section 3.
Notation and preliminary results {#sec2}
================================
It is well known ([@cazenave]) that when ${\mathcal G}={{\mathbb R}}$, the ground-state energy level function, as defined in , has a sharp transition from $0$ to $-\infty$, corresponding to a special value ${\mu_{{\mathbb R}}}$ of the mass, known as the *critical mass*: $$\label{Rlevel}
{{\mathcal E}}_{{\mathbb R}}(\mu) = \begin{cases}
0 & \text{ if } \mu \le \mu_{{\mathbb R}}\\
-\infty & \text{ if } \mu > \mu_{{\mathbb R}}\end{cases}\quad\qquad \left({\mu_{{\mathbb R}}}= \pi\sqrt 3 /2\right).$$ Furthermore, the infimum ${{\mathcal E}}_{{\mathbb R}}(\mu)$ is attained (i.e. a ground state exists in ) [*if and only if*]{} $\mu = \mu_{{\mathbb R}}$. Thus every ground state $u$ (necessarily of mass ${\mu_{{\mathbb R}}}$) satisfies $E(u,{{\mathbb R}}) =0$. The ground states, called *solitons*, form a quite large family: up to phase multiplication and translations, they can be written as $$\label{solitoni}
\phi_\lambda(x) = \sqrt\lambda \phi(\lambda x), \qquad \lambda>0,$$ where $$\label{solit}
\phi (x) = \hbox{\rm{sech}}^{1/2}\left(2x/{\sqrt 3}\right).$$
When ${\mathcal G}={{\mathbb R}}^+$ (the positive half-line), the situation is similar, but with the proper critical mass ${\mu_{{{\mathbb R}}^+}}={\mu_{{\mathbb R}}}/2$: $${{\mathcal E}}_{{{\mathbb R}}^+}(\mu) =
\begin{cases} 0 & \text{ if } \mu \le {\mu_{{{\mathbb R}}^+}}\\
-\infty & \text{ if } \mu > {\mu_{{{\mathbb R}}^+}}\end{cases}
\quad\qquad \left({\mu_{{{\mathbb R}}^+}}= \pi\sqrt 3 /4\right).$$ Again, ground states of mass $\mu$ exist if and only if $\mu={\mu_{{{\mathbb R}}^+}}$. They are called “half-solitons”, as they are the restrictions to ${{\mathbb R}}^+$ of the family $\phi_\lambda$.
Thus, when ${\mathcal G}$ is ${{\mathbb R}}$ or ${{\mathbb R}}^+$, problem is trivialized by these sharp transitions.
For a general noncompact graph ${\mathcal G}$, the behavior of ${{\mathcal E}}_{\mathcal G}(\mu)$ is strictly related to the Gagliardo–Nirenberg inequality $$\label{GN}
\| u\|_{L^6({\mathcal G})}^6 \le K_{\mathcal G}\,\| u\|_{L^2({\mathcal G})}^4\, \| u'\|_{L^2({\mathcal G})}^2,\quad
\forall u\in H^1({\mathcal G})$$ valid for every noncompact graph (see [@ast2; @tentarelli]). The number $K_{\mathcal G}$ is the *best constant* that one can put in , namely, $$\label{defKG}
K_{\mathcal G}=\sup_{u\in H^1({\mathcal G}) \atop u\not\equiv 0}
\frac{\| u\|_{L^6({\mathcal G})}^6}{\|u\|_{L^2({\mathcal G})}^4\cdot\|u'\|_{L^2({\mathcal G})}^2}
=\sup_{u\in H_\mu^1({\mathcal G})}\; \frac {\Vert u\Vert_{L^6({\mathcal G})}^6}
{\mu^2\cdot\Vert u'\Vert_{L^2({\mathcal G})}^2},$$ where the last equality follows from homogeneity.
The role of this constant, in connection with the behavior of ${{\mathcal E}}_{\mathcal G}(\mu)$, is clear: recalling and the definition of $H^1_\mu({\mathcal G})$ in , using we have, for every $u\in H_\mu^1({\mathcal G})$, $$E(u,{\mathcal G})\geq \frac12 \|u'\|_{L^2({\mathcal G})}^2 -\frac16 K_{\mathcal G}\mu^2 \|u'\|_{L^2({\mathcal G})}^2 =
\frac16 \Vert u'\Vert_{L^2({\mathcal G})}^2 \bigr(3-K_{\mathcal G}\,\mu^2\bigr).$$ We then see that $$\label{elevelpos}
\mu^2 \le 3/K_{\mathcal G}\implies E(u,{\mathcal G}) \ge 0 \qquad\hbox{for all}\quad u\in H_\mu^1({\mathcal G}).$$ Note also that $$\label{Epos}
\mu^2 < 3/K_{\mathcal G}\implies E(u,{\mathcal G}) > 0\qquad\hbox{for all}\quad u\in H_\mu^1({\mathcal G}).$$ On the other hand, if $\mu^2 >3/K_{\mathcal G}$, say $\mu^2 =3(1+\delta)/K_{\mathcal G}$ for some $\delta>0$, we can take $u \in H_\mu^1({\mathcal G})$ close to optimality in , say $$\| u\|_{L^6({\mathcal G})}^6 > \frac{K_{\mathcal G}}{1+\delta} \,\mu^2\, \| u'\|_{L^2({\mathcal G})}^2,$$ to obtain $$\label{dopo}
E(u,{\mathcal G})
< {\frac}1 6 \Vert u'\Vert_{L^2({\mathcal G})}^2 \bigr(3-\frac{K_{\mathcal G}}{1+\delta} \,\mu^2\bigr) = 0.$$ This shows that $$\label{elevelneg}
\mu^2 > 3/K_{\mathcal G}\implies E(u,{\mathcal G}) < 0 \qquad\hbox{for some}\quad u\in H_\mu^1({\mathcal G}).$$ Now and justify the following definition.
\[critmass\] The [*critical mass*]{} for a noncompact metric graph ${\mathcal G}$ is the number $${\mu_{\mathcal G}}= \sqrt{3/ K_{\mathcal G}}.$$
This definition, of course, gives the correct critical mass when ${\mathcal G}$ is ${{\mathbb R}}$ or ${{\mathbb R}}^+$. In general, from and , we see that ${\mu_{\mathcal G}}$ is the precise mass threshold, after which the ground-state energy level ${{\mathcal E}}_{\mathcal G}(\mu)$ becomes negative (possibly $-\infty$).
\[remSC\] Any noncompact metric graph ${\mathcal G}$ has (at least) one unbounded edge which, in turn, contains arbitrarily large intervals. Therefore, any function $v\in H^1({{\mathbb R}})$ having *compact support* can be regarded as an element of $H^1({\mathcal G})$, by placing the support of $v$ inside a half-line of ${\mathcal G}$, and setting $v\equiv 0$ outside. Thus, in a sense, $H^1({\mathcal G})$ “contains” a dense subset of $H^1({{\mathbb R}})$.
Next we notice that any noncompact ${\mathcal G}$ is, in a way, intermediate between ${{\mathbb R}}^+$ and ${{\mathbb R}}$ in the sense of the following statement.
\[intermediate\] Let ${\mathcal G}$ be a noncompact graph, and let ${\mu_{\mathcal G}}$ be the critical mass for ${\mathcal G}$. Then $$\label{intermu}
{\mu_{{{\mathbb R}}^+}}\, \le\, {\mu_{\mathcal G}}\, \le \,{\mu_{{\mathbb R}}}$$ or, equivalently, $$\label{interK}
K_{{\mathbb R}}\,\le \, K_{\mathcal G}\,\le \,K_{{{\mathbb R}}^+}.$$ Moreover, we also have $$\label{interE}
{{\mathcal E}}_{{{\mathbb R}}^+}(\mu) \,\le \, {{\mathcal E}}_{\mathcal G}(\mu)\,\le \,{{\mathcal E}}_{{{\mathbb R}}}(\mu),\quad
\forall \mu>0.$$
Given $u \in H^1({\mathcal G})$ (assume $u\geq 0$ and $u\not\equiv 0$), let $u^* \in H^1({{\mathbb R}}^+)$ be its decreasing rearrangement on ${{\mathbb R}}^+$ (for properties of rearrangements on graphs see [@ast; @friedlander]). Since $$\| (u^*)'\|_{L^2({{\mathbb R}}^+)}^2\leq\| u'\|_{L^2({\mathcal G})}^2,
\quad
\| u^*\|_{L^p({{\mathbb R}}^+)}^p=\| u\|_{L^p({\mathcal G})}^p
\quad\forall p,$$ the quotient for $u$ in does not exceed the same quotient for $u^*$ (over ${{\mathbb R}}^+$): since the latter is bounded by $K_{{{\mathbb R}}^+}$, we obtain that $K_{\mathcal G}\le K_{{{\mathbb R}}^+}$. In the same way, we have $E(u^*,{{\mathbb R}}^+)\leq E(u,{\mathcal G})$, and hence ${{\mathcal E}}_{{{\mathbb R}}^+}(\mu)\leq E(u,{\mathcal G})$, where $\mu:=\| u\|_{L^2({\mathcal G})}^2=\| u^*\|_{L^2({{\mathbb R}}^+)}^2$. By the arbitrariness of $u$, we obtain the first inequality in .
Finally, let $H^1_{\mu,c}$ denote the set of all $u\in H^1_\mu({{\mathbb R}})$ with compact support. By a density argument, we have $${{\mathcal E}}_{{\mathbb R}}(\mu)
=\inf_{u\in H^1_{\mu,c}({{\mathbb R}})} E(u,{{\mathbb R}})
\geq \inf_{u\in H^1_\mu({\mathcal G})} E(u,{\mathcal G})
={{\mathcal E}}_{\mathcal G}(\mu)$$ (the inequality follows from Remark \[remSC\]). This proves the second inequality in ; the first inequality in is proved in the same way, working with the supremum in .
After this discussion, we summarize in the next proposition the properties that hold for a generic noncompact graph, without any additional assumption.
\[banali\] Let ${\mathcal G}$ be a noncompact metric graph.
1. If $\mu\le {\mu_{\mathcal G}}$, then ${{\mathcal E}}_{\mathcal G}(\mu) = 0$, and is never attained when $\mu<{\mu_{\mathcal G}}$.
2. If $\mu >{\mu_{\mathcal G}}$, then ${{\mathcal E}}_{\mathcal G}(\mu) < 0$ (possibly $-\infty$).
3. If $\mu > {\mu_{{\mathbb R}}}$, then ${{\mathcal E}}_{\mathcal G}(\mu) = -\infty$.
When $\mu \le {\mu_{\mathcal G}}$, shows that ${{\mathcal E}}_{\mathcal G}(\mu) \ge 0$. On the other hand, we infer from that ${{\mathcal E}}_{\mathcal G}(\mu)\leq {{\mathcal E}}_{{\mathbb R}}(\mu)$, and the latter is zero due to , since $\mu\leq {\mu_{\mathcal G}}\leq {\mu_{{\mathbb R}}}$ by . Moreover, by we see that ${{\mathcal E}}_{\mathcal G}(\mu)$ is not attained when $\mu < {\mu_{\mathcal G}}$. This proves (i).
By , one immediately obtains (ii).
Finally, to prove (iii), observe that ${{\mathcal E}}_{\mathcal G}(\mu)\leq {{\mathcal E}}_{{\mathbb R}}(\mu)=-\infty$, due to and .
\[cor1\] A necessary condition for the existence of a ground state of mass $\mu$ in is that $\mu\in[{\mu_{\mathcal G}},{\mu_{{\mathbb R}}}]$.
Statement of the main results
=============================
In this section we state the main results of the paper (Theorems \[teobaffo\]–\[quartocaso\]), thus providing a precise and formal setting for the four possible cases [(a)]{}–[(d)]{}, that were informally described in the Introduction.
The following theorem covers case [(a)]{}, represented in Fig. \[figbaffo\].
By a “terminal point” (or “tip”) we mean a point $x$, in the metric graph ${\mathcal G}$, that corresponds to a *vertex of degree one* in the underlying (combinatorial) graph. Usually, $x$ is one of the two endpoints of a *bounded* edge attached to the rest of ${\mathcal G}$ only at the other endpoint, as a pendant: the only exception is when ${\mathcal G}$ consists of exactly one unbounded edge (i.e. when ${\mathcal G}={{\mathbb R}}^+$), in which case the tip $x$ is the origin of the half-line. We point out that the $\infty$-point of any half-line of ${\mathcal G}$ (though being a vertex of degree one in the underlying combinatorial graph) is *not* a terminal point, since it is not a point of ${\mathcal G}$ (as a metric graph).
\[teobaffo\] Let ${\mathcal G}$ be a noncompact metric graph having at least one terminal point (a tip). Then ${\mu_{\mathcal G}}= {\mu_{{{\mathbb R}}^+}}$. When $\mu \in ({\mu_{{{\mathbb R}}^+}}, {\mu_{{\mathbb R}}}]$, ${{\mathcal E}}_{\mathcal G}(\mu) = -\infty$. When $\mu = {\mu_{{{\mathbb R}}^+}}$, ${{\mathcal E}}_{\mathcal G}(\mu)=0$ and it is attained if and only if ${\mathcal G}$ is isometric to a half-line.
The next theorem covers case [(b)]{}, when ${\mathcal G}$ admits a cycle covering (see Fig. \[figH\]).
Here and throughout, by “cycle” we mean either a *loop* (a homeomorphic image of $\mathcal S^1$) or, by extension, an unbounded path that joins two (necessarily distinct) $\infty$-points of ${\mathcal G}$. In the former case the cycle corresponds to a closed path in the underlying combinatorial graph (i.e. it is a “cycle” in the usual sense of graph theory) whereas, in the latter case, it does not (the two notions would essentially coincide, however, if properly reformulated for the one-point compactification of ${\mathcal G}$). Alternatively, in the underlying combinatorial graph, one might identify all the $\infty$-points of ${\mathcal G}$ into a unique, special vertex (of degree equal to the number of half-lines of ${\mathcal G}$): in this way, also cycles of the second type would be usual cycles in the graph-theoretic sense.
The existence of a cycle covering is equivalent (see [@ast; @ast2]) to a property of ${\mathcal G}$, called “assumption (H)”, first identified in [@ast] as a topological obstruction to the existence of ground states in the subcritical cases. In particular, this assumption is incompatible with the presence of tips and forces the graph to have at least two half-lines.
\[teoH\] Let ${\mathcal G}$ be a noncompact metric graph that admits a cycle covering. Then ${\mu_{\mathcal G}}= {\mu_{{\mathbb R}}}$. The infimum ${{\mathcal E}}_{\mathcal G}(\mu)$ is attained if and only if $\mu={\mu_{{\mathbb R}}}$ and ${\mathcal G}$ is ${{\mathbb R}}$ or a “tower of bubbles” (one of the special graphs described in Example 2.4 of [@ast]).
A different behaviour occurs if one considers graphs without terminal points and with one half-line only: the critical mass turns out to coincide with $\mu_{{{\mathbb R}}^+}$, but the ground state energy level remains finite if the mass does not exceed $\mu_{{\mathbb R}}$. Furthermore, in the interval $(\mu_{{{\mathbb R}}^+}, \mu_{{\mathbb R}}]$ a ground state exists and it has negative energy.
\[teounasemi\] Let ${\mathcal G}$ be a noncompact metric graph having exactly one half-line and no terminal point. Then ${\mu_{\mathcal G}}={\mu_{{{\mathbb R}}^+}}$. The infimum ${{\mathcal E}}_{\mathcal G}(\mu)$ is attained if and only if $\mu \in({\mu_{{{\mathbb R}}^+}},{\mu_{{\mathbb R}}}]$.
The last theorem deals with the remaining cases, under the additional hypothesis $\mu_{\mathcal G}< \mu_{{\mathbb R}}$. Notice that, for such graphs, the interval of masses where a ground state exists, is closed.
\[quartocaso\] Let ${\mathcal G}$ be a noncompact metric graph with no terminal point, having at least two half-lines and admitting no cycle covering. If $\mu_{\mathcal G}<\mu_{{\mathbb R}}$, then for every $\mu\in [{\mu_{\mathcal G}},{\mu_{{\mathbb R}}}]$, the infimum ${{\mathcal E}}_{\mathcal G}(\mu)$ is attained.
The class of graphs satisfying the assumptions of Theorem \[quartocaso\] is not empty, since it contains, for instance, the “signpost" graph ${\mathcal G}$ of Fig. \[figD\]. To see this we take a soliton $\phi$ (necessarily of mass ${\mu_{{\mathbb R}}}$) as defined in and we apply the procedure detailed in Sec. 3 of [@ast2]. This produces a function $u \in H^1_{\mu_{{\mathbb R}}} ({\mathcal G})$ such that $E(u, {\mathcal G}) < E (\phi, {{\mathbb R}}) =0$. Therefore, ${{\mathcal E}}_{\mathcal G}({\mu_{{\mathbb R}}}) < 0$ and, by Proposition \[banali\], we conclude that ${\mu_{\mathcal G}}<{\mu_{{\mathbb R}}}$.
\[remGN\] The last four theorems also provide an answer to the question of existence of extremal functions for the Gagliardo-Nirenberg inequality . The key observation is that, for any noncompact graph ${\mathcal G}$, the following two conditions are in fact equivalent:
- there exists $u\in H^1({\mathcal G})$, $u\not\equiv 0$, achieving equality in ;
- the infimum ${{\mathcal E}}_{\mathcal G}({\mu_{\mathcal G}})$ is attained by a ground state of mass ${\mu_{\mathcal G}}$.
Indeed, by homogeneity, a function extremal for can be supposed to have mass ${\mu_{\mathcal G}}$: on the other hand, since $K_{\mathcal G}\,{\mu_{\mathcal G}}^2=3$, optimality in combined with $\Vert u\Vert_{L^2}^2={\mu_{\mathcal G}}$ is equivalent to $E(u,{\mathcal G})=0$, i.e. to $u$ being a ground state, by Proposition \[banali\].
We end this section with the short proofs of the first two theorems. The last two theorems are, on the contrary, much more involved, and to their proofs are devoted the last sections of the paper.
Let $\mu >{\mu_{{{\mathbb R}}^+}}$. For every ${\varepsilon}>0$, there exists $u \in H_\mu^1({{\mathbb R}}^+)$ with compact support such that $$\label{nearly}
\frac {\| u\|_{L^6({{\mathbb R}}^+)}^6} {\mu^2\cdot\| u'\|_{L^2({{\mathbb R}}^+)}^2} \ge K_{{{\mathbb R}}^+} -{\varepsilon}.$$ Replacing $u$ by $u_\lambda(x) = \sqrt \lambda u(\lambda x)$ with $\lambda$ large, we can assume that the support of $u_\lambda$ is contained in an interval shorter than a terminal edge of ${\mathcal G}$. Then the function $u_\lambda$ can be seen as an element of $H_\mu^1({\mathcal G})$ (place the support of $u$ on a terminal edge of ${\mathcal G}$, and set $u_\lambda\equiv 0$ elsewhere on ${\mathcal G}$). Consequently, the quotient in the preceding inequality does not exceed $K_{\mathcal G}$. Thus, for every ${\varepsilon}>0$, $K_{\mathcal G}\ge K_{{{\mathbb R}}^+} - {\varepsilon}$ and, recalling , we obtain $K_{\mathcal G}= K_{{{\mathbb R}}^+}$, that is, ${\mu_{\mathcal G}}= {\mu_{{{\mathbb R}}^+}}$.
To prove that ${{\mathcal E}}_{\mathcal G}(\mu)= -\infty$, notice that since $K_{\mathcal G}= K_{{{\mathbb R}}^+}$, readily implies, for ${\varepsilon}$ small, that $E(u_\lambda,{\mathcal G}) <0$ (as in ). Therefore $$E(u_\lambda,{\mathcal G}) = E(u_\lambda,{{\mathbb R}}^+) = \lambda^2 E(u,{{\mathbb R}}^+) \to -\infty$$ as $\lambda \to +\infty$. Thus, ${{\mathcal E}}_{\mathcal G}(\mu) = -\infty$.
Finally, let $\mu = {\mu_{{{\mathbb R}}^+}}$. If ${\mathcal G}={{\mathbb R}}^+$, plainly ${{\mathcal E}}_{{{\mathbb R}}^+}(\mu_{{{\mathbb R}}^+})$ is attained by the half-solitons. Conversely, assume that there exists $u \in H^1_{\mu_{{{\mathbb R}}^+}}({{\mathcal G}})$ such that $E(u,{\mathcal G}) = {{\mathcal E}}_{\mathcal G}({\mu_{{{\mathbb R}}^+}}) = 0$. Its decreasing rearrangement $u^*$ is in $H^1_ {\mu_{{{\mathbb R}}^+}}({{{\mathbb R}}^+})$ and satisfies $E(u^*,{{\mathbb R}}^+) \le E(u,{\mathcal G}) = 0$. Since ${{\mathcal E}}_{{{\mathbb R}}^+}({\mu_{{{\mathbb R}}^+}}) = 0$, the function $u^*$ must be a half-soliton. From $E(u,{\mathcal G}) = E(u^*,{{\mathbb R}}^+)$ we deduce (via Proposition 3.1 of [@ast]) that almost every point in the range of $u$ has exactly one preimage. As $u$ and $u^*$ are equimeasurable, it follows that $u$ is injective, which means that $u$ is supported on a subset of a half-line. But as $E(u,{\mathcal G}) = 0$, also $u$ must be a half-soliton and this is possible only if there are no vertices other than the ends of the half-line, i.e., ${\mathcal G}$ is (isometric to) ${{\mathbb R}}^+$.
Take any nonnegative $v\in H^1({\mathcal G})\setminus \{0\}$, and let $\widehat v\in H^1({{\mathbb R}})$ denote its symmetric rearrangement on ${{\mathbb R}}$. Since ${\mathcal G}$ admits a cycle covering (namely, it satisfies assumption (H)), we have (see Proposition 3.1 in [@ast]) $$\| \widehat{v}'\|_{L^2({{\mathbb R}})} \le \| v'\|_{L^2({\mathcal G})}$$ and therefore $$\frac {\| v\|_{L^6({\mathcal G})}^6} {\| v\|_{L^2({\mathcal G})}^4 \cdot \| v'\|_{L^2({\mathcal G})}^2} \le
\frac {\|\widehat{v}\|_{L^6({{\mathbb R}})}^6} {\|\widehat{v}\|_{L^2({{\mathbb R}})}^4 \cdot \| \widehat{v}'\|_{L^2({{\mathbb R}})}^2}
\le K_{{\mathbb R}},$$ showing that $K_{\mathcal G}\le K_{{\mathbb R}}$. By we obtain $K_{\mathcal G}= K_{{\mathbb R}}$, namely $\mu_{\mathcal G}= {\mu_{{\mathbb R}}}$.
If $\mu\ne {\mu_{{\mathbb R}}}={\mu_{\mathcal G}}$, Proposition \[banali\] shows that ${{\mathcal E}}_{\mathcal G}(\mu)$ is not attained.
If $\mu = {\mu_{{\mathbb R}}}$, a soliton can be placed on ${{\mathbb R}}$ or on any tower of bubbles (see [@ast]), showing that those graphs carry a ground state. Conversely, assume that there exists $u \in H^1_{{\mu_{{\mathbb R}}}}({{\mathcal G}})$ such that $E(u,{\mathcal G}) = {{\mathcal E}}_{\mathcal G}({\mu_{{\mathbb R}}}) = 0$. Since ${\mathcal G}$ satisfies assumption (H), the symmetric rearrangement $\widehat u$ is in $H^1_ {{\mu_{{\mathbb R}}}}({{\mathbb R}})$ and satisfies $E(\widehat u,{{\mathbb R}}) \le E(u,{\mathcal G}) = 0$. Since ${{\mathcal E}}_{{{\mathbb R}}}({\mu_{{\mathbb R}}}) = 0$, the function $\widehat u$ is a soliton. Furthermore, since $E(u,{\mathcal G}) = E(\widehat u, {{\mathbb R}})$, almost every point in the range of $u$ has exactly two preimages. These features show, as in Theorem 2.5 of [@ast], that ${\mathcal G}$ must be one of the special graphs of [@ast].
A general existence argument
============================
We now present a general existence result for ground states, which is the common core of the proofs of Theorems \[teounasemi\] and \[quartocaso\], as long as ${\mu_{\mathcal G}}<\mu\leq{\mu_{{\mathbb R}}}$.
\[mainprop\] Let ${\mathcal G}$ be a noncompact metric graph with no terminal point, such that $\mu_{\mathcal G}<\mu_{{\mathbb R}}$. For every $\mu \in ({\mu_{\mathcal G}},{\mu_{{\mathbb R}}}]$ the infimum ${{\mathcal E}}_{\mathcal G}(\mu)$ is attained.
The proof is quite involved, and will rely on the following three lemmas.
\[weakzero\] Assume ${\mathcal G}$ is noncompact and let $v_n\in H^1({\mathcal G})$ be a sequence of functions such that $v_n \rightharpoonup 0$ in $H^1({\mathcal G})$. Then, as $n \to \infty$, $$\label{weaklevel}
E(v_n,{\mathcal G}) \ge \frac12\left( 1 - \frac{\|v_n\|_{L^2({\mathcal G})}^4}{\mu_{{\mathbb R}}^2}\right)\|v_n'\|_{L^2({\mathcal G})}^2 +o(1).$$
Let ${\mathcal K}$ be the compact core of ${\mathcal G}$ (i.e., the compact metric graph obtained from ${\mathcal G}$ by removing the interior of every halfline) and set $${\varepsilon}_n := \max_{x\in {\mathcal K}} |v_n(x)|,
\qquad
w_n(x) = \begin{cases}
v_n(x)+{\varepsilon}_n &\text{if $v_n(x)\leq-{\varepsilon}_n$}\\
0&\text{if $|v_n(x)|<{\varepsilon}_n$}\\
v_n(x)-{\varepsilon}_n &\text{if $v_n(x)\geq{\varepsilon}_n$}
\end{cases}$$ Since $v_n \rightharpoonup 0$ in $H^1({\mathcal G})$, we have $v_n \to 0$ in $L^{\infty}_{\rm loc}({\mathcal G})$ and hence ${\varepsilon}_n\to 0$. Moreover, as $\Vert v_n-w_n\Vert_{L^\infty}\leq{\varepsilon}_n$, from equiboundedness in $L^2({\mathcal G})$ we have $$\label{cof}
\|v_n -w_n\|_{L^6({\mathcal G})} \to 0.$$ Now let ${\mathcal H}$ be an arbitrary halfline of ${\mathcal G}$. By choosing a coordinate $t\geq 0$ on it, we can identify ${\mathcal H}$ with $[0,+\infty)$ and, since $w_n\equiv 0$ on ${\mathcal K}$ and ${\mathcal H}$ is attached to ${\mathcal K}$ at $t=0$, we have $w_n(0)=0$. Setting $w_n(t)\equiv 0$ for $t<0$, the restriction of $w_n$ to ${\mathcal H}$ can be seen as a function in $H^1({{\mathbb R}})$ and, as such, it must obey the following Gagliardo-Nirenberg iequality: $$\|w_n\|_{L^6({\mathcal H})}^6
\le 3 \frac{\| w_n\|_{L^2({\mathcal H})}^4}{\mu_{{\mathbb R}}^2}
\| w_n'\|_{L^2({\mathcal H})}^2
\leq
3 \frac{\| w_n\|_{L^2({\mathcal G})}^4}{\mu_{{\mathbb R}}^2}
\| w_n'\|_{L^2({\mathcal H})}^2.$$ Summing over all the halflines of ${\mathcal G}$, since $w_n\equiv 0$ on ${\mathcal K}$ we obtain $$\|w_n\|_{L^6({\mathcal G})}^6
\le 3 \frac{\| w_n\|_{L^2({\mathcal G})}^4}{\mu_{{\mathbb R}}^2}
\| w_n'\|_{L^2({\mathcal G})}^2$$ (just as it would be if we had ${\mu_{\mathcal G}}={\mu_{{\mathbb R}}}$). Since $\| w_n'\|_{L^2({\mathcal G})}^2 \le \| v_n'\|_{L^2({\mathcal G})}^2$ and $\| w_n\|_{L^2({\mathcal G})}^2 \le \| v_n\|_{L^2({\mathcal G})}^2$, from and the previous inequality we obtain, $$\|v_n\|_{L^6({\mathcal G})}^6 +o(1) = \|w_n\|_{L^6({\mathcal G})}^6 \le 3 \frac{\|v_n\|_{L^2({\mathcal G})}^4 }{\mu_{{\mathbb R}}^2}
\| v_n'\|_{L^2({\mathcal G})}^2,$$ and follows immediately from the definition of $E(v_n,{\mathcal G})$.
\[limit\] Let ${\mathcal G}$ be a noncompact graph, and take $\mu \in [\mu_{\mathcal G}, \mu_{{\mathbb R}}]$. be a nonnegative minimizing sequence for $E(\, \cdot\,,{\mathcal G})$ such that $
u_n \rightharpoonup u\quad\text {in } H^1({\mathcal G})
$, for some $u\in H^1({\mathcal G})$.
If $u\not\equiv 0$, then $u\in H^1_\mu({\mathcal G})$ and $u$ is a minimizer.
Passing to a subsequence, we may assume that $u_n(x) \to u(x)$ a.e. in ${\mathcal G}$. By a standard use of the Brezis–Lieb Lemma ([@BL]), as in [@ast2], $$E(u_n,{\mathcal G}) = E(u_n-u,{\mathcal G}) + E(u,{\mathcal G}) + o(1)$$ as $n\to \infty$. Since $u_n-u \rightharpoonup 0$ in $H^1({\mathcal G})$, from Lemma \[weakzero\] applied with $v_n=u_n-u$ we obtain $$E(u_n-u,{\mathcal G}) \ge \frac12\left(1 - \frac{\| u_n - u\|_{L^2({\mathcal G})}^4}{\mu_{{\mathbb R}}^2}\right) \| u_n' - u'\|_{L^2({\mathcal G})}^2 + o(1)$$ as $n\to \infty$. Therefore $$E(u_n,{\mathcal G}) \ge E(u, {\mathcal G}) + o(1),$$ that is, $$E(u,{\mathcal G}) \le {{\mathcal E}}_{\mathcal G}(\mu).$$ Now, by semicontinuity, we have $m:=\Vert u\Vert_{L^2({\mathcal G})}^2\leq\mu$, and $m>0$ by assumption. If $m<\mu$, then $$E\left(\sqrt{\frac{\mu}{m}}u,{\mathcal G}\right) =\frac{\mu}{m}\frac12\int_{\mathcal G}|u'|^2{{\,dx}}- \left(\frac{\mu}{m}\right)^3\frac16 \int_{\mathcal G}|u|^6{{\,dx}}<
\frac{\mu}{m}E(u,{\mathcal G})\le {{\mathcal E}}_{\mathcal G}(\mu),$$ since $\mu/m>1$, $\Vert u\Vert_{L^6({\mathcal G})}>0$ and ${{\mathcal E}}_{\mathcal G}(\mu) \le 0$. This contradicts the definition of ${{\mathcal E}}_{\mathcal G}(\mu)$. Then it must be $m=\mu$, namely $u$ is the required minimizer. It is also easy to see that the convergence of $u_n$ to $u$ is strong in $H^1({\mathcal G})$.
The next lemma establishes a crucial modification of the Gagliardo–Nirenberg inequality. Its proof is quite long, and is therefore split in a series of steps.
\[modGN\] Assume ${\mathcal G}$ is noncompact and has no terminal point, and let $u\in H^1_\mu({\mathcal G})$ for some $\mu\in (0,{\mu_{{\mathbb R}}}]$. Then there exists a number ${\theta}\in [0,\mu]$ such that $$\label{sharpineq}
\|u\|_{L^6({\mathcal G})}^6 \le 3\left(\frac{\mu-{\theta}}{{\mu_{{\mathbb R}}}}\right)^2 \|u'\|_{L^2({\mathcal G})}^2 +C{\theta}^{1/2},$$ where $C>0$ is a constant that depends only on ${\mathcal G}$.
Replacing $u$ with $|u|$, we may assume that $u\geq 0$ and $u\not\equiv 0$.
*Step 1.* There exist $\ell>0$ (depending only on ${\mathcal G}$) and a function $\psi\in H^1((-\infty,\ell])$, such that
1. $\int_{-\infty}^\ell |\psi|^2{{\,dx}}= \int_{\mathcal G}|u|^2{{\,dx}}= \mu$
2. $\int_{-\infty}^\ell |\psi|^6{{\,dx}}= \int_{\mathcal G}|u|^6{{\,dx}}$, $\int_{-\infty}^\ell |\psi'|^2 {{\,dx}}\le \int_{\mathcal G}|u'|^2 {{\,dx}}$;
3. $\psi$ is nonnegative on $(-\infty,\ell]$, and nonincreasing on $[0,\ell]$;
If ${\mathcal G}$ has at least a loop (a homeomorphic image of $\mathcal S^1$), let $2\ell$ denote the length of the shortest loop; otherwise, let $\ell:=1$. Let us denote by $M$ the maximum of $u$ on ${\mathcal G}$, and by $x_0$ a point of ${\mathcal G}$ such that $u(x_0) = M$: since ${\mathcal G}$ is noncompact and connected, there is a path $\Gamma$ in ${\mathcal G}$ that joins an $\infty$-point of ${\mathcal G}$ to $x_0$.
Since $x_0$ is not a terminal point of ${\mathcal G}$, the path $\Gamma$ can be prolonged beyond $x_0$, to a longer path that crosses $x_0$. Two cases are possible: (i) $\Gamma$ can be prolonged beyond $x_0$ for a length $\ell$, or (ii) a self intersection occurs (i.e. a loop is created) before an extra length of $\ell$ has been traveled.
In case (i), let $\gamma$ denote the new path (of length $\ell$, starting at $x_0$) used to prolong $\Gamma$ (note: $\gamma$ shares with $\Gamma$ only the point $x_0$). We then define the function $\psi:[0,\ell]\to{{\mathbb R}}$ as the decreasing rearrangement of $u_\gamma$ (i.e. $u$ restricted to $\gamma$), so that $$\psi(0)=M,\quad \int_0^\ell |\psi'|^2\,dx\leq
\int_\gamma |u'|^2\,dx,
\quad
\int_0^\ell |\psi|^p\,dx=
\int_\gamma |u|^p\,dx\quad\forall p.$$ Now assume, for a while, that the metric graph $\overline{{\mathcal G}\setminus\gamma}$ (the closure of ${\mathcal G}\setminus\gamma$) is *connected*: in this case, we can consider the function $u^*:[0,+\infty)\to{{\mathbb R}}$, defined as the decreasing rearrangement of the restricted function $u_{\overline{{\mathcal G}\setminus\gamma}}$, and observe that, as before, $$u^*(0)=M,\quad \int_0^\infty |(u^*)'|^2\,dx\leq
\int_{{\mathcal G}\setminus \gamma} |u'|^2\,dx,
\quad
\int_0^\infty |u^*|^p\,dx=
\int_{{\mathcal G}\setminus\gamma} |u|^p\,dx\quad\forall p.$$ Then the construction of $\psi$ is easily completed, by letting $\psi(x)=u^*(-x)$ for every $x<0$.
In general, though, $\overline{{\mathcal G}\setminus\gamma}$ can be disconnected (which would prevent the use of the monotone rearrangement): one of its connected component ${\mathcal G}_0$, however, contains the original path $\Gamma$ and, since $u\geq 0$ and $\Gamma$ contains a half-line along which $u$ tends to zero, the range of $u_\Gamma$ ($u$ restricted to $\Gamma$) is the interval $[0,M]$, that is, the full range of $u$. Therefore, by a graph–surgery procedure, every other connected component ${\mathcal G}_j$ ($j>0$) of $\overline{{\mathcal G}\setminus\gamma}$ can still be “reattached” to $\Gamma$ (hence to ${\mathcal G}_0$) as follows: if ${\mathcal G}_j$ was originally attached to $\gamma$ at a point $y_j$ (not necessarily unique), take $z_j\in\Gamma$ such that $u(z_j)=u(y_j)$, and attach $y_j$ at $z_j$ (i.e., identify $y_j$ with $z_j$). By this trick (which preserves the continuity of $u$, its integral norms and those of $u'$) the restricted function $u_{\overline{{\mathcal G}\setminus\gamma}}$ can now be seen as if defined on a connected graph, and the theory of rearrangements becomes available: then, the construction of $\psi$ can be completed as before.
In case (ii), let $x_1\in{\mathcal G}$ be the point where the self-intersection occurs, and let $\gamma$ denote the new-added curve, from $x_0$ to $x_1$. The length of $\gamma$ is at most $\ell$, otherwise we would be in case (i), and hence (since ${\mathcal G}$ has no loop of length smaller than $2\ell$) we see that $x_1\in\Gamma$ (in other words, the self-intersection occurs along $\Gamma$, not along $\gamma$, which is a simple arc). For the same reason, the length of the complementary arc $\gamma'\subset\Gamma$, from $x_1$ to $x_0$, is *at least* $\ell$. But then, observing that $\Gamma':=(\Gamma\setminus\gamma')\cup\gamma$ is still a path from an $\infty$-point of ${\mathcal G}$ to $x_0$, the proof can be completed as in case (i), with $\Gamma'$ and (a suitable portion of) $\gamma'$ (the latter prolonging the former, for a length of $\ell$) now playing the roles of the original $\Gamma$ and $\gamma$.
*Step 2.* There exist $x_0 \in [\ell/2,\ell)$ and $v\in H^1({{\mathbb R}}^+)$ such that, defining $$\label{defM}{\theta}:= \frac12\int_{x_0}^\ell \psi^2{{\,dx}},$$ there hold:
- $v(0) = \psi(0)$;
- $\displaystyle\int_0^\infty |v|^2{{\,dx}}= \int_0^\ell |\psi|^2{{\,dx}}- {\theta}$;
- $\displaystyle \int_0^\infty |v'|^2{{\,dx}}\le \int_0^\ell |\psi'|^2{{\,dx}}+ C{\theta}^{1/2}$;
- $\displaystyle \int_0^\infty |v|^6{{\,dx}}\ge \int_0^\ell |\psi|^6{{\,dx}}- C{\theta}$;
the constant $C>0$ depends only on ${\mathcal G}$.
Here we shall work with $\psi$ restricted to the interval $[0,\ell]$.
We first prove the existence of $x_0\in [\ell/2,\ell)$ such that $$\label{eq55}
|\psi(x_0)|^4 \leq \frac {64 m^{1/2}}{\ell^2}\left(\int_{x_0}^\ell |\psi|^2\,dx\right)^{3/2},\qquad
m:=\int_{\ell/2}^\ell |\psi|^2\,dx.$$ To see this, let $F(x)=\int_x^\ell |\psi|^2{{\,dx}}$. If the inequality in were false for every $x_0\in [\ell/2,\ell)$, we would have $$-F'(x) = \psi(x)^2 > \frac {8 m^{1/4}}{\ell} \left(\int_x^\ell |\psi|^2\,dx\right)^{3/4}= \frac {8 m^{1/4}}{\ell} F(x)^{3/4}$$ for every $ x\in [\ell/2,\ell)$. Therefore, $$-\left(F(x)^{1/4}\right)' >\frac {2m^{1/4}}{\ell} \qquad\forall x\in [\ell/2,\ell)$$ and, since $F(\ell)=0$, integration over $(\ell/2,\ell)$ yields $$F(\ell/2)^{1/4}> \frac {2 m^{1/4}}{\ell} \cdot \frac \ell 2 = m^{1/4},$$ which is clearly a contradiction due to how $m$ and $F$ were defined.
We now take a point $x_0$ satisfying , and define ${\theta}$ as in . If ${\theta}=0$, then $\psi$ vanishes on $[x_0,\ell]$; in this case we define $v$ by extending $\psi$ to $0$ on $[\ell,+\infty)$. If $\theta \ne 0$, we define $v: [0,+\infty) \to {{\mathbb R}}$ as $$v(x)=\begin{cases}
\psi(x) & \text{if $0\leq x \leq x_0$,}\\
\psi(x_0)e^{-\lambda (x-x_0)}&\text{if $x>x_0$}
\end{cases}$$ where $$\lambda:=\frac {|\psi(x_0)|^2}{2{\theta}}.$$ Clearly $v\in H^1({{\mathbb R}}^+)$ and $v(0) = \psi(0)$, so that $i)$ is satisfied. Next, $$\begin{aligned}
\int_0^\infty |v|^2 {{\,dx}}&= \int_0^{x_0} |\psi|^2 {{\,dx}}+ |\psi(x_0)|^2 \int_{x_0}^\infty e^{-2\lambda(x-x_0)} {{\,dx}}= \int_0^{x_0} |\psi|^2 {{\,dx}}+ \frac {|\psi(x_0)|^2}{2\lambda}\nonumber \\
&= \int_0^{x_0} |\psi|^2 {{\,dx}}+ {\theta}= \int_0^\ell |\psi|^2 {{\,dx}}- {\theta},\end{aligned}$$ and $ii)$ is proved. Similarly, $$\begin{aligned}
\int_0^\infty |v'|^2 {{\,dx}}&= \int_0^{x_0} |\psi'|^2 {{\,dx}}+ \lambda^2|\psi(x_0)|^2 \int_{x_0}^\infty e^{-2\lambda(x-x_0)} {{\,dx}}= \int_0^{x_0} |\psi'|^2 {{\,dx}}+ \frac {\lambda|\psi(x_0)|^2}{2} \nonumber \\
&= \int_0^{x_0} |\psi'|^2 {{\,dx}}+ \frac{|\psi(x_0)|^4}{4{\theta}} \le \int_0^\ell |\psi'|^2 {{\,dx}}+\frac{32 (2m)^{1/2}}{\ell^2} {\theta}^{1/2} \nonumber \\
&\le \int_0^\ell |\psi'|^2 {{\,dx}}+ C{\theta}^{1/2}\end{aligned}$$ since $m\le {\mu_{{\mathbb R}}}= \pi\sqrt 3 /2$ by , while $\ell$ depends only on ${\mathcal G}$. This proves $iii)$. Finally, $$\begin{aligned}
\label{v6}
\int_0^\infty |v|^6 {{\,dx}}&= \int_0^{x_0} |\psi|^6 {{\,dx}}+ |\psi(x_0)|^6 \int_{x_0}^\infty e^{-6\lambda(x-x_0)} {{\,dx}}= \int_0^{x_0} |\psi|^6 {{\,dx}}+ \frac {|\psi(x_0)|^6}{6\lambda} \nonumber \\
& \ge \int_0^\ell |\psi|^6 {{\,dx}}- \int_{x_0}^\ell |\psi|^6{{\,dx}}.\end{aligned}$$ Since $\psi$ is decreasing (on $[0,\ell]$) and $x_0\geq \ell/2$, we have $$\int_{x_0}^\ell |\psi|^6\,dx \le |\psi(x_0)|^4 \int_{x_0}^\ell |\psi|^2{{\,dx}}\le 2{\theta}|\psi(\ell/2)|^4$$ and $$|\psi(\ell/2)|^2\leq \frac 2 {\ell} \int_0^{\ell/2} |\psi|^2\,dx
< \frac {2\mu}\ell \le C$$ as above, which plugged into the previous inequality gives $$\int_{x_0}^\ell |\psi|^6\,dx \le C{\theta}.$$ Therefore reads $$\int_0^\infty |v|^6 {{\,dx}}\ge \int_0^\ell |\psi|^6 {{\,dx}}-C{\theta},$$ and this concludes the proof.
*Step 3.* Combining $\psi$ and $v$, we now define $$w(x)=\begin{cases}
\psi(x) & \text{if $x \le 0$,}\\
v(x) &\text{if $x>0$.}
\end{cases}$$ Clearly $w\in H^1({{\mathbb R}})$ and, by the properties of $\psi$ and $v$, $$\int_{{\mathbb R}}|w|^2{{\,dx}}= \int_{-\infty}^0 |\psi|^2{{\,dx}}+ \int_0^\infty |v|^2{{\,dx}}= \int_{-\infty}^\ell |\psi|^2{{\,dx}}- {\theta}=
\int_{\mathcal G}|u|^2{{\,dx}}-{\theta}=\mu - {\theta}.$$
By the Gagliardo–Nirenberg inequality , $$\begin{aligned}
\label{GNw}
\|w\|_{L^6({{\mathbb R}})}^6 &\le K_{{\mathbb R}}\left(\mu -{\theta}\right)^2 \|w'\|_{L^2({{\mathbb R}})}^2 = K_{{\mathbb R}}{\mu_{{\mathbb R}}}^2 \left(\frac{\mu-{\theta}}{{\mu_{{\mathbb R}}}}\right)^2 \|w'\|_{L^2({{\mathbb R}})}^2\nonumber \\
& = 3\left(\frac{\mu-{\theta}}{{\mu_{{\mathbb R}}}}\right)^2 \|w'\|_{L^2({{\mathbb R}})}.\end{aligned}$$ Now, still from the properties of $\psi$ and $v$, $$\begin{aligned}
\label{stima6}
\|w\|_{L^6({{\mathbb R}})}^6 &= \int_{-\infty}^0 |\psi|^6{{\,dx}}+ \int_0^\infty |v|^6{{\,dx}}\nonumber \\
&\ge \int_{-\infty}^0 |\psi|^6{{\,dx}}+ \int_0^\ell |\psi|^6 {{\,dx}}- C{\theta}= \|u\|_{L^6({\mathcal G})}^6 -C{\theta}\end{aligned}$$ and $$\begin{aligned}
\label{stimader}
\|w'\|_{L^2({{\mathbb R}})}^2 &= \int_{-\infty}^0 |\psi'|^2{{\,dx}}+ \int_0^\infty |v'|^2{{\,dx}}\nonumber \\
& \le \int_{-\infty}^0 |\psi'|^2{{\,dx}}+ \int_0^\ell |\psi'|^2 {{\,dx}}+ C{\theta}^{1/2}
\le \|u'\|_{L^2({\mathcal G})}^2 +C{\theta}^{1/2}.\end{aligned}$$ Inserting and in we obtain $$\|u\|_{L^6({\mathcal G})}^6 -C{\theta}\le 3\left(\frac{\mu-{\theta}}{{\mu_{{\mathbb R}}}}\right)^2 \left( \|u'\|_{L^2({\mathcal G})}^2 +C{\theta}^{1/2}\right).$$ Rearranging terms and observing that ${\theta}\le {\theta}^{1/2}{\mu_{{\mathbb R}}}^{1/2} = C{\theta}^{1/2}$, one obtains .
We are now in a position to prove Proposition \[mainprop\].
Fix a mass $\mu\in({\mu_{\mathcal G}},{\mu_{{\mathbb R}}}]$. We know from Proposition \[banali\] that ${{\mathcal E}}_{\mathcal G}(\mu)<0$ (possibly $-\infty$, until proven otherwise), thus we only consider functions $u\in H^1_\mu({\mathcal G})$ such that $E(u,{\mathcal G}) \le -\alpha $ for some fixed $\alpha>0$ that depends on $\mu$. Let $u$ be any of these functions: since $\mu \le {\mu_{{\mathbb R}}}$, Lemma \[modGN\] applies, and yields $$\|u\|_{L^6({\mathcal G})}^6 \le 3\left(1-\frac{{\theta}_u}{{\mu_{{\mathbb R}}}}\right)^2 \|u'\|_{L^2({\mathcal G})}^2 +C{\theta}_u^{1/2}$$ for some ${\theta}_u\in (0,\mu)$. We have denoted by ${\theta}_u$ the constant ${\theta}$ appearing in to stress its dependence on $u$. Rearranging terms we obtain $$3\frac{{\theta}_u}{{\mu_{{\mathbb R}}}}\left(2-\frac{{\theta}_u}{\mu_{{\mathbb R}}}\right)\|u'\|_{L^2({\mathcal G})}^2-C{\theta}_u^{1/2} \le 6E(u,{\mathcal G}) \le -6\alpha,$$ and (since ${\theta}_u <\mu\leq{\mu_{{\mathbb R}}}$) this shows that ${\theta}_u$ is bounded away from zero (in terms of $C$ and $\alpha$, uniformly with respect to $u$). Once this is established, the same inequality also shows that $ \|u'\|_{L^2({\mathcal G})}$ is bounded from above, in terms of $C$ and $\alpha$. We have thus proved that whenever $E(u,{\mathcal G}) \le \alpha <0$, the $L^2$ norm of $u'$ on ${\mathcal G}$ is uniformly bounded (the bounding constant depends only on ${\mathcal G}$ and $\mu$, via $C$ and $\alpha$). By the Gagliardo–Nirenberg inequality the same holds for $\|u\|_{L^p({\mathcal G})}$, for every $p\in [2,+\infty]$. In particular, ${{\mathcal E}}_{\mathcal G}(\mu)$ is finite.
Finally, we show that ${{\mathcal E}}_{\mathcal G}(\mu)$ is attained.
Let $u_n \in H_{\mu}^1({\mathcal G})$ be a minimizing sequence for $E(\,\cdot\, , {\mathcal G})$. By the preceding argument we can assume that $\|u_n'\|_{L^2({\mathcal G})}$, $\|u_n\|_{L^6({\mathcal G})}$ and $\|u_n\|_{L^\infty({\mathcal G})}$ are bounded independently of $n$. Up to subsequences, $u_n\rightharpoonup u$ in $H^1({\mathcal G})$, as well as $u_n\to u$ in $L^q_{\rm loc}({\mathcal G})$ for every $q\in [2,\infty]$.
If $u \equiv 0$, by Lemma \[weakzero\], $$E(u_n, {\mathcal G}) \ge \frac12\left(1 - \frac{\mu^2}{\mu_{{\mathbb R}}^2}\right) \| u_n'\|_{L^2({\mathcal G})}^2 + o(1) \ge o(1),$$ since $\mu \le \mu_{{\mathbb R}}$. This implies ${{\mathcal E}}_{\mathcal G}(\mu) \ge 0$, which is false. Thus, the weak limit $u$ is not identically zero and, by Lemma \[limit\], is the required minimizer.
Proof of the main results
=========================
In this section, building on Proposition \[mainprop\], we present the proofs of Theorems \[teounasemi\] and \[quartocaso\].
Let $\phi$ be the function defined by , thought of as a half-soliton on ${{\mathbb R}}^+$ and notice that $\phi(0) = 1$. Identify ${\mathcal H}$, the unique half-line of ${\mathcal G}$, with the interval $[0,+\infty)$ and for ${\varepsilon}>0$ set $$u_{\varepsilon}(x) = \begin{cases} \sqrt {\varepsilon}\phi ({\varepsilon}x) & \text{ if $x\in{\mathcal H}$ (i.e. if $x \in [0,+\infty)$)} \\
\sqrt {\varepsilon}& \text{ elsewhere on } {\mathcal G}. \end{cases}$$ Observe that ${\mathcal G}\setminus{\mathcal H}\ne\emptyset$, because ${\mathcal G}$ has no terminal point. Since ${\mathcal G}$ is connected and ${\mathcal H}$ is attached to ${\mathcal G}\setminus{\mathcal H}$ at $x=0$, we see that $u_{\varepsilon}\in H^1 ({\mathcal G})$ and hence $$K_{\mathcal G}\ge\frac {\| u_{\varepsilon}\|_{L^6 ({\mathcal G})}^6}{\| u_ {\varepsilon}\|_{L^2 ({\mathcal G})}^4 \| u_ {\varepsilon}' \|_{L^2 ({\mathcal G})}^2 } =
\frac {\| \phi \|_{L^6 ({{\mathbb R}}^+)}^6 + \ell{\varepsilon}^3 }
{(\| \phi \|_{L^2 ({{\mathbb R}}^+)}^2 + \ell{\varepsilon})^2 \| \phi' \|_{L^2 ({{\mathbb R}}^+)}^2 },$$ where $\ell$ is the measure (i.e. the total length) of ${\mathcal G}\setminus {\mathcal H}$. Since the last quotient tends to $K_{{{\mathbb R}}^+}$ as ${\varepsilon}\to 0$, we have $K_{\mathcal G}\ge K_{{{\mathbb R}}^+}$ and, in fact, equality occurs, by . Thus ${\mu_{\mathcal G}}={\mu_{{{\mathbb R}}^+}}$.
Now the fact that ${{\mathcal E}}_{\mathcal G}(\mu)$ is attained when $\mu \in ({\mu_{{{\mathbb R}}^+}},{\mu_{{\mathbb R}}}]$ is the content of Proposition \[mainprop\].
Finally, since ${\mu_{{{\mathbb R}}^+}}= {\mu_{\mathcal G}}$, ${{\mathcal E}}_{\mathcal G}({\mu_{{{\mathbb R}}^+}})=0$. If $u\in H_{{\mu_{{{\mathbb R}}^+}}}^1({\mathcal G})$ is such that $E(u,{\mathcal G}) = 0$, then its decreasing rearrangement $u^*$ on ${{\mathbb R}}^+$ satisfies $E(u^*,{{\mathbb R}}^+) \le 0$. Then, exactly as in the last part of the proof of Theorem \[teobaffo\], this implies that ${\mathcal G}$ is (isometric to) ${{\mathbb R}}^+$. But this is impossible, since ${\mathcal G}$ has no terminal point.
To complete the proof of Theorem \[quartocaso\] we need the following lemma.
\[bridge\] Assume ${\mathcal G}$ is noncompact, and let ${\mathcal B}$ denote the union of all those edges that do not belong to any cycle. Then, for every $u\in H_\mu^1({\mathcal G})$, $$\label{brdoub}
\int_{\mathcal G}|u|^6\,dx +
3\int_{\mathcal B}|u|^6\,dx\leq
3\left(\frac {\mu+3\mu_{\mathcal B}}{{\mu_{{\mathbb R}}}}\right)^2
\int_{\mathcal G}|u'|^2\,dx,$$ where $$\label{defmuB}
\mu_{\mathcal B}:= \int_{\mathcal B}|u|^2\,dx.$$
If ${\mathcal B}=\emptyset$, then ${\mathcal G}$ admits a cycle covering and, by Theorem \[teoH\], ${\mu_{\mathcal G}}={\mu_{{\mathbb R}}}$: then $\mu_{\mathcal B}=0$ and reduces to the Gagliardo-Nirenberg inequality. So we may assume ${\mathcal B}\not=\emptyset$: the idea behind the proof is that is still the Gagliardo-Nirenberg inequality, but for a modified function $\widetilde u$ on a modified graph $\widetilde{{\mathcal G}}$ where we force a cycle covering.
We construct $\widetilde{{\mathcal G}}$ from ${\mathcal G}$, as follows: for every edge $e\in{\mathcal B}$, we stretch $e$ by a factor $2$, and we *duplicate* the resulting edge, so that, if the original $e$ had length $\ell$, there are now *two* edges, each of length $2\ell$, joining the same two vertices (or emanating from the same vertex, if $e$ is a half-line and $\ell=+\infty$). These two new edges now form a cycle, so that the resulting graph $\widetilde{{\mathcal G}}$ admits a cycle covering.
Given $u\in H^1({\mathcal G})$, we construct $\widetilde{u}\in H^1(\widetilde{{\mathcal G}})$ as follows. First, we let $\widetilde u\equiv u$ on ${\mathcal G}\setminus{\mathcal B}$. Then, for every edge $e\in{\mathcal B}$, we duplicate $u$ (stretched horizontally by a factor $2$) on each of the two copies of the stretched edge $2 e$ of $\widetilde{\mathcal G}$. Choosing a coordinate $x\in [0,\ell]$ on every $e\in{\mathcal B}$ of length $\ell$, this amounts to replacing $u(x)$ over $[0,\ell]$ with *two copies* of $u(x/2)$, over $[0,2 \ell]$ (these intervals are to be replaced with $[0,+\infty)$ when $e$ is a half-line). It is then clear that for every $p\ge 1$, $$\label{eq58}
\int_{\widetilde {\mathcal G}} \left|\widetilde{u}\right|^p\,dx=
\int_{{\mathcal G}\setminus{\mathcal B}} \left|u\right|^p\,dx
+
4 \int_{{\mathcal B}} \left|u\right|^p\,dx = \int_{{\mathcal G}} \left|u\right|^p\,dx
+
3 \int_{{\mathcal B}} \left|u\right|^p\,dx$$ (in particular, when $p=2$ and when $p=6$), and, similarly, $$\label{eq59}
\int_{\widetilde {\mathcal G}} \left|\widetilde{u}'\right|^2\,dx =
\int_{{\mathcal G}\setminus{\mathcal B}} \left|u'\right|^2\,dx + \int_{{\mathcal B}} \left|u'\right|^2\,dx
= \int_{{\mathcal G}} \left|u'\right|^2\,dx.$$ On the other hand, since $\widetilde{{\mathcal G}}$ is covered by cycles, Theorem \[teoH\] gives $\mu_{\widetilde {\mathcal G}}={\mu_{{\mathbb R}}}$, so that $\widetilde u$ is subject to a Gagliardo-Nirenberg inequality that can be written $$\int_{\widetilde {\mathcal G}} \left|\widetilde{u}\right|^6\,dx\leq
3\left(\frac {\int_{\widetilde {\mathcal G}} \left|\widetilde{u}\right|^2\,dx}{{\mu_{{\mathbb R}}}}\right)^2
\int_{\widetilde {\mathcal G}} \left|\widetilde{u}'\right|^2\,dx,$$ and follows immediately using , and .
We recall that ${\mu_{\mathcal G}}<{\mu_{{\mathbb R}}}$ by assumption. When $\mu \in ({\mu_{\mathcal G}},{\mu_{{\mathbb R}}}]$, the fact that ${{\mathcal E}}_{\mathcal G}(\mu)$ is attained follows from Proposition \[mainprop\].
On the other hand, the proof of Proposition \[mainprop\] cannot be adapted to the case where $\mu = {\mu_{\mathcal G}}$, because it is based on the inequality ${{\mathcal E}}_{\mathcal G}(\mu) <0$, that now is replaced by ${{\mathcal E}}_{\mathcal G}({\mu_{\mathcal G}}) =0$. This is not a weakness of the proof: now, indeed, any sequence $u_n\in H^1_\mu({\mathcal G})$ such that $\Vert u_n'\Vert_{L^2}\to 0$ is, by virtue of , a minimizing sequence, so that minimizing sequences are in general noncompact. To get compactness, one has to carefully select the minimizing sequence, as follows.
Let $u_n \in H_{\mu_{\mathcal G}}^1({\mathcal G})$ be a maximizing sequence for the Gagliardo–Nirenberg inequality, namely a sequence such that $$\label{uptoK}
\frac{\|u_n\|_{L^6({\mathcal G})}^6}{\|u_n'\|_{L^2({\mathcal G})}^2}\;
\to\; K_{\mathcal G}\mu_{\mathcal G}^2 = 3$$ as $n\to \infty$. Applying inequality to $u_n$ keeping in mind that $\mu = {\mu_{\mathcal G}}$ and ${\theta}\le \mu \le C$ we can get rid of ${\theta}$ and write (for some other $C$) $$\label{newstima}
\|u_n\|_{L^6({\mathcal G})}^6 \le 3\frac{{\mu_{\mathcal G}}^2}{{\mu_{{\mathbb R}}}^2} \|u_n'\|_{L^2({\mathcal G})}^2 +C.$$ We first notice that $\|u_n'\|_{L^2({\mathcal G})}$ (and hence also $\|u_n\|_{L^6({\mathcal G})}^6$) must be bounded. Indeed, if this is not the case, along a subsequence, $$\lim_n \frac{\|u_n\|_{L^6({\mathcal G})}^6}{\|u_n'\|_{L^2({\mathcal G})}^2} \le \lim_n \left(3\frac{{\mu_{\mathcal G}}^2}{{\mu_{{\mathbb R}}}^2} +\frac{C}{\|u_n'\|_{L^2({\mathcal G})}^2}\right) = 3\frac{{\mu_{\mathcal G}}^2}{{\mu_{{\mathbb R}}}^2} < 3,$$ by , contradicting .
Now since ${\mathcal G}$ has at least two half-lines, by we have
$$\int_{\mathcal G}|u_n|^6\,dx + 3\int_{\mathcal B}|u_n|^6\,dx\leq
3\left(\frac {\mu_{\mathcal G}+3\mu_n}{\mu_{{\mathbb R}}}\right)^2
\int_{\mathcal G}|u_n'|^2\,dx, \quad \mu_n:= \int_{\mathcal B}|u_n|^2\,dx,$$ where ${\mathcal B}$ is defined as in Lemma \[bridge\]. This allows us to show that $\|u_n\|_{L^6({\mathcal G})}$ and $\|u_n'\|_{L^2({\mathcal G})}$ are bounded away from zero. If (for some subsequence) the two norms tend to zero, then clearly also $\mu_n \to 0$. Dividing the preceding inequality by $\|u_n'\|_{L^2({\mathcal G})}$ we obtain $$\frac{\|u_n\|_{L^6({\mathcal G})}^6}{\|u_n'\|_{L^2({\mathcal G})}^2} \le 3\left(\frac {\mu_{\mathcal G}+3\mu_n}{\mu_{{\mathbb R}}}\right)^2 = 3\frac{{\mu_{\mathcal G}}^2}{{\mu_{{\mathbb R}}}^2} +o(1),$$ and then $$\lim_n \frac{\|u_n\|_{L^6({\mathcal G})}^6}{\|u_n'\|_{L^2({\mathcal G})}^2} <3,$$ contradicting again .
Finally, since $u_n$ is bounded in $H^1({\mathcal G})$, writing $$6E(u_n,{\mathcal G}) = 3 \|u_n'\|_{L^2({\mathcal G})}^2 - \|u_n\|_{L^6({\mathcal G})}^6 = \|u_n'\|_{L^2({\mathcal G})}^2\left( 3 - \frac{\|u_n\|_{L^6({\mathcal G})}^6}{\|u_n'\|_{L^2({\mathcal G})}^2}\right)$$ we see by that $E(u_n,{\mathcal G}) \to 0 = {{\mathcal E}}_{\mathcal G}(\mu_{\mathcal G})$. We have thus constructed a minimizing sequence for $E(\,\cdot\,,{\mathcal G})$ which is bounded and uniformly away from zero.
We may then assume that $u_n\rightharpoonup u$ in $H^1({\mathcal G})$, as well as the other usual convergence properties. If $u\equiv 0$, by Lemma \[weakzero\], $$E(u_n, {\mathcal G}) \ge \frac12 \left(1- \frac{{\mu_{\mathcal G}}^2}{{\mu_{{\mathbb R}}}^2}\right) \|u_n'\|_{L^2({\mathcal G})}^2 + o(1).$$ Since $E(u_n,{\mathcal G}) \to 0$, this contradicts the construction of the sequence $u_n$. Therefore the weak limit $u$ does not vanish identically, and by Lemma \[limit\], it is the required minimizer.
[99]{}
Adami, R., Cacciapuoti, C., Finco, D., Noja D.: [Fast solitons on star graphs]{}. Rev. Math. Phys. [**23**]{}(4), 409–451 (2011)
Adami, R., Cacciapuoti, C., Finco, D., Noja D.: On the structure of critical energy levels for the cubic focusing NLS on star graphs. J. Phys. A [**45**]{}(19), 192001 (2012)
Adami, R., Cacciapuoti, C., Finco, D., Noja, D.: [Variational properties and orbital stability of standing waves for NLS equation on a star graph]{}. J. Diff. Eq. [**257**]{}(10), 3738–3777 (2014)
Adami, R., Cacciapuoti, C., Finco, D., Noja, D.: [Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy]{}. J. Diff. Eq. [**260**]{} (10), 7397–7415 (2016)
Adami, R., Golse, F., Teta, A.: [Rigorous derivation of the cubic NLS in dimension one]{}. J. Stat. Phys. [**127**]{}, 1193–1220 (2007)
Adami, R., Serra, E., Tilli, P.: [NLS ground states on graphs]{}. Calc. Var. and PDEs [**54**]{} (1), 743–761 (2015)
Adami, R., Serra, E., Tilli, P.: Threshold phenomena and existence results for NLS ground states on metric graphs. J. Func. An. [**271**]{} (1), 201–223 (2016)
Ali Mehmeti, F.: [Nonlinear waves in networks]{}. Akademie Verlag Berlin (1994)
Below, J. von: [An existence result for semilinear parabolic network equations with dynamical node conditions]{}. In: Pitman Research Notes in Mathematical Series 266, pp. 274–283. Longman, Harlow Essex (1992)
Benedikter, N., de Oliveira, G., Schlein, B.: [Quantitative Derivation of the Gross-Pitaevskii Equation]{}. Comm. Pure App. Math. [**68**]{} (8), 1399–1482 (2015)
Berkolaiko, G., Kuchment, P.: [Introduction to quantum graphs]{}, Mathematical Surveys and Monographs 186. AMS, Providence, RI (2013)
Bona, J., Cascaval, R.C.: [Nonlinear dispersive waves on trees]{}. Can. J. App. Math [**16**]{}, 1–18 (2008)
Brezis, H., Lieb, E.H.: [A relation between pointwise convergence of functions and convergence of functionals]{}. Proc. Amer. Math. Soc. [**88**]{}(3), 486–490 (1983)
Cacciapuoti, C., Finco, D., Noja, D.: [Topology induced bifurcations for the NLS on the tadpole graph]{}. Phys. Rev. E [**91**]{} (1), 013206 (2015)
Caudrelier, V.: [On the Inverse Scattering Method for Integrable PDEs on a Star Graph]{}. Commun. Math. Phys. [**338**]{}(2), 893–917 (2015)
Cazenave, T.: [Semilinear Schrödinger Equations]{}, Courant Lecture Notes 10. American Mathematical Society, Providence, RI (2003)
Chen T., Pavlovic, N.: [The quintic NLS as the mean field limit of a boson gas with three-body interactions]{}. J. Func. An. [**260**]{}(4), 959–997 (2011)
Chen, X., Holmer, J.: [Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schršdinger Equation]{}. Arch. Rat. Mech. An. [**221**]{}(2), 631–676 (2016)
Donley, E.A., Claussen, N.R., Cornish, S.L., Roberts, J.L., Cornell, E.A., Wieman, C.E.: [Dynamics of collapsing and exploding Bose-Einstein condensates]{}, Nature [**412**]{}, 295–299 (2001)
Erdős, L., Schlein, B., Yau, H.-T.: [Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems]{}. Invent. Math. [**167**]{} , 515–614 (2007)
Erdős, L., Schlein, B., Yau, H.-T.: Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential. J. Am. Math. Soc. [**22**]{}(4), 1099–1156 (2009)
Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. Ann. of Math. [**172**]{}(1), 291–370 (2010)
Exner, P., Keating, J.P., Kuchment, P., Sunada, T., Teplyaev, A.: [Analysis on graphs and its applications]{}. Proc. Sympos. Pure Math. 77., Amer. Math. Soc., Providence, RI (2008)
Friedlander, L.: [Extremal properties of eigenvalues for a metric graph]{}. Ann. Inst. Fourier (Grenoble) [**55**]{}(1),199–211 (2005)
Gnutzmann, S., Waltner, D.: [Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory]{}. Phys. Rev. E [**93**]{}(3), 032204 (2016)
Lieb, E.H., Seiringer, R.: [Proof of Bose-Einstein condensation for dilute trapped gases]{}. Phys. Rev. Lett. [**88**]{}, 170409 (2002)
Lieb, E.H., Seiringer, R., Yngvason, J.: [Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional]{}. Phys. Rev. A [**61**]{}, 043602 (2000)
Lieb, E.H., Yngvason, J.: [Ground state energy of the low density Bose gas]{}. Phys. Rev. Lett. [**80**]{}, 2504–2507 (1998)
Lorenzo, M., Luccia, M., Merlo, V., Ottaviani, I., Salvato, M., Cirillo, M., Müller, M., Weimann, T., Castellano, M.G., Chiarello, F., Torrioli, G.: [On Bose-Einstein condensation in Josephson junctions star graph arrays]{}. Phys. Lett. A [**378**]{}(7-8), 655–658 (2014)
Marzuola, J., Pelinovsky, D.: [Ground state on the dumbbell graph]{}. App. Math. Res. EX. [**1**]{}, 98–145 (2016)
Noja, D., Pelinovsky, D., Shaikhova, G.: [Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph]{}. Nonlinearity [**28**]{} (7), 243–278 (2015)
Pelinovsky, D., Schneider, G.: [Bifurcations of standing localized waves on periodic graphs]{}. http://arxiv.org/abs/1603.05463 (2016)
Pickl, P.: [Derivation of the time dependent Gross-Pitaevskii equation with external fields]{}. Rev. Math. Phys. [**27**]{}(1), 1550003 (2015)
Post, O.: [Spectral analysis on graph-like spaces]{}. Lecture Notes in Mathematics 2039. Springer, Heidelberg (2012)
Sabirov, K., Sobirov, Z., Babajanov, D., Matrasulov, D.: [Stationary nonlinear Schrödinger equation on simplest graphs]{}. Phys. Lett. A [**377**]{} (12), 860–865 (2013)
Sobirov, Z., Matrasulov, D., Sabirov, K., Sawada, S., Nakamura, K.: [Integrable nonlinear Schrödinger equation on simple networks: Connection formula at vertices]{}. Phys. Rev. E [**81**]{}, 066602 (2010)
Tentarelli, L.: [NLS ground states on metric graphs with localized nonlinearities]{}. J. Math. An. App. [**433**]{}(1), 291–304 (2016)
[^1]: Author partially supported by the FIRB 2012 project “Dispersive dynamics: Fourier Analysis and Variational Methods".
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---
abstract: 'An averaging method is applied to derive effective approximation to the following singularly perturbed nonlinear stochastic damped wave equation $$\nu u_{tt}+u_t={\Delta}u+f(u)+\nu^\alpha\dot{W}$$ on an open bounded domain $D\subset{{\mathbb R}}^n$, $1\leq n\leq 3$. Here $\nu>0$ is a small parameter characterising the singular perturbation, and $\nu^\alpha$, $0\leq \alpha\leq 1/2$, parametrises the strength of the noise. Some scaling transformations and the martingale representation theorem yield the following effective approximation for small $\nu$, $$u_t={\Delta}u+f(u)+\nu^\alpha\dot{W}$$ to an error of ${\ensuremath{o\big(\nu^\alpha\big)}}$.'
author:
- 'Yan Lv[^1]'
- 'A. J. Roberts[^2]'
title: Averaging approximation to singularly perturbed nonlinear stochastic wave equations
---
#### Keywords
stochastic nonlinear wave equations, averaging, tightness, martingale.
#### Mathematics Subject Classifications (2000)
60F10, 60H15, 35Q55.
Introduction {#sec:intro}
============
Wave motion is one of the most commonly observed physical phenomena, and typically described by hyperbolic partial differential equations. Nonlinear wave equations also have been studied a great deal in many modern problems such as sonic booms, bottlenecks in traffic flows, nonlinear optics and quantum field theory [@ReedSimon; @White e.g.]. However, for many problems, such as wave propagation through the atmosphere or the ocean, the presence of turbulence causes random fluctuations. More realistic models must account for such random fluctuations. Hence we study stochastic wave equations [@Chow02; @ChowB e.g.].
Here we study an effective approximation, in the sense of distribution, for the following nonlinear wave stochastic partial differential equation ([<span style="font-variant:small-caps;">spde</span>]{}). The [<span style="font-variant:small-caps;">spde</span>]{} is a singularly perturbed problem on a bounded open domain $D\subset{{\mathbb R}}^n$, $1\leq n\leq3$, $$\label{e:SWE}
\nu u^\nu_{tt}+u^\nu_t={\Delta}u^\nu+f(u^\nu)+\nu^\alpha\dot{W}\,, \quad
u^\nu(0)=u_0\,,\quad u^\nu_t(0)=u_1\,,$$ with zero Dirichlet boundary on $D$. Here $\nu^\alpha$ with $0<\nu\leq1$ and $0\leq\alpha\leq1/2$ parametrises the strength of noise, and ${\Delta}$ is the Laplace operator in ${{\mathbb R}}^n$. The noise $W(t)$ is an infinite dimensional Q-Wiener process which is detailed in section \[sec:basic\]. The [<span style="font-variant:small-caps;">spde</span>]{} (\[e:SWE\]) also describes the motion of a small particle with mass $\nu$ and an infinite number of degrees of freedom [@CF05; @CF06]. We are concerned with the effective approximation of the solution to the [<span style="font-variant:small-caps;">spde</span>]{} (\[e:SWE\]) for small $\nu>0$.
For $\alpha=1/2$, the limit of the random dynamics of [<span style="font-variant:small-caps;">spde</span>]{} (\[e:SWE\]) as $\nu\rightarrow 0$ has been studied by Lv and Wang [@LW08; @WL10]. The random attractor and measure attractor of [<span style="font-variant:small-caps;">spde</span>]{} (\[e:SWE\]) are approximated by those of the deterministic [<span style="font-variant:small-caps;">pde</span>]{}$$\label{1HE}
u_t=\Delta u+f(u)$$ as $\nu\rightarrow 0$ in the almost sure sense [@LW08] and weak topology [@WL10] respectively.
The important case of $\alpha=0$, which is an infinite dimensional version of the Smolukowski–Kramers approximation, is studied by analysing the structure of solution of linear stochastic wave equations [@CF05; @CF06]. For any $T>0$, the solution $u(t)$ to the [<span style="font-variant:small-caps;">spde</span>]{} (\[e:SWE\]) is approximated in probability by that of the stochastic system $$u_t=\Delta u+f(u)+\dot{W}$$ as $\nu\rightarrow 0$ in space $C(0, T; L^2(D))$.
Here we extend the approximating result to the case when $0\leq
\alpha\leq 1/2$ and derive a higher order approximation in the sense of distribution. Recently, the stochastic averaging approach was developed to study the effective approximation to slow-fast <span style="font-variant:small-caps;">spde</span>s [@CerFre09; @WangRoberts08] in the following form $$\begin{aligned}
u^\nu_t&=&\Delta u^\nu+f(u^\nu, v^\nu)+ \sigma_1\dot{W}_1\,,\\
v^\nu_t&=&\frac{1}{\nu}\left[\Delta
v^\nu+g(u^\nu,v^\nu)\right]+\frac{\sigma_2}{\sqrt{\nu}}\dot{W}_2 \,,\end{aligned}$$ where $f$ and $g$ are nonlinear terms, $\sigma_1$ and $\sigma_2$ are some constants, and $W_1$ and $W_2$ are Wiener processes. Notice that upon introducing $v^\nu=u^\nu_t$, the [<span style="font-variant:small-caps;">spde</span>]{} (\[e:SWE\]) is rewritten as $$\begin{aligned}
u^\nu_t&=&v^\nu,\quad u^\nu(0)=u_0\,,\\
v^\nu_t&=&\frac{1}{\nu}\left[-v^\nu+\Delta
u^\nu+f(u^\nu)\right]+\frac{1}{\nu^{1-\alpha}}\dot{W},\quad
v^\nu(0)=u_1\,,\end{aligned}$$ which are also in the form of slow-fast <span style="font-variant:small-caps;">spde</span>s. Then we can follow the stochastic averaging approach to derive an effective averaging approximation of $u^\nu$, the solution of [<span style="font-variant:small-caps;">spde</span>]{} (\[e:SWE\]) as $\nu\rightarrow 0$ for all $0\leq \alpha \leq 1/2$. Here the case $\alpha=1/2$ is the most important case because all cases of $\alpha\in [0,1/2]$ can be transformed to the case $\alpha=1/2$, see section \[sec:alpha=0\] and section \[sec:alpha=1\]. By an averaging approach and martingale representation theorem, we prove that for small $\nu>0$ with $0\leq \alpha\leq 1/2$ the solution of [<span style="font-variant:small-caps;">spde</span>]{} (\[e:SWE\]) is approximated in the sense of distribution by $\bar{u}^\nu$ which solves $$\label{e:1-bar-u}
\bar{u}^\nu_t=\Delta \bar{u}^\nu+f(\bar{u}^\nu)+\nu^\alpha
\dot{\bar{W}}\,, \quad \bar{u}^\nu(0)=u_0 \,,$$ where $\bar{W}(t)$ is a Wiener process distributes same as $W(t)$. This result shows that for any small $\nu>0$ with $0\leq \alpha\leq
1/2$ the term $\nu u^\nu_t(t)$ is a higher order term than the random force term $\nu^\alpha W(t)$.
Section \[sec:sqrt-nu\] gives the approximation for the important case that $\alpha=1/2$. Previous research [@LW08] gives an approximation which is a deterministic equation. However, our new approximation shows that for small $\nu\neq 0$, there is a small fluctuation which distributes same as $\sqrt{\nu}W(t)$, see (\[e:1-bar-u\]). This gives a more effective approximation.
Section \[ssmcf\] explores a parameter regime where a nonlinear coordinate transformation underlies the existence of a stochastic slow manifold for the case $\alpha=0$. The stochastic slow manifolds of both the [<span style="font-variant:small-caps;">spde</span>]{} and the model have the same evolution in the parameter regime and so provide evidence of the stronger result of pathwise approximation therein.
Preliminaries {#sec:basic}
=============
Let $D\subset {{\mathbb R}}^n$, $1\leq n\leq 3$, be a regular domain with boundary $\Gamma$. Denote by $L^2(D)$ the Lebesgue space of square integrable real valued functions on $D$, which is a Hilbert space with inner product $$\langle u, v\rangle=\int_Du(x)v(x)\,dx\,, \quad u, v\in L^2(D)\,.$$ Write the norm on $L^2(D)$ by $\|u\|_0=\langle u, u\rangle^{1/2}$. Define the following abstract operator $$Au=-{\Delta}u\,,\quad u\in \text{Dom}(A)=\{u\in L^2(D): A u\in L^2(D)\,,\
u|_{\Gamma}=0\}\,.$$ Denote by $\{\lambda_k\}$ the eigenvalues of $A$ with $0<\lambda_1\leq \lambda_2\leq \cdots\leq\lambda_k\leq \cdots$, $\lambda_k\rightarrow\infty$ as $k\rightarrow\infty$. For any $s\in{{\mathbb R}}$, introduce the space $H^{s}_0(D)=\text{Dom}(A^{s/2})$ endowed with the norm $$\|u\|_s=\|A^{s/2}u\|_0\,, \quad u\in H^s_0(D).$$ Consider the following singularly perturbed stochastic wave equation with cubic nonlinearity on $D$: $$\begin{aligned}
\label{SWE1}
\nu u^\nu_{tt}+u^\nu_t&=&{\Delta}u^\nu+\beta u^\nu-(u^\nu)^3+\nu^\alpha\dot{W}(t),\\
u^\nu(0)&=&u_0\,,\quad u^\nu_t(0)=u_1\,,\label{SWE2}\\
u^\nu|_\Gamma&=&0\,,\label{SWE3}\end{aligned}$$ with $0<\nu<1$ and $0\leq \alpha\leq 1/2$. Here $\{W(t)\}_{t\in {{\mathbb R}}}$ is an $L^2(D)$-valued two sided Wiener process defined on a complete probability space ($\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq 0},\mathbb{P}$) with covariance operator $Q$ such that $$Qe_k=b_k e_k\,, \quad k=1,2,\ldots\,,$$ where $\{e_k\}$ is a complete orthonormal system in $L^2(D)$, $b_k$ is a bounded sequence of non-negative real numbers. Then $$W(t)=\sum_{k=1}^\infty \sqrt{b_k} e_k w_k(t),$$ where $w_k$ are real mutually independent Brownian motions [@PZ92]. Further, we assume $$\label{Q}
B_0=\sum^\infty_{k=1} b_k<\infty
\quad\text{and}\quad
B_1=\sum^\infty_{k=1}\lambda_k b_k<\infty\,.$$ Then by a standard method [@Chow07], for any $(u_0, u_1)\in
H^{s+1}_0(D)\times H^s(D)$, $s\in{{\mathbb R}}$, there is a unique solution $u^\nu$ to (\[SWE1\])–(\[SWE3\]), $$\begin{aligned}
\label{sol1}
&&u^\nu\in L^2(\Omega, C(0, T; H_0^{s+1}(D))), \\
&&u^\nu_t\in L^2(\Omega, C(0, T; H^s(D))).\label{sol2}\end{aligned}$$ In the following we write $f(u)=\beta u-u^3$ and $F(u)=\int_0^uf(r)\,dr$.
For our purpose we need the following lemma.
\[embedding\] Assume $E$, $E_0$ and $E_1$ be Banach spaces such that $E_1\Subset
E_0$, the interpolation space $(E_0, E_1)_{\theta,1}\subset E$ with $\theta\in (0, 1)$ and $E\subset E_0$ with $\subset$ and $\Subset$ denoting continuous and compact embedding respectively. Suppose $p_0,p_1\in [1,\infty]$ and $T>0$, such that $$\begin{aligned}
&&
\mathcal{V} \text{ is a bounded set in } L^{p_1}(0, T;
E_1),\quad\text{and}
\\&&
{\partial}\mathcal{V}:=\{{\partial}v: v\in \mathcal{V}\} \text{ is
a bounded set in } L^{p_0}(0, T; E_0).\end{aligned}$$ Here ${\partial}$ denotes the distributional derivative. If $1-\theta>1/p_\theta$ with $$\frac{1}{p_\theta}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}\,,$$ then $\mathcal{V}$ is relatively compact in $C(0, T; E)$.
In the following, for any $T>0$, we denote by $C_T$ a generic positive constant which is independent of $\nu$.
The case of $\alpha=1/2$ {#sec:sqrt-nu}
========================
We first consider the special case of $\alpha=1/2$ which was recently studied by a direct approximation method [@LW08; @WL10]. Here we apply an averaging method to give more effective approximation to equation (\[SWE1\])–(\[SWE3\]). We rewrite (\[SWE1\])–(\[SWE3\]) in the form of slow-fast <span style="font-variant:small-caps;">spde</span>s: $$\begin{aligned}
du^{\nu}&=&v^{\nu}\,dt\,,\quad u^\nu(0)=u_0\,,\label{3SWE1}\\
dv^{\nu}&=&-\frac{1}{\nu}\left[v^{\nu}-{\Delta}u^{\nu}-f(u^{\nu})\right]dt+\frac{1}{\sqrt{\nu}}\,dW(t)\,, \quad v^\nu(0)=u_1\,.\label{3SWE2}\end{aligned}$$ Notice that the slow part $u^\nu$ and fast part $v^\nu$ are linearly coupled. For simplicity we consider $(u_0, u_1)\in (H^2(D)\cap
H_0^1(D))\times H^1(D)$. Then (\[SWE1\])–(\[SWE3\]) has a unique solution in $L^2(\Omega, C(0, T; (H^2(D)\cap H_0^1(D))\times
H^1(D)))$.
Tightness of solutions
----------------------
Let $(u^{\nu},v^{\nu})$ be a solution to (\[3SWE1\])–(\[3SWE2\]) with $\nu>0$. In order to pass to the limit $\nu\rightarrow 0$ in the averaging approach, we need some a priori estimates on the solutions.
\[thm:estimate\] Assume $B_1<\infty$. For any $T>0$, there is a positive constant $C_T$ such that $$\mathbb{E}\left[\max_{0\leq t\leq T}\|u^\nu(t)\|_2^2+\max_{0\leq
t\leq T}\| v^\nu(t)\|^2_0 \right]\leq C_T \,,$$ and for any integer $m>0$ $$\mathbb{E}\int_0^T\|u^{\nu}(t)\|_1^{2m}dt\leq C_T\,.$$
The result on $\|u^\nu(t)\|_2$ is found by a simple energy estimate [@WL10]. Now we give the estimate on $\|v^\nu(t)\|_0$. By equation (\[3SWE2\]), $$\begin{aligned}
v^{\nu}(t)&=&e^{-{t}/{\nu}}u_1
+\frac{1}{\nu}\int_0^te^{-({t-s})/{\nu}}\left[{\Delta}u^{\nu}(s)+
f(u^{\nu}(s))\right]ds
\\&&{}
+\frac{1}{\sqrt{\nu}}
\int_0^te^{-({t-s})/{\nu}}\,dW(s).\end{aligned}$$ Noticing assumption (\[Q\]), by the estimate on $\|u^\nu(t)\|_2$ and maximal inequality of stochastic convolution [@PZ92 Lemma 7.2], $$\mathbb{E}\left[\max_{0\leq t\leq T}\|v^\nu(t)\|_0^2\right]\leq C_T$$ for some positive constant $C_T$. The last inequality of the theorem is obtained by the same method [@WL10] and Poincaré inequality. This completes the proof.
Now by the above estimates and Lemma \[embedding\], we have the following theorem.
For any $T>0$, $\{\mathcal{L}(u^\nu)\}_{0<\nu\leq 1}$ the distribution of $u^\nu$ is tight in the space $C(0, T; H_0^1(D))$.
By the above tightness result, to determine the limit of $u^\nu$ we can pass to the limit $\nu\rightarrow 0$ in a weak sense; that is, for any ${\varphi}\in C_0^\infty(D)$, we consider the limit of $u^{\nu,{\varphi}}(t)=\langle u^\nu(t), {\varphi}\rangle$ in the space $C(0, T)$ as $\nu\rightarrow 0$.
Limit of $u^{\nu, {\varphi}}$ in $C(0, T)$
------------------------------------------
Now we pass to the limit $\nu\rightarrow 0$ in $\{u^{\nu,{\varphi}}\}$ in the space $C(0, T)$ for any $T>0$. First, by equations (\[3SWE1\])–(\[3SWE2\]), $$\begin{aligned}
du^{\nu,{\varphi}}&=&v^{\nu,{\varphi}}\,dt\,,\label{phi-SWE1}\\
dv^{\nu,{\varphi}}&=&-\frac{1}{\nu}\left[v^{\nu,{\varphi}}+\langle
\nabla u^{\nu}, \nabla{\varphi}\rangle-\langle f(u^{\nu}), {\varphi}\rangle\right]dt+
\frac{1}{\sqrt{\nu}}\,dW^{\varphi}(t),\label{phi-SWE2}\end{aligned}$$ with $ u^{\nu,{\varphi}}(0)=\langle u_0, {\varphi}\rangle$ and $
v^{\nu,{\varphi}}(0)=\langle u_1, {\varphi}\rangle$ where $v^{\nu,{\varphi}}=\langle v^\nu, {\varphi}\rangle$ and $W^{\varphi}(t)=\langle
W(t), {\varphi}\rangle$. In the following we also write $v^{\nu,{\varphi}}$ as $v^{\nu,{\varphi}, u(t)}$ which shows the dependence of $v^{\nu, {\varphi}}$ on the slow part $u^\nu$.
Second, for any fixed $u\in H^2(D)\cap H_0^1(D)$ we consider the fast equation $$\label{u-SWE2}
dv^{\nu, u}=-\frac{1}{\nu}\left[ v^{\nu,u}-\Delta u- f(u) \right]dt
+\frac{1}{\sqrt{\nu}}\,dW(t)\,.$$ Equation (\[u-SWE2\]) has a unique stationary solution $\bar{v}^{\nu, u}$. Moreover, the stationary solution $\bar{v}^{\nu,
u}$ is exponentially mixing and the distribution of $\bar{v}^{\nu, u}$ is the normal distribution $\mathcal{N}\left( \Delta u+f(u), Q/2 \right)$ [@CF05].
Now for any $u\in H^2(D)\cap H_0^1(D)$ define $$H^\nu(u, t)=\nu\left[v^{\nu,u}(t)-v^{\nu,u}(0)\right]+
\int_0^t\left[v^{\nu, u}(s)-\Delta u-f(u)\right]\,ds\,.$$ Then $u^{\nu,{\varphi}}$ solves the following equation $$\begin{aligned}
u^{\nu, {\varphi}}(t)&=&\langle u_0, {\varphi}\rangle-\int_0^t[\langle \nabla
u^{\nu}(s), \nabla{\varphi}\rangle-\langle f(u^{\nu}(s)),
{\varphi}\rangle]\,ds\nonumber\\&& {} +\langle H^\nu(u^\nu(t),t), {\varphi}\rangle -\nu\left\langle v^{\nu, u}(t)-v^{\nu,
u}(0),{\varphi}\right\rangle\,.\label{e:phi-u}\end{aligned}$$ Third, we study the behaviour of $\langle H^\nu(u^\nu(t),t), {\varphi}\rangle$ for small $\nu$. Let $H^{\nu,{\varphi}}(u, t)=\langle H^\nu(u, t), {\varphi}\rangle $, then define $$\begin{aligned}
\label{e:M-e}
M_t^{\nu,\varphi}= \frac{1}{\sqrt{\nu}}H^{\nu,\varphi}(u^\nu(t),
t)\,.
\end{aligned}$$ By the definition of $H^{\nu,{\varphi}}(u, t)$ and equation (\[u-SWE2\]), $M_t^{\nu,\varphi}$ is a martingale with respect to $\{\mathcal{F}_t: t\geq 0\}$, and the quadratic covariance is $\langle M^{\nu,{\varphi}} \rangle_t=t\langle Q{\varphi}, {\varphi}\rangle$.
Now define $R^{\nu,{\varphi}}(t)=-\left\langle v^{\nu, u}(t)-v^{\nu,
u}(0),{\varphi}\right\rangle$, then rewrite (\[e:phi-u\]) as $$\label{e:phi-SWE}
u^{\nu, {\varphi}}(t)=\langle u_0,
{\varphi}\rangle-\int_0^t\left[\langle\nabla u^{\nu}(s),
\nabla{\varphi}\rangle-\langle f(u^{\nu}(s)),
{\varphi}\rangle\right]ds+\sqrt{\nu}M_t^{\nu,{\varphi}}+\nu R^{\nu,{\varphi}}(t)\,.$$
Invoking Theorem \[thm:estimate\], $$\label{e:R}
\lim_{\nu\rightarrow 0}\mathbb{E}\left[\max_{0\leq t\leq
T}\sqrt{\nu}\left| R^{\nu, {\varphi}}(t)\right|\right]=0\,.$$ Then define the process $$\label{e:cal-M-nu}
\mathcal{M}_t^{\nu,{\varphi}}=\frac{1}{\sqrt{\nu}}\left\{
u^{\nu,{\varphi}}(t)-\langle u_0, {\varphi}\rangle +\int_0^t\big
[\langle\nabla u^{\nu}(s), \nabla{\varphi}\rangle-\langle f(u^{\nu}(s)),
{\varphi}\rangle\big] ds \right\}.$$ By the definition of $H^{\nu,{\varphi}}(u, t)$ and (\[e:R\]) we have the tightness of $\mathcal{M}_t^{\nu,{\varphi}}$ in space $C(0, T)$ for any $T>0$. Let $P$ be a limit point of the family of probability measures $\{\mathcal{L}(\mathcal{M}_t^{\nu,{\varphi}})\}_{0<\nu\leq
1}$ and denote by $\mathcal{M}_t^{\varphi}$, a $C(0, T)$-valued random variable with distribution $P$. Let $\Psi$ be a continuous bounded function on $C(0, T)$. Set $\Psi^\nu(s)=\Psi( u^{\nu, {\varphi}}(s) )$, then noticing (\[e:R\]), $$\mathbb{E}\left[(\mathcal{M}^{\nu,{\varphi}}_t-\mathcal{M}^{\nu,{\varphi}}_s) \Psi^\nu(s)\right]=
\mathbb{E}\left[\sqrt{\nu} (R^{\nu, {\varphi}}(t)-R^{\nu,{\varphi}}(s)) \Psi^\nu(s)\right]\rightarrow 0\,, \quad \nu\rightarrow 0 \,,$$ which yields that the process $\{\mathcal{M}_t^{\varphi}\}_{0\leq t\leq
T}$ is a $P$-martingale with respect to the Borel $\sigma$-filter of $C(0, T)$.
We consider the quadratic covariation of the martingale $\mathcal{M}_t^{\varphi}$. By the definition of $\mathcal{M}^{\nu,{\varphi}}_t$, passing to the limit $\nu\rightarrow 0$ in (\[e:cal-M-nu\]), we derive $\mathcal{M}_t^{\varphi}$ is a square integrable martingale with the associated quadratic covariation process is $\langle
Q{\varphi}, {\varphi}\rangle t$. Then by the representation theorem for martingales [@IW81], without changing the distributions of $\mathcal{M}^{\nu,{\varphi}}_t$ and $\mathcal{M}_t^{\varphi}$, one extends the original probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and chooses a new Wiener process $\hat{W}^{\varphi}(t)$ such that $\mathcal{M}_t^{\varphi}=\sqrt{Q} \hat{W}^{\varphi}(t)$, which is unique in the sense of distribution.
By the definition of $\mathcal{M}_t^{\nu, {\varphi}}$, $\hat{W}^{\varphi}$ can be chosen as $\langle \hat{W}, {\varphi}\rangle$ where $\hat{W}$ is a cylindrical Wiener process. Then from (\[e:cal-M-nu\]) we have in the sense of distribution $$\begin{aligned}
&&\langle u^\nu(t), {\varphi}\rangle\\&=&\langle u_0, {\varphi}\rangle
-\int_0^t[\langle \nabla u^\nu(s), \nabla{\varphi}\rangle-\langle
f(u^\nu(s)), {\varphi}\rangle] ds
+\sqrt{\nu}\mathcal{M}_t^{\varphi}+{\ensuremath{o\big(\sqrt{\nu}\big)}}\\
&=&\langle u_0, {\varphi}\rangle -\int_0^t[\langle \nabla u^\nu(s),
\nabla{\varphi}\rangle-\langle f(u^\nu(s)), {\varphi}\rangle] ds
+\sqrt{\nu}\sqrt{Q}\langle\hat{W},
{\varphi}\rangle+{\ensuremath{o\big(\sqrt{\nu}\big)}}\end{aligned}$$ for any ${\varphi}\in C_0^\infty(D)$. Then by discarding the higher order term and the tightness of $u^\nu$, we have the following approximating equation $$\label{e:3-bar-u}
d\bar{u}^\nu=[\Delta \bar{u}^\nu+f(\bar{u}^\nu)]dt+\sqrt{\nu} \,
d\bar{W}^Q \,,$$ where $\bar{W}^Q$ is some an $L^2(D)$ valued Q-Wiener process.
Assume $B_1<\infty$ and $\alpha=1/2$. For small $\nu>0$, there is a new probability space $(\bar{\Omega}, \bar{\mathcal{F}},
\bar{\mathbb{P}})$, an extension of the original probability space $(\Omega, \mathcal{F}, \mathbb{P})$, such that for any $T>0$, the solution $u^\nu$ to (\[3SWE1\])–(\[3SWE2\]) is approximated by $\bar{u}^\nu$ which solves (\[e:3-bar-u\]), to an error of ${\ensuremath{o\big(\sqrt{\nu}\big)}}$, in the space $C(0, T; H_0^1(D))$ for almost all $\omega\in\bar{\Omega}$.
The above [<span style="font-variant:small-caps;">spde</span>]{} (\[e:3-bar-u\]) is more effective than the limit [<span style="font-variant:small-caps;">pde</span>]{} (\[1HE\]) [@LW08] as it incorporates fluctuations for small $\nu>0$. This result also implies that the singular term $\nu u^\nu_{t}(t)$ is a higher order term than $\sqrt{\nu}W(t)$ for small $\nu>0$ in the sense of distribution at least. The following two sections show that $\nu
u^\nu_t(t)$ is always a higher order term than $\nu^\alpha W(t)$ for any $0\leq \alpha\leq 1/2$.
The case of $\alpha=0$ {#sec:alpha=0}
======================
Next we consider the case of $\alpha=0$; that is, consider the following [<span style="font-variant:small-caps;">spde</span>]{}$$\begin{aligned}
\label{4-SWE1}
\nu u^\nu_{tt}+u^\nu_t&=&{\Delta}u^\nu+\beta u^\nu-(u^\nu)^3+\dot{W}(t),\\
u^\nu(0)&=&u_0\,,\quad u^\nu_t(0)=u_1\,,\label{4-SWE2}\\
u^\nu|_\Gamma&=&0\label{4-SWE3}\,.\end{aligned}$$ First we have the following a priori estimates on $u^\nu$ in the space $C(0, T; H_0^1(D))$.
\[CF06\] Assume $B_1<\infty$. For any $T>0$, there is a positive constant $C_T$ such that $$\mathbb{E}\left[\max_{0\leq t\leq T}\|u^\nu(t)\|_1^2\right]\leq C_T\,.$$
We follow the approach for the case of $\alpha=1/2$. For this we introduce the scalings $\tilde{u}^\nu=\sqrt{\nu}u^\nu$ and $\tilde{v}^\nu=\sqrt{\nu}u^\nu_t$. Then $$\begin{aligned}
d \tilde{u}^\nu&=&\tilde{v}^\nu dt\,,\quad \tilde{u}^\nu(0)=\sqrt{\nu}u_0\,,\\
d\tilde{v}^\nu&=&-\frac{1}{\nu}\left[\tilde{v}^\nu- \Delta
\tilde{u}^\nu-\sqrt{\nu}f\left(\frac{\tilde{u}^\nu}{\sqrt{\nu}}
\right)\right] dt+ \frac{1}{\sqrt{\nu}} \, dW(t),
\quad \tilde{v}^\nu(0)=\sqrt{\nu}u_1\,.\end{aligned}$$ By standard energy estimates [@WL10], by a similar discussion to that in Section \[sec:sqrt-nu\], and by Theorem \[CF06\], we have the following theorem.
Assume $B_1<\infty$. For any $T>0$, there is a positive constant $C_T$ such that $$\mathbb{E}\left[ \max_{0\leq t\leq
T}\|\tilde{u}^\nu(t)\|_2^2+\max_{0\leq t\leq T}\|
\tilde{v}^\nu(t)\|^2_0\right]\leq C_T \,,$$ and for any integer $m>0$ $$\mathbb{E}\int_0^T\|\tilde{u}^\nu(t)\|_1^{2m}dt\leq C_T\,.$$ Moreover, the distribution of $\tilde{u}^\nu$ is tight in space $C(0, T; H_0^1(D))$.
We consider the asymptotic approximation of $\tilde{u}^\nu$ for small $\nu>0$. For any ${\varphi}\in C_0^\infty(D)$, let $\tilde{u}^{\nu, {\varphi}}=\langle \tilde{u}^\nu, {\varphi}\rangle $, $\tilde{v}^{\nu, {\varphi}}=\langle \tilde{v}^\nu, {\varphi}\rangle$ and $W^{\varphi}(t)=\langle W(t), {\varphi}\rangle$. Then $$\begin{aligned}
d\tilde{u}^{\nu,{\varphi}}&=&\tilde{v}^{\nu,{\varphi}} dt\,,\\
d\tilde{v}^{\nu,{\varphi}}&=&-\frac{1}{\nu}\left[\tilde{v}^{\nu, {\varphi}}+
\langle \nabla \tilde{u}^\nu, \nabla{\varphi}\rangle-\sqrt{\nu}\langle
f(\tilde{u}^\nu/\sqrt{\nu}), {\varphi}\rangle \right]
dt+\frac{1}{\sqrt{\nu}}\,dW^{\varphi}(t),\end{aligned}$$ with $\tilde{u}^{\nu,{\varphi}}(0)=\langle \tilde{u}^\nu(0), {\varphi}\rangle $ and $\tilde{v}^{\nu,{\varphi}}(0)=\langle \tilde{v}^\nu(0), {\varphi}\rangle $.
We also consider the following fast [<span style="font-variant:small-caps;">spde</span>]{} for fixed $\nu$ and $\tilde{u}\in H^2(D)\cap H_0^1(D)$: $$\label{e:v-tilde}
d\tilde{v}^{\nu, \tilde{u}}=-\frac{1}{\nu}\left[
\tilde{v}^{\nu,\tilde{u}}- \Delta \tilde{u}-
\sqrt{\nu}f\left(\tilde{u}/\sqrt{\nu}\right) \right]dt
+\frac{1}{\sqrt{\nu}}\,dW(t)\,.$$ For fixed $\nu\in (0,1]$ and $\tilde{u}\in H^2(D)\cap H_0^1(D)$, [<span style="font-variant:small-caps;">spde</span>]{} (\[e:v-tilde\]) has a unique stationary solution with the normal distribution $\mathcal{N}\left( \Delta \tilde{u}+\sqrt{\nu}f(\tilde{u}/\sqrt{\nu}), \; Q/2 \right)$ [@CF05]. Now for any $\tilde{u}\in H^2(D)\cap H_0^1(D)$ define $$\widetilde{H}^\nu(\tilde{u},t)=\nu\left[\tilde{v}^{\nu,\tilde{u}}(t)-\tilde{v}^{\nu,\tilde{u}}(0)\right]+
\int_0^t\left[\tilde{v}^{\nu, \tilde{u}}(s)-\Delta
\tilde{u}-\sqrt{\nu}f(\tilde{u}/\sqrt{\nu})\right]\,ds\,.$$ Thus we can follow the same discussion in last section for the case of $\alpha=1/2$. We write $$\begin{aligned}
\tilde{u}^{\nu, {\varphi}}(t)&=&\sqrt{\nu}\langle u_0,
{\varphi}\rangle-\int_0^t\langle\nabla\tilde{u}^\nu(s), \nabla{\varphi}\rangle ds+\sqrt{\nu}\int_0^t\langle
f\left(\tilde{u}^\nu(s)/\sqrt{\nu}\right), {\varphi}\rangle
ds\nonumber\\&&{} +\sqrt{\nu}\tilde{\mathcal{M}}^{\nu, {\varphi}}_t \,,
\label{e:tilde-u}\end{aligned}$$ where $\sqrt{\nu}\tilde{\mathcal{M}}^{\nu, {\varphi}}_t$ is the remainder term. By a similar discussion to that of the last section, $\tilde{\mathcal{M}}^{\nu, {\varphi}}_t$ is tight in space $C(0, T)$ for any $T>0$. Let $\tilde{P}$ be a limit point of the family of probability measures $\mathcal{L}\{\tilde{\mathcal{M}}^{\nu,
{\varphi}}_t\}_{0<\nu\leq1}$ in space $C(0, T)$. Let $\tilde{\mathcal{M}}^{\varphi}_t$ be a $C(0, T)$-valued random variable with distribution $\tilde{P}$. Then we have the following lemma.
For any ${\varphi}\in C_0^\infty(D)$, the process $\tilde{\mathcal{M}}_t^{\varphi}$ defined on the probability space $(C(0,
T), \mathcal{B}(C(0, T)), \tilde{P})$ is a square integrable martingale with the associated quadratic covariation process $\langle Q{\varphi}, {\varphi}\rangle t$.
By the representation theorem for martingales [@IW81], without changing the distributions of $\tilde{\mathcal{M}}^{\nu,{\varphi}}_t$ and $\tilde{\mathcal{M}}_t^{\varphi}$ one can extend the original probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and choose a new cylindrical Wiener process $\tilde{W}(t)$ such that $\tilde{\mathcal{M}}_t^{\varphi}=\sqrt{Q} \langle \tilde{W}, {\varphi}\rangle$, which is unique in the sense of distribution.
Then in the sense of distribution by (\[e:tilde-u\]) we write out $$\begin{aligned}
\langle \tilde{u}^\nu(t), {\varphi}\rangle&=&\sqrt{\nu}\langle u_0,
{\varphi}\rangle-\int_0^t\langle \nabla \tilde{u}^\nu(s),
\nabla{\varphi}\rangle ds
+\sqrt{\nu}\int_0^t\langle
f\left(\tilde{u}^\nu(s)/\sqrt{\nu} \right), {\varphi}\rangle ds
\nonumber\\&&{}
+\sqrt{\nu}\tilde{\mathcal{M}}_t^{\varphi}+{\ensuremath{o\big(\sqrt{\nu}\big)}}\nonumber\\
&=&\sqrt{\nu}\langle u_0, {\varphi}\rangle-\int_0^t\langle \nabla
\tilde{u}^\nu(s), \nabla{\varphi}\rangle ds
+\sqrt{\nu}\int_0^t\langle
f\left(\tilde{u}^\nu(s)/\sqrt{\nu} \right), {\varphi}\rangle
ds
\nonumber\\&&{}
+\sqrt{\nu}\sqrt{Q}\langle \tilde{W},
{\varphi}\rangle+{\ensuremath{o\big(\sqrt{\nu}\big)}}\end{aligned}$$ for any ${\varphi}\in C_0^\infty(D)$. Then we have, noticing that $\tilde{u}^\nu=\sqrt{\nu}u^\nu$, the following approximating [<span style="font-variant:small-caps;">spde</span>]{} for small $\nu>0$: $$\label{e:4-bar-u}
d\bar{u}^\nu=[\Delta\bar{u}^\nu+f(\bar{u}^\nu)]dt+d\bar{W}^Q, \quad
\bar{u}^\nu(0)=u_0\,,$$ where $\bar{W}^Q$ is some $L^2(D)$ valued Q-Wiener process. Then we infer the following result.
Assume $B_1<\infty$ and $\alpha=0$. Then for small $\nu>0$, there is a new probability space $(\bar{\Omega}, \bar{\mathcal{F}},
\bar{\mathbb{P}})$ which is an extension of the original probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that for any $T>0$, the solution $u^\nu$ to (\[4-SWE1\])–(\[4-SWE3\]) is approximated by $\bar{u}^\nu$ which solves (\[e:4-bar-u\]), to an error of ${\ensuremath{o\big(1\big)}}$, in the space $C(0, T; H_0^1(D))$ for almost all $\omega\in\bar{\Omega}$.
The case of $0<\alpha<1/2$ {#sec:alpha=1}
==========================
Now we consider the case of $0< \alpha < 1/2$; that is, consider the following [<span style="font-variant:small-caps;">spde</span>]{}$$\begin{aligned}
\label{5-SWE1}
\nu u^\nu_{tt}+u^\nu_t&=&{\Delta}u^\nu+\beta u^\nu-(u^\nu)^3+\nu^\alpha\dot{W}(t),\\
u^\nu(0)&=&u_0\,,\quad u^\nu_t(0)=u_1\,,\label{5-SWE2}\\
u^\nu|_\Gamma&=&0\label{5-SWE3}\,.\end{aligned}$$
First, by the same analysis as Theorem \[CF06\], we also have the following result on the a priori estimates on $u^\nu$.
\[cfthm\] Assume $B_1<\infty$. For any $T>0$, there is a positive constant $C_T$ such that $$\mathbb{E}\left[\max_{0\leq t\leq T}\|u^\nu(t)\|_1^2\right]\leq C_T\,.$$
We also apply the method in Section \[sec:sqrt-nu\]. Make the following scaling transformation $\tilde{u}^\nu=\nu^{1/2-\alpha} u^\nu$ and $\tilde{v}^\nu=\nu^{1/2-\alpha} v^\nu$. Then $$\begin{aligned}
d \tilde{u}^\nu&=&\tilde{v}^\nu dt\,,\\
d\tilde{v}^\nu&=&-\frac{1}{\nu}\left[\tilde{v}^\nu- \Delta
\tilde{u}^\nu-\nu^{1/2-\alpha}f\left(\frac{\tilde{u}^\nu}{\nu^{1/2-\alpha}}
\right)\right] dt+ \frac{1}{\sqrt{\nu}} \, dW(t),\\
\tilde{u}^\nu(0)&=&\nu^{1/2-\alpha}u_0\,,\quad \tilde{v}^\nu(0)=\nu^{1/2-\alpha}u_1\,.\end{aligned}$$ By a direct energy estimate or the scaling transformation and Theorem \[cfthm\] we deduce the following theorem.
Assume $B_1<\infty$. For any $T>0$, there is a positive constant $C_T$ such that $$\mathbb{E}\left[ \max_{0\leq t\leq
T}\|\tilde{u}^\nu(t)\|_2^2+\max_{0\leq t\leq T}\|
\tilde{v}^\nu(t)\|^2_0\right]\leq C_T \,,$$ and for any integer $m>0$ $$\mathbb{E}\int_0^T\|\tilde{u}^\nu(t)\|_1^{2m}dt\leq C_T\,.$$ Moreover, the distribution of $\tilde{u}^\nu$ is tight in space $C(0,
T; H_0^1(D))$.
Then we can follow the same discussion of Section \[sec:alpha=0\] and have the following result.
Assume $B_1<\infty$ and $0<\alpha<1/2$. For small $\nu>0$, there is a new probability space $(\bar{\Omega}, \bar{\mathcal{F}},
\bar{\mathbb{P}})$ which is an extension of the original probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that for any $T>0$, the solution $u^\nu$ to (\[5-SWE1\])–(\[5-SWE3\]) is approximated by $\bar{u}^\nu$ which solves $$\label{e:5-bar-u}
d\bar{u}^\nu=[\Delta\bar{u}^\nu+f(\bar{u}^\nu)]dt+\nu^\alpha
d\bar{W}^Q, \quad \bar{u}^\nu(0)=u_0\,,$$ to an error of ${\ensuremath{o\big(\nu^\alpha\big)}}$, in the space $C(0, T; H_0^1(D))$ for almost all $\omega\in\bar{\Omega}$.
A stochastic slow manifold compares the SPDEs for the case of $\alpha=0$ {#ssmcf}
========================================================================
This section shows the long time effectiveness of the averaged model by comparing it to the original via their stochastic slow manifolds.
We compare the [<span style="font-variant:small-caps;">spde</span>]{} and its model [<span style="font-variant:small-caps;">spde</span>]{} in a parameter regime where both have an accessible stochastic slow manifold. Consider the [<span style="font-variant:small-caps;">spde</span>]{} restricted to one spatial dimension as $$\nu u_{tt}+u_t=u_{xx}+f(u)+\sigma\dot W
\quad\text{where}\quad
f=(1+\beta')u-u^3.
\label{eq:sde}$$ Consider this [<span style="font-variant:small-caps;">spde</span>]{} on the non-dimensional domain $D=(0,\pi)$ with boundary conditions $u=0$ on $x=0,\pi$. The parameter $\sigma$ here explicitly measures the overall size of the Q-Wiener process $W(t)$ which by is finite. The small parameter $\beta'$ measures the distance from the stochastic bifurcation that occurs near $\beta'=0$. In this domain there will be a stochastic slow manifold of the [<span style="font-variant:small-caps;">spde</span>]{} that matches the slow dynamics in the approximating [<span style="font-variant:small-caps;">spde</span>]{} . This section compares the stochastic slow manifolds.
The [<span style="font-variant:small-caps;">spde</span>]{} has a technically challenging spectrum. However, the construction of its stochastic slow manifold is easiest by embedding the [<span style="font-variant:small-caps;">spde</span>]{} as the $\gamma=1$ case of the following slow-fast system of [<span style="font-variant:small-caps;">spde</span>]{}s $$\begin{aligned}
u_t={}&u_{xx}+u+v\,,
\label{eq:u}
\\
\nu v_t={}& -v -\gamma \nu\left( \partial _{xx}+1\right)u_t
+\beta' u-u^3
+\sigma\dot W\,.
\label{eq:v}\end{aligned}$$ The parameter $\gamma$ controls the homotopy: from a tractable base when $\gamma=0$ as then all linear modes in the very fast $v$ equation decay at the same rate $1/\nu$ (and the slow $u$ modes of $\sin kx$ have decay rates $1-k^2$); to the original [<span style="font-variant:small-caps;">spde</span>]{} when $\gamma=1$ (upon eliminating $v$).
#### A stochastic slow manifold appears
On the non-dimensional interval $(0,\pi)$, with Dirichlet boundary conditions on $u$, the eigenmodes must be proportional to $\sin kx$ for integer wavenumber $k$. Neglecting noise temporarily, $\sigma=0$ in this sentence, for all $\nu<1$ and all homotopy parameter $0\leq\gamma\leq1$ there is one zero eigenvalue and all the rest of the eigenvalues have negative real part; the slow subspace corresponding to the neutral mode is spanned by $(u,v)\propto (\sin x,0)$ (local in $(u,v,\sigma)$, but global in $\nu$ and $\gamma$). By stochastic center manifold theory [@Arnold03; @Boxler89], and supported by stochastic normal form transformations [@Arnold98; @Roberts06k; @Roberts07d], when the noise spectrum truncates and the nonlinearity is small enough, the dynamics of the [<span style="font-variant:small-caps;">spde</span>]{}s – are essentially finite dimensional and a stochastic slow manifold exists which is exponentially quickly attractive to all nearby trajectories.
#### Computer algebra constructs the stochastic slow manifold
We seek the stochastic slow manifold as a systematic perturbation of the slow subspace $u=a\sin x$. The intricate algebra necessary to handle the multitude of nonlinear noise interactions is best left to a computer [@Roberts05c; @Roberts07d e.g.]. However, the following expressions may be checked by substituting into the governing [<span style="font-variant:small-caps;">spde</span>]{}s – and confirming the order of the residuals is as small as quoted—albeit tedious, this check is considerably easier than the derivation. The evolution on the stochastic slow manifold may be written $$\begin{aligned}
\dot a={}& \beta' a-{{\textstyle\frac{3}{4}}}a^3
+\left[ 1-2\nu\beta'
+{{\textstyle\frac{9}{2}}}\nu a^2
-{{\textstyle\frac{9}{1024}}}a^4
\right]b_1\dot w_1
\nonumber\\&{}
+\left[({{\textstyle\frac{3}{32}}}+{{\textstyle\frac{3}{128}}}\beta')a^2
-{{\textstyle\frac{21}{1024}}}a^4\right]b_3\dot w_3
+{{\textstyle\frac{5}{1024}}}a^4b_5\dot w_5
+{\ensuremath{o\big(\nu^2+{\beta'}^2+a^4,\sigma\big)}}
\label{eq:ssme}\end{aligned}$$ The stochastic slow manifold itself involves Ornstein–Uhlenbeck processes written as convolutions over the past history of the noise processes: define ${e^{-\mu t}{\star}}\dot w=\int_{-\infty}^t\exp[-\mu(t-s)]dw_s$ for decay rates $\mu_k=k^2-1$ characteristic of the $k$th mode. Then the stochastic slow manifold is $$\begin{aligned}
u={}& a\sin x +{{\textstyle\frac{1}{32}}}a^3\sin3x
-{{\textstyle\frac{3}{32}}}a^2\left[b_3{e^{-8 t}{\star}}w_3\sin x+b_1{e^{-8 t}{\star}}w_1\sin 3x\right]
\nonumber\\&{}
+\sum_{k\geq2}b_k
\left[1+\mu_k\nu+\gamma\nu(\mu_k-\mu_k^2{e^{-\mu_k t}{\star}})\right]
{e^{-\mu_k t}{\star}}\dot w_k\sin kx
\nonumber\\&{}
-\sum_{k\geq1}b_k{e^{- t/\nu}{\star}}\dot w_k\sin kx
+\beta'\sum_{k\geq2}b_k{e^{-\mu_k t}{\star}}{e^{-\mu_k t}{\star}}\dot w_k\sin kx
\nonumber\\&{}
+{{\textstyle\frac{3}{4}}}\sum_{k\geq2}\left\{
b_{k+2}{e^{-\mu_k t}{\star}}{e^{-\mu_{k+2} t}{\star}}\dot w_{k+2}\sin kx
-2b_k{e^{-\mu_k t}{\star}}{e^{-\mu_{k} t}{\star}}\dot w_{k}\sin kx
\right.\nonumber\\&\left.\qquad{}
+b_{k}{e^{-\mu_{k+2} t}{\star}}{e^{-\mu_{k} t}{\star}}\dot w_{k}\sin[(k+2)x]
\right\}
+{\ensuremath{\mathcal O\big(\nu^2+{\beta'}^2+a^4,\sigma^2\big)}},
\label{eq:ssmu}
$$ and a correspondingly complicated expression for the field $v(x,t)$. Observe that the slow [<span style="font-variant:small-caps;">sde</span>]{} does not contain any fast time convolutions from the Ornstein–Uhlenbeck processes: it would be incongruous to have such fast processes in a supposedly slow model. We keep fast time convolutions out of the slow [<span style="font-variant:small-caps;">sde</span>]{} by introducing carefully crafted terms in the slow mode $\sin x$ in the parametrization of the stochastic slow manifold : here the amplitude of the slow mode $\sin x$ is approximately $a-{{\textstyle\frac{3}{32}}}a^2b_3{e^{-8 t}{\star}}w_3-b_1{e^{- t/\nu}{\star}}\dot w_1$. Other methods which do not adjust the slow mode either average over such adjustments and so are weak models, or invoke fast processes in the slow model.
Note that the homotopy parameter $\gamma$ affects the stochastic slow manifold shape , but only weakly. To this order the homotopy has no effect on the evolution on the stochastic slow manifold .
#### Compare with SPDE
The corresponding stochastic slow manifold of the [<span style="font-variant:small-caps;">spde</span>]{} , in this parameter regime, is straightforward to construct, via the web server [@Roberts07d] for example. For stochastic slow manifold $\bar u\approx \bar a\sin x$ one finds the corresponding slow [<span style="font-variant:small-caps;">sde</span>]{}$$\begin{aligned}
\dot {\bar a}={}& \beta' \bar a-{{\textstyle\frac{3}{4}}}\bar a^3 +\left[ 1-{{\textstyle\frac{9}{1024}}}a^4
\right]\bar b_1\dot{\bar w}_1
\nonumber\\&{}
+\left[({{\textstyle\frac{3}{32}}}+{{\textstyle\frac{3}{128}}}\beta')\bar a^2
-{{\textstyle\frac{21}{1024}}}a^4\right]\bar b_3\dot{\bar w}_3
+{{\textstyle\frac{5}{1024}}}a^4\bar b_5\dot{\bar w}_5
+{\ensuremath{o\big({\beta'}^{2}+\bar a^4,\sigma\big)}}.
\label{eq:ssmm}\end{aligned}$$ This slow [<span style="font-variant:small-caps;">sde</span>]{} is symbolically identical with the [<span style="font-variant:small-caps;">sde</span>]{} , one just removes the overbars. We conclude that these stochastic slow manifolds confirm the modeling of the [<span style="font-variant:small-caps;">spde</span>]{} by its model [<span style="font-variant:small-caps;">spde</span>]{} ; at least in the regime of one space dimension with small amplitude $a$, bifurcation parameter $\beta'$, and finite truncation to the noise.
#### Acknowledgements
The research was supported by the NSF of China grant No. 10901083, Zijin star of Nanjing University of Science and Technology, and the ARC grant DP0988738.
[99]{}
Ludwig Arnold. . Springer Monographs in Mathematics. Springer, June 2003.
Ludwig Arnold and Peter Imkeller. Normal forms for stochastic differential equations. , 110:559–588, 1998. [<http://dx.doi.org/10.1007/s004400050159>]{}.
P. Boxler. A stochastic version of the centre manifold theorem. , 83:509–545, 1989.
S. Cerrai & M. Freidlin, On the Smoluchowski–Kramers approximation for a system with an infinite number of degrees of freedom, *Prob. Th. and Relat. Fields* **135** (2006), 363–394.
S. Cerrai & M. Freidlin, Smoluchowski–Kramers approximation for a general class of SPDEs, *J. Evol. Equa.* **6** (2006), 657–689.
S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, *Probab. Theory Relat. Fields* **144** (2009), 137–177.
P. L. Chow, Stochastic wave equation with polynomial nonlinearity, *Ann. of Appl. Prob.* **12(1)** (2002), 361–381.
P. L. Chow, *Stochastic Partial Differential Equations*. Chapman & Hall/CRC, New York, 2007.
P. L. Chow, W. Kohler, & G. Papanicolaou, *Multiple Scattering and Waves in Random Media*, North–Holland, Amsterdam, 1981.
N. Ikeda and S. Watanabe, *Stochastic Differential Equations and Diffusion Processes, Vol. 24 of North-Holland Mathematical Library*, North-Holland Publishing Co., Amsterdam, 1981.
Y. Lv & W. Wang, Limit dynamics for stochastic wave equations, *J. Diff. Equa.* **244** (2008),1–23.
G. Da Prato & J. Zabczyk, *Stochastic Equations in Infinite Dimensions*, Cambridge University Press, 1992.
M. Reed & B. Simon, *Methods of Modern Mathematical Physics II*, Academic Press, New York, 1975.
A. J. Roberts. Resolving the multitude of microscale interactions accurately models stochastic partial differential equations. , 9:193–221, 2006. <http://www.lms.ac.uk/jcm/9/lms2005-032>.
A. J. Roberts. Normal form transforms separate slow and fast modes in stochastic dynamical systems. , 387:12–38, 2008.
A. J. Roberts. Normal form of stochastic or deterministic multiscale differential equations. Technical report, <http://www.maths.adelaide.edu.au/anthony.roberts/sdenf.html>, 2009. Revised April 2011.
J. Simon, Compact sets in the space $L^p(0, T; B)$, *Ann. Mat. Pura Appl.*, **146** (1987), 65–96.
W. Wang and Y. Lv, Limit behavior of nonlinear stochastic wave equations with singular perturbation, *Disc. and Cont. Dyna. Syst. B*, **13(1)** (2010) 175–193.
W. Wang and A. J. Roberts, Average and deviation for slow–fast stochastic partial differential equations, preprinted, 2009.
G. Whitham, *Linear and Nonlinear Waves*, Wiley, New York, 1974.
[^1]: School of Science, Nanjing University of Science & Technology, Nanjing, 210094, <span style="font-variant:small-caps;">China</span>. [mailto:lvyan1998@yahoo.com.cn](mailto:lvyan1998@yahoo.com.cn)
[^2]: School of Mathematics, University of Adelaide, South Australia, <span style="font-variant:small-caps;">Australia</span>. [mailto:anthony.roberts@adelaide.edu.au](mailto:anthony.roberts@adelaide.edu.au)
|
---
abstract: 'Testing uniformity on the $p$-dimensional unit sphere is arguably the most fundamental problem in directional statistics. In this paper, we consider this problem in the framework of *axial* data, that is, under the assumption that the $n$ observations at hand are randomly drawn from a distribution that charges antipodal regions equally. More precisely, we focus on axial, rotationally symmetric, alternatives and first address the problem under which the direction ${{\pmb\theta}}$ of the corresponding symmetry axis is specified. In this setup, we obtain Le Cam optimal tests of uniformity, that are based on the sample covariance matrix (unlike their non-axial analogs, that are based on the sample average). For the more important unspecified-${{\pmb\theta}}$ problem, some classical tests are available in the literature, but virtually nothing is known on their non-null behavior. We therefore study the non-null behavior of the celebrated Bingham test and of other tests that exploit the single-spiked nature of the considered alternatives. We perform Monte Carlo exercises to investigate the finite-sample behavior of our tests and to show their agreement with our asymptotic results.'
address:
- |
$^{\dagger * \ddagger}$Université libre de Bruxelles\
ECARES and Département de Mathématique\
Avenue F.D. Roosevelt, 50\
ECARES, CP114/04\
B-1050, Brussels\
Belgium\
- |
$^*$Université Toulouse Capitole\
Toulouse School of Economics\
21, Allée de Brienne\
31015 Toulouse Cedex 6\
France\
author:
- '$^\dagger$, and $^\ddagger$'
bibliography:
- 'Paper.bib'
title: On the power of axial tests of uniformity on spheres
---
Introduction {#sec:intro}
============
Directional statistics are concerned with data taking values on the unit hypersphere $\mathcal{S}^{p-1}:=\{{\mathbf{x}}\in{\mathbb R}^p:\|{\mathbf{x}}\|^2:={\mathbf{x}}'{\mathbf{x}}=1\}$ of ${\mathbb R}^p$. Classical applications, that most often relate to the circular case ($p=2$) or spherical one ($p=3$), involve wind and animal migration data or belong to fields such as geology, paleomagnetism, or cosmology. We refer, e.g., to [@Fish87], [@MarJup2000], and [@LV17book] for book-length treatments of the topic and for further applications.
Arguably the most fundamental problem in directional statistics is the problem of testing for uniformity, which, for a random sample ${\mathbf{X}}_{1},\ldots,{\mathbf{X}}_{n}$ at hand, consists in testing the null hypothesis that the observations are sampled from the uniform distribution over $\mathcal{S}^{p-1}$. This is a very classical problem in multivariate analysis that can be traced back to [@Bernoulli1735]. As explained in the review paper [@GV19], the topic has recently received a lot of attention: to cite only a few contributions, [@Ju08] proposed data-driven Sobolev tests, [@CuA09] and [@GNC19] proposed tests based on random projections, [@LPthanh14] studied the problem for noisy data, [@PaiVer2016] obtained the high-dimensional limiting behavior of some classical test statistics under the null hypothesis while [@GPV19] transformed some uniformity tests into tests of rotational symmetry.
A classical uniformity test dates back to [@Ray1919] and rejects the null hypothesis for large values of $\|\bar{{\mathbf{X}}}_n\|$, where $\bar{{\mathbf{X}}}_n=\frac{1}{n}\sum_{i=1}^n {\mathbf{X}}_i$ is the sample mean of the observations. This test will detect only alternatives whose mean vectors are non-zero, hence is therefore typically used when possible deviations from uniformity are suspected to be asymmetric. In particular, the Rayleigh test will show no power when the common distribution of the ${\mathbf{X}}_i$’s is an antipodally symmetric distribution on the sphere, that is, when this distribution attributes the same probability to antipodal regions. Far from being the exception, such antipodally symmetric distributions are actually those that need be considered when practitioners are facing *axial data*, that is, when one does not observe genuine locations on the sphere but rather axes (a typical example of axial data relates to the directions of optical axes in quartz crystals; see, e.g., [@MarJup2000]). Models and inference for axial data have been considered a lot in the literature: to cite a few, [@Tyl87ang] and more recently [@PRV19] considered inference on the distribution of the spatial sign of a Gaussian vector, [@Watson65], [@Bi07] and [@Sra13] considered inference for Watson distributions (see the next section), [@Dryden05] obtained distributions on high-dimensional spheres while [@AndSte1972], [@Bin1974] and [@JU01] considered uniformity tests against antipodally symmetric alternatives.
Now, while the literature offers both axial and non-axial tests of uniformity, axial procedures unfortunately remain much less well understood than their non-axial counterparts, particularly so when it comes to their non-null behaviors. Strong results have been obtained for non-axial tests of uniformity regarding their asymptotic power under suitable local alternatives to uniformity and even regarding their optimality (we refer to [@PaiVer17a] and to the references therein), but virtually nothing is known in that direction for axial tests of uniformity. This provides the main motivation for the present work, that intends to fill an important gap by studying the non-null behavior of some classical (and less classical) axial tests of uniformity. Quite naturally, we will do so in the semiparametric distributional framework that has been classically considered for non-axial tests of uniformity, namely the framework of *rotationally symmetric distributions* indexed by a finite-dimensional parameter $(\kappa, {{\pmb\theta}}) \in {\mathbb R}^+ \times {\cal S}^{p-1}$ and an infinite-dimensional parameter $f \in {\cal F}$ (a family of functions we define in the next section). Within this semiparametric model, the null hypothesis of uniformity takes the form ${\cal H}_0: \kappa=0$. In this paper, we first derive the shape of tests that are locally and asymptotically optimal under any $f \in {\cal F}$ for the specified-${{\pmb\theta}}$ problem. We then focus on the unspecified-${{\pmb\theta}}$ problem and discuss its connection with the specified-${{\pmb\theta}}$ problem. We derive the limiting behavior, under sequences of contiguous alternatives (for any $f \in {\cal F}$), of the tests provided in [@AndSte1972] and [@Bin1974]. Doing so, we obtain in particular the limiting behavior, under local alternatives, of the extreme eigenvalues of the spatial sign covariance matrix, which is a result of independent interest; see [@DuTyl16] and the references therein for a recent study of these eigenvalues.
The outline and contribution of the paper are as follows. In Section \[sec:axial\], we introduce the class of (rotationally symmetric) alternatives to uniformity that will be considered in this work, which deviate from uniformity along a direction ${{\pmb\theta}}$ and have a severity controlled by a concentration parameter $\kappa$. In this framework, we identify the sequences $(\kappa_n)$ that make the corresponding sequences of alternatives contiguous to the null hypothesis of uniformity. In Section \[sec:TestSpeci\], we tackle the problem of testing uniformity under specified ${{\pmb\theta}}$ and show that the resulting model is *locally asymptotically normal* (LAN). We define the resulting optimal tests of uniformity and determine their asymptotic powers under contiguous alternatives. In Section \[sec:Bingham\], we turn to the unspecified-${{\pmb\theta}}$ problem, we show that our LAN result naturally leads to the [@Bin1974] test of uniformity, and we study the limiting behavior of this test under contiguous alternatives. In Section \[sec:eigentest\], we turn our attention to tests that take into account the “single-spiked" structure of the considered alternatives. We characterize the asymptotic behavior of these tests both under the null hypothesis and under sequences of contiguous alternatives. While all results above are confirmed by suitable numerical exercises in Sections \[sec:TestSpeci\]–\[sec:eigentest\], we specifically conduct, in Section \[sec:Simu\], Monte-Carlo simulations in order to compare the finite-sample powers of the various tests. We provide final comments and perspectives for future research in Section \[sec:Final\]. Finally, an appendix contains all proofs.
Axial rotationally symmetric distributions {#sec:axial}
==========================================
In this section, we set the notation and describe the class of axial distributions we will use as alternatives to uniformity. A celebrated class of axial distributions on the sphere is the one that collects the Watson distributions, which admit a density (throughout, densities over $\mathcal{S}^{p-1}$ are with respect to the surface area measure) of the form $$\label{densWatson}
{\mathbf{x}}\mapsto
\frac{c_{p,\kappa} \Gamma(\frac{p-1}{2})}{2\pi^{(p-1)/2}}
\exp(\kappa\, ({\mathbf{x}}'{{\pmb\theta}})^2)
,
\qquad
{\mathbf{x}}\in\mathcal{S}^{p-1}
,$$ where ${{\pmb\theta}}$ belongs to $\mathcal{S}^{p-1}$, $\kappa$ is a real number, and where the value of the normalizing constant $c_{p,\kappa}$ can be obtained from (\[normconst\]); as usual, $\Gamma$ denotes Euler’s Gamma function. Since this density is a symmetric function of ${\mathbf{x}}$, it attributes the same probability to antipodal regions on the sphere, hence is indeed suitable for axial data. The Watson distribution is rotationally symmetric about ${{\pmb\theta}}$, in the sense that if ${\mathbf{X}}$ has density (\[densWatson\]), then ${\bf O}{\mathbf{X}}$ and ${\mathbf{X}}$ share the same distribution for any $p\times p$ orthogonal matrix ${\bf O}$ such that ${\bf O}{{\pmb\theta}}={{\pmb\theta}}$; consequently, ${{\pmb\theta}}$ will be considered a *location* parameter. The Watson distributions are the rotationally symmetric (or single-spiked) [@Bin1974] distributions. The parameter $\kappa$ is a *concentration* parameter: the larger $|\kappa|$, the more the probability mass will be concentrated—symmetrically about the poles $\pm{{\pmb\theta}}$ for positive values of $\kappa$ (bipolar case) or symmetrically about the hyperspherical equator $\mathcal{S}^\perp_{{{\pmb\theta}}}:=\{{\mathbf{x}}\in\mathcal{S}^{p-1}:{\mathbf{x}}'{{\pmb\theta}}=0\}$ for negative values of $\kappa$ (girdle case). Of course, the value $\kappa=0$ corresponds to the uniform distribution over the sphere.
In this paper, we consider a natural semiparametric extension of the class of Watson distributions, namely the class of axial distributions admitting a density of the form $$\label{densconc}
{\mathbf{x}}\mapsto
\frac{c_{p,\kappa,f} \Gamma(\frac{p-1}{2})}{2\pi^{(p-1)/2}}
f(\kappa\, ({\mathbf{x}}'{{\pmb\theta}})^2)
,
\qquad
{\mathbf{x}}\in\mathcal{S}^{p-1},$$ where ${{\pmb\theta}}$ and $\kappa$ are as in (\[densWatson\]), $f$ belongs to the class of functions $\mathcal{F}:=\{f:{\mathbb R}\to{\mathbb R}^+: f \textrm{ monotone increasing, twice differentiable at }0, \textrm{ with } f(0)=f'(0)=1\}$, and where $$\label{normconst}
c_{p,\kappa,f}
=
1\ \Big/ \int_{-1}^1 (1-s^2)^{(p-3)/2}f(\kappa s^2)\,ds
.$$ The parameter $\kappa$ is still a concentration parameter, that shares the same interpretation as for Watson distributions. The restriction to $\mathcal{F}$ above is made for identifiability purposes. If $\kappa\neq 0$, then $f$ and the pair $\{\pm{{\pmb\theta}}\}$ are identifiable, but ${{\pmb\theta}}$ itself is not (which is natural for axial distributions). For $\kappa=0$, the uniform distribution over the sphere is obtained (irrespective of $f$), in which case the location parameter ${{\pmb\theta}}$ is unidentifiable, even up to a sign. The corresponding normalizing constant is $c_p :=\lim_{\kappa\to 0} c_{p,\kappa,f}$. The distribution associated with (\[densconc\]) is rotationally symmetric about ${{\pmb\theta}}$, and if ${\mathbf{X}}$ is a random vector with this distribution, then ${\mathbf{X}}'{{\pmb\theta}}$ has density $s\mapsto c_{p,\kappa,f}(1-s^2)^{(p-3)/2}f(\kappa s^2)\mathbb{I}[|s|\leq 1]$, which explains the expression (\[normconst\]). In the present axial case, this density is of course symmetric with respect to zero. The semiparametric class of distributions just introduced will be used in the paper as alternatives to uniformity on the sphere. We will consider the following hypotheses and asymptotic scenarios. For ${{\pmb\theta}}\in\mathcal{S}^{p-1}$, a sequence $(\kappa_n)$ in ${\mathbb R}_0$ and $f\in\mathcal{F}$, we will denote as ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$ the hypothesis under which ${\mathbf{X}}_{n1},\ldots,{\mathbf{X}}_{nn}$ form a random sample from the density ${\mathbf{x}}\mapsto c_{p,\kappa_n,f} f(\kappa_n\, ({\mathbf{x}}'{{\pmb\theta}})^2)$ over $\mathcal{S}^{p-1}$. This triangular array framework will allow us to consider local alternatives, associated with suitable sequences $(\kappa_n)$ converging to zero. The null hypothesis of uniformity will be denoted as ${\rm P}{^{(n)}}_{0}$. The sequence of hypotheses ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$ determines a sequence of alternatives to uniformity: the larger $|\kappa_n|$, the more severe the corresponding alternative, whereas the sign of $\kappa_n$ determines the type of alternatives considered, i.e., *bipolar* (for $\kappa_n>0$) or *girdle-like* (for $\kappa_n<0$). At places, it will be of interest to compare our results with those obtained in the non-axial case, that is, in the case where ${\mathbf{X}}_{n1},\ldots,{\mathbf{X}}_{nn}$ have a common density proportional to $f(\kappa_n {\mathbf{x}}'{{\pmb\theta}})$ still with $f$ monotone increasing (rather than $f(\kappa_n({\mathbf{x}}'{{\pmb\theta}})^2)$).
Our first result identifies the sequences of alternatives ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$ that are contiguous to the sequence of null hypotheses ${\rm P}{^{(n)}}_{0}$.
\[TheorContig\] Fix $p\in\{2,3,\ldots\}$, ${{\pmb\theta}}\in\mathcal{S}^{p-1}$, and $f\in\mathcal{F}$. Let $(\kappa_n)$ be a sequence in ${\mathbb R}_0$ that is $O(1/\sqrt{n})$. Then, the sequence of alternative hypotheses ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$ and the sequence of null hypotheses ${\rm P}{^{(n)}}_{0}$ are mutually contiguous.
In other words, if $\kappa_n=O(1/\sqrt{n})$, then no test for $\mathcal{H}_{0n}:\{{\rm P}{^{(n)}}_{0}\}$ against $\mathcal{H}_{1n}:\{{\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}\}$ can be consistent. Actually, it will follow from Theorem \[TheorLAN\] in the next section that $1/\sqrt{n}$ is the *contiguity rate*, in the sense that if $\kappa_n=\tau/\sqrt{n}$ (for some non-zero real constant $\tau$), then there exist tests for $\mathcal{H}_{0n}:\{{\rm P}{^{(n)}}_{0}\}$ against $\mathcal{H}_{1n}:\{{\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}\}$ showing non-trivial asymptotic powers (that is, asymptotic powers in $(\alpha,1)$, where $\alpha$ denotes the nominal level). The contiguity rate in the axial case thus coincides with the one obtained in the non-axial case; see Theorems 2.1 and 3.1 in [@PaiVer17a].
Tests of uniformity under specified location {#sec:TestSpeci}
============================================
In this section, we consider the problem of testing uniformity over $\mathcal{S}^{p-1}$ against the class of alternatives introduced in the previous section, in a situation where the location ${{\pmb\theta}}$ is specified. In other words, this corresponds to cases where it is known in which direction the possible deviation from uniformity would materialize. Depending on the exact type of alternatives we want to focus on (bipolar, girdle-type, or both), we will then consider, for a fixed ${{\pmb\theta}}$, the problem of testing $\mathcal{H}_0{^{(n)}}:\{{\rm P}_0{^{(n)}}\}$ against (i) $\mathcal{H}_1{^{(n)}}:\cup_{\kappa>0}\cup_{f\in\mathcal{F}} \{{\rm P}_{{{\pmb\theta}},\kappa,f}{^{(n)}}\}$, against (ii) $\mathcal{H}_1{^{(n)}}:\cup_{\kappa<0}\cup_{f\in\mathcal{F}} \{{\rm P}_{{{\pmb\theta}},\kappa,f}{^{(n)}}\}$, or against (iii) $\mathcal{H}_1{^{(n)}}:\cup_{\kappa\neq 0}\cup_{f\in\mathcal{F}} \{{\rm P}_{{{\pmb\theta}},\kappa,f}{^{(n)}}\}$. Optimal testing may be based on the following Local Asymptotic Normality (LAN) result.
\[TheorLAN\] Fix $p\in\{2,3,\ldots\}$, ${{\pmb\theta}}\in\mathcal{S}^{p-1}$, and $f\in\mathcal{F}$. Let $\kappa_n=\tau_n p/\sqrt{n}$, where the real sequence $(\tau_n)$ is $O(1)$ but not $o(1)$. Then, letting $$\Delta_{{{\pmb\theta}}}{^{(n)}}:=
\frac{p}{\sqrt{n}}
\sum_{i=1}^n \Big\{
({\mathbf{X}}_{ni}{^{\prime}}{{\pmb\theta}})^2-\frac{1}{p}
\Big\}
\quad\textrm{and}\quad
\Gamma_p
:=
\frac{2(p-1)}{p+2}
\,
,$$ we have that, as $n\to\infty$ under ${\rm P}_{0}{^{(n)}}$, $$\label{LAN}
\Lambda_n
=
\log \frac{d{\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}}{d{\rm P}{^{(n)}}_{0}}
=
\tau_n
\Delta_{{{\pmb\theta}}}{^{(n)}}-
\frac{\tau_n^2}{2} \Gamma_p
+
o_{\rm P}(1)
,$$ where $\Delta_{{{\pmb\theta}}}{^{(n)}}$ is asymptotically normal with mean zero and variance $\Gamma_p$. In other words, the sequence $(\{ {\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa,f} : \kappa\in{\mathbb R}\})$ $($where we let ${\rm P}{^{(n)}}_{{{\pmb\theta}},0,f}:={\rm P}{^{(n)}}_0)$ is locally asymptotically normal at $\kappa=0$ with central sequence $\Delta_{{{\pmb\theta}}}{^{(n)}}$, Fisher information $\Gamma_p$, and contiguity rate $1/\sqrt{n}$.
This result confirms that $1/\sqrt{n}$ is the contiguity rate when testing uniformity against the considered axial alternatives. Note also that the central sequence rewrites $$\Delta_{{{\pmb\theta}}}{^{(n)}}=
\sqrt{n}
\big(
p
\,
{{\pmb\theta}}'
{\mathbf{S}}_n
{{\pmb\theta}}-
1
\big)
,$$ where ${\mathbf{S}}_n:=n^{-1}\sum_{i=1}^n {\mathbf{X}}_{ni}{\mathbf{X}}_{ni}{^{\prime}}$ is the sample covariance matrix of the observations (with respect to a fixed location, namely the origin of ${\mathbb R}^{p}$). Consequently, optimal testing of uniformity for axial data will be based on $\mathbf{S}_n$. This is to be compared with the non-axial case considered in [@PaiVer17a], where optimal testing of uniformity is rather based on $\bar{{\mathbf{X}}}_n=n^{-1}\sum_{i=1}^n {\mathbf{X}}_{ni}$. This will have important consequences when considering the unspecified-${{\pmb\theta}}$ case we turn to in Sections \[sec:Bingham\]–\[sec:eigentest\].
More importantly, the optimal axial tests of uniformity in the specified location case directly result from the LAN property above. More precisely, Theorem \[TheorLAN\] entails that, for the problem of testing $\mathcal{H}_0{^{(n)}}:\{{\rm P}_0{^{(n)}}\}$ against $\cup_{\kappa>0}\cup_{f\in\mathcal{F}} \{{\rm P}_{{{\pmb\theta}},\kappa,f}{^{(n)}}\}$, the test $\phi_{{{\pmb\theta}}+}^{(n)}$ rejecting the null hypothesis at asymptotic level $\alpha$ whenever $$\label{Testright}
T{^{(n)}}_{{{\pmb\theta}}}
:=
\frac{\Delta_{{{\pmb\theta}}}{^{(n)}}}{\sqrt{\Gamma_p}}
>
z_{\alpha}$$ is locally asymptotically most powerful; here, $z_\alpha=\Phi^{-1}(1-\alpha)$ denotes the upper of the standard normal distribution. A routine application of Le Cam’s third lemma shows that, under ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$ with $\kappa_n=\tau p/\sqrt{n}$ ($\tau>0$), $T{^{(n)}}_{{{\pmb\theta}}}$ is asymptotically normal with mean $\Gamma_p^{1/2}\tau$ and variance one. Therefore, the corresponding asymptotic power of $\phi_{{{\pmb\theta}}+}^{(n)}$ is $$\label{eq:right-sided-power}
\lim_{n\to\infty}
{\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}[T{^{(n)}}_{{{\pmb\theta}}}>z_{\alpha}]
=
1-\Phi\big(z_{\alpha}-\Gamma_p^{1/2}\tau\big)
.$$ Note that this asymptotic power does not converge to $\alpha$ as $p$ diverges to infinity. This may be surprising at first since departures from uniformity here are of a *single-spiked* nature, that is, only materialize in a single direction out of the $p$ directions in $\mathcal{S}^{p-1}$. The fact that this asymptotic power does not fade out for larger dimensions is actually explained by the fact that we did not consider local alternatives associated with $\kappa_n=\tau/\sqrt{n}$ but rather with $\kappa_n=\tau p/\sqrt{n}$, which properly scales local alternatives for different dimensions $p$.
Optimal tests for the other one-sided problem and for the two-sided problem are obtained in a similar way. More precisely, for the problem of testing $\mathcal{H}_0{^{(n)}}:\{{\rm P}_0{^{(n)}}\}$ against $\cup_{\kappa<0}\cup_{f\in\mathcal{F}} \{{\rm P}_{{{\pmb\theta}},\kappa,f}{^{(n)}}\}$, the test $\phi_{{{\pmb\theta}}-}^{(n)}$ rejecting the null hypothesis of uniformity at asymptotic level $\alpha$ whenever $T{^{(n)}}_{{{\pmb\theta}}}<-z_\alpha$ is locally asymptotically most powerful and has asymptotic power $$\lim_{n\to\infty}
{\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}[T{^{(n)}}_{{{\pmb\theta}}}< -z_{\alpha}]
=
\Phi(-z_{\alpha}-\Gamma_p^{1/2}\tau)$$ under ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$ with $\kappa_n=\tau p/\sqrt{n}$ ($\tau<0$). The corresponding two-sided test, $\phi_{{{\pmb\theta}}\pm}^{(n)}$ say, rejects the null hypothesis at asymptotic level $\alpha$ whenever $|T{^{(n)}}_{{{\pmb\theta}}}|>z_{\alpha/2}$. This test is locally asymptotically maximin for the two-sided problem and has asymptotic power $$\lim_{n\to\infty}{\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}[|T{^{(n)}}_{{{\pmb\theta}}}|>z_{\alpha/2}]
=
2-\Phi(z_{\alpha/2}-\Gamma_p^{1/2}\tau)-\Phi(z_{\alpha/2}+\Gamma_p^{1/2}\tau)$$ under ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$ with $\kappa_n=\tau p/\sqrt{n}$ ($\tau\neq 0$). Again, the local asymptotic powers of these tests do not fade out for larger dimensions $p$ but rather converge to a constant larger than $\alpha$.
We conducted the following Monte Carlo exercise in order to check the validity of our asymptotic results. For any combination $(n,p)$ of sample size $n\in\{100,1\:\!000\}$ and dimension $p\in\{3,10\}$, we generated collections of $5\:\!000$ independent random samples of size $n$ from the Watson distribution with location ${{\pmb\theta}}=(1,0,\ldots,0)'\in{\mathbb R}^{p}$ and concentration $\kappa_n=\tau p/\sqrt{n}$, for $\tau=-2,-1,0,1,2$; see Section \[sec:axial\]. The value $\tau=0$ corresponds to the null hypothesis of uniformity over $\mathcal{S}^{p-1}$, whereas the larger the non-zero value of $|\tau|$ is, the more severe the alternative is. Kernel density estimates of the resulting values of the test statistic $T{^{(n)}}_{{{\pmb\theta}}}$ in (\[Testright\]) are provided in Figure \[Fig1\], that further plots the densities of the corresponding asymptotic distributions (for the null case $\tau=0$, histograms of the values of $T{^{(n)}}_{{{\pmb\theta}}}$ are also shown). Clearly, our asymptotic results are confirmed by these simulations (yet, unsurprisingly, larger dimensions require larger sample sizes for asymptotic results to materialize).
![ Plots of the kernel density estimates (solid curves) of the values of the test statistic $T{^{(n)}}_{{{\pmb\theta}}}$ in (\[Testright\]) obtained from $M=5\:\!000$ independent random samples, of size $n=100$ (top) or $1\:\!000$ (bottom), from the Watson distribution with location ${{\pmb\theta}}=(1,0,\ldots,0)'\in{\mathbb R}^p$ and concentration $\kappa=\tau p/\sqrt{n}$, with $\tau=-2,-1,0,1,2$ and with $p=3$ (left) or $p=10$ (right); for $\tau=0$, histograms of the values of $T{^{(n)}}_{{{\pmb\theta}}}$ are shown. The densities of the corresponding asymptotic ($\mathcal{N}(\Gamma_p^{1/2}\tau,1)$) distributions are also plotted (dashed curves). Throughout this paper, kernel density estimates are obtained from the R command `density` with default parameter values. []{data-label="Fig1"}](Figs/Fig1.pdf){width="\textwidth"}
The unspecified location case: the Bingham test {#sec:Bingham}
===============================================
We now turn to the version of the testing problems considered in the previous section. We focus first on the one-sided problem of testing $\mathcal{H}_0{^{(n)}}:\{{\rm P}_0{^{(n)}}\}$ against $\mathcal{H}_1{^{(n)}}:\cup_{{{\pmb\theta}}\in\mathcal{S}^{p-1}}\cup_{\kappa>0}\cup_{f\in\mathcal{F}} \{{\rm P}_{{{\pmb\theta}},\kappa,f}{^{(n)}}\}$. It is convenient to reparameterize the submodel associated with $\kappa\geq 0$ by writing ${{\pmb\vartheta}}=\sqrt{\kappa} {{\pmb\theta}}$. In this new parametrization (which, unlike the original curved one, is flat), the testing problem writes $\mathcal{H}_0{^{(n)}}:\{{\rm P}_{\bf 0}{^{(n)}}={\rm P}_0{^{(n)}}\}$ against $\cup_{{{\pmb\vartheta}}\in{\mathbb R}^{p}\setminus\{0\}}\cup_{f\in\mathcal{F}} \{{\rm P}_{{{\pmb\vartheta}},f}{^{(n)}}\}$, that is, simply consists in testing $\mathcal{H}_0{^{(n)}}:{{\pmb\vartheta}}={\bf 0}$ against $\mathcal{H}_1{^{(n)}}:{{\pmb\vartheta}}\neq {\bf 0}$. Theorem \[TheorLANunspec\] below then describes the asymptotic behavior of the corresponding local log-likelihood ratios.
To be able to state the result, we need to introduce the following notation. For a matrix ${\mathbf{A}}$, we will write ${\rm vec}\, {\mathbf{A}}$ for the vector obtained by stacking the columns of ${\mathbf{A}}$ on top of each other. We let ${\mathbf{J}}_p:=({\rm vec}\, \mathbf{I}_p)({\rm vec}\, \mathbf{I}_p)'$, where $\mathbf{I}_\ell$ is the $\ell\times \ell$ identity matrix. Finally, with the usual Kronecker product $\otimes$, the $p^2\times p^2$ commutation matrix is ${\bf K}_p:=\sum_{i,j=1}^p ({\bf e}_i {\bf e}_j')\otimes ({\bf e}_j {\bf e}_i')$, where $e_\ell$ is the $\ell$th vector of the canonical basis of ${\mathbb R}^p$. We then have the following result.
\[TheorLANunspec\] Fix $p\in\{2,3,\ldots\}$ and $f\in\mathcal{F}$. Let ${{\pmb\vartheta}}_n=(p/\sqrt{n})^{1/2}{{\pmb\tau}}_n$, where $({{\pmb\tau}}_n)$ is a sequence in ${\mathbb R}^p$ that is $O(1)$ but not $o(1)$. Then, letting $${{\pmb \Delta}}{^{(n)}}:=
p\sqrt{n}
\,
{\rm vec}\bigg(
{\mathbf{S}}_n-\frac{1}{p} \mathbf{I}_p
\bigg)
\quad\textrm{and}\quad
{{\pmb \Gamma_p}}:=
\frac{p}{p+2} \Big( \mathbf{I}_{p^2} + \mathbf{K}_p -\frac{2}{p} \mathbf{J}_p \Big)
\,
,$$ we have that, as $n\to\infty$ under ${\rm P}_{0}{^{(n)}}$, $$\label{LANvec}
\log \frac{d{\rm P}{^{(n)}}_{{{\pmb\vartheta}}_n,f}}{d{\rm P}{^{(n)}}_{0}}
=
({\rm vec}({{\pmb\tau}}_n{{\pmb\tau}}_n'))'
{{\pmb \Delta}}{^{(n)}}-
\frac{1}{2} ({\rm vec}({{\pmb\tau}}_n{{\pmb\tau}}_n'))'{{\pmb \Gamma_p}}{\rm vec}({{\pmb\tau}}_n{{\pmb\tau}}_n')
+
o_{\rm P}(1)
,$$ where ${{\pmb \Delta}}{^{(n)}}$ is, still under ${\rm P}_{0}{^{(n)}}$, asymptotically normal with mean vector zero and covariance matrix ${{\pmb \Gamma_p}}$.
Theorem \[TheorLANunspec\] shows that the contiguity rate for ${{\pmb\vartheta}}$ is $n^{-1/4}$, which corresponds to the contiguity rate $n^{-1/2}$ obtained for $\kappa$ in Theorem \[TheorLAN\] (recall that ${{\pmb\vartheta}}=\sqrt{\kappa} {{\pmb\theta}}$); however, as we will explain below, the limiting experiment in Theorem \[TheorLANunspec\] is non-standard. Writing ${\mathbf{A}}^-$ for the Moore-Penrose generalized inverse of ${\mathbf{A}}$, a natural test of uniformity is the test rejecting the null hypothesis at asymptotic level $\alpha$ whenever $$\label{Binghamtest}
Q{^{(n)}}:=
({{\pmb \Delta}}{^{(n)}})' {\pmb\Gamma}_p^- {{\pmb \Delta}}{^{(n)}}=
\frac{np(p+2)}{2}
\bigg(\text{tr}[{\mathbf{S}}_n^2]-\frac{1}{p}\bigg)
>
\chi^2_{d_p,1-\alpha}
,$$ where we denoted as $\chi^2_{d_p,1-\alpha}$ the upper $\alpha$-quantile of the chi-square distribution with $d_p:=p(p+1)/2-1$ degrees of freedom. This test, which rejects the null hypothesis when the sample variance of the eigenvalues $\hat{\lambda}_{n1},\ldots,\hat{\lambda}_{np}$ of ${\mathbf{S}}_n$ is too large, also addresses the problem of testing uniformity against the one-sided alternatives associated with $\kappa<0$ or against the two-sided alternatives associated with $\kappa\neq 0$. This procedure, which is known as the [@Bin1974] test (hence will be denoted as $\phi_{\rm Bing}$ in the sequel), is often regarded as the simplest test of uniformity for axial data; see Section 10.7 in [@MarJup2000]. When applied to the unit vectors ${\mathbf{X}}_{ni}={\mathbf{Z}}_{ni}/\|{\mathbf{Z}}_{ni}\|$, $i=1,\ldots,n$, obtained from Euclidean data ${\mathbf{Z}}_{ni}$, $i=1,\ldots,n$, this test is also the sign test of sphericity from [@HalPai2006], and it follows from that paper that, as an axial test for uniformity on the sphere, the Bingham test is optimal against *angular Gaussian alternatives* (see [@Tyl87ang]), that is, against projections of elliptical distributions on the sphere.
Local asymptotic powers of the Bingham test can be obtained from the LAN result in Theorem \[TheorLAN\] and Le Cam’s third lemma. We have the following result.
\[TheorBinghamHD\] Fix $p\in\{2,3,\ldots\}$, ${{\pmb\theta}}\in\mathcal{S}^{p-1}$, and $f\in\mathcal{F}$. Let $\kappa_n=\tau_n p/\sqrt{n}$, where the real sequence $(\tau_n)$ converges to $\tau$. Then, under ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$, $$\label{Binghamtestncp}
Q{^{(n)}}\stackrel{\mathcal{D}}{\to}
\chi^2_{d_p}
\bigg(
\frac{2(p-1)\tau^2}{p+2}
\bigg)
,$$ where $\chi^2_\ell(\delta)$ denotes the non-central chi-square distribution with $\ell$ degrees of freedom and non-centrality parameter $\delta$. Under the same sequence of alternatives, the asymptotic power of the Bingham test is therefore $$\label{eq:right-sided-powerBingham}
\lim_{n\to\infty}
{\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}[
Q{^{(n)}}>
\chi^2_{d_p,1-\alpha}]
=
1
-
\Psi_{d_p}\bigg(
\chi^2_{d_p,1-\alpha}
;
\frac{2(p-1)\tau^2}{p+2}
\bigg)
,$$ where $\Psi_{\ell}(\cdot;\delta)$ is the cumulative distribution function of the $\chi^2_\ell(\delta)$ distribution.
This result in particular shows that the Bingham test is a two-sided procedure, as the asymptotic power in (\[eq:right-sided-powerBingham\]) exhibits a symmetric pattern with respect to girdle-type alternatives ($\tau<0$) and bipolar alternatives ($\tau>0$). This power, unlike the powers of the specified-${{\pmb\theta}}$ tests in the previous section, converges to $\alpha$ as $p$ diverges to infinity, which materializes the fact that, for larger dimensions, the Bingham test severely suffers (even asymptotically) from the unspecification of ${{\pmb\theta}}$. Note also that since the Bingham test is invariant with respect to rotations, its limiting power naturally does not depend on the location parameter ${{\pmb\theta}}$ under the alternative.
We conducted the following simulation exercise to check the validity of the asymptotic results of this section. In dimension $p=3$, we generated $5\:\!000$ mutually independent random samples of size $n=2\:\!000$ from the Watson distribution with location ${{\pmb\theta}}=(1,0,\ldots,0)'\in{\mathbb R}^{p}$ and concentration $\kappa_n=\tau p/\sqrt{n}$, for $\tau=-4,-3,0,3,4$; see Section \[sec:axial\]. We did the same in dimension $p=10$, with sample size $n=10\:\!000$. For both dimensions $p$, Figure \[Fig2\] reports kernel density estimates of the resulting values of the Bingham test statistic $Q{^{(n)}}$. They perfectly match with the corresponding asymptotic distribution in (\[Binghamtestncp\]). The results also confirm the two-sided nature of the Bingham test, that, irrespective of $\tau_0$, asymptotically behaves in the exact same way under $\tau=\pm\tau_0$.
![ (Left:) Plots of the kernel density estimates (solid curves) of the values of the Bingham test statistic $Q{^{(n)}}$ in (\[Binghamtest\]) obtained from $M=5\:\!000$ independent random samples of size $n=2\:\!000$ from the Watson distribution with location ${{\pmb\theta}}=(1,0,\ldots,0)'\in{\mathbb R}^p$ and concentration $\kappa=\tau p/\sqrt{n}$, with $\tau=-4,-3,0,3,4$ and with $p=3$; for $\tau=0$, histograms of the values of $Q{^{(n)}}$ are shown. The density of the corresponding asymptotic distributions in (\[Binghamtestncp\]), which do not depend on the sign of $\tau$, are also plotted (dashed curves). (Right:) The corresponding results for $p=10$ and $n=10\:\!000$. []{data-label="Fig2"}](Figs/Fig2.pdf){width="\textwidth" height="70mm"}
Our results indicate that the Bingham test shows non-trivial asymptotic powers under the contiguous alternatives identified in the previous section. However, a key point is the following: if $({{\pmb\tau}}_n)\to{{\pmb\tau}}$ in the LAN result of Theorem \[TheorLANunspec\], then, under ${\rm P}_{{{\pmb\vartheta}}_n,f}{^{(n)}}$ with ${{\pmb\vartheta}}_n=(p/\sqrt{n})^{1/2}{{\pmb\tau}}_n$, ${{\pmb \Delta}}{^{(n)}}$ is asymptotically normal with mean ${\bf s}_{{\pmb\tau}}={{\pmb \Gamma_p}}{\rm vec}({{\pmb\tau}}{{\pmb\tau}}')$ and covariance matrix ${{\pmb \Gamma_p}}$, so that the sequence of asymptotic experiments at hand does converge to a Gaussian shift experiment (${{\pmb \Delta}}\sim\mathcal{N}({\bf s}_{{\pmb\tau}},{{\pmb \Gamma_p}})$) involving a *constrained* shift ${\bf s}_{{\pmb\tau}}$. As a result, the Bingham test is not Le Cam optimal for the considered problem: this test, which would rather be Le Cam optimal (more precisely, locally asymptotically maximin) for an unconstrained shift ${\bf s}\in{\mathbb R}^{p^2}$, is here “wasting" power against multi-spiked alternatives that are incompatible with the present single-spiked axial model. This is in line with the fact that the Bingham test, which rejects the null hypothesis of uniformity when the sample variance of the eigenvalues $\hat{\lambda}_{n1},\ldots,\hat{\lambda}_{np}$ of ${\mathbf{S}}_n$ is too large, uses these eigenvalues in a permutation-invariant way (in the considered single-spiked models, it would be more natural to consider specifically $\hat{\lambda}_{n1}$ and/or $\hat{\lambda}_{np}$ to detect possible deviations from uniformity).
The unspecified location case: single-spiked tests {#sec:eigentest}
==================================================
A natural question is then: how to construct a test that is more powerful than the Bingham test? We now describe two constructions that actually lead to the same test(s). Focusing again at first on the one-sided problem involving the bipolar alternatives, we saw in Section \[sec:TestSpeci\] that, in the specified location case, Le Cam optimal tests of uniformity reject $\mathcal{H}_0{^{(n)}}:\{{\rm P}_0{^{(n)}}\}$ in favor of $\cup_{\kappa>0}\cup_{f\in\mathcal{F}} \{{\rm P}_{{{\pmb\theta}},\kappa,f}{^{(n)}}\}$ for large values of $\Delta_{{{\pmb\theta}}}{^{(n)}}=
\sqrt{n}
\big(
p
\,
{{\pmb\theta}}'
{\mathbf{S}}_n
{{\pmb\theta}}-
1
\big)$. In the unspecified location case, it is then natural, following [@Dav1977; @Dav1987; @Dav2002], to consider the test, $\phi_+^{(n)}$ say, rejecting the null hypothesis of uniformity at asymptotic level $\alpha$ when $$\label{Teststatlambda1}
T_+^{(n)}
:=
\sup_{{{\pmb\theta}}\in\mathcal{S}^{p-1}}
\Delta_{{{\pmb\theta}}}{^{(n)}}=
\sqrt{n}
\big(
p
\hat{\lambda}_{n1}
-
1
\big)
>
c_{p,\alpha,+}
,$$ where $\hat{\lambda}_{n1}$ still denotes the largest eigenvalue of ${\mathbf{S}}_n$ and $c_{p,\alpha,+}$ is such that this test has asymptotic size $\alpha$ under the null hypothesis.
A similar rationale yields natural tests for the other one-sided problem and for the two-sided problem: since Le Cam optimal tests of uniformity reject $\mathcal{H}_0{^{(n)}}:\{{\rm P}_0{^{(n)}}\}$ in favor of $\cup_{\kappa<0}\cup_{f\in\mathcal{F}} \{{\rm P}_{{{\pmb\theta}},\kappa,f}{^{(n)}}\}$ for large values of $-\Delta_{{{\pmb\theta}}}{^{(n)}}=
\sqrt{n}
\big(
p
\,
{{\pmb\theta}}'
{\mathbf{S}}_n
{{\pmb\theta}}-
1
\big)$, the resulting unspecified-${{\pmb\theta}}$ test, $\phi_-^{(n)}$ say, will reject the null hypothesis of uniformity at asymptotic level $\alpha$ when $$\label{Teststatlambdap}
T_-^{(n)}
:=
\sup_{{{\pmb\theta}}\in\mathcal{S}^{p-1}}
(-\Delta_{{{\pmb\theta}}}{^{(n)}})
=
-
\sqrt{n}
\big(
p
\hat{\lambda}_{np}
-
1
\big)
>
c_{p,\alpha,-}
,$$ where $c_{p,\alpha,-}$ is such that this test has asymptotic size $\alpha$ under the null hypothesis. Finally, since Le Cam optimal tests of uniformity reject $\mathcal{H}_0{^{(n)}}:\{{\rm P}_0{^{(n)}}\}$ in favor of $\cup_{\kappa\neq 0}\cup_{f\in\mathcal{F}} \{{\rm P}_{{{\pmb\theta}},\kappa,f}{^{(n)}}\}$ for large values of $|\Delta_{{{\pmb\theta}}}{^{(n)}}|=
\sqrt{n}
\big|
p
\,
{{\pmb\theta}}'
{\mathbf{S}}_n
{{\pmb\theta}}-
1
\big|$, the resulting unspecified-${{\pmb\theta}}$ test, $\phi_\pm^{(n)}$ say, will reject the null hypothesis of uniformity at asymptotic level $\alpha$ when $$\label{Teststatlambda1p}
T_{\pm}^{(n)}
:=
\sup_{{{\pmb\theta}}\in\mathcal{S}^{p-1}}
|\Delta_{{{\pmb\theta}}}{^{(n)}}|
=
\sqrt{n}
\max\big(
|p\hat{\lambda}_{n1}-1|
,
|p\hat{\lambda}_{np}-1|
\big)
>
c_{p,\alpha,\pm}
,$$ where $c_{p,\alpha,\pm}$ is still such that this test has asymptotic size $\alpha$ under the null hypothesis of uniformity.
Another rationale for considering the above tests is the following. For the sake of brevity, let us focus on the one-sided problem involving the bipolar alternatives, that is, the ones associated with $\kappa>0$. A natural idea to obtain an unspecified-${{\pmb\theta}}$ test is to replace ${{\pmb\theta}}$ in the corresponding optimal specified-${{\pmb\theta}}$ test $\phi_{{{\pmb\theta}}+}^{(n)}$ with an estimator $\hat{{{\pmb\theta}}}_n$. Now, under ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa,f}$, we have ${\rm E}[{\mathbf{X}}_{n1}]={\bf 0}$ and $${\rm E}[{\mathbf{X}}_{n1}{\mathbf{X}}_{n1}']
=
g_f(\kappa)
{{\pmb\theta}}{{\pmb\theta}}'
+
\frac{1-g_f(\kappa)}{p-1}
(\mathbf{I}_p-{{\pmb\theta}}{{\pmb\theta}}')$$ (see, e.g., Lemma B.3(i) in [@PaiVer17a]), with $$g_f(\kappa)
:=
{\rm E}_{{{\pmb\theta}},\kappa,f}[({\mathbf{X}}_{n1}'{{\pmb\theta}})^2]
=
c_{p,\kappa,f} \int_{-1}^1 (1-s^2)^{(p-3)/2} s^2 f(\kappa s^2)\,ds
.$$ It is easy to check that, for any $f\in\mathcal{F}$, the function $\kappa\mapsto g_f(\kappa)$ is differentiable at $0$, with derivative $g_f'(0)={\rm Var}{^{(n)}}_{0}[({\mathbf{X}}_{n1}'{{\pmb\theta}})^2]>0$, where ${\rm Var}{^{(n)}}_{0}$ still denotes variance under ${\rm P}{^{(n)}}_0$. Consequently, for $\kappa>0$ small, we have $
g_f(\kappa)
>
g_f(0)
=
1/p
$, so that ${{\pmb\theta}}$ is, up to an unimportant sign (recall that only the pair $\{\pm{{\pmb\theta}}\}$ is identifiable), the leading unit eigenvector of ${\rm E}[{\mathbf{X}}_{n1}{\mathbf{X}}_{n1}']$ (for many functions $f$, including the Watson one $f(z)=\exp(z)$, this remains true for any $\kappa>0$). Therefore, a moment estimator of ${{\pmb\theta}}$ is the leading eigenvector $\hat{{{\pmb\theta}}}_n$ of ${\mathbf{S}}_n=\frac{1}{n} \sum_{i=1}^n {\mathbf{X}}_{ni}{\mathbf{X}}_{ni}'$. Note that in the Watson parametric submodel $\{{\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa,\exp}:\kappa>0\}$, this estimator $\hat{{{\pmb\theta}}}_n$ is also the MLE of ${{\pmb\theta}}$. The resulting test then rejects the null hypothesis of uniformity for large values of $$\Delta{^{(n)}}_{\hat{{{\pmb\theta}}}_n}
:=
\sqrt{n}
\big(
p
\hat{{{\pmb\theta}}}_n'
{\mathbf{S}}_n
\hat{{{\pmb\theta}}}_n
-
1
\big)
=
T_+^{(n)}
,$$ hence coincides with the test $\phi_+{^{(n)}}$ in (\[Teststatlambda1\]). A similar reasoning for the other one-sided problem leads to the test $\phi_{-}^{(n)}$.
The critical values in (\[Teststatlambda1\])–(\[Teststatlambda1p\]) above can of course be obtained from the asymptotic distribution of the corresponding test statistics under the null hypothesis. For $p=3$, the asymptotic null distributions of $T{^{(n)}}_+$ and $T{^{(n)}}_-$ were obtained in [@AndSte1972], where the corresponding one-sided tests $\phi{^{(n)}}_+$ and $\phi{^{(n)}}_-$ were first proposed. We extend their result to the two-sided test statistic $T{^{(n)}}_\pm$ and, more importantly, to the non-null case. The key to do so is the following result.
\[thlambda1asympt\] Fix $p\in\{2,3,\ldots\}$ and $f\in\mathcal{F}$. Let ${\mathbf{Z}}$ be a $p\times p$ random matrix such that ${\rm vec}\,{\mathbf{Z}}\sim\mathcal{N}
(
{\bf 0},
{\bf V}_p
)
$, with $
{\bf V}_p
=
(p/(p+2))
(\mathbf{I}_{p^2}+{\bf K}_p)
-
(2/(p+2))
\mathbf{J}_{p}
.
$ Then, (i) under ${\rm P}_0{^{(n)}}$, $$\bigg(
\begin{array}{c}
\sqrt{n} (p\hat{\lambda}_{n1}-1) \\
\sqrt{n} (p\hat{\lambda}_{np}-1)
\end{array}
\bigg)
\stackrel{\mathcal{D}}{\to}
\bigg(
\begin{array}{c}
L^{\rm max}_{p} \\
L^{\rm min}_{p}
\end{array}
\bigg)
,$$ where $L_{p,{\rm max}}$ (resp., $L_{p,{\rm min}}$) is the largest (resp., smallest) eigenvalue of ${{\mathbf{Z}}}$; (ii) under ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$, where $\kappa_n=\tau_n p/\sqrt{n}$ involves a real sequence $(\tau_n)$ converging to $\tau$, $$\bigg(
\begin{array}{c}
\sqrt{n} (p\hat{\lambda}_{n1}-1) \\
\sqrt{n} (p\hat{\lambda}_{np}-1)
\end{array}
\bigg)
\stackrel{\mathcal{D}}{\to}
\bigg(
\begin{array}{c}
L^{\rm max}_{p,\tau} \\
L^{\rm min}_{p,\tau}
\end{array}
\bigg)
,$$ where $L^{\rm max}_{p,\tau}$ (resp., $L^{\rm min}_{p,\tau}$) is the largest (resp., smallest) eigenvalue of ${{\mathbf{Z}}}_\tau:={\mathbf{Z}}+(2\tau/(p+2)){\mathbf{W}}_\tau$, with ${{\mathbf{W}}}_\tau:={\rm diag}(p-1,-1,\ldots,-1)$ for $\tau\geq 0$ and ${{\mathbf{W}}}_\tau:={\rm diag}(-1,\ldots,-1,p-1)$ for $\tau<0$.
A direct consequence of Theorem \[thlambda1asympt\](i) is that simulations can be used to obtain arbitrarily precise estimates of the asymptotic critical values needed to implement the tests $\phi_+^{(n)}$, $\phi_-^{(n)}$ and $\phi_\pm^{(n)}$. For instance, the test $\phi_+^{(n)}$ will reject the null hypothesis of uniformity at asymptotic level $\alpha$ whenever $T_+^{(n)}
=
\sqrt{n}
\big(
p
\hat{\lambda}_{n1}
-
1
\big)
>
\hat{c}^{(m)}_{p,\alpha,+}$, where $\hat{c}^{(m)}_{p,\alpha,+}$ denotes the upper $\alpha$-quantile of $m$ independent realizations of $L_{p}^{\rm max}$. Interestingly, the following corollary shows that simulations can actually be avoided in dimensions $p=2$ and $p=3$, as the asymptotic null distribution of $T{^{(n)}}_+$, $T{^{(n)}}_-$ and $T{^{(n)}}_\pm$ can be explicitly determined for these values of $p$ (the result for $T{^{(n)}}_+$ and $T{^{(n)}}_-$ in dimension $p=3$ in (\[cdfp3a\]) below agrees with the one from [@AndSte1972]).
\[Corollambda1asympt\] (i) Under the null hypothesis of uniformity over $\mathcal{S}^{1}$, the test statistics $T_+{^{(n)}}$, $T_-{^{(n)}}$, and $T_\pm{^{(n)}}$ converge weakly to $L^{\rm max}_{2}$, where $L^{\rm max}_{2}$ has cumulative distribution function $$\label{cdfp2}
\ell\mapsto (1-\exp(-\ell^2)) \, \mathbb{I}[\ell>0]
;$$ (ii) under the null hypothesis of uniformity over $\mathcal{S}^{2}$, the test statistics $T_+{^{(n)}}$ and $T_-{^{(n)}}$ converge weakly to $L^{\rm max}_{3}$, where $L^{\rm max}_{3}$ has cumulative distribution function $$\label{cdfp3a}
\ell
\mapsto
\big\{\Phi(\sqrt{5}\ell)+\Phi({\textstyle{\frac{\sqrt{5}\ell}{2}}})+3\Phi''({\textstyle{\frac{\sqrt{5}\ell}{2}}})-1\big\} \, \mathbb{I}[\ell>0]
,$$ whereas the test statistic $T_\pm{^{(n)}}$ converges weakly to $L_3:=\max(L^{\rm max}_{3},-L^{\rm min}_{3})$, where $L_{3}$ has cumulative distribution function $$\label{cdfp3b}
\ell
\mapsto
\big\{
2\Phi({\textstyle{\frac{\sqrt{5}\ell}{2}}})
+
6\Phi''({\textstyle{\frac{\sqrt{5}\ell}{2}}})
-
2\sqrt{3} \Phi''({\textstyle{\frac{\sqrt{5}\ell}{\sqrt{3}}}})-1
\big\}
\, \mathbb{I}[\ell>0]$$ $($here, $\Phi''$ is the second derivative of the standard normal distribution function $\Phi)$.
Writing $\lambda_\ell({\mathbf{A}})$ for the $\ell$th largest eigenvalue of the $p\times p$ matrix ${\mathbf{A}}$ and denoting as $\stackrel{\mathcal{D}}{=}$ equality in distribution, Theorem \[thlambda1asympt\] entails that, under the null hypothesis, $$T_+{^{(n)}}\stackrel{\mathcal{D}}{\to}
L_p^{\rm max}
=
\lambda_1({\mathbf{Z}})
\stackrel{\mathcal{D}}{=}
\lambda_1(-{\mathbf{Z}})
=
-\lambda_p({\mathbf{Z}})
=
-L_p^{\rm min}
\stackrel{\mathcal{D}}{\leftarrow}
T_-{^{(n)}}.$$ This shows that, for any dimension $p$, the test statistics $T_+{^{(n)}}$ and $T_-{^{(n)}}$ share the same weak limit under the null hypothesis, which is confirmed in dimensions $p=2,3$ by Corollary \[Corollambda1asympt\]. Maybe surprisingly, this corollary further implies that, for $p=2$, the two-sided test statistic $T_\pm{^{(n)}}$ has the same asymptotic null distribution as $T_+{^{(n)}}$ and $T_-{^{(n)}}$. This can be explained as follows: since ${\mathbf{S}}_n$ has trace one almost surely, its eigenvalues $\hat{\lambda}_{n\ell}$, $\ell=1,\ldots,p$ sum up to one almost surely (incidentally, this implies that the quantities $\sqrt{n}(p\hat{\lambda}_{n\ell}-1)$, $\ell=1,\ldots,p$, do not admit a joint density, not even asymptotically so, which makes the proof of Theorem \[thlambda1asympt\] rather challenging). For $p=2$, it follows that $T_+{^{(n)}}=T_-{^{(n)}}=T_\pm{^{(n)}}$ almost surely, which of course entails that these three tests statistics share the same weak limit, not only under the null hypothesis but under *any* sequence of hypotheses. In line with this, $L_{2,\tau}^{\rm max}=-L_{2,\tau}^{\rm min}=\max(L_{2,\tau}^{\rm max},-L_p^{\rm min})$ almost surely for any $\tau\in{\mathbb R}$.
To check the validity of Theorem \[thlambda1asympt\] and Corollary \[Corollambda1asympt\], we conducted the following numerical exercises in dimensions $p=3$ and $p=10$. We generated $5\:\!000$ mutually independent random samples of size $n=2\:\!000$ (for $p=3$) and $n=10\:\!000$ (for $p=10$) from the Watson distribution with location ${{\pmb\theta}}=(1,0,\ldots,0)'\in{\mathbb R}^{p}$ and concentration $\kappa_n=\tau p/\sqrt{n}$, for $\tau=-4,-3,0,3,4$. Figure \[Fig3\] plots kernel density estimates of the resulting values of $T{^{(n)}}_+$, $T{^{(n)}}_-$ and $T{^{(n)}}_\pm$, along with the densities of the corresponding asymptotic distributions; for $p=3$ and $\tau=0$, these densities are those associated with the distribution functions in (\[cdfp3a\])–(\[cdfp3b\]), whereas, in all other cases, they are kernel density estimates obtained from $10^6$ independent realizations of $L^{\rm max}_{p,\tau}$, $-L^{\rm min}_{p,\tau}$, and $\max(L^{\rm max}_{p,\tau},-L^{\rm min}_{p,\tau})$, respectively; see Theorem \[thlambda1asympt\]. Clearly, the results support our asymptotic findings. It is seen that the one-sided test $\phi_+{^{(n)}}$ not only shows power against the bipolar alternatives it is designed for (those associated with $\tau>0$) but also against girdle-type ones (those associated with $\tau<0$), which is actually desirable. The same can be said about the one-sided test $\phi_-{^{(n)}}$, but each of these tests, of course, shows higher powers against the alternatives it was designed for. In contrast, the two-sided test $\phi_\pm{^{(n)}}$ shows a symmetric power pattern for positive and negative values of $\tau$.
![ (Left:) Plots of the kernel density estimates (solid curves) of the values of $T_+{^{(n)}}=\sqrt{n}(p\hat{\lambda}_{n1}-1)$ (top), $T_-{^{(n)}}=-\sqrt{n}(p\hat{\lambda}_{np}-1)$ (middle) or $T_\pm{^{(n)}}=\max(T_+{^{(n)}},T_-{^{(n)}})$ (bottom), obtained from $5\,000$ independent random samples of size $n=2\,000$ from the Watson distribution with location ${{\pmb\theta}}=(1,0,\ldots,0)'\in{\mathbb R}^p$ and concentration $\kappa=\tau p/\sqrt{n}$, with $\tau=-4,-3,0,3,4$ and with $p=3$; for $\tau=0$, histograms of the corresponding test statistics are shown. The density of the corresponding asymptotic distributions are also plotted (dashed curves). (Right:) The corresponding results for $p=10$ and $n=10\,000$. See Section \[sec:eigentest\] for details. []{data-label="Fig3"}](Figs/Fig3.pdf){width=".95\textwidth"}
Finite-sample comparisons {#sec:Simu}
=========================
In the previous sections, we conducted Monte-Carlo exercises in order to check correctness of our null and non-null asymptotic results, but it is of course of primary importance to compare the power behaviors of the various tests considered in this work. In this section, we therefore study the finite-sample powers of the Bingham test $\phi{^{(n)}}_{\rm Bing}$ and of the test $\phi_+{^{(n)}}$ (we could similarly consider the tests $\phi_+{^{(n)}}$ and $\phi_\pm{^{(n)}}$), and we compare them with those of the optimal specified-${{\pmb\theta}}$ test $\phi_{{{\pmb\theta}}+}{^{(n)}}$. Our asymptotic results further allow us to complement these finite-sample comparisons with comparisons of the corresponding asymptotic powers.
We conducted the following Monte Carlo experiment. For any combination $(n,p)$ of sample size $n\in\{200,20\:\!000\}$ and dimension $p\in\{3,10\}$, we generated collections of $2\:\!000$ independent random samples of size $n$ from the Watson distribution on $\mathcal{S}^{p-1}$ with location ${{\pmb\theta}}=(1,0,\ldots,0)'\in{\mathbb R}^p$ and concentration $\kappa_n =\tau_\ell p/\sqrt{n}$, with $\tau_\ell=0.8\ell$, $\ell=0,1,\ldots,5$. The value $\ell=0$ corresponds to the null hypothesis of uniformity, whereas $\ell=1,\ldots,5$ provide increasingly severe bipolar alternatives. In each sample, we performed three tests at asymptotic level $\alpha=5\%$, namely the specified-${{\pmb\theta}}$ test $\phi_{{{\pmb\theta}}+}^{(n)}$ in (\[Testright\]), the Bingham test $\phi{^{(n)}}_{\rm Bing}$ in (\[Binghamtest\]), and the test $\phi{^{(n)}}_+$ in (\[Teststatlambda1\]); for $p=3$, the asymptotic critical value for $\phi_{+}^{(n)}$ was obtained from Corollary \[Corollambda1asympt\](ii), whereas, for $p=10$, an approximation of the corresponding critical value was obtained from $10\:\!000$ independent realizations of $L_p^{\rm max}$ in Theorem \[thlambda1asympt\].
Figure \[Fig4\] shows the resulting empirical powers along with their theoretical asymptotic counterparts (for any given $p$ and $\tau_\ell$, the asymptotic power of $\phi_{+}^{(n)}$ was obtained from $10\,000$ independent copies of the random variable $L_{p,\tau_\ell}^{\rm max}$ in Theorem \[thlambda1asympt\]). The results show that, as expected, the optimal specified-${{\pmb\theta}}$ test outperforms both unspecified-${{\pmb\theta}}$ tests. The test $\phi{^{(n)}}_+$ dominates the Bingham test $\phi{^{(n)}}_{\rm Bing}$ and this dominance, quite intuitively, increases with the dimension $p$. Clearly, rejection frequencies agree very well with our asymptotic results for large sample sizes.
![ Rejection frequencies (solid curves) of three axial tests of uniformity over $\mathcal{S}^{p-1}$ obtained from $2\,000$ mutually independent random samples of size $n$ from the Watson distribution with location ${{\pmb\theta}}=(1,0,\ldots,0)'\in{\mathbb R}^p$ and concentration $\kappa=\tau_\ell p/\sqrt{n}$, with $\ell=0$ (null hypothesis of uniformity) and $\tau_\ell=0.8,1.6,2.4,3.2,4.0$ (increasingly severe bipolar alternatives). The tests considered are the test $\phi_{{{\pmb\theta}}+}^{(n)}$ in (\[Testright\]), the Bingham test $\phi{^{(n)}}_{\rm Bing}$ in (\[Binghamtest\]), and the test $\phi_+{^{(n)}}$ in (\[Teststatlambda1\]). The corresponding asymptotic powers (dashed curves) are also provided; see Section \[sec:Simu\] for details. []{data-label="Fig4"}](Figs/Fig4.pdf){width="\textwidth"}
Final comments and perspectives for future research {#sec:Final}
===================================================
Practitioners often face axial data, which explains that statistical procedures for such data are presented in most directional statistics monographs and have been the topic of numerous research papers; we refer to [@Fish87], [@MarJup2000], [@LV17book], and to the references therein. However, the non-null properties of axial tests of hypotheses are barely known, compared to those of their non-axial counterparts. Since this is in particular the case for axial tests of uniformity, we systematically studied in this work the asymptotic power of several axial tests of uniformity. In particular, we derived the asymptotic powers of the Bingham and Anderson tests under contiguous rotationally symmetric alternatives. Our results identify the underlying contiguity rate and allow for theoretical power comparisons. Throughout, our asymptotic findings were confirmed through simulations. Far from being of academic interest only, our results may be useful to practitioners who, when rejection occurs, will get some insight on the underlying distribution by combining the outcomes of the various tests of uniformity (that is, they will get hints on the single-spiked vs multi-spiked nature of the distribution, on its bipolar vs girdle nature, etc.)
The non-null asymptotic analysis conducted in this work essentially settles the low-dimensional case. Perspectives for future research therefore mainly relate to the high-dimensional framework. It is rather straightforward to extend the contiguity and LAN results in Theorems \[TheorContig\]–\[TheorLAN\] to the case where the dimension $p=p_n$ diverges to infinity with $n$ at an arbitrary rate. However, in high dimensions, it is extremely challenging to derive the non-null asymptotic powers of the Bingham and Anderson tests under suitable local alternatives. For the Anderson tests, for instance, this is due to the fact that eigenvalues of sample covariance matrices suffer complicated phase transition phenomenons which, close to uniformity, results in a lack of consistency. These challenging questions are of course beyond the scope of the present work, hence are left for future research.
Proofs of Theorems \[TheorContig\] to \[TheorBinghamHD\] {#proofcontigth}
========================================================
These proofs require the following preliminary result.
\[lemmeA1\] Let $g:{\mathbb R}\to{\mathbb R}$ be twice differentiable at 0, $(\kappa_n)$ be a sequence in ${\mathbb R}_0$, and $(p_n)$ be a sequence in $\{2,3,\ldots\}$. Assume that $\kappa_n$ is $o(p_n)$ as $n\to\infty$. Then, $$\begin{aligned}
R_n(g)
&\!\!:=\!\!&
c_{p_n}\int_{-1}^1(1-s^2)^{(p_n-3)/2}g(\kappa_n s^2)ds
\\[2mm]
&\!\!=\!\!&
g(0)+\frac{\kappa_n}{p_n}g'(0)+\frac{3\kappa_n^2}{2p_n(p_n+2)}g''(0)+o\left(\frac{\kappa_n^2}{p_n^2}\right)
,\end{aligned}$$ with $c_p=1/\int_{-1}^1 (1-s^2)^{(p-3)/2}\,dt$.
[Proof of lemma \[lemmeA1\].]{} Recall that if ${\mathbf{X}}$ is uniformly distributed over $\mathcal{S}^{p_n-1}$, then, for any ${{\pmb\theta}}\in\mathcal{S}^{p_n-1}$, we have that ${{\pmb\theta}}'{\mathbf{X}}$ has density $c_{p_n} (1-s^2)^{(p_n-3)/2}\mathbb{I}[|s|\leq 1]$, that is, the distribution of ${{\pmb\theta}}'{\mathbf{X}}$ is symmetric about zero and such that $({{\pmb\theta}}'{\mathbf{X}})^2\sim {\rm Beta}(1/2,(p_n-1)/2)$. Consequently, $$\label{foref1}
c_{p_n}
\int_{-1}^1 s^2 (1-s^2)^{(p_n-3)/2} \,ds
=
{\rm E}[({{\pmb\theta}}'{\mathbf{X}})^2]
=
\frac{1}{p_n}
,$$ and $$\label{foref2}
c_{p_n}
\int_{-1}^1 s^4 (1-s^2)^{(p_n-3)/2} \,ds
=
{\rm E}[({{\pmb\theta}}'{\mathbf{X}})^4]
=
\frac{3}{p_n(p_n+2)}
\cdot$$ From (\[foref1\]), we can write $$R_n(g)-g(0)-\frac{\kappa_n}{p_n}g'(0)
=
c_{p_n}
\int_{-1}^1 (1-s^2)^{(p_n-3)/2} \left(g(\kappa_n s^2)-g(0)-\kappa_n s^2 g'(0)\right) \,ds
.$$ Letting $t=|\kappa_n|^{1/2} s$ and using (\[foref1\])–(\[foref2\]) then provides $$R_n(g)-g(0)-\frac{\kappa_n}{p_n}g'(0)
=
\frac{3\kappa_n^2}{p_n(p_n+2)}
\int_{-\infty}^{\infty}
h_n(t)
\left(
\frac{g\big(\frac{\kappa_n}{|\kappa_n|}t^2\big)-g(0)-\frac{\kappa_n}{|\kappa_n|}t^2g'(0)}{t^4}
\right)
\,dt
,$$ or, equivalently, $$\begin{aligned}
\lefteqn{
\frac{R_n(g)-g(0)-\frac{\kappa_n}{p_n}g'(0)-\frac{3\kappa_n^2}{2p_n(p_n+2)}g''(0)}{\frac{3\kappa_n^2}{p_n(p_n+2)}}
}
\nonumber
\\[2mm]
& &
\hspace{13mm}
=
\int_{-\infty}^{\infty}
h_n(t)
\left(
\frac{g\big(\frac{\kappa_n}{|\kappa_n|}t^2\big)-g(0)-\frac{\kappa_n}{|\kappa_n|}t^2g'(0)}{t^4}
\right)
\,dt
-
\frac{1}{2}g''(0)
,
\label{prequeA1}\end{aligned}$$ where $h_n$ is defined through $$t
\mapsto
h_n(t)
=
\frac{
\Big(\frac{t}{|\kappa_n|^{1/2}}\Big)^4
\Big(1-\Big(\frac{t}{|\kappa_n|^{1/2}}\Big)^2\Big)^{(p_n-3)/2}
\mathbb{I}\big[ |t|\leq |\kappa_n|^{1/2} \big]
}{
\int_{-|\kappa_n|^{1/2}}^{|\kappa_n|^{1/2}}
\Big(\frac{t}{|\kappa_n|^{1/2}}\Big)^4
\Big(1-\Big(\frac{t}{|\kappa_n|^{1/2}}\Big)^2\Big)^{(p_n-3)/2}\,dt}
\cdot$$ Since $\kappa_n$ is $o(p_n)$, it can be checked that the sequence $(h_n)$ is an *approximate* , in the sense that $
\int_{-\infty}^\infty h_n(t)\,dt=1
$ for any $n$ and $
\int_{-\varepsilon}^\varepsilon h_n(t)\, dt \to 1
$ for any $\varepsilon>0$. Hence, $$\lim_{n\to \infty}
\int_{-\infty}^{\infty}
h_n(t)
\left(
\frac{g\big(\frac{\kappa_n}{|\kappa_n|}t^2\big)-g(0)-\frac{\kappa_n}{|\kappa_n|}t^2g'(0)}{t^4}
\right)
\,dt
=
\lim_{t\to 0}
\frac{g\big(\frac{\kappa_n}{|\kappa_n|}t^2\big)-g(0)-\frac{\kappa_n}{|\kappa_n|}t^2g'(0)}{t^4}
,$$ which, by using L’Hôpital’s rule, is equal to $$\lim_{t\to 0} \frac{ \frac{2\kappa_n}{|\kappa_n|}t g'\left(\frac{\kappa_n}{|\kappa_n|}t^2\right)-\frac{2\kappa_n}{|\kappa_n|}tg'(0)}{4t^3}
=
\frac{1}{2}
\lim_{t\to 0} \frac{ g'\big(\frac{\kappa_n}{|\kappa_n|}t^2\big)-g'(0)}{\frac{\kappa_n}{|\kappa_n|} t^2}
=
\frac{1}{2} g''(0)
.$$ The result thus follows from (\[prequeA1\]). [$\square$]{}
The proof of Theorems \[TheorContig\]–\[TheorLAN\] actually only requires the particular case of lemma \[lemmeA1\] corresponding to $p_n\equiv p$. We still presented this more general version of the lemma as it allows one to extend Theorems \[TheorContig\]–\[TheorLAN\] to high-dimensional asymptotic scenarios where $p_n$ would diverge to infinity with $n$ (this would prove the claims in high dimensions provided in the last paragraph of Section \[sec:axial\]).
[Proof of theorem \[TheorContig\].]{} In this proof, all expectations and variances are taken under the null of uniformity ${\rm P}_{0}{^{(n)}}$ and all stochastic convergences and $o_{\rm P}$’s are as $n\to\infty$ under ${\rm P}_{0}{^{(n)}}$. Throughout, we write $\ell_{f,k}:=(\log f)^k$ and $E_{nk}:={\rm E}[ \ell_{f,k}(\kappa_n ({\mathbf{X}}_{ni}{^{\prime}}{{\pmb\theta}})^2)]$.
Consider the local log-likelihood ratio $$\begin{aligned}
\Lambda_n
:=
\log \frac{d{\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}}{d{\rm P}{^{(n)}}_{0}}
&=
\sum_{i=1}^n
\,
\log \frac{c_{{p},\kappa_n,f} f(\kappa_n ({\mathbf{X}}_{ni}{^{\prime}}{{\pmb\theta}})^2)}{c_{{p}}}\\
&=
n
\left(\log \frac{c_{{p},\kappa_n,f}}{c_{{p}}}+E_{n1}\right)
+
\sum_{i=1}^n
\left(
\log f(\kappa_n ({\mathbf{X}}_{ni}{^{\prime}}{{\pmb\theta}})^2) - E_{n1}
\right)\\
&=:
L_{n1}+L_{n2}.\end{aligned}$$ Lemma \[lemmeA1\] readily yields $$\begin{aligned}
\log \frac{c_{{p},\kappa_n,f}}{c_{{p}}}
&=
- \log
\left(
c_{{p}}\int_{-1}^1 (1-s^2)^{({p}-3)/2} f(\kappa_n s^2)\,ds
\right)
\nonumber\\[2mm]
&=
-\log
\left(
1
+
\frac{\kappa_n}{{p}}
+
\frac{3\kappa_n^2}{2{p}({p}+2)}
f''(0)
+
o(\kappa^2_n)
\right)
\nonumber\\[2mm]
&=
-\frac{\kappa_n}{{p}}
-
\frac{3\kappa_n^2}{2{p}({p}+2)}f''(0)
+
\frac{\kappa_n^2}{2{p}^2}
+
o(\kappa^2_n)
.
\label{expandlogconst}\end{aligned}$$ Similarly, $$\begin{aligned}
E_{n1}
&=
c_{{p}}
\int_{-1}^1 (1-s^2)^{({p}-3)/2} \ell_{f,1}(\kappa_n s^2) \,ds
\nonumber\\[2mm]
&=
\frac{\kappa_n}{{p}} \ell_{f,1}'(0)
+
\frac{3\kappa^2_n}{2{p}({p}+2)} \ell_{f,1}''(0)
+
o(\kappa^2_n)
\nonumber\\[2mm]
&=
\frac{\kappa_n}{{p}}+\frac{3\kappa^2_n}{2{p}({p}+2)} (f''(0)-1)
+
o(\kappa^2_n)
.
\label{expandEk}\end{aligned}$$ Combining (\[expandlogconst\]) and (\[expandEk\]) provides $$L_{n1}
=
\frac{n\kappa_n^2}{2{p}^2}-\frac{3n\kappa^2_n}{2{p}({p}+2)}
+
o(n\kappa^2_n)
=
-\frac{n\kappa_n^2({p}-1)}{{p}^2({p}+2)}
+
o(n\kappa^2_n)
.$$
Turning to $L_{n2}$, write $$L_{n2}
=
\sqrt{n V_n}
\,
\sum_{i=1}^n
W_{ni}
:=
\sqrt{n V_n}
\,
\sum_{i=1}^n
\frac{\log f(\kappa_n ({\mathbf{X}}_{ni}{^{\prime}}{{\pmb\theta}})^2) - E_{n1}}{\sqrt{n V_n}}
,$$ where we let $
V_n
:=
{\rm Var}\big[ \log f(\kappa_n ({\mathbf{X}}_{ni}{^{\prime}}{{\pmb\theta}})^2) \big]
.
$ First note that, since $$\begin{aligned}
E_{n2}
&=
c_{{p}}
\int_{-1}^1 (1-s^2)^{({p}-3)/2} \ell_{f,2}(\kappa_n s^2) \,ds
\nonumber\\[2mm]
&=
\frac{\kappa_n}{{p}} \ell_{f,2}'(0)
+
\frac{3\kappa^2_n}{2{p}({p}+2)} \ell_{f,2}''(0)
+
o(\kappa^2_n)
\nonumber\\[2mm]
&=
\frac{3\kappa^2_n}{{p}({p}+2)}
+
o(\kappa^2_n)
,\end{aligned}$$ we have $$\label{expandV}
nV_n
=
n\big(E_{n2} - E_{n1}^2\big)
=
\frac{2n\kappa_n^2({p}-1)}{{p}^2({p}+2)}
+
o(n\kappa^2_n)
,$$ which leads to $$\label{quasicont}
\Lambda_n
=
-\frac{n\kappa_n^2({p}-1)}{{p}^2({p}+2)}
+
\sqrt{\frac{2n\kappa_n^2({p}-1)}{{p}^2({p}+2)}
+
o(n\kappa^2_n)}
\,
\sum_{i=1}^n
W_{ni}
+
o(n\kappa^2_n)
.$$ Since $W_{ni}$, $i=1,\ldots,n$, are mutually independent with mean zero and variance $1/n$, we obtain that $$\label{L2norm}
{\rm E}
\big[
\Lambda_n^2
\big]
=
\left(
{\rm E}
[
\Lambda_n
]
\right)^2
+
{\rm Var}
[
\Lambda_n
]
=
\frac{n^2\kappa_n^4({p}-1)^2}{{p}^4({p}+2)^2}
+
o(n^2\kappa^4_n)
+
\frac{2n\kappa_n^2({p}-1)}{{p}^2({p}+2)}
+
o(n\kappa^2_n)
.$$
If $\kappa_n=o(1/\sqrt{n})$, then (\[L2norm\]) implies that $\exp(\Lambda_n)=1+o_{\rm P}(1)$, so that Le Cam’s first lemma yields that ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$ and ${\rm P}{^{(n)}}_{0}$ are mutually contiguous.
We may therefore assume that $\kappa_n=\tau_n/\sqrt{n}$, where ($\tau_n$) is $O(1)$ but not $o(1)$, or, equivalently, that $\kappa_n=\tau_n {p}/\sqrt{n}$ with the same sequence ($\tau_n$). Then, (\[quasicont\]) rewrites $$\Lambda_n
=
-
\frac{({p}-1)\tau_n^2}{{p}+2}
+
\sqrt{
\frac{2({p}-1)\tau_n^2}{{p}+2}
+
o(1)
}
\,
\sum_{i=1}^n
W_{ni}
+
o(1).$$
Applying the Cauchy-Schwarz inequality and the Chebychev inequality, then using Lemma \[lemmeA1\] and (\[expandV\]), yields that, for some positive constant $C$ and any $\varepsilon>0$, $$\begin{aligned}
\lefteqn{
\sum_{i=1}^n
{\rm E}[
W_{ni}^2
\mathbb{I}[ |W_{ni}| >\varepsilon]]
}
\\[1mm]
& &
\hspace{1mm}
\leq
n
\sqrt{{\rm E}[ W_{ni}^4] {\rm P}[|W_{ni}|>\varepsilon]}
\leq
\frac{n}{\varepsilon}
\sqrt{{\rm E}[ W_{ni}^4] {\rm Var}[W_{ni}]}
=
\frac{1}{\varepsilon}
\sqrt{n{\rm E}[ W_{ni}^4]}
\leq
\frac{C\sqrt{n E_{n4}}}{\varepsilon nV_n}
\\[2mm]
& &
\hspace{1mm}
=
\frac{C
\Big(
\frac{n\kappa_n}{{p}} \ell_{f,4}'(0)
+
\frac{3n\kappa_n^2}{2{p}({p}+2)} \ell_{f,4}''(0)
+
o(n\kappa^2_n)
\Big)^{1/2}
}{
\varepsilon
\Big(
\frac{2n\kappa_n^2({p}-1)}{{p}^2({p}+2)}
+
o(n\kappa^2_n)
\Big)
}
=
\frac{
o(\tau_n)
}{
\varepsilon
\Big(
\frac{2({p}-1)\tau_n^2}{{p}+2}
+
o(\tau_n^2)
\Big)
}
=
o(1)
,\end{aligned}$$ where we have used the fact that $\ell_{f,4}'(0)=\ell_{f,4}''(0)=0$. This shows that $\sum_{i=1}^n W_{ni}$ satisfies the classical Levy-Lindeberg condition, hence is asymptotically standard normal (as already mentioned, $W_{ni}$, $i=1,\ldots,n$, are mutually independent with mean zero and variance $1/n$). For any subsequence $(\exp(\Lambda_{n_m}))$ converging in distribution, the weak limit must then be $
\exp(Y)
$, with $
Y
\sim
\mathcal{N}(
- \eta
,
2\eta
)
,
$ where we let $\eta:=((p-1)/(p+2)) \lim_{m\to\infty} \tau_{n_m}^2$. Mutual contiguity of ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$ and ${\rm P}{^{(n)}}_{0}$ then follows from the fact that ${\rm P}[\exp(Y)=0]=0$ and ${\rm E}[\exp(Y)]=1$. [$\square$]{}
[Proof of Theorem \[TheorLAN\].]{} As in the proof of Theorem \[TheorContig\], all expectations and variances in this proof are taken under the null of uniformity ${\rm P}_{0}{^{(n)}}$ and all stochastic convergences and $o_{\rm P}$’s are as $n\to\infty$ under ${\rm P}_{0}{^{(n)}}$. Recall that we have obtained in the proof of Theorem \[TheorContig\] that $$\begin{aligned}
\Lambda_n
&\!=\!&
-
\frac{({p}-1)\tau_n^2}{{p}+2}
+
\sqrt{
\frac{2({p}-1)\tau_n^2}{{p}+2}
+
o(1)
}
\,
\sum_{i=1}^n
W_{ni}
+
o(1)
\\[2mm]
&\!=\!&
-
\frac{\tau_n^2}{2} \Gamma_p
+
|\tau_n| \Gamma_p^{1/2}
\,
\sum_{i=1}^n
W_{ni}
+
o_{\rm P}(1)
,\end{aligned}$$ where $
\sum_{i=1}^n W_{ni}
=
(1/\sqrt{nV_n})
\sum_{i=1}^n
(\log f(\kappa_n ({\mathbf{X}}_{ni}{^{\prime}}{{\pmb\theta}})^2) - E_{n1})
$ is asymptotically standard normal. Since $(\tau_n\Gamma_p^{1/2})$ is $O(1)$, it is therefore sufficient to show that $$d_n
:=
{\rm E}\bigg[
\bigg( \Gamma_p^{-1/2}\Delta_{{{\pmb\theta}}}{^{(n)}}- \frac{\tau_n}{|\tau_n|}\sum_{i=1}^n W_{ni} \bigg)^2
\bigg]
=
o(1)
.
\label{EM2a}$$ To do so, write $$\begin{aligned}
\lefteqn{
\hspace{-0mm}
\Gamma_p^{-1/2}\Delta_{{{\pmb\theta}}}{^{(n)}}- \frac{\tau_n}{|\tau_n|}\sum_{i=1}^n W_{ni}
}
\nonumber
\\[2mm]
& &
\hspace{2mm}
=
\frac{1}{\sqrt{nV_n}}
\sum_{i=1}^n
\bigg(
{p}\sqrt{V_n}
\Gamma_p^{-1/2} \Big(({\mathbf{X}}_{ni}{^{\prime}}{{\pmb\theta}})^2-\frac{1}{{p}}\Big)
-
\frac{\tau_n}{|\tau_n|}
(\log f(\kappa_n ({\mathbf{X}}_{ni}{^{\prime}}{{\pmb\theta}})^2) - E_{n1})
\bigg)
\nonumber
\\[2mm]
& &
\hspace{2mm}
=:
\frac{M_n}{\sqrt{nV_n}}
\cdot
\label{EM2b}\end{aligned}$$
Then using the fact that ${\rm E}[({\mathbf{X}}_{n1}{^{\prime}}{{\pmb\theta}})^2]=1/{p}$ and ${\rm E}[({\mathbf{X}}_{n1}{^{\prime}}{{\pmb\theta}})^4]=3/({p}({p}+2))$ (see the proof of Lemma \[lemmeA1\]) provides ${\rm Var}[({\mathbf{X}}_{n1}{^{\prime}}{{\pmb\theta}})^2]=\Gamma_p/{p}^2$, we obtain $$\begin{aligned}
{\rm E}\big[M_n^2\big]&=
n
{\rm E}
\left[
\left(
{p}\sqrt{V_n}
\Gamma_p^{-1/2}
\left(({\mathbf{X}}_{n1}{^{\prime}}{{\pmb\theta}})^2-\frac{1}{{p}}\right)
-
\frac{\tau_n}{|\tau_n|}
(\log f(\kappa_n ({\mathbf{X}}_{n1}{^{\prime}}{{\pmb\theta}})^2) - E_{n1})
\right)^2
\right]
\\[2mm]
&=
2nV_n
-
2n{p} \sqrt{V_n} \Gamma_p^{-1/2}
\frac{\tau_n}{|\tau_n|}
{\rm E}
\left[
\left(({\mathbf{X}}_{n1}{^{\prime}}{{\pmb\theta}})^2-\frac{1}{{p}}\right)
(\log f(\kappa_n ({\mathbf{X}}_{n1}{^{\prime}}{{\pmb\theta}})^2) - E_{n1})
\right]
,\end{aligned}$$ which, letting $g(x):=x\log f(x)$, rewrites $$\begin{aligned}
{\rm E}\big[M_n^2\big]
&\!\!=\!\!&
2nV_n
-
2n{p} \sqrt{V_n} \Gamma_p^{-1/2}
\frac{\tau_n}{|\tau_n|}
\Big(
{\rm E}
\big[
({\mathbf{X}}_{ni}{^{\prime}}{{\pmb\theta}})^2 \log f(\kappa_n ({\mathbf{X}}_{n1}{^{\prime}}{{\pmb\theta}})^2)
\big]
- \frac{E_{n1}}{{p}}
\Big)
\nonumber
\\[2mm]
&\!\!=\!\!&
2nV_n
-
2 \sqrt{nV_n} \Gamma_p^{-1/2}
\frac{\tau_n}{|\tau_n|}
\Big(
\frac{\sqrt{n}{p}}{\kappa_n}
{\rm E}
\big[
g(\kappa_n ({\mathbf{X}}_{n1}{^{\prime}}{{\pmb\theta}})^2)
\big]
-
\sqrt{n} E_{n1}
\Big)
.
\label{EM2}\end{aligned}$$ Lemma \[lemmeA1\] provides $${\rm E}[
g(\kappa_n ({\mathbf{X}}_{n1}{^{\prime}}{{\pmb\theta}})^2)
]
=
c_{{p}}
\int_{-1}^1 (1-s^2)^{({p}-3)/2} g(\kappa_n s^2) \,ds
=
\frac{3\kappa^2_n}{{p}({p}+2)}
+
o(\kappa^2_n)
.$$ Using this jointly with (\[expandEk\]) and (\[expandV\]), it follows from (\[EM2b\])–(\[EM2\]) that $$\begin{aligned}
d_n
&\!\!=\!\!&
2
-
\frac{
2\tau_n
\Big(
\frac{3\sqrt{n}\kappa_n}{{p}+2}
+
o\left(\frac{\sqrt{n}\kappa_n}{{p}} \right)
-
\Big\{
\frac{\sqrt{n}\kappa_n}{{p}}+\frac{3\sqrt{n}\kappa^2_n}{2{p}({p}+2)} (f''(0)-1)
+
o(\sqrt{n}\kappa^2_n)
\Big\}
\Big)
}{
\sqrt{\Gamma_p}|\tau_n|
\left(
\frac{n\kappa_n^2\Gamma_p}{{p}^2}
+
o(n\kappa^2_n)
\right)^{1/2}
}
\\[2mm]
&\!\!=\!\!&
2
-
\frac{
2\tau_n
\big(
\Gamma_p \tau_n
+
o(1)
\big)
}{
\sqrt{\Gamma_p}|\tau_n|
\big(
\Gamma_p\tau_n^2
+
o(1)
\big)^{1/2}
}
=
o(1)
,\end{aligned}$$ as was to be shown. [$\square$]{}
[Proof of Theorem \[TheorLANunspec\].]{} First note that by proceeding strictly along the same lines as in the proof of Theorem \[TheorLAN\], one can show that, for any sequence $({{\pmb\theta}}_n)$ in $\mathcal{S}^{p-1}$, and any real sequence $(\tau_n)$ that is $O(1)$ but not $o(1)$, $$\label{LANthetan}
\log \frac{d{\rm P}{^{(n)}}_{{{\pmb\theta}}_n,\kappa_n,f}}{d{\rm P}{^{(n)}}_{0}}
=
\tau_n
\Delta_{{{\pmb\theta}}_n}{^{(n)}}-
\frac{\tau_n^2}{2} \Gamma_p
+
o_{\rm P}(1)
,$$ under ${\rm P}_0{^{(n)}}$, with $\kappa_n=\tau_n p/\sqrt{n}$ (that is, the stochastic second-order expansion in (\[LAN\]) remains true if one replaces the fixed location ${{\pmb\theta}}$ with an $n$-dependent value ${{\pmb\theta}}_n$). Therefore, noting that the parameter value ${{\pmb\vartheta}}_n=(p/\sqrt{n})^{1/2}{{\pmb\tau}}_n$ in the statement of the theorem corresponds to ${{\pmb\theta}}_n={{\pmb\tau}}_n/\|{{\pmb\tau}}_n\|$ and $\kappa_n=p\|{{\pmb\tau}}_n\|^2/\sqrt{n}$, we have $$\log \frac{d{\rm P}{^{(n)}}_{{{\pmb\vartheta}}_n,f}}{d{\rm P}{^{(n)}}_{0}}
=
\log \frac{d{\rm P}{^{(n)}}_{{{\pmb\tau}}_n/\|{{\pmb\tau}}_n\|,p\|{{\pmb\tau}}_n\|^2/\sqrt{n},f}}{d{\rm P}{^{(n)}}_{0}}
=
\|{{\pmb\tau}}_n\|^2
\Delta_{{{\pmb\tau}}_n/\|{{\pmb\tau}}_n\|}{^{(n)}}-
\frac{\|{{\pmb\tau}}_n\|^4}{2} \Gamma_p
+
o_{\rm P}(1)$$ under ${\rm P}_0{^{(n)}}$. Now, since the central sequence in Theorem \[TheorLAN\] rewrites $$\begin{aligned}
\lefteqn{
\Delta_{{{\pmb\theta}}}{^{(n)}}=
\frac{p}{\sqrt{n}}\sum_{i=1}^n \Big\{({\mathbf{X}}_{ni}{^{\prime}}{{\pmb\theta}})^2-\frac{1}{p}\Big\}
=
p\sqrt{n}
\,
{{\pmb\theta}}'
\bigg(
{\mathbf{S}}_n
-\frac{1}{p}\mathbf{I}_{p}
\bigg)
{{\pmb\theta}}}
\nonumber
\\[2mm]
& &
\hspace{18mm}
=
p\sqrt{n}
\,
({{\pmb\theta}}\otimes {{\pmb\theta}})'
{\rm vec}\bigg(
{\mathbf{S}}_n-\frac{1}{p} \mathbf{I}_p
\bigg)
=
({\rm vec}({{\pmb\theta}}{{\pmb\theta}}'))'
{{\pmb \Delta}}{^{(n)}},
\label{centralrewr}\end{aligned}$$ this rewrites $$\log \frac{d{\rm P}{^{(n)}}_{{{\pmb\vartheta}}_n,f}}{d{\rm P}{^{(n)}}_{0}}
=
({\rm vec}({{\pmb\tau}}_n{{\pmb\tau}}_n'))'
{{\pmb \Delta}}{^{(n)}}-
\frac{\|{{\pmb\tau}}_n\|^4}{2} \Gamma_p
+
o_{\rm P}(1)$$ under ${\rm P}_0{^{(n)}}$. By using the identities $({\rm vec}\,{\mathbf{A}})'({\rm vec}\,{\mathbf{B}})={\rm tr}[{\mathbf{A}}'{\mathbf{B}}]$ and ${\bf K}_p({\rm vec}\,{\mathbf{A}})={\rm vec}({\mathbf{A}}')$, straightforward calculations show that $\|{{\pmb\tau}}_n\|^4 \Gamma_p=({\rm vec}({{\pmb\tau}}_n{{\pmb\tau}}_n'))'{{\pmb \Gamma_p}}{\rm vec}({{\pmb\tau}}_n{{\pmb\tau}}_n')$, which establishes (\[LANvec\]). Since the asymptotic normality result readily follows from the multivariate central limit theorem, the theorem is proved. [$\square$]{}
[Proof of Theorem \[TheorBinghamHD\].]{} Denoting as ${\rm E}_0{^{(n)}}$ and ${\rm Var}_0{^{(n)}}$ expectation and variance under ${\rm P}_0{^{(n)}}$, one has (see (\[centralrewr\])) $$\begin{aligned}
\lim_{n\to\infty}
{\rm E}_0{^{(n)}}\big[
\Delta_{{{\pmb\theta}}}{^{(n)}}{{\pmb \Delta}}{^{(n)}}\big]
&\!=\!&
\lim_{n\to\infty}
{\rm E}_0{^{(n)}}\Bigg[
{{\pmb \Delta}}{^{(n)}}\bigg(
p\sqrt{n}
\,
({{\pmb\theta}}\otimes {{\pmb\theta}})'
{\rm vec}\bigg(
{\mathbf{S}}_n
-\frac{1}{p}\mathbf{I}_{p}
\bigg)
\bigg){^{\prime}}\Bigg]
\\[2mm]
&\!=\!&
\lim_{n\to\infty}
{\rm Var}_0{^{(n)}}\big[
{{\pmb \Delta}}{^{(n)}}\big]
({{\pmb\theta}}\otimes {{\pmb\theta}})
=
{{\pmb \Gamma_p}}({{\pmb\theta}}\otimes {{\pmb\theta}})
.\end{aligned}$$ Therefore, Le Cam’s third lemma implies that, under ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$ with $\kappa_n=\tau p/\sqrt{n}$, ($\tau\neq 0$), ${{\pmb \Delta}}{^{(n)}}$ is asymptotically normal with mean vector $\tau{{\pmb \Gamma_p}}({{\pmb\theta}}\otimes {{\pmb\theta}})$ and covariance matrix ${{\pmb \Gamma_p}}$, so that, under the same sequence of hypotheses, $$Q{^{(n)}}=
({{\pmb \Delta}}{^{(n)}})' {\pmb\Gamma}_p^- {{\pmb \Delta}}{^{(n)}}\stackrel{\mathcal{D}}{\to}
\chi^2_{d_p}(\delta)
,$$ with $$\delta
=
\tau^2
({{\pmb\theta}}\otimes {{\pmb\theta}})'
{{\pmb \Gamma_p}}{\pmb\Gamma}_p^-{{\pmb \Gamma_p}}({{\pmb\theta}}\otimes {{\pmb\theta}})
=
\frac{p\tau^2}{p+2}\Big( 2-\frac{2}{p}\Big)
=
\frac{2(p-1)\tau^2}{p+2}
\cdot$$ The asymptotic power in (\[eq:right-sided-powerBingham\]) readily follows. [$\square$]{}
Proofs of Theorem \[thlambda1asympt\] and Corollary \[Corollambda1asympt\] {#proofeigentests}
==========================================================================
[Proof of Theorem \[thlambda1asympt\].]{} Since ${\rm E}[{\mathbf{X}}_{n1}{\mathbf{X}}_{n1}']=(1/p)\mathbf{I}_p$ and ${\rm E}[{\rm vec}({\mathbf{X}}_{n1}{\mathbf{X}}_{n1}')
({\rm vec}({\mathbf{X}}_{n1}{\mathbf{X}}_{n1}'))']
\linebreak
=
1/(p(p+2))(\mathbf{I}_{p^2}+{\bf K}_p+{\bf J}_p)$ under ${\rm P}_0{^{(n)}}$ (see, e.g., Lemma A.2 in [@PaiVer2016]), the multivariate central limit theorem yields $$\sqrt{n}
\,
{\rm vec}\bigg(
{\mathbf{S}}_n-\frac{1}{p} \mathbf{I}_p
\bigg)
\stackrel{\mathcal{D}}{\to}
\mathcal{N}
\bigg(
{\bf 0}
,
\frac{1}{p^2}{\bf V}_p
\bigg)
.$$ Now, by using (\[centralrewr\]), we obtain that, under ${\rm P}_0{^{(n)}}$, $$\begin{aligned}
\lefteqn{
\hspace{-13mm}
{\rm E}\bigg[
\sqrt{n}
\,
{\rm vec}\bigg(
{\mathbf{S}}_n-\frac{1}{p} \mathbf{I}_p
\bigg)
\Delta_{{{\pmb\theta}}}{^{(n)}}\bigg]
=
np
{\rm E}\bigg[
{\rm vec}\bigg(
{\mathbf{S}}_n-\frac{1}{p} \mathbf{I}_p
\bigg)
{\rm vec}'\bigg(
{\mathbf{S}}_n-\frac{1}{p} \mathbf{I}_p
\bigg)
\bigg]
{\rm vec}({{\pmb\theta}}{{\pmb\theta}}')
}
\\[2mm]
& &
\hspace{3mm}
=
p
\bigg(
\frac{1}{p^2} {\bf V}_p
\bigg)
{\rm vec}({{\pmb\theta}}{{\pmb\theta}}')
=
\frac{2}{p+2} {\rm vec}({{\pmb\theta}}{{\pmb\theta}}')
- \frac{2}{p(p+2)} {\rm vec}(\mathbf{I}_p)
,\end{aligned}$$ so that Le Cam’s third lemma shows that, under ${\rm P}{^{(n)}}_{{{\pmb\theta}},\kappa_n,f}$, where $\kappa_n=\tau_n p/\sqrt{n}$ is based on a sequence $(\tau_n)$ converging to $\tau$, $$\sqrt{n}
\,
{\rm vec}\bigg(
{\mathbf{S}}_n-\frac{1}{p} \mathbf{I}_p
\bigg)
\stackrel{\mathcal{D}}{\to}
\mathcal{N}
\bigg(
\frac{2\tau}{p+2} {\rm vec}({{\pmb\theta}}{{\pmb\theta}}')
- \frac{2\tau}{p(p+2)} {\rm vec}(\mathbf{I}_p)
,
\frac{1}{p^2}
{\bf V}_p
\bigg)
,$$ which rewrites $$\label{TCLvic}
\sqrt{n}
\,
{\rm vec}\big(
{\mathbf{S}}_n
-
{{\pmb \Sigma}}_n
\big)
\stackrel{\mathcal{D}}{\to}
\mathcal{N}
\bigg(
{\bf 0},
\frac{1}{p^2}
{\bf V}_p
\bigg)
,$$ where $$\begin{aligned}
{{\pmb \Sigma}}_n
&\!\!\!:=\!\!\!&
\bigg(
\frac{1}{p} - \frac{2\tau}{\sqrt{n}p(p+2)}
\bigg)
\mathbf{I}_p
+
\frac{2\tau}{\sqrt{n}(p+2)} {{\pmb\theta}}{{\pmb\theta}}'
\\[2mm]
&\!\!\!=\!\!\!&
\bigg(
\frac{1}{p} + \frac{2(p-1)\tau}{\sqrt{n}p(p+2)}
\bigg)
{{\pmb\theta}}{{\pmb\theta}}'
+
\bigg(
\frac{1}{p} - \frac{2\tau}{\sqrt{n}p(p+2)}
\bigg)
(\mathbf{I}_p-{{\pmb\theta}}{{\pmb\theta}}')
.\end{aligned}$$ We need to consider the cases (a) $\tau\geq 0$ and (b) $\tau<0$ separately.
In Case (a), ${{\pmb \Sigma}}_n$ has eigenvalues $$\label{eigenvic}
\lambda_{n1}
=
\frac{1}{p} + \frac{2(p-1)\tau}{\sqrt{n}p(p+2)}
\quad
\textrm{ and }
\quad
\lambda_{n2}
=
\ldots
=
\lambda_{np}
=
\frac{1}{p} - \frac{2\tau}{\sqrt{n}p(p+2)}
\cdot$$ Fix then arbitrarily ${{\pmb\theta}}_2, \ldots, {{\pmb\theta}}_p$ such that the $p \times p$ matrix ${\bf G}_p:=({{\pmb\theta}},{{\pmb\theta}}_2, \ldots, {{\pmb\theta}}_p)$ is orthogonal. Letting ${{\pmb \Lambda}}_n:={\rm diag}(\lambda_{n1},\lambda_{n2}, \ldots, \lambda_{np})$, we have ${{\pmb \Sigma}}_n {\bf G}_p={\bf G}_p{{\pmb \Lambda}}_n$, so that ${\bf G}_p$ is an eigenvectors matrix for ${{\pmb \Sigma}}_n$. Clearly, $\xi_{n1}:=\sqrt{n}p (\hat{\lambda}_{n1}-\lambda_{n1})$ is the largest root of the polynomial $P_{n1}(h):={\rm det}({\sqrt n}p({\bf S}_n-\lambda_{n1} {\bf I}_p)- h {\bf I}_p)$, whereas $\xi_{np}:=\sqrt{n}p (\hat{\lambda}_{np}-\lambda_{np})$ is the smallest root of $P_{np}(h):={\rm det}({\sqrt n}p({\bf S}_n-\lambda_{np} {\bf I}_p)- h {\bf I}_p)$. Letting $$\label{defZn}
{\bf Z}_n
:=
\sqrt{n}p({\bf G}_p' {\bf S}_n {\bf G}_p- {{\pmb \Lambda}}_n)
=
{\bf G}_p' \sqrt{n}p({\bf S}_n- {{\pmb \Sigma}}_n) {\bf G}_p
,$$ rewrite these polynomials as $
P_{nj}(h)
=
{\rm det}(\sqrt{n}p{\bf G}_p' {\bf S}_n {\bf G}_p- \sqrt{n}p\lambda_{nj} {\bf I}_p - h {\bf I}_p)
=
{\rm det}({{\mathbf{Z}}}_n+\sqrt{n}p({{\pmb \Lambda}}_n-\lambda_{nj} {\bf I}_p)- h {\bf I}_p)
,
$ $j=1,p$, which gives $$P_{n1}(h)
=
{\rm det}({{\mathbf{Z}}}_n+{\rm diag}(0, -v_\tau, \ldots, -v_\tau)- h {\bf I}_p)
\nonumber$$ and $$P_{np}(h)
=
{\rm det}({{\mathbf{Z}}}_n+{\rm diag}(v_\tau,0, \ldots,0)- h {\bf I}_p)
\nonumber
,$$ where we wrote $$v_\tau
:=
\sqrt{n}p
\bigg[
\frac{2(p-1)\tau}{\sqrt{n}p(p+2)}
+
\frac{2\tau}{\sqrt{n}p(p+2)}
\bigg]
=
\frac{2p\tau}{p+2}
\cdot$$
Note that (\[TCLvic\]) readily implies that ${\rm vec}\,{\mathbf{Z}}_n
=
\sqrt{n}p
({\bf G}_p\otimes {\bf G}_p)'
\,
{\rm vec}\big(
{\mathbf{S}}_n
-
{{\pmb \Sigma}}_n
\big)
$ converges weakly to ${\rm vec}\,{\mathbf{Z}}\sim\mathcal{N}
(
{\bf 0},
{\bf V}_p
)
$. It readily follows that $(\xi_{n1},\xi_{np})'$ converges weakly to $(\xi_{1},\xi_{p})'$, where $\xi_{1}$ is the largest root of the polynomial ${\rm det}({{\mathbf{Z}}}+ {\rm diag}(0, -v_\tau, \ldots, -v_\tau)- h {\bf I}_p)$ and $\xi_{p}$ is the smallest root of the polynomial ${\rm det}({{\mathbf{Z}}}+ {\rm diag}(v_\tau,0, \ldots, 0)- h {\bf I}_p)$. This implies that $$\begin{aligned}
\bigg(
\begin{array}{c}
\sqrt{n} (p\hat{\lambda}_{n1}-1) \\
\sqrt{n} (p\hat{\lambda}_{np}-1)
\end{array}
\bigg)
&\! = \!&
\bigg(
\begin{array}{c}
\xi_{n1}\\
\xi_{np}
\end{array}
\bigg)
+
\frac{2\tau}{p+2}
\bigg(
\begin{array}{c}
p-1 \\
-1
\end{array}
\bigg)
\\[2mm]
&\! \stackrel{\mathcal{D}}{\to} \!&
\bigg(
\begin{array}{c}
\xi_{1}\\
\xi_{p}
\end{array}
\bigg)
+
\frac{2\tau}{p+2}
\bigg(
\begin{array}{c}
p-1 \\
-1
\end{array}
\bigg)
=:
\bigg(
\begin{array}{c}
\eta_1 \\
\eta_p
\end{array}
\bigg)
.\end{aligned}$$ Clearly, $
\eta_1
$ is the largest root of the polynomial ${\rm det}({{\mathbf{Z}}}+ {\rm diag}(0, -v_\tau, \ldots, -v_\tau) + (2(p-1)\tau/(p+2)) {\bf I}_p - h {\bf I}_p)$, that is the largest eigenvalue of ${{\mathbf{Z}}}+(2\tau/(p+2)){\rm diag}(p-1,-1,\ldots,-1)$, whereas $
\eta_p
$ is the smallest root of the polynomial ${\rm det}({{\mathbf{Z}}}+ {\rm diag}(v_\tau,0, \ldots, 0) - (2\tau/(p+2)) {\bf I}_p - h {\bf I}_p)$, that is, the smallest eigenvalue of ${{\mathbf{Z}}}+(2\tau/(p+2)){\rm diag}(p-1,-1,\ldots,-1)$. This proves the result for $\tau\geq 0$.
We turn to Case (b), for which ${{\pmb \Sigma}}_n$ has eigenvalues $$\label{eigenvic2}
\lambda_{n1}
=
\ldots
=
\lambda_{n,p-1}
=
\frac{1}{p} - \frac{2\tau}{\sqrt{n}p(p+2)}
\quad
\textrm{ and }
\quad
\lambda_{np}
=
\frac{1}{p} + \frac{2(p-1)\tau}{\sqrt{n}p(p+2)}
\cdot$$ Here, we accordingly fix arbitrarily ${{\pmb\theta}}_1, \ldots, {{\pmb\theta}}_{p-1}$ such that the $p \times p$ matrix $\tilde{{\bf G}}_p:=({{\pmb\theta}}_1, \ldots, {{\pmb\theta}}_{p-1},{{\pmb\theta}})$ is orthogonal. Still with ${{\pmb \Lambda}}_n:={\rm diag}(\lambda_{n1},\lambda_{n2}, \ldots, \lambda_{np})$, we have ${{\pmb \Sigma}}_n \tilde{{\bf G}}_p=\tilde{{\bf G}}_p{{\pmb \Lambda}}_n$, so that $\tilde{{\bf G}}_p$ is an eigenvectors matrix for ${{\pmb \Sigma}}_n$. As above, $\xi_{n1}:=\sqrt{n}p (\hat{\lambda}_{n1}-\lambda_{n1})$ is the largest root of the polynomial $P_{n1}(h):={\rm det}({\sqrt n}p({\bf S}_n-\lambda_{n1} {\bf I}_p)- h {\bf I}_p)$, whereas $\xi_{np}:=\sqrt{n}p (\hat{\lambda}_{np}-\lambda_{np})$ is the smallest root of $P_{np}(h):={\rm det}({\sqrt n}p({\bf S}_n-\lambda_{np} {\bf I}_p)- h {\bf I}_p)$. Letting $$\label{defZn2}
{\bf Z}_n
:=
\sqrt{n}p(\tilde{{\bf G}}_p{^{\prime}}{\bf S}_n \tilde{{\bf G}}_p - {{\pmb \Lambda}}_n)
=
\tilde{{\bf G}}_p{^{\prime}}\sqrt{n}p({\bf S}_n- {{\pmb \Sigma}}_n) \tilde{{\bf G}}_p
,$$ rewrite these polynomials as $
P_{nj}(h)
=
{\rm det}(\sqrt{n}p\tilde{{\bf G}}_p{^{\prime}}{\bf S}_n \tilde{{\bf G}}_p - \sqrt{n}p\lambda_{nj} {\bf I}_p - h {\bf I}_p)
=
{\rm det}({{\mathbf{Z}}}_n+\sqrt{n}p({{\pmb \Lambda}}_n-\lambda_{nj} {\bf I}_p)- h {\bf I}_p)
,
$ $j=1,p$, which here gives $$P_{n1}(h)
=
{\rm det}({{\mathbf{Z}}}_n+{\rm diag}(0, \ldots,0, v_\tau)- h {\bf I}_p)
\nonumber$$ and $$P_{np}(h)
=
{\rm det}({{\mathbf{Z}}}_n+{\rm diag}(-v_\tau,\ldots,-v_\tau,0)- h {\bf I}_p)
\nonumber
,$$ with the same $v_\tau$ as above. Of course, we still have that ${\rm vec}\,{\mathbf{Z}}_n
=
\sqrt{n}p
(\tilde{{\bf G}}_p\otimes \tilde{{\bf G}}_p)'
\,
{\rm vec}\big(
{\mathbf{S}}_n
-
{{\pmb \Sigma}}_n
\big)
$ converges weakly to ${\rm vec}\,{\mathbf{Z}}\sim\mathcal{N}
(
{\bf 0},
{\bf V}_p
)
$. It readily follows that $(\xi_{n1},\xi_{np})'$ converges weakly to $(\xi_{1},\xi_{p})'$, where $\xi_{1}$ is the largest root of the polynomial ${\rm det}({{\mathbf{Z}}}+ {\rm diag}(0, \ldots, 0,v_\tau) - h {\bf I}_p)$ and $\xi_{p}$ is the smallest root of the polynomial ${\rm det}({{\mathbf{Z}}}+{\rm diag}(-v_\tau, \ldots, -v_\tau,0)- h {\bf I}_p)$. This still implies that $$\begin{aligned}
\bigg(
\begin{array}{c}
\sqrt{n} (p\hat{\lambda}_{n1}-1) \\
\sqrt{n} (p\hat{\lambda}_{np}-1)
\end{array}
\bigg)
&\!\!=\!\!&
\bigg(
\begin{array}{c}
\xi_{n1}\\
\xi_{np}
\end{array}
\bigg)
+
\frac{2\tau}{p+2}
\bigg(
\begin{array}{c}
-1 \\
p-1
\end{array}
\bigg)
\\[2mm]
&\!\!\stackrel{\mathcal{D}}{\to}\!\!&
\bigg(
\begin{array}{c}
\xi_{1}\\
\xi_{p}
\end{array}
\bigg)
+
\frac{2\tau}{p+2}
\bigg(
\begin{array}{c}
-1 \\
p-1
\end{array}
\bigg)
=:
\bigg(
\begin{array}{c}
\eta_1 \\
\eta_p
\end{array}
\bigg)
.\end{aligned}$$ Clearly, $
\eta_1
$ is the largest root of the polynomial ${\rm det}({{\mathbf{Z}}}+ {\rm diag}(0,\ldots,0,v_\tau) - (2\tau/(p+2)) {\bf I}_p - h {\bf I}_p)$, that is the largest eigenvalue of ${{\mathbf{Z}}}+(2\tau/(p+2)){\rm diag}(-1,\ldots,-1,p-1)$, whereas $
\eta_p
$ is the smallest root of the polynomial ${\rm det}({{\mathbf{Z}}}+ {\rm diag}(-v_\tau, \ldots,-v_\tau, 0) + (2(p-1)\tau/(p+2)) {\bf I}_p - h {\bf I}_p)$, that is, the smallest eigenvalue of ${{\mathbf{Z}}}+(2\tau/(p+2)){\rm diag}(-1,\ldots,-1,p-1)$. This establishes the result for $\tau<0$. [$\square$]{}
[Proof of Corollary \[Corollambda1asympt\].]{} According to Theorem \[thlambda1asympt\], $L_p^{\rm max}$ is equal in distribution to the first marginal of ${\pmb \ell}^{(p)}=(\ell^{(p)}_{1},\ldots,\ell^{(p)}_{p})'$, where $\ell^{(p)}_{1}\geq \ldots \geq \ell^{(p)}_{p}$ are the eigenvalues of ${\mathbf{Z}}=(Z_{ij})$, with ${\rm vec}\,{\mathbf{Z}}\sim\mathcal{N}({\bf 0},{\bf V}_p)$. Note that $Z_{ij}=Z_{ji}$ almost surely for any $1\leq i<j\leq p$, so ${\rm vec}\,{\mathbf{Z}}$ may not have a density with respect to the Lebesgue measure on ${\mathbb R}^{p^2}$, but ${\rm vech}\,{\mathbf{Z}}$ might in principle have a density with respect to the Lebesgue measure on ${\mathbb R}^{p(p+1)/2}$. If ${\rm vech}\,{\mathbf{Z}}$ indeed has such a density, then Theorem 13.3.1 of [@And2003] can be used to obtain the density of ${\pmb \ell}^{(p)}$ from that of ${\rm vech}\,{\mathbf{Z}}$. However, since ${\rm tr}[{\mathbf{Z}}]=({\rm vec}\,{\bf I}_p)'({\rm vec}\,{\mathbf{Z}})=0$ almost surely, ${\rm vech}\,{\mathbf{Z}}$ does not admit a density and the eigenvalues $\ell^{(p)}_{1},\ldots,\ell^{(p)}_{p}$ add up to zero almost surely (and thus do not admit a joint density over ${\mathbb R}^p$ either).
We solve this issue by considering a sequence of $p\times p$ random matrices ${\mathbf{Z}}_{\delta_k}$ ($k=1,2,\ldots$), with $\delta_k(> 0)$ converging to zero as $k$ goes to infinity, and such that ${\rm vec}\,{\mathbf{Z}}_\delta\sim\mathcal{N}
(
{\bf 0},
{\bf V}_{p,\delta}
)$, with $${\bf V}_{p,\delta}
:=
\frac{p}{p+2} (\mathbf{I}_{p^2}+{\bf K}_p) - \frac{2-\delta}{p+2} \mathbf{J}_{p}
.$$ Of course, ${\mathbf{Z}}_{\delta_k}$ converges weakly to ${\mathbf{Z}}$, so that the continuous mapping theorem ensures that ${\pmb \ell}^{(p)}_{\delta_k}=(\ell^{(p)}_{1\delta_k},\ldots,\ell^{(p)}_{p\delta_k})'$ (here, $\ell^{(p)}_{1\delta}\geq \ldots \geq \ell^{(p)}_{p\delta}$ are the eigenvalues of ${\mathbf{Z}}_\delta$) also converges weakly to ${\pmb \ell}^{(p)}=(\ell^{(p)}_{1},\ldots,\ell^{(p)}_{p})'$. Let then ${\mathbf{D}}_p$ be the $p$-dimensional duplication matrix, that is such that ${\mathbf{D}}_p({\rm vech}\,{\mathbf{A}})={\rm vec}\,{\mathbf{A}}$ for any $p\times p$ symmetric matrix ${\mathbf{A}}$. Write ${\mathbf{W}}_\delta:={\rm vech}\,{\mathbf{Z}}_\delta={\mathbf{D}}_p^-({\rm vec}\,{\mathbf{Z}}_\delta)$, where ${\mathbf{D}}_p^-=({\mathbf{D}}_p'{\mathbf{D}}_p)^{-1}{\mathbf{D}}_p'$ is the Moore-Penrose inverse of ${\mathbf{D}}_p$. By definition of ${\mathbf{Z}}_\delta$, the random vector ${\mathbf{W}}_\delta$ has density $${\bf w}\mapsto
h_\delta({\bf w})
=
\frac{a_p}{\sqrt{\delta}}
\exp\Big(-\frac{1}{2}\, {\bf w}'
\big(
{\mathbf{D}}_p^-
\big\{
{\textstyle{\frac{p}{p+2}}}
({\bf I}_{p^2}+ {\bf K}_p)
-
{\textstyle{\frac{2-\delta}{p+2}}}
{\mathbf{J}}_p
\big\}
({\mathbf{D}}_p^-)'
\big)^{-1}
{\bf w} \Big)
,$$ where $$a_p
:=
\frac{(p+2)^{p(p+1)/4}}{2^{(p^2+3p-2)/4} (\pi p)^{p(p+1)/4}}$$ is a normalizing constant. By using the identities ${\bf K}_p{\bf D}_p={\bf D}_p$ and ${\mathbf{D}}_p^-({\rm vec}\,\mathbf{I}_p)={\mathbf{D}}_p'({\rm vec}\,\mathbf{I}_p)={\rm vech}\,\mathbf{I}_p$, it is easy to check that $$\begin{aligned}
\lefteqn{
\big
({\mathbf{D}}_p^-
\big\{
{\textstyle{\frac{p}{p+2}}}
({\bf I}_{p^2}+ {\bf K}_p)
-
{\textstyle{\frac{2-\delta}{p+2}}}
{\mathbf{J}}_p
\big\}
({\mathbf{D}}_p^-)'
\big)^{-1}
}
\\[2mm]
& &
\hspace{3mm}
=
\frac{p+2}{2p}\,
{\mathbf{D}}_p'
\bigg\{
\frac{1}{2} ({\bf I}_{p^2}+ {\bf K}_p)
+
\frac{\frac{2-\delta}{p+2}}{\frac{2p}{p+2}-\frac{p(2-\delta)}{p+2}}
{\mathbf{J}}_p
\bigg\}
{\mathbf{D}}_p
\\[2mm]
& &
\hspace{3mm}
=
\frac{p+2}{2p}\,
{\mathbf{D}}_p'
\bigg\{
\frac{1}{2} ({\bf I}_{p^2}+ {\bf K}_p)
+
\frac{2-\delta}{\delta p}
{\mathbf{J}}_p
\bigg\}
{\mathbf{D}}_p
.\end{aligned}$$ Using the identities $({\rm vec}\,{\mathbf{A}})'({\rm vec}\,{\mathbf{B}})={\rm tr}[{\mathbf{A}}'{\mathbf{B}}]$ and ${\bf K}_p({\rm vec}\,{\mathbf{A}})={\rm vec}({\mathbf{A}}')$, the resulting density for ${\mathbf{Z}}_\delta$ is therefore $$\begin{aligned}
\lefteqn{
{\mathbf{z}}\mapsto f({\mathbf{z}})=h_\delta({\rm vech}\,{\mathbf{z}})
}
\\[2mm]
& &
\hspace{3mm}
=
\frac{a_p}{\sqrt{\delta}}
\exp\Big(
-
\frac{p+2}{4p}\,
({\rm vech}\,{\mathbf{z}})'
{\mathbf{D}}_p'
\bigg\{
\frac{1}{2} ({\bf I}_{p^2}+ {\bf K}_p)
+
\frac{2-\delta}{\delta p}
{\mathbf{J}}_p
\bigg\}
{\mathbf{D}}_p
({\rm vech}\,{\mathbf{z}})
\Big)
\\[2mm]
& &
\hspace{3mm}
=
\frac{a_p}{\sqrt{\delta}}
\exp\Big(
-
\frac{p+2}{4p}\,
\,
({\rm vec}\,{\mathbf{z}})'
\Big\{
{\bf I}_{p^2}
+
\frac{2-\delta}{\delta p}
{\mathbf{J}}_p
\Big\}
{\rm vec}\,{\mathbf{z}}\Big)
\\[2mm]
& &
\hspace{3mm}
=
\frac{a_p}{\sqrt{\delta}}
\exp\Big(
-
\frac{p+2}{4p}\,
\,
\Big\{
({\rm tr}\,{\mathbf{z}}^2)
+
\frac{2-\delta}{\delta p}
({\rm tr}\,{\mathbf{z}})^2
\Big\}
\Big)
.\end{aligned}$$ Theorem 13.3.1 from [@And2003] then implies that ${\pmb \ell}^{(p)}_{\delta}=(\ell^{(p)}_{1\delta},\ldots,\ell^{(p)}_{p\delta})'$ has density $$\begin{aligned}
\lefteqn{
\hspace{-10mm}
(\ell_1, \ldots, \ell_p)'
\mapsto
\frac{b_p}{\sqrt{\delta}}
\exp
\bigg(
-
\frac{p+2}{4p}
\,
\bigg\{
\bigg( \sum_{j=1}^p \ell_j^2 \bigg)
+
\frac{2-\delta}{\delta p}
\bigg(\sum_{j=1}^p \ell_j\bigg)^2
\bigg\}
\bigg)
}
\\[2mm]
& &
\hspace{40mm}
\times
\bigg( \prod_{1\leq k<j \leq p} (\ell_k- \ell_j) \bigg)
\,
\mathbb{I}[\ell_1\geq\ldots\geq \ell_p]
,
\nonumber\end{aligned}$$ with $$b_p
:=
\frac{(p+2)^{p(p+1)/4}}{2^{(p^2+3p-2)/4} p^{p(p+1)/4}\prod_{j=1}^{p} \Gamma(\frac{j}{2})}
\cdot$$ We now turn to the particular cases (i) $p=2$ and (ii) $p=3$ considered in the statement of the corollary. (i) Our general derivation above states that ${\pmb \ell}^{(2)}_{\delta}=(\ell^{(2)}_{1\delta},\ell^{(2)}_{2\delta})'$ has density $(\ell_1,\ell_2)'
\mapsto \mathcal{I}(\ell_1,\ell_2)
\mathbb{I}[\ell_1\geq\ell_2]$, with $$\mathcal{I}(\ell_1,\ell_2)
:=
\frac{1}{\sqrt{2\pi\delta}}
(\ell_1-\ell_2)
e^{
-
\frac{1}{2}
\{
( \ell_1^2+\ell_2^2 )
+
\frac{2-\delta}{2\delta}
(\ell_1+\ell_2)^2
\}
}
.$$ Direct computations allow checking that $\mathcal{I}(\ell_1,\ell_2)$ is the derivative of the function $$\ell_2
\mapsto
\frac{\sqrt{2\delta}}{\sqrt{\pi}(2+\delta)}
e^{
-
\frac{1}{2}
\{
( \ell_1^2+\ell_2^2 )
+
\frac{2-\delta}{2\delta}
(\ell_1+\ell_2)^2
\}
}
+
\frac{2^{5/2} \ell_1}{(2+\delta)^{3/2}}
e^{
-\frac{2\ell^2_1}{2+\delta}
}
\Phi
\bigg(
\frac{(2-\delta)\ell_1+(2+\delta)\ell_2}{\sqrt{2\delta(\delta+2)}}
\bigg)
.
$$ It follows that $\ell^{(2)}_{1\delta}$ has density $$\begin{aligned}
\lefteqn{
\hspace{0mm}
\ell_1
\mapsto
\int_{-\infty}^{\ell_1}
\mathcal{I}(\ell_1,\ell_2)
\,d\ell_2
}
\\[2mm]
& &
\hspace{-4mm}
=
\bigg[
\frac{\sqrt{2\delta}}{\sqrt{\pi}(2+\delta)}
e^{
-
\frac{1}{2}
\{
( \ell_1^2+\ell_2^2 )
+
\frac{2-\delta}{2\delta}
(\ell_1+\ell_2)^2
\}
}
+
\frac{2^{5/2} \ell_1}{(2+\delta)^{3/2}}
e^{
-\frac{2\ell^2_1}{2+\delta}
}
\Phi
\bigg(
\frac{(2-\delta)\ell_1+(2+\delta)\ell_2}{\sqrt{2\delta(\delta+2)}}
\bigg)
\bigg]_{-\infty}^{\ell_1}
\\[2mm]
& &
\hspace{-4mm}
=
\frac{\sqrt{2\delta}}{\sqrt{\pi}(2+\delta)}
e^{
-\frac{2\ell^2_1}{\delta}
}
+
\frac{2^{5/2} \ell_1}{(2+\delta)^{3/2}}
e^{
-\frac{2\ell^2_1}{2+\delta}
}
\Phi
\bigg(
\frac{4\ell_1}{\sqrt{2\delta(\delta+2)}}
\bigg)
.\end{aligned}$$ Taking the limit as $\delta\to 0$, we obtain that $\ell^{(2)}_{1}(\stackrel{\mathcal{D}}{=}L_2^{\rm max})$ has density $
\ell_1
\mapsto
2\ell_1
\exp(-\ell^2_1)
\mathbb{I}[\ell_1>0]
$. From Theorem \[thlambda1asympt\], this is the density of the asymptotic distribution of $T_+{^{(n)}}=\sqrt{n} (p\hat{\lambda}_{n1}-1)$. Part (i) of the result then follows from the fact that $T_+{^{(n)}}=T_-{^{(n)}}=T_\pm{^{(n)}}$ almost surely for $p=2$ (see the discussion below the corollary).
\(ii) For $p=3$, the density of ${\pmb \ell}^{(3)}_{\delta}=(\ell^{(3)}_{1\delta},\ell^{(3)}_{2\delta},\ell^{(3)}_{3\delta})'$ is $(\ell_1,\ell_2,\ell_3)' \mapsto \mathcal{J}(\ell_1,\ell_2,\ell_3)
\mathbb{I}[\ell_1\geq\ell_2\geq\ell_3]$, with $$\mathcal{J}(\ell_1,\ell_2,\ell_3)
:=
\frac{125}{216\pi\sqrt{\delta}}
(\ell_1-\ell_2)
(\ell_1-\ell_3)
(\ell_2-\ell_3)
e^{
-
\frac{5}{12}
\,
\{
( \ell_1^2+\ell_2^2+\ell_3^2 )
+
\frac{2-\delta}{3\delta}
(\ell_1+\ell_2+\ell_3)^2
\}
}
.$$ Lengthy yet straightforward computations show that $\mathcal{J}(\ell_1,\ell_2,\ell_3)$ is the derivative of the function $\ell_3
\mapsto \mathcal{K}(\ell_1,\ell_2,\ell_3)$, with $$\begin{aligned}
\lefteqn{
\hspace{0mm}
\mathcal{K}(\ell_1,\ell_2,\ell_3)
:=
}
\\[2mm]
& &
\hspace{-4mm}
-
\frac{25\sqrt{\delta}(\ell_1-\ell_3)}{48\pi(1+\delta)^2}
\{2(2\ell_1+2\ell_3-\ell_2)+\delta(\ell_1+\ell_3-2\ell_2)\}
e^{
-
\frac{5}{12}
\,
\{
( \ell_1^2+\ell_2^2+\ell_3^2 )
+
\frac{2-\delta}{3\delta}
(\ell_1+\ell_2+\ell_3)^2
\}
}
\\[2mm]
& &
\hspace{-4mm}
-
\frac{5\sqrt{10}(\ell_1-\ell_3)}{288\sqrt{\pi}(1+\delta)^{5/2}}
\{20(2\ell^1_1+5\ell_1\ell_3+2\ell_3^2)-2\delta(5(\ell_1-\ell_3)^2-18)-\delta^2(5(\ell_1-\ell_3)^2-36)\}
\\[2mm]
& &
\hspace{10mm}
\times
e^{
-
\frac{20(\ell_1^2+\ell_1\ell_3+\ell_3^2)+5\delta (\ell_1-\ell_3)^2)}{24(1+\delta)}
}
\Phi
\bigg(
\frac{\sqrt{5}(2(\ell_1+\ell_2+\ell_3)-\delta(\ell_1+\ell_3-2\ell_2))}{6\sqrt{\delta(1+\delta)}}
\bigg)
.\end{aligned}$$ Therefore, $(\ell^{(3)}_{1\delta},\ell^{(3)}_{3\delta})'$ has density $$(\ell_1,\ell_3)
\mapsto
\bigg(
\int_{\ell_3}^{\ell_1}
\mathcal{J}(\ell_1,\ell_2,\ell_3)
\,d\ell_2
\bigg)
\mathbb{I}[\ell_1\geq\ell_3]
=
(\mathcal{K}(\ell_1,\ell_1,\ell_3)-\mathcal{K}(\ell_1,\ell_3,\ell_3))
\mathbb{I}[\ell_1\geq\ell_3]
,$$ with $$\begin{aligned}
\lefteqn{
\hspace{-2mm}
\mathcal{K}(\ell_1,\ell_1,\ell_3)-\mathcal{K}(\ell_1,\ell_3,\ell_3)
=
}
\\[2mm]
& &
\hspace{-3mm}
-
\frac{25\sqrt{\delta}(\ell_1-\ell_3)}{48\pi(1+\delta)^2}
\{2(\ell_1+2\ell_3)+\delta(\ell_3-\ell_1)\}
e^{
-
\frac{5}{12}
\,
\{
( 2\ell_1^2+\ell_3^2 )
+
\frac{2-\delta}{3\delta}
(2\ell_1+\ell_3)^2
}
\\[2mm]
& &
\hspace{-3mm}
+
\frac{25\sqrt{\delta}(\ell_1-\ell_3)}{48\pi(1+\delta)^2}
\{2(2\ell_1+\ell_3)+\delta(\ell_1-\ell_3)\}
e^{
-
\frac{5}{12}
\,
\{
( \ell_1^2+2\ell_3^2 )
+
\frac{2-\delta}{3\delta}
(\ell_1+2\ell_3)^2
}
\\[2mm]
& &
\hspace{-3mm}
-
\frac{5\sqrt{10}(\ell_1-\ell_3)}{288\sqrt{\pi}(1+\delta)^{5/2}}
e^{
-
\frac{20(\ell_1^2+\ell_1\ell_3+\ell_3^2)+5\delta (\ell_1-\ell_3)^2)}{24(1+\delta)}
}
\\[2mm]
& &
\hspace{4mm}
\times
\{20(2\ell^1_1+5\ell_1\ell_3+2\ell_3^2)-2\delta(5(\ell_1-\ell_3)^2-18)-\delta^2(5(\ell_1-\ell_3)^2-36)\}
\\[2mm]
& &
\hspace{4mm}
\times
\bigg[
\Phi
\bigg(
\frac{\sqrt{5}(2(2\ell_1+\ell_3)-\delta(\ell_3-\ell_1))}{6\sqrt{\delta(1+\delta)}}
\bigg)
-
\Phi
\bigg(
\frac{\sqrt{5}(2(\ell_1+2\ell_3)-\delta(\ell_1-\ell_3))}{6\sqrt{\delta(1+\delta)}}
\bigg)
\bigg]
.\end{aligned}$$ Taking the limit as $\delta\to 0$ shows that the density of $(\ell^{(3)}_{1},\ell^{(3)}_{3})'$ is $$\label{jointlambdap3}
(\ell_1,\ell_3)
\mapsto
-
\frac{100\sqrt{10}}{288\sqrt{\pi}}
(\ell_1-\ell_3)(2\ell^2_1+5\ell_1\ell_3+2\ell_3^2)
e^{
-
\frac{5(\ell_1^2+\ell_1\ell_3+\ell_3^2)}{6}
}
\mathbb{I}[-2\ell_1< \ell_3\leq -\ell_1/2]
.$$ From Theorem \[thlambda1asympt\], the density of the asymptotic null distribution of $T_+{^{(n)}}=\sqrt{n} (p\hat{\lambda}_{n1}-1)$ coincides with the density of $\ell_1^{(3)}(\stackrel{\mathcal{D}}{=}L_3^{\rm max})$. After marginalization in (\[jointlambdap3\]), this last density is seen to be $$\begin{aligned}
\lefteqn{
\ell_1
\mapsto
\bigg\{
\sqrt{\frac{5}{2\pi}} e^{-\frac{5\ell_1^2}{2}}
+
\sqrt{\frac{5}{8\pi}} e^{-\frac{5\ell_1^2}{8}}
+
\frac{3}{4}\sqrt{\frac{5}{8\pi}} (5\ell_1^2-4) e^{-\frac{5\ell_1^2}{8}}
\bigg\}
\mathbb{I}[\ell_1\geq 0]
}
\\[2mm]
& &
\hspace{3mm}
=
\bigg\{
\frac{d}{d\ell_1} \Phi(\sqrt{5}\ell_1)
+
\frac{d}{d\ell_1} \Phi({\textstyle{\frac{\sqrt{5}\ell_1}{2}}})
+
3\frac{d}{d\ell_1} \Phi''({\textstyle{\frac{\sqrt{5}\ell_1}{2}}})
\bigg\}
\mathbb{I}[\ell_1\geq 0]
,
\end{aligned}$$ which proves the result for $T_+{^{(n)}}$, hence also for $T_-{^{(n)}}$ (recall from the discussion below the corollary that $T_+{^{(n)}}$ and $T_-{^{(n)}}$ share the same weak limit in any dimension $p$). Finally, the result for $T_\pm{^{(n)}}$ follows from (\[jointlambdap3\]) by using the fact that $T_\pm{^{(n)}}$ converges weakly to $\max(\ell_1^{(3)},-\ell_3^{(3)})$. [$\square$]{}
Acknowledgement {#acknowledgement .unnumbered}
===============
Davy Paindaveine’s research is supported by a research fellowship from the Francqui Foundation and by the Program of Concerted Research Actions (ARC) of the Université libre de Bruxelles. Thomas Verdebout’s research is supported by the ARC Program of the Université libre de Bruxelles and by the Crédit de Recherche J.0134.18 of the FNRS (Fonds National pour la Recherche Scientifique), Communauté Française de Belgique.
|
---
abstract: |
Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is equipped with a positive $\ZZ$-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials.
More precisely, in the case of $\ZZ$-grading, $\ZZ^2$ can be splitted into a finite number of regions such that each region corresponds to a polynomial that depending to the degree $(\mu, t)$, $\dim_k \left(\operatorname{Tor}_i^S(I^t, k)_{\mu} \right)$ is equal to one of these polynomials in $(\mu, t)$. This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers of ideals in [@Ko1].
Our main statement treats the case of a power products of homogeneous ideals in a $\ZZ^d$-graded algebra, for a positive grading, in the sense of [@ms].
author:
- Amir Bagheri
- Kamran Lamei
title: Graded Betti numbers of powers of ideals
---
[^1]
Introduction
============
The study of homological invariants of powers of ideals goes back, at least, to the work of Brodmann in the 70’s and attracted a lot of attention in the last two decades.
One of the most important results in this area is the result on the asymptotic linearity of Castelnuovo-Mumford regularity obtained by Kodiyalam [@Ko] and Cutkosky, Herzog and Trung [@CHT], independently. The proof of Cutkosky, Herzog and Trung further describes the eventual linearity in $t$ of $\operatorname{end}\left(\operatorname{Tor}_i^S(I^t, k)\right) :=$ max$\{\mu | \operatorname{Tor}_i^S(I^t, k)_{\mu} \neq 0 \} $.
It is natural to concern the asymptotic behavior of Betti numbers $\beta_i (I^t):=\dim_k \operatorname{Tor}_i^S(I^t, k)$ as $t$ varies. In [@NR], Northcott and Rees already investigated the asymptotic behavior of $\beta_1^k(I^t)$. Later, using the Hilbert-Serre theorem, Kodiyalam [@Ko1 Theorem 1] proved that for any non-negative integer $i$ and sufficiently large $t$, the $i$-th Betti number, $\beta_i^{k}(I^t)$, is a polynomial $Q_i$ in $t$ of degree at most the analytic spread of $I$ minus one.
Recently, refining the result of [@CHT] on $\operatorname{end}\left(\operatorname{Tor}_i^S(I^t, k)\right)$, [@BCH] gives a precise picture of the set of degrees $\gamma$ such that $\operatorname{Tor}_i^S(I^t, A)_\gamma \not= 0$ when $t$ runs over $\NN$. In [@BCH], the authors consider a polynomial ring $S=A[x_1, \ldots, x_n]$ over a Noetherian ring $A$ graded by a finitely generated abelian group $G$ (see [@BCH Theorem 4.6]).
When $A=k$ is a field and the ideal is generated in a single degree $d\in G$, it is proved that for any $\gamma$ and any $j$, the function $$\dim_k\operatorname{Tor}_i^S(I^t, k)_{\gamma +td}$$ is a polynomial in $t$ for $t\gg 0$ (see [@BCH Theorem 3.3] and [@Si]).
We are here interested in the behavior of $\dim_k\operatorname{Tor}_i^S(I^t, k)_{\gamma}$ when $I$ is an arbitrary graded ideal and $S=k[x_1, \ldots, x_n]$ is a $\ZZ^p$-graded polynomial ring over a field $k$, for a positive grading in the sense of [@ms].
In the case of a positive $\ZZ$-grading, our result takes the following form:
**Theorem**(See Theorem \[main res\]). *Let $S=k[x_1, \ldots, x_n]$ be a positively graded polynomial ring over a field $k$ and let $I$ be a homogeneous ideal in $S$.* *There exist, $t_0,m,D\in \ZZ$, linear functions $L_i(t)=a_i t+b_i$, for $i=0,\ldots ,m$, with $a_i$ among the degrees of the minimal generators of $I$ and $b_i\in \ZZ$, and polynomials $Q_{i,j}\in \QQ [x,y]$ for $i=1,\ldots ,m$ and $j\in 1,\ldots ,D$, such that, for $t\geq t_0$,*\
*(i) $L_i(t)<L_j(t)\ \Leftrightarrow\ i<j$,\
(ii) If $\mu <L_0(t)$ or $\mu >L_m(t)$, then $\operatorname{Tor}_i^S(I^t, k)_{\mu}=0$.\
(iii) If $L_{i-1} (t)\leq \mu \leq L_{i}(t)$ and $a_i t-\mu \equiv j\mod (D)$, then $$\dim_k\operatorname{Tor}_i^S(I^t, k)_{\mu}=Q_{i,j}(\mu ,t).$$*
Our general result, Theorem \[tor-general case\], involves a finitely generated graded module $M$ and a finite collection of graded ideals $I_1,\ldots ,I_s$. The grading is a positive $\ZZ^p$-grading and a special type of finite decomposition of $\ZZ^{p+s}$ is described in such a way that in each region $\dim_k (\operatorname{Tor}_i^S(MI_1^{t_1}...I_s^{t_s}, k)_{\gamma })$ is a polynomial in $(\gamma ,t_1,\ldots ,t_s)$ (i. e. is a quasi-polynomial) with respect to a lattice defined in terms of the degrees of generators of the ideals.
The central object in this study is the Rees modification $M\R_I$. This graded object admits a graded free resolution over a polynomial extension of $S$ from which we deduce a $\ZZ^{p+s}$-grading on Tor modules as in [@BCH]. Investigating Hilbert series of modules for such a grading, using vector partition functions, leads to our results.
This paper, is organized as follows. In the next Section, we provide some definitions and terminology that we will need. In Section 3, we will discuss Hilbert functions of non-standard graded rings and in the last section, we prove the main theorem.
Preliminaries {#sec.prel}
=============
In this section, we are going to collect some necessary notations and terminology used in the paper. For basic facts in commutative algebra, we refer the reader to [@Ei; @bh].
Hilbert series
--------------
In this part we will recall the definition and some important properties of Hilbert functions and Hilbert series.
Let $S=k[x_1, \ldots, x_n]$ be a polynomial ring over field $k$. We first make clear our definition of grading.
Let $G$ be an abelian group. A $G$-grading of $S$ is a group homomorphism $\deg : \ZZ^n \longrightarrow G $. We set $\deg(x^{u}) :=\deg(u)$ for a monomial $x^{u}= x_{1}^{u_1}... x_{n}^{u_n} \in S $. An element $\sum c_u x^{u}\in S$ is homogeneous of degree $\mu\in G$ if $\deg (u)=\mu$ whenever $c_u\not= 0$. The set of elements of degree $\mu$ in $S$, $S_\mu$, is called the homogeneous component of $S$ in degree $\mu$. The grading is called positive if furthermore $G$ is torsion-free and $S_0=k$.
We also recall that for any $\mu\in G$, $S(\mu)$ is again a $G$-graded ring such that $S(\mu)_n=S_{\mu+n}$. Obviously it is isomorphic to $S$ and it is obtained just by a shift on $S$.
From now on, $S$ will be a positively graded polynomial ring. Criterions of positivity are given in [@ms 8.6]. One defines graded ideals and modules similarly to the classical $\ZZ$-graded case. When $G=\ZZ^d$ and the grading is positive, (generalized) power series are associated to finitely generated graded modules:
By [@ms 8.8], if $S$ is positively graded by $\ZZ^d$, then the semigoup $Q=\deg(\NN^n)$ can be embedded in $\NN^d$. Hence, after such a change of embedding, the above Hilbert series are power series in the usual sense.
Let $M$ be a finitely generated $\ZZ^d$-graded $S$-module. It admits a finite minimal graded free $S$-resolution $$\mathbb{F}_\bullet: 0\rightarrow F_u \rightarrow \ldots \rightarrow F_1\rightarrow F_0\rightarrow M\rightarrow 0.$$ Writing $$F_i=\oplus_\mu S(-\mu)^{\beta_{i,\mu}(M)},$$ the minimality of resolution shows that $\beta_{i,\mu}(M)=\dim_k\left(\operatorname{Tor}_i^S(M,k)\right)_\mu$, as the maps of $\mathbb{F}_\bullet\otimes_S k$ are zero. We also recall that the support of a $\ZZ^d$-graded module $N$ is $$\operatorname{Supp}_{\ZZ^d} (N):=\{\mu\in \ZZ^d | N_\mu\neq0\}.$$
The Hilbert function of a finitely generated module $M$ over a positively graded polynomial ring is the map $$\begin{array}{llll}
HF(M; -):&\ZZ^d&\longrightarrow& \NN\\
&\mu&\longmapsto&\dim_k(M_\mu).
\end{array}$$ Moreover, the Hilbert series of $M$ is the power series $$H(M;{\bold t})=\sum_{{\bf \mu}\in \ZZ^d}\dim_k(M_{\mu}){\bold t}^{\mu}$$ where ${\bold t}:=(t_1, \ldots, t_d)$. Also we write ${\bold t}\gg 0$ if for all $i=1, \ldots, d$ we have $t_i\gg 0$.
\[Hilbert nonstandard\] Let $S=k[x_1, \ldots, x_n]$ be a positively graded $\ZZ^d$-graded polynomial ring over a field $k$. Then the followings hold.
1. The Hilbert series of $S$ is the development in power series of the rational function $$H(S(-\mu ); {\bold t})= \frac{{\bold t}^\mu}{\prod_{i=1}^n(1-{\bold t}^{\mu_i})}$$ where $\mu_i = \deg (x_i)$.
2. If $M$ is a finitely generated graded $S$-module, setting $\Sigma_M:=\cup_{\ell}\operatorname{Supp}_{\ZZ^d}(\operatorname{Tor}^R_\ell (M,k))$ and $$\kappa_M ({\bold t}):=\sum_{a\in \Sigma_M}\left( \sum_\ell (-1)^\ell \dim_k (\operatorname{Tor}^R_\ell (M,k))_a\right) {\bold t}^a,$$ one has $H(M;{\bold t}) = \kappa_M (t)H(S; {\bold t})$.
\(1) is a simple calculation and (2) follows from the existence of graded minimal free resolutions.
Vector partition function
-------------------------
We first recall the definition of quasi-polynomials. Let $d\geq 1$ and $\Lambda$ be a lattice in $\ZZ^d$.
[@Bar] A function $f:\ZZ^d\to\QQ$ is a quasi-polynomial with respect to $\Lambda$ if there exists a list of polynomials $Q_i\in\QQ[T_1, \ldots, T_d]$ for $i\in \ZZ^d /\Lambda$ such that $f(s)=Q_i(s)$ if $s\equiv i\mod \Lambda$.
In fact, a quasi-polynomial is a polynomial that its coefficients are replaced by periodic functions.
Notice that $\ZZ^d/\Lambda$ has $\vert \det (\Lambda )\vert $ elements, and when $d=1$, then $\Lambda =q\ZZ$ for some $q>0$. In this case $f$ is called a quasi-polynomial of period $q$.
Let $A=(a_{i,j})$ be a $d\times n$-matrix of rank $d$ with entries in $\NN$. Let $a_j:=(a_{1j}, \ldots, a_{dj})$ be the $j$-th column of $A$ and the mapping $\varphi$ acts as below: $$\begin{array}{llll}
\varphi_A:&\NN^d&\longrightarrow&\NN\\
&u&\longrightarrow&\#\left\{ \lambda\in \NN^n \vert A.\lambda =u \right\}.
\end{array}$$ Then $\varphi$ is called the **vector partition function** corresponding to the matrix $A$. Equivalently, $\varphi_A(u)$ is the coefficient of ${\bold t}^u$ in the formal power series $\prod_{i=1}^n \frac{1}{(1-{\bold t}^{a_i})}$.
Notice that $\varphi_A$ vanishes outside of $\operatorname{Pos}(A):=\{\sum\lambda_ia_i\in\RR^n | \lambda_i\geq 0, 1\leq i\leq n\}$. The type of this function and how this function works is very important. Indeed the values of $\varphi_A$ come from a quasi-polynomial. Blakley showed in [@B] that $\NN^d$ can be decomposed into a finite number of parts, called chambers, in such a way that $\varphi_A$ is a quasi-polynomial of degree $n-d$ in each chamber. Later, Sturmfels in [@St] investigated these decompositions and the differences of polynomials from one piece to another.
Here we briefly introduce the basic facts and necessary terminology of vector partition functions, specially the chambers and the polynomials (quasi-polynomials) obtained from vector partition functions corresponding to a matrix $A$. For more details about the vector partition function, we refer the reader to [@B; @BV; @St].
### Polyhedral Sets
(See also [@Zie]) A polyhedral complex $\Im$ is a finite collection of polyhedra in $\RR^d$ such that
1. the empty polyhedron is in $\Im$,
2. if $P \in \Im$, then all the faces of $P$ are also in $\Im$,
3. the intersection $P \cap Q$ of two polyhedra $P,Q \in \Im $ is a face of both of $P$ and of $Q$.
The support $|\Im|$ of a polyhedral complex $\Im$ is the union of the polyhedra in $\Im$.
If $\Im_1$ and $\Im_{2}$ are polyhedral complexes such that the support of $\Im_2$ contains the support of $\Im_1$ and such that for every $P \in \Im_1$ there is a $Q \in \Im_2$ with $Q \subset P$, then $\Im_2$ is a refinement of $\Im_1$.The common refinement of a family of polyhedra is a polyhedral complex such that its support is the union of the polyhedra and such that each member is the intersection of some of the polyhedra.
### Chamber complex of a vector partition function.\
\
Let $\sigma\subseteq \{1, \ldots, n\}$. We will say that $\sigma$ is independent if for all $i\in \sigma$, the vectors $a_i$ are linearly independent. If $\sigma$ is independent, we set $A_\sigma:=(a_{i})_{i\in\sigma}$ and denote by $\Lambda_\sigma$ the $\ZZ$-module generated by the columns of $A_\sigma$ as its base and $\partial Pos(A_{\sigma})$ the boundary of $ \operatorname{Pos}(A_{\sigma})$. When $\sigma$ has $d$ elements ([*i. e.*]{} is a maximal independent set), $\Lambda_\sigma$ is a sublattice of $\ZZ^d$.
Let $\sum_{A}$ be the set of all simplicial cones whose extremal rays are generated by $d$-linearly independent column vectors of $A$.
The chamber complex of a vector partition function as defined is the common refinement of all simplicial cones $Pos(A_{\sigma})$. The maximal chambers $C$ of the chamber complex of $A$ are the connected components of $Pos (A) - \bigcup_{ \ell \in \sum_{A}} \partial \ell$. These chambers are open and convex.
Associated to each maximal chamber $C$ there is an index set $$\Delta(C):= \{\sigma\subset \{1, \ldots, n\}\; |\; C\subseteq \operatorname{Pos}(A_\sigma)\}.$$ $\sigma\in \Delta(C)$ is called non-trivial if $G_\sigma:=\ZZ^d/\Lambda_\sigma\neq0$, equivalently if $\det (\Lambda_\sigma )\not= \pm 1$ ($G_\sigma$ is finite because $C\subseteq \operatorname{Pos}(A_\sigma)).$
\[ex4vect\] Let $\varphi$ be the vector partition function corresponding to the matrix $A = \left(\begin{array}{cccc}2 & 3 & 6 & 7 \\1 & 1 & 1 & 1\end{array}\right)$ i.e., $a_1=(2,1)$ , $a_2= (3,1)$ , $a_3=(6,1)$ , $a_4 = (7,1)$. Figure \[figure:chamber.decomposition\] shows the chamber decomposition associated to the vector partition function $\varphi$ and since any arbitrary vectors $a_{i} ,a_{i+1}$ make an independent set, the common refinement consists of disjoint union of open convex polyhedral cones generated by $a_{i} ,a_{i+1}$. Figure \[figure:chamber.complex\] shows the common refinement of all simplicial cones $A_\sigma$.\
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If $A$ is an arbitrary matrix, it is not easy to explain precisely its corresponding chamber complex. Here, in the following lemma we describe the chamber complex associated to a special kind of $2\times n$-matrix that its columns are $(d_i, 1)$, $d_i\in \NN$ which it plays an important role in the proof of proposition \[hilbert bigraded\].
\[chamber of n vector\] Let $$A=\left(\begin{array}{lll}
d_1&\ldots&d_n\\
1&\ldots&1\end{array}\right)$$ be a $2\times n$-matrix with entries in $\NN$ such that $d_1\leq\ldots \leq d_n$. Then the chambers corresponding to $\operatorname{Pos}(A)$ are positive polyhedral cones $\Delta$ where $\Delta$ is generated by $\{(d_i, 1), (d_{i+1}, 1)\}$ for all $d_i\neq d_{i+1}$ and $i$ runs over $\{1, \ldots, n\}$.
Now, we are ready to state the vector partition function theorem, which relies on the chamber decomposition of $\operatorname{Pos}(A)\subseteq \NN^d$.
\[vector partition\](See [@St Theorem 1]) For each chamber $C$ of maximal dimension in the chamber complex corresponding to matrix $A$, there exist a polynomial $P$ of degree $n-d$, a collection of polynomials $Q_\sigma$ and functions $\Omega_\sigma: G_\sigma\setminus\{0\}\rightarrow \QQ$ indexed by non-trivial $\sigma\in \Delta(C)$ such that, if $u\in \NN A\cap \overline{C}$, $$\varphi_A(u)=P(u)+\sum\{\Omega_\sigma([u]_\sigma).Q_\sigma(u) : \sigma\in \Delta(C) , [u]_\sigma\neq 0\}$$ where $[u]_\sigma$ denotes the image of $u$ in $G_\sigma$. Furthermore, $\deg (Q_\sigma )=\#\sigma-d$.\
\[vector partition2\] For each chamber $C$ of maximal dimension in the chamber complex of $A$, there exists a collection of polynomials $Q_\tau$ for $\tau\in \ZZ^d/\Lambda$ such that $$\varphi_A(u)=Q_\tau (u), \ \hbox{if}\ u\in \NN A\cap \overline{C}\ \hbox{and}\ u \in \tau+\Lambda_{C}$$ where $\Lambda_{C} = \cap_{\sigma \in \Delta(C)} \Lambda_{\sigma}.$
The class $\tau$ of $u$ modulo $\Lambda$ determines $[u]_\sigma$ in $G_\sigma=\ZZ^d/\Lambda_\sigma$. The term of the right-hand side of the equation in the above theorem is a polynomial determined by $[u]_\sigma$, hence by $\tau$.
Notice that setting $\Lambda$ for the intersection of the lattices $\Lambda_\sigma$ with $\sigma$ maximal, the class of $u$ mod $\Lambda$ determines the class of $u$ mod $\Lambda_{C}$. Thus, the corollary holds with $\Lambda$ in place of $\Lambda_{C}$.\
In the next section, we will use this theorem to show that the Hilbert function of a graded ring is determined by a finite collection of polynomials.
Hilbert functions of non-standard bigraded rings
================================================
Let $S=k[y_1, \ldots, y_m]$ be a $\ZZ^{d-1}$-graded polynomial ring over a field and let $I=(f_1,\ldots ,f_n)$ be a graded ideal such that its generators $f_i$ are homogeneous of degree $d_i$. To get information about the behavior of $i$-syzygy module of $I^t$ as $t$ varies, we pass to Rees algebra $\R_I=\oplus_{t\geq 0}I^t$. $\R_I$ has a $(\ZZ^{d-1}\times\ZZ )$-graded algebra structure such that $\left(\R_I\right)_{(\mu, n)}=(I^n)_\mu$.
Recall that $\R_I$ is a graded quotient of $R:=S[x_1, \ldots, x_n]$ with grading extended from the one of $S$ by setting $\deg (a):=(\deg (a), 0)$ for $a\in S$ and $\deg (x_j):=(d_j ,1)$ for all $j= 1, \ldots, n$. As noticed in [@BCH], if $\G_\bullet$ is a $\ZZ^d$-graded free $R$-resolution of $\R_I$, then by setting $$B:=k[x_1, \ldots, x_n]=R/(y_1, \ldots, y_m)$$ we will have the following equality: $$\operatorname{Tor}_i^S(I^t, k)_\mu=H_i(\G_\bullet\otimes_R B)_{(\mu,t)}.$$
The complex $\G_\bullet\otimes_R B$ is a $\ZZ^d$-graded complex of free $S$-modules. Its homology modules are therefore finitely generated $\ZZ^d$-graded $S$-modules, on which we will apply results derived from the ones on vector partition functions describing the Hilbert series of $S$.
Now we are ready to prove the main result of this section.
\[hilbert bigraded\] Let $B=k[T_1, \ldots, T_n]$ be a bigraded polynomial ring over field $k$ with $\deg(T_i)=(d_i, 1)$. Assume that the number of distinct $d_i$’s is $r\geq 2$. Then there exists a finite index sublattice $\Lambda$ of $\ZZ^2$ and collections of polynomials $Q_{ij}$ of degree $n-2 $ for $1\leq i\leq r-1$ and $1\leq j\leq s$ such that for any $(\mu, \nu)\in \ZZ^2\cap R_i$ and $\overline{\left(\mu, \nu\right)}\equiv g_j \mod \Lambda$ in ${\ZZ^2}/{\Lambda}:=\{g_1, \ldots, g_s\}$, $$HF(B, (\mu, \nu))= Q_{ij}(\mu, \nu)$$ where $R_i$ is the convex polyhedral cone generated by linearly independent vectors $\{(d_i, 1), (d_{i+1}, 1)\}$. Furthermore, $Q_{ij}(\mu, \nu) = Q_{ij}(\mu^{\prime}, \nu^{\prime}) $ if $ \mu - \nu d_{i} \equiv \mu^{\prime}-\nu^{\prime}d_i \mod (\det (\Lambda))$.
Let $$A=\left(\begin{array}{lll}
d_1 & \ldots & d_n\\
1 & \ldots & 1\end{array}\right)$$ be a $2\times n$-matrix corresponding to degrees of $T_i$.
The Hilbert function in degree ${\bf u}=(\mu, \nu)$ is the number of monomials $T_1^{\alpha_1}\ldots T_n^{\alpha_n}$ such that $\alpha_1(d_1, 1)+\ldots+\alpha_n(d_n, 1)=(\mu, \nu)$. This equation is equivalent to the system of linear equations $$A.\left(\begin{array}{c}\alpha_1\\
\vdots\\
\alpha_n\end{array}\right)=(\begin{array}{ll}
\mu & \nu\end{array}).$$ In this sense $HF(B, (\mu, \nu))$ will be the value of vector partition function at $(\mu, \nu)$. Assume that $(\mu, \nu)$ belongs to the chamber $R_i$ which is the convex polyhedral cone generated by $\{(d_i, 1), (d_{i+1}, 1)\}$. By \[vector partition2\], we know that for $(\mu, \nu)\in R_i$ and $(\mu, \nu)\equiv g_j\mod(\det \Lambda)$, $$\label{1}
\varphi_A(\mu, \nu)=Q_{ij}(\mu, \nu).$$
Notice that in Proposition \[hilbert bigraded\], if moreover we suppose that $d_i\neq d_j$ for all $i\neq j$, then the Hilbert function in degree $(\mu, \nu)$ will also be the number of lattice paths from $(0,0)$ to $(\mu, \nu)$ in the lattice $S = \{(d_i, 1) | 1 \leqslant i \leqslant n \}$, but this can fail if some of the $d_i$s are equal. For example if one has $d_i=d_{i+1}<d_{i+2}$, so the independent sets of vectors $\{(d_i, 1), (d_{i+2}, 1)\}$ and $\{(d_{i+1}, 1), (d_{i+2}, 1)\}$ generate the same chamber and the number of lattice paths from $(0, 0)$ to $(\mu, \nu)$ via lattice points is less than $\operatorname{HF}(B, (\mu, \nu))$.
Notice that although in this section the Hilbert function of special bigraded polynomial rings are obtained, this result can be given for every arbitrary $\ZZ^d$-graded polynomial rings using the concept of vector partition function theorem.
Betti numbers of powers of ideals
==================================
We now turn to the main result on Betti numbers of powers of ideals. Without additional effort, we treat the following more general situation of a collection of graded ideals and include a graded module $M$. Hence we will study the behaviour of $\dim_k \operatorname{Tor}_i^R(MI_1^{t_1}\cdots I_{s}^{t_s},k)_{\mu}$ for $\mu\in \ZZ^p$ and ${\bold t}\gg 0$. To this aim we first use the important fact that the module $$B_i:=\oplus_{t_1,\ldots ,t_s}\operatorname{Tor}_i^R(MI_1^{t_1}\cdots I_{s}^{t_s},k)$$ is a finitely generated $(\ZZ^p\times \ZZ^s)$-graded ring, over $k[T_{i,j}]$ setting $\deg (T_{i,j})=(\deg (f_{i,j}),e_i)$ with $e_i$ the $i$-th canonical generator of $\ZZ^s$ and for fixed $i$ the $f_{i,j}$ form a set of minimal generators of $I_i$. Hence $\operatorname{Tor}_i^R(MI_1^{t_1}\cdots I_{s}^{t_s},k)_\mu =(B_i)_{\mu ,t_1e_1+\cdots +t_se_s}$.
The following result applied to $B_i$ will then give the asymptotic behaviour of Betti numbers. In particular case if we have a single $\ZZ$-graded ideal, we will use it to give a simple description of this eventual behaviour.
The general case
-----------------
Let $\varphi : \ZZ^n \rightarrow \ZZ^d $ with $\varphi(\NN^n) \subseteq \NN^d$ be a positive $\ZZ^d$-grading of $R:=k[T_{i,j}]$. Set $\ZZ^n:=\sum_{i=1}^n\ZZ e_i$, let $E$ be the set of $d$-tuples $e=(e_{i_1},...,e_{i_d})$ such that $(\varphi(e_{i_1}),...,\varphi(e_{i_d}))$ generates a lattice $\Lambda_e$ in $\ZZ^d$, and set $$\Lambda :=\cap_{e\in E} \Lambda_e,\quad\quad \xymatrix{s_\Lambda : \ZZ^d \ar^{can}[r]&\ZZ^d/\Lambda\\}.$$
Denote by $C_i$, $i\in F$, the maximal cells in the chamber complex associated to $\varphi$. One has $$\overline{C_i} = \{ \xi\ \vert\ H_{i,j}(\xi ) \geqslant 0, \ 1\leqslant j \leqslant d \}$$ where $H_{i,j}$ is a linear form in $\xi \in \ZZ^d$.
Let $S=k[y_1, \ldots, y_m]$ be a $\ZZ^{p}$-graded polynomial ring over a field. Assume that $\deg (y_j)\in \NN^p$ for any $j$, and let $I_i=(f_{i,1},\ldots ,f_{i,r_i})$ be ideals, with $f_{i,j}$ homogeneous of degree $d_{i,j}$.
Consider $R:=k[T_{i,j}]_{1\leq i\leq s,\,1\leq j\leq r_i}$, set $\deg (T_{i,j})=(\deg (f_{i,j}),e_i)$, with $e_i$ the $i$-th canonical generator of $\ZZ^s$ and the induced grading $\varphi : \ZZ^{r_1+\cdots +r_s} \rightarrow \ZZ^d :=\ZZ^p\times \ZZ^s$ of $R$.
Denote as above by $\Lambda$ the lattice in $\ZZ^d$ associated to $\varphi$, by $s_\Lambda : \ZZ^d \to \ZZ^d/\Lambda$ the canonical morphism and by $C_i$, for $i\in F$, the maximal cells in the the chamber complex associated to $\varphi$. One has $\overline{C_i} = \{ (\mu ,t)\ \vert\ H_{i,j}(\mu ,t ) \geqslant 0, 1\leqslant j \leqslant d \}$ where $H_{i,j}$ is a linear form in $(\mu ,t) \in \ZZ^p\times \ZZ^s=\ZZ^d$.
\[general case\] With notations as above, let $B$ be a finitely generated $\ZZ^d$-graded $R$-module. There exist a finite set $U$ and convex sets of dimension $d$ in $\RR^d$ of the form $$\Delta_u = \{ x \ \vert\ H_{i,j}(x) \geqslant a_{u,i,j}, \forall (i,j)\in G_u \} \subseteq \RR^d$$ for $u \in U$, with $a_{u,i,j}=H_{i,j}(a)$ for $a\in \cup_{\ell}\operatorname{Supp}_{\ZZ^d}(\operatorname{Tor}^R_\ell (B,k))$, $G_u\subset F\times \{ 1,\ldots ,d\}$ and polynomials $P_{u,\tau}$ for $u\in U$ and $\tau\in \ZZ^d/\Lambda$ such that for all $\xi$ belonging to $\Delta_u$ we have $$\dim_k (B_{\xi }) = P_{u,s_{\Lambda} (\xi )} (\xi )$$ and for all $\xi$ out of $\bigcup_{u\in U}\Delta_u$ we have $$\dim_k (B_{\xi }) = 0.$$
By Proposition \[Hilbert nonstandard\], there exists a polynomial $\kappa_B (t_1,\ldots ,t_d)$ with integral coefficients such that $$H(B;{\bold t}) =\kappa_B ({\bold t})H(R;{\bold t})$$ and $\kappa_B({\bold t})=\sum_{a\in A} c_a{\bold t}^a$ with $A\subset \cup_{\ell}\operatorname{Supp}_{\ZZ^d}(\operatorname{Tor}^R_\ell (B,k))$. Let $D_i:=\cup_j \{ x\ \vert\ H_{i,j}(x)=0\}$ be the minimal union of hyperplanes containing the border of $C_i$. The union $C$ of the convex sets $\overline{C_i}+a$ can be decomposed into a finite union of convex sets $\Delta_u$, each $u\in U$ corresponding to one connected component of $C\setminus\bigcup_{i,a}(D_i+a)$. (Notice that $\RR^d \setminus \bigcup_{i,a}(D_i+a)$ has finitely many connected components, which are convex sets of the form of $\Delta_u$, and that we may drop the ones not contained in $C$ as the dimension of $B_\xi$ is zero for $\xi$ not contained in any $\overline{C_i}+a$.) We define $u$ as the set of pairs $(i,a)$ such that $(C_i+a)\bigcap \Delta_u\neq \emptyset$, and remark that if $(i,a)\in u$ then $(j,a)\not\in u$ for $j\not= i$.
If $\xi\not\in \bigcup_i \overline{C_i}+a$, then $\dim_k R_{\xi -a}=0$, while if $\xi\in\overline{C_i}+a$ then it follows from Corollary \[vector partition2\] that there exist polynomials $Q_{i,\tau}$ such that $$\dim_k R_{\xi -a}=Q_{i,\tau }(\xi -a)\quad \hbox{if}\ \xi -a\equiv\tau\mod (\Lambda).$$ Hence, setting $Q'_{i,a,\tau }(\xi ):=Q_{i,\tau +a}(\xi -a)$, one gets the conclusion with $$P_{u,\tau }=\sum_{(i,a)\in u} c_aQ'_{i,a,\tau}.$$
\[severalmodules\] The above proof shows that if one has a finite collection of modules $B_i$, setting $A:=\cup_{i,\ell}\operatorname{Supp}_{\ZZ^d}\operatorname{Tor}_\ell^R(B_i,k)$, there exist convex polyhedral cones $\Delta_u$ as above on which any $B_i$ has its Hilbert function given by a quasi-polynomial with respect to the lattice $\Lambda$.
We now turn to the main result of this article. The more simple, but important, case of powers of an ideal in a positively $\ZZ$-graded algebra over a field will be detailed just after.
\[tor-general case\] In the situation above, there exist a finite set $U$ and a finite number of polyhedral convex cones $$\Delta_u = \{(\mu,t) | H_{i,j}(\mu,t)\geqslant a_{u,i,j}, (i,j)\in G_u \} \subseteq \RR^d,$$ polynomials $P_{\ell ,u,\tau}$ for $u\in U$ and $\tau\in \ZZ^d/\Lambda$ such that, for any $\ell$, $$\dim_k (\operatorname{Tor}_\ell^S(MI_1^{t_1}...I_s^{t_s}, k)_{\mu }) = P_{\ell,u,s_{\Lambda}} (\mu,t),\quad \forall (\mu,t)\in \Delta_u,$$ and $\dim_k (\operatorname{Tor}_\ell^S(MI_1^{t_1}...I_s^{t_s}, k)_{\mu}) = 0$ if $(\mu,t)\not\in \cup_{u\in U}\Delta_u$.
Furthermore, for any $(u,i,j)$, $a_{u,i,j}=H_{i,j}(b)$, for some $$b\in
\bigcup_{i,\ell}\operatorname{Supp}_{\ZZ^d}\operatorname{Tor}_\ell^R(\operatorname{Tor}^S_i(M\R_{I_1,\ldots ,I_s},R),k).$$
We know from [@BCH] that $B_i:=\oplus_{t_1,\ldots ,t_t}\operatorname{Tor}_i^S(MI_1^{t_1}\cdots I_{s}^{t_s},k)$ is a finitely generated $\ZZ^d$-graded module over $R$. As $B_i\not= 0$ for only finitely many $i$, the conclusion follows from Proposition \[general case\] and Remark \[severalmodules\].
The above results tell us that $\RR^d$ can be decomposed in a finite union of convex polyhedral cones $\Delta_u$ on which, for any $\ell$, the dimension of $\operatorname{Tor}_\ell^S(MI_1^{t_1}...I_s^{t_s}, k)_{\mu}$, as a function of $(\mu ,t)\in \ZZ^{p+s}$ is a quasi-polynomial with respect to a lattice determined by the degrees of the generators of the ideals $I_1,\ldots ,I_s$.
This general finiteness statement may lead to pretty complex decompositions in general, that depends on the number of ideals and on arithmetic properties of the sets of degrees of generators. This complexity is reflected both by the covolume of $\Lambda$ as defined above and by the number of simplicial chambers in the chamber complex associated to $\varphi$.
The case of one graded ideal on a positively $\ZZ$-graded ring
--------------------------------------------------------------
In the case that the polynomial ring is multigraded and we have finitely many ideals, the results are done in the Theorem \[tor-general case\]. To make the result more clear and maybe more useful, let us explain the details of this theorem in an important and simple case when we have one ideal in a positively $\ZZ$-graded polynomial ring over a field.
We begin by an elementary lemma.
\[intersection\] For a strictly increasing sequence $d_1<\cdots <d_r$, and points of coordinates $(\beta_1^j, \beta_2^j)\in \RR^2$ for $1\leq j\leq N$, consider the half-lines $$L_i^j:= \{ (\beta_1^j, \beta_2^j)+\lambda (d_i, 1),\ \lambda\in \RR_{\geq 0} \}$$ and set $L_i^j(t):=L_i^j\cap \{ y=t\}$. Then there exist a positive integer $t_0$ and permutations $\sigma_i$, for $i=1,\cdots, r$ in the permutation group $S_N$ such that for all $t\geq t_0$, the following properties are satisfied:
\(1) $L_{i}^{\sigma_i (1)}(t) \leqslant L_{i}^{\sigma_i (2)}(t) \leqslant \dots \leqslant L_{i}^{\sigma_i (N)}(t)$ for $1 \leqslant i\leqslant r$,
\(2) $L_{i}^{\sigma_i (N)} (t)\leqslant L_{i+1}^{\sigma_{i+1} (1)}(t)$.
Moreover $t_0$ can be taken as the biggest second coordinate of the intersection points of all pairs of half-lines(See Figure \[figure:shifts\] and Figure \[figure:t-large\]).
If two half-lines $L_i^j$ and $L_u^v$ intersect at a unique point $A(x_A, y_A)$, then\
\
$$y_A=\frac{\det\left(\begin{array}{cc}\beta_1^v&d_u\\ \beta_2^v& 1\end{array}
\right)-\det\left(\begin{array}{cc}\beta_1^j& d_i\\
\beta_2^j& 1\end{array}\right)}{d_i-d_u}.$$ Choose $t_0$ as the max of $y_A$, $A$ running over the intersection points. Only for $t\in [ t_0,+\infty [$ the ordering of the intersection points $L_i^j(t)$ on the line $\{ y=t\}$ is independent of $t$. Furthermore, as the $d_i$’s are strictly increasing (2) holds, which shows (1) as the ordering is independent of $t$.\
\
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(7,4) node\[left\] – (14,7) node\[right\][$L_{3}^{1}$]{};
(0,4) node\[left\] [$n \gg 0$]{} – (14,4) node\[right\][$$]{};
Now we are ready to give a specific description of $\operatorname{Tor}^S_i(I^t, k)$ in the case of a $\ZZ$-graded ideal. Let $E:=\{ e_1,\ldots ,e_s\}$ with $e_1<\cdots <e_s$ be a set of positive integers. For $\ell$ from $1$ up to $s-1$, let $$\Omega_\ell :=\{ a{{e_\ell}\choose{1}}+b{{e_{\ell +1}}\choose{1}},\ (a,b)\in \RR_{\geq 0}^2\}$$ be the closed cone spanned by ${{e_\ell}\choose{1}}$ and ${{e_{\ell +1}}\choose{1}}$. For integers $i\neq j$, let $\Lambda_{i,j}$ be the lattice spanned by ${{e_i}\choose{1}}$ and ${{e_{j}}\choose{1}}$ and $$\Lambda_\ell :=\cap_{i\leq \ell < j}\Lambda_{i,j}.$$ Also we set $$\Lambda :=\cap_{i < j}\Lambda_{i,j} \hspace*{2mm} with \hspace*{2mm} \Delta = \det(\Lambda).$$
In the case $E:=\{ d_1,\ldots ,d_r\}$, $e_1=d_1$ and $e_s=d_r$ and if $s\geq 2$, it follows from Theorem \[vector partition\] that
- $\dim_k B_{\mu ,t}=0$ if $(\mu ,t)\not\in \varOmega:= \bigcup_\ell \Omega_\ell$,
- $\dim_k B_{\mu ,t}$ is a quasi-polynomial with respect to the lattice $\Lambda_\ell$ for $(\mu ,t)\in \Omega_\ell$.
Notice further that $\Lambda :=\cap_{i < j}\Lambda_{i,j}$ is a sublattice of $\Lambda_\ell$ for any $\ell$.
\[z-graded\]
In the above situation, if $M$ is a finitely generated graded $B$-module, there exist $t_0$, $N$ and half-lines $L_i(t):=a_i t+b_i$ for $i=1,\ldots ,N$ with $b_i\in \ZZ$ and $\{ a_1,\ldots ,a_N\}=E$ such that for $t\geq t_0$:
- $L_i (t)<L_j (t)\ \Leftrightarrow i<j$,
- $M_{\mu ,t}=0$ if $\mu<L_1 (t)$ or $\mu >L_N (t)$,
- For $t\geq t_0$ and $L_{i} (t)\leq \mu \leq L_{i+1}(t)$ $1\leq i < N $, $\dim_k (M_{\mu ,t})$ is a quasi-polynomial $Q_{i}(\mu ,t)$ with respect to the lattice $\Lambda$.
By Proposition \[Hilbert nonstandard\], there exists a polynomial $P(x,y)$ with integral coefficients such that $$H(M;(x,y)) =P(x,y)H(B;(x,y))$$ of the form $P(x,y)=\sum_{(a,b)\in A} c_{a,b}x^ay^b$ with $A\subset \cup_{\ell}\operatorname{Supp}_{\ZZ^2}(\operatorname{Tor}^R_\ell (M,k))$. Let $$A=\{(\beta_1^{1}, \beta_2^{1}), \ldots, (\beta_1^{N}, \beta_2^{N})\}.$$ Now let $L_i^{j}(t):=d_i t+b_j$ be the half-line parallel to the vector $(d_i, 1)$ and passing through the point $(\beta_1^j, \beta_2^j)$ for $1\leq i, j\leq N$. Then item(i) follows directly from Lemma \[intersection\] (i) and item(ii) from the fact that $M_{(\mu,t)}= 0$ unless $(\mu,t)\in \bigcup_{i=1}^{N} (\beta_1^{i}, \beta_2^{i})+\varOmega$.\
To prove (iii), following \[intersection\] we can consider two type of intervals as below: $$I_{i}^{j}:= [L_{i}^{\sigma_{i}(j)}(n),L_{i}^{\sigma_{i}(j+1)}(n)] \hspace*{2mm} for \hspace*{2mm} j<N$$ and $$I_{i}^{N}:= [ L_{i}^{\sigma_{i}(N)}(n),L_{i+1}^{\sigma_{i+1}(1)}(n)] \hspace*{2mm} for \hspace*{2mm} i<r$$ We write $I_{N_{p}+q}:=I_{p}^{\sigma_{p}(q)}$, $L_{N_{p}+q} = L_{p}^{\sigma_{p}(q)}$for $0 \leq q < N$. Then for any degree $(\alpha,n)$ in the support of $M$ there is two cases:
- if $\alpha \in I_{i}^{j}$, then $(\alpha,n)$ belongs to i-th chamber (i.e., chamber where $L_i^{j}(t)$ and $L_{i+1}^{j}(t)$ are its boundary.) of shifts $\{(\beta_1^{\sigma_j(1)}, \beta_2^{\sigma_j(1)}), \ldots, (\beta_1^{\sigma_i(j)}, \beta_2^{\sigma_i(j)})\}$ and for the other shifts $(\alpha, n)$ belongs to $(i-1)$-st chambers.\
- if $\alpha \in I_{i}^{N}$, then $(\alpha, n)$ belongs to $i$-th chamber for all of the shifts.
Then by Proposition \[hilbert bigraded\] there exist polynomials $Q_{ij}$ such that $\dim B_{(\mu,t)} = Q_{ij}((\mu,t))$,\
if $\mu-td_{i} \equiv j \mod(\Delta)$.\
By setting $\widetilde{Q}_{ik}^{j} = c_{(\beta_1^{j}, \beta_2^{j})}Q_{i,(k-\beta_1^{j}+\beta_2^{j}d_i)}(x-\beta_1^{j},y- \beta_2^{j})$ one can conclude that if $\alpha \in I_{i}^{j},$ then $$\dim_k(M_{\alpha,t})= \sum_{c=1}^{j} \widetilde{Q}_{i,(\alpha-td_i)}^c (\alpha,n) + \sum_{c=j+1}^{N} \widetilde{Q}_{(i-1),(\alpha-td_{i-1})}^c (\alpha,t).$$
\[main res\] Let $S=k[x_1, \ldots, x_n]$ be a positively graded polynomial ring over a field $k$ and let $I$ be a homogeneous ideal in $S$.
There exist, $t_0,m,D\in \ZZ$, linear functions $L_i(t)=a_i t+b_i$, for $i=0,\ldots ,m$, with $a_i$ among the degrees of the minimal generators of $I$ and $b_i\in \ZZ$, and polynomials $Q_{i,j}\in \QQ [x,y]$ for $i=1,\ldots ,m$ and $j\in 1,\ldots ,D$, such that, for $t\geq t_0$,
\(i) $L_i(t)<L_j(t)\ \Leftrightarrow\ i<j$,
\(ii) If $\mu <L_0(t)$ or $\mu >L_m(t)$, then $\operatorname{Tor}_i^S(I^t, k)_{\mu}=0$.
\(iii) If $L_{i-1} (t)\leq \mu \leq L_{i}(t)$ and $a_i t-\mu \equiv j\mod (D)$, then $$\dim_k\operatorname{Tor}_i^S(I^t, k)_{\mu}=Q_{i,j}(\mu ,t).$$
We know from [@BCH] that $M:=\operatorname{Tor}_i^S(I^t, k)$ is a finitely generated $\ZZ^2$-graded module over $R$. Then it follows from Proposition \[z-graded\].
Let $I=(f_1, \ldots, f_r)\subseteq S=k[x_0, \ldots, x_n]$ be a homogeneous complete intersection ideal. It is well known that minimal graded free resolution of $I$ and its power is again by koszul complex and Eagon-Northcott complex respectively, but by inspiration of above theorem we can see that Betti table of powers of $I$ (for sufficiently large powers) encoded by finite number of numerical polynomials. In the following example we give the explicit formulas in the case the ideal is generated by three forms.
Let $I=(f_1, f_2, f_3)$ be a complete intersection homogeneous ideal in the polynomial ring $S=k[x_0, \ldots, x_n]$ over a field $k$, where $\deg f_1=2$, $\deg f_2=3$ and $\deg f_3=6$. Let $R=S[T_1, T_2, T_3]$ be $\ZZ^2$-graded by setting $\deg_{\mathbb{Z}^2}(a)=(\deg a, 0)$ for any $a\in S$ and $\deg_{\mathbb{Z}^2}(T_1)=(2, 1)$, $\deg_{\mathbb{Z}^2}(T_2)=(3, 1)$ and $\deg_{\mathbb{Z}^2}(T_3)=(6, 1)$. Then a bigraded resolution of the Rees algebra of $I$, $\R_I$, over $R$ has the following form $$0\rightarrow \begin{array}{c}R(-11, -2)\\
\oplus \\ R(-11, -1)\end{array}\xrightarrow{\left(\begin{array}{lll}
T_1 &T_2 &T_3\\
f_1 & f_2 & f_3
\end{array}\right)} \begin{array}{c}R(-9, -1)\\
\oplus \\ R(-8, 1)\\
\oplus \\
R(-5, -1)\end{array}\xrightarrow{
\left(\begin{array}{l}
f_2T_3-f_3T_2\\
f_3T_1-f_1T_3\\
f_1T_2-f_2T_1
\end{array}\right)} R\rightarrow \R_I\rightarrow 0.$$ It follows that $\operatorname{Tor}_0^S(\R_I, k)=B$, $\operatorname{Tor}_2^S(\R_I, k)=B(-11, -1)$ and the minimal free $B$-resolution of $\operatorname{Tor}_1^S(\R_I, k)$ is
$$0\rightarrow B(-11, -2)\xrightarrow{
\left(\begin{array}{lll}
T_1 &T_2 &T_3\\
\end{array}\right)} \begin{array}{c}B(-9, -1)\\
\oplus \\ B(-8, 1)\\
\oplus \\
B(-5, -1)\end{array}\xrightarrow{} \operatorname{Tor}_1^S(\R_I, k) \rightarrow 0$$ where $B=k[T_1, T_2, T_3]$ is a non-standard graded polynomial ring over $k$. To compute the numerical polynomials expresing the Hilbert function of $\operatorname{Tor}$-modules of powers of $I$, we should first determine the Hilbert function of $B$. To this end, we first calculate the Hermite Normal form (HNF) of the matrix of degrees of $B$ which is $$A=\left(\begin{array}{lll}
2&3&6\\
1&1&1\end{array}\right).$$ $$H:=HNF(A)=\left(\begin{array}{lll}
1&0&0\\
0&1&0\end{array}\right)$$ and $$U=\left(\begin{array}{lll}
-1&3&3\\
1&-2&-4\\
0&0&1\end{array}\right).$$ Here $U$ is a unimodular matrix such that $H=AU$. Now $H$ gives us a transformed polytope $Q$, which is an interval in this case, such that the Hilbert function of $B$ at the point $(\mu, t)$ is equal to the number of lattice points on $Q$. The following inequalities give us the description of $Q$: $$\left\{\begin{array}{r}
(\mu-3t)-3\lambda_1\leq 0,\\
(2t-\mu)+ 4\lambda_1\leq 0,\\
-\lambda_1\leq 0.\end{array}\right.$$ From Lemma \[chamber of n vector\], one considers two chambers $C_1$ and $C_2$ in $\ZZ^2$, defined as follows.
- $(\mu, t)\in C_1, \quad$ if $\left\{\begin{array}{l}
\mu-2t \geqslant 0\\
3t - \mu \geqslant 0\end{array}\right.$
- $(\mu, t)\in C_2, \quad$ if $\left\{\begin{array}{l}
6t-\mu \geqslant 0\\
\mu-3t \geqslant 0\end{array}\right.$.
The Hilbert function of $B$ written as $$H(B, (\mu, t))=\left\{\begin{array}{ll}
[\frac{\mu-2t}{4}] + 1&(\mu, t)\in C_1,\\\\
Ž[\frac{\mu-2t}{4}] - \frac{\mu-3t}{3} +1& (\mu, t)\in C_2 \hspace{5mm} and \hspace{5mm} \frac{\mu-3t}{3} \in \ZZ,\\\\
Ž [\frac{\mu- 2t}{4}] - [\frac{\mu-3t}{3}]& (\mu, t)\in C_2 \hspace{5mm} and \hspace{5mm} \frac{\mu-3t}{3} \notin \ZZ .
\end{array}\right.$$
The function $P(\mu,t)$ is given by polynomials depending on the value of $\mu-2t$ mod $4$. $$P(\mu,t) = P_{i}(\mu,t)=\frac{\mu-2t}{4} - \frac{i}{4} + 1 \hspace{5mm} if \hspace{5mm} \mu-2t \equiv i \mod 4$$
and the function $Q(\mu,t) $ is given by polynomials depending on the values of $\mu-3t$ mod $3$ and $\mu-2t$ mod $4$.
$$Q(\mu,t)=\left\{\begin{array}{ll}
\frac{6t-\mu +4j-3i}{12} \hspace{5mm} if \hspace{5mm} \mu-2t \equiv i \hspace{1mm}, \hspace{1mm} \mu-3t \equiv j \hspace{3mm} and \hspace{3mm} \frac{\mu-3t}{3} \notin \ZZ,\\\\
\frac{6t-\mu +4j-3i}{12} + 1 \hspace{5mm} if \hspace{5mm} \mu-2t \equiv i \hspace{1mm}, \hspace{1mm} \mu-3t \equiv j \hspace{3mm} and \hspace{3mm} \frac{\mu-3t}{3} \in \ZZ.
\end{array}\right.$$ Now Hilbert function of the graded module $\operatorname{Tor}_0^S(\R_I, k)$ can be written as
$$\beta_{0\mu}^{t}=\left\{\begin{array}{ll}
P(\mu,t)&2t\leqslant \mu \leqslant 3t,\\\\
Q(\mu,t)&3t< \mu \leqslant 6t\\\\
0 & Otherwise.
\end{array}\right.$$ For the Hilbert function of the graded module $\operatorname{Tor}_1^S(\R_I, k)$ we need to write new polynomials (see Figure \[figure:Tor\_1\]):\
$$P_1 = P(\mu-5,t-1), \hspace{2mm} P_2 = P(\mu-8,t-1), \hspace{2mm} P_3 = P(\mu-9,t-1), \hspace{2mm} P_4 = P(\mu-11,t-2), \hspace{2mm} P_5 = P(\mu-11,t-1).$$ $$Q_1 = Q(\mu-5,t-1), \hspace{2mm} Q_2 = Q(\mu-8,t-1), \hspace{2mm} Q_3 = Q(\mu-9,t-1), \hspace{2mm} Q_4 = Q(\mu-11,t-2), \hspace{2mm} Q_5 = Q(\mu-11,t-1).$$
$$\beta_{1\mu}^{t}=\left\{\begin{array}{ll}
P_1(\mu,t)&2t+3\leqslant \mu < 2t+6,\\\\
(P_1 +P_2)(\mu,t)&2t+6\leqslant \mu < 2t+7,\\\\
(P_1 +P_2 + P_3 - P_4)(\mu,t)&2t+7\leqslant \mu < 3t+2,\\\\
(Q_1 +P_2 + P_3 - P_4)(\mu,t)&3t+2\leqslant \mu < 3t+5,\\\\
(Q_1 +Q_2 + P_3 - P_4)(\mu,t)&3t+5\leqslant \mu < 3t+6,\\\\
(Q_1 +Q_2 + Q_3 - Q_4)(\mu,t)&3t+6\leqslant \mu < 6t-1,\\\\
(Q_2 +Q_2)(\mu,t)&6t-1\leqslant \mu < 6t+2,\\\\
Q_3(\mu,t)&6t+2< \mu \leqslant 6t+3,\\\\
0 & Otherwise.
\end{array}\right.$$
![Regions and corresponding polynomials of $\operatorname{Tor}_1^S(\R_I, k)$.[]{data-label="figure:Tor_1"}](Pic_Tor1.pdf){width="140mm"}
Finally the Hilbert function of graded module $\operatorname{Tor}_2^S(\R_I, k)$ can be written as
$$\beta_{2\mu}^{t}=\left\{\begin{array}{ll}
P_5(\mu,t)&2t+9\leqslant \mu < 3t+8,\\\\
Q_5(\mu,t)&3t+8\leqslant \mu \leqslant 6t+5,\\\\
0 & Otherwise.
\end{array}\right.$$
This example shows that, even in a very simple situation, quite many polynomials are involved to give a full discription of the Betti tables of powers of ideals.
Acknowledgement
===============
This work was done as a part of the second author’s Ph.D. thesis. The authors gratefully acknowledge the support and help of Marc Chardin without whose knowledge and assistance, the present study could not have been completed. Special thanks are due to Michele Vergne, Siamak Yassemi and Jean-Michel Kantor for their very useful mathematical discussions and valuable comments regarding this work.
[100]{} T. V. Alekseevskaya, I. M. Gel’fand and A. V. Zelevinskii. [*An arrangement of real hyperplanes and the partition function connected with it*]{}. Soviet Math. Dokl. [*3*6]{} (1988), 589-593. K. Baclawski and A.M. Garsia. [*Combinatorial decompositions of a class of rings*]{}. Adv. in Math. **39** (1981), 155-184. A. Barvinok. [*A course in convexity*]{}, American Mathematical Society. 2002, 366 pages. A. Bagheri, M. Chardin and H.T. Hà. [*The eventual shape of the Betti tables of powers of ideals*]{}. To appear in Math. Research Letters. D. Berlekamp. [*Regularity defect stabilization of powers of an ideal*]{}. Math. Res. Lett. 19 (2012), no. [**1**]{}, 109-119. L. J. Billera, I. M. Gelfand and B. Sturmfels . [*Duality and minors of secondary polyhedra*]{}. J. Combin. Theory Ser. B 57 (1993), no. 2, 258Ð268. S. Blakley. [*Combinatorial remarks on partitions of a multipartite number*]{}. Duke Math. J. [**31**]{}(1964), 335-340. M. Brion and M. Vergne. [*Residue formulae, vector partition functions and lattice points in rational polytopes.*]{} J. Amer. Math. Soc.. [**10**]{}(1997), 797-833. W. Bruns, C. Krattenthaler, and J. Uliczka. [*Stanley decompositions and Hilbert depth in the Koszul complex.*]{} J. Commutative Algebra, **2** (2010), no. 3, 327-357. W. Bruns and J. Herzog, [*Cohen-Macaulay rings*]{}. Cambridge Studies in Advanced Mathematics, **39**. Cambridge University Press, Cambridge, 1993. M. Chardin. [*Powers of ideals and the cohomology of stalks and fibers of morphisms*]{}. Preprint. [arXiv:1009.1271]{}. S.D. Cutkosky, J. Herzog and N.V. Trung. [*Asymptotic behaviour of the Castelnuovo-Mumford regularity.*]{} Composito Mathematica, **118** (1999), 243-261. D. Loera, Jesœs A.; Rambau, Jšrg; Santos, Francisco. [*Triangulations Structures for algorithms and applications. Algorithms and Computation in Mathematics*]{}, 25. Springer-Verlag, Berlin, 2010. D. Loera, Jesœs A. [*The many aspects of counting lattice points in polytopes*]{}. Math. Semesterber. 52 (2005), no. 2, 175Ð195. D. Eisenbud. [*Commutative Algebra: with a View Toward Algebraic Geometry*]{}. Springer-Verlag, New York, 1995. D. Eisenbud and J. Harris. [*Powers of ideals and fibers of morphisms*]{}. Math. Res. Lett. **17** (2010), no. 2, 267-273. D. Eisenbud and B. Ulrich. [*Stabilization of the regularity of powers of an ideal*]{}. Preprint. [arXiv:1012.0951]{}. V. Kodiyalam. [*Homological invariants of powers of an ideal.*]{} Proceedings of the American Mathematical Society, **118**, no. 3, (1993), 757-764. V. Kodiyalam. [*Asymptotic behaviour of Castelnuovo-Mumford regularity.*]{} Proceedings of the American Mathematical Society, **128**, no. 2, (1999), 407-411. E. Miller and B. Sturmfels. [*Combinatorial commutative algebra*]{}. Graduate Texts in Mathematics, 227. Springer-Verlag, New York, 2005 D. G. Northcott and D. Rees. [*Reductions of ideals in local rings*]{}. Proc. Cambridge Philos. Soc. [*5*0]{} (1954). 145-158. P. Singla. [*Onvex-geometric, homological and combinatorial properties of graded ideals*]{}. genehmigte Dissertation, Universit" at Duisburg-Essen, December 2007. R. Stanley. [*Combinatorics and commutative algebra*]{}. Birkhäuser, Boston, 1983. R. Stanley. [*Enumerative Combinatorics*]{}. Vol 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997, With a foreward by Gian-Carlo Rota, Corrected reprint of the 1986 original. B. Sturmfels. [*On vector partition functions*]{}. J. Combinatorial Theory, Series A **72**(1995), 302-309. G. Whieldon. [*Stabilization of Betti tables*]{}. Preprint. [arXiv:1106.2355]{}. G.M. Ziegler.[*Lectures on Polytopes*]{}. Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. Z. Xu, [*An explicit formulation for two dimensional vector partition functions, Integer points in polyhedra-geometry, number theory, representation theory, algebra, optimization, statistics*]{}. Contemporary Mathematics, 452[(**2008)**]{}, 163-178.\
\
\
\
\
[2]{} Amir BAGHERI\
School of Mathematics, Institute for\
Research in Fundamental Sciences(IPM),\
P.O. Box:19395-5746, Tehran, Iran.\
Email: abagheri@ipm.ir\
Marand technical college, University of\
Tabriz, Tabriz, Iran\
Email : a\_bageri@tabrizu.ac.ir
[Kamran LAMEI\
Institut de Mathématiques de Jussieu\
UPMC,Boite thésard,\
4,place Jussieu,F-75252 Paris Cedex,\
France\
Email : kamran.lamei@imj-prg.fr]{}
[^1]: The research of Amir Bagheri was in part supported by a grant from IPM (No. 93130024).
|
---
abstract: 'Variations in the pulsation arrival time of five independent pulsation frequencies of the DB white dwarf EC 20058-5234 individually imitate the effects of reflex motion induced by a planet or companion but are inconsistent when considered in unison. The pulsation frequencies vary periodically in a $12.9$ year cycle and undergo secular changes that are inconsistent with simple neutrino plus photon cooling models. The magnitude of the periodic and secular variations increase with the period of the pulsations, possibly hinting that the corresponding physical mechanism is located near the surface of the star. The phase of the periodic variations appears coupled to the sign of the secular variations. The standards for pulsation timing based detection of planetary companions around pulsating white dwarfs, and possibly other variables such as subdwarf B stars, should be reevaluated. The physical mechanism responsible for this surprising result may involve a redistribution of angular momentum or a magnetic cycle. Additionally, variations in a supposed combination frequency are shown to match the sum of the variations of the parent frequencies to remarkable precision, an expected but unprecedented confirmation of theoretical predictions.'
author:
- 'J. Dalessio, D. J. Sullivan, J. L. Provencal, H. L. Shipman, T. Sullivan, D. Kilkenny, L. Fraga, and R. Sefako'
bibliography:
- 'bibliography.bib'
title: 'Periodic Variations in the $O-C$ Diagrams of 5 Pulsation Frequencies of the DB White Dwarf EC 20058-5234 [^1]'
---
Introduction and Formalism {#ocsect}
==========================
The orbital motion of a planet hosting star about the system’s mass center, often referred to as a “wobble”, will cause variations in the time it takes for light from the star to reach external observers. If the star provides some predictable repetitive behavior, like regular pulsations, the wobble will cause systematic differences between the time the behavior is predicted to be observed and when it is actually observed. This phenomenon has resulted in the discovery of the first known exoplanet and the rest of the pulsar planets [@1992Natur.355..145W; @1993ApJ...412L..33T], confirmation and detection of a growing number of planets with Kepler transit timing variations (see @2010Sci...330...51H and subsequent Kepler publications), a planet around a sub-dwarf B star [@2007Natur.449..189S] as well as strong detection limits on several pulsating white dwarfs [@2008ApJ...676..573M; @2010AIPC.1273..446H] and one tentative white dwarf planet candidate [@2009ApJ...694..327M]. However, changes to the actual frequency of the repetitive behavior will also manifest as differences between the expected and actual time the behavior is observed. To illustrate this, imagine observations of two distant ticking clocks. Clock one keeps poor time. Everyday the period of the ticks changes in such a way that clock one appears a second slow at noon and a second fast at midnight. Clock two keeps perfect time but is moving in a light second orbit in the plane of the observer with a period of a day. While these clocks are fundamentally very different, the two clocks are indistinguishable in terms of timing alone. It has been asserted via lex parsimoniae (Occum’s Razor), specifically for white dwarf pulsators, that any sinusoidal or “planet like” variations in the timing of the pulsations are most likely due to orbital motion as there is no known mechanism that would conspire to make the actual pulsation frequency vary in such a manner. This paper presents conclusive evidence to the contrary: frequency variations in a white dwarf pulsator that can mimic the effect of a planetary companion.
The dominant cooling mechanism for the hot pulsating DB white dwarfs is thought to be the production of plasmon neutrinos. The frequency of the pulsations are determined in part by the temperature, so cooling should cause a slow secular change in frequency. This effect has been well studied with the far cooler, photon cooling dominated DA white dwarf G117-B15A [@2005ApJ...634.1311K]. If the rate of frequency change of a hot pulsating DB white dwarf was measured to sufficient precision, the neutrino production rate could be extracted and used to constrain the Standard Model [@2004ApJ...602L.109W]. This paper presents evidence that there are other processes that cause large secular changes to multiple pulsation frequencies in a white dwarf.
The diagnostic tool used to measure variations in pulsation arrival time and pulsation frequency is called the $O-C$, or “observed minus calculated” diagram. Depending on the implementation, “observed minus calculated” can be a misnomer. In many cases it is a plot of successive measurements of the absolute phase of a periodic variation assuming constant frequency. The ordinate and abscissa of the $O-C$ diagram are both expressed in units of time, but are sometimes normalized by the typical period of the behavior into units of cycles (often labeled “epochs” on the abscissa), radians, or degrees. The $O-C$ diagram can reveal frequency changes that are difficult to detect by direct measurement. In addition to the detection of planets around pulsars and other pulsating stars, it was the primary diagnostic tool that helped earn Hulse and Taylor earn the Nobel Prize in Physics for their indirect detection of gravitational wave emission [@1979Natur.277..437T]. The $O-C$ diagram remains a critical tool across time domain astronomy, see e.g. [@2012MNRAS.419..959O; @2011MNRAS.414.3434B; @2008ApJ...676..573M; @2012ApJ...757L..21H].
For pulsating white dwarf stars and other short period variables it is convenient to describe $O-C$ as a time varying absolute phase. Given some fixed observed frequency, $f_{obs}$, the instantaneous amplitude of a pulsation can be rewritten in terms of the time varying absolute phase, $\tau(t)$ $$\label{e2}
H(t) = A \sin (2 \pi f_{obs}(t - \tau(t)) )$$
$\tau(t)$ can be mathematically described as (for a complete derivation see @Dalessio)
$$\boxed{\tau(t) = t_0 - \frac{\delta f_{obs}}{f_{obs}} t - \frac{1}{f_{obs}} \int^t \gamma(t^{\prime}) d t^{\prime} + \frac{q(t)}{c} } \label{main}$$
where $t_0$ is an arbitrary offset depending on choice of $t=0$, $f_{obs}$ is the observed frequency, $\delta f_{obs}$ is the difference between the observed and actual, time averaged, Doppler shifted frequency, $\gamma(t)$ is the perturbation to the frequency, and $q(t)$ is the perturbation to the distance between the object and observer. Note that $\gamma(0)=q(0)=0$.
Except in the case of continuous observation where the number of elapsed cycles is explicitly known, all points in the $O-C$ diagram suffer from the ambiguity that they can be shifted up or down by any integer times the pulsation period. A common assumption is that $O-C$ has been sampled often enough that $\gamma(t)$ and $q(t)$ change slowly and smoothly during the observational gaps. Under this assumption, the location of the points on an $O-C$ diagram can be constrained if there is a single unambiguous configuration that creates a smooth, continuous trend.
Observations
============
EC 20058-5234 [@1995MNRAS.277..913K] has been identified as a prime candidate for measurement of the neutrino production rate in a white dwarf (see Section \[ocsect\]). This motivated many of the over $250,000$ PMT (photo-multiplier tube) and CCD (charge-coupled device) photometric measurements spanning 1994-2011. EC 20058-5234 was a primary target of the Whole Earth Telescope campaign XCOV15 in 1997 [@2008MNRAS.387..137S], followed by seasonal high speed photometry from the 1m McLellan telescope at Mt. John Observatory and the 36 inch SMARTS telescope at CTIO (see e.g. @2005ASPC..334..495S). It was also a tertiary target during XCOV27 and was observed at Magellan in 2003 [@2007ASPC..372..629S]. A summary of the observations is shown in Table \[obs\_table\].
[|cccc|]{}
\
& & &\
\
& & &\
\
SAAO & 1994/05/14 - 1994/06/02 & PMT & 5535\
SAAO & 1994/07/06 - 1994/07/12 & PMT & 6881\
SAAO & 1994/10/03 - 1994/10/03 & PMT & 1304\
SAAO & 1994/06/12 - 1994/06/12 & PMT & 1378\
XCOV15& 1997/07/02 - 1997/07/11 & PMT & 46289\
Mt. John & 1997/10/05 - 1997/10/06 & PMT & 3162\
Mt. John & 1998/07/24 - 1998/07/25 & PMT & 7057\
Mt. John & 1998/08/14 - 1998/08/14 & CCD & 1153\
Mt. John & 1999/09/09 - 1999/09/13 & PMT & 1517\
Mt. John & 2000/07/06 - 2000/07/08 & PMT & 8025\
Mt. John & 2000/09/05 - 2000/09/05 & PMT & 770\
Mt. John & 2001/03/30 - 2001/04/01 & PMT & 2101\
Mt. John & 2001/09/21 - 2001/09/24 & PMT & 7710\
Mt. John & 2002/04/12 - 2002/04/15 & PMT & 8872\
Mt. John & 2002/08/01 - 2002/08/06 & PMT & 12672\
Mt. John & 2002/09/05 - 2002/09/09 & PMT & 3814\
Magellan & 2003/07/11 - 2003/07/13 & CCD & 2317\
Magellan & 2003/07/25 - 2003/07/31& CCD & 8865\
Mt. John & 2003/08/27 - 2003/09/02 & PMT & 12216\
Mt. John & 2003/09/22 - 2003/09/23 & PMT & 4694\
Mt. John & 2003/10/31 - 2003/10/31 & PMT & 1229\
Mt. John & 2004/04/23 - 2004/04/25 & PMT & 2474\
Mt. John & 2004/05/16 - 2004/05/16 & PMT & 1599\
Mt. John & 2004/06/09 - 2004/06/16 & PMT & 14554\
Mt. John & 2004/07/09 - 2004/07/14 & PMT & 17240\
Mt. John & 2004/08/07 - 2004/08/10 & PMT & 5072\
CTIO SMARTS & 2004/08/24 - 2004/08/28 & CCD & 2320\
CTIO SMARTS & 2005/09/21 - 2005/09/25& CCD & 979\
CTIO SMARTS & 2006/08/31 - 2006/09/13& CCD & 5019\
CTIO SMARTS & 2007/08/17 - 2007/09/06& CCD & 14117\
Mt. John & 2007/07/16 - 2007/07/17& PMT & 6281\
CTIO SMARTS & 2008/08/29 - 2008/09/01 & CCD & 1736\
Mt. John & 2008/08/30 - 2008/08/31 & CCD & 3092\
XCOV27[^2]& 2009/05/18 - 2009/05/26 & CCD & 4609\
Mt. John & 2010/07/05 - 2010/07/11 & CCD & 20201\
CTIO SMARTS & 2011/09/08 - 2011/09/15 & CCD & 3584\
Results and Analysis
====================
Data Reduction and Preparation
------------------------------
The PMT photometry obtained at SAAO in 1994 was reduced as described in [@1995MNRAS.277..913K], while the 1997 XCOV15 photometry and the Mt. John PMT photometry was reduced using the methods outlined in [@2008MNRAS.387..137S]. All the Mt. John PMT photometry collected between July 1998 and July 2007 employed a three channel photometer [@2000BaltA...9..425S] attached to the one meter McLellan telescope. All CCD image calibration and aperture photometry was performed with Maestro [@2010AAS...21545209D]. WQED [@2009JPhCS.172a2081T] was used to remove spurious points and to perform the barycentric correction (for a review see @2010PASP..122..935E) for the CCD data. The barycentric correction applied to the PMT data were performed using the same implementation as used in WQED . The photometric measurements were divided into 36 individual lightcurves, one for each observing campaign in Table \[obs\_table\]. The following analysis was also performed by dividing the points into lightcurves for each individual night and observing season. The differences were not significant and do not merit further discussion.
Testing for Stability
---------------------
The frequency and amplitude of all pulsation frequencies with amplitudes above $0.1\%$ were extracted from two of the largest data sets, XCOV15 in 1997 and the 2007 CTIO SMARTS observations. The results are summarized in Table \[stable\_table\]. The same spectrum of frequencies appears in both years (see the Fourier spectrum from @2008MNRAS.387..137S), with the exception that pulsation frequencies F, J, and L have amplitudes below the noise threshold in 2007 and that pulsation frequency E appears to have changed at a statistically significant level. The systematically higher pulsation amplitudes in 2007 are an observational effect due to two companions within several arc seconds of EC 20058-5234 (see the Figure in @1995MNRAS.277..913K). The PMT observations included all three objects as it was impractical in the typically $2\arcsec$ seeing conditions at Mt. John to separate flux from the target only. However the CCD data, especially when combined with the better seeing conditions at CTIO and the Magellan site, permitted extraction of just the flux from EC 20058-5234 using synthetic aperture techniques. This effect makes it non-trivial to search for amplitude modulation, which, particularly for RR Lyrae stars, is known to be coupled to phase modulation (the Blazhko effect, see e.g. @2011ApJ...731...24B). However, if the pulsation amplitudes are intrinsically constant, the ratio of amplitudes from one set to another should be constant. The ratio of the pulsation amplitudes from 2007 to the pulsation amplitudes in 1997 is shown in the last column of Table \[stable\_table\]. The ratio is consistent with there being no amplitude variation with a reduced $\chi^2$ value of $2.3$ with a statistical likelihood around $10\%$. This is the worst reduced $\chi^2$ value for the model of constant amplitude between any two of the data sets in Table \[obs\_table\]. This value of reduced $\chi^2$ is slightly higher than expected for a constant amplitude. The removal of atmospheric extinction and other observational effects could introduce small changes in observed amplitude so we do not find this slight disagreement particularly alarming. Even if the amplitude of these pulsations frequencies are intrinsically changing, the changes are relatively small and only barely distinguishable between our two best data sets.
[|c|c|ll|ll|l|]{} Label & Period (s) & &\
& (approx) & & & & &\
A & 134 & 7452.18(.05) & 7452.27(.02) & 1.67 & 2.2 & 1.3(.1)\
B & 195 & 5128.53(.03) & 5128.71(.02) & 2.51 & 3.3 & 1.31(.09)\
C & 204 & 4902.14(.05) & 4902.14(.02) & 1.47 & 2.6 & 1.8(.2)\
D & 205 & 4887.84(.03) & 4887.91(.02) & 2.62 & 3.2 & 1.22(.09)\
E & 257 & 3893.142(.009) & 3893.239(.005) & 8.37 & 11.3 & 1.35(.03)\
F & 275 & 3640.37(.07) & & 1.13 & &\
G & 281 & 3559.032(.009) & 3559.033(.005) & 8.45 & 10.5 & 1.24(.03)\
H & 287 & 3489.07(.05) & 3489.05(.02) & 1.58 & 2.2 & 1.4(.1)\
I & 333 & 2998.67(.03) & 2998.71(.01) & 3.01 & 4.3 & 1.43(.08)\
i J & 350 & 2852.36(.09) & & 1.40 & &\
K & 525 & 1903.61(.04) & 1903.49(.02) & 1.83 & 2.44 & 1.3(.1)\
L & 540 & 1852.53(.04) & & 1.86 & &\
Extracting Precise Frequencies
------------------------------
Precise measurements of the pulsation frequencies were made using the “bootstrapping” procedure. The process is described as follows. The frequencies above the noise threshold in the July 2004 dataset were extracted. A sum of sinusoids (one sinusoid at each frequency) were fit to the July 2004 dataset using nonlinear least squares and allowing each frequency to converge to the optimum value. After the fit converged, additional data were added to the set, ensuring that each of the frequencies had been measured to sufficient precision so that if the frequency had remained constant, there would be no ambiguity in the number of cycles between any two data points in the combined set. This ensures that the linear term of equation \[main\] will not dominate the $O-C$ diagram and that the frequencies are clearly separated from the aliases of the combined set. After the new data were added, the frequencies were re-fitted. This process was repeated until most of the 2003 and 2004 data were combined into a single set. The amplitudes of pulsation frequencies F and L were well below the noise threshold and were not considered. The combined set of all data between September 2003 and August 2004 gave precise enough frequency measurements to merge the entire data set. However, merging the September 2003 - August 2004 data set with data from August 2003 or 2005 decreased the precision of our frequency measurements. This is an indication that the phase/frequency of the pulsation frequencies are changing by a substantial amount over that time span. Further combination was abandoned beyond the September 2003 - August 2004 pulsation frequency measurements. The results are summarized in Table \[bootstrap\_table\].
[|c|c|cc|]{} Label & Period \[s\] & Frequency \[uhz\] & $\frac{1}{\sigma_f}$ \[yrs\]\
A & 134 & 7452.249(.001) & 32\
B & 195 & 5128.598(.001) & 32\
C & 204 & 4902.176(.0009) & 35\
D & 205 & 4887.8491(.0009) & 35\
E & 257 & 3893.2482(.0002) & 160\
G & 281 & 3559.0017(.0003) & 106\
H & 287 & 3489.056(.001) & 29\
I & 333 & 2998.7144(.0007) & 45\
J & 350 & 2852.439(.002) & 19\
K & 525 & 1903.464(.001) & 24\
Building the $O-C$ Diagrams
---------------------------
A sum of sinusoids at the 10 frequencies in Table \[bootstrap\_table\] were linearly fit to each data set. The calculated absolute phase was plotted as $O-C$ for each pulsation frequency. The $O-C$ diagram of all ten pulsation frequencies showed structure that was not consistent with scatter. It was also apparent that for some frequencies, $O-C$ was changing by a large fraction of the period, and that *none* of the $O-C$ diagrams were consistent with the parabolic trend of a simple neutrino plus photon cooling model [@2008CoAst.154...16B].
It was assumed (see Section \[ocsect\]) that for each pulsation frequency, $\gamma(t)$ and $q(t)$ change slowly and smoothly during the observational gaps. An unambiguous smooth trend in the $O-C$ diagrams of pulsation frequencies A, D, E, G, H, and I was clearly discernible and the $O-C$ measurements were fixed to an appropriate location on the $O-C$ diagram. Due to the large observational gap between 1994 and 1997, the measurements from 1994 remained unconstrained. The appropriate location in the $O-C$ diagram of the 1994 $O-C$ measurements is further addressed in section \[model\_section\]. No unambiguous trend was apparent in the $O-C$ diagrams of pulsation frequencies C, B, J, and K and they were removed from analysis. The $O-C$ diagrams of pulsation frequencies A, D, E, G, H, and I are shown in Figures \[ocA\]-\[ocI\].
![$O-C$ of pulsation frequency A. The dashed lines are located at $0$, $P$, and $-P$. The solid line is the best model fit for $\Pi=12.9 yrs$. The dotted lines indicate the boundaries of the one sigma likelihood prediction of the model not including the 1994 data. Note that a first order polynomial has been removed from the data.[]{data-label="ocA"}](Abw.eps){width="1\columnwidth"}
![$O-C$ of pulsation frequency D. The dashed lines are located at $0$, $P$, and $-P$. The solid line is the best model fit for $\Pi=12.9 yrs$. The dotted lines indicate the boundaries of the one sigma likelihood prediction of the model not including the 1994 data. Note that a first order polynomial has been removed from the data.[]{data-label="ocD"}](Dbw.eps){width="1\columnwidth"}
{width="1\columnwidth"}
![$O-C$ of pulsation frequency G. The dashed lines are located at $0$, $P$, and $-P$. The solid line is the best model fit for $\Pi=12.9 yrs$. The dotted lines indicate the boundaries of the one sigma likelihood prediction of the model not including the 1994 data. Note that a first order polynomial has been removed from the data.[]{data-label="ocG"}](Gbw.eps){width="1\columnwidth"}
![$O-C$ of pulsation frequency H. The dashed lines are located at $0$, $P$, and $-P$. The solid line is the best model fit for $\Pi=12.9 yrs$. The dotted lines indicate the boundaries of the one sigma likelihood prediction of the model not including the 1994 data. Note that a first order polynomial has been removed from the data. []{data-label="ocH"}](Hbw.eps){width="1\columnwidth"}
![$O-C$ of pulsation frequency I. The dashed lines are located at $0$, $P$, $-P$, and $2P$. The solid line is the best model fit for $\Pi=12.9 yrs$. The dotted lines indicate the boundaries of the one sigma likelihood prediction of the model not including the 1994 data. Note that a first order polynomial has been removed from the data.[]{data-label="ocI"}](Ibw.eps){width="1\columnwidth"}
![$O-C$ of pulsation frequency A. The dashed lines are located at $0$, $P$, and $-P$. The blue line is the best model fit for $\Pi=12.9 yrs$. The red lines indicate the boundaries of the one sigma likelihood prediction of the model not including the 1994 data. Note that a first order polynomial has been removed from the data.[]{data-label="ocA"}](A.eps){width="1\columnwidth"}
![$O-C$ of pulsation frequency D. The dashed lines are located at $0$, $P$, and $-P$. The blue line is the best model fit for $\Pi=12.9 yrs$. The red lines indicate the boundaries of the one sigma likelihood prediction of the model not including the 1994 data. Note that a first order polynomial has been removed from the data.[]{data-label="ocD"}](D.eps){width="1\columnwidth"}
![$O-C$ of pulsation frequency E. The dashed lines are located at $0$, $P$, and $-P$. The blue line is the best model fit for $\Pi=12.9 yrs$. The red lines indicate the boundaries of the one sigma likelihood prediction of the model not including the 1994 data. Note that a first order polynomial has been removed from the data.[]{data-label="ocE"}](E.eps){width="1\columnwidth"}
![$O-C$ of pulsation frequency G. The dashed lines are located at $0$, $P$, and $-P$. The blue line is the best model fit for $\Pi=12.9 yrs$. The red lines indicate the boundaries of the one sigma likelihood prediction of the model not including the 1994 data. Note that a first order polynomial has been removed from the data.[]{data-label="ocG"}](G.eps){width="1\columnwidth"}
![$O-C$ of pulsation frequency H. The dashed lines are located at $0$, $P$, and $-P$. The blue line is the best model fit for $\Pi=12.9 yrs$. The red lines indicate the boundaries of the one sigma likelihood prediction of the model not including the 1994 data. Note that a first order polynomial has been removed from the data.[]{data-label="ocH"}](H.eps){width="1\columnwidth"}
![$O-C$ of pulsation frequency I. The dashed lines are located at $0$, $P$, $-P$, and $2P$. The blue line is the best model fit for $\Pi=12.9 yrs$. The red lines indicate the boundaries of the one sigma likelihood prediction of the model not including the 1994 data. Note that a first order polynomial has been removed from the data.[]{data-label="ocI"}](I.eps){width="1\columnwidth"}
Modeling the $O-C$ Variations {#model_section}
-----------------------------
All six $O-C$ diagrams (Figures \[ocA\]-\[ocI\]) show some sort of oscillatory behavior. A sinusoid plus an offset, linear, and parabolic term were chosen as a suitable model. The model is mathematically described as
$$\label{model}
\tau(t) = t_0 - \frac{\delta f_{obs}}{f_{obs}} t + \frac{\dot{P}}{2 P_{obs}} t^2 +\frac{\alpha \Pi}{2\pi f_{obs}} \sin \big(\frac{2 \pi}{\Pi}t-\phi \big)$$
Instead of fitting this nonlinear model with some initial guesses to $t_0$, $\delta f_{obs}$, the rate of period change $\dot{P}$, the amplitude of the sinusoidal frequency variation $\alpha$, the period of the sinusoidal frequency variation $\Pi$, and the absolute phase of the sinusoidal frequency variation $\phi$, a grid of linear fits can be calculated for a range of $\Pi$. By removing any dependence on initial guesses for the parameters, it is guaranteed that the absolute minimum in $\chi^2$ over the chosen range of $\Pi$ will be found. A Monte Carlo simulation was performed to calculate the statistical likelihood of the possible locations of the 1994 measurements given the model and the 1997-2012 measurements. The one sigma likelihood boundaries are indicated in Figures \[ocA\]-\[ocI\]. It was concluded that for all frequencies there was only one reasonable choice for the possible values of the 1994 $O-C$ measurements and that all ambiguity could be considered resolved for the given model. The 1994 $O-C$ measurements were then fixed to these values in each $O-C$ diagram. Fits to Equation \[model\] were then calculated over a range of $\Pi$ for each of the $O-C$ diagrams. A plot of reduced $\chi^2$ for the fit of each of the six $O-C$ diagrams as a function of $\Pi$ is shown in Figure \[pfit\]. Four of the six $O-C$ diagrams have a strong periodicity at around 13 years while the other two are not particularly sensitive to the choice of period, but are consistent with a periodicity of 13 years. The minima of the total reduced $\chi^2$ for all six $O-C$ diagrams is at $\Pi = 12.9$ years. This value of $\Pi$ was used to fit each $O-C$ diagram. The best fits are overlayed in Figures \[ocA\] - \[ocI\] and Table \[fit\_table\] summarizes the resultant parameters. The large value of reduced $\chi^2$ for some of the fits is disturbing, but the ability of the model to reproduce the *overall* trend in the data is excellent. To further investigate, $\Pi$ was then allowed to vary for each individual $O-C$. This lead to a worse overall reduced $\chi^2$, an indicator that the oscillatory behavior in each $O-C$ is represented appropriately by the same $\Pi$. Fitting the data with high order polynomials did not result in a dramatic improvement in reduced $\chi^2$. An 8th order polynomial, for example, actually increased reduced $\chi^2$ for two of the $O-C$ diagrams and only led to an overall improvement of $7\%$ over Equation \[model\]. Equation \[model\] is dramatically more constrained than a polynomial with a similar number of degrees of freedom so this strongly supports the ability of Equation \[model\] to represent the overall trends in the data. The high values of reduced $\chi^2$ mean that either the error bars in the data are somehow underestimated or there are high frequency variations in $O-C$ that are undersampled. To investigate, the Fourier transforms of the averaged residuals and absolute averaged residuals (see Figure \[res\]) were calculated. There was no statistically significant or notable power at any frequency, although the residuals appear correlated. It is concluded that there are likely unresolved variations in phase, but these variations do not dilute the effectiveness of Equation \[model\] in modeling the overall variations.
![Reduced $\chi^2$ as a function of $\Pi$ (see Equation \[model\]) for each of the coherent $O-C$ diagrams. The average minimum is at $12.9$ years. []{data-label="pfit"}](Xbw.eps){width="1\columnwidth"}
![Reduced $\chi^2$ as a function of $\Pi$ (see Equation \[model\]) for each of the coherent $O-C$ diagrams. The average minimum is at $12.9$ years. []{data-label="pfit"}](X.eps){width="1\columnwidth"}
[|c|c|cccc|]{} Label & Period \[s\] & $\dot{P}$ \[$10^{-14}$\] & $\alpha$ \[$10^{-9}\frac{1}{s}$\] & $\Phi$ \[Deg\] & $\chi^2_{Red}$\
A & 134 & -28(1) & 4.0(.1) & 160(2) & 3.3\
D & 205 & -1(2) & 1.33(.09) & 187(4) & 6.6\
E & 257 & 13.0(.8) & 2.48(.02) & 0(1)& 14\
G & 281 & -131(1) & 6.44(.03)& 166.6(.2) & 26\
H & 287 & 88(6) & 6.8(.1) & 1(1) & 2.8\
I & 333 & 313(4) & 7.31(.07) & -1(1) & 13\
![Residuals of the fit of Equation \[model\] to the $O-C$ diagrams of D,E,G,H and I. The open circles are the weighted average residuals for each data set while the filled circles are the weighted average of the absolute value of the residuals.[]{data-label="res"}](Rbw.eps){width="1\columnwidth"}
![Residuals of the fit of Equation \[model\] to the $O-C$ diagrams of D,E,G,H and I. The black points are the weighted average residuals for each data set while the red points are the weighted average of the absolute value of the residuals.[]{data-label="res"}](R.eps){width="1\columnwidth"}
Discussion
==========
Planetary Detection with the Pulsation Timing Method
----------------------------------------------------
The sinusoidal component of the variations shown in Figures \[ocA\]-\[ocI\] do not share the same amplitude and phase, so are clearly not due to the effects of a planetary companion. Had only one of the pulsation frequencies in this star been analyzed, this paper could very well be announcing the first detection of a planet around a white dwarf. The discovery that multiple $O-C$ diagrams can show similar periodic behavior when not in the presence of a planetary companion is alarming. Pulsation timing based white dwarf planet detection should now require supplemental confirmation, even when similar pulsation timing variations are observed in multiple pulsation frequencies. Until the physical mechanisms behind these variations are identified as specific to white dwarfs, these observations cast at least some shadow of doubt on the effectiveness of pulsation timing to *reliably* detect planetary companions to subdwarf B pulsators, and raise at least some skepticism as to the existence the subdwarf B planet V391 Pegasi b. We note that this does not affect the ability of the pulsation timing method to *exclude* the existence of planets around white dwarfs and other variables. The lack of known white dwarf planets is gaining statistical significance and further timing observations of pulsating white dwarfs could place strong constraints on post main sequence stellar evolution.
The Combination Frequency A {#comb}
---------------------------
[@2008MNRAS.387..137S] identified the frequency of A to be an additive combination of the frequencies of E and G, a well known occurrence in white dwarf pulsators (see e.g. @2009ApJ...693..564P). Any perturbation to the frequency of E or G should result in an identical perturbation to the frequency of A. The frequency perturbations found by fitting Equation \[model\] to the $O-C$ diagram of E, G, and A can be compared, see Table \[AEG\]. The agreement is outstanding. This is strong evidence that Equation \[model\] is effectively modeling the the frequency perturbations and that the parabolic and sinusoidal components of the perturbation are linearly independent. It is also the most conclusive evidence that combination frequencies are indeed exactly equal to the additive frequency of the parents as predicted by [@2001MNRAS.323..248W].
[|c|c|c|]{} Parameter & A & E + G\
$\dot{f} = -\frac{\dot{P}}{P^2}$ \[$\frac{10^{-18}}{s^2}$\] & 14.6(.2) & 15.3(.5)\
$\alpha$ \[$\frac{10^{-9}}{s}$\] & 4.0(.1) & 4.07(.03)\
$\phi$ \[deg\]& 160(2) & 158.7(.2)\
An Asteroseismologic Interpretation of the Results {#astero}
--------------------------------------------------
The frequency of these pulsations are dependent on many physical quantities, including the star’s rotational velocity, magnetic field strength, and magnetic field geometry (for an in depth review, see @1979nos..book.....U). Any redistribution of angular momentum or change in magnetic field strength (or geometry) would change the observed frequencies. The sign of the first order frequency perturbation caused by changes in rotation is well known to depend solely on whether the pulsations are moving with or against rotation. In other words, if one pulsation is moving with rotation and another against it, a redistribution of angular momentum would perturb the frequency of the two pulsations in the opposite direction. The variations in pulsation frequencies E, H, and I have positive $\dot{P}$ and $\phi\approx0$ degrees while pulsation frequencies D and G have negative $\dot{P}$ and $\phi$ close to 180 degrees. We speculate that one of these groups of pulsation frequencies could be moving with rotation and the other against rotation and that the observed variations in $O-C$ may be caused by changes in the star’s rotation profile. Additionally, the long period g-mode pulsation frequencies in white dwarfs are well known to be more affected by physical properties at the surface of the star than short period g-mode pulsation frequencies. The observed frequency perturbations appear to affect the longer period pulsations more, a hint that the corresponding physical processes responsible for these frequency variations may be near the surface of the star. We again emphasize the speculative nature of these interpretations. The possible effects of magnetic field variations and quantitative modeling of these frequency variations will be investigated in the primary author’s doctoral dissertation.
[^1]: Based on observations obtained at the Southern Astrophysical Research (SOAR) telescope, which is a joint project of the Ministério da Ciência, Tecnologia, e Inovação (MCTI) da República Federativa do Brasil, the U.S. National Optical Astronomy Observatory (NOAO), the University of North Carolina at Chapel Hill (UNC), and Michigan State University (MSU).
[^2]: XCOV27 included observations from SOAR, SAAO, and the CTIO SMARTS .9m.
|
---
abstract: 'It has been demonstrated that one of the most striking features of the nervous system, the so called ’plasticity’ (i.e high adaptability at different structural levels) is primarily based on Hebbian learning which is a collection of slightly different mechanisms that modify the synaptic connections between neurons. The changes depend on neural activity and assign a special dynamic behavior to the neural networks. From a structural point of view, it is an open question what network structures may emerge in such dynamic structures under ’sustained’ conditions when input to the system is only noise. In this paper we present and study the ‘HebbNets’, networks with random noise input, in which structural changes are exclusively governed by neurobiologically inspired Hebbian learning rules. We show that Hebbian learning is able to develop a broad range of network structures, including scale-free small-world networks.'
address: 'Department of Information Systems, Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest Hungary,H-1117'
author:
- 'G. Szirtes'
- 'Zs. Palotai'
- 'A. L[ő]{}rincz'
bibliography:
- 'hebbnet\_physicaD.bib'
title: 'HebbNets: Dynamic network with Hebbian learning rule'
---
small world, Hebbian learning, central nervous system, scale-free network
\[s:intro\]Introduction
=======================
In the last few years research on complex interactive systems (CISs) has become one of the most fascinating areas. One generally applied way to describe such systems is based on graphs with nodes (vertices) and (directed) edges, representing constituents of the system and their interactions. Classification of CISs is grounded on their structural and dynamic network properties. Similar network structures may be found in many different fields spanning from social connection systems to biochemical processes [@Watts98Collective; @kleinberg98authoritative; @albert99diameter; @barabasi99emergence; @barabasi00scalefree; @marchioria00harmony; @Latora01Efficient; @Bohland01Efficient; @FerrerCancho01Optimization; @albert02statistical]. Due to the recognition of many common characteristics in both natural and human (artificial) CISs, several general models have been designed to describe the emergence of such structures, e.g., by random restructuring of the links among a finite number of ‘nodes’ [@Watts98Collective] or by ’preferential attachment’ [@albert99diameter; @barabasi99emergence], or by optimizing the link structure of finite systems [@FerrerCancho01Optimization].
Probably the most complex network is inside us: the most exciting properties of our brain have a lot to do with the special connection system among its units. It is widely accepted that activity correlation between the computing units (i.e. different forms of the so called Hebbian learning mechanism) plays a fundamental role in forming the complex neural structures and maintaining its intrinsic plasticity. Its essence is that the connection strength between the communicating units is modified according to the simultaneous activity correlation of the signal sender and receiver.
It is worth noting that the concept of Hebbian learning has undergone revolutionary changes in the last few years. The original suggestion of Hebb [@Hebb49Organization] has been modified by recent findings [@markram97regulation; @magee97synaptically; @bell97synaptic]. For a review, see, e.g., [@abbott00synaptic]. A unifying description is called spike-time dependent synaptic plasticity (STDP) and it allows different time shift patterns between the units’ activities.
\[s:model\]Description of HebbNet
=================================
In this letter we examine what network structures may emerge in a simplistic neural system by applying **pure** Hebbian dynamics without any special additional constraints. This neuronal network model will be referred as to *HebbNet*. We assume that the network is *sustained* by inputs with no spatio-temporal structure; the input is random noise. Our models consist of $N$ number of simplified integrate-and-fire like ‘neurons’ or nodes. The dynamics of the internal activity is written as $$\frac{\Delta a_i}{\Delta t} =\sum_{j} w_{ij}a_j^s+x_i^{(ext)},
\label{e:int_and_fire}$$ for $i=1,2,\ldots ,N$. (N was 200 in our simulations.) Variable $x^{(ext)} \in {(0,1)}^N$ denotes the randomly generated input from the environment, $a_i$ is the internal activity of neuron $i$, $w_{ij}$ is $ij^{th}$ element of matrix $\mathbf{W}$, i.e., the connection strength from neuron $j$ to neuron $i$. If $\Delta
t = 1$ then we have a discrete-time network and each parameter has a time index, or if $\Delta t$ is infinitesimally small then Eq. \[e:int\_and\_fire\] becomes a set of coupled differential equations. The neuron $j$ outputs a spike (neuron $j$ fires) when $a_j$ exceeds a certain level, the threshold parameter $\theta$. Spiking means that the output of the neuron $a_j^s$ (superscript $s$ stands for ’spiking’) is set to 1. Otherwise, $a_j^s=0$. Amount of excitation received by neuron $i$ from neuron $j$ is $w_{ij} a_j^s$ when neuron $j$ fires. After firing, $a_j$ is set to zero at the next time step. For continuous case $a_j$ is set to zero after a very small time interval. Equation \[e:int\_and\_fire\] describes the simplest form of ‘integrate–and–fire’ network models which is still plausible from a neurobiological point of view. No temporal integration occurs for the discrete case provided that the left hand side of Eq. \[e:int\_and\_fire\] is replaced by $a^{+}_i$ where superscript $+$ denotes time shifting. In this limiting case, and if the threshold is high enough, ‘binary neurons’ emerge. This model resembles the original model of McCullough and Pitts [@McCullough43Logical].
We examined the effect of local activity threshold and global activity constraint (selection of a given percent of nodes with the highest activity). The former one is more realistic biologically, while the latter one is more convenient: in this way the ratio of active units is always known and fixed. For these two cases, computer simulations showed negligible differences. Synaptic strengths were modified as follows: $$\frac{\Delta w_{ij}}{\Delta t}
=\sum_{(t_i,t_j)}K(t_j-t_i)a_i^{t_i,s}a_j^{t_j,s},
\label{e:pot}$$ where $K$ is a kernel function which defines the influence of the temporal activity correlation on synaptic efficacy and $\Delta
w_{ij} / \Delta t$ may be taken over discrete or over infinitesimally small time intervals. Possible examples are depicted in Fig. \[f:kernelf\]. The kernel is a function of the time differences. When the input is made of noise, as in our studies, only the ratio of the positive (strengthening) and the negative (weakening) parts of the kernel function should count. This is the result of the lack of temporal correlations in the input. Temporal grouping and reshaping of the kernel would not modify our results as long as the said ratio is kept constant. In turn, our results concern both types of kernels depicted in Fig. \[f:kernelf\].
![**Kernel functions** Two temporal kernels as a function of time difference between spiking time of neuron $i$ and $j$ ($t_i-t_j$). Relevant parameter of the shape for noise-sustained systems is the ratio ($r_{A^{+}/A^{-}}$) of the areas/sums of positive and negative parts/components of the kernel, $A^{+}$ and $A^{-}$, respectively ($r_{A^{+}/A^{-}}=A^{+}/A^{-}$). []{data-label="f:kernelf"}](fig1.eps){width="6cm"}
In the first place, we have been interested in the emerging local and global connectivity structure of $\mathbf{W}$. Instead of using global structural property ($L$, characteristic path length which is the average number of edges on the shortest paths) and the clustering coefficient ($C$) proposed by Watts and Strogatz [@Watts98Collective] we applied the so called connectivity length measure based on the concept of *network efficiency* [@Latora01Efficient]. This single measure is more appropriate for weighted networks [@marchioria00harmony], equally well applicable for describing global and local properties and offers a unified theoretical background to characterize our system. According to the definition [@marchioria00harmony; @Bohland01Efficient], local efficiency between nodes $i$ and $j$ in a weighted network with connectivity matrix $\mathbf{W}$ is $\epsilon_{ij}= 1/d_{ij}$, where $d_{ij}=
\min_{n,\,k_1, \ldots k_n} \left( 1/w_{ij}, 1/w_{ik_1} + \ldots +
1/w_{k_{n-1}k_n} + 1/w_{k_n j}\right)$ ($k_m \in (1,2, \ldots N)$ for every $1 \leq m < N-1 $ and $1 < n \leq N$). For graphs with connection strengths of values 0 or 1, $d_{ij}$ corresponds to the *shortest distance* between nodes $i$ and $j$. The average of these values ($E[d_{ij}] =\frac{1}{N(N-1)}\sum_{i\neq
j}\epsilon_{ij}$) characterizes the efficiency of the whole network. The local harmonic mean *distance* for node $i$ is defined as $$D_h(i) =
\frac{n^{(i)}}{\sum_{j : w_{ij} > 0}\epsilon_{ij}},$$ where $n^{(i)}$ is the number of neurons around neuron $i$ with $w_{ij}>0$. In terms of efficiency, this inverse of this value describes how good the local communication is amongst the first neighbors of node $i$ with node $i$ removed. It is a measure of the fault tolerance of the system. The mean *global distance* in the network is defined by the following quantity: $$D_h = \frac{N(N-1)}{\sum_{i,j}\epsilon_{ij}}.$$ Global distance provides a measure for the *size* (or the diameter) of the network, which influences the average time of information transfer. According to [@marchioria00harmony; @Bohland01Efficient] local harmonic mean distance measure behaves like $1/C$ (inverse of the clustering coefficient), whereas the global value corresponds to $L$. It can be shown that L is a good approximation of $D_h$ (or $1/L$ for the global efficiency) under certain conditions [@Bohland01Efficient].
Results and Discussion {#s:results}
======================
These connectivity length measures allowed us to study the emerging network structures as the function of the following parameters: (i) the magnitude of the external excitation (defined by the average percentage of neurons receiving excitation from the environment and (ii) the strengthening–weakening area ratio of the kernel, $K$. The binary neuron model was also investigated. Figures \[f:nofeedback\] and \[f:loglog\] summarize our findings in different parameter regions. The figure displays the appearance of scale free nets as a function of the excitation level and $r_{A^+/A^-}$. The length of the scale-free regions was determined by first plotting the distribution of the sum of the weights of outgoing connections (averaged over 10000 samples taken from 20 networks) for every parameter set studied. Results were depicted on loglog plot. Supposing a power-law distribution ($P(k^*)\approx k^{*-\gamma}e^{-k^*/\xi}$, where $k^*$ denotes the discretized values of the connection strength), a linear fitting was made to approximate $\gamma$. The width of the scale-free region was estimated by the length of the region with power-law distribution relative to the full length covered on the log scale. Maximum error of the linear fit was set to $10^{-3}$ STD. That is, for 100 discretization points, the width of a region spreading an order of magnitude on the loglog plot is equal to 0.5.
![**Scale-free region with negligible interaction** **Left:** *exponent of the power law*, **right:** *relative percentage of the power-law domain* as a function of $r_{A^+/A^-}$ and $r_{ex}$ (the ratio of excited neurons). Contribution of other neurons to the neuronal inputs is negligibly small. Difference between binary and integrate-and-fire neurons disappears in this limiting case. Results are averaged over 20 runs, all sampled 50 times, $\theta=0.5$. Stripes denote unstable region: components of matrix $\mathbf{W}$ may vanish. Log-log plots corresponding to points (a)–(d) are shown in Fig. \[f:loglog\]. Power-law with negative (positive) exponent: cases (a) and (d) (case (c)). Positive exponents are thresholded to zero on the figure. For visualization purposes, the data have been interpolated between the calculated grid points. []{data-label="f:nofeedback"}](fig2.eps){width="8cm"}
![**Log-log plots for different parameters** The four diagrams display typical distributions ($P(k^*)$) for parameters shown in Fig. \[f:nofeedback\] by (a), (b), (c) and (d). Cases (a) and (d) are arbitrary examples from the power law region. []{data-label="f:loglog"}](fig3.eps){width="8cm"}
Fig. \[f:Harmdis\] displays the emerging connections of a HebbNet for two different parameter sets. We compared the resulting HebbNet structures with a random net, in which the same weights of the dynamic network have been randomly assigned to different node pairs. The two inlets show the HebbNet connection matrices. While inlet (c) belonging to case (c) in Fig. \[f:nofeedback\] resembles a random connection matrix, inlet (d) belonging to case (d) in Fig. \[f:nofeedback\] represents a sparse structure. (Note that most elements are not zero, but very small.)
Fig. \[f:Harmdis\] highlights clearly the emerging small-world properties, i.e., small local connectivity values (high clustering coefficients) for case (d). Although the global connectivity length was almost the same for all HebbNets and their corresponding random nets, local distances are much smaller in case (d). That is, connectivity structure is sparse but information flow is still fault tolerant and efficient.
![**Harmonic mean distances** Local harmonic mean distances in ascending order are shown. For better visualization not all data points are marked and the points are connected with a solid line. Lines with upward triangle markers: STDP learning. Lines with circles: same but randomly redistributed weights. Line with empty (solid) markers: HebbNet of case (c) (case (d)). Global harmonic mean distances for the original and for the randomized networks in case (c) of Fig. \[f:nofeedback\] (case (d) of Fig. \[f:nofeedback\]) are about the same $D_h\approx D_h^r \approx 5.5$ ($D_h \approx D_h^r \approx 10$). The two inlets show the resulting connection matrices. []{data-label="f:Harmdis"}](fig4.eps){width="6cm"}
The robustness of the network to the external excitation is illustrated on the next figure.
![**Average local distance vs. excitation ratio** A: $r_{A^+/A^-}=0.1$, B: $r_{A^+/A^-}=0.6$. Diamonds: average local distances for the evolving network. Circles: average local distances for the corresponding random net. []{data-label="f:robust"}](fig5.eps){width="6cm"}
By increasing the excitation level, the average local connectivity length of the random net is drastically increasing, whereas the efficiency of the small-world network does not change too much in the same region. For the network with parameters $r_{A^+/A^-}=0.1$ (Fig. \[f:robust\](A)), there is a sharp cut-off around excittion level 0.55, where local distances suddenly drop, due to the high ratio of excitation. Qualitatively similar behavior can be seen for $r_{A^+/A^-}=0.6$ (Fig. \[f:robust\](B)), but the cut-off is around $r_{ex}=0.9$.
For networks with significant interaction we have experienced a convergence of the exponent of the power-law distribution to -1. The width of the scale-free region was relatively broad (see, Fig. \[f:feedback\]).
![**Power-law with significant interaction** **Left:** *exponent of the power law*, **right:** *relative percentage of the power-law domain* as a function of $r_{ex}$ and excitation threshold $\theta$. $r_{A^+/A^-}=0.1$ Results are averaged over 700 steps. Input from other neurons could exceed the external inputs by a factor of 10. The exponent of the power-law approximates -1 for broad regions of $\theta$ and $r_{ex}$. Outside this region the network may vanish or may start to oscillate. For visualization purposes, the data have been interpolated between the calculated grid points. []{data-label="f:feedback"}](fig6.eps){width="8cm"}
Summary
=======
In summary, we have demonstrated that small-world architectures with scale-free domains may emerge in sustained networks under STDP Hebbian learning rule without any other specific constraints on the evolution of the net. Although one always has to remember that results from simplified models may not carry over to biophysically realistic networks, we feel that some intriguing conjectures can be made based on our findings. The role of noise in the central nervous system [@Ferster96Neuralnoise; @Miller02Neuralnoise] is unclear. The existence of such ‘HebbNets’ may support the speculative view of Kandel et al. [@Kandel92Adult] that structural development and learning plasticity in CNS may have a common basis. According to our results, evolution and plasticity of the networks may be maintained by noise randomly generated within the CNS. We conjecture that the sustained nature of noise and the competition imposed by small $r_{A^+/A^-}$ values are the two relevant components of plasticity and learning. It might be equally important that exponents of HebbNets with significant interaction amongst neurons are similar in a broad range of parameters.
As far as other evolving networks are considered, the profound implication of our result is that local (Hebbian) learning rules may be sufficient to form and maintain an efficient network in terms of information flow. This feature differs from existing models, such as the model on preferential attachment [@barabasi99emergence], the global optimization scheme [@FerrerCancho01Optimization], and also from the original Watts and Strogatz model [@Watts98Collective].
Acknowledgements
================
This work was partially supported by the Hungarian National Science Foundation, under Grant No. OTKA 32487.
|
---
abstract: 'The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996. From that moment on, many research articles are devoted to the study of the properties of the inverse integrating factor and its relation with limit cycles and their bifurcations. This paper is a summary of all the results about this topic. We include a list of references together with the corresponding related results aiming at being as much exhaustive as possible. The paper is, nonetheless, self-contained in such a way that all the main results on the inverse integrating factor are stated and a complete overview of the subject is given. Each section contains a different issue to which the inverse integrating factor plays a role: the integrability problem, relation with Lie symmetries, the center problem, vanishing set of an inverse integrating factor, bifurcation of limit cycles from either a period annulus or from a monodromic $\omega$-limit set and some generalizations.'
author:
- '[Isaac A. Garc'' ia$^{\ (1)}$ & Maite Grau$^{\ (1)}$]{}'
title: 'A survey on the inverse integrating factor.[^1]'
---
The Euler integrating factor \[sect1\]
======================================
The method of integrating factors is, in principle, a means for solving ordinary differential equations of first order and it is theoretically important. The use of integrating factors goes back to Leonhard Euler.
Let us consider a first order differential equation and write the equation in the Pfaffian form $$\label{survey-1}
\omega = P(x,\,y)\,dy-Q(x,\,y)\,dx = 0 \ .$$ We assume that the functions $P$ and $Q$ are of class $\mathcal{C}^1$ in a region $\mathcal{U} \subseteq \mathbb{R}^2$. If there is a solution of (\[survey-1\]) which may be expressed in the form $H(x,\,y) = h$ with $H$ having continuous partial derivatives in $\mathcal{U}$ and with $h$ an arbitrary constant, then it is not difficult to see that such an $H$ satisfies the linear partial differential equation $$\label{survey-2}
P \frac{\partial H}{\partial x} + Q \frac{\partial H}{\partial y} = 0 \ .$$ Conversely, every non-constant solution $H$ of (\[survey-2\]) gives also a solution $H(x,\,y) = h$ of (\[survey-1\]). Thus, solving (\[survey-1\]) and solving (\[survey-2\]) are equivalent tasks.
It is straightforward to show that if $H_0(x,\,y)$ is a non-constant solution of equation (\[survey-2\]), then all solutions of this equation are of the form $F(H_0(x,\,y))$ where $F$ is a freely chosen function with continuous derivative. The connection between equations (\[survey-1\]) and (\[survey-2\]) may be presented also in another form. Suppose that $H(x,\,y) = h$ is any solution of (\[survey-1\]). Then (\[survey-2\]) implies $$\frac{\partial H / \partial y}{P} = -\frac{\partial H / \partial x}{Q} \ .$$ If we denote the common value of these two ratios by $\mu(x,\,y)$, then we have $\partial H / \partial y = \mu\, P$ and $\partial H /
\partial x = - \mu \, Q$. This gives to the differential of the function $H$ the expression $d\,H(x,\,y) =
\mu(x,\,y)(P(x,\,y)\,dy-Q(x,\,y)\,dx)$. Hence, $\mu(x,\,y)$ is called the integrating factor of the given differential equation (\[survey-1\]) because the left hand side of (\[survey-1\]) turns, when multiplied by $\mu(x,\,y)$, to be an exact differential.
Conversely, any integrating factor $\mu$ of (\[survey-1\]), i.e. such that $\mu(x,\,y)(P(x,\,y)\,dy-Q(x,\,y)\,dx)$ is the differential of some function $H$, is easily seen to determine the solutions of the form $H(x,\,y) = h$ of (\[survey-1\]). Altogether, solving the differential equation (\[survey-1\]) is equivalent to finding an integrating factor of the equation.
When an integrating factor $\mu$ of (\[survey-1\]) is available, the function $H$ can be obtained from the line integral $$H(x,\,y) = \int_{(x_0,\,y_0)}^{(x,\,y)} \mu(x,\,y)(P(x,\,y)\,dy-Q(x,\,y)\,dx)$$ along any curve connecting an arbitrarily chosen point $(x_0,\,y_0)$ and the point $(x,\,y)$ in the region $\mathcal{U}$. We remark that this line integral might not be well-defined if the region $\mathcal{U}$ is not simply-connected. When we know an integrating factor $\mu$ of (\[survey-1\]), we have a first integral well-defined in each simply-connected subcomponent of the region $\mathcal{U}$.
The inverse integrating factor
==============================
Let us consider a real planar autonomous differential system $$\dot{x} \, = \, P(x,y), \qquad \dot{y} \, = \, Q(x,y), \label{eq1}$$ where $P(x,y)$ and $Q(x,y)$ are of class $\mathcal{C}^1(\mathcal{U})$ and $\mathcal{U} \subseteq
\mathbb{R}^2$ is an open set. The dot denotes derivation with respect to the independent variable $t$ usually called [*time*]{}, that is $\dot{}=\frac{d}{dt}$.
As usual, we associate to system (\[eq1\]) the vector field $\mathcal{X} \, = \, P(x,y)
\partial_x \, + \, Q(x,y) \partial_y$. Notice that the ordinary differential equation $\omega = 0$ given in (\[survey-1\]) is just the differential equation of the orbits of system (\[eq1\]).
A function $V : \mathcal{U} \to \mathbb{R}$ is said to be an inverse integrating factor of system [(\[eq1\])]{} if it is of class $\mathcal{C}^1(\mathcal{U})$, it is not locally null and it satisfies the following partial differential equation: $$\label{def-V}
P(x,y) \, \frac{\partial V(x,y)}{\partial x} \, + \,
Q(x,y) \, \frac{\partial V(x,y)}{\partial y} \, = \, \left(
\frac{\partial P(x,y)}{\partial x} \, + \, \frac{\partial
Q(x,y)}{\partial y} \right) \, V(x,y).$$
In short notation, an inverse integrating factor $V$ of system (\[eq1\]) satisfies $\mathcal{X} V = V {\rm div} \mathcal{X}$, where ${\rm div} \mathcal{X} = \frac{\partial P}{\partial x} \, +
\, \frac{\partial Q}{\partial y}$ stands for the divergence of the vector field $\mathcal{X}$.
Of course, the computation of an inverse integrating factor for a concrete system is a delicate matter whose difficulty is comparable to solving the system itself.
If $V$ is an inverse integrating factor of a $\mathcal{C}^1$ vector field $\mathcal{X}$, then the zero set of $V$, $V^{-1}(0)
:=\{(x,y) \mid V(x,y)=0 \}$, is composed of trajectories of $\mathcal{X}$. For by the equation (\[def-V\]) that defines $V$, $\mathcal{X}$ is orthogonal to the gradient vector field $\nabla
V$ along the zero set of $V$.
The name “inverse integrating factor” arises from the fact that if $V$ solves equation (\[def-V\]), then its reciprocal $1/V$ is an integrating factor for $\mathcal{X}$ on $\mathcal{U} \setminus
V^{-1}(0)$.
Local nontrivial Lie symmetries and inverse integrating factors \[sect3\]
=========================================================================
Roughly speaking, a symmetry group of a system of differential equations is a continuous group which transforms solutions of the system to other solutions. Simple typical examples are groups of translations, rotations and scalings, but these certainly do not exhaust the range of possibilities. Once one has determined the symmetry group of a system of differential equations, a number of applications become available.
More precisely, a symmetry of system (\[eq1\])in $\mathcal{U}$, where $\mathcal{U} \subseteq \mathbb{R}^2$ is an open set, is a 1–parameter Lie group of diffeomorphisms $\Phi_\epsilon$ acting in $\mathcal{U}$ that maps the set of orbits of (\[eq1\]) into itself. When $\Phi_\epsilon(x,y) = (\bar{x}(x,y; \epsilon),
\bar{y}(x,y; \epsilon))$, the symmetry condition of (\[eq1\]) reads for $\dot{\bar{x}} = P(\bar{x}, \bar{y})$, $\dot{\bar{y}} =
Q(\bar{x}, \bar{y})$ for all $\epsilon$ close to zero. Let the $\mathcal{C}^1(\mathcal{U})$ vector field $\mathcal{Y} = \xi(x, y)
\partial_x + \eta(x, y) \partial_y$ be the infinitesimal generator of the 1–parameter Lie group $\Phi_\epsilon$, that is, $\bar{x}(x,y;\epsilon) = x+ \epsilon \xi(x,y) + O(\epsilon^2)$, $\bar{y}(x,y;\epsilon) = y+ \epsilon \eta(x,y) + O(\epsilon^2)$. Denoting by $\mathcal{X} \, = \, P(x,y) \partial_x \, + \, Q(x,y)
\partial_y$ the vector field associated to system (\[eq1\]), it is well known that a characterization of the Lie symmetries of (\[eq1\]) is given by the relation $[\mathcal{X}, \mathcal{Y}] =
\mu(x,y) \mathcal{X}$ for certain scalar function $\mu :
\mathcal{U} \to \mathbb{R}$. In this expression we have used the [*Lie bracket*]{} of two $\mathcal{C}^1$-vector fields ${\cal X}$ and ${\cal Y}$ defined as $[{\cal X}, {\cal Y} ]:= {\cal X} {\cal
Y} - {\cal Y} {\cal X}$. Using coordinates we have [$$[{\cal X}, {\cal Y} ] = \left( P {\partial \xi \over \partial x}- \xi {\partial P \over \partial x}+ Q {\partial \xi \over \partial y} - \eta {\partial P \over \partial y} \right) \partial_x + \left( P {\partial \eta \over \partial x}- \xi {\partial Q \over \partial x}+ Q {\partial \eta \over \partial y} - \eta {\partial Q \over \partial y} \right) \partial_y \ .
\label{sim2.4.1}$$ ]{} When beginning students first encounter ordinary differential equations, they are presented with a variety of special techniques designed to solve certain particular types of equations, such as separable, homogeneous or exact. Indeed, this was the state of the art around the middle of the nineteenth century, when Sofus Lie made the profound discovery that these special methods were, in fact, all special cases of a general integration procedure based on the invariance of the differential equation under a continuous group of symmetries. This observation at once unified and significantly extended the available integration techniques.
[||c|c||]{}\
[Differential Equation]{} & [Lie Symmetry ]{}\
$dy/dx = f(x)g(y)$ & ${\cal Y} = g(y) \partial_y$\
$dy/dx = f(a x+b y)$ & ${\cal Y} = b \partial_x + a \partial_y$\
$dy/dx =
\frac{y+x f(\sqrt{x^2+y^2})}{x-y f(\sqrt{x^2+y^2})}$ & ${\cal Y} =
y \partial_x - x \partial_y$\
$dy/dx =
f(y/x)$ & ${\cal Y} = x \partial_x + y \partial_y$\
$dy/dx = P(x) y+Q(x)$ & ${\cal Y} = \exp\left( \int P(x) dx \right)
\partial_y$\
$dy/dx = P(x) y+Q(x) y^n$ & ${\cal Y} = y^n \exp\left[ (1-n) \int
P(x) dx \right] \partial_y$\
Consider now a $\mathcal{C}^1$ vector field ${\cal X} = P(x,y)
\partial_x + Q(x,y) \partial_y$ defined in an open connected subset $\mathcal{U} \subseteq \mathbb{R}^2$. In the case of a single first order ordinary differential equation $d y / d x =
Q(x,y) / P(x,y)$, the Lie symmetries method provides by quadrature an explicit formula for the general solution. In fact, one can easily see that if we know a Lie symmetry in $\mathcal{U}$ with infinitesimal generator ${\cal Y} = \xi(x,y) \partial_x +
\eta(x,y) \partial_y$ then we construct an inverse integrating factor $V = \det \{{\cal X}, {\cal Y}\} = P \eta - Q \xi$ defined in $\mathcal{U}$, but the converse is not always true. To see that, assume now the existence of an inverse integrating factor $V$ of $\mathcal{X}$ in a simply connected domain $\mathcal{U}$ and we look for an infinitesimal generator $\mathcal{Y} = \xi(x,y) \partial_x + \eta(x,y)
\partial_y$ of a Lie symmetry of $\mathcal{X}$ well defined in $\mathcal{U}$. We recall that a singular point $p \in U$ of $\mathcal{X}$ is called [*weak*]{} if ${{\textrm{div}}} \mathcal{X}(p) = 0$. If there is no weak singularity of $\mathcal{X}$ in $\mathcal{U}$, then we can do at least one of the following constructions:
(i)
: Prescribe the function $\xi(x,y)$ and solve $\eta(x,y)$ from $V= P \eta- Q \xi$.
(ii)
: Prescribe the function $\eta(x,y)$ and solve $\xi(x,y)$ from $V= P \eta- Q \xi$.
(iii)
: Take the rescaled hamiltonian vector field $$\mathcal{Y}= \frac{1}{{{\textrm{div}}} \mathcal{X}} \
(-\frac{\partial{V}}{\partial{y}}\partial_x +
\frac{\partial{V}}{\partial{x}}\partial_y) \ ,$$ defined in $\mathcal{U} \backslash \{ (x,y) \in \mathcal{U} : {{\textrm{div}}}
\mathcal{X} = 0 \}$.
Therefore, the equivalence between inverse integrating factors and Lie symmetries for planar vector fields $\mathcal{X}$ is not true, in general, in neighborhoods of weak singular points of $\mathcal{X}$. Of course, some special situations can appear giving the equivalence when $\mathcal{X}$ possesses an analytic first integral in these neighborhoods as the nondegenerate center singular point shows.
Importance of inverse integrating factors arises from the fact that the differential 1-form $\omega / V = (P \ dy -
Q \ dx) / V$ is closed ($d (\omega/V) =0$) in $\mathcal{U}
\backslash V^{-1}(0)$. Then in the case in which $\mathcal{U}
\backslash V^{-1}(0)$ is simply-connected, the 1-form $\omega / V$ is [*exact*]{} ($\omega/V = d H$), and therefore a ${\cal C}^2$ first integral $H(x,y)$ of the differential equation is immediately constructed. As a consequence, the vector field $\mathcal{X} \, = \, P(x,y)
\partial_x \, + \, Q(x,y) \partial_y$ is topologically equivalent, in $\mathcal{U}$, to the hamiltonian vector field $\mathcal{X} / V
\, = \, \frac{\partial H}{\partial y}
\partial_x \, - \, \frac{\partial H}{\partial x} \partial_y$.
Making a pause in this exposition we now present an example. Let us consider the following cubic system $$\dot{x}= P(x,y) = -y-x(x^2+y^2-1) \ , \ \ \dot{y}= Q(x,y)= x-y(x^2+y^2-1) \ .
\label{ej}$$ An inverse integrating factor for system (\[ej\]) is given by $V(x,y)=(x^2+y^2)(x^2+y^2-1)$. Associated to him one has the first integral $$H(x,y)={(x^2+y^2-1) \over (x^2+y^2)} \, \exp\left\{ 2 \, \arctan
\left({y \over x}\right) \right\} ,$$ which is not continuous in $(0,0)$. On the other hand, since the polar form of the system is $\dot{r} = 2 r^2 (r^2-1)$, $\dot{\varphi} = 1$ it is easy to check that the unit circle $x^2+y^2-1=0$ is the unique limit cycle of system (\[ej\]). Let ${\cal X}=P(x,y) \partial / \partial x + Q(x,y) \partial /
\partial y$ be the vector field associated with system (\[ej\]). From the symmetries point of view, since ${\cal Y}=y \partial_x
-x \partial_y$ satisfies $[ {\cal X} , {\cal Y} ] \equiv 0$ we have that ${\cal Y}$ is the infinitesimal generator of a Lie group admitted by system (\[ej\]) which is just the $SO(2)$ rotation group $\bar{x} = x \cos\epsilon - y \sin\epsilon$, $\bar{y} = x
\sin\epsilon + y \cos\epsilon$. Hence $V(x,y) = \det \{{\cal X},
{\cal Y}\}$ is an inverse integrating factor of system (\[ej\]). Notice that the only common integral curves for the vector fields ${\cal X}$ and ${\cal Y}$ are included in $V^{-1}(0)$ and are just the separatrices of ${\cal X}$. This behavior will be explained in future sections.
On the integrability problem \[sectinteg\]
==========================================
The integrability problem is mainly related to planar [**polynomial**]{} differential systems of the form $$\dot{x}=P(x,y), \quad \dot{y}=Q(x,y), \label{0eq1}$$ where $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are coprime polynomials, that is, there is no non-constant polynomial which divides both $P$ and $Q$. We call [d]{} the maximum degree of $P$ and $Q$ and we say that system (\[0eq1\]) is of degree [d]{}. When ${\rm d}=2$, we say that (\[0eq1\]) is a [*quadratic system*]{}.
If $p$ is a point such that $P(p)=Q(p)=0$, then we say that $p$ is a [*singular point*]{} of system (\[0eq1\]).
As we have already defined in Section \[sect1\], a ${\mathcal{C}}^j$ function $H: {\mathcal{U}} \to \mathbb{R}$ such that it is constant on each trajectory of (\[0eq1\]) and it is not locally constant is called a [*first integral*]{} of system (\[0eq1\]) of class $j$ defined on $\mathcal{U} \subseteq
\mathbb{R}^2$. The equation $H(x,y)=h$ for a fixed $h \in
\mathbb{R}$ gives a set of trajectories of the system, but in an implicit way. When $j \geq 1$, these conditions are equivalent to $P(x,y) \frac{\partial H}{\partial x} + Q(x,y) \frac{\partial
H}{\partial y} = 0$ and $H$ not locally constant. The problem of finding such a first integral and the functional class it must belong to is what we call the [*integrability problem*]{}.
To find an integrating factor or an inverse integrating factor for system (\[0eq1\]) is closely related to finding a first integral for it. When considering the integrability problem we are also addressed to study whether an (inverse) integrating factor belongs to a certain given class of functions.
When a first integral $H$ of system (\[0eq1\]) is known, all the orbits of the system are contained in its domain of definition are given by the level sets $H(x,y)=h$. Thus, a natural strategy is to look for the determination of some of the orbits of the system and try to build a first integral with them. In particular, and since system (\[0eq1\]) is polynomial, those orbits which are algebraic will be of special interest.
An [*invariant curve*]{} is a curve given by $f(x,y)=0$, where $f: \mathcal{U}
\subseteq \mathbb{R}^2 \to \mathbb{R}$ is a $\mathcal{C}^1$ function in the open set $\mathcal{U}$, non locally constant and such that there exists a $\mathcal{C}^1$ function in $\mathcal{U}$, denoted by $k(x,y)$ and called [*cofactor*]{}, which satisfies: $$P(x,y) \, \frac{\partial f}{\partial x} (x,y) \, + \, Q(x,y) \,
\frac{\partial f}{\partial y}(x,y) \, = \, k(x,y) \, f(x,y),
\label{0ci}$$ for all $(x,y) \in \mathcal{U}$. The notion of invariant curve was first introduced in [@GarciaGine]. The identity (\[0ci\]) can be rewritten by $\mathcal{X} f = k f$. We recall that $\mathcal{X} f$ denotes the scalar product of the vector field $\mathcal{X}$ and the gradient vector $\nabla f$ related to $f(x,y)$, that is, $\nabla f(x,y)=( \frac{\partial f}{\partial x}
(x,y) , \frac{\partial f}{\partial y} (x,y) )$. We will denote by $\frac{d f}{d t}$ or by $\dot{f}$ the function $\mathcal{X} f$ once evaluated on a solution of system (\[0eq1\]). In case $f(x,y)=0$ defines a curve in the real plane, this definition implies that the function $\mathcal{X} f$ is equal to zero on the points such that $f(x,y)=0$. In the article [@GarciaGine] an invariant curve is defined as a $\mathcal{C}^1$ function $f(x,y)$ defined in the open set $\mathcal{U} \subseteq \mathbb{R}^2$, such that, the function $\mathcal{X} f$ is zero in all the points $\{
(x,y) \in \mathcal{U} \, | \, f(x,y)=0 \}$. We notice that our definition of invariant curve is a particular case of the previous one but, for the sake of our results, the cofactor is very important and that’s why we always assume its existence.
When the cofactor $k(x,y)$ is a polynomial, we say that $f(x,y)=0$ is an invariant curve with polynomial cofactor. We only admit invariant curves with polynomial cofactor of degree lower or equal than ${\rm d}-1$, that is $\deg k(x,y) \leq {\rm d}-1$, where ${\rm d}$ is the degree of system (\[0eq1\]).
The notion of invariant curve is a generalization of the notion of invariant algebraic curve. An [*invariant algebraic curve*]{} is an algebraic curve $f(x,y)=0$, where $f(x,y) \in \mathbb{C}[x,y]$, which is invariant by the flow of system (\[0eq1\]). This condition equals to $\mathcal{X} f = k f$, where the cofactor of an invariant algebraic curve is always a polynomial of degree $\deg k(x,y) \leq {\rm d}-1$.
We cite [@Llibre; @Schlomiuk1; @Schlomiuk2] as compendiums of the results on invariant algebraic curves. For instance, in [@Llibre], it is shown that if $f(x,y)=0$ and $g(x,y)=0$ are two invariant algebraic curves of system (\[0eq1\]) with cofactors $k_f(x,y)$ and $k_g(x,y)$, respectively, then the product of the two polynomials gives rise to the curve $(f g)(x,y)
=0$ which is also an invariant algebraic curve of system (\[0eq1\]) and whose cofactor is $k_f(x,y) + k_g(x,y)$.
In order to state the known results of integrability using invariant algebraic curves, we need to consider complex algebraic curves $f(x,y)=0$, where $f(x,y) \in \mathbb{C}[x,y]$. Since system (\[0eq1\]) is defined by real polynomials, if $f(x,y)=0$ is an invariant algebraic curve with cofactor $k(x,y)$, then its conjugate $\bar{f}(x,y)=0$ is also an invariant algebraic curve with cofactor $\bar{k}(x,y)$. Hence, its product $f(x,y)
\bar{f}(x,y) \in \mathbb{R}[x,y]$ gives rise to a real invariant algebraic curve with a real cofactor $k(x,y)+ \bar{k}(x,y)$. For a sake of simplicity, we consider invariant algebraic curves defined by polynomials in $\mathbb{C}[x,y]$, although we always keep in mind the previous observation. In $\mathbb{R}^2$, the curve given by $f(x,y)=0$, where $f(x,y)$ is a real function, may only contain a finite number of isolated singular points or be the null set.
An algebraic curve $f(x,y)=0$ is called [*irreducible*]{} when $f(x,y)$ is an irreducible polynomial in the ring $\mathbb{C}[x,y]$. We can assume, without loss of generality, that $f(x,y)$ is an irreducible polynomial in $\mathbb{C}[x,y]$, because if $f(x,y)$ is reducible, then all its proper factors give rise to invariant algebraic curves. Given an algebraic curve $f(x,y)=0$, we can always assume that the polynomial $f(x,y)$ has no multiple factors, that is, its decomposition in the ring $\mathbb{C}[x,y]$ is of the form $f(x,y) = f_1(x,y) f_2(x,y)
\ldots f_{\ell}(x,y)$, where $f_i(x,y)$ are irreducible polynomials and $f_i(x,y) \neq c f_j(x,y)$ if $i \neq j$ and for any $c \in \mathbb{C}$. The assumption that given an algebraic curve $f(x,y)=0$, the polynomial $f(x,y)$ has no multiple factors is mainly used to ensure that we do not consider “false” singular points. If $p$ is a point such that $f(p)=0$ and $\nabla
f(p)=0$, and $f(x,y)$ has no multiple factors, then $p$ is a singular point of the curve $f(x,y)=0$. But, if $f(x,y)$ has multiple factors, for instance, $f(x,y)=f_1(x,y)^2$ where $f_1(x,y)$ is an irreducible polynomial in $\mathbb{C}[x,y]$, then all the points of the curve $\{ p \, | \, \, f_1(p)=0 \}$ satisfy the property that $f(p)=0$ and $\nabla f(p) =0$ although they are not all singular points.
We recall that if $p$ is a singular point of an invariant algebraic curve $f(x,y)=0$ of a system (\[0eq1\]), then $p$ is a singular point of the system. Given an algebraic curve $f(x,y)=0$, we will always assume that the decomposition of $f(x,y)$ in the ring $\mathbb{C}[x,y]$ has no multiple factors. We want to generalize this property to invariant curves, that’s why we will always assume that, given an invariant curve $f(x,y)=0$, if $p \in
\mathcal{U}$ is such that $f(p)=0$ and $\nabla f(p)=0$, then $p$ is a singular point of system (\[0eq1\]). This technical hypothesis generalizes the notion of not having multiple factors for algebraic curves. In [@seiden4], a set of necessary conditions for a system (\[0eq1\]) to have an irreducible invariant algebraic curve is given.
Invariant algebraic curves are the main objects used in the Darboux theory of integrability. In [@Darboux], G. Darboux gives a method for finding an explicit first integral for a system (\[0eq1\]) in case that ${\rm d}({\rm d}+1)/2 + 1$ different irreducible invariant algebraic curves are known, where [d]{} is the degree of the system. In this case, a first integral of the form $H=f_1^{\lambda_1}f_2^{\lambda_2} \ldots f_s^{\lambda_s}, $ where each $f_i(x,y)=0$ is an invariant algebraic curve for system (\[0eq1\]) and $\lambda_i \in \mathbb{C}$ not all of them null, for $i=1,2,\ldots,s$, $s \in \mathbb{N}$, can be constructed. The functions of this type are called [*Darboux functions*]{}.
As we have already stated, given an invariant algebraic curve $f(x,y)=0$ whose imaginary part is not null, then its conjugate is also an invariant algebraic curve. Moreover, as system (\[0eq1\]) is real, if $f(x,y)$ appears in the expression of a first integral of the form given by Darboux with exponent $\lambda$, then $\bar{f}(x,y)$ appears in the same expression with exponent $\bar{\lambda}$. We call ${\rm Re} f$ the real part of the polynomial $f$ and by ${\rm Im} f$ its imaginary part. Analogously, let us call ${\rm Re} \lambda$ the real part of the complex number $\lambda$ and by ${\rm Im} \lambda$ its imaginary part. We call $\mathbf{i}=\sqrt{-1}$ and we use the following formula for complex numbers: $$\arctan(z) = \log \left[ \left( \frac{1 - \mathbf{i} z}{1 + \mathbf{i} z} \right)
^{ \mathbf{i}/2} \right], \quad z \in \mathbb{C},$$ to show that $$\begin{aligned}
f^{\lambda} \bar{f}^{\bar{\lambda}} & = & \left( {\rm Re} f + {\rm
Im} f \, \mathbf{i} \right) ^{{\rm Re} \lambda + {\rm Im} \lambda
\, \mathbf{i}} \ \left( {\rm Re} f - {\rm Im} f\, \mathbf{i}
\right) ^{{\rm Re} \lambda - {\rm Im} \lambda \, \mathbf{i}}
\\ & = & \left( ({\rm Re} f)^2 + ({\rm Im} f)^2 \right)^{{\rm Re} \lambda} \ \exp \left\{
-2\, {\rm Im} \lambda \, \arctan \left( \frac{{\rm Im} f}{{\rm
Re} f} \right) \right\}.\end{aligned}$$ We deduce that the product $f(x,y)^{\lambda}
\bar{f}(x,y)^{\bar{\lambda}}$ is a real function and so it is any Darboux function $H=f_1^{\lambda_1}f_2^{\lambda_2} \ldots
f_s^{\lambda_s}$.
We have that the Darboux function $H$ can be defined in the open set $\mathbb{R}^2 \setminus \Sigma$, where $ \Sigma = \{ (x,y) \in
\mathbb{R}^2 \mid (f_1 \cdot f_2 \cdot \dots \cdot f_r)(x,y) = 0
\}.$ We remark that, particularly, if $\lambda_i \in \mathbb{Z}$ , $\forall i=1,2,\ldots,r$, $H$ is a [*rational first integral*]{} for system (\[0eq1\]). In this sense J. P. Jouanoulou [@Jouanolou], showed that if at least ${\rm d}({\rm d}+1) + 2$ different irreducible invariant algebraic curves are known, then there exists a rational first integral.
The main fact used to prove Darboux’s theorem (and Jouanoulou’s improvement) is that the cofactor corresponding to each invariant algebraic curve is a polynomial of degree $\leq {\rm d}-1$. Invariant curves with polynomial cofactor can also be used in order to find a first integral for the system. This observation enables a generalization of the Darboux’s theory which is given in [@GarciaGine1], where, for instance, non-algebraic invariant curves with an algebraic cofactor for a polynomial system of degree $4$ are presented. In [@rocky], other examples are given of such invariant curves with polynomial cofactor for some families of systems and the way they are used to construct explicit first integrals and inverse integrating factors for the corresponding systems. As a continuation of [@rocky], in [@rocky2] we study when a planar differential system polynomial in one variable linearizes in the sense that it has an inverse integrating factor which can be constructed by means of the solutions of linear differential equations and we describe some families of differential systems which are Darboux integrable and whose inverse integrating factor is constructed using the solutions of a second–order linear differential equation defining a family of orthogonal polynomials.
Some generalizations of the classical Darboux theory of integrability may be found in the literature. For instance, independent singular points can be taken into account to reduce the number of invariant algebraic curves necessary to ensure the Darboux integrability of the system, see [@ChLlSoto]. A good summary of many of these generalizations can be found in [@Chara] and a survey on the integrability of two-dimensional systems can be found in [@flows]. One of the most important definitions in this sense is the notion of exponential factor which is given by C. Christopher in [@Christopher1], when he studies the multiplicity of an invariant algebraic curve. The notion of exponential factor is a particular case of invariant curve for system (\[0eq1\]). Given two coprime polynomials $h,g \in \mathbb{R}[x,y]$, the function $e^{h/g}$ is called an [*exponential factor*]{} for system (\[0eq1\]) if for some polynomial $k$ of degree at most ${\rm
d}-1$, where [d]{} is the degree of the system, the following relation is fulfilled: $$P \left( \frac{\partial\, e^{h/g}}{\partial x} \right) + Q
\left( \frac{\partial\, e^{h/g}}{\partial y} \right) = k(x,y) \,
\, e^{h/g}.$$ As before, we say that $k(x,y)$ is the [*cofactor*]{} of the exponential factor $e^{h/g}$.
The next proposition, proved in [@Christopher1], gives the relationship between the notion of invariant algebraic curve and exponential factor.
[[@Christopher1]]{} If $F=e^{h/g}$ is an exponential factor and $g$ is not a constant, then $g=0$ is an invariant algebraic curve, and $h$ satisfies the equation $P
\frac{\partial h}{\partial x} + Q \frac{\partial h}{\partial y}= h
\, k_g + g\, k_F$ where $k_g$ and $k_F$ are the cofactors of $g$ and $F$, respectively.
The notion of exponential factor is very important in the Darboux theory of integrability since it does not only allow the construction of first integrals following the same method described by Darboux, but it also explains the meaning of the multiplicity of an invariant algebraic curve in relation with the differential system (\[0eq1\]). A complete work on this subject can be found in [@ChLlPe].
In the same way as with invariant algebraic curves, given an exponential factor $F=\exp\{ h/g \}$, since system (\[0eq1\]) is a real system, there is no lack of generality in considering that $h(x,y), g(x,y) \in \mathbb{R}[x,y]$. If $F=\exp\{ h/g \}$ is an exponential factor with non-null imaginary part, then its complex conjugate, $\bar{F} = \exp \{ \bar{h} / \bar{g} \}$ is also an exponential factor, as it can be easily checked by its defining equation. Moreover, the product $F \, \bar{F}= \exp \{ h/g +
\bar{h}/\bar{g} \}$ is a real exponential factor with a real cofactor.
Since the notion of exponential factor is the most current generalization in the Darboux theory of integrability, any function of the form: $$f_1^{\lambda_1} f_2^{\lambda_2} \cdots f_r^{\lambda_r} \left( \exp
\left( \frac{h_1}{g_1^{n_1}} \right)\right)^{\mu_1} \left( \exp
\left( \frac{h_2}{g_2^{n_2}} \right)\right)^{\mu_2} \cdots \left(
\exp \left( \frac{h_\ell}{g_\ell^{n_\ell}}
\right)\right)^{\mu_\ell}, \label{0gdf}$$ where $r, \ell \in \mathbb{N}$, $f_i(x,y)=0$ ($1 \leq i \leq r$) and $g_j(x,y)=0$ ($1 \leq j \leq \ell$) are invariant algebraic curves of system (\[0eq1\]), $h_j(x,y)$ ($1 \leq j \leq \ell$) are polynomials in $\mathbb{C}[x,y]$, $\lambda_i$ ($1 \leq i \leq
r$) and $\mu_j$ ($1 \leq j \leq \ell$) are complex numbers and $n_j$ ($1 \leq j \leq \ell$) are non-negative integers, is called a [*(generalized) Darboux function*]{}.
Let us present a short survey about the Darboux method and its improvements. Let us recall that a singular point $(x_0, y_0)$ of system (\[0eq1\]) is called [*weak*]{} if the divergence, ${\rm div}\mathcal{X}$, of system (\[0eq1\]) at $(x_0, y_0)$ is zero. We recall that $\mathcal{X}$ denotes the vector field associated to system (\[0eq1\]). We denote by $\mathbb{C}_{\rm d-1} [x,y]$ the set of polynomials in $\mathbb{C}[x,y]$ of degree lower than ${\rm d}$. We say that $s$ points $(x_k, y_k)\in \mathbb{C}^2$, $k=1,2,\ldots,s$, are [*independent*]{} with respect to $\mathbb{C}_{\rm d-1} [x,y]$ if the intersection of the $s$ hyperplanes $$\left\{ \left( a_{ij}\right) \in \mathbb{C}^{\rm d(d+1)/2} \, :
\, \sum_{i+j=0}^{\rm d-1} x_k^i y_k^i a_{ij} \, = \,
0\right\}_{k=1,2,\ldots,s}$$ is a linear subspace of $\mathbb{C}^{\rm d(d+1)/2}$ of dimension ${\rm d(d+1)/2} -s > 0$.
The main results about the Darboux method and its improvements are summarized in the following theorem, which can be found in [@LlibPan04], see also [@Chara].
\[thLlibPan04\] Suppose that a polynomial differential system [(\[0eq1\])]{} of degree ${\rm d}$ admits $r$ irreducible invariant algebraic curves $f_i = 0$ with cofactors $K_i$ for $i = 1,2,\ldots,r$; $\ell$ exponential factors $exp(h_j/g_j^{n_j})$ with cofactors $L_j$ for $j =1,2,\ldots, \ell$; and $s$ independent singular points $(x_k, y_k)$ such that $f_i(x_k, y_k) \neq 0$ for $i =
1,2,\ldots,r$ and for $k=1,2,\ldots,s$. Moreover, the irreducible factors of the polynomials $g_j$ are some $f_i$’s.
- There exist $\lambda_i, \mu_j\in \mathbb{C}$ not all zero such that $\sum_{i=1}^{r} \lambda_i K_i + \sum_{j=1}^{\ell} \mu_j L_j=0$, if and only if the (multi–valued) function [(\[0gdf\])]{} is a first integral of system [(\[0eq1\])]{}.
- If $r + \ell + s = [{\rm d}({\rm d} + 1)/2] + 1$, then there exist $\lambda_i, \mu_j\in \mathbb{C}$ not all zero such that $\sum_{i=1}^{r} \lambda_i K_i + \sum_{j=1}^{\ell} \mu_j
L_j=0$.
- If $r + \ell + s \geq [{\rm d}({\rm d} + 1)/2] + 2$, then system [(\[0eq1\])]{} has a rational first integral, and consequently all trajectories of the system are contained in invariant algebraic curves.
- There exist $\lambda_i, \mu_j\in \mathbb{C}$ not all zero such that $\sum_{i=1}^{r} \lambda_i K_i + \sum_{j=1}^{\ell} \mu_j L_j={\rm
div}\, \mathcal{X}$ if and only if the function [(\[0gdf\])]{} is an inverse integrating factor of system [(\[0eq1\])]{}.
- If $r + \ell + s = {\rm d}({\rm d} + 1)/2$ and $s$ independent singular points are weak, then the function [(\[0gdf\])]{} for convenient $\lambda_i, \mu_j\in \mathbb{C}$ not all zero is a first integral or an inverse integrating factor of system [(\[0eq1\])]{}.
Introducing the notion of multiplicity of the invariant algebraic hypersurfaces of a polynomial vector field in $\mathbb{C}^n$, the results of Darboux integrability theory of Theorem \[thLlibPan04\] have been generalized to systems in $\mathbb{C}^n$, where $n \geq 2$, see [@Lli-Zha] and the references therein.
An improvement of the previous Darboux theorem is presented in [@ChaGiaGin99] when the system has a center. As usual $\lfloor
q \rfloor$ means the integer part of the real number $q$.
[[@ChaGiaGin99]]{} \[thChaGiaGin99\] Consider a polynomial system [(\[0eq1\])]{} of degree ${\rm
d}$, with a center at the origin and with an arbitrary linear part. Suppose that this system admits ${\rm d}({\rm d} + 1)/2 -
\lfloor ({\rm d} + 1)/2\rfloor$ invariant algebraic curves or exponential factors. Then this system has a Darboux inverse integrating factor.
In the following section we present several relations between the existence of an inverse integrating factor and the center problem.
We recall that the integrability problem consists in finding the class of functions a first integral of a given system (\[0eq1\]) must belong to. We have system (\[0eq1\]) defined in a certain class of functions, in this case, the polynomials with real coefficients $\mathbb{R}[x,y]$, and we consider the problem whether there is a first integral in another, possibly larger, class. For instance in [@Poin97], H. Poincaré stated the problem of determining when a system (\[0eq1\]) has a rational first integral. The works of M.J. Prelle and M.F. Singer [@PrelleSinger] and M.F. Singer [@Singer] go on this direction since they give a characterization of when a polynomial system (\[0eq1\]) has an elementary or a Liouvillian first integral. An important fact of their results is that invariant algebraic curves play a distinguished role in this characterization. Moreover, this characterization is expressed in terms of the inverse integrating factor.
Roughly speaking, an [*elementary function*]{} is a function constructed from rational functions by using algebraic operations, composition and exponentials, applied a finite number of times, and a [*Liouvillian function*]{} is a function constructed from rational functions by using algebraic operations, composition, exponentials and integration, applied a finite number of times. A precise definition of these classes of functions is given in [@PrelleSinger; @Singer]. We are mainly concerned with Liouvillian functions but we will state some results related to integration of a system (\[0eq1\]) by means of elementary functions.
We recall that $\mathbb{C}(x,y)$ denotes the quotient field associated to the ring of polynomials with complex coefficients, that is, $\mathbb{C}(x,y)$ is the field of rational functions with complex coefficients.
[[@PrelleSinger]]{} If the system [(\[0eq1\])]{} has an elementary first integral, then there exist $w_0, w_1, \ldots,
w_n$ algebraic over the field $\mathbb{C}(x,y)$ and $c_1, c_2,
\ldots, c_n$ in $\mathbb{C}$ such that the elementary function $$H = w_0 + \sum_{i=1}^{n} c_i \ln (w_i) \label{0ef}$$ is a first integral of system [(\[0eq1\])]{}. \[0thprellesinger0\]
The existence of an elementary first integral is intimately related to the existence of an algebraic inverse integrating factor, as the following result shows.
[[@PrelleSinger]]{} If the system [(\[0eq1\])]{} has an elementary first integral, then there is an inverse integrating factor of the form $$V= \left(
\frac{A(x,y)}{B(x,y)} \right)^{1/N} ,$$ where $A, B \in
\mathbb{C}[x,y]$ and $N$ is an integer number. \[0thprellesinger\]
The paper [@CGGLl2] is devoted to study which is the form of the inverse integrating factor of a polynomial planar system (\[0eq1\]) with a Darboux first integral $H$ of the form (\[0gdf\]). This work is an improvement of the results of Prelle and Singer in [@PrelleSinger] where it is shown that these Darboux integrable vector fields have a rational inverse integrating factor (see Theorem 7 of [@PrelleSinger]). In [@CGGLl2], another proof of this result is presented.
[[@CGGLl2]]{} If the system [(\[0eq1\])]{} has a (generalized) Darboux first integral of the form [(\[0gdf\])]{}, then there is a rational inverse integrating factor, that is, an inverse integrating factor of the form: $$V= \frac{A(x,y)}{B(x,y)} ,$$ where $A, B \in
\mathbb{C}[x,y]$. \[0thcggll2\]
Unfortunately, not all the elementary functions of the form (\[0ef\]) are of (generalized) Darboux type. That’s why, we can find systems with an elementary first integral and without a rational inverse integrating factor. The following example is of this type. The system appears in the works of Jean Moulin-Ollagnier [@Moulin1; @Moulin], although he does not give an explicit expression for the first integral. The Lotka-Volterra system: $$\dot{x}= x \left( 1 - \frac{x}{2} + y \right), \quad \dot{y}= y
\left( - 3 + \frac{x}{2} - y \right), \label{0cemo}$$ has the irreducible invariant algebraic curves $x=0$, $y=0$ and $f(x,y)=0$, where $f(x,y):= (x-2)^2 - 2 x y$. Applying the results described in [@seiden4], it can be shown that this system has no other irreducible invariant algebraic curve. The function $
V(x,y) = x^{-1/2} y^{1/2} f(x,y)$ is the only algebraic inverse integrating factor of system (\[0cemo\]) (modulus multiplication by non null constants). Since there is no rational inverse integrating factor, we deduce, by Theorem \[0thcggll2\], that there is no (generalized) Darboux first integral. An elementary first integral for this system, which is of the form (\[0ef\]), is given by: $$H(x,y):= \sqrt{2} \sqrt{x} \sqrt{y} + \ln ( x-2 + \sqrt{2} \sqrt{x}
\sqrt{y}) - \ln ( x-2 - \sqrt{2} \sqrt{x} \sqrt{y}).$$
We remark that both Theorems \[0thprellesinger\] and \[0thcggll2\] give a necessary condition to have an elementary or (generalized) Darboux, respectively, first integral. The reciprocals to the statements of Theorems \[0thprellesinger\] and \[0thcggll2\] are not true. A result to clarify the easiest functional class of the first integral once we know the inverse integrating factor appears in [@ChaGaSo06], see also [@FeLliMa], where the following theorem is stated:
[[@ChaGaSo06]]{} If the system [(\[0eq1\])]{} has a rational inverse integrating factor, then the system has a (generalized) Darboux first integral. \[0thfer\]
In any case, the following Theorem \[0thsinger\] ensures that given an algebraic inverse integrating factor, there is a Liouvillian first integral. The Liouvillian class of functions contains the rational, algebraic, Darboux and elementary classes of functions.
M.F. Singer shows in [@Singer] the characterization of the existence of a Liouvillian first integral for a system (\[0eq1\]) by means of its invariant algebraic curves.
System [(\[0eq1\])]{} has a Liouvillian first integral if, and only if, there is an inverse integrating factor of the form $ V=
\exp \left\{ \int_{(x_0,y_0)}^{(x,y)} \eta \right\} , $ where $\eta$ is a rational $1$–form such that $d\eta \equiv 0$. \[0thsinger\]
We recall that when $1$–form $\eta$ is such that $d\eta \equiv
0$, we say that it is [*closed*]{} and if there exists a function $\varphi$ such that $\eta \, = \, d \varphi$, we say that $\eta$ is [*exact*]{}.
Taking into account Theorem \[0thsinger\], C. Christopher in [@Christopher2] gives the following result, which makes precise the form of the inverse integrating factor.
[[@Christopher2]]{} If the system [(\[0eq1\])]{} has an inverse integrating factor of the form $\exp
\left\{ \int_{(x_0,y_0)}^{(x,y)} \eta \right\} , $ where $\eta$ is a rational $1$–form such that $d\eta \equiv 0$, then there exists an inverse integrating factor of system [(\[0eq1\])]{} of the form $$V= \exp \{D/E \} \prod C_i^{l_i},$$ where $D$, $E$ and the $C_i$ are polynomials in $x$ and $y$ and $l_i \in \mathbb{C}$. \[0thchris\]
We notice that $C_i=0$ are invariant algebraic curves and $\exp \{
D/E \}$ is an exponential factor for system (\[0eq1\]). In fact, since system (\[0eq1\]) is a real system, we can assume, without loss of generality, that $V$ is a real function.
Theorem \[0thchris\] states that the search for Liouvillian first integrals can be reduced to the search of invariant algebraic curves and exponential factors. Therefore, if we characterize the possible cofactors, we have the invariant algebraic curves of a system and, hence, its Liouvillian or non Liouvillian integrability.
Several works study the relation between the existence of invariant algebraic curves and the integrability of the system. The existence of an inverse integrating factor and the functional class it belongs to is crucial in the resolution of the integrability problem, as Theorems \[0thcggll2\], \[0thprellesinger\] and \[0thchris\] show. A number high enough of invariant algebraic curves of system (\[0eq1\]) implies its integrability in one of the rational, elementary or Liouvillian class, due to Darboux’s theorem and Jouanoulou’s improvement, see also Theorems \[Main-CGGLl2\] and \[Teo-V-KCZ\]. The degree of an invariant algebraic curve is not necessarily related with the integrability class of the system, see [@ChaGra03; @Moulin1] and the references therein.
We conclude this part with a theorem that summarizes some relations between inverse integrating factors and first integrals of polynomial vector fields.
\[thResum\] Let $\mathcal{X}$ be a planar polynomial vector field.
- If $\mathcal{X}$ has a Liouvillian first integral, then it has a Darboux inverse integrating factor.
- If $\mathcal{X}$ has a Darboux first integral, then it has a rational inverse integrating factor.
- If $\mathcal{X}$ has a polynomial first integral then it has a polynomial inverse integrating factor.
Statement (i) of Theorem \[thResum\] was proved in [@Singer] and [@Christopher2] and statements (ii) and (iii) in [@CGGLl2].
Another problem related with the inverse integrating factor and the integrability problem is an inverse problem: given a function $V(x,y)$, the question is to find (all the) planar differential systems with $V(x,y)$ as inverse integrating factor. In the case of searching for a Darboux inverse integrating factor, a very exhaustive approach to this problem is given in [@CLPW08; @LlibPan04; @Chara]. The main result of [@CLPW08] establishes, under two generic conditions, [**all**]{} the planar polynomial differential systems with an inverse integrating factor of the form $V(x,y)=f_1^{\lambda_1}f_2^{\lambda_2} \ldots
f_s^{\lambda_s}, $ where each $f_i(x,y)=0$ is an invariant algebraic curve of the system and $\lambda_i \in \mathbb{C}$, for $i=1,2,\ldots,s$, $s \in \mathbb{N}$. We do not reproduce the main result of [@CLPW08] because a lot of notation would need to be introduced.
In [@ChaGiaGin0] another method to construct systems with a given inverse integrating factor is described. In fact, in 1997 the function $V$ was named [*null divergence factor*]{}. This method is a generalization of the classical Darboux method to generate integrable systems. One of the main results in this paper is the following one.
[[@ChaGiaGin0]]{} \[thChaGiaGin0\] Let $\mathcal{X}_i = P_i(x,y) \partial_x + Q_i(x,y) \partial_y$, with $i = 1,2, \ldots ,n$, be $\mathcal{C}^1$ vector fields defined in an open subset $\mathcal{U} \subseteq
\mathbb{R}^2$, which have $\mathcal{C}^2$ inverse integrating factors $V_i(x, y)$, respectively. Then, the vector field $\mathcal{X} = P(x,y) \partial_x + Q(x,y) \partial_y$ with $$\begin{aligned}
P & = & \displaystyle \lambda_0 \frac{\partial V}{\partial y} \, +
\, \sum_{i=1}^{n} \lambda_i \left( \prod_{j=1,j\neq i}^{n}
V_j\right) P_i, \vspace{0.2cm} \\ Q & = & \displaystyle -\lambda_0
\frac{\partial V}{\partial x} \, + \, \sum_{i=1}^{n} \lambda_i
\left( \prod_{j=1,j\neq i}^{n} V_j\right) Q_i,\end{aligned}$$ where $\lambda_i$ are arbitrary real numbers for $i=0,1,2,\ldots,n$, has the inverse integrating factor $V(x,y)$ given by $ \displaystyle V(x,y) \, = \, \prod_{i=1}^{n} V_i(x,y).$
Indeed, if two systems have the same inverse integrating factor, a more general system which has such inverse integrating factor can be constructed, as it is shown in the following proposition.
[[@ChaGiaGin0]]{} \[propChaGiaGin0\] Let $\mathcal{X}_i = P_i(x,y) \partial_x + Q_i(x,y) \partial_y$ with $i=1,2$, be two $\mathcal{C}^1$ vector fields defined in an open subset $\mathcal{U} \subseteq
\mathbb{R}^2$, which have the same inverse integrating factor $V(x,y)$. Then, the vector field $\mathcal{X}_1+ \lambda \mathcal{X}_2$ has also the function $V (x, y)$ as an inverse integrating factor, for arbitrary values of the real parameter $\lambda$.
This proposition establishes that the set of vector fields with the same inverse integrating factor forms a $\mathbb{R}$ vector space.
A polynomial inverse integrating factor allows the study of the dynamics of system (\[0eq1\]), because a first integral can be computed, but it is not so involving as looking for a polynomial first integral. Indeed, once the degree of a polynomial inverse integrating factor is fixed, by an ansatz for instance, the problem of looking for it is reduced to a system of linear equations on its coefficients. Many authors have used this idea to find families of planar polynomial differential systems of the form (\[0eq1\]) for which all the dynamics can be determined through an inverse integrating factor.
In [@ChaGiaGin1], necessary conditions for a planar polynomial vector field to have a polynomial inverse integrating factor are obtained, see also [@FeLliMa]. All the quadratic systems with a polynomial inverse integrating factor are determined in [@CollFeLli] and all the quadratic systems with a polynomial first integral are given in [@ChaGLPR].
In [@CaLli1] all polynomial first integrals of the non-homogeneous two–dimensional Lotka–Volterra system of ordinary differential equations are determined and the role of polynomial inverse integrating factors is emphasized. Indeed, new first integrals of this class of systems having a polynomial inverse integrating factor is presented. The Liouvillian integrability of Lotka-Volterra systems has been studied in [@Moulin; @CaGiLli03].
In the work [@ChGaGi01], planar differential systems of the form (\[0eq1\]) and defined by the sum of homogeneous vector fields are studied. In particular systems with degenerate infinity are taken into account. Let us denote by $P_{\rm d}(x,y)$ and $Q_{\rm d}(x,y)$ the terms of the highest degree ${\rm d}$ in system (\[0eq1\]). We say that system (\[0eq1\]) is [*of degenerate infinity*]{} if $x Q_{\rm d}(x,y) - y P_{\rm d}(x,y)
\equiv 0$. We remark that when a system (\[0eq1\]) with degenerate infinity is embedded into a compact space (either by the Poincaré compactification into an sphere or when it is embedded in the complex projective plane) the line at infinity is filled with singular points.
We recall that a real function $H(x,y)$ is said to be $p$-degree homogeneous if $H(\lambda \, x ,
\lambda \, y) \, = \, \lambda^p \, H(x,y)$ for all $(x,y)$ in the domain of definition of $H(x,y)$ and for all $\lambda \in
\mathbb{R}$, where $p \in \mathbb{Z}$.
One of the main results in [@ChGaGi01] is the following one.
[[@ChGaGi01]]{} \[thChGaGi01\] Let us consider the following planar polynomial differential system $$\dot{x} \, = \, P_n(x,y) \, + \, x \, A_{\rm d -1}(x,y), \quad \dot{y} \, = \, Q_n(x,y) \, + \, y \, A_{\rm d
-1}(x,y), \label{eqthChGaGi01+}$$ where $P_n(x,y)$ and $Q_n(x,y)$ are homogeneous real polynomials of degree $n$, $A_{\rm d -1}(x,y)$ is a real homogeneous polynomial of degree ${\rm d}-1$ and ${\rm d}>n \geq 1$. Let us also consider the related homogeneous polynomial differential system: $$\dot{x} \, = \, P_n(x,y), \quad \dot{y} \, = \, Q_n(x,y). \label{eqthChGaGi010}$$ Then, the following statements hold.
- If $H(x,y)$ is a $p$-degree homogeneous first integral of system [(\[eqthChGaGi010\])]{}, then $H(x,y)$ is a particular solution of system [(\[eqthChGaGi01+\])]{}.
- The homogeneous function $V_{n+1}(x,y) \, := \, x
Q_n(x,y)-yP_n(x,y)$ is an inverse integrating factor of system [(\[eqthChGaGi010\])]{}.
- The homogeneous function $V_{n+1}(x,y) \, := \, x
Q_n(x,y)-yP_n(x,y)$ is a particular solution of system [(\[eqthChGaGi01+\])]{}.
- If $H(x,y)$ is a $p$-degree homogeneous first integral of system [(\[eqthChGaGi010\])]{}. Then, the function $$\left(x
Q_n(x,y)-yP_n(x,y)\right) H(x,y)^{\frac{{\rm d}-n}{p}}$$ is a (generalized) Darboux inverse integrating factor of system [(\[eqthChGaGi01+\])]{}.
The degree of a polynomial inverse integrating factor in relation with the degree ${\rm d}$ of the system can be bounded under certain conditions. The conditions established in the following result come from the embedding of a planar vector field in $\mathbb{C}P^2$, see [@seiden4] for the complete definition of this embedding. Consider the polynomial differential system (\[0eq1\]) with $P$ and $Q$ coprime polynomials of maximum degree ${\rm d}$. Extending system (\[0eq1\]) to a differential equation in the complex projective plane $\mathbb{C}P^2$, a point $(X_0:Y_0:0) \in \mathbb{C}P^2$ is termed [*infinite singular point*]{} of system (\[0eq1\]) if $(X_0,Y_0) \in \mathbb{C}^2$ is a root of the homogeneous polynomial $y P_{{\rm d}}(x,y) - x Q_{{\rm
d}}(x,y)$. Here $P_{{\rm d}}$ and $Q_{{\rm d}}$ denote the highest homogeneous components of $P$ and $Q$ of degree ${\rm d}$. Following Seidenberg, a singular point is called [*simple*]{} if the eigenvalues $\lambda, \mu \in \mathbb{C}$ associated to its linear part satisfy $\lambda \neq \mu \neq 0$ and $\lambda / \mu
\not\in \mathbb{Q}^+$, where $\mathbb{Q}^+$ stands for the positive rational numbers. Given a polynomial $F(x,y)$ of degree $n$, we denote by $\tilde{F}(X,Y,Z)$ its projectivization, that is, the homogeneous polynomial $\tilde{F}(X,Y,Z) = Z^n
F(X/Z,Y/Z)$. Taking into account these definitions we can state the following result of Walcher proved in [@Walcher].
[[@Walcher]]{} \[teopoincare2.2\] Let $V(x,y)$ be a polynomial inverse integrating factor of a polynomial system $\dot{x} = P(x,y)$, $\dot{y} = Q(x,y)$ of degree ${\rm d}$ with $P$ and $Q$ coprime. Assume that the highest homogeneous components of $P$ and $Q$ of degree ${\rm d}$ are coprime. If there is an infinite simple critical point of the system, then the degree of $V$ is exactly ${\rm d}+1$.
We observe that Theorem \[teopoincare2.2\] is also proved in [@ChGa-Poincare] with the additional assumption that the singularity at infinity $p$ satisfies $\tilde{V}(p) = 0$ where $\tilde{V}$ is the projectivization of $V$. We remark that the approach introduced in [@Walcher] uses analytical techniques such as the Poincaré–Dulac normal form and the proof given in [@ChGa-Poincare] is completely algebraic and based on the extension of differential equations to the complex projective plane and the results of Seidenberg about the reduction of singularities. The structure of polynomial inverse integrating factors is also studied by Walcher in [@Walcher2; @Walcher3].
In relation with rational first integrals and in order to state the main result of [@CGGLl2], we need to introduce some preliminary concepts, see also [@FeLli07]. Let $H = f /g$ be a rational first integral of a polynomial system (\[0eq1\]). According to Poincaré [@Poin97] we say that $c \in
\mathbb{C} \cup \{\infty \}$ is a [*remarkable value*]{} of $H$ if $f +c g$ is a reducible polynomial in $\mathbb{C}[x,y]$ (here, $c=\infty$ means that $f +c g$ denotes just $g$). In the work [@CGGLl2] it is proved that there are finitely many remarkable values for a given rational first integral $H$.
Let now $H$ be a polynomial first integral of degree $n$ of a polynomial system (\[0eq1\]). We say that the degree of $H$ is [*minimal*]{} between all the degrees of the polynomial first integrals of (\[0eq1\]) if any other polynomial first integral of (\[0eq1\]) has degree $\geq n$.
Assume $H =f / g$ to be a rational first integral. Hence, we say that $H$ has [*degree*]{} $n$ if $n$ is the maximum of the degrees of $f$ and $g$. Moreover, we say that the degree of $H$ is minimal between all the degrees of the rational first integrals of system (\[0eq1\]) if any other rational first integral of (\[0eq1\]) has degree $\geq n$.
Now suppose that $c \in
\mathbb{C}$ is a remarkable value of a rational first integral $H=f/g$ and that $\prod_{i=1}^r u_i^{\alpha_i}$ is the factorization of the polynomial $f + c g$ into irreducible factors in $\mathbb{C}[x,y]$. If some of the $\alpha_i$ is larger than 1, then we say that $c$ is a [*critical remarkable value*]{} of $H$ and that $u_i = 0$ having $\alpha_i > 1$ is a critical remarkable invariant algebraic curve of (\[0eq1\]) with exponent $\alpha_i$.
Finally, let $f$ be a polynomial. We denote by $\tilde{f}$ the homogeneous part of $f$ of highest degree and this notation is also used for a Darboux functions like (\[0gdf\]).
The main result of [@CGGLl2] is the following one.
[[@CGGLl2]]{} \[Main-CGGLl2\] Suppose that a complex polynomial vector field $\mathcal{X} =
P(x,y) \partial_x + Q(x,y) \partial_y$ of degree ${\rm d}$ with $P$ and $Q$ coprime has a Darboux first integral $H$ given by [(\[0gdf\])]{} where the polynomials $f_i$ and $g_i$ are irreducible and the polynomials $g_i$ and $h_i$ are coprime in $\mathbb{C}[x, y]$. Then the following statements hold.
- The inverse integrating factor $V_{\log H}$ associated to the first integral $\log H$ is a rational function, and it can be written in the form $V_{\log H} = \prod_{i=1}^m
u_i^{k_i}(x,y)$ with $u_i \in \mathbb{C}[x, y]$ irreducible and $k_i \in \mathbb{Z}$. Moreover, if $\mathcal{X}$ has no rational first integrals, then $V_{\log H}$ is the unique rational inverse integrating factor of $\mathcal{X}$.
- Assume that $H$ is a minimal polynomial first integral. Then there exists a polynomial inverse integrating factor.
- Suppose that $H =f / g$ is a minimal rational first integral of $\mathcal{X}$ and that $\mathcal{X}$ has no polynomial first integrals. It is not restrictive to assume that $f$ and $g$ are irreducible. Then,
- the rational function $$V_{f/g} = \frac{g^2}{\prod_i u_i^{\alpha_i-1}}$$ where the product runs over all critical remarkable invariant algebraic curves $u_i = 0$ having exponent $\alpha_i$ is an inverse integrating factor; and
- $\mathcal{X}$ has a polynomial inverse integrating factor if and only if $H$ has at most two critical remarkable values.
Additionally, if we assume for the first integral [(\[0gdf\])]{} that $f_i$ (respectively $g_j$) are different for $i
=1, \ldots, r$ (respectively $j = 1,\ldots,\ell$), and that it is complete (i.e. the unique algebraic invariant curves of system $\mathcal{X}$ are the $f_i = 0$ and the $g_j = 0$), then the following two statements hold.
- If $\mathcal{X}$ has no rational first integrals, then the inverse integrating factor $V_{\log H}$ associated to the first integral $\log H$ is the polynomial $$V_{\log H} = \prod_
{i=1}^r f_i \prod_ {j=1}^\ell g_j^{n_j+1}.$$
- If $\tilde{H}$ is a multi–valued function and $\exp(h_j/g_j)$ are exponential factors of $\mathcal{X}$ for $j = 1, \ldots,
\ell$, then $V_{\log H} = \prod_ {i=1}^r f_i \prod_ {j=1}^\ell
g_j^{n_j+1}$ is a polynomial of degree ${\rm d} + 1$.
In the particular case that $\mu_i = 0$ for $i=1,\ldots, \ell$ in the expression of (\[0gdf\]), statement (d) of Theorem \[Main-CGGLl2\] can be thought as a generalization of following result due to Kooij and Christopher [@Koo-Ch1] and independently to Żoladek [@Zol1].
[[@Koo-Ch1; @Zol1]]{} \[Teo-V-KCZ\] Consider a polynomial vector field $\mathcal{X} = P(x,y)
\partial_x + Q(x,y) \partial_y$ of degree ${\rm d}$ with $P, Q \in
\mathbb{C}[x,y]$ (resp. $P, Q \in \mathbb{R}[x,y]$) having $q$ invariant algebraic curves $f_i = 0$ such that the polynomials $f_i$ are irreducible in $\mathbb{C}[x,y]$ (resp. $\mathbb{R}[x,y]$ and satisfy that no more than two curves meet at any point of the plane $\mathbb{C}^2$ (resp. $\mathbb{R}^2$) and are not tangent at these points, no two curves have a common factor in their highest order terms and the sum of the degrees of the curves is ${\rm d}+ 1$. Then, $\prod_{i=1}^r f_i$ is an inverse integrating factor of $\mathcal{X}$ and $\prod_{i=1}^r
f_i^{\lambda_i}$ for convenient $\lambda_i \in \mathbb{C}$ (resp. $\lambda_i \in \mathbb{R}$) is a first integral of $\mathcal{X}$.
$ $From statement (d) of Theorem \[Main-CGGLl2\], the following result easily follows.
[[@CGGLl2]]{} \[Cor1-CGGLl2\] Suppose that a real polynomial vector field $\mathcal{X} = P(x,y)
\partial_x + Q(x,y) \partial_y$ of degree ${\rm d}$ with $P$ and $Q$ coprime has a Darboux first integral $H$ given by [(\[0gdf\])]{} where the polynomials $f_i$ and $g_j$ are irreducible, $f_i \neq g_j$, the polynomials $g_j$ and $h_j$ are coprime in $\mathbb{R}[x,y]$, $\exp(h_j/g_j^{n_j})$ are exponential factors of $\mathcal{X}$, the $\lambda_i$ and $\mu_j$ are either real numbers, or if some of them is complex then it appears its conjugate. If $H$ is complete and $\tilde{H}$ is multi–valued, then $V_{\log H}= \prod_ {i=1}^r f_i \prod_
{j=1}^\ell g_j^{n_j+1}$ is a polynomial of degree ${\rm d} + 1$. If the system has foci or limit cycles, these are contained in the set $\{ V_{\log H} \, = \, 0 \}$.
Using Corollary \[Cor1-CGGLl2\] particularized to quadratic (${\rm d}=2$) polynomial vector fields, in [@CGGLl2] it is obtained the next result.
[[@CGGLl2]]{} \[Cor2-CGGLl2\] Under the assumptions of Corollary [\[Cor1-CGGLl2\]]{} there are no real quadratic polynomial vector fields with a Darboux first integral [(\[0gdf\])]{} and a limit cycle.
Some examples of polynomial systems satisfying the assumptions of Corollary \[Cor1-CGGLl2\] are the following ones, see again [@CGGLl2]:
- $\dot{x}=-y - x f_1(x,y)$, $\dot{y}=x - y f_1(x,y)$ where $f_1(x,y) = x^2+y^2-1 = 0$ is an invariant circle which becomes an algebraic limit cycle. The origin is a focus and $f_2(x,y)=
x^2+y^2$ is another invariant algebraic curve. The system possesses the inverse integrating factor $V=f_1 f_2$.
- The system $\dot{x}=y - 4 x y$, $\dot{y}= -x + x^2 + 2 x y - y^2$ has the invariant algebraic curves $f_1(x,y) = 1-4 x$, $f_2(x,y) =
\sqrt{2} y + (x+y-1) i$ and $f_3(x,y) = \sqrt{2} y - (x+y-1) i$ with $i^2=-1$. The function $V(x,y) = f_1 f_2 f_3$ is an inverse integrating factor. Notice that the system has a center at $(0,0)$ and a unstable focus at $(1,0)$
The recent work [@FeLli07] is also devoted to study the properties of remarkable values. The polynomial $R(x,y):= \prod_i
u_i^{\alpha_i-1}$ defined as the product of all remarkable curves powered to their respective exponent minus one, is called the [*remarkable factor*]{}. From Theorem \[Main-CGGLl2\], if $H$ is a polynomial first integral, then the remarkable factor $R$ is a polynomial integrating factor of $\mathcal{X}$. Moreover $R$ divides the product $\prod(H+c_i)$ where $c_i$ are all the critical remarkable values of $H$. Thus the polynomial $V_R =
\prod(H+c_i)/R$ is an inverse integrating factor of the system. The following theorem gives some relations between the degree of a system with a polynomial first integral, the degree of its inverse integrating factor $V_R(x,y)$ and the number of critical remarkable values.
[[@FeLli07]]{} Let $\mathcal{X}$ be a polynomial vector field of degree ${\rm d}$ and let $H$ be a minimal polynomial first integral of $\mathcal{X}$. Consider the remarkable factor $R$ and the polynomial inverse integrating factor $V_R$. Let $k$ be the number of critical remarkable values. Then,
- $
k(k+{\rm d}) \leq \deg V_R = k \deg H - \deg R \leq k(\deg
H-1) \leq \deg R( \deg R + {\rm d})$ and
- $\deg V_R < \deg H$ if and only if $k=1$. Moreover in this case $\deg V_R = {\rm d}+1$.
\[thFeLLib07\]
On the center problem
=====================
One of the classical problems in the qualitative theory of planar analytic differential systems is to characterize the local phase portrait near an isolated singular point. By using the blow-up technique, see [@Dumor], this problem can be solved except when the singularity is monodromic, that is, it is either a focus or a center. The problem of distinguishing between a center or a focus is called the [*center problem*]{}. Another interesting problem is to know whether there exists or not a local analytic first integral defined in a neighborhood of a singular point. These two problems are equivalent when the singularity has associated nonvanishing complex conjugated eigenvalues. In this case, translating the singular point at the origin, after a linear change of variables and a rescaling of the time variable, the system can be written into the form: $$\label{V-centernondeg}
\dot{x} = -y + f(x,y) \ , \ \dot{y} = x + g(x,y) \ ,$$ where $f(x, y)$ and $g(x, y)$ are analytic functions near the origin without constant nor linear terms. It is well known since Poincaré and Liapunov that system (\[V-centernondeg\]) has a center at the origin if and only if there exists a local analytic first integral of the form $H(x,y) = x^2 + y^2 + F(x, y)$ defined in a neighborhood of the origin, where $F$ starts with terms of order higher than 2. We recall here that the [*Poincaré–Liapunov constants*]{} are the values $V_{2k}$ defined from the formal power series $H (x, y) = \sum_{n=2}^\infty H_n(x,
y)$, where $H_2(x, y) =(x^2+y^2)/2$ and $H_n(x,y)$ are homogeneous polynomials of degree $n$ satisfying $\mathcal{X} H =
\sum_{k=2}^\infty V_{2k}(x^2 + y^2)^k$. The origin is a center of (\[V-centernondeg\]) if and only if all the Poincaré–Liapunov constants vanish. When $V_{2j} \, = \, 0$ for $j=2,3, \ldots, k-1$ and $V_{2k} \neq 0$, we say that the origin of system (\[V-centernondeg\]) is a focus [*of order $k$*]{}.
The existence of invariant algebraic curves is strongly related with the origin of system (\[V-centernondeg\]) being a center, as it is explained in [@Christopher1; @Schlomiuk1; @Schlomiuk2].
The proof of the following result is a particular case of a theorem that was given by Reeb in [@Reeb] (see also Mattei and Moussu [@MaMo1] and Moussu [@Mo]). For a proof using elementary methods see [@Coll].
[[@Reeb]]{} \[Teo-V-reeb\] System [(\[V-centernondeg\])]{} has a center at the origin if and only if there is a nonzero analytic inverse integrating factor in a neighborhood of the origin.
In fact, given a system (\[V-centernondeg\]), the computational problems of looking for a first integral $H(x,y) = x^2 + y^2 +
\cdots$ or for an inverse integrating factor $V(x,y) = 1 +
\cdots$, where the dots denote terms of higher order, are of the same difficulty. Thus, the inverse integrating factor offers an alternative to the solution of the center problem. In [@flows], it has been noticed that for many systems of type (\[V-centernondeg\]) having a center at the origin there is an inverse integrating factor $V$ with very simple properties which can be globally defined in all $\mathbb{R}^2$ and which is usually a polynomial. By contrary, the first integral is, in general, a complicated expression that can not be written in terms of elementary functions.
In particular, when in system (\[V-centernondeg\]) the functions $f$ and $g$ are both quadratic then there exists a polynomial inverse integrating factor of degree $3$ or $5$, see [@Cha94], whereas the first integrals are far more complicated, see [@lunkevichsibirskii]. When the functions $f$ and $g$ in system (\[V-centernondeg\]) are both cubic homogeneous polynomials and the origin is a center, there exists a polynomial inverse integrating factor of degree at most $10$, as it is also shown in [@Cha94]. In [@GiaNdi96n], the authors study cubic systems of the form (\[V-centernondeg\]) and give some sufficient conditions for the origin to be a center. This conditions come from the imposition to the system to have an inverse integrating factor.
The work [@ChaSab99] is a survey on isochronous centers, that is, centers of the form (\[V-centernondeg\]) such that all the periodic orbits surrounding the origin have the same period. Many families of isochronous systems are listed and an explicit expression of an inverse integrating factor is given in each case. We include here a couple of results of the ones appearing in [@ChaSab99] which we have chosen for being the most known examples of isochronous centers in the literature.
The quadratic systems with a isochronous center at the origin are characterized in the following result.
\[thChaSab99a\] The origin is an isochronous center of a quadratic system [(\[V-centernondeg\])]{} if, and only if, the system can be brought by means of an affine change of coordinates and a rescaling of time, to one of the following four systems. For each case in the list we include the corresponding inverse integrating factor $V(x,y)$.
- $\dot{x} \, = \, -y + x^2-y^2$, $\dot{y} \, =
\, x(1+2y)$, with $V(x,y)=(1+2y)^2$.
- $\dot{x} \, = \, -y + x^2$, $\dot{y} \, =
\, x(1+y)$, with $V(x,y)=(1+y)^3$.
- $\dot{x} \, = \, -y -\frac{4}{3} x^2$, $\dot{y} \, =
\, x(1-\frac{16}{3}y)$, with $V(x,y)=(3-16y)(9-24y+32x^2)$.
- $\dot{x} \, = \, -y +\frac{16}{3} x^2-\frac{4}{3} y^2$, $\dot{y} \, =
\, x(1+\frac{8}{3}y)$, with $V(x,y)=(3+8y)(9+96y-256x^2+128y^2)$.
Let us consider a cubic polynomial system of the form (\[V-centernondeg\]) and let us assume that it contains no quadratic terms, that is, it is the sum of a linear system and a cubic homogeneous system. We say that such a system is cubic and with homogeneous nonlinearities. The following results characterizes which of these systems have an isochronous center at the origin.
\[thChaSab99b\] The origin is an isochronous center of a cubic system with homogeneous nonlinearities of the form [(\[V-centernondeg\])]{} if, and only if, the system can be brought by means of an affine change of coordinates and a rescaling of time, to one of the following four systems. For each case in the list we include the corresponding inverse integrating factor $V(x,y)$.
- $\dot{x} \, = \, -y + x^3-3xy^2$, $\dot{y} \, =
\, x+3x^2y-y^3$, with $V(x,y)=(x^2+y^2)^3$.
- $\dot{x} \, = \, -y + x^3-xy^2$, $\dot{y} \, =
\, x+x^2y-y^3$, with $V(x,y)=(1+2xy)^2$.
- $\dot{x} \, = \, -y +3x^2y$, $\dot{y} \, =
\, x-2x^3+9xy^2$, with $V(x,y)=(1-3x^2)^4$.
- $\dot{x} \, = \, -y -3x^2y$, $\dot{y} \, =
\, x+2x^3-9xy^2$, with $V(x,y)=(1+3x^2)^4$.
In the works [@Cha94; @Cha95], Chavarriga writes system (\[V-centernondeg\]) in polar coordinates and studies the existence of inverse integrating factors polynomial in the radial variable. In [@ChaGi96; @ChaGi97; @ChaGi98], the authors look for possible inverse integrating factors for polynomial vector fields of the form $$\label{V-integ-center}
\mathcal{X} = -y \partial_x + x \partial_y + \mathcal{X}_s \ ,$$ where $\mathcal{X}_s$ is a polynomial homogeneous vector field of degree $s \geq 2$. In particular, they use the quasi–polar coordinates $(R, \varphi)$ where $R = r^{s-1}$ and $(r, \varphi)$ are the polar coordinates, that is, $x=r \cos\varphi$, $y =r
\sin\varphi$. Next, it is assumed the existence of an inverse integrating factor $V(R, \varphi)$ of (\[V-integ-center\]) which is polynomial in the variable $R$, that is, of the form $V(R ,
\varphi) = \sum_{i=0}^p V_i(\varphi) R^i$ with $V_0(\varphi)
\equiv 1$ and where $V_i(\varphi)$ are homogeneous trigonometrical polynomials of degree $i (s-1)$. This assumption is clearly equivalent to impose an inverse integrating factor of the form $V(R, \varphi) = \prod_{i=1}^p (1+ x_i(\varphi) R)^{\alpha_i}$, with $\alpha_i \in \mathbb{R}$. The authors try to solve the system of equations for the unknown functions $x_i(\varphi)$ in the cases $p=1,2,3$. The case $p=1$ is totally solved. If $p = 2$, only is solved the case $\alpha_1 = \alpha_2 = (s + 1)/(s - 1) \pm
1 /2$ with arbitrary $s$. Finally, when $p = 3$ the following two particular cases are investigated: $\alpha_1 = \alpha_2 = \alpha_3
= 5/3$ and either $s = 2$ or $s =3$.
In [@ChaGiGra1], some invariants are determined from which a formal first integral for system (\[V-integ-center\]) can be computed. Moreover, this technique is applied to the problem of determining the centers of polynomial vector fields (\[V-integ-center\]). Recall that a complete classification of such centers is known when $s = 2, 3$ but only partial results are known in the cases $s = 4$ and $s =
5$.
Theorem \[Teo-V-reeb\] is used in [@Gine2] to find conditions to have a center. In this work, Giné proposes a formal power series $V (x, y) = \sum_{n=0}^\infty \bar{V}_n(x,
y)$, where $\bar{V}_0(x, y) \equiv 1$ and $\bar{V}_n(x,y)$ are homogeneous polynomials of degree $n$ such that $\mathcal{X} V - V
{\rm div} \mathcal{X} = \sum_{k=2}^\infty v_{2k}(x^2 + y^2)^k$, where $\mathcal{X}$ is given by (\[V-integ-center\]) and the constants $v_{2k}$ are called the [*inverse integrating factor constants*]{}. Using the above mentioned quasi–polar coordinates $(R, \varphi)$ it is shown that, if the Poincaré–Liapunov constants $V_k = 0$ for $k = 1, \ldots, m$ and $V_{m+1} \neq 0$, then $v_{m+1} = -((m+ 1)(s - 1)+ 2) V_{m+1}$. In this spirit, the paper [@Lin] is concerned with the existence of a formal integrating factor of planar analytic system having a non degenerate focus or center at the origin and gives an algorithm to calculate the Poincaré–Liapunov constants of any order.
Given a real analytic planar vector field $\mathcal{X}_0$ with a center at $p_0 \in \mathbb{R}^2$, in [@GiGiLlib1] the authors say that this center is [*limit of a linear type center*]{} if there exists a 1–parameter family $\mathcal{X}_\epsilon$ of analytic planar vector fields with $\epsilon \geq 0$, defined in a neighborhood of $p_\epsilon \in \mathbb{R}^2$ and having a non degenerate center at $p_\epsilon$ for all $\epsilon > 0$ sufficiently small. The main results of [@GiGiLlib1] are summarized as follows.
[[@GiGiLlib1]]{} \[Teo-limitcenters\] Let $\mathcal{X}_0$ be a real analytic planar vector field with a center at $p_0 \in \mathbb{R}^2$. Then, the following holds:
- If $p_0$ is a nilpotent center, then it is limit of a linear type center.
- If $p_0$ is a Hamiltonian degenerate center, then it is limit of a linear type Hamiltonian center.
- If $p_0$ is a time–reversible degenerate center, then it is limit of a time–reversible linear type center.
In the work [@Gine1], Giné continues the study of the analytic centers which are limit of linear type centers. It is proved that if a degenerate center has an analytic inverse integrating factor $V(x, y)$ which does not vanish near the center, then this degenerate center is also the limit of a linear type center (changing the time variable). The idea is as follows. Assume $V(x,y)$ is an analytic inverse integrating factor of the analytic vector field $\mathcal{X}_0 = P(x,y) \partial_x + Q(x,y)
\partial_y$ such that $(0,0)$ is a degenerate center and $V(0,0)
\neq 0$. Hence, the rescaled vector field $\mathcal{X}_0 /V$ is hamiltonian near the origin with analytic first integral $H(x,y)$. Therefore, since the perturbed vector field $\mathcal{X}_\epsilon
= \mathcal{X}_0 /V + \epsilon (-y \partial_x + x \partial_y)$ is Hamiltonian too, the origin becomes a linear type center of $\mathcal{X}_\epsilon$ for all $\epsilon \neq 0$.
Limit cycles
============
Let $V(x,y)$ be an inverse integrating factor in the open set $\mathcal{U} \subset \mathbb{R}^2$ of a $\mathcal{C}^1(\mathcal{U})$ planar vector field $\mathcal{X}$. That is, the vector field $\mathcal{X} / V$ has zero divergence, where defined. If $\mathcal{W}$ is any simply connected component of $\mathcal{U} \setminus V^{-1}(0)$, then the condition ${\rm
div}(\mathcal{X} / V) \equiv 0$ implies that $\mathcal{X} / V$ is *Hamiltonian* on $\mathcal{W}$ with $\mathcal{C}^2$ single–valued hamiltonian function $H:\mathcal{W} \to
\mathbb{R}$. Since Hamiltonian systems are area–preserving, hence have no limit cycles, and $\mathcal{X}$ and $\mathcal{X} / V$ are topologically equivalent, it follows immediately that, in the presence of an inverse integrating factor $V$, any limit cycle of $\mathcal{X}$ lies either in $V^{-1}(0)$ or in a component of $\mathcal{U} \setminus V^{-1}(0)$ that is not simply connected. Using the machinery of de Rham cohomology, Giacomini, Llibre, and Viano eliminated the latter possibility in [@GLV]. Hence, they prove the following theorem.
[[@GLV]]{} \[Teo-vanulaciclo\] Let $\gamma$ be a limit cycle of a $\mathcal{C}^1$ real planar vector field $\mathcal{X}$ and let $V$ be any inverse integrating factor of $\mathcal{X}$ defined in some neighborhood of $\gamma$. Then, $\gamma \subset V^{-1}(0)$.
A different proof of Theorem \[Teo-vanulaciclo\] can be found in [@BerroneGiacomini2; @Ga-Sh]. We would like to recall here that M.V. Dolov in [@Dolov0] studies the existence of a single valued regular integrating factor in a neighborhood of a limit cycle and presents some connections between an integrating factor and a limit cycle. Moreover, in the works [@Dolov; @Dolov2] of Dolov and coauthors, published before the proof of Theorem \[Teo-vanulaciclo\], it is shown that vector fields with a Darboux inverse integrating factor of the form $V=\exp(R)$ with rational $R$ cannot have limit cycles.
Theorem \[Teo-vanulaciclo\] has been applied in many papers to study limit cycles of a system as we will see in forthcoming sections. As an example where this theorem is applied, we would like to recall the result of Llibre and Rodríguez in [@LliRod04] where it is shown that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field. Moreover, the realization can be made by algebraic limit cycles, and an explicit polynomial vector field exhibiting any given finite configuration of limit cycles is provided. The proof of this realization makes use of the inverse integrating factor and, in particular, of Theorem \[Teo-vanulaciclo\]. A generalization of the result of Llibre and Rodríguez is given in [@Peralta05] for systems in higher dimension, that is, it is shown that any finite configuration of (smooth) cycles in $\mathbb{R}^n$ can be realized (up to global diffeomorphism) as hyperbolic and asymptotically stable limit cycles of a polynomial vector field.
$ $From Theorem \[Teo-vanulaciclo\] we have that the knowledge of an inverse integrating factor for a planar differential system (\[eq1\]) implies the knowledge of the number (and location) of the limit cycles of the system. Many authors have treated the problem of the existence of an inverse integrating factor. For a polynomial system (thus defined in the whole $\mathbb{R}^2$), the knowledge of a polynomial inverse integrating factor solves the question of the number and location of limit cycles of the polynomial system, see Section \[sectinteg\]. In [@GaGiSo1], the authors study the problem of existence of a polynomial inverse integrating factor in several cases of quadratic vector fields $\mathcal{X}$. If such an integrating factor $V(x,y)$ exists, then from Theorem \[Teo-vanulaciclo\] the curve $V = 0$ is invariant for $\mathcal{X}$ and any limit cycle of $\mathcal{X}$ lies in this curve. Therefore, in [@GaGiSo1], the authors study planar quadratic polynomial vector fields that can have limit cycles and study the nonexistence of invariant algebraic curves, polynomial inverse integrating factors and algebraic limit cycles of arbitrary degree for these systems. Ye Yian-Qian [@Yian-Qian] classified real quadratic systems that can have limit cycles in the following three families $$\dot{x} = \delta x - y + \ell x^2 + M x y + N y^2 \ , \ \
\dot{y}=x (1+a x+ b y) \ ,$$ according to: family $(I)$ if $a=b=0$; family $(II)$ if $a \neq 0$ and $b=0$; family $(III)$ if $b \neq 0$. In [@GaGiSo1] it is proved that there are not algebraic limit cycles except for $\ell N \delta
\neq 0$ and $M^2-4\ell N \geq 0$ in family $(I)$ (this result is improved in [@ChaGaSo] where it is proved that there is no algebraic limit cycle for family $(I)$). Moreover, they also prove that the polynomial inverse integrating factors into families $(I)$, $(II)_{N=0}$, $(III)_{a=0}$ and $(III)_{N=0}$ generically have at most degree $3$. So, in the studied cases, the existence of polynomial inverse integrating factor implies the nonexistence of limit cycles or at most the existence of a circle as a unique limit cycle.
Another interesting example of application of Theorem \[Teo-vanulaciclo\] is given in the proof of several extensions to the Bendixson–-Dulac Criterion to study of the number of limit cycles of planar differential systems, see [@GasGia02; @GasGia09; @GasGiaLli]. An open set $\mathcal{U} \subseteq
\mathbb{R}^2$ with smooth boundary is said to be $\ell$–connected if its fundamental group, $\pi_1(\mathcal{U})$ is $Z*
\ldots^{(\ell)} *Z$, or in other words if $\mathcal{U}$ has $\ell$ gaps. The classical Bendixson–Dulac Criterion is the following proposition, see [@GasGia02] for the statement and a short proof.
[(Bendixson–Dulac Criterion)]{} \[propgasgia02\] Let $\mathcal{U}$ be an open $\ell$–connected subset of $\mathbb{R}^2$ with smooth boundary. Let $\mathcal{X} \, = \, P(x,y) \, \partial_x \, + \, Q(x,y) \,
\partial_y$ be a vector field of class $\mathcal{C}^1$ in $\mathcal{U}$. Let $g:\mathcal{U} \to \mathbb{R}$ be a $\mathcal{C}^1$ function such that $$M\, := \, {\rm div}(g\mathcal{X}) \, = \, P \, \frac{\partial
g}{\partial x} \, + \, Q\, \frac{\partial g}{\partial y} \, + \, g
\left( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial
y}\right)$$ does not change sign in $\mathcal{U}$ and vanishes only on a null measure Lebesgue set, such that $\{ M= 0\} \cap \{
g=0\}$ does not contain periodic orbits of $\mathcal{X}$. Then the maximum number of periodic orbits of $\mathcal{X}$ contained in $\mathcal{U}$ is $\ell$. Furthermore, each one of them is a hyperbolic limit cycle that does not cut $\{g=0\}$ and its stability is given by the sign of $gM$ over it.
The zero set of inverse integrating factors
===========================================
In Theorem \[Teo-vanulaciclo\] it is shown that limit cycles $\gamma$ of a $\mathcal{C}^1$ real planar vector field $\mathcal{X}$ belong to the zero set of any inverse integrating factor of $\mathcal{X}$ defined near $\gamma$, that is, $\gamma
\subset V^{-1}(0)$.
In addition to containing any limit cycle of $\mathcal{X}$ lying in $\mathcal{U}$, the zero set of $V$ is also often connected to the separatrices of critical points of $\mathcal{X}$ in $\mathcal{U}$. To understand why, recall that integral curves of $\mathcal{X}$ that map to themselves under the action of a Lie group are *invariant solutions* for the Lie group. Recall that when $\mathcal{Y} = \xi(x,y) \partial_x + \eta(x,y)
\partial_y$ is the infinitesimal generator of a nontrivial local Lie group of symmetries of $\mathcal{X}$ then the function $V(x,y)
= \det\{\mathcal{X}, \mathcal{Y}\}$ is an inverse integrating factor of $\mathcal{X}$, as it has already been stated in Section \[sect3\]. It is obvious that every solution of $\mathcal{X}$ which remains invariant under the action of the group with infinitesimal generator $\mathcal{Y}$ must satisfy $V(x,y) = 0$. In other words, inverse integrating factors must vanish on invariant solutions.
Based on these ideas, Bluman and Anco [@Bluman-Anco] argue heuristically that separatrices should also lie in $V^{-1}(0)$. Of course, any saddle loop in a Hamiltonian system is composed of separatrices not lying in the zero set of the trivial inverse integrating factor $V \equiv 1$. Nevertheless, the idea has merit and we expect the zero set of $V$ to play a role in the dynamics of $\mathcal{X}$ and it is very surprising that this fact was not completely accomplished until recent times. We repeat verbatim the following historical development on this issue given in [@BerroneGiacomini].
> “To our best knowledge, J. M. Page was the first author in making an observation of this kind. Concretely, the idea is developed in [@Pa] of using Lie groups in the computation of singular solutions to the implicit first order differential equation $ F(x,
> y, y') = 0$. The same idea is gathered in pgs. 113 and ss. of [@Pa], where several examples of calculation of envelopes are given, and later quoted without variations in [@Co], pgs. 66 and ss.. In pg. 111 of his classical textbook [@In], first published in 1926, E. L. Ince rescue Page’s observation on envelopes but no other material is added. It would took several decades until some advance along this line of thinking might be registered. In this regard, the works of W. H. Steeb, C. E. Wulfman and G. D. Bluman and S. Kumei must be cited. In [@Steeb], Steeb discussed the connection between limit cycles of two-dimensional systems and one-parameter groups of transformations. In [@Wulfman], Wulfman stated apparently general conditions on the infinitesimal generator of a Lie group admitted by a system of autonomous differential equations in order that an invariant solution is a limit cycle of the system. However, the argument he offers to support these conditions rests on a heuristic more than rigorous basis. In turn, chapter 3 of [@Bluman-Kumei] contains a section (Section 3.6) devoted to discuss the relationships existing between invariant solutions on one hand and “exceptional paths” on the other. Even though the developments in this section of the book seems to remain also on a semi-heuristic level, several examples and exercises are provided showing how the technique works in particular systems.”
In addition to this exhaustive historical description, we also would like to add the work of González–Gascón [@Gonz-Ga] where it is pointed out that if there is an infinitesimal generator of a Lie symmetry $\mathcal{Y}$ of a vector field $\mathcal{X}$ in $\mathbb{R}^n$, then on the limit cycles (periodic isolated orbits) of $\mathcal{X}$ it follows that $\mathcal{X}$ and $\mathcal{Y}$ are parallel. This implies, in the particular case of planar fields that the associated inverse integrating factor $V = \det\{ \mathcal{X}, \mathcal{Y} \}$ vanishes on the limit cycle.
In [@BerroneGiacomini] Berrone and Giacomini showed that, under mild additional hypotheses, the separatrices of *hyperbolic* saddle–points lying in $\mathcal{U}$ are contained in $V^{-1}(0)$, and extended this result by showing that if $\Gamma$ is a compact limit set all of whose critical points are hyperbolic saddle–points, then under mild conditions $\Gamma
\subset V^{-1}(0)$ holds. Now, we summarize the results in [@BerroneGiacomini].
It is easy to see that isolated vanishing points of an inverse integrating factor are singular points of the vector field. Moreover, for non–degenerate singularities (singularities $p_0
\in \mathbb{R}^2$ of $\mathcal{X}$ with non vanishing Jacobian determinant $\det (D \mathcal{X} (p_0)) \neq 0$) one has the following result.
[[@BerroneGiacomini]]{} \[Teo1-V-BG\] Let $p_0$ be a non–degenerate critical point of a $\mathcal{C}^1$ vector field $\mathcal{X}$ and let $V$ be an inverse integrating factor defined in a neighborhood of $p_0$ and satisfying $V(p_0)
\neq 0$. If $\det (D \mathcal{X} (p_0)) > 0$ then $p_0$ is a center. On the contrary, when $\det (D \mathcal{X} (p_0)) < 0$, $p_0$ is a saddle–point.
Next theorem is concerned with the stability of isolated zeroes of an inverse integrating factor.
[[@BerroneGiacomini]]{} \[Teo1-V-BG-2\] Let $p_0$ be an isolated zero of a non–negative inverse integrating factor $V$ of a $\mathcal{C}^1$ vector field $\mathcal{X}$ defined in a neighborhood $\mathcal{U}$ of $p_0$. Then $p_0$ is a stable (resp. unstable) singular point of $\mathcal{X}$ provided that ${\rm div} \mathcal{X}|_{\mathcal{U}}
\leq 0$ (resp. $\geq$). Furthermore, $p_0$ is asymptotically stable (resp. unstable) provided that ${\rm div}
\mathcal{X}|_{\mathcal{U}} < 0$ (resp. $> 0$).
When a singularity $p_0$ of $\mathcal{X}$ is a non–isolated zero of an inverse integrating factor the following result holds. Here, given an orbit $\gamma_0$ of $\mathcal{X}$, we denote by $\omega(\gamma_0)$ and $\alpha(\gamma_0)$ its $\omega$–limit set and $\alpha$–limit set respectively.
[[@BerroneGiacomini]]{}\[Teo1-V-BG-3\] Let $p_0$ be a non–isolated zero of an inverse integrating factor $V$ of a $\mathcal{C}^1$ vector field $\mathcal{X}$. Then, one of the following two possibilities may occur:
- There exists at least an orbit $\gamma_0$ of $\mathcal{X}$ (different of $p_0$) such that $\omega(\gamma_0)=p_0$ or $\alpha(\gamma_0) = p_0$ and $V|_{\gamma_0} \equiv 0$.
- There exists a infinite sequence $\{ \gamma_n
\}_{n \in \mathbb{N}}$ of periodic orbits of $\mathcal{X}$ accumulating at $p_0$ such that $V|_{\gamma_n} \equiv 0$.
A singularity $p_0$ of the vector field $\mathcal{X}$ is called [*strong*]{} if ${\rm div} \mathcal{X}(p_0) \neq 0$. Otherwise, when ${\rm div} \mathcal{X}(p_0) = 0$, it is called [*weak*]{}. For a linear strong saddle points, it is easy to see that every inverse integrating factor must vanish on all four separatrix curves of the saddle. As it is established by the next theorem, the situation with nonlinear hyperbolic saddle points of $\mathcal{C}^1$ systems is entirely analogous to the linear case. The proof is based on the normal form of $\mathcal{X}$ near the hyperbolic saddle $p_0$ and the Stable Manifold Theorem.
[[@BerroneGiacomini]]{} \[Teo1-V-BG-4\] Let $p_0$ be a hyperbolic saddle-point of a $\mathcal{C}^1$ vector field $\mathcal{X}$ and $V$ an inverse integrating factor defined in a neighborhood $\mathcal{U}$ of $p_0$. Then $V$ vanishes on all four separatrix curves of the saddle provided that one of the following conditions holds: (i) $p_0$ is strong; (ii) $p_0$ is weak and $V(p_0) = 0$.
In the work [@GaGiGr], the previous theorem is slightly improved. If $p_0$ is a hyperbolic saddle point of a $\mathcal{C}^{k+1}$ vector field $\mathcal{X}$ whose $k^{th}$ saddle quantity is not zero and $V$ is an inverse integrating factor defined in a neighborhood of $p_0$, then $V(p_0)=0$ (and, thus, $V$ vanishes on all four separatrix curves of the saddle). For a full definition of saddle quantities see Subsection \[sect82\] in relation with system (\[normal-3.0\]).
As a corollary of Theorem \[Teo1-V-BG-4\], one can ensure the vanishing of an inverse integrating factor defined near certain saddle connections. Recall that a saddle connection is a union of saddle points and orbits connecting them.
[[@BerroneGiacomini]]{} \[Teo1-V-BG-5\] Let $V$ be an inverse integrating factor defined in a region containing a saddle connection $\Gamma$ whose critical points are $p_i$ for $i=1,\ldots, n$. If $V$ vanishes at a certain singular point $p_k$, then $V|_{\Gamma} \equiv 0$.
A [*graphic*]{} $\bar{\Gamma} = \cup_{i=1}^k \phi_i(t) \cup \{
p_1,\ldots,p_k \}$ is formed by $k$ singular points $p_1, \ldots,
p_k$, $p_{k+1} =p_1$ and $k$ oriented regular orbits $\phi_1(t),
\ldots, \phi_k(t)$, connecting them such that $\phi_i(t)$ is an unstable characteristic orbit of $p_i$ and a stable characteristic orbit of $p_{i+1}$. A graphic may or may not have associated a Poincaré return map. In case it has one, it is called a [*polycycle*]{}.
Now, let us suppose that $\Gamma$ is a [*graphic*]{}, that is, $\Gamma$ is a limit set which differs from a critical point or a periodic orbit.
[[@BerroneGiacomini]]{}\[Teo1-V-BG-6\] Let $V$ be an inverse integrating factor defined in a region containing a compact graphic $\Gamma$. Then, the following holds:
- $V$ vanishes at a critical point at least of $\Gamma$.
- If all the critical points on $\Gamma$ are non–degenerate, then $V|_{\Gamma} \equiv 0$.
The main results of the paper [@Ga-Sh] are generalizations and extensions of the previous results stated in [@BerroneGiacomini]. A key ingredient in the proof of the results of [@Ga-Sh] is the concept of an integral invariant, introduced by Poincaré in [@Poincare] for arbitrary dimension, and its relation to inverse integrating factors. We denote by $\phi(t; (x_0,y_0))$ the solution of (\[eq1\]) passing through the point $(x_0,y_0) \in \mathcal{U}$ at $t=0$; $\phi(t;
D)$ will denote the image of a domain $D \subset \mathcal{U}$ under the time–$t$ map of the flow generated by the solutions of system (\[eq1\]).
\[difig1\] Let $\mu : \mathcal{U} \subset \mathbb{R}^2 \to \mathbb{R}$ be a non–zero integrable function on $\mathcal{U}$. The integral $$\label{ifig2}
\int_{\phi(t; D)} \mu(x,y) \ dx dy$$ is an *integral invariant* of system [(\[eq1\])]{} if for any measurable set $D \subset \mathcal{U}$ the integral is independent of $t$.
The function $\mu$ is called the *density* of the integral invariant, based on the obvious hydrodynamic interpretation. Various versions of the following result can be found in textbooks, see for instance [@Andronov]. We state it in a form suited to our needs. In [@Ga-Sh] is also provided a short proof.
[[@Poincare]]{} \[lifig1\] Let $\mathcal{U}$ be an open subset of $\mathbb{R}^2$, let $V :
\mathcal{U} \to \mathbb{R}$ be a $\mathcal{C}^1$ function, and define a $\mathcal{C}^1$ function $\mu: \mathcal{U} \setminus
V^{-1}(0) \to \mathbb{R}$ by $\mu = 1 / V$. Then $V$ is an inverse integrating factor of system [(\[eq1\])]{} in $\mathcal{U}$ if and only if the integral [(\[ifig2\])]{} is an integral invariant for system [(\[eq1\])]{} on $\mathcal{U} \setminus
V^{-1}(0)$.
By using the relationship between inverse integrating factors and integral invariants given in Lemma \[lifig1\], it is easy to see the next result. The definition of parabolic or elliptic sector can be found, for instance, in [@DuLlAr].
[[@Ga-Sh]]{} \[limitpt\] Let $p_0$ be any critical point of system [(\[eq1\])]{} at which there is an elliptic or parabolic sector. If $V$ is any inverse integrating factor of [(\[eq1\])]{} defined on a neighborhood of $p_0$, then $V(p_0)=0$.
In order to state the next result, we recall that a function $f$ is called a [*Morse function*]{} if all its critical points are nondegenerate, i.e., the associated Hessian matrix has maximal rank at all the critical points. For Morse functions it is well known, see [@Hirsch] for instance, that the set of critical points is discrete, that is, has no accumulation points.
[[@Ga-Sh]]{} \[limitset\] Let $\Gamma$ be any compact $\alpha$– or $\omega$–limit set of system [(\[eq1\])]{} that contains a regular point, and let $V$ be any inverse integrating factor of [(\[eq1\])]{} defined in some neighborhood of $\Gamma$. Depending on the smoothness of $V$, the following statements hold.
- There exists a point $p$ in $\Gamma$ such that $V(p)=0$.
- If $V$ is $\mathcal{C}^2$, then either $\Gamma$ contains a point that is an accumulation point of isolated critical points of $V$ or $\Gamma \subset V^{-1}(0)$.
- If $V$ is real analytic or Morse, then $\Gamma \subset
V^{-1}(0)$.
Theorem \[Teo1-V-BG-4\] does not hold, in general, for non–hyperbolic singularities. But it does generalize for saddle or saddle–node singularities with exactly one non–zero eigenvalue, as the next result shows.
[[@Ga-Sh]]{} \[teoonenuleigen\] Suppose $p_0$ is an isolated singularity of a $\mathcal{C}^1$ vector field $\mathcal{X}$, and that $V$ is an inverse integrating factor for $\mathcal{X}$ defined in a neighborhood of $p_0$. If the linear part $D \mathcal{X}(p_0)$ has exactly one zero eigenvalue, then $V$ vanishes along any separatrix of $\mathcal{X}$ at $p_0$.
We finish this section by stating a corollary of Theorems \[Teo1-V-BG-4\] and \[teoonenuleigen\].
[[@Ga-Sh]]{} Let $\Gamma$ be a polycycle (or graphic which need not be a limit set) of system [(\[eq1\])]{} and let $V$ be any inverse integrating factor of [(\[eq1\])]{} defined in some neighborhood of $\Gamma$. Assume that the critical points of [(\[eq1\])]{} that belong to $\Gamma$ are hyperbolic saddles $p_1,
p_2, \ldots, p_n$ or saddles and saddle–nodes $q_1, q_2, \ldots,
q_m$ with exactly one zero eigenvalue. If the separatrices of $\Gamma$ are such that they always connect either $p_k$ with $p_j$ and $V(p_k)=0$ or $p_k$ with $q_j$ or $q_k$ with $q_j$ then $\Gamma \subset V^{-1}(0)$.
As the authors of [@Ga-Sh] remark, the hypothesis in Theorem \[limitset\] that $V$ be real analytic does not seem to be essential. Thus, in [@Ga-Sh] it is conjectured that only in the class $\mathcal{C}^1$ for $V$, Theorem \[limitset\] remains valid. This conjecture was solved positively in [@EncPer]. In summary, in that paper the authors prove that there always exists a smooth inverse integrating factor in a neighborhood of a limit cycle and obtain a necessary and sufficient condition for the existence of an analytic one. This condition is expressed in terms of the Ecalle–Voronin modulus of the associated Poincaré map. We recall that a germ of a map in the set of real analytic diffeomorphisms near the origin of $\mathbb{R}$ is analytically embeddable, i.e., it is the time-one map of an analytic vector field on the line, if and only if its Ecalle–Voronin modulus is trivial. The embedding properties of the Poincaré map are crucial for the proof of the next theorem.
[[@EncPer]]{} \[Teo-daniel-1\] Let $\gamma$ be a limit cycle of the analytic planar vector field $\mathcal{X}$. Then there exists a neighborhood $\mathcal{U}$ of $\gamma$ and a function $V \in \mathcal{C}^\infty(\mathcal{U})$ which is an inverse integrating factor of $\mathcal{X}$ and vanishes exactly on $\gamma$. Moreover, $V$ can be chosen analytic if and only if the Ecalle–Voronin modulus of the germ of the Poincaré map of $\mathcal{X}$ along the limit cycle $\gamma$ is trivial.
[[@EncPer]]{} \[Co-daniel-1\] If $\gamma$ is a hyperbolic limit cycle of an analytic vector field $\mathcal{X}$, then $\mathcal{X}$ admits an analytic inverse integrating factor in a neighborhood of $\gamma$.
In addition, in [@EncPer] it is also proved that a $\mathcal{C}^1$ inverse integrating factor of a $\mathcal{C}^1$ planar vector field must vanish identically on the polycycles which are limit sets of its flow. We recall that a polycycle of a $\mathcal{C}^1$ vector field is a compact invariant set which contains both regular and singular points.
[[@EncPer]]{} \[Teo-daniel-2\] Let $\mathcal{X}$ be a $\mathcal{C}^1$ vector field defined in a domain $\mathcal{U} \subseteq \mathbb{R}^2$. Suppose that $\Gamma
\subset \mathcal{U}$ is a polycycle which is a limit set of $\mathcal{X}$ and $\mathcal{X}$ has a finite number of singular points in $\Gamma$. Then if $\mathcal{X}$ admits a $\mathcal{C}^1$ inverse integrating factor $V$ in $\mathcal{U}$, then $\Gamma
\subset V^{-1}(0)$.
The main idea of the proof of Theorem \[Teo-daniel-2\] is to pull back the vector field $\mathcal{X}$ and the inverse integrating factor $V$ to the universal cover of $\mathcal{U}
\backslash \{ V \mathcal{X} = 0 \}$ and exploit the fact that $\mathcal{X}/V$ lifts to a Hamiltonian vector field in the covering space.
The existence of inverse integrating factors in a neighborhood of an elementary singularity is also established in [@EncPer]. The regularity of the inverse integrating factor depends on the kind of singularity and the proof makes crucial use of the theory of normal forms for planar vector fields. This considerably extends previous results in [@flows], where the authors prove for analytic vector fields the existence of a unique analytic inverse integrating factor in a neighborhood of a strong focus, or a non–resonant hyperbolic node, or a Siegel hyperbolic saddle.
The following result, which is stated in [@Hopf], is a summary and a generalization of several results on the existence of a smooth and non–flat inverse integrating factor $V_0(x,y)$ in a neighborhood of an isolated singular point, see [@flows; @EncPer; @GiaVia].
\[thpoint\] Let the origin be an isolated singular point of [(\[eq1\])]{} and let $\lambda, \mu \in \mathbb{C}$ be the eigenvalues associated to the linear part of [(\[eq1\])]{}. If $\lambda
\neq 0$, then there exists a smooth and non–flat inverse integrating factor $V(x,y)$ in a neighborhood of the origin.
In [@EncPer] the existence of an analytic inverse integrating factor in a neighborhood of a non-degenerate monodromic singular point of an analytic system is characterized. If the origin is a non–degenerate center or a strong focus, there exists an analytic inverse integrating factor. If the origin is a weak focus, by Theorem \[thpoint\] we have the existence of a smooth and non-flat inverse integrating factor, and there exists an analytic inverse integrating factor if and only if the Ecalle–Voronin modulus of the associated Poincaré map is trivial, see also Theorem \[Teo-daniel-1\]. In [@EncPer] the first known examples of real planar analytic vector fields not admitting an analytic inverse integrating factor in any neighborhood of either a limit cycle or an isolated singularity are given.
In [@Hopf], we show the existence of an inverse integrating factor in a neighborhood of some degenerate singular points.
[[@Hopf]]{} \[thHopfpunts\] There exists an inverse integrating factor $V(x,y)$, of class at least $\mathcal{C}^1$, in a neighborhood of the following two types of singular points: a degenerate focus without characteristic directions and a nilpotent focus.
Bifurcations
============
The inverse integrating factor has been shown to be very useful in many bifurcation problems. The books [@HaleKocak; @Roussarie] contain the main concepts and ideas of this theory in the framework of ordinary differential equations.
Consider system (\[eq1\]) and take a parametric family of systems of the form $$\label{qq1}
\dot{x} \, = \, \mathcal{P}(x, y, \varepsilon), \quad \dot{y} \, =
\, \mathcal{Q}(x, y, \varepsilon),$$ where $\mathcal{P}(x, y, \varepsilon)$ and $\mathcal{Q}(x, y,
\varepsilon)$ are analytic functions in $(x,y)$ in the same open set as $P(x,y)$ and $Q(x,y)$ (or an open set we are interested in), are analytic for $\varepsilon$ near the origin and coincide with $P(x, y) $ and $Q(x, y)$ when $\varepsilon = 0$, that is, $\mathcal{P}(x, y, 0) \, = \, P(x, y) $ and $\mathcal{Q}(x, y, 0)
\, = \, Q (x, y)$. The parameter $\varepsilon$ is called [*bifurcation parameter*]{} and we assume it is defined in a neighborhood of the origin of $\mathbb{R}^k$, with $k \in
\mathbb{N}$; in many cases we consider that $\varepsilon$ is a real one-dimensional parameter $(k=1)$. For small values of the norm of $\varepsilon$, we say that the family of systems (\[qq1\]) is a perturbation of system (\[eq1\]). When $\varepsilon$ takes values near the origin $0<|\varepsilon|<<1$, the qualitative behavior of system (\[qq1\]) can change with respect to the one of system (\[eq1\]) for $\varepsilon = 0$. In this case, we say that a bifurcation has occurred. Bifurcation theory aims at characterizing under which conditions on system (\[eq1\]) and its perturbations, this bifurcations eventually happen and which are their properties. For example, consider a singular point $p$ of system (\[eq1\]) and denote by $\lambda$ and $\mu$ the eigenvalues of the linearization of the system around $p$. If $\lambda, \mu \in \mathbb{R}$ and $\lambda \, \cdot
\, \mu \, <\, 0$, then we say that $p$ is a hyperbolic saddle and a classical result states that any perturbation of the system in a neighborhood of this point has the same qualitative behavior, that is, we have a saddle singular point that, when $\varepsilon$ tends to zero, tends to $p$. We give the adjective [*hyperbolic*]{} to those objects which maintain their qualitative nature under perturbations. In contrast, if $p$ is a singular point of center type, i.e. it has a neighborhood filled with periodic orbits, then a perturbation of the system usually breaks these orbits and the point can be transformed, for instance, into a singular point of focus type, i.e. surrounded by orbits that spiral towards (or from) it. In this case, we say that $p$ is a bifurcation point. When the considered family (\[qq1\]) shows all the possible sample of qualitative behaviors that might occur when perturbing an object of system (\[eq1\]), we say that it is an [*unfolding*]{}. The minimum number of parameters needed to have an unfolding is called the [*codimension*]{}.
Bifurcation theory is one of the most current tools used when trying to solve $16^{th}$ Hilbert problem, part b. This problem was proposed in 1900 by D. Hilbert and asks for the maximum number and possible configurations of limit cycles that a polynomial system of the form (\[0eq1\]) of degree ${\rm d}$ may have, only depending on the degree ${\rm d}$. For a fixed system, Écalle (1992) and Il’yashenko (1991) have demonstrated, in a different and independent way, that the number of limit cycles that the system may have is finite. However, the problem of determining whether there exists an upper bound on the number of limit cycles that a polynomial system of the form (\[0eq1\]) can have, only depending on the degree ${\rm d}$ of the system, is still open.
As R. Roussarie defines in [@Roussarie], given a family of systems of the form (\[qq1\]), a [*limit periodic set*]{} is a compact and nonempty subset $\Gamma$ of points so that there exists some succession $(\varepsilon_n)_n$ which tends to $\varepsilon_*$ when $n \to + \infty$ such that for every $\varepsilon_n$, the corresponding system (\[qq1\]) has a limit cycle $\gamma_{\varepsilon_n}$ which tends to $\Gamma$, in the sense of the Haussdorf distance, when $n \to + \infty$. In this context, it is assumed that the parameters take values in a compact set. Following an analogous argument to the one used to prove Poincaré–Bendixson Theorem, the structure of limit periodic sets can be determined. Given a limit periodic set of the family (\[qq1\]), we define its [*cyclicity*]{} as the maximum number of limit cycles which can be bifurcated from $\Gamma$ in this family. In [@R3], see also [@Roussarie], R. Roussarie showed that the existence of a uniform upper bound in the number of limit cycles of an analytic family (\[qq1\]) is equivalent to that each of its limit periodic sets $\Gamma$ has finite cyclicity. This equivalence and the fact that all the limit periodic sets in a family (\[qq1\]) can be determined shows how bifurcation theory allows to tackle $16^{th}$ Hilbert problem. In [@DRR], Dumortier, Roussarie and Rousseau established a list of 121 cases which are all the possible limit periodic sets that can appear within the family of quadratic systems and proposed a program, currently unfinished, to study all these graphics to demonstrate that there is a uniform upper bound for the number of limit cycles of polynomial systems of degree $2$.
The knowledge of inverse integrating factors for particular systems has simplified its study and has allowed the understanding of several bifurcations. In [@ChaGiaGin2], for instance, the following family of cubic systems $$\begin{array}{lll}
\dot{x} & = & \displaystyle \lambda x - y + \lambda m_1 x^3 + (m_2
- m_1 + m_1 m_2)x^2y+\lambda m_1 m_2 x y^2 + m_2 y^3, \\ \dot{y} &
= & \displaystyle x + \lambda y - x^3 + \lambda m_1 x^2 y + (m_1
m_2 - m_1 -1) x y^2 + \lambda m_1 m_2 y^3, \end{array}$$ where $\lambda$, $m_1$ and $m_2$ are arbitrary real parameters, is considered. The fact of knowing an inverse integrating factor $$V(x,y) \, := \, (x^2+y^2)(1+m_1 x^2+ m_1 m_2 y^2)$$ for this family of systems allows the determination of all the bifurcations within the family.
Indeed, inverse integrating factors allow the understanding of the bifurcation of limit cycles from many limit periodic sets, as we explain in this section. The main result used in this context is Theorem \[Teo-vanulaciclo\] as it states that any inverse integrating factor defined in a neighborhood of a limit cycle needs to vanish on it. We recall that the zero set of a limit cycle is formed by orbits of the system.
We split this section in three subsections depending on the considered limit periodic sets.
Bifurcation from a period annulus
---------------------------------
In this subsection we consider planar differential systems of the form (\[eq1\]) with a singular point of center type. The set of periodic orbits surrounding this point is called its [*period annulus*]{}. A perturbation of the system usually breaks these periodic orbits but some of them might be maintained as limit cycles for the perturbed system. We say that this periodic orbits have bifurcated from the period annulus.
There are several methods to determine how many limit cycles bifurcate from the periodic orbits of a period annulus. These methods are based upon different tools: the Poincaré return map, see for instance [@BlowsPerko]; the Poincaré–-Pontrjagin-–Melnikov integrals, see for instance [@GH]; averaging theory, see [@SV]; and the inverse integrating factor, see [@GLV2; @GLV3; @VLG]. This last method also gives the shape of the bifurcated limit cycles up to any order of the perturbation parameter.
We are going to explain the method described in [@GLV2]. Let us consider a Hamiltonian planar differential system with a center at the origin: $$\dot{x} \, = \, \frac{\partial H}{\partial y}, \quad \dot{y} \,
= \, - \, \frac{\partial H}{\partial x},$$ where $H(x,y)$ is the Hamiltonian function and it is analytic in a neighborhood of the origin. We denote by $\mathcal{P}$ the period annulus of the center at the origin. Any analytic (nonzero) function of the Hamiltonian is an inverse integrating factor of the system. In particular, any (nonzero) constant function is an inverse integrating factor.
Leu us consider an analytic perturbation of the previous Hamiltonian system: $$\label{hamp}
\dot{x} \, = \, \mathcal{P}(x,y,\varepsilon), \quad \dot{y} \, =
\, \mathcal{Q}(x,y,\varepsilon),$$ where $$\mathcal{P}(x,y,\varepsilon) \, : = \,
\frac{\partial H}{\partial y} \, + \, \sum_{k=1}^{\infty}
\varepsilon^k \, f_k(x,y), \quad \mathcal{Q}(x,y,\varepsilon) \,
:= \, - \, \frac{\partial H}{\partial x} \, + \,
\sum_{k=1}^{\infty} \varepsilon^k \, g_k(x,y),$$ and where $\varepsilon$ is a small real parameter and $f_k(x,y)$, $g_k(x,y)$ are analytic functions in $\mathcal{P} \cup \{(0,0)\}$. Let us look for an analytic solution $$V(x,y,\varepsilon) \, = \, \sum_{k=0}^{\infty} \varepsilon^k \,
V_k(x,y),$$ of the partial differential equation $$\mathcal{P} \, \frac{\partial V}{\partial x} \, + \, \mathcal{Q} \, \frac{\partial V}{\partial
y}\, = \, \left( \frac{\partial \mathcal{P}}{\partial x} \, + \,
\frac{\partial \mathcal{Q}}{\partial y} \right) V.$$ This partial differential equation gives a succession of linear differential equations for the functions $V_k(x,y)$ which can be solved recursively.
The equation of order $0$ in $\varepsilon$ implies that $V_0(x,y)$ needs to be an inverse integrating factor for the unperturbed system. We have, thus, that $V_0 = V_0(H)$ is a function of the Hamiltonian $H(x,y)$.
The equation of order $1$ in $\varepsilon$ gives a linear differential equation in $V_1(x,y)$ whose non-homogeneous term contains the function $V_0(x,y)$. Imposing that the function $V$ needs to be periodic when evaluated on the unperturbed periodic orbits, it can be shown that $V_0(h)$ needs to be $$V_0(h) \, = \, \lambda \, \int_{\{H=h\}} g_1(x,y) \, dx \, - \, f_1(x,y) \,
dy,$$ where $\lambda$ is a nonzero real constant, and $\{H=h\}$ denotes the periodic orbit of $\mathcal{P}$ contained in the $h$–level set of the Hamiltonian $H(x,y)$. Therefore, $V_0(h)$ is the first Poincaré–-Pontrjagin–-Melnikov integral associated to system (\[hamp\]). Once we take this expression of $V_0$, we can solve the linear differential equation for $V_1$ which is determined up to the sum of an arbitrary function of the Hamiltonian $H$.
By induction on $k$ it can be shown that $V_{k}$ is determined up to the sum of an arbitrary function of the Hamiltonian $H$ which we denote by $W_k(h)$. When solving the linear differential equation of order $k+1$ in $\varepsilon$, and imposing that the function $V$ needs to be periodic on the orbits of $\mathcal{P}$, it can be shown that $W_k(h)$ corresponds to the $k+1$ Poincaré–-Pontrjagin–-Melnikov integral associated to system (\[hamp\]).
Indeed, in [@GLV2], the authors show that, fixed a small value of $|\varepsilon|$, the zero sets of the functions $\sum_{k=0}^{n} \varepsilon^k \, V_k(x,y)$ give approximations up to order $\varepsilon^n$ of the limit cycles of system (\[hamp\]) which bifurcate from $\mathcal{P}$. When increasing the value of $n$, better approximations of these limit cycles are obtained and, thus, their shape is determined.
In [@GLV3], this method is generalized to non-Hamiltonian centers. The paper [@VLG] purports a better understanding of this method as it studies this problem when the first $\ell-1$ Poincaré–-Pontrjagin–-Melnikov functions are identically zero. The main result in this paper is that, in this case, $V_0(h)$ is the first non identically zero Poincaré–-Pontrjagin–-Melnikov function.
Most of these ideas are also used in [@GiaLliVia03] to determine semistable limit cycles that bifurcate from $\mathcal{P}$. Moreover, the method is applied to study the limit cycles which bifurcate from a Liénard system.
We remark that this method is not only an alternative to the other methods as it shows how the inverse integrating factor is linked to bifurcation problems. This method is computationally as difficult as any other method but, moreover, it provides the shape of the bifurcated limit cycles.
Bifurcation from monodromic $\omega$-limit sets \[sect82\]
----------------------------------------------------------
The work [@GaGiGr] is concerned with planar real analytic systems (\[eq1\]) with an analytic inverse integrating factor defined in a neighborhood of a regular orbit $\phi(t)$. First of all it is shown that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit, see equation (\[eqvpi\]). Taking two transversal sections $\Sigma_1$ and $\Sigma_2$ based on $\phi(t)$, it is studied the transition map of the flow of $\mathcal{X}$ in a neighborhood of $\phi(t)$. This transition map is studied by means of the [*Poincaré map*]{} $\Pi: \Sigma_1 \to \Sigma_2$. Given a point in $\Sigma_1$, we consider the orbit of (\[eq1\]) with it as initial point and we follow this orbit until it first intersects $\Sigma_2$.
Let $(\varphi(s), \psi(s)) \in \mathcal{U}$, with $s \in
\mathcal{I} \subseteq \mathbb{R}$ be a parameterization of the regular orbit $\phi(t)$ between the base points of $\Sigma_1$ and $\Sigma_2$. Given a point $(x,y)$ in a sufficiently small neighborhood of the orbit $(\varphi(s), \psi(s))$, we can always encounter values of the [*curvilinear coordinates*]{} $(s, n)$ that realize the following change of variables: $x(s,n)=\varphi(s)- n \psi'(s)$, $y(s,n) =\psi(s) + n
\varphi'(s)$. We remark that the variable $n$ measures the distance perpendicular to $\phi(t)$ from the point $(x,y)$ and, therefore, $n=0$ corresponds to the considered regular orbit $\phi(t)$. We can assume, without loss of generality, that the transversal section $\Sigma_1$ corresponds to $\Sigma_1 \, := \,
\left\{ s=0 \right\}$ and $\Sigma_2$ to $\Sigma_2 \, := \,
\left\{ s=L \right\}$, for a certain real number $L>0$. We perform the change to curvilinear coordinates $(x,y) \mapsto (s,n)$ in a neighborhood of the regular orbit $n=0$ with $s \in \mathcal{I} \,
= \, [0, L]$. Then, system (\[eq1\]) is written as the following ordinary differential equation: $$\frac{dn}{ds} \, = \, F(s,n) \ . \label{eq2.0}$$ We denote by $\Psi(s;n_0)$ the flow associated to the equation (\[eq2.0\]) with initial condition $\Psi(0;n_0)=n_0$. In these coordinates, the Poincaré map $\Pi: \Sigma_1 \to \Sigma_2$ between these two transversal sections is given by $\Pi(n_0)=\Psi(L;n_0)$.
We assume the existence of an analytic inverse integrating factor $V(x,y)$ in a neighborhood of the considered regular orbit $\phi(t)$ of the analytic system (\[eq1\]). In fact, when $\Sigma_1 \neq
\Sigma_2$ and no return is involved, there always exists such an inverse integrating factor. The change to curvilinear coordinates gives us an inverse integrating factor for equation (\[eq2.0\]), denoted by $\tilde{V}(s,n)$ and which satisfies $$\frac{\partial \tilde{V}}{\partial s} \, + \,
\frac{\partial \tilde{V}}{\partial n} \, F(s,n) \, = \,
\frac{\partial F}{\partial n} \, \tilde{V}(s,n). \label{eq3.0}$$
Now, we can state one of the main results of [@GaGiGr].
[[@GaGiGr]]{}\[thvpi\] We consider a regular orbit $\phi(t)$ of the analytic system [(\[eq1\])]{} which has an inverse integrating factor $V(x,y)$ of class $\mathcal{C}^1$ defined in a neighborhood of it and we consider the Poincaré map associated to the regular orbit between two transversal sections $\Pi: \Sigma_1 \to \Sigma_2$. We perform the change to curvilinear coordinates and we consider the ordinary differential equation [(\[eq2.0\])]{} with the inverse integrating factor $\tilde{V}(s,n)$ which is obtained from $V(x,y)$. In these coordinates, the transversal sections can be taken such that $\Sigma_1 \, := \, \left\{ s=0\right\}$ and $\Sigma_2 \, := \, \left\{ s=L\right\}$, for a certain real value $L>0$. We parameterize $\Sigma_1$ by the real value of the coordinate $n$. The following identity holds. $$\tilde{V} \left(L,\Pi(n)\right) \, = \, \tilde{V} \left(0,n\right)
\Pi'(n). \label{eqvpi}$$
Theorem \[thvpi\] is the key point to prove Theorems \[th-mult-limv\] and \[th-mult-loop0\].
Further, in [@GaGiGr] the authors consider regular orbits whose Poincaré map is a return map and take profit from the result stated in Theorem \[thvpi\] in order to study the Poincaré map associated to a limit cycle or to a homoclinic loop, in terms of the inverse integrating factor. To do that, the following definition of [*vanishing multiplicity*]{} of an analytic inverse integrating factor $V(x,y)$ of the analytic system (\[eq1\]) over a regular orbit $\phi(t)$ is needed.
[[@GaGiGr]]{} Let $V(x,y)$ be an analytic inverse integrating factor of the analytic system [(\[eq1\])]{} and $\phi(t)$ a regular orbit of it parameterized by $(\varphi(s), \psi(s)) \in \mathcal{U}$, with $s \in \mathcal{I} \subseteq \mathbb{R}$. Consider the local change of coordinates $x(s,n)=\varphi(s) - n \psi'(s)$, $y(s,n)
=\psi(s) + n \varphi'(s)$ defined in a neighborhood of the considered regular orbit $n=0$ and take the following Taylor development around $n=0$: $$\label{eq-vnonula.0}
V(x(s,n), y(s,n)) \, = \, n^{m} \, v(s) \, + \, O(n^{m+1}) ,$$ where $m$ is an integer with $m \geq 0$ and the function $v(s)$ is not identically null, we say that $V$ has multiplicity $m$ on $\phi(t)$.
In fact, in [@GaGiGr] it is proved that $v(s) \neq 0$ for any $s \in \mathcal{I}$, and thus, the vanishing multiplicity of $V$ on $\phi(t)$ is well–defined over all its points.
Let us consider as regular orbit a limit cycle $\gamma$ and we use the parameterization of $\gamma$ in curvilinear coordinates $(s,n)$ with $s \in [0, L)$. Thus, the Poincaré map associated to $\gamma$ is $\Pi(n_0) \, = \, \Psi(L;n_0)$. It is well known that $\Pi$ is analytic in a neighborhood of $n_0=0$. We recall that the periodic orbit $\gamma$ is a limit cycle if, and only if, the Poincaré return map $\Pi$ is not the identity. If $\Pi$ is the identity, we have that $\gamma$ belongs to a period annulus. We recall the definition of multiplicity of a limit cycle: $\gamma$ is said to be a limit cycle of [*multiplicity*]{} $1$ if $\Pi'(0) \neq 1$ and $\gamma$ is said to be a limit cycle of multiplicity $m$ with $m \geq 2$ if $\Pi(n_0) \, = \, n_0 \, + \,
\beta_m \, n_0^m \, + \, O(n_0^{m+1})$ with $\beta_m \neq 0$. Then, one has the following result for limit cycles.
[[@GaGiGr]]{} \[th-mult-limv\] Let $\gamma$ be a periodic orbit of the analytic system [(\[eq1\])]{} and let $V$ be an analytic inverse integrating factor defined in a neighborhood of $\gamma$.
- If $\gamma$ is a limit cycle of multiplicity $m$, then $V$ has vanishing multiplicity $m$ on $\gamma$.
- If $V$ has vanishing multiplicity $m$ on $\gamma$, then $\gamma$ is a limit cycle of multiplicity $m$ or it belongs to a continuum of periodic orbits.
Since the Poincaré map of a periodic orbit is an analytic function and the multiplicity of a limit cycle is a natural number, the following corollary is obtained.
[[@GaGiGr]]{} \[cor-mult-limv\] Let $\gamma$ be a periodic orbit of the analytic system [(\[eq1\])]{} and let $V$ be an inverse integrating factor of class $\mathcal{C}^1$ defined in a neighborhood of $\gamma$. We take the change to curvilinear coordinates $x(s,n)=\varphi(s) - n
\psi'(s)$, $y(s,n) =\psi(s) + n \varphi'(s)$ defined in a neighborhood of $\gamma$. If we have that the leading term in the following development around $n=0$: $$V(x(s,n), y(s,n)) \, = \, n^{\rho} \, v(s) \, + \, o(n^{\rho}) ,$$ where $v(s) \not\equiv 0$ is such that either $\rho=0$ or $\rho
> 1$ and $\rho$ is not a natural number, then $\gamma$ belongs to a continuum of periodic orbits.
A regular orbit $\phi(t) = (x(t), y(t))$ of (\[eq1\]) is called a [*homoclinic orbit*]{} if $\phi(t) \to p_0$ as $t \to \pm
\infty$ for some singular point $p_0$. A [*homoclinic loop*]{} is the union $\Gamma = \phi(t) \cup \{p_0\}$. We assume that $p_0$ is a hyperbolic saddle, that is, a critical point of system (\[eq1\]) such that the eigenvalues of the Jacobian matrix $D
\mathcal{X}(p_0)$ are both real, different from zero and of contrary sign. We remark that this type of graphics always has associated (maybe only its inner or outer neighborhood) a Poincaré return map $\Pi : \Sigma \to \Sigma$ with $\Sigma$ any local transversal section through a regular point of $\Gamma$. We will assume that $\Gamma$ is a compact invariant set. A goal in [@GaGiGr] is to study the cyclicity of the described homoclinic loop $\Gamma$ in terms of the vanishing multiplicity of an inverse integrating factor. Roughly speaking, the [*cyclicity*]{} of $\Gamma$ is the maximum number of limit cycles which bifurcate from it under an analytic perturbation of the analytic system (\[eq1\]). Before state the result for homoclinic loops, we recall briefly that the first saddle quantity is $\alpha_1 \, = \, {\rm div\, } \mathcal{X}(p_0)$ and it classifies the point $p_0$ between being strong (when $\alpha_1
\neq 0$) or weak (when $\alpha_1=0$). If $p_0$ is a weak saddle point, the saddle quantities are the obstructions for it to be analytically orbitally linearizable. In order to define the next saddle quantities associated to $p_0$, we translate the saddle–point $p_0$ to the origin of coordinates and we make a linear change of variables so that its unstable (resp. stable) separatrix has the horizontal (resp. vertical) direction at the origin. Let $p_0$ be a weak hyperbolic saddle point situated at the origin of coordinates and whose associated eigenvalues are taken to be $\pm 1$ by a rescaling of time, if necessary. Then, it is well known the existence of an analytic near–identity change of coordinates that brings the system into: $$\label{normal-3.0}
\begin{array}{lll}
\dot{x} & = & \displaystyle x \, + \, \sum_{i=1}^{k-1} a_i \,
x^{i+1} y^i \, + \, a_k \, x^{k+1} y^k \, + \, \cdots \ ,
\vspace{0.2cm} \\ \dot{y} & = & \displaystyle -y \, - \,
\sum_{i=1}^{k-1} a_i \, x^{i} y^{i+1} \, - \, b_k \, x^k y^{k+1}
\, + \, \cdots \ , \end{array}$$ with $a_k - b_k \neq 0$ and where the dots denote terms of higher order. The first non–vanishing saddle quantity is defined by $\alpha_{k+1} := a_k-b_k$.
[[@GaGiGr]]{} \[th-mult-loop0\] Let $\Gamma$ be a compact homoclinic loop through the hyperbolic saddle $p_0$ of the analytic system [(\[eq1\])]{} whose Poincaré return map is not the identity. Let $V$ be an analytic inverse integrating factor defined in a neighborhood of $\Gamma$ with vanishing multiplicity $m$ over $\Gamma$. Then, $m \geq 1$ and the first possible non–vanishing saddle quantity is $\alpha_m$. Moreover,
- the cyclicity of $\Gamma$ is $2m-1$, if $\alpha_m \, \neq \, 0$,
- the cyclicity of $\Gamma$ is $2m$, otherwise.
In addition, in [@GaGiGr] it is described one obstruction to the existence of an analytic inverse integrating factor defined in a neighborhood of certain homoclinic loops. First of all, we recall some concepts. By an affine change of coordinates, in a neighborhood of a hyperbolic saddle, any analytic system can be written as $ \dot{x} = \lambda x + f(x,y)$, $\dot{y} = \mu y +
g(x,y)$, where $f$ and $g$ are analytic in a neighborhood of the origin with lowest terms at least of second order and $\mu<0<\lambda$. This hyperbolic saddle is analytically [*orbitally linearizable*]{} if there exists an analytic near–identity change of coordinates transforming the system to $\dot{x} =
\lambda x h(x,y)$, $\dot{y} = \mu y h(x,y)$ with $h(0,0)=1$. On the other hand, when $\mu/\lambda = -q/p \in \mathbb{Q}^{-}$ with $p$ and $q$ natural and coprime numbers, the saddle is called $p:q$ resonant.
[[@GaGiGr]]{} \[prop-noV0\] Suppose that the analytic system [(\[eq1\])]{} has a homoclinic loop $\Gamma$ through the hyperbolic saddle point $p_0$ which is not orbitally linearizable, $p:q$ resonant and strong $(p
\neq q)$. Then, there is no analytic inverse integrating factor $V(x,y)$ defined in a neighborhood of $\Gamma$.
Generalized Hopf Bifurcation
----------------------------
Let us consider a planar real system (\[eq1\]), $\dot{x} =
P(x,y)$, $\dot{y} = Q(x,y)$ and suppose that it is analytic near an isolated monodromic singular point $p_0$ which we assume to be at the origin. We associate to system (\[eq1\]) the vector field $\mathcal{X}_0 \, = \, P(x,y) \partial_x \, + \, Q(x,y)
\partial_y$. We consider an analytic perturbation of system (\[eq1\]) of the form: $$\dot{x} \, = \, P(x,y) \, + \, \bar{P}(x,y,\varepsilon), \qquad
\dot{y} \, = \, Q(x,y) \, + \, \bar{Q}(x,y,\varepsilon),
\label{eq2}$$ where $\varepsilon \in \mathbb{R}^p$ is the perturbation parameter, $0<\|\varepsilon\|<<1$ and the functions $\bar{P}(x,y,\varepsilon)$ and $\bar{Q}(x,y,\varepsilon)$ are analytic for $(x,y) \in \mathcal{U}$, analytic in a neighborhood of $\varepsilon=0$ and $\bar{P}(x,y,0)= \bar{Q}(x,y,0)\equiv 0$. We associate to this perturbed system (\[eq2\]) the vector field $\mathcal{X}_\varepsilon \, = \, (P(x,y) \, + \,
\bar{P}(x,y,\varepsilon))\partial_x \, + \, (Q(x,y) \, + \,
\bar{Q}(x,y,\varepsilon)) \partial_y$.
We say that a limit cycle $\gamma_\varepsilon$ of system (\[eq2\]) [*bifurcates from the origin*]{} if it tends to the origin (in the Hausdorff distance) as $\varepsilon \to 0$. We are interested in giving a sharp upper bound for the number of limit cycles which can bifurcate from the origin $p_0$ of system (\[eq1\]) under any analytic perturbation with a finite number $p$ of parameters. The word sharp means that there exists a system of the form (\[eq2\]) with exactly that number of limit cycles bifurcating from the origin, that is, the upper bound is realizable. This sharp upper bound is called the [*cyclicity*]{} of the origin $p_0$ of system (\[eq1\]) and will be denoted by ${\rm Cycl}(\mathcal{X}_\varepsilon,p_0)$ all along this section.
In [@Hopf] we consider systems of the form (\[eq1\]) where the origin $p_0$ is a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. The results of [@Hopf] do not establish that the cyclicity of this type of singular points is finite but give an effective procedure to study it. In the three mentioned types of focus points, we will consider a change to (generalized) polar coordinates which embed the neighborhood $\mathcal{U}$ of the origin into a cylinder $C\, = \, \left\{
(r,\theta) \in \mathbb{R} \times \mathcal{S}^1 \, : \, |r|<\delta
\right\}$ for a certain sufficiently small value of $\delta>0$. This change to polar coordinates is a diffeomorphism in $\mathcal{U} - \{(0,0)\}$ and transforms the origin of coordinates to the circle of equation $r=0$. In these new coordinates, system (\[eq1\]) can be seen as a differential equation over the cylinder $C$ of the form: $$\frac{dr}{d\theta} \, = \, \mathcal{F}(r,\theta), \label{eq3}$$ where $\mathcal{F}(r,\theta)$ is an analytic function in $C$. We consider an inverse integrating factor $V(r,\theta)$ of equation (\[eq3\]), that is, a function $V: C \to \mathbb{R}$ of class $\mathcal{C}^1(C)$, which is non locally null and which satisfies the following partial differential equation: $$\frac{\partial V(r,\theta)}{\partial \theta} \, + \, \frac{\partial
V(r,\theta)}{\partial r} \, \mathcal{F}(r,\theta) \, = \,
\frac{\partial \mathcal{F}(r,\theta)}{\partial r} \, V(r,\theta).$$ We remark that since $V(r,\theta)$ is a function defined over the cylinder $C$ it needs to be $T$–periodic in $\theta$, where $T$ is the minimal positive period of the variable $\theta$, that is, we consider the circle $\mathcal{S}^1 \, = \, \mathbb{R} /
[0,T]$. The function $V(r,\theta)$ is smooth ($\mathcal{C}^{\infty}$) and non–flat in $r$ in a neighborhood of $r=0$.
Let us consider the Taylor expansion of the function $V(r,\theta)$ around $r=0$: $ V(r,\theta) \, = \, v_m(\theta) \, r^m \, + \,
\mathcal{O}(r^{m+1}), $ where $v_m(\theta) \not\equiv 0$ for $\theta \in \mathcal{S}^1$ and $m$ is an integer number with $m
\geq 0$. We say that $m$ is the [*vanishing multiplicity*]{} of $V(r,\theta)$ on $r=0$. The uniqueness of $V(r,\theta)$ implies that the number $m$ corresponding to the vanishing multiplicity of $V(r,\theta)$ on $r=0$ is well–defined.
We consider a system (\[eq1\]) of the form: $$\label{eqdeg}
\dot{x} \, = \, p_d(x,y)\, +\, P_{d+1}(x,y), \qquad \dot{y} \, =
\, q_d(x,y)\, +\, Q_{d+1}(x,y),$$ where $d \geq 1$ is an odd number, $p_d(x,y)$ and $q_d(x,y)$ are homogeneous polynomials of degree $d$ and $P_{d+1}(x,y),
Q_{d+1}(x,y) \in \mathcal{O}(\|(x,y)\|^{d+1})$. We assume that $p_d^2(x,y)+q_d^2(x,y) \not\equiv 0$. A [*characteristic direction*]{} for the origin of system (\[eqdeg\]) is a linear factor in $\mathbb{R}[x,y]$ of the homogeneous polynomial $xq_d(x,y)-yp_d(x,y)$. If there are no characteristic directions, then the origin is a monodromic singular point of system (\[eqdeg\]), that is, it is either a center or a focus.
In relation with system (\[eq2\]), an analytic perturbation field $(\bar{P}(x,y,\varepsilon), \bar{Q}(x,y,\varepsilon))$ is said to have subdegree $s$ if $(\bar{P}(x,y,\varepsilon), \bar{Q}(x, y,
\varepsilon)) = \mathcal{O}( \|(x,y)\|^s )$. In this case, we denote by $\mathcal{X}_\varepsilon^{[s]}\, = \, (P(x,y) \, + \,
\bar{P}(x,y,\varepsilon))\partial_x \, + \, (Q(x,y) \, + \,
\bar{Q}(x,y,\varepsilon)) \partial_y$ the vector field associated to such a perturbation.
[[@Hopf]]{} \[thdeg\] We assume that the origin $p_0$ of system [(\[eqdeg\])]{} is monodromic and without characteristic directions. Take polar coordinates $x= r \cos\theta$, $y= r \sin\theta$ and let $V(r,\theta)$ be an inverse integrating factor of the corresponding equation [(\[eq3\])]{} which has a Laurent expansion in a neighborhood of $r=0$ of the form $ V(r, \theta) \,
= \, v_{m}(\theta) \, r^m \, + \, \mathcal{O}(r^{m+1}), $ with $v_m(\theta) \not\equiv 0$ and $m \in \mathbb{Z}$.
- If $m \leq 0$ or $m$ is even, then the origin of system [(\[eqdeg\])]{} is a center.
- If the origin of system [(\[eqdeg\])]{} is a focus, then $m \geq 1$, $m$ is an odd number and the cyclicity ${\rm
Cycl}(\mathcal{X}_\varepsilon,p_0)$ of the origin of system [(\[eqdeg\])]{} satisfies ${\rm Cycl}(\mathcal{X}_\varepsilon,p_0)
\geq (m+d)/2-1$. In this case $m$ is the vanishing multiplicity of $V(r,\theta)$ on $r=0$.
- If, moreover, the focus is non–degenerate $(d=1)$, then the aforementioned lower bound is sharp, that is, ${\rm
Cycl}(\mathcal{X}_\varepsilon,p_0) = (m-1)/2$.
- If only perturbations whose subdegree is greater than or equal to $
d$ are considered, then the maximum number of limit cycles which bifurcate from the origin is $(m-1)/2$, that is, ${\rm
Cycl}(\mathcal{X}_\varepsilon^{[d]},p_0) = (m-1)/2$.
\[remdeg\] From the proof of Theorem [\[thdeg\]]{}, it follows that if there exists an inverse integrating factor $V_0(x,y)$ of system [(\[eqdeg\])]{} such that $V_0(r \cos
\theta, \, r \sin \theta)/r^d$ has a Laurent expansion in a neighborhood of $r=0$, then the exponents of the leading terms of $V_0(r \cos \theta, \, r \sin \theta)/r^d$ and $V(r,\theta)$ coincide. Thus, the vanishing multiplicity $m$ can be computed without passing the system to polar coordinates.
We assume that the origin of system (\[eqdeg\]) is a focus without characteristic directions and that the vanishing multiplicity of an inverse integrating factor on it is $m$. If system (\[eqdeg\]) is written as $\dot{x} \, = \, P(x,y)$ and $\dot{y} \, = \, Q(x,y)$, then the system: $$\label{eqdegp1} \dot{x} \, = \, P(x,y) \, + \, x
\, K(x,y,\varepsilon), \qquad \dot{y} \, = \, Q(x,y) \, + \, y \,
K(x,y,\varepsilon),$$ where $L \,
: =\, (m+d)/2-1$ and $\displaystyle K(x,y,\varepsilon) \, = \,
\sum_{i=0}^{k-1} \varepsilon^{k-i} \, a_i \, (x^2+y^2)^{i+
\frac{d-1}{2}}$, has at least $(m+d)/2-1$ limit cycles bifurcating from the origin for convenient values of the real parameters $a_i$. We recall that both $m$ and $d$ are odd and $d\geq 1$, $m
\geq 1$.
We say that the origin of system (\[eq1\]) is a [*nilpotent singular point*]{} if it is a degenerate singularity that can be written as: $$\label{eqnil}
\dot{x} = y + P_2(x,y) \ , \ \dot{y} = Q_2(x,y) \ ,$$ with $P_2(x,y)$ and $Q_2(x,y)$ analytic functions near the origin without constant nor linear terms. The following theorem is due to Andreev [@Andreev] and it solves the monodromy problem for the origin of system (\[eqnil\]).
\[thandreev\] [[@Andreev]]{} Let $y=F(x)$ be the solution of $y + P_2(x,y)
= 0$ passing through $(0, 0)$. Define the functions $f(x) = Q_2(x,
F(x)) = a x^\alpha + \cdots$ with $a \neq 0$ and $\alpha \geq 2$ and $\phi(x) = (\partial P_2/\partial x \, + \, \partial Q_2 /
\partial y)(x,F(x))$. We have that either $\phi(x) = b x^\beta + \cdots$ with $b \neq 0$ and $\beta \geq 1$ or $\phi(x) \equiv 0$. Then, the origin of [(\[eqnil\])]{} is monodromic if, and only if, $a < 0$, $\alpha = 2 n-1$ is an odd integer and one of the following conditions holds: (i) $\beta > n-1$; (ii) $\beta = n-1$ and $b^2+4 a n < 0$; (iii) $\phi(x) \equiv 0$.
\[defnil\] We consider a system of the form [(\[eqnil\])]{} with the origin as a monodromic singular point. We define its [*Andreev number*]{} $n \geq 2$ as the corresponding integer value given in Theorem [*\[thandreev\]*]{}.
We consider system (\[eqnil\]) and we assume that the origin is a nilpotent monodromic singular point with Andreev number $n$. Then, the change of variables $$\label{change1}
(x,y) \, \mapsto \, (x, y-F(x)),$$ where $F(x)$ is defined in Theorem \[thandreev\], and the scaling $$\label{change2}
(x,y) \, \mapsto \, (\xi \, x, -\xi \, y),$$ with $\xi = (-1/a)^{1/(2-2n)}$, brings system (\[eqnil\]) into the following analytic form for monodromic nilpotent singularities $$\label{eqnil2}
\dot{x}\, = \, y\, (-1 + X_1(x,y)), \quad \dot{y} \, =\, f(x) + y
\, \phi(x) + y^2\, Y_0(x,y),$$ where $X_1(0,0)=0$, $f(x) = x^{2n-1} + \cdots$ with $n \geq 2$ and either $\phi(x) \equiv 0$ or $\phi(x) = b x^\beta + \cdots$ with $\beta \geq n-1$. We remark that we have relabelled the functions $f(x)$, $\phi(x)$ and the constant $b$ with respect to the ones corresponding to system (\[eqnil\]).
We are going to transform system (\[eqnil2\]) to an equation over a cylinder of the form (\[eq3\]). The transformation depends on the Andreev number $n$ and it is given through the [*generalized trigonometric functions*]{} defined by Lyapunov as the unique solution $x(\theta) = {\rm Cs} \,
\theta$ and $y(\theta) = {\rm Sn}\, \theta$ of the following Cauchy problem $$\label{Cauchy}
\frac{d x}{d \theta} \, =\, -y , \ \frac{d y}{d \theta}\, =\,
x^{2n-1}, \qquad x(0)=1 , \, y(0)=0 .$$ We introduce in $\mathbb{R}^2 \backslash \{(0,0)\}$ the change to [*generalized polar coordinates*]{}, $\, (x,y) \mapsto
(r,\theta)$, defined by $$\label{change} x\, =\, r
\, {\rm Cs}\, \theta , \qquad y\, =\, r^n \, {\rm Sn}\, \theta.$$
We consider the following definition, which will be used in the following Theorem \[thnil\].
\[defnilqh\] An analytic perturbation vector field $(\bar{P}(x,y,\varepsilon),
\bar{Q}(x,y,\varepsilon))$ is said to be [*$(1,n)$–quasihomogeneous of weighted subdegrees $(w_x,w_y)$*]{} if $\bar{P}(\lambda x,\lambda^n y,\varepsilon) \, = \,
\mathcal{O}(\lambda^{w_x})$ and $\bar{Q}(\lambda x,\lambda^n
y,\varepsilon) \, = \, \mathcal{O}(\lambda^{w_y})$. In this case, we denote by $\mathcal{X}_\varepsilon^{[w_x,w_y]}\, = \, (P(x,y)
\, + \, \bar{P}(x,y,\varepsilon))\partial_x \, + \, (Q(x,y) \, +
\, \bar{Q}(x,y,\varepsilon)) \partial_y$ the vector field associated to such a perturbation.
The following theorem is one of the main results of [@Hopf]. The symbol $\lfloor x \rfloor$ denotes the integer part of $x$.
[[@Hopf]]{} \[thnil\] We assume that the origin of system [(\[eqnil\])]{} is monodromic with Andreev number $n$. Take generalized polar coordinates (\[change\]) and let $V(r,\theta)$ be an inverse integrating factor of the corresponding equation [(\[eq3\])]{} which has a Laurent expansion in a neighborhood of $r=0$ of the form $ V(r, \theta) \, = \, v_{m}(\theta) \, r^m \, +
\, \mathcal{O}(r^{m+1}), $ with $v_m(\theta) \not\equiv 0$ and $m
\in \mathbb{Z}$.
- If $m \leq 0$ or $m+n$ is odd, then the origin of system [(\[eqnil\])]{} is a center.
- If the origin of system [(\[eqnil\])]{} is a focus, then $m \geq 1$, $m+n$ is even and its cyclicity ${\rm
Cycl}(\mathcal{X}_\varepsilon,p_0)$ satisfies $\, {\rm
Cycl}(\mathcal{X}_\varepsilon,p_0) \, \geq \, (m+n)/2\, -\, 1 $. In this case, $m$ is the vanishing multiplicity of $V(r,\theta)$ on $r=0$.
- If the origin of system [(\[eqnil2\])]{} is a focus and if only analytic perturbations of $(1,n)$–quasihomogeneous weighted subdegrees $(w_x,w_y)$ with $w_x \geq n$ and $w_y \geq 2n-1$ are taken into account, then the maximum number of limit cycles which bifurcate from the origin is $\lfloor (m-1)/2 \rfloor$, that is, ${\rm
Cycl}(\mathcal{X}_\varepsilon^{[n,2n-1]},p_0)\, = \, \lfloor
(m-1)/2 \rfloor$.
\[remnil\] The proof of this theorem shows that if there exists an inverse integrating factor $V_0^{*}(x,y)$ of system [(\[eqnil2\])]{} such that $V_0^{*}(r \, {\rm Cs}\, \theta, \, r^n \, {\rm Sn}\,
\theta)/r^{2n-1}$ has a Laurent expansion in a neighborhood of $r=0$, then the exponents of the leading terms of $V_0^{*}(r \,
{\rm Cs}\, \theta, \, r^n \, {\rm Sn}\, \theta)/r^{2n-1}$ and $V(r,\theta)$ coincide. Therefore, the value of $m$ can be determined without performing the transformation of the system to generalized polar coordinates.
We assume that the origin of system (\[eqnil2\]) is a focus with Andreev number $n$ and that the vanishing multiplicity of an inverse integrating factor on it is $m$. If system (\[eqnil2\]) is written as $\dot{x} \, = \, P(x,y)$ and $\dot{y} \, = \,
Q(x,y)$, then the system: $$\label{eqnilp1} \dot{x} \, = \, P(x,y) \, + \, x \, K(x,y,\varepsilon),
\qquad \dot{y} \, = \, Q(x,y) + n y K(x,y, \varepsilon),$$ where $$K(x,y , \varepsilon)\, = \, \displaystyle \sum_{i=0}^{L-1}
\varepsilon^{L-i} \, a_i \, x^{2i}$$ and $L=(m+n)/2-1$, has at least $(m+n)/2-1$ limit cycles bifurcating from the origin for convenient values of the real parameters $a_i$. We recall that $m$ and $n$ have the same parity.
The following corollary establishes a necessary condition for system (\[eqnil\]) to have an analytic inverse integrating factor $V_0(x,y)$ defined in a neighborhood of the origin.
\[cornil\] We assume that the origin of system [(\[eqnil\])]{} is a nilpotent focus with Andreev number $n$, and that there exists an inverse integrating factor $V_0(x,y)$ of [(\[eqnil\])]{} which is analytic in a neighborhood of the origin. Then, $n$ is odd.
Singular perturbations
----------------------
As some recent research papers show, see [@DRP], limit periodic sets containing an infinite number of critical points may have a cyclicity higher than expected. Due to the narrow relationship between limit cycles and the inverse integrating factor, the context of singular perturbations is a brand new and very interesting place to apply properties of the inverse integrating factor in order to detect limit cycles. As we have seen, for other limit periodic sets, the inverse integrating factor does not only give an alternative way to study the cyclicity but contains more information: location of limit cycles, direct computation of the cyclicity of the object under study, an explicit partial differential equation (\[def-V\]) which gathers all the information, …
As far as we know, the only work where the inverse integrating factor is related with a singular perturbation problem is [@LlibMedSil], where one-parameter families of vector fields $\mathcal{X}_\varepsilon$ in $\mathbb{R}^2$ of the form $\mathcal{X}_\varepsilon \, = \, f(x,y,\varepsilon)\, \partial_x
\, + \, \varepsilon g(x,y,\varepsilon) \, \partial_y$, where $\varepsilon \geq 0$ and $f,g$ are analytic functions, are taken into account. The aim of the singular perturbation problems is to study the phase portrait, for $\varepsilon$ sufficiently small, near the set of singular points of $\mathcal{X}_0$, that is, $\Sigma \, = \, \{(x,y) \in \mathbb{R}^2 \, : \, f(x,y,0)=0\}$. In particular, the question is to decide if $\mathcal{X}_\varepsilon$ has a limit cycle which tends to a singular orbit of $\mathcal{X}_0$ when $\varepsilon \searrow 0$. A singular orbit (also denoted as [*slow-fast cycle*]{}, see [@DRP; @DRR]) is a limit periodic set of the system $\mathcal{X}_0$. For the vector field $\mathcal{X}_0$, we say that a point $n \in \Sigma$ is [*normally hyperbolic*]{} if $(\partial f/\partial x)(n,0) \, \neq \,
0$. The system of differential equations associated to $\mathcal{X}_\varepsilon$ is $$\dot{x} \, = \, f(x,y,\varepsilon), \quad \dot{y} \, = \,
\varepsilon \, g(x,y,\varepsilon),$$ where the dot denotes derivation with respect to the time $t$. We call this system the [*fast system*]{}. By the time rescaling $\tau \, = \, \varepsilon
t$, we get the [*slow system*]{}: $$\varepsilon x' \, = \, f(x,y,\varepsilon), \quad y' \, = \,
g(x,y,\varepsilon),$$ where $'$ denotes derivation with respect to $\tau$. The [*reduced problem*]{} is defined by the slow system taking $\varepsilon=0$, which gives one differential equation constrained to the slow manifold or critical curve $\Sigma$, that is, the reduced problem is $$f(x,y,0)=0, \quad y'\, = \, g(x,y,\varepsilon).$$ The only singular orbits taken into account in [@LlibMedSil] are the ones consisting of three pieces of smooth curves; an orbit of the reduced problem starting at a normally hyperbolic point $n_1 \in \Sigma$, an orbit of the reduced problem ending at a normally hyperbolic point $n_2 \in \Sigma$ and an orbit of the fast problem connecting the two previous ones. The main results are the following.
[[@LlibMedSil]]{} \[thLlibMedSil\] Consider $\varepsilon_0>0$ and $V_\varepsilon(x,y)$ an inverse integrating factor of $X_\varepsilon$, that is $\mathcal{X}_\varepsilon(V_\varepsilon(x,y)) \, = \, {\rm div}
\mathcal{X}_\varepsilon \, V_\varepsilon(x,y)$, defined in an open set $\mathcal{U} \subseteq \mathbb{R}^2$ for any $0\leq
\varepsilon \leq \varepsilon_0$. Let $\Gamma \subset \mathcal{U}$ be a singular orbit and $\Gamma_\varepsilon$ be a limit cycle of $\mathcal{X}_\varepsilon$ in $\mathcal{U}$ for $\varepsilon \in
(0,\varepsilon_0)$, with $\Gamma_\varepsilon \to \Gamma$, according to the Hausdorff distance. Then $V_0(\Gamma)=0$.
[[@LlibMedSil]]{} \[corLlibMedSil\] Consider $V_\varepsilon(x,y)$ an inverse integrating factor of $X_\varepsilon$ as in Theorem [\[thLlibMedSil\]]{}. If the level zero of the function $V_0(x,y)$ does not contain a closed curve, then there exists $\varepsilon_0>0$ such that $\mathcal{X}_\varepsilon$ does not present a limit cycle for $0<\varepsilon<\varepsilon_0$ in $\mathcal{U}$.
As an application of these results, the following examples are given in [@LlibMedSil]. The following vector fields present no limit cycles because the corresponding inverse integrating factors have no closed curves in their level zero sets.
- The vector field $\mathcal{X}_\varepsilon \, = \,
(y^2-x^2)\partial_x \, + \,\varepsilon \, x^2\, \partial_y$ has the inverse integrating factor $V_\varepsilon(x,y)=y^3-yx^2-x^3\varepsilon$.
- The vector field $\mathcal{X}_\varepsilon \, = \,
(y-x^2)\partial_x \, + \,\varepsilon \, x\, \partial_y$ has the inverse integrating factor $V_\varepsilon(x,y)=-y+x^2+(1/2)\varepsilon$.
- The vector field $\mathcal{X}_\varepsilon \, = \,
(-y+x^2)\partial_x \, + \,\varepsilon \, x\, \partial_y$ has the inverse integrating factor $V_\varepsilon(x,y)=y-x^2+(1/2)\varepsilon$.
Some generalizations
====================
The inverse Jacobi multiplier
-----------------------------
Inverse Jacobi multipliers are a natural generalization of inverse integrating factors to $n$-dimensional dynamical systems with $n
\geq 3$. In [@BerroneGiacomini2], it is developed the theory of inverse Jacobi multiplier from its beginning in the formal methods of integration of ordinary differential equations to modern applications.
In this section we will assume that $\mathcal{X} = \sum_{i=1}^n
X_i(x_1,\ldots, x_n) \partial_{x_i}$ is a $\mathcal{C}^1$ vector field defined in the open set $\mathcal{U} \subseteq
\mathbb{R}^n$. A $\mathcal{C}^1$ function $V : \mathcal{U} \to
\mathbb{R}$ is said to be an inverse Jacobi multiplier for the vector field $\mathcal{X}$ in $\mathcal{U}$ when $V$ solves in $\mathcal{U}$ the linear first order partial differential equation $\mathcal{X} V = V {\rm div} \mathcal{X}$. The first appearance of these multipliers occurs in the works of C.G.J. Jacobi, about the middle of the past century. Many properties of inverse integrating factors for the planar case ($n=2$) are inherited by inverse Jacobi multiplier. We list some of them:
- If the change of coordinates $y = \phi(x)$ is introduced, then $W(y) = (V \circ \phi^{-1})(y) \det\{ D \phi(\phi^{-1}(y))
\}$ is an inverse multiplier of the transformed vector field $\phi_*\mathcal{X}$.
- Let $V_1$ and $V_2$ be two linearly independent inverse Jacobi multipliers of $\mathcal{X}$ defined in $\mathcal{U}$. If $V_1(x) \neq 0$ for all $x \in \mathcal{U}$, then the ratio $V_2/V_1$ is a first integral of $\mathcal{X}$ in $\mathcal{U}$.
- One can use local Lie groups of transformations to find inverse Jacobi multipliers as follows. Assume $\mathcal{X}$ admits in $\mathcal{U}$ a $(n - 1)$–parameter local Lie group of transformations with infinitesimal generators $\{ \mathcal{Y}_1,
\ldots, \mathcal{Y}_{n-1} \}$. Then, an inverse Jacobi multiplier $V$ for $\mathcal{X}$ in $\mathcal{U}$ is furnished by the determinant $V= \det \{ \mathcal{X}, \mathcal{Y}_1, \ldots,
\mathcal{Y}_{n-1} \}$.
- Let $\{ \mathcal{Y}_1, \ldots, \mathcal{Y}_{n-1} \}$ be the generators of $n - 1$ local Lie groups of symmetries admitted by $\mathcal{X}$ in $\mathcal{U}$. Then, the inverse multiplier $V= \det \{ \mathcal{X}, \mathcal{Y}_1, \ldots,
\mathcal{Y}_{n-1} \}$ vanishes on every invariant solution of $\mathcal{X}$ contained in $\mathcal{U}$. Recall here that an invariant solution of $\mathcal{X}$ corresponding to the group $G$ is defined to be an integral curve of $\mathcal{X}$ which is invariant under the action of $G$.
- Let $p_0 \in \mathcal{U}$ be an isolated zero of an inverse Jacobi multiplier $V$ such that $V \geq 0$ in a neighborhood $\mathcal{N}$ of $p_0$. Then, $p_0$ is a stable (resp. unstable) singular point of $\mathcal{X}$ provided that ${\rm div} \mathcal{X} \leq 0$ (resp. $\geq 0$) in $\mathcal{N}$. Furthermore, the stability (resp. unstableness) of $p_0$ is asymptotic stability (resp. unstableness) provided that ${\rm div}
\mathcal{X} < 0$ (resp. $> 0$) in $\mathcal{N}$.
In the following, we summarize some of the results obtained in [@BerroneGiacomini2]. By a [*limit cycle*]{} $\gamma$ of $\mathcal{X}$ we mean a $T$–periodic orbit which is $\alpha$ or $\omega$–limit set of another orbit of $\mathcal{X}$. Let $V$ be an inverse Jacobi multiplier defined in a region containing $\gamma$. If $\gamma = \{ \gamma(t) \in \mathcal{U} : 0 \leq t
\leq T \}$, we define $$\Delta(\gamma) = \int_0^T {\rm div} \mathcal{X} \circ \gamma(t) \ dt \ .$$ As it is well known, $\Delta(\gamma)$ is the sum of the characteristic exponents of the limit cycle $\gamma$. We recall that if $\Delta(\gamma) > 0$ then $\gamma$ is not orbitally stable. We will say that $\gamma$ is a [*strong*]{} limit cycle when $\Delta(\gamma) \neq 0$. If, on the contrary, $\Delta(\gamma)
= 0$, then we say that $\gamma$ is a [*weak*]{} limit cycle.
[[@BerroneGiacomini2]]{} Let $V$ be an inverse Jacobi multiplier of $\mathcal{X}$ defined in a region containing a limit cycle $\gamma$ of $\mathcal{X}$. Then, $\gamma$ is contained in $V^{-1}(0)$ in the following cases: [(i)]{} if $\gamma$ is a strong limit cycle, or [(ii)]{} if $\gamma$ is asymptotically orbitally stable (unstable).
[[@BerroneGiacomini2]]{} Let $V$ be a Jacobi inverse multiplier defined in a neighborhood of a limit cycle $\gamma$ of the vector field $\mathcal{X}$. If $\gamma$ is a strong limit cycle, then
- $V$ vanishes on $W^s(\gamma)$, the stable manifold of $\gamma$, provided that $\Delta(\gamma) > 0$;
- $V$ vanishes on $W^u(\gamma)$, the unstable manifold of $\gamma$, provided that $\Delta(\gamma) < 0$.
The following example appears in [@BerroneGiacomini2]. Consider the cubic polynomial vector field in $\mathbb{R}^3$ $$\dot x \, = \, \lambda (-y + x f(x,y)), \quad \dot y \, =\,
\lambda (x + y f(x,y)), \quad \dot z \,=\, z , \label{9-naLJM}$$ where $f(x,y) = 1-x^2-y^2$ and $\lambda > 0$ is a real parameter. The circle $\gamma = \{ f(x,y)= 0 \} \cap \{ z=0 \}$ is a limit cycle of system (\[9-naLJM\]) of period $T = 2 \pi / \lambda$. In fact, $\gamma(t) = (\cos\lambda t, \sin\lambda t, 0)$. It is easy to compute that $$\Delta(\gamma) = \int_0^T {\rm div} \mathcal{X} \circ \gamma(t) dt = \frac{2(1-2 \lambda)}{\lambda} \pi \ .$$ An inverse Jacobi multiplier for this system is $$V_1(x,y,z) = f(x,y) (x^2+y^2) z \ .$$ In addition, when $\lambda = -1/2$, then $V_2(x,y,z) =
(x^2+y^2)^2$ is another inverse Jacobi last multiplier for system (\[9-naLJM\]).
As usual, a hyperbolic singular point $p_0$ of a $\mathcal{C}^1$ vector field $\mathcal{X}$ is named a [*saddle point*]{} when the matrix $D\mathcal{X}(p_0)$ has eigenvalues with both positive and negative real parts. Assuming that $k$ of these real parts are positive and the remaining $n - k$ are negative, the stable manifold theorem ensures the existence of two invariant $\mathcal{C}^1$ manifolds $W^u(p_0)$ and $W^s(p_0)$ with dimensions $\dim W^u(p_0) = k$ and $\dim W^s(p_0) = n-k$, such that they intersect transversally one each other in $p_0$.
[[@BerroneGiacomini2]]{} Let $p_0$ be a nondegenerate strong singular point of the $\mathcal{C}^1$ vector field $\mathcal{X}$ having an inverse Jacobi multiplier $V$ defined in a neighborhood of $p_0$. Then $V$ vanishes on $W^u(p_0)$ [(]{}resp. $W^s(p_0)$[)]{} provided that ${\rm div} \mathcal{X}(p_0) < 0$ [(]{}resp. ${\rm div}
\mathcal{X}(p_0) > 0$[)]{}.
Time–dependent inverse integrating factors
------------------------------------------
In [@GaGiMa], the authors consider autonomous second order differential equations $$\label{Lie-Su-31}
\ddot{x} = w(x, \dot{x}) \ ,$$ with $w \in \mathcal{C}^\infty(\mathcal{U})$ and $\mathcal{U}
\subseteq \mathbb{R}^2$ an open set. They associated to (\[Lie-Su-31\]) the first order planar system defined on $\mathcal{U}$ in the usual way $$\label{Lie-Su-32}
\dot{x} = y \ , \ \dot{y} = w(x, y) \ .$$ Moreover, it is associated to equations (\[Lie-Su-31\]) and (\[Lie-Su-32\]) the vector fields $\mathcal{X} = \partial_t + \dot{x} \partial_x +
w(x, \dot{x}) \partial_{\dot{x}}$ and $\bar{\mathcal{X}} = y
\partial_x + w(x, y) \partial_{y}$, respectively. A $\mathcal{C}^1$ nonconstant function $I(t,x, y)$ is called an [*invariant*]{} (or non–autonomous first integral) of system (\[Lie-Su-32\]) in $\mathcal{U}$ if it is constant along the solutions of (\[Lie-Su-32\]). In other words, $\mathcal{X} I \equiv 0$ must be satisfied in $\mathcal{U}$. Of course, we can find at most two functionally independent invariants of (\[Lie-Su-32\]). Notice that an invariant provides information about the asymptotic behavior of the orbits.
A symmetry of (\[Lie-Su-31\]) is a diffeomorphism $\Phi : (t,x)
\mapsto (\bar{t}, \bar{x})$ that maps the set of solutions of (\[Lie-Su-31\]) into itself. Therefore, the symmetry condition for (\[Lie-Su-31\]) is just $\bar{x}'' = w(\bar{x},
\bar{x}')$, where the prime denotes the derivative $'= d / d
\bar{t}$. When the symmetry is a 1–parameter Lie group of point transformations $\Phi_\epsilon$, then $\bar{t} = t + \epsilon \xi(t,x) + O(\epsilon^2)$, $\bar{x} = x
+ \epsilon \eta(t,x) + O(\epsilon^2)$, for $\epsilon$ close to zero, and the vector field $\mathcal{Y} =
\xi(t,x) \partial_t + \eta(t,x) \partial_x$ is called the [*infinitesimal generator*]{} of the 1–parameter Lie group of point transformations $\Phi_\epsilon$. It is well known that the [*determining equations*]{} for Lie point symmetries can be obtained from the linearized condition $$\label{Lie-Su-7}
\mathcal{Y}^{[2]} (\ddot{x} - w(x, \dot{x})) = 0 \ \mbox{when} \
\ddot{x} = w(x, \dot{x}) \ ,$$ where $\mathcal{Y}^{[2]} = \mathcal{Y} + \eta^{[1]}(t,x,\dot{x})
\partial_{\dot{x}} + + \eta^{[2]}(t,x,\dot{x}, \ddot{x})
\partial_{\ddot{x}}$ is the so–called [*second prolongation*]{} of the infinitesimal generator $\mathcal{Y}$ and $
\eta^{[1]}(t,x,\dot{x}) = D_t \eta - \dot{x} D_t \xi$, $
\eta^{[2]}(t,x,\dot{x},\ddot{x}) = D_t \eta^{[1]} - \ddot{x} D_t
\xi$ where $D_t = \partial_t + \dot{x} \partial_x + \ddot{x}
\partial_{\dot{x}}$ is the operator total derivative with respect to $t$. Of course, since (\[Lie-Su-31\]) is autonomous, it always admits the generator $\mathcal{Y} = \partial_t$ of a Lie point symmetry. Let $\mathcal{L}_r$ denote the set of all infinitesimal generators of 1–parameter Lie groups of point symmetries of the differential equation (\[Lie-Su-31\]). It is known that $\mathcal{L}_r$ is a finite dimensional real Lie algebra, where we denote $r = \dim
\mathcal{L}_r$. Moreover, for autonomous second order differential equation we have $r \in \{
1,2,3,8 \}$.
For any $\mathcal{Y}_i = \xi_i(t,x) \partial_t + \eta_i(t,x)
\partial_x \in \mathcal{L}_r$, easily one can check that the Lie bracket $[\mathcal{X}, \mathcal{Y}_i^{[1]}] = \mu_i(t,x,\dot{x})
\mathcal{X}$ where $\mu_i(t,x, \dot{x}) = \mathcal{X} \xi_i$ and $\mathcal{Y}_i^{[1]} = \mathcal{Y}_i + \eta_i^{[1]}(t,x,\dot{x})
\partial_{\dot{x}}$ is the first prolongation of $\mathcal{Y}$. If $r \geq 2$, we define the functions $$\label{Lie-Su-51}
V_{ij}(t,x,\dot{x}) = \det\{\mathcal{X}, \mathcal{Y}_i^{[1]},
\mathcal{Y}_j^{[1]} \} = \left| \begin{array}{ccc} 1 & \dot{x} & w(x,\dot{x}) \\
\xi_i(t,x) & \eta_i(t ,x) & \eta_i^{[1]}(t,x,\dot{x}) \\
\xi_j(t,x) & \eta_j(t ,x) & \eta_j^{[1]}(t,x,\dot{x})
\end{array}\right|$$ for $i,j \in \{ 1, \ldots, r\}$ with $1 \leq i < j \leq r$. The aim of the work [@GaGiMa] is to generalize the concept of inverse integrating factor $V(x,y)$ of system (\[Lie-Su-32\]) via the functions $V_{ij}(t,x,y)$ defined in (\[Lie-Su-51\]). In fact, in the autonomous particular case $\partial V_{ij} /
\partial t \equiv 0$, we get that $V_{ij}$ is just an inverse integrating factor of (\[Lie-Su-32\]). On the contrary, when $\partial V_{ij} / \partial t \not\equiv 0$, in [@GaGiMa] it is proved that the zero–sets $V^{-1}(0)$ and $V_{ij}^{-1}(0)$ have similar properties. The next result provides the connection between inverse integrating factors of system (\[Lie-Su-32\]) and the functions $V_{ij}(t,x,y)$.
\[Lie-Su-53\] Assume that system [(\[Lie-Su-32\])]{} possesses an $r$–dimensional Lie point symmetry algebra with $r \geq 2$ and define the functions $V_{ij}(t,x,\dot{x})$ as in [(\[Lie-Su-51\])]{}.
- $V_{ij}$ satisfies the linear partial differential equation $\mathcal{X} V_{ij} = V_{ij} \ {\rm div} \mathcal{X}$, where $\mathcal{X} = \partial_t + \dot{x} \partial_x + w(x,
\dot{x}) \partial_{\dot{x}}$.
- If $r \geq 3$ then, the ratio of any two nonzero $V_{ij}$ is either a constant or an invariant of [(\[Lie-Su-32\])]{}.
- If $V_{ij} \equiv 0$, then $(\eta_i-y
\xi_i)/(\eta_j-y \xi_j)$ is an invariant of system [(\[Lie-Su-32\])]{}.
The next theorem is about the invariant curves of $\bar{\mathcal{X}}$ contained in $V_{ij}^{-1}(0)$ and give us an extension of Theorem 9 in [@GLV] for a case with $\partial V_{ij} / \partial t \not\equiv 0$. We put special emphasis on periodic orbits of (\[Lie-Su-32\]) of any kind (isolated and, therefore, limit cycles or non-isolated and so belonging to a period annulus). Recall here that a limit cycle $\gamma := \{ (x(t),y(t)) \in \mathcal{U} :
0 \leq t < T \}$ is [*hyperbolic*]{} if $\oint_\gamma {\rm
div} \bar{\mathcal{X}}(x(t),y(t)) dt \neq 0$. On the other hand, a $\mathcal{C}^1$ curve $f(x,y)=0$ defined on $\mathcal{U}$ is invariant for $\bar{\mathcal{X}}$ if $\bar{\mathcal{X}} f = K
f$ for some function $K(x,y)$ called [*cofactor*]{}.
[[@GaGiMa]]{} \[Lie-Su-36\] Let $\mathcal{U} \subset \mathbb{R}^2$ be an open set and assume that $\ddot{x} = w(x, \dot{x})$ with $w$ smooth in $\mathcal{U}$ admits an $r$–dimensional Lie point symmetry algebra $\mathcal{L}_r$ with $r \geq 2$. Consider the functions $V_{ij}(t,x,\dot{x})$ defined in [(\[Lie-Su-51\])]{} for $i,j
\in \{ 1, \ldots, r\}$ with $1 \leq i < j \leq r$. Suppose that $\gamma=(x(t), y(t)) \subset \mathcal{U}$ is a $T$–periodic orbit of [(\[Lie-Su-32\])]{}. Then the next statements hold:
- If $V_{ij}(t,x,\dot{x}) = V(x,\dot{x}) \not\equiv 0$, with $V \in \mathcal{C}^1(\mathcal{U})$, then $V(x,y)$ is an inverse integrating factor of system [(\[Lie-Su-32\])]{} in $\mathcal{U}$. In particular, if $\gamma$ is a limit cycle, then $\gamma \subset \{ V(x,y) = 0 \}$.
- If $V_{ij}(t,x,\dot{x}) = F(t) G(x,\dot{x}) \not\equiv
0$ with non–constants $F$ and $G \in \mathcal{C}^1(\mathcal{U})$, then $\dot{F} = \alpha F$ with $\alpha \in \mathbb{R} \backslash
\{0\}$ and $G(x,y)=0$ is an invariant curve of system [(\[Lie-Su-32\])]{}. Moreover, we have:
- If $\gamma \subset \{ G = 0 \}$ and $G$ is analytic on $\mathcal{U}$, then $G$ is not square–free, i.e., $G(x,y) =
g^n(x,y) u(x,y)$ with a positive integer $n > 1$ and $g$ and $u$ are analytic functions on $\mathcal{U}$ satisfying $\gamma \subset
\{ g = 0 \}$ and $\gamma \not\subset \{ u = 0 \}$.
- If $\gamma \not\subset \{ G = 0 \}$ then $\gamma$ is hyperbolic and $\alpha T = \oint_\gamma {\rm div}
\bar{\mathcal{X}} (x(t), y(t)) dt$.
An immediate consequence is obtained.
[[@GaGiMa]]{} \[Lie-Su-36-1\] Assume that $\ddot{x} = w(x, \dot{x})$, with $w$ smooth in the open set $\mathcal{U} \subseteq \mathbb{R}^2$, admits an $r$–dimensional Lie point symmetry algebra $\mathcal{L}_r$ with $r \geq 2$. Consider the functions $V_{ij}(t,x,\dot{x})$ for $i,j
\in \{ 1, \ldots, r\}$ with $1 \leq i < j \leq r$. If there is one $V_{ij}(t,x,y) = F(t) G(x,y) \not\equiv 0$ with non–constants $F$ and $G \in \mathcal{C}^1(\mathcal{U})$, then system [(\[Lie-Su-32\])]{} does not have period annulus in $\mathcal{U}$.
In the sequel, we concentrate our attention in the 2–dimensional case $\mathcal{L}_2$. In [@GaGiMa] it is proved that, if $\partial_t \in \mathcal{L}_2$, then the autonomous or separate time–variable forms of $V_{ij}(t,x,\dot{x})$ given in Theorem \[Lie-Su-36\] are the only possibilities. Moreover, defining the domain of definition of the infinitesimal generators as the unbounded open strip $\Xi = \{ (t,x) \in \mathbb{R} \times
\mathbb{X} \} \subset \mathbb{R}^2$, one has the following result.
[[@GaGiMa]]{} \[Lie-Su-42\] Assume that $\ddot{x} = w(x,\dot{x})$ with $w$ smooth in $\mathcal{U} \subset \mathbb{R}^2$ admits a 2–dimensional Lie point symmetry algebra $\mathcal{L}_2$ spanned by the $\mathcal{C}^1(\Xi)$ vector fields $\mathcal{Y}_1 = \partial_t$ and $\mathcal{Y}_2$ such that $[\mathcal{Y}_1, \mathcal{Y}_2]=c_1
\mathcal{Y}_1 + c_2 \mathcal{Y}_2$.
- If $c_2=0$ and $\mathcal{Y}_2 \in \mathcal{C}^2(\Xi)$, then $V_{12}(t,x,\dot{x}) = G(x,\dot{x})$ with $G(x,y) = y^2 [c_1+y
\alpha'(x)-\beta'(x)] + \beta(x) w(x,y)$ an inverse integrating factor of $\bar{\mathcal{X}}$ in $W = \mathcal{U} \cap
\{\mathbb{X} \times \mathbb{R}\}$ provided that $G \not\equiv 0$. Moreover, for analytic vector fields $\mathcal{Y}_2$ in $\Xi$, $\bar{\mathcal{X}}$ has no limit cycles in $W$.
- If $c_2 \neq 0$ then, changing the basis of $\mathcal{L}_2$ such that $[\bar{\mathcal{Y}}_1,
\bar{\mathcal{Y}}_2] = \bar{\mathcal{Y}}_1$, we have that $\bar{V}_{12}(t,x,\dot{x}) = \exp(c_2 t) \bar{G}(x,\dot{x})$ with $\bar{G}(x,\dot{x}) = \dot{x} [c_2 \dot{x} \alpha(x)-c_2 \beta(x)+
\dot{x}^2 \alpha'(x)-\dot{x} \beta'(x)] + \beta(x) w(x,\dot{x})$. In addition, $\partial w/ \partial x \equiv 0$ or $\beta(x)\equiv
0$. If $\bar{G} \not\equiv 0$ and $\mathcal{U}$ is a simply connected domain, then $\bar{\mathcal{X}}$ has no periodic orbits in $\mathcal{U}$ and all the $\alpha$ or $\omega$–limit sets of $\bar{\mathcal{X}}$ are contained in the invariant curve $\bar{G}(x,y) = 0$ of $\bar{\mathcal{X}}$.
As an application of these results to polynomial Liénard systems, in [@GaGiMa] it is proved the next theorem.
[[@GaGiMa]]{} \[Lie-Su-44\] The polynomial Liénard differential equation $\ddot{x} + f(x)
\dot{x} + g(x) = 0$ with $f, g \in \mathbb{R}[x]$ having a $r$–dimensional Lie point symmetry algebra $\mathcal{L}_r$ with $r \geq 2$ has no limit cycles in $\mathbb{R}^2$.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Professor Héctor Giacomini, from Université de Tours (France), for his useful comments on this survey and for encouraging us to study the inverse integrating factor.
[99]{}
,[* Investigation on the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular point*]{}, Translation Amer. Math. Soc. [**8**]{} (1958), 187–207.
, John Wiley and Sons, New York, 1973.
, [* On the vanishing set of inverse integrating factors*]{}, Qual. Th. Dyn. Systems [**1**]{} (2000), 211–230.
, [* Inverse Jacobi multipliers*]{}, Rend. Circ. Mat. Palermo (2) [**52**]{} (2003), 77–130.
, [* Bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems*]{}, SIAM Rev. [**36**]{} (1994) 341–-376.
Applied Math. Sciences [**154**]{}, 2002 (New York: Springer)
, [* Symmetries and Differential Equations*]{}, Springer, New York, 1989.
,[* Liouvillian first integrals for the planar Lotka-Volterra system.*]{} Rend. Circ. Mat. Palermo (2) [**52**]{} (2003), 389–418.
, [* Integrability of the $2D$ Lotka-Volterra system via polynomial first integrals and polynomial inverse integrating factors*]{}, J. Phys. A [**33**]{} (2000), 2407–2417.
, [* Integrable systems in the plane with a center type linear part*]{}, Appl. Math. (Warsaw) [**22**]{} (1994), 285–-309.
, [* A class of integrable polynomial vector fields.*]{} Appl. Math. (Warsaw) [**23**]{} (1995), 339–350.
, [*The Poincaré Problem in the Non-Resonant Case: An Algebraic Approach*]{}, Differ. Geom. Dyn. Syst. [**8**]{} (2006), 54–68.
, [*On the integrability of differential equations defined by the sum of homogeneous vector fields with degenerate infinity*]{}, Internat. J. Bifur. Chaos Appl. Sci. Engrg. [**11**]{} (2001), 711–722.
, [* Polynomial first integrals of quadratic vector fields*]{}, J. Differential Equations [**230**]{} (2006), 393–421.
, [*Resolution of the Poincaré problem and nonexistence of algebraic limit cycles in family (I) of Chinese classification*]{}, Chaos Solitons Fractals [**24**]{} (2005), 491–499.
, [*Non-nested configuration of algebraic limit cycles in quadratic systems*]{}, J. Differential Equations [**225**]{} (2006), 513–527.
, [*The null divergence factor*]{}, Publicacions Matemàtiques [**41**]{} (1997), 41–-56.
, [*An improvement to Darboux integrability theorem for systems having a center.*]{} Appl. Math. Lett. [**12**]{} (1999), 85–89.
, [*On a new type of bifurcation of limit cycles for planar cubic systems*]{}, Nonlinear Anal. [**36**]{}, (1999), 139–149.
, [*Polynomial inverse integrating factors*]{}, Ann. Differential Equations [**16**]{} (2000), 320–329.
, [*On the integrability of two–dimensional flows*]{}, J. Differential Equations [**157**]{} (1999), 163–182.
,[* Darboux integrability and the inverse integrating factor.*]{} J. Differential Equations [**194**]{} (2003), 116–139.
, [* Necessary conditions for the existence of invariant algebraic curves for planar polynomial systems*]{}, Bull. Sci. Math., [**129**]{} (2005), 99–126.
, [* Integrability of a linear center perturbed by a fourth degree homogeneous polynomial.*]{} Publ. Mat. [**40**]{} (1996), 21–39.
, [*Integrability of a linear center perturbed by a fifth degree homogeneous polynomial.*]{} Publ. Mat. [**41**]{} (1997), 335–356.
, [*Integrable systems via inverse integrating factor*]{}, Extracta Math. [**13**]{} (1998), 41–60.
, [*Integrable systems via polynomial inverse integrating factors*]{}, Bull. Sci. Math. [**126**]{} (2002), 315–331.
, [* A family of non-Darboux-integrable quadratic polynomial differential systems with algebraic solutions of arbitrarily high degree.*]{} Appl. Math. Lett. [**16**]{} (2003), 833–837.
, [* Algebraic solutions for polynomial systems with emphasis in the quadratic case.*]{} Exposition. Math. [**15**]{} (1997), 161–173.
,[*A survey of isochronous centers.*]{} Qual. Theory Dyn. Syst. [**1**]{} (1999), 1–70.
,[* Invariant algebraic curves and conditions for a centre.*]{} Proc. Roy. Soc. Edinburgh Sect. A [**124**]{} (1994), 1209–1229.
, [* Liouvillian first integrals of second order polynomial differential systems.*]{}, Electron. J. Differential Equations, Vol. [**1999**]{}(1999), 1–7.
,[* Darboux integrating factors: Inverse problems.*]{} Preprint, 2008.
,[* Multiplicity of invariant algebraic curves in polynomial vector fields.*]{} Pacific J. Math. [**229**]{} (2007), 63–117.
, [* An Introduction to the Lie Theory of One-Parameter Groups, with Applications to the Solutions of Differential Equations*]{}, D. C. Heath, New York, 1911.
, [*Polynomial inverse integrating factors of quadratic differential systems.*]{} Preprint, 2008.
, [*Conditions for a center in a simple class of cubic systems*]{}, Differential Integral Equations [**10**]{} (2) (1997), 333-–356.
,[* Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges)*]{}, Bull. Sci. Math. [**32**]{} (1878), 60–96; 123–144; 151–200.
, [* The structure of a single-valued integrating factor neaar a cycle*]{}. (Russian) Differentsial’nye Uravneniya [**17**]{} (1981), 1490–1492.
, [*Algebraic differential equations with an integrating factor of Darboux type*]{}. (Russian) Izv. Akad. Nauk Respub. Moldova Mat. (1993), 96–106.
, [*On the absence of limit cycles in dynamical systems with an integrating factor of a special type*]{}, Differentsial’nye Uravneniya [**30**]{} (1994), 947–954; translation in Differential Equations [**30**]{} (1994), 876–883.
, [* Formal integrating factors as a method of distinguishing between a center and a focus.*]{} J. Math. (Wuhan) [**17**]{} (1997), 231–239.
, [* Singularities of vector fields on the plane.*]{} J. Differential Equations [**23**]{} (1977), 53–106.
, [*Qualitative theory of planar differential systems.*]{} Universitext, Springer–Verlag, Berlin Heidelberg, 2006.
,[* More limit cycles than expected in Liénard equations.*]{} Proc. Amer. Math. Soc. [**135**]{} (2007), 1895–1904 (electronic).
, [* Hilbert’s 16th problem for quadratic vector fields.*]{} J. Differential Equations [**110**]{} (1994), 86–133.
,[* Existence and vanishing set of inverse integrating factors for analytic vector fields*]{}, preprint, 2008.
, [* On the remarkable values of the rational first integrals of polynomial vector fields.*]{} J. Differential Equations [**241**]{} (2007), 399–417.
, [*Polynomial inverse integrating factors for polynomial vector fields.*]{} Discrete Contin. Dynam. Systems, [**17**]{} (2007) 387–-395.
, [* The inverse integrating factor and the Poincaré map*]{}, Trans. Amer. Math. Soc., to appear. `arXiv:0710.3238v1 [math.DS]`
, [* Generalized Hopf bifurcation for planar vector fields via the inverse integrating factor*]{}. Preprint, 2009. `arXiv:0902.0681v1 [math.DS]`
,[* Generalized cofactors and nonlinear superposition principles,*]{} Appl. Math. Lett. [**16**]{} (2003), 1137–1141.
,[* Non-algebraic invariant curves for polynomial planar vector fields.*]{} Discrete Contin. Dyn. Syst. [**10**]{} (2004), 755–768.
, [*Periodic Solutions of 2nd Order Differential Equations with 2–Dimensional Lie Point Symmetry Algebra*]{}, Preprint 2009.
, [*On the existence of polynomial inverse integrating factors in quadratic systems with limit cycles*]{}, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. [**12**]{} (2005), 151–173.
, [* Integral invariants and limit sets of planar vector fields*]{}, J. Differential Equations [**217**]{} (2005), 363–376.
,[* A new criterion for controlling the number of limit cycles of some generalized Liénard equations*]{}, J. Differential Equations [**185**]{} (2002), 54–73.
,[*Upper bounds for the number of limit cycles of some planar polynomial differential systems,*]{} Preprint 2008.
,[* New criteria for the existence and non-existence of limit cycles in Liénard differential systems.*]{} Dynamical Systems: An International Journal, 2008 pp. 1–-15, 2008.
, [* Integrability of planar polynomial differential systems through linear differential equations*]{}, Rocky Mountain J. Math., [**36**]{} (2006), 457–485.
, [*Linearizable planar differential systems via the inverse integrating factor*]{}, J. Phys. A: Math. Theor. [**41**]{} (2008), 135–205.
, [*The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems*]{}, J. Differential Equations [**227**]{} (2006), 406–426.
, [* On the nonexistence, existence, and uniqueness of limit cycles*]{}, Nonlinearity [**9**]{} (1996), 501–516.
, [* On the shape of limit cycles that bifurcate from Hamiltonian centers*]{}, Nonlinear Anal. [**41**]{}, (2000), 523–537.
, [* The shape of limit cycles that bifurcate from non-Hamiltonian centers*]{}, Nonlinear Anal. [**43**]{} (2001), no. 7, Ser. A: Theory Methods, 837–859.
,[*New sufficient conditions for a center and global phase portraits for polynomial systems.*]{} Publ. Mat. [**40**]{} (1996), 351–372.
, [*Determination of limit cycles for two–dimensional dynamical systems*]{}, Phys. Rev. E [**52**]{} (1995), 222–228.
, [* Semistable limit cycles that bifurcate from centers*]{}, Internat. J. Bifur. Chaos Appl. Sci. Engrg. [**13**]{} (2003), 3489–3498.
, [* The nondegenerate center problem and the inverse integrating factor*]{}, Bull. Sci. Math. [**130**]{} (2006), 152–161.
, [* On the centers of planar analytic differential systems*]{}, Internat. J. Bifur. Chaos Appl. Sci. Engrg. [**17**]{} (2007), 3061–3070.
, [* Limit cycles and symmetries of dynamical systems*]{}, Phys. Lett. A [**76**]{} (1980), 205–208.
, [* Nonlinear oscillations, dynamical systems, and bifurcations of vector fields*]{}, in: Applied Mathematical Sciences, vol. [**42**]{}, Springer–Verlag, New York, 1986.
,[* Dynamics and bifurcations*]{}, Texts in Applied Mathematics, [**3**]{}. Springer–Verlag, New York, 1991.
, New York: Springer–Verlag, 1976.
, [* Ordinary Differential Equations*]{}, Dover, New York, 1956.
,[* Équations de Pfaff algébriques.*]{} Lecture Notes in Mathematics, [**708**]{}. Springer, Berlin, 1979.
, [*Algebraic invariant curves and the integrability of polynomial systems*]{}, Appl. Math. Lett. [**6**]{} (1993) 51-–53.
,[* Integrability of polynomial differential systems.*]{} Handbook of differential equations, 437–532, Elsevier/North-Holland, Amsterdam, 2004.
,[* Limit cycles for singular perturbation problems via inverse integrating factor.*]{} Bol. Soc. Parana. Mat., to appear.
,[* Polynomial differential systems having a given Darbouxian first integral*]{}, Bull. Sci. Math. [**128**]{} (2004), 775–788.
,[*Configurations of limit cycles and planar polynomial vector fields.*]{} J. Differential Equations [**198**]{} (2004), 374–380.
,[*Darboux theory of integrability in $\mathbb{C}^n$ taking into account the multiplicity.*]{} J. Differential Equations [**246**]{} (2009), 541-–551.
, [* Integrals of a general quadratic differential system in cases of a center.*]{} Diff. Equations [**18**]{} (1982), 563–568.
, [*Holonomie et intégrales premières*]{}, Ann. Sci. École Norm. Sup. [**13**]{} (1980) 469-–523.
,[* About a conjecture on quadratic vector fields.*]{} J. Pure Appl. Algebra [**165**]{} (2001), 227–234.
,[* Liouvillian integration of the Lotka-Volterra system.*]{}, Qual. Theory Dyn. Syst. [**2**]{} (2001), 307–358.
, [* Sur l’existence d’integrales premières pour un germe de forme de Pfaff*]{}, Ann. Inst. Fourier (Grenoble) [**26**]{} (1976) 171-–220.
, [* Note on singular solutions*]{}, Am. J. of Math. XVIII, (1896), 95–97.
, [* Ordinary Differential Equations with an Introduction to Lies’s Theory of the Group of One Parameter*]{}, MacMillan, London, 1897.
, [* Inverse problems of the Darboux theory of integrability for planar polynomial differential systems.*]{} Doctoral thesis, Universitat Autònoma de Barcelona, July $2004$.
, [* Note on a paper of J. Llibre and G. Rodríguez concerning algebraic limit cycles.*]{} J. Differential Equations [**217**]{} (2005), 249–256.
,[* Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré,*]{} Rend. Circ. Mat. Palermo [**11**]{} (1897), 193–239.
, III, Paris, 1899.
, [* Elementary first integrals of differential equations.*]{}, Trans. Amer. Math. Soc. [**279**]{} (1983), 215–229.
, [* Sur certaines propiétés topologiques des variétés feuilletées*]{}, in: W.T. Wu, G. Reeb (Eds.), Sur les espaces fibrés et les variétés feuilletées, Tome XI, in: Actualités Sci. Indust., vol. 1183, Hermann et Cie, Paris, 1952.
, [* A note on finite cyclicity property and Hilbert’s 16th problem.*]{} Dynamical systems, Valparaiso 1986, 161–168, Lecture Notes in Math., [**1331**]{}, Springer, Berlin, 1988.
,[* Bifurcation of planar vector fields and Hilbert’s sixteenth problem.*]{} Progress in Mathematics, [**164**]{}. Birkhäuser Verlag, Basel, 1998.
, [* Averaging methods in nonlinear dynamical systems*]{}, in: Applied Mathematical Sci., vol. [**59**]{}, Springer–Verlag, New York, 1985.
,[* Algebraic particular integrals, integrability and the problem of center.*]{}, Trans. Amer. Math. Soc. [**338**]{} (1993), 799–841.
,[* Algebraic and geometric aspects of the theory of polynomial vector fields.*]{} Bifurcations and Periodic Orbits of Vector Fields, D. Schlomiuk (ed.), (1993), 429–467.
, [* Liouvillian first integrals of differential equations.*]{}, Trans. Amer. Math. Soc. [**333**]{} (1992), 673–688.
, [* Nonlinear autonomous dynamic systems, limit cycles and oneparameter groups of transformations*]{}, Letters in Math. Phys. [**2**]{}, (1977), 171–174.
, [*Arbitrary order bifurcations for perturbed Hamiltonian planar systems via the reciprocal of an integrating factor*]{}, Nonlinear Anal. [**48**]{} (2002), Ser. A: Theory Methods, 117–136.
, [*On the Poincaré problem*]{}, J. Differential Equations [**166**]{} (2000), 51–78.
, [*Plane polynomial vector fields with prescribed invariant curves*]{}, Proc. Roy. Soc. Edinburgh Sect. A [**130**]{} (2000), 633–649.
, [*Local integrating factors*]{}, J. Lie Theory [**13**]{} (2003), 279–289.
, [*Limit cycles as invariant functions of Lie groups*]{}, J. Phys. A12, (1979), L73-L75.
, [*Theory of limit cycles*]{}, Translations of Math. Monographs [**66**]{}, Amer. Math. Soc., Providence, 1986.
, [* On algebraic solutions of algebraic Pfaff equations*]{}, Stud. Math. [**114**]{} (1995) 117–-126.
[**Addresses and e-mails:**]{}\
$^{\ (1)}$ Departament de Matemàtica. Universitat de Lleida.\
Avda. Jaume II, 69. 25001 Lleida, SPAIN.\
[E–mails:]{} [garcia@matematica.udl.cat]{}, [mtgrau@matematica.udl.cat]{}
[^1]: The authors are partially supported by a MCYT/FEDER grant number MTM2008-00694
|
---
abstract: 'We establish an explicit upper bound $B(p,l,m)$, depending on $p,l,m$, on the number of conjugacy classes of order $p^2$ torsion elements $u$ of type $\langle l,m\rangle$ of the Nottingham group defined over the prime field of characteristic $p >0$. In the cases where $l < p $, the number of conjugacy classes of type $\langle l,m\rangle$ coincides with $B(p,l,m)$. Moreover, we give a criterion on when $u$ and $u^n$ are conjugate.'
author:
- Chun Yin Hui and Krishna Kishore
title: Torsion elements of the Nottingham group of order $p^2$
---
[^1]
Introduction {#intro}
============
This paper is a continuation of [@Ki] where the second author classified elements of the Nottingham group over the *prime* field $\F_p$ that are of order $p^2$ and type $\langle 2,m \rangle$. In order to discuss some preliminary notions and to state our main results we need some notation. Let $\kappa$ be a finite field of characteristic $p > 0$, let $K := \kappa(\!( t )\!)$ be the field of Laurent-series in one variable $t$ over $\kappa$ and ${\mathcal{O}}_K$ its ring of integers, and ${\mathfrak{M}}$ the maximal ideal of ${\mathcal{O}}_K$ generated by the uniformizer $t$. Consider the group of automorphisms of $K$ that fix $\kappa$, and in turn consider the subgroup of wild automorphisms namely those which map the uniformizer $t$ to the product $tz$ for some $z$ in the principal unit group $U_1 := 1 + {\mathfrak{M}}$. The set $\Set{t z | z \in U_1}$ corresponding to this subgroup equipped with composition of power series is a group, called the *Nottingham group* ${\mathfrak{N}}_\kappa$ over $\kappa$; for a detailed description about the Nottingham group see [@Ca] and [@Ca1].
The classification of elements of order $p$ over $\kappa$ is due to Klopsch [@Kl]. On the other hand, by associating conjugacy classes of torsion elements of order $p^n$ in ${\mathfrak{N}}_\kappa$ with continuous surjective characters $\chi: U_1 \to \Z/p^n \Z$ up to *strict equivalence*, Lubin [@Lu] deduced the result of Klopch as a particular case, but most importantly constructed a framework to understand torsion elements of high order. This is due to the fact that each surjective $\chi$, in turn, is associated with a sequence of integers $\langle b^{(0)}, \ldots, b^{(n-1)} \rangle$, called *the break sequence of $\chi$ or the *type* of $\chi$*, where $b^{(j-1)}$ is defined as the largest positive integer $b$ such that $$\chi(1 + {{\mathfrak{M}}}^b) \not \subset p^j \Z/p^n \Z.$$ Since the type of $\chi$ is invariant under strict equivalence, the analysis of the conjugacy classes of torsion elements of order $p^n$ reduces to the analysis of the strict equivalence classes of a given type.
Now, let us restrict our attention to the case where $\kappa = \F_p$, the prime finite field of characteristic $p > 0$. We drop the subscripts $\kappa$ from the notation, for instance instead of ${\mathfrak{N}}_{\F_p}$ we simply denote it as ${\mathfrak{N}}$. The continuous characters ${\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}(U_1, \Z/p^n \Z)$ form a $\Z_p$-module equipped with the action of the Nottingham group ${\mathfrak{N}}$ that acts on the left in a manner compatible with $\Z_p$-module structure, namely as $
{}_u \chi(f(t)) := \chi( f \circ u(t))
$ for $f(t) \in {U_1}= 1 + t {\F_p[\![t ]\!]}$ and $u \in {\mathfrak{N}}$. Two such characters $\chi, \psi \in {\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}(U_1, \Z/p^n \Z) $ are said to be *strictly equivalent*, denoted $\chi \simeq \psi$, if there exists an element $u \in {\mathfrak{N}}$ such that $\psi = {{}_{u}} \chi$ and $u(t)/t \in \ker \chi$; if only the former condition holds then they are said to be *weakly equivalent*, denoted $\chi \sim \psi$. Both relations $\simeq$ and $\sim$ are equivalent relations on ${\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}(U_1, \Z/p^n \Z)$; see [@Lu]. Let us note here that, given a surjective character $\chi\in {\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}(U_1, \Z/p^n \Z)$, characters strictly (resp. weakly) equivalent to $\chi$ are surjective and are of the same type.
In an unpublished work, Lubin classified torsion elements of order $p^2$ of type $\langle 1, m \rangle$ over *any* finite field; see Theorem $3.6$ in [@Ki]. On the other hand, the second author classified weak equivalence classes of order $p^2$ of type $\langle 2,m \rangle$ over any prime finite field and gave bounds on the number of strict equivalent classes; see Theorem \[prior\_thm\_3\]. The main goal of this article is to give bounds on the number of conjugacy classes of torsion elements of order $p^2$ of any type over any *prime* finite field.
Let $\Set{E_k : = 1 + t^k}_{p \nmid k}$ be a topological basis of the principal unit group $U_1 = 1 + {\mathfrak{M}}$, and $\{{\mathfrak{Z}}_i\}_i$ be the basis of ${\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}(1 + {\mathfrak{M}}, \Z/p^2 \Z)$ dual to $E_i$. Then any character $\chi \in {\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}(1 + {\mathfrak{M}}, \Z/p^2 \Z)$ of type $\langle l, m \rangle$ has the following expansion, called the *standard expansion*: $$\label{intro_eq}
\chi = \sum_{1 \leq i \leq l, p \nmid i} x_i {\mathfrak{Z}}_i + \sum_{\substack{1 \leq j \leq m, p \nmid j}} a_j. p {\mathfrak{Z}}_j,$$ where $x_i, a_j$ belong to $\{0,1,...,p-1\}$ for all $i,j$. The coefficient $x_l$ is nonzero, and if $p \nmid m$ then $a_m$ is nonzero too. Now we can state some of our main results.\
We show that every strict equivalence class contains a character of a special form.
\[main\_thm\] Let $[\chi]$ be a strict equivalence class of type $\langle l, m \rangle$. Then there exists a character in the class $[\chi]$ whose standard expansion is $$\label{intro_reduction}
x_l {\mathfrak{Z}}_l + \sum_{ m-l \leq j \leq m, p \nmid j} b_j \cdot p {\mathfrak{Z}}_j,$$ where $x_l \neq 0$ and if $p \nmid m$, then $b_m \neq 0$. The character in is said to be in *reduced form*.
Let $B(p,l,m)$ be the number of type $\langle l, m \rangle$ reduced forms . In the case where $l < p$, $B(p,l,m)$ is equal to the number of strict equivalence classes of type $\langle l, m \rangle$.
\[exact\] Suppose $l < p$. Then each strict equivalence class of type $\langle l, m \rangle$ has precisely one representative in the reduced form in Theorem \[main\_thm\].
The condition $l<p$ of Theorem \[exact\] is necessary. We will provide a counterexample for $p=2$ and type $\langle 5, 15 \rangle$. Finally, by combining Theorems \[main\_thm\], \[exact\], and Lubin [@Lu], we obtain our main result on the torsion elements of ${\mathfrak{N}}$ of order $p^2$.
\[intro\_final\_thm\] Let $d_{l,m}$ denote the number of conjugacy classes of elements of the Nottingham group ${\mathfrak{N}}$ that are of order $p^2$ and type $\langle l, m \rangle$. Then $$\label{ineq}
d_{l,m} \leq B(p,l,m)=p^k (p-1)^\epsilon$$ where $k$ is the number of integers in $[m-l,m-1]$ incongruent to $0$ modulo $p$, and $\epsilon$ is equal to $1$ if $m$ is congruent to $0$ modulo $p$, and equal to $2$ otherwise. The inequality is an equality if $l<p$.
Let $n\in\N$ and $u\in{\mathfrak{N}}$ be a torsion element of order $p$. It is not difficult to see (e.g., by Theorem \[prior\_thm\_1\]) that $u$ and $u^n$ are conjugate in ${\mathfrak{N}}$ if and only if $u=u^n$. As an application of the above results, the following group theoretic result related to the Nottingham group is of independent interest.
\[conjugate\] Let $n\in\N$ and $u\in{\mathfrak{N}}$ be a torsion element of order $p^2$ and type $\langle l,m\rangle$. If $u$ and $u^n$ are distinct elements, then they are conjugate in ${\mathfrak{N}}$ if and only if $n\equiv 1 {\pmod{p}}$ and $(p,l,m)\neq (2,l,2l)$.
The organization of the paper is as follows. In §\[not\] we establish the notation and conventions that will be adapted throughout the paper, and also state some prior results so that the reader may appreciate the results established in this paper better. In §\[prelim\_comp\] we perform some preliminary computations that will be useful in §\[classification\]. In §\[classification\] we prove Theorems \[main\_thm\] and \[exact\] and also justify the condition $l < p$ by providing a counterexample. In §\[torsion\_elements\] we prove our main results (Theorems \[intro\_final\_thm\] and \[conjugate\]) on torsion elements of the Nottingham group ${\mathfrak{N}}$ of order $p^2$.
Notation, conventions, and prior results {#not}
========================================
We adapt the following notation in this article. The letter $p$ always denotes a prime number, and $\F_p$ the prime field of characteristic $p >0$. By $\kappa$ we mean a characteristic $p$ finite field, $K$ is the field of Laurent-series ${\kappa(\!(t )\!)}$ over $\kappa$, and ${\mathcal{O}}_K := {\kappa[\![t ]\!]}$ its ring of integers with maximal ideal ${\mathfrak{M}}= t {\mathcal{O}}_K$ where $t$ is a fixed uniformizer of ${\mathcal{O}}_K$. Then the group of principal units $ 1 + {\mathfrak{M}}$ is denoted by $U_1$, and its higher unit subgroups $1 + {\mathfrak{M}}^j$, $j \geq 1$, by $U_j$.
The Nottingham group over $\kappa$, denoted by ${\mathfrak{N}}_\kappa$, is the subgroup of elements of $\mathrm{Aut}_\kappa(K)$ mapping $t$ to $u(t)\in t(1+{\mathfrak{M}})$. An element of ${\mathfrak{N}}_\kappa$ is determined and represented by the power series $u(t)$. We drop the subscript $\kappa$ in the case where $\kappa = \F_p$, the prime finite field of characteristic $p > 0$. So, for example ${\mathfrak{N}}_{\F_p}$ is simply written as ${\mathfrak{N}}$.
In this article, we deal *only* with continuous characters $\chi : U_1 \to \Z/p^n \Z$, where $U_1$ equipped with induced topology from that of the topological group $K^\times$, and $\Z/p^n \Z$ with the discrete topology. So we omit the adjective ‘continuous’ while referring to the characters in the ${\Z_p}$-module ${\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}({U_1}, \Z/p^n \Z)$.
The definitions of strict equivalence $\simeq$ (resp. weak equivalence $\sim$) on ${\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}({U_1}, \Z/p^n \Z)$ and the break sequence (or the type) of a surjective character are defined in the same ways as the $\kappa=\F_p$ case (see $\mathsection1$) so that a strictly (resp. weakly) equivalent class of surjective characters have the same type. When we say a character has certain type, the character is assumed to be a surjective character.
We now describe the main results of Lubin on torsion elements of ${\mathfrak{N}}_\kappa$. Given an order $p^n$ element $u\in{\mathfrak{N}}_\kappa\subset\mathrm{Aut}_\kappa(K)$, the subset $F$ of $K={\kappa(\!(t )\!)}$ fixed by $u$ is a subfield of $K$ for which $K/F$ is a degree $p^n$ totally ramified abelian extension. Local class field theory produces a canonical continuous group homomorphism onto the subgroup generated by $u$: $$\rho^K_F:F^*\to \langle u\rangle.$$ Since $K/F$ is totally ramified, the norm $N_{K/F}(t)$ of $t$ is a uniformizer of $F$. Hence, there is a unique $\kappa$-isomorphism $K\cong F$ of fields mapping $t$ to $N_{K/F}(t)$. On the other hand, there is a unique group isomorphism $\langle u\rangle\cong \Z/p^n\Z$ mapping $u$ to $1$. Therefore, we obtain, by compositions of maps, a surjective character $\chi$ in ${\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}({U_1}, \Z/p^n \Z)$:
$$\chi: {U_1}\hookrightarrow K^*\stackrel{\cong}{\longrightarrow} F^*
\stackrel{\rho^K_F}{\longrightarrow}\langle u\rangle \stackrel{\cong}{\longrightarrow}\Z/p^n\Z.$$
\[prior\_thm\_1\] [@Lu Theorem 2.2] Let $n$ be a natural number. The above association $u\mapsto \chi$ induces a bijective correspondence between the conjugacy classes of order $p^n$ elements of the Nottingham group ${\mathfrak{N}}_\kappa$ and the strictly equivalent classes of surjective characters in ${\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}({U_1}, \Z/p^n \Z)$.
By Theorem \[prior\_thm\_1\], the conjugacy classes of order $p^n$ elements of ${\mathfrak{N}}_\kappa$ can be further classified by the types $\langle b^{(0)}, \ldots, b^{(n-1)} \rangle$ of their corresponding strictly equivalent classes of characters. Hence, it makes sense to talk about the type of a torsion element.
\[prior\_thm\_2\] [^2] Let $q$ be the size of the finite field $\kappa$. Let $d_m$ denote the conjugacy classes of order $p$ elements of ${\mathfrak{N}}_\kappa$ of type $\langle m\rangle$ and $d_{1,m}$ denote the number of conjugacy classes of order $p^2$ elements of ${\mathfrak{N}}_\kappa$ of type $\langle 1, m \rangle$. Then $d_m=q-1$ and
$$d_{1,m} =
\begin{cases}
p(q-1) & \; \textrm{if} \; m \equiv 0 {\pmod{p}}. \\
(q-1)^2 & \; \textrm{if} \; m \equiv 1 {\pmod{p}}.\\
p(q-1)^2 & \; \textrm{otherwise}. \\
\end{cases}$$
When $\kappa=\F_p$, the second author proved the following.
\[prior\_thm\_3\] [@Ki][^3] Let $d_{2,m}$ denote the number of conjugacy classes of order $p^2$ elements of ${\mathfrak{N}}$ of type $\langle 2, m \rangle$ and $d_{2,m}^{\textrm{weak}}$ denote the number of weakly equivalent classes of surjective characters in ${\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}({U_1}, \Z/p^2 \Z)$ of type $\langle 2,m\rangle$. Then $$d_{2,m}^{\textrm{weak}} \leq d_{2,m} \leq p d_{2,m}^{\textrm{weak}},$$ where $$d_{2,m}^{\textrm{weak}} =
\begin{cases}
p(p-1) & \; \textrm{if} \; m \equiv 0 {\pmod{p}}. \\
(p-1)^2 & \; \textrm{if} \; m \equiv 1 {\pmod{p}}.\\
p(p-1)^2 & \; \textrm{otherwise}. \\
\end{cases}$$
In this article, our main result Corollary \[main\_cor\] improves the result of the second author above, namely that we obtain $d_{2,m} = p d_{2,m}^{\textrm{weak}}$.
Preliminary computations {#prelim_comp}
========================
From now on till the end of the paper, we assume $K={\F_p(\!(t )\!)}$ and work on the Nottingham group ${\mathfrak{N}}$ over $\F_p$. Consider the $\Z/p^2 \Z$-module ${\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}(U_1, \Z/p^2 \Z)$ of characters. The set $$\{E_j:=1+t^j\in U_1: \hspace{.1in} j\in\N\}$$ contains the subset $$\{E_j:=1+t^j\in U_1: \hspace{.1in} j\in\N, \hspace{.1in} p\nmid j\},$$ which is a topological $\Z_p$-basis of $U_1$ and the dual basis is denoted by $$\{{\mathfrak{Z}}_j\in {\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}(U_1, \Z/p^2 \Z):\hspace{.1in} j\in\N,\hspace{.1in} p\nmid j \},$$ i.e., ${\mathfrak{Z}}_j( E_i) = \delta_{ij}$, the Kronecker delta function. Let ${\mathcal{C}}= \{ 0, 1 \ldots, p^2 -1 \}$ be a fixed choice of representatives of $\Z/p^2 \Z$. Then any surjective character $\chi\in {\operatorname{Hom}}_{\Z_p}^{\textrm{cont}}(U_1, \Z/p^2 \Z)$ with break sequence $\langle l,m \rangle$ has the expression of the form $$\label{basic}
\chi = \sum_{ \substack{1 \leq j \leq m, p \nmid j}} c_j {\mathfrak{Z}}_j,$$ where the coefficients $c_j \in {\mathcal{C}}$.
\[bs\_lemma\] (Lubin [@Lu]) Let $\chi : {U_1}\to \Z/p^n \Z$ be a surjective continuous character. Let $\langle b^{(0)}, \ldots, b^{(n-1)} \rangle$ be its break sequence. Then the following conditions hold:
(a) \[pri\_cond\] $\gcd(p,b^{(0)}) =1$;
(b) \[ine\_cond\] for each $i >0$, $b^{(i)} \geq p b^{(i-1)}$, and
(c) \[str\_cond\] if the above inequality is strict, then $\gcd(p,b^{(i)}) =1$.
Conversely, every sequence $\langle b^{(0)}, \ldots, b^{(n-1)} \rangle$ satisfying the above three conditions is the break sequence of some character $\chi$ on ${U_1}$. There are only finitely many different characters $\chi$ with the break sequence $\langle b^{(0)}, \ldots, b^{(n-1)} \rangle$, and a , only finitely many strict equivalence classes of such characters.
Let $\chi$ be the character in . It follows from Proposition \[bs\_lemma\] that
1. $l$ is relative prime to $p$;
2. $c_l$ is relative prime to $p$;
3. for all $l+1 \leq j \leq m$ such that $p \nmid j$, the coefficient $c_j \in \Set{0,p,2p, \ldots, (p-1)p}$;
4. if $m$ is relative prime to $p$, then $c_m \neq 0$.
Writing $c_1,\ldots, c_l$ in the form $x + p . a$ where $x,a \in \{ 0, 1 \ldots, p-1 \}$, and changing the notation, the above expansion takes the following form $$\label{sta_expansion}
\chi = \sum_{1 \leq i \leq l, p \nmid i} x_i {\mathfrak{Z}}_i + \sum_{\substack{1 \leq j \leq m, p \nmid j}} a_j. p {\mathfrak{Z}}_j,$$ where now $x_1, x_2, \ldots x_l, a_1, \ldots, a_m \in \{ 0, 1, \ldots p-1 \}$, and $x_l \neq 0$, and if $(m,p) =1 $ then $a_m \neq 0$ too. We call the *standard expansion* of $\chi$.
Let $\psi$ be a character with break sequence $\langle l \rangle$. Then $\psi$ is trivial on $1 + {\mathfrak{M}}^{l+1}$ but not on $1 + {\mathfrak{M}}^{l}$, and that $\psi$ restricts to a nonzero linear functional ${\widetilde{\psi}}$ on the one dimensional $\F_p$-vector space $(1 + {\mathfrak{M}}^l)/ (1 + {\mathfrak{M}}^{l+1})$.
\[lubin\_lemma\] (Lubin) Let $\chi,\psi: {U_1}\to \Z/p\Z$ be characters both of type $\langle l \rangle$. Then $\chi \simeq \psi$ if and only ${\widetilde{\chi}} = {\widetilde{\psi}}$.
For a proof see [@Lu Theorem 4.2]
\[lubin\_lemma\_2\] [^4] Let $\chi, \psi : {U_1}\to {\Z/p^n\Z}$ be two characters. Then $\chi$ is strictly equivalent to $\psi$ if and only if there exists $u \in {\mathfrak{N}}$ such that $\psi = {}_u \chi$ and $\chi( u(t)/t ) \equiv 0 \pmod{p^{n-1}}$.
For a proof see [@Ki Lemma 4.2].
\[degenerate\] A variant of the following lemma can be found in [@Ki Lemma 5.1]. As a preliminary, we note that if $m \equiv 0 {\pmod{p}}$, then $m = l p$ by Proposition \[bs\_lemma\] .
\[evaluation\] Let $\chi, \psi$ be two characters of type $\langle l, m \rangle$ with standard expansions as in equation . Let $u(t) = t(1 + \alpha t + \beta t^2 \cdots) \in {\mathfrak{N}}$. The following assertions hold.
(a) \[12\] ${{}_{u}} \chi(E_l) \equiv x_l \equiv \chi(E_l)$ ${\pmod{p}}$.
(b) \[m\] If $p \nmid m$ then ${{}_{u}} \chi(E_m) = p\cdot a_m $.
<!-- -->
(a) Since $E_l \circ u(t) = 1 + u(t)^l = 1 + t^l(1 + \alpha t + \beta t^2 + \cdots)^l = 1 + t^l +l \alpha t^2 + \cdots = (1 + t^l) \cdots$ we have, modulo $p$, $$\begin{aligned}
{{}_{u}} \chi (E_l)
&= \chi(E_l \circ u(t)) \equiv \chi((1+t)^l) \\
&= \chi(E_l) \equiv x_l. \end{aligned}$$
(b) If $p \nmid m$ then we have $$\begin{aligned}
{}_{u} \chi (E_m ) = \chi( E_m \circ u(t)) = \chi (1 + t^m) = \chi(E_m) = p\cdot a_m .\end{aligned}$$
Classification of strict equivalence classes {#classification}
============================================
\[lem7\] Let $\chi$ be a character of type $\langle l , m \rangle$ with the standard expansion . Then $\chi$ is strictly equivalent to the character with standard expansion $$\label{row2}
x_l {\mathfrak{Z}}_l + \sum_{\substack{1 \leq j \leq m, p \nmid j}} b_j. p {\mathfrak{Z}}_j,$$ where $b_j \in \{0,1,...,p-1\}$. In addition, if $p \nmid m$ then $b_m = a_m$.
By Theorem \[lubin\_lemma\], $\chi {\pmod{p}}\simeq x_l {\mathfrak{Z}}_\ell{\pmod{p}}$, so that, by definition, there exists an element $u\in{\mathfrak{N}}$ such that $x_l {\mathfrak{Z}}_\ell\equiv {}_u \chi{\pmod{p}}$ and $\chi(u(t)/t)\equiv 0{\pmod{p}}$. By Lemma \[lubin\_lemma\_2\] for $n=2$, $\chi$ is strictly equivalent to ${}_u \chi$ and the latter is of the form . The last assertion follows from Lemma \[evaluation\](b).
The form is strictly equivalent to a even more simple form.
\[prop7\] Let $\chi$ be a character of type $\langle l , m \rangle$ with the standard expansion . Then $\chi$ is strictly equivalent to a character with the standard expansion $$\label{row20}
x_l {\mathfrak{Z}}_l + \sum_{ m-l \leq j \leq m, p \nmid j} b_j \cdot p {\mathfrak{Z}}_j,$$ where the coefficients $x_l$ and $b_j$ for $m-l \leq j \leq m$, $p \nmid j$ are the coefficients in .
Let $\chi$ be the character of type $\langle l , m \rangle$ in . Then there exists $b_m\in\{1,...,p-1\}$ such that $$b_m\cdot p =\chi(E_m).$$ By induction, it suffices to find, for each index $j$ starting from $1$ to $m-l-1$ with $m-l-j$ relative prime to $p$, an element $u_j:=u_j(t)\in{\mathfrak{N}}$ satisfying three conditions:
(a) ${}_{u_j} \chi(E_{m-l-j})=0;$
(b) $\chi(u_j(t)/t)=0$;
(c) $\chi(1+t^k)=\chi(1+u_j^k)$ if $m-l-j<k\leq m$.
Consider $$u_j :=u_j(t)=t(1+t^{l+j})^d(1+t^m)^e\in{\mathfrak{N}},$$ where $d$ and $e$ are integers to be chosen later. Then, we obtain $$\begin{aligned}
1 + u_j^{m-l-j}
&= 1 + t^{m-l-j}(1 + t^{l+j})^{d (m-l-j)} (1 + t^m)^{e (m-l-j)} \\
&= 1 + t^{m-l-j} + d (m-l-j)t^m \pmod{U_{m+1}} \\
&= (1 + t^{m-l-j})(1 + t^m)^{d (m-l-j)} \pmod{U_{m+1}}\end{aligned}$$
Therefore $$\begin{aligned}
\label{compute}
\begin{split}
{{}_{u_j}} \chi(E_{m-l-j}) &= \chi(1 + u_j^{m-l-j}) \\
& = p\cdot (b_{m-l-j} + d(m-l-j) b_m )
\end{split}\end{aligned}$$ Since $b_m$ and $m-l-j$ are both relative prime to $p$, there exists some $d\in\N$ such that is zero in $\Z/p^2\Z$, i.e., (a) is fulfilled. On the other hand, as $l+j>l$ we have $$\begin{aligned}
\label{compute2}
\begin{split}
\chi(u_j(t)/t) &= \chi((1 + t^{l+j})^d (1 + t^m)^e) \\
&= p\cdot( d b_{l+j} + e b_m).
\end{split}\end{aligned}$$ Again since $b_m$ is prime to $p$, there exists $e\in\N$ such that is zero in $\Z/p^2\Z$, i.e., (b) is fulfilled. Finally, one checks easily that (c) is also fulfilled by the definition of $u_j$. We are done.
[\[main\_thm\]]{} Let $[\chi]$ be a strict equivalence class of type $\langle l, m \rangle$. Then there exists a character in the class $[\chi]$ whose standard expansion is $$x_l {\mathfrak{Z}}_l + \sum_{ m-l \leq j \leq m, p \nmid j} b_j \cdot p {\mathfrak{Z}}_j,
\tag{\ref{intro_reduction}}$$ where $x_l \neq 0$ and if $p \nmid m$, then $b_m \neq 0$. The character in is said to be in *reduced form*.
The assertion follows directly from Lemma \[lem7\] and Proposition \[prop7\].
\[main\_cor\] Let $B(p,l,m)$ be the number of type $\langle l, m \rangle$ reduced forms . The number of strict equivalence classes of type $\langle l, m \rangle$ is at most $B(p,l,m)=p^k (p-1)^\epsilon$, where $k$ is the number of integers in $[m-l,m-1]$ incongruent to $0$ modulo $p$, and $\epsilon$ is equal to $1$ if $m$ is congruent to $0$ modulo $p$, and equal to $2$ otherwise.
If $m$ is incongruent to $0$ modulo $p$, then each of $x_l, b_m$, being nonzero, can be chosen in $p-1$ different ways, otherwise when $m$ is congruent to $0$ modulo $p$ the coefficient $a_m$ does not appear in the standard expansion and so that only $x_l$ needs to be chosen, which can be done in $p-1$ ways. The assertion about the other factor $p^k$, with restriction on $k$, is evident.
[\[exact\]]{} Suppose $l < p$. Then each strict equivalence class of type $\langle l, m \rangle$ has precisely one representative in the reduced form in Theorem \[main\_thm\].
Suppose $l<p$ and $\chi$ is in reduced form . Let $u=u(t)\in{\mathfrak{N}}$ satisfy $$\label{kern}
\chi(u(t)/t)=0$$ and that $$\label{long}
\chi' = {{}_{u}} \chi = x'_l {\mathfrak{Z}}_l + \sum_{ m-l \leq j \leq m, p \nmid j} b_j' \cdot p {\mathfrak{Z}}_j$$ is also in reduced form but distinct from $\chi$. By Lemma \[evaluation\](a) and (b), we obtain $x_l = x'_l$ and if $p \nmid m$, then $b_m = b_m'$. If $\widetilde{j}$ denotes the the largest index $j$ such that $b_j\neq b'_j$, then $\widetilde{j}$ is strictly less than $m$, so that $0<m-\widetilde{j}\leq l$. Write $$u(t)=t\prod_{k=1}^\infty (1+t^k)^{n_k}$$ where $n_k\in\{0,1,...,p-1\}$. If $\widetilde{k}$ denotes the smallest index $k$ such that $n_k \neq 0$, then the inequalities $0<\widetilde{k}\leq m-\widetilde{j}$ hold; indeed if $\widetilde{k} > m - \widetilde{j}$, then $b_{\widetilde{j}} = b'_{\widetilde{j}}$ which contradicts the definition of $\widetilde{j}$. Hence, we conclude that $$0<\widetilde{k}\leq m-\widetilde{j}\leq l<p.$$
The equality $\widetilde{k}=l$ is impossible since it contradicts . It follows that $0< l-\widetilde{k}<p$, and together with $x_l,n_{\widetilde{k}}\in \{1,...,p-1\}$, we obtain $$\begin{aligned}
\begin{split}
\chi'(E_{l-\widetilde{k}})={{}_{u}}
\chi(1+t^{l-\widetilde{k}})
&=\chi(1+t^{l-\widetilde{k}}(1+n_{\widetilde{k}}t^{\widetilde{k}}+\cdots)^{l-\widetilde{k}})\\
&=\chi(1+t^{l-\widetilde{k}}+n_{\widetilde{k}}(l-\widetilde{k}) t^l+\cdots)\\
&=\chi((1+t^{l-\widetilde{k}})(1+ t^l)^{n_{\widetilde{k}}(l-\widetilde{k})}\cdots)\\
&= n_{\widetilde{k}}(l-\widetilde{k})x_l\neq 0 ~\text{mod $p$},
\end{split}\end{aligned}$$ contrary to the hypothesis that $\chi'$ is in reduced form.
Consider $p=2$ and the type $\langle 5,15\rangle$ character $\chi$ in reduced form: $$\chi= {\mathfrak{Z}}_5 + 2{\mathfrak{Z}}_{15}.$$ Take $u(t)=t(1+t^3+t^4)(1+t^{15})^e \in{\mathfrak{N}}$ so that $\chi(u(t)/t)=0$. Then ${}_u \chi$ is of the form such that ${}_u \chi(E_{11})=2$. By Proposition \[prop7\], ${}_u \chi$ is strictly equivalent to a reduced form $\psi$ with $\psi(E_{11})=2\neq 0= \chi(E_{11})$. But $\psi$ and $\chi$ are strictly equivalent and both are in reduced forms.
Torsion elements of ${\mathfrak{N}}$ of order $p^2$ {#torsion_elements}
===================================================
[\[intro\_final\_thm\]]{} \[final\_thm\] Let $d_{l,m}$ denote the number of conjugacy classes of elements of the Nottingham group ${\mathfrak{N}}$ that are of order $p^2$ and type $\langle l, m \rangle$. Then $$d_{l,m} \leq B(p,l,m)=p^k (p-1)^\epsilon
\tag{\ref{ineq}}$$ where $k$ is the number of integers in $[m-l,m-1]$ incongruent to $0$ modulo $p$, and $\epsilon$ is equal to $1$ if $m$ is congruent to $0$ modulo $p$, and equal to $2$ otherwise. The inequality is an equality if $l<p$.
The result is an immediate consequence of Theorem \[prior\_thm\_1\] of Lubin, Corollary \[main\_cor\], and Theorem \[exact\].
Since $l<p$ always holds when $l=1,2$, Theorem \[intro\_final\_thm\] recovers the formula of $d_{1,m}$ in Theorem \[prior\_thm\_2\] when $\kappa=\F_p$ and is an improvement over Theorem \[prior\_thm\_3\] of the second author in which only the upper bound on $d_{2,m}$ was established.
[\[conjugate\]]{} Let $n\in\N$ and $u\in{\mathfrak{N}}$ be a torsion element of order $p^2$ and type $\langle l,m\rangle$. If $u$ and $u^n$ are distinct elements, then they are conjugate in ${\mathfrak{N}}$ if and only if $n\equiv 1 {\pmod{p}}$ and $(p,l,m)\neq (2,l,2l)$.
Let $\chi$ be a surjective character of type $\langle l,m\rangle$ associated to $u$.\
($\Rightarrow$) Suppose $u$ and $u^n$ are conjugate in ${\mathfrak{N}}$. It follows that $n$ is prime to $p$. Then one sees from the correspondence in $\mathsection2$ that $n\cdot\chi$ is a surjective character associated to $u^n$. Let $x_l\neq0$ be $\chi(E_l){\pmod{p}}$. As $u$ and $u^n$ are conjugate, $\chi$ and $n\cdot\chi$ are strictly equivalent. Hence by Lemma \[evaluation\](a), we have $$x_l\equiv n x_l{\pmod{p}}$$ which implies $n\equiv 1 {\pmod{p}}$.
On the other hand, assume $(p,l,m)=(2,l,2l)$ for some odd $l$, we need to prove that $u$ and $u^n$ are not conjugate. Let $\chi_1$ be a reduced form of $u$. Since $m-l=l$ is prime to $p=2$, the character $n\cdot \chi_1$ is a reduced form of $u^n$. If $u$ and $u^n$ are conjugate, then $n$ is odd. Since $u\neq u^n$ and $p^2=4$, we may assume $n=3$. It follows that the reduced forms $\chi_1 \neq 3\chi_1$ differ only at the coefficient $b_l$ (see ) and ${}_w \chi_1 = 3\chi_1$ for some $w\in{\mathfrak{N}}$ satisfying
(a) $w:=w(t)\neq t$;
(b) $\chi_1(w(t)/t)=0$.
The conditions (a) and (b) imply that $0<k< l$ if we write $w(t)=t(1+t^k+\cdots)\in{\mathfrak{N}}$. If $k$ is odd, then one computes by using $\chi_1 (E_{2l})\neq 0$ and $\chi_1$ is a reduced form that $${}_w \chi_1 (E_{2l-k})=\chi_1 (E_{2l-k}) + \chi_1 (E_{2l})\neq \chi_1 (E_{2l-k})=3\chi_1 (E_{2l-k}).$$ If $k$ is even, then one computes by using $\chi_1$ is a reduced form that $${}_w \chi_1 (E_{l-k})\equiv
\chi_1 (E_{l})\not\equiv 0 \equiv 3\chi_1 (E_{l-k}) ~~(\mathrm{mod}~2).$$ Since both cases contradict the equation ${}_w \chi_1 = 3\chi_1$, $u$ and $u^n$ are not conjugate.\
($\Leftarrow$) Without loss of generality, assume $\chi$ is in reduced form. Then $n\cdot \chi$ is in the form of . By Proposition \[bs\_lemma\] and Remark \[degenerate\], $(p,l,m)\neq (2,l,2l)$ is equivalent to $m-l>l$. Apply Proposition \[prop7\] to $n\cdot \chi$ and we obtain a reduced form $\psi\simeq n\cdot \chi$. Since $m-l>l$ and $n\equiv 1 {\pmod{p}}$, it follows that $\psi$ is identical to $\chi$, which implies that $n\cdot \chi\simeq \chi$. Therefore, $u^n$ and $u$ are conjugate in ${\mathfrak{N}}$.
[9]{} R. Camina: ‘Subgroups of the Nottingham Group’, [*Journal of Algebra*]{} **196** (1997) 101–113.
R. Camina: ‘The Nottingham Group’. In: du Sautoy M., Segal D., Shalev A. (eds) New Horizons in pro-p Groups. Progress in Mathematics, vol **184**. Birkh$\mathrm{\ddot{a}}$user, Boston, MA.
K. Kishore: ‘Torsion elements in the Nottingham group of order $p^2$ and type $\langle 2,m \rangle$’, https://arxiv.org/abs/1710.09194, [*to appear in the Journal of the Ramanujan Mathematical Society.*]{}
B. Klopsch: ‘Automorphisms of the Nottingham group’, [*Journal of Algebra*]{} **223** (2000) 37–56.
J. Lubin: ‘Torsion in the Nottingham group’, [*Bull. London Math. Soc.*]{}, (2011) 547–560.
----------------------------------- ----------------------------------------
Chun Yin Hui Krishna Kishore
Yau Mathematical Sciences Center Dept. of Mathematics
Tsinghua University Indian Institute of Technology-Dharwad
Haidian District, Beijing 100084, WALMI Campus, Dharwad
China Karnataka 580011, India
----------------------------------- ----------------------------------------
[^1]: C.Y. Hui is supported by China’s Thousand Talents Plan: The Recruitment Program for Young Professionals.
[^2]: The result on $d_m$ is due to Klopsch [@Kl] and the result on $d_{1,m}$ is an unpublished work of Lubin.
[^3]: In [@Ki], the notation $d_{2,m}$ and $d_{2,m}^{\textrm{weak}}$ are respectively denoted by $d_m$ and $d_{m}^{\textrm{weak}}$.
[^4]: The second author thanks Jonathan Lubin for sharing the proof with him.
|
---
abstract: |
We construct a positive constant curvature space by identifying some points along a Killing vector in a de Sitter Space. This space is the counterpart of the three-dimensional Schwarzschild-de Sitter solution in higher dimensions. This space has a cosmological event horizon, and is of the topology ${\cal
M}_{D-1}\times S^1$, where ${\cal M}_{D-1}$ denotes a $(D-1)$-dimensional conformal Minkowski spacetime.
address: |
Institute of Theoretical Physics, Chinese Academy of Sciences,\
P.O. Box 2735, Beijing 100080, China
author:
- 'Rong-Gen Cai[^1]'
title: A Note on the Positive Constant Curvature Space
---
In recent years people have been paying particular interest on the constant curvature spacetimes. For the case of negative constant curvature space, namely the anti-de Sittter (AdS) space, it should be mainly attributed to the celebrated AdS/CFT (conformal field theory) correspondence [@AdS], which states that the string or M theory on an AdS space times a compact space is dual to a strong coupling conformal field theory residing on the boundary of the AdS. For the case of positive constant curvature space, namely the de Sitter (dS) space, one of the reasons responsible for the particular interest is that defined in a manner analogous to the AdS/CFT correspondence, an interesting proposal, which says that there is a dual between quantum gravity on a dS space and a Euclidean CFT on a boundary of the dS space, the so-called dS/CFT correspondence, has been suggested recently [@Strom]. Another reason is that our universe may be an asympotically dS spacetime [@Cos]. However, different from the cases for the asymptotically flat or AdS spacetimes, we have not yet a fundamental description of asymptotically dS spacetimes in the sense of quantum gravity: there is presently no fully satisfactory embedding of dS space into string theory [@BDM].
In $D(\ge 3)$ dimensions, the AdS space has the topology $S_1
\times R^{D-1}$. By identifying points in the universal covering space of the AdS space in a some manner, one can construct various spacetimes which are locally equivalent to the AdS space. It is well-known that the BTZ black hole [@BTZ] just belongs to this kind of spacetime, which is constructed by identifying points along a boost Killing vector in a three dimensional AdS space. Its counterparts are discussed in four dimensions [@Amin] and in higher dimensions [@Bana], respectively. The topology structure of the BTZ black hole is ${\cal M}_2 \times S^1$, where ${\cal M}_n$ denotes a conformal Minkowski spacetime in $n$ dimensions. So the topology of the BTZ black hole is not peculiar, but its higher dimensional counterparts have the topology ${\cal
M}_{D-1} \times S^1$ in the case of $D$ dimensions, which is quite different from the usual one ${\cal M}_2 \times S^{D-2}$. These black holes look strange in the sense that the exterior of black holes is time-dependent and there is no globally defined timelike Killing vector in the geometry of these black holes [@Holst]. So it is quite difficult to discuss the thermodynamics associated with the black hole. In spite of the peculiar feature of the black hole geometry, in [@Cai] we applied the surface counterterm approach to a five dimensional constant curvature black hole, and obtained the stress-energy tensor of dual conformal field theory. For other methods to study the constant curvature black holes, see [@Bana; @Mann].
On the other hand, a $D(\ge 3)$-dimensional dS space has the topology $R_1\times S^{D-1}$. Different from the AdS case, the dS space has a cosmological horizon and associated Hawking temperature and thermodynamic entropy $S$ [@Gibbons]: $$\label{1eq1}
S=\frac{\Omega_{D-2}}{4G}l^{D-2},$$ where $G$ is the gravitational constant in $D$ dimensions, $l$ is the radius of dS space and $\Omega_{D-2}$ denotes the volume of a $(D-2)$-dimensional unit sphere. Due to this, it is argued that the dS space should be described by a theory with a finite number $e^S$ of independent quantum states; the cosmological constant $\Lambda$ should be understood as a direct consequence of the finite number of states $e^N$ in the Hilbert space describing the world [@Banks; @Bousso; @Fischler]. Thus a natural consequence is that a necessary condition for a (coarse-gained) spacetime to be described by a theory with $e^N$ states is that the entropy accessible to any observer in that spacetime must obey [@BDM] $$S\le N.$$ According to the “$\Lambda \sim N $ correspondence" [@Bousso], the theory with $e^N$ states may describe the set [**all**]{}$(\Lambda(N))$ of spacetimes: the class of spacetimes with positive cosmological constant $\Lambda(N)$, irrespective of asymptotic conditions and types of the matter present [@BDM]. Applying the condition (\[1eq1\]) to the candidate set [**all**]{}$(\Lambda(N))$, a suggestion named “N bound" has been proposed in [@Bousso], which in four dimensions states: [*In any universe with a positive cosmological constant $\Lambda$ (as well as arbitrary additional matter that may well dominate at all times) the observable entropy $S$ is bounded by $N=3\pi/G\Lambda$*]{}. Roughly specking, the “N bound" means that the observable entropy in the spacetimes with a positive cosmological constant must be less than or equal to the entropy (\[1eq1\]). This “N bound" leads naturally to the Bekenstein entropy bound of matter in dS spaces [@Bousso2]. The geometry of the dS space looks simple, but this space has much non-trivial physics.
Similar to the negative constant curvature case, in this note we construct a positive constant curvature spacetime by identifying points along a rotation Killing vector in a dS space. Our solutions turns out to be counterparts of the three-dimensional Schwarzschild-dS solution in higher dimensions, and have an associated cosmological horizon. There is a parameter in the solution, which can be explained as the size of cosmological horizon.
Let us begin with the five dimensional case. A five dimensional dS space can be understood as a hypersurface embedded into a six dimensional flat space, satisfying $$\label{2eq1}
-x_0^2+x_1^2 +x_2^2 +x_3^2 +x_4^2 +x_5^2 =l^2,$$ where $l$ is the radius of the dS space. This dS space has fifteen Killing vectors, five boosts and ten rotations. Consider a rotation Killing vector $\xi =(r_+/l)(x_4\partial _5
-x_5\partial_4)$ with norm $\xi^2 = r_+^2/l^2 (x_4^2 +x_5^2)$, where $r_+$ is an arbitrary real constant. The norm is always positive or zero. In terms of the norm, the hypersurface (\[2eq1\]) can be expressed as $$\label{2eq2}
x_0^2 =x_1^2 +x_2^2 +x_3^2 +l^2 (\xi^2/r_+^2-1).$$ From (\[2eq2\]) we see that when $\xi^2=r_+^2$, the hypersurface reduces to a null one $$\label{2eq3}
x_0^2=x_1^2 +x_2^2 +x_3^2,$$ while $\xi^2 =0$, it goes to a hyperboloid $$\label{2eq4}
x_0^2 =x_1^2 +x_2^2 +x_3^2 -l^2.$$ The null surface (\[2eq3\]) has two pointwise connected brances, called $H_f$ and $H_p$, described by $$\begin{aligned}
H_f:&& x_0=+\sqrt{x_1^2 +x_2^2 +x_3^2}, \nonumber \\
H_p:&& x_0=-\sqrt{x_1^2 +x_2^2 +x_3^2}.
\end{aligned}$$ The hyperboloid (\[2eq4\]) has two connected surfaces, named $S_f$ and $S_p$, defined as $$\begin{aligned}
S_f:&& x_0 =+\sqrt{x_1^2 +x_2^2 +x_3^2-l^2}, \nonumber \\
S_p:&& x_0 =-\sqrt{x_1^2 +x_2^2 +x_3^2-l^2}.
\end{aligned}$$ We plot the dS space in Fig. 1. Each point in the figure represent a pair $x_4$ and $x_5$ with $x_4^2 +x_5^2$ fixed. On the surfaces $S_f$ and $S_p$, we have $\xi ^2=0$, while $\xi^2 =r_+^2$ on the surfaces $H_f$ and $H_p$ of the two cones. In the region between $S$ and $H$, one has $0<\xi^2<r_+^2$, and $\xi^2 >r_+^2 $ inside the two cones. Thus an observer located at the surface $S_f$ or $S_p$ cannot see events which happen inside the cone with null surface $H_f$ or $H_p$. That is, there does not exist any communication between the inside the cones and outside the cones. Therefore, the surface $H_f$ can be regarded as the future cosmological event horizon and $H_p$ as the past one.
Identifying points along the Killing vector $\xi$, another one-dimensional manifold becomes compact and isomorphic to $S^1$. Thus we finally obtain a spacetime with cosmological horizon and with the topology ${\cal M}_4\times S^1$. The Penrose diagram of the positive constant curvature spacetime is plotted in Fig. 2. Similiar to the case of negative constant curvature [@Bana], we may describe the spacetime in the region with $0\le \xi^2 \le
r_+^2$ by introducing six dimensionless local coordinates $(y_i,
\phi)$, $$\begin{aligned}
\label{2eq7}
&& x_i=\frac{2ly_i}{1+y^2}, \ \ \ i=0,1,2,3 \nonumber \\
&& x_4=\frac{lr}{r_+}\sin\left(\frac{r_+\phi}{l}\right),
\nonumber \\
&& x_5=\frac{lr}{r_+}\cos\left(\frac{r_+\phi}{l}\right),
\end{aligned}$$ where $$r=r_+\frac{1-y^2}{1+y^2}, \ \ \ y^2=-y_0^2+y_1^2 +y_2^2+y_3^2.$$ Here the coordinate range is $-\infty <y_i <+\infty$, and $-\infty
<\phi <+\infty$ with the restriction $-1<y^2<1$ in order to keep $r$ positive. In coordinates (\[2eq7\]), the induced metric is $$\label{2eq9}
ds^2=\frac{l^2(r+r_+)^2}{r_+^2}(-dy_0^2+dy_1^2+dy_2^2+dy_3^2)
+r^2 d\phi^2,$$ which has the same form as the case of negative constant curvature [@Bana]. However, it should be pointed out here that the coordinates (\[2eq7\]) and the definition of $r$ are different from the corresponding ones in the constant curvature black holes. In these coordinates it is evident that the Killing vector is $\xi =\partial_{\phi}$ with norm $\xi^2=r^2$. Thus in these coordinates one has $r=0$ on the surfaces $S_f$ and $S_p$ and $r=r_+$ on the horizons $H_f$ and $H_p$. With the identification $\phi \sim \phi +2\pi n$, the solution has obviously the topology ${\cal M}_4 \times S^1$.
It is trivial to generalize the five dimensional case to other dimensions. For the case of an arbitrary dimensions ($D \ge 3$), the dS space is a hypersurface satisfying $$-x_0^2 +x_1^2 +\cdots +x_{D-1}^2 +x_D^2=l^2,$$ in a $(D+1)$-dimensional flat spacetime. Consider a rotation Killing vector $\xi=r_+/l(x_D\partial_{D-1} -x_{D-1}\partial_{D})$ with norm $\xi^2=r_+^2/l^2(x^2_{D-1}+x^2_D)$ and identify points along the orbit of this Killing vector, we can obtain a positive constant curvature space with the topology ${\cal M}_{D-1}\times
S^1$. Introducing $(D+1)$ dimensionless coordinates like (\[2eq7\]), one has the induced metric $$ds^2 =\frac{l^2(r+r_+)^2}{r_+^2}\eta_{ij}dy^idy^j +r^2d\phi^2,$$ where $\eta_{ij}={\rm diag}(-1,1,\cdots,1)$ and $i,j=0, 1, \cdots,
D-2$.
Like the dS space, we can also introduce Schwarzschild coordinates to describe the solution. Using local “spherical" coordinates $(t,r,\theta,\chi)$ defined as $$\begin{aligned}
\label{2eq12}
&& y_0=f\cos\theta\sinh(r_+t/l), \ ~~~~~ y_1=f\cos\theta
\cosh(r_+t/l), \nonumber \\
&& y_2=f\sin\theta \sin\chi, \ ~~~~~ y_3=f\sin\theta \cos\chi,
\end{aligned}$$ where $f=[(r_+-r)/(r+r_+)]^{1/2}$, and the coordinate range is $ 0< \theta <\pi/2$, $0<\chi <2\pi$ and $0<r<r_+$, we find that the solution can be expressed as $$\label{2eq13}
ds^2 =l^2 N^2d\Omega_3 +N^{-2}dr^2 +r^2d\phi^2,$$ where $N^2=(r_+^2-r^2)/l^2$ and $$\label{2eq14}
d\Omega_3= -\sin^2\theta dt^2 +\frac{l^2}{r_+^2}(d\theta^2
+\cos^2\theta d\chi^2).$$ In these coordinates $r=r_+$ is the cosmological horizon. This solution is just the counterpart of a five dimensional constant curvature black hole in the Schwarzschild coordinates [@Bana]. The only difference is that $N^2=(r^2-r_+^2)/l^2$ there is replaced by $N^2=(r_+^2-r^2)/l^2$ here. In three dimensions, the corresponding induced metric is $$\label{2eq15}
ds^2=-(r_+^2-r^2)dt^2 +\frac{l^2}{r_+^2-r^2}dr^2 +r^2d\phi^2,$$ After a suitable rescaling of coordinates it can be transformed to usually three-dimensional Schwarzschild-de Sitter solution [@Park]. In (\[2eq14\]), if $\theta=\pi/2$, it also reduces to the three-dimensional Schwarzschild-dS solution.
Within the cosmological horizon dS space is time-independent in the static coordinates. The solution (\[2eq13\]) looks also static, but it does not cover the whole region within the cosmological horizon, which can be seen from the definition of coordinates (\[2eq12\]) because they must obey the constraint: $y_1^2-y_0^2=f^2\cos^2\theta \ge 0$,
As the black hole case, there is a set of coordinates which cover the whole region within the cosmological horizon. They are [@Cai] $$\begin{aligned}
\label{2eq16}
&& y_0=f\sinh(r_+t/l), \ ~~~~~~ y_1=f\cos\theta
\cosh(r_+t/l),
\nonumber \\
&& y_2=f\sin\theta \cos\chi \cosh(r_+t/l), \ ~~~~~~
y_3=f\sin\theta \sin\chi \cosh(r_+t/l).\end{aligned}$$ In terms of these coordinates, the solution is described by $$\label{2eq17}
ds^2=l^2N^2\tilde {d\Omega_3} +N^{-2}dr^2 +r^2d\phi^2,$$ where $N^2=(r_+^2-r^2)/l^2$ and $$\label{2eq18}
\tilde {d\Omega_3}=-dt^2
+\frac{l^2}{r_+^2}\cosh^2(r_+t/l)(d\theta^2+\sin^2\theta
d\chi^2).$$ It can be seen from (\[2eq18\]) that when $\theta =\chi =0$, the solution will reduce to the three-dimensional Schwarzschild-dS solution (\[2eq15\]). Therefore the solution we constructed here is the counterpart of the three-dimensional Schwarzschild-dS spacetime in five dimensions. Further it should be emphasized here that both sets of coordinates (\[2eq12\]) and (\[2eq16\]) can be used within the cosmological horizon only. This can be seen from the metrics (\[2eq13\]) and (\[2eq17\]): beyond the horizon the signature of the solution changes. It is certainly of interest to find a set of coordinates describing the outer region of cosmological horizon.
In $D$ dimensions, the solution is $$ds^2=l^2N^2d\tilde \Omega_{D-2} +N^{-2}dr^2 +r^2 d\phi^2,$$ where $N^2$ is still the one given before, and $$d\tilde\Omega_{D-2}=-dt^2
+\frac{l^2}{r_+^2}\cosh^2(r_+t/l)d\Omega_{D-3}.$$ Here $d\Omega_{D-3}$ denotes the line element of a $(D-3)$-dimensional unit sphere.
The Euclidean sector of the solution can be obtained via the transformation, $t\to -i (\tau +\pi l/2r_+)$, in the solution (\[2eq17\]). In that case, the line element (\[2eq18\]) is changed to $$\label{2eq19}
d\tilde \Omega_3 =d\tau^2 +\frac{l^2}{r_+^2}
\sin^2(r_+\tau/l)(d\theta^2 +\sin^2\theta d\chi^2).$$ In order the $d\tilde \Omega_3$ to describe a regular three-sphere, the $\tau$ must have a range $0 \le \tau \le \beta$ with $$\label{2eq20}
\beta =\pi l/r_+,$$
In the coordinates (\[2eq12\]), the Euclidean sector of the solution is (\[2eq13\]) with $$d\Omega_3= \sin^2\theta d\tau^2 +\frac{l^2}{r_+^2}(d\theta^2
+\cos^2\theta d\chi^2),$$ here $\tau$ has the range $0 \le \tau \le \tilde \beta $ with $$\tilde \beta =2 \pi l/r_+,$$ which differs from the value of $\beta$ (\[2eq20\]). Although the solution expressed in terms of the coordinates (\[2eq12\]) does not cover the whole interior of cosmological horizon, the Euclidean sector of solution is complete in both sets of coordinates (\[2eq12\]) and (\[2eq16\]). Further, regarding $\beta$ or $\tilde \beta$ as the inverse temperature of the cosmological horizon seems problematic since calculating the surface gravity $\kappa$ of the cosmological horizon shows it does not satisfy the usual thermodynamic relation $\beta =2\pi
/\kappa$. Obviously, it is of great interest to discuss the thermodynamics associated with the cosmological horizon in the constructed spacetime.
Next we further consider the Euclidean solution in the coordinates (\[2eq19\]). Making a coordinate transformation $r^2=r_+^2(1-R^2/l^2)$ and $\tau =l\varphi /r_+$, one can find that the Euclidean solution (\[2eq17\]) with (\[2eq19\]) can be expressed as $$\label{e27}
ds^2=\left(1-\frac{R^2}{l^2}\right) d(r_+\phi)^2
+\left(1-\frac{R^2}{l^2}\right)^{-1}dR^2 +R^2(d\varphi^2
+\sin^2\varphi (d\theta^2 +\sin^2\theta d\chi^2)).$$ This is evidently the Euclidean solution of de Sitter space in the static coordinates with the Euclidean time $r_+\phi$. Since the $\phi$ has the period $2\pi$, so the solution (\[e27\]) does not describe a regular instanton. The regular instanton requires the Euclidean time $r_+\phi$ has a period $2\pi l$. This implies that the solution (\[2eq17\]) has a conical singularity along the circle $\phi$ if $r_+\ne l$ with a deficit angle $2\pi (1-r_+/l)$. When $r_+ <l$, the deficit angle is less than $2\pi$, otherwise it becomes negative. When $r_+=l$, the solution is regular without any singularity. This situation is the same as the case of three dimensional Schwarzschild-dS solution [@Park].
In summary we have constructed a positive constant curvature space by identifying points along a rotation Killing vector in a dS space. This space is the counterpart of the three-dimensional Schwarzschild-dS solution in higher dimensions. Also this space can be viewed as corresponding counterpart of the negative constant curvature black hole constructed in [@Bana]. Unlike the dS space, the positive constant curvature space constructed in this note has the topology ${\cal M}_{D-1}\times S^1$. As the dS space, however, it still has a cosmological horizon $r_+$. The solution has a conical singularity along the circle $\phi$ if $r_+\ne l$. It should be interesting to discuss the thermodynamics of cosmological horizon and dual conformal field theory in the spirit of dS/CFT correspondence.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks H.Y. Guo and J.X. Lu for useful discussions, and C.T. Shi for help in drawing the figures. This work was supported in part by a grant from Chinese Academy of Sciences.
J. M. Maldacena, Adv. Theor. Math. Phys. [**2**]{}, 231 (1998) \[Int. J.Theor. Phys. [**38**]{}, 1113 (1999)\] \[arXiv:hep-th/9711200\]; S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B [**428**]{}, 105 (1998) \[arXiv:hep-th/9802109\]; E. Witten, Adv. Theor. Math. Phys. [**2**]{}, 253 (1998) \[arXiv:hep-th/9802150\]. A. Strominger, arXiv:hep-th/0106113; M. Spradlin, A. Strominger and A. Volovich, arXiv:hep-th/0110007. A. G. Riess [*et al.*]{} \[Supernova Search Team Collaboration\], Astron. J. [**116**]{}, 1009 (1998) \[arXiv:astro-ph/9805201\]; S. Perlmutter [*et al.*]{} \[Supernova Cosmology Project Collaboration\], Astrophys. J. [**483**]{}, 565 (1997) \[arXiv:astro-ph/9608192\]; R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. [**80**]{}, 1582 (1998) \[arXiv:astro-ph/9708069\]; P. M. Garnavich [*et al.*]{}, Astrophys. J. [**509**]{}, 74 (1998) \[arXiv:astro-ph/9806396\]. R. Bousso, O. DeWolfe and R. C. Myers, arXiv:hep-th/0205080. M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. [**69**]{}, 1849 (1992) \[arXiv:hep-th/9204099\]; M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D [**48**]{}, 1506 (1993) \[arXiv:gr-qc/9302012\]. S. Aminneborg, I. Bengtsson, S. Holst and P. Peldan, Class. Quant. Grav. [**13**]{}, 2707 (1996) \[arXiv:gr-qc/9604005\].
M. Banados, Phys. Rev. D [**57**]{}, 1068 (1998) \[arXiv:gr-qc/9703040\]; M. Banados, A. Gomberoff and C. Martinez, Class. Quant. Grav. [**15**]{}, 3575 (1998) \[arXiv:hep-th/9805087\]. S. Holst and P. Peldan, Class. Quant. Grav. [**14**]{}, 3433 (1997) \[arXiv:gr-qc/9705067\]. R. G. Cai, Phys. Lett. B [**544**]{}, 176 (2002) \[arXiv:hep-th/0206223\].
J. D. Creighton and R. B. Mann, Phys. Rev. D [**58**]{}, 024013 (1998) \[arXiv:gr-qc/9710042\]. G. W. Gibbons and S. W. Hawking, Phys. Rev. D [**15**]{} (1977) 2738. T. Banks, arXiv:hep-th/0007146. R. Bousso, JHEP [**0011**]{}, 038 (2000) \[arXiv:hep-th/0010252\]. W. Fischler, 2000, unpublished; [*Taking de Sitter seriously*]{}. talking given at [*Role of scaling Laws in Physics and Biology (Celebrating the 60th birthday of Geoffrey West)*]{}, Santa Fe, Dec. 2000; also see the review paper hep-th/0203101 by R. Bousso.
R. Bousso, JHEP [**0104**]{}, 035 (2001) \[arXiv:hep-th/0012052\]; R. G. Cai, Y. S. Myung and N. Ohta, Class. Quant. Grav. [**18**]{}, 5429 (2001) \[arXiv:hep-th/0105070\]. S. Deser and R. Jackiw, Annals Phys. [**153**]{}, 405 (1984); M. I. Park, Phys. Lett. B [**440**]{}, 275 (1998) \[arXiv:hep-th/9806119\].
[^1]: Email address: cairg@itp.ac.cn
|
---
abstract: 'We experimentally demonstrate a broadband and an ultra-broadband spectral bandwidth polarization rotators. Both polarization rotators have modular design, that is, they are comprised of an array of half-wave plates rotated to a given angle. We show that the broadband and ultra-broadband performance of the polarization rotators is due to the adiabatic nature of the light polarization evolution. In this paper we experimentally investigate the performance of broadband and ultra-broadband polarization rotators comprising of ten multi-order half-wave plates or ten commercial achromatic half-wave plates, respectively. The half-wave plates in the arrays are rotated gradually with respect to each other starting from an initial alignment between the fast polarization axis of the first one and the incoming linearly polarized light, to the desired polarization rotation angle.'
address:
- '$^1$ Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussée, 1784 Sofia, Bulgaria'
- '$^2$ Department of Physics, Sofia University, James Bourchier 5 blvd., 1164 Sofia, Bulgaria'
- '$^3$ Engineering Product Development, Singapore University of Technology and Design, 8 Somapah Road, 487372 Singapore'
author:
- 'Emiliya Dimova$^1$, Andon Rangelov$^2$$^,$$^3$, and Elica Kyoseva$^3$'
title: 'Broadband and ultra-broadband polarization rotators with adiabatic modular design'
---
*Keywords*: broadband polarization rotator, adiabatic evolution, piecewise adiabatic passage, modular design.
Introduction
============
The analogy between the Poincaré space, which is used for the description of light polarization, and the Bloch space, which is used for the description of quantum-state dynamics of two-level systems, is well-known [@Kubo1980; @Kubo1981; @Kubo1983; @Kuratsuji1998; @Kuratsuji2007; @Ardavan; @Rangelov2010]. This has allowed a transfer of concepts from the field of quantum optics [@Allen; @Shore] and nuclear magnetic resonance to the field of polarization optics [@Azzam; @Goldstein]. For instance, the concept of composite pulses was successfully used to create broadband half- and quarter-wave plates [@Ivanov; @Peters; @Dimova2014] as well as tunable polarization rotators [@Rangelov2015; @Dimova2015]. Another example is the application of adiabatic techniques developed for quantum systems in polarization optics [@Zapasskii; @Rangelov2011; @Berent; @Shore2015] to design various robust and achromatic polarization retarders and optical isolators.
In this paper we report an experimental demonstration of a broadband and an ultra-broadband polarization rotators, designed by an analogy to piecewise adiabatic passage technique [@Shapiro2007; @Shapiro2008; @Zhdanovich]. The broadband and ultra-broadband polarization rotators are assembled according to the recent theoretical work of Shore et al. [@Shore2015] as a sequence of ten multi-order half-wave plates and ten commercial achromatic half-wave plates, respectively. For both designs, the half-wave plates are rotated gradually from an alignment between the fast-polarization axis with the initial linearly polarized light to an alignment of the fast-polarization axis with the desired final linear polarization orientation.
Theory
======
We first consider the propagation of a plane electromagnetic wave along the z-axis through a lossless medium. In this case polarization evolution is given with the torque equation for the Stokes vector [Kuratsuji1998,Kuratsuji2007,Sala,Seto]{}: $$\frac{\,\text{d}}{\,\text{d}z}\mathbf{S}(z)=\mathbf{\Omega }(z)\times
\mathbf{S}(z), \label{Stokes equation}$$where $z$ is the distance along the propagation direction, $\mathbf{S}%
(z)=[S_{1}(z),S_{2}(z),S_{3}(z)]$ is the Stokes vector, and the driving torque $\mathbf{\Omega }(z)=[\Omega _{1}(z),\Omega _{2}(z),\Omega _{3}(z)]$ is the birefringence vector of the medium. When the medium does not possess optical activity and is uniaxial with the slow and fast axes in the $xy$ plane, then the components of the birefringence vector $\mathbf{\Omega }(z)$ are given explicitly as $$\begin{aligned}
\Omega _{1}(z) &=&\Omega _{0}\cos \left( 2\varphi \right) , \\
\Omega _{2}(z) &=&\Omega _{0}\sin \left( 2\varphi \right) , \\
\Omega _{3}(z) &=&0, \\
\Omega _{0} &=&\frac{2\pi }{\lambda }\left( n_{e}-n_{o}\right) .\end{aligned}$$Here $n_{e}$ and $n_{o}$ are the refractive indices along the fast and slow axes, $\lambda $ is the light wavelength, $\Omega _{0}$ is the rotary power, and $\varphi $ is the angle of rotation between the fast (slow) axis and the $x$ ($y$) Cartesian axis.
Now if the Stokes vector $\mathbf{S}$ is initially parallel to the birefringence vector $\mathbf{\Omega }$, such that $\mathbf{\Omega }\times\mathbf{S=0}$, and we change slowly (adiabatically) the angle $\varphi $, then the Stokes vector will also evolve adiabatically with it. For instance, if linearly polarized light is initially in the horizontal plane, i.e., $\mathbf{S}\left( z_{i}\right) =\left( 1,0,0\right) $, and $\varphi \left( z_{i}\right) =0$, which corresponds to aligning the $x$ and $y$ axes of the Cartesian coordinate system to coincide with the fast and slow optical axes, then the birefringence vector $\mathbf{\Omega }$ is initially aligned with the Stokes vector. Following an adiabatic evolution, the final angle of linear polarization will be $\varphi \left( z_{f}\right) $. This adiabatic approach has the advantages of being robust with respect to large intervals of values for $\Omega_{0}$ and is therefore, broadband.
In this work we propose an alternative scheme for the adiabatic evolution for the Stokes vector by means of discrete modular design of the polarization rotators [@Shore2015; @Shapiro2007; @Shapiro2008; @Zhdanovich]. That is, we consider that the rotation of the angle $\varphi $ takes place in a sequence of discrete steps, rather than continuously. Experimentally, we realize this by a sequence of $N=10$ birefringent crystals, each rotated at an angle $%
\varphi _{m}$ with respect to the chosen Cartesian coordinate system (illustrated schematically in Figure \[fig:fig1\]). For such a scheme the overall adiabatic evolution is achieved if the individual rotation angles change gradually and if there are sufficient number of steps ($\varphi _{m}=\frac{\varphi
}{N}\ll \varphi $). It is necessary that each of the birefringent crystals drives not more than a "Rabi cycle, in other words, that each module is not more than a half-wave plate.
![(Color online) Schematic illustration of a discrete adiabatic light evolution, which consists of a series of birefringent modules, separated by free-space propagation. The fast polarization axes of the wave plates are represented by dashed lines, while the solid red lines represent the orientation of linear polarization.[]{data-label="fig:fig1"}](fig1.eps){width="0.8\columnwidth"}
Experiment
==========
Optical setup {#Optical setup}
-------------
We performed several experiments to verify our theoretical predictions for the performance of both broadband and ultra-broadband adiabatic polarization rotators. The linear polarization of white light beam was rotated at 45$^{0}$, 60$^{0}$, 75$^{}$ and 90$^{0}$ degrees passing through a set of ten half-wave plates.
![(Color online) Experimental setup. The source $S$, irises $I$, lens $L_{1}$, lens $L_{2}$ and polarizer $P_{1}$ form a collimated beam of white polarized light. Analyzer $P_{2}$ and lens $L_{3}$ focus the beam of output light onto the entrance $F$ of an optical fibre connected to a spectrometer. The stack of ten wave plates rotates the linear polarization of the light in the broadband and ultra-broadband spectra. []{data-label="fig:fig2"}](fig2.eps){width="0.9\columnwidth"}
A 10 W Halogen-Bellaphot (Osram) lamp with DC power supply was applied as a light source with continuous spectrum in the range from 450 nm to 1100 nm (Figure \[fig:fig2\]). A sequence of an iris, two lenses and a polarizer were used for production of collimated light beam. The iris $I$ was placed in the focus of plano-convex lens $L_{1}$ ($f$=35mm) imitating a point light source. The received beam was additionally collimated by a second lens $L_{2}$ with $f$=150 mm. Polarizer $P_{1}$ (Glan-Tayler, 210-1100 nm, borrowed from a Lambda-950 spectrometer) linearly polarized the white light in the horizontal plane.
We used two types of half-wave plates for our experiments. In the first experiment a set of ordinary multi-order quarter-wave plates (WPMQ10M-780, Thorlabs) which perform as half-wave plates at $\lambda$=763 nm were used while in the second experiment achromatic half-wave plates (WRM053-mica, 700-1100 nm, aperture 20 mm) were exercised. Each wave plate (aperture of $1^{\prime\prime}$) was assembled onto a separate RSP1 rotation mount which realizes a $360^{0}$ rotation. The scale marked at $2^{0}$ increments allows for precise, repeatable positioning and fine angular adjustment.
We analyzed the characteristics of the transmitted light through the ten wave plates by a polarizer $P_{2}$ (the same type as $P_{1}$) and a grating monochromator (Model AvaSpec-3648 Fiber Optic Spectrometer with controlling software AvaSoft 7.5). We used a plano-convex lens $L_{3}$ ($f$=20 mm) and a two-axis micro-positioner to focus the light beam onto the optical fibre entrance $F$ of the monochromator.
![(Color online) Broadband linear polarization rotator consisting of ten multi-order half-wave plates. The curves corresponded to different rotation angles $\alpha$ (a) and $\alpha+90^{0}$ (b) as follow: red dashed line $90^{0}$, green short dash line $75^{0}$, blue dashed dotted line $60^{0}$ and black solid line $45^{0}$.[]{data-label="fig:fig3"}](fig3.eps){width="0.8\columnwidth"}
Measurement procedure
---------------------
Our aim was to demonstrate broad and ultra-broad spectral bandwidth linear polarization rotators composed by two sets of ten identical wave plates. Each set was assembled by multi-order wave plates or by achromatic wave plates, respectively, as described above in Subsection \[Optical setup\]. The wave plates were slightly tilted to reduce unwanted reflections [@Peters]. For the analysis of the experimental data we used a single beam spectrometer. To account for noise and losses due to light transmission, reflection and absorption in different media, we measured the light and dark spectra for all experiments. Furthermore, we used the dark spectrum, which is measured with the light path completely blocked, for correction of hardware offsets. The reference spectrum is usually taken with the light source on and a blank sample instead of the sample of interest measured. In our case, however, we measured the transmission spectrum of the already assembled polarization rotator, but the axes of the polarizer $P_{1}$, the analyzer $P_{2}$ and the fast axis of each single wave plates were all set parallel. We used the measured light spectrum as a reference for the subsequent measurements.
A linearly polarized light beam passed by the sequences of $N$ wave plates. Each of them was adjusted at the estimated angle by the formula: $\varphi_{m}=\left(m-1\right)\frac{\alpha}{\left(N-1\right)}$, where $m$ is the sequential number of the wave plate and $N=10$ in the present experiment. The rotation effect was investigated for each of the arbitrary chosen angles $\alpha=\left\{45^{0}, 60^{0}, 75^{0}, 90^{0}\right\}$. The data were taken with analyzer $P_{2}$ revolved at angle $\alpha$ and $\alpha+90^{0}$.
Experimental results
--------------------
![(Color online) Ultra-broadband linear polarization rotator built by ten achromatic half-wave plates. The curves corresponded to different rotation angle $\alpha$ (a) and $\alpha+90^{0}$ (b) as follow: red dashed line $90^{0}$, green short dash line $75^{0}$, blue dashed dotted line $60^{0}$ and black solid line $45^{0}$. The black dashed dot dot curve shows the spectrum of single achromatic wave plate.[]{data-label="fig:fig4"}](fig4.eps){width="0.8\columnwidth"}
We experimentally tested and proved development of a broad spectral bandwidth polarization rotator settled by ten multi-order half-wave plates. The transmittance spectra were taken for four arbitrary chosen angles $\alpha$. The obtained results are presented on the Figure \[fig:fig3\] a. The experimental curves show flat maximum in wide spectral diapason what is kept at different angles $\alpha$. The experimental curves at angles $\alpha+90^{0}$ of the analyzer $P_{2}$ were also presented on Figure \[fig:fig3\] b. The blow out the ray at $\alpha+90^{0}$ proves that the polarization of the light is orthogonal to the one at angle $\alpha$.
Even broader effect was reached by using ten achromatic half-wave plates. We created ulta-broad bandwidth linear polarization rotator following the above explained algorithm. We used the same rotation angles $\alpha$. Each of the wave plate in the sequences was the same type and with axis rotated at the angle $\varphi_{m}$. The experimental results are shown on the Figure \[fig:fig4\] a. Again for proving that the developed set of wave plates was working as a linear polarization rotator, the transmittance spectra at angles $\alpha+90^{0}$ are presented (see Figure \[fig:fig4\] b). The shown experimental curves not exceed 1000 nm because of the lack of the signal after this spectral region.
Conclusion
==========
In this paper we demonstrated experimental application of piecewise adiabatic passage, concept introduced by Shapiro et al. in quantum optics [@Shapiro2007; @Shapiro2008; @Zhdanovich], for an optical broadband and ultra-broadband polarization rotators. The broadband and ultra-broadband polarization rotators are modular designed as stacks of multi order half-wave plates or commercial achromatic half-wave plates, which are rotated smoothly from alignment between the fast-polarization axes with initial linear polarized light to the alignment of the fast-polarization axes with desired linearly polarization orientation. The experimental results show visible broadening effects which are kept the same at different angles of the chosen sequence of wave plates.
Acknowledgements {#acknowledgements .unnumbered}
================
E. D acknowledges financial support by Bulgarian National Science Fund Grant: DRila 1/04. A.R and E.K acknowledge financial support by SUTD start-up Grant No. SRG-EPD-2012-029, SUTD-MIT International Design Centre (IDC) Grant No. IDG31300102.
[99]{}
Kubo H and Nagata R 1980 *Opt. Commun.* **34** 306
Kubo H and Nagata R 1981 *J. Opt. Soc. Am.* **71** 327
Kubo H and Nagata R 1983 *J. Opt. Soc. Am.* **73** 1719
Kuratsuji H and Kakigi S 1998 *Phys. Rev. Lett.* **80** 1888
Kuratsuji H, Botet R and Seto R 2007 *Prog. Theor. Phys.* **117** 195
Ardavan A 2007 *New J. Phys.* **9** 24
Rangelov A A, Gaubatz U and Vitanov N V 2010 *Opt. Commun.* **283** 3891
Allen L and Eberly J H 1987 *Optical Resonance and Two-Level Atoms* (Dover, New York)
Shore B W 1990 *The Theory of Coherent Atomic Excitation* (John Wiley & Sons, New York)
Levitt M H and Freeman R 1979 *J. Magn. Reson.* **33** 473
Levitt M H 1986 *Prog. Nucl. Magn. Reson. Spectrosc.* **18** 61
Freeman R 1997 *Spin Choreography* (Spektrum, Oxford)
Azzam M A and Bashara N M 1977 *Ellipsometry and Polarized Light* (North Holland, Amsterdam)
Goldstein D and Collett E 2003 *Polarized Light* (Marcel Dekker, New York)
Ivanov S S, Rangelov A A, Vitanov N V, Peters T and Halfmann T 2012 *J. Opt. Soc. Am. A* **29** 265
Peters T, Ivanov S S, Englisch D, Rangelov A A, Vitanov N V and Halfmann T 2012 *Appl. Opt.* **51** 7466
Dimova E, Ivanov S S, Popkirov G and Vitanov N V 2014 *J. Opt. Soc. Am. A* **31** 952
Rangelov A A and Kyoseva E 2015 *Opt. Commun.* **338** 574
Dimova E, Rangelov A and Kyoseva E *to be published* (arXiv:1502.00747)
Zapasskii V S and Kozlov G G 1999 *Phys. Usp.* **42** 817
Rangelov A A 2011 *Opt. Lett.* **36** 2716
Berent M, Rangelov A A and Vitanov N V 2013 *J. Opt.* **15** 085401
Shore B W, Rangelov A A, Vitanov N V and Bergmann K *to be published*
Shapiro E A, Milner V, Menzel-Jones C and Shapiro M 2007 *Phys. Rev. Lett.* **99** 033002
Shapiro E A, Pe’er A, Ye J, Shapiro M 2008 *Phys. Rev. Lett.* **101** 023601
Zhdanovich S, Shapiro E A, Shapiro M, Hepburn J W and Milner V 2008 *Phys. Rev. Lett.* **100** 103004
Sala K L 1984 *Phys. Rev. A* **29** 1944
Seto R, Kuratsuji H and Botet R 2005 *Europhys. Lett.* **71** 751
|
---
abstract: |
We will present and study an algebra describing a mixed paraparticle model, known in the bibliography as “The Relative Parabose Set (<span style="font-variant:small-caps;">Rpbs</span>)”. Focusing in the special case of a single parabosonic and a single parafermionic degree of freedom $P_{BF}^{(1,1)}$, we will construct a class of Fock–like representations of this algebra, dependent on a positive parameter $p$ a kind of *generalized parastatistics order*. Mathematical properties of the Fock–like modules will be investigated for all values of $p$ and constructions such as ladder operators, irreducibility (for the carrier spaces) and $(\mathbb{Z}_{2} \times
\mathbb{Z}_{2})$–gradings (for both the carrier spaces and the algebra itself) will be established.
address: |
$^1$ Instituto de Física y Matemáticas (<span style="font-variant:small-caps;">Ifm</span>), Universidad Michoacana de San Nicolás de Hidalgo (<span style="font-variant:small-caps;">Umsnh</span>), Edificio C-3, Cd. Universitaria, CP 58040, Morelia, Michoacán, <span style="font-variant:small-caps;">Mexico</span>\
$^2$ School of Physics, Nuclear and Elementary Particle Physics Department, Aristotle [University]{} of Thessaloniki (<span style="font-variant:small-caps;">Auth</span>), 54124, Thessaloniki, <span style="font-variant:small-caps;">Greece</span>
author:
- 'Konstantinos Kanakoglou$^{1,2}$, Alfredo Herrera–Aguilar$^1$'
title: 'Graded Fock–like representations for a system of algebraically interacting paraparticles'
---
Introduction {#sect1}
============
Our central object of study in this short letter, will be the *Relative Parabose Set algebra* $P_{BF}^{(1,1)}$ in a single parabosonic and a single parafermionic degree of freedom. It belongs to the general family of paraparticle algebras intimately related to the Wigner quantization scheme [@Wi]. The “*free*" paraparticle algebras (parabosonic and parafermionic algebras) have been -implicitly- introduced in the early ’50’s [@Green] while the “*mixed*" paraparticle models such as $P_{BF}$ (and a couple of others as well) have been first introduced in [@GreeMe].
The algebra $P_{BF}^{(1,1)}$ is generated (as an algebra over $\mathbb{C}$) by the four generators $b^{+}, b^{-}, f^{+}, f^{-}$ subject to the usual trilinear relations of the *free parabosonic* and the *free parafermionic* algebras which can be compactly summarized as $$\label{equat1}
\begin{array}{ccc}
\big[ \{ b^{\xi}, b^{\eta}\}, b^{\epsilon} \big] = (\epsilon - \eta)b^{\xi} + (\epsilon - \xi)b^{\eta} & , &
\big[ [ f^{\xi}, f^{\eta} ], f^{\epsilon} \big] = \frac{1}{2}(\epsilon - \eta)^{2} f^{\xi} - \frac{1}{2}(\epsilon - \xi)^{2} f^{\eta}
\end{array}$$ for all values $\xi, \eta, \epsilon = \pm$, together with the *mixed trilinear relations* $$\label{equat2}
\begin{array}{ccc}
\big[ \{ b^{\xi}, b^{\eta}\}, f^{\epsilon} \big] = \big[ [ f^{\xi}, f^{\eta} ], b^{\epsilon} \big] = 0, & \big[ \{ f^{\xi}, b^{\eta}\}, b^{\epsilon} \big] = (\epsilon - \eta) f^{\xi}, & \big\{ \{ b^{\xi}, f^{\eta}\}, f^{\epsilon} \big\} = \frac{1}{2}(\epsilon - \eta)^{2} b^{\xi}
%\\
%\big[ \{ f^{\xi}, b^{\eta}\}, b^{\epsilon} \big] = (\epsilon - \eta) f^{\xi}, \ \ \ \
%\big\{ \{ b^{\xi}, f^{\eta}\}, f^{\epsilon} \big\} = \frac{1}{2}(\epsilon - \eta)^{2} b^{\xi}
\end{array}$$ for all values $\xi, \eta, \epsilon = \pm$, which represent a kind of algebraically established interaction between parabosonic and parafermionic degrees of freedom and *characterize the relative parabose set*.
It is easy for one to see that when all combinations of the $\xi,
\eta, \epsilon = \pm$ are taken into account, the first of equations produces $6$ relations, the second of eq. produces $2$ relations and equations produce $24$ relations. One can easily see (see also the discussion in [@KaHa3]) that not all of these $24+6+2=32$ are algebraically independent. However, we keep the relations in the form given in equations , and we do not proceed in further simplifying them, because of the compact notational and computational advantages of this presentation. (See also [@KaDaHa1; @KaHa3; @Ya1]). The purpose of this letter, will be to present the construction of a class of representations for the relations , . We will present an infinite class of irreducible, $(\mathbb{Z}_{2} \times \mathbb{Z}_{2})$–graded representations of these relations. This class will be parametrized by the values of a positive integer $p$ (similarly to the way that the free parabosonic and parafermionic algebra Fock–like spaces are parametrized by the values of a positive integer). The parameter $p$ will be called *generalized parastatistics order*. It will be shown that the $b^{\pm}, f^{\pm}$ generators act as “ladder" operators (*Creation/Annihilation* operators) in a kind of two–dimensional “ladder", generalizing thus the way that the usual Canonical Commutation–Anticommutation relations (CCR–CAR) are represented in their Fock spaces and also the way that the single degree of freedom free parabosonic and free parafermionic algebras act in their own Fock–spaces [@LiStVdJ] respectively.
Structure of the Fock–like spaces {#sect2}
=================================
In this section we are going to briefly review the results of [@Ya2] on the construction of the Fock–like spaces, which will serve as carrier spaces for the Fock–like representations of $P_{BF}^{(1,1)}$.
An old conjecture [@GreeMe] on the study of the representations of the paraparticle algebras, states that if we consider representations of $P_{BF}^{(1,1)}$, satisfying the adjointness conditions $(b^{-})^{\dagger} = b^{+}$ and $(f^{-})^{\dagger} =
f^{+}$, on a complex, infinite dimensional, pre–Hilbert [^1] space, possessing a unique vacuum vector $|0 \rangle$ satisfying $b^{-} |0
\rangle = f^{-} |0 \rangle = b^{-} f^{+} |0 \rangle = f^{-} b^{+} |0
\rangle = 0$ and $\langle0|0\rangle=1$ then the following additional conditions (where $p$ may be an arbitrary positive integer) $$\label{singleoutFock}
b^{-} b^{+} |0 \rangle = f^{-} f^{+} |0 \rangle = p |0 \rangle$$ single out an irreducible representation which is unique up to unitary equivalence. In other words the above statement, tells us that for any positive integer $p$ there is an irreducible representation of $P_{BF}^{(1,1)}$ uniquely specified (up to unitary equivalence) by the above relations. We now proceed in summarizing the results of [@Ya2]: The carrier spaces of the Fock–like representations of $P_{BF}^{(1,1)}$ are $\bigoplus_{n=0}^{p} \bigoplus_{m=0}^{\infty} \mathcal{V}_{m,n}$ where $p$ is an arbitrary (but fixed) positive integer. The subspaces $\mathcal{V}_{m,n}$ are 2–dim except for the cases $m =
0$, $n = 0, p$ i.e. except the subspaces $\mathcal{V}_{0,n}$, $\mathcal{V}_{m,0}$, $\mathcal{V}_{m,p}$ which are 1–dim. Let us see how the corresponding vectors look like:
$\bullet$ , then the subspace $\mathcal{V}_{m,n}$ is spanned by all vectors of the form [$$\label{spanvect}
\Big| \begin{array}{c}
m_{1}, m_{2}, ..., m_{l} \\
n_{0},n_{1}, n_{2}, ..., n_{l}
\end{array} \Big\rangle \equiv (f^{+})^{n_{0}} (b^{+})^{m_{1}} (f^{+})^{n_{1}} (b^{+})^{m_{2}} (f^{+})^{n_{2}} ... (b^{+})^{m_{l}} (f^{+})^{n_{l}} |0 \rangle$$ ]{} where $m_{1} + m_{2} + ... + m_{l} = m$ , $n_{0} + n_{1} + n_{2} + ... + n_{l} = n$ and $m_{i} \geq 1$ (for $i = 1, 2, ... , l$), $n_{i} \geq 1$ (for $i = 1, 2, ..., l-1$) and $n_{0}, n_{l} \geq 0$.
For any specific combination of values $(m,n)$ the corresponding subspace $\mathcal{V}_{m,n}$ has a basis consisting of the two vectors ($R^{\eta} = \frac{1}{2} \{ b^{\eta}, f^{\eta} \}$ for $\eta = \pm$) [$$\label{subspbase}
\begin{array}{cc}
|m,n,\alpha\rangle\equiv(f^{+})^{n}(b^{+})^{m}|0\rangle=
\Big|\!\begin{array}{c}
m \\
n,0
\end{array}\!\Big\rangle, &
|m,n,\beta\rangle\equiv(f^{+})^{(n-1)}(b^{+})^{(m-1)}
R^{+}|0\rangle=\frac{1}{2}\Big|\!\begin{array}{c}
m \\
n-1,1
\end{array}\!\Big\rangle+
\frac{1}{2}\Big|\!\begin{array}{c}
m-1,1 \\
n-1,1,0
\end{array}\!\Big\rangle
\end{array}$$ ]{}
$\bullet$ , the vectors $|0, n, \beta \rangle$ and $|m, 0, \beta \rangle$ are (by definition) zero and furthermore, for $m\neq0$, the vector $|m, p, \beta \rangle$ becomes parallel to $|m, p, \alpha \rangle$: $|m, p, \beta \rangle=\frac{1}{p} |m, p, \alpha \rangle$
$\bullet$ , all basis vectors of vanish.
*Note 1:* The above described subspaces of $\bigoplus_{n=0}^{p} \bigoplus_{m=0}^{\infty} \mathcal{V}_{m,n}$ can be visualized as follows:
$\bullet$ Inside each one of the above presented $2$–dim subspaces $\mathcal{V}_{m,n}$ (with $m\neq0$, $n\neq0,p$) the vectors of are linearly independent and constitute a basis. However, these vectors are neither orthogonal nor normalized. We can show that an orthonormal set of basis vectors can be obtained as\
$
\begin{array}{cccc}
| m, n, + \rangle = c_{+} | m, n, \alpha \rangle & & & | m, n, - \rangle = -c_{-} \Big( | m, n, \alpha \rangle - p | m, n, \beta \rangle \Big)
\end{array}
$\
where $c_{\pm}$ are suitable normalization factors [@Ya2]. Now we can show orthonormalization $\langle m,n,s|m^{'},n^{'},s^{'}\rangle=\delta_{m,m^{'}}\delta_{n,n^{'}}\delta_{s,s^{'}}$ and completeness $\sum_{m=0}^{\infty} \sum_{n=0}^{p} \sum_{s=\pm}|m,n,s\rangle \langle m,n,s| = 1$.
*Note 2:* If we consider the Hermitian operators $N_{s} = \frac{1}{p} \big( N_{f}^{2} -(p+1)N_{f} + f^{+}f^{-} + \frac{p}{2} \big)$, $N_{f}=\frac{1}{2}[f^{+},f^{-}]+\frac{p}{2}$ and $N_{b} = \frac{1}{2}\{b^{+},b^{-} \}-\frac{p}{2}$ we can show that they constitute a *Complete Set of Commuting Observables* (*C.S.C.O.*): We have $[N_{b}, N_{f}]=[N_{b}, N_{s}]=[N_{f}, N_{s}]=0$; their common eigenvectors are exactly the elements of the orthonormal basis formerly described. Any vector $|m,n,s\rangle$ is uniquely determined as an eigenvector of $N_{b}$, $N_{f}$, $N_{s}$ by its eigenvalues $0 \leq m$, $0 \leq n \leq p$ and $s=\pm\frac{1}{2}$ respectively.
Main results: Construction of the Fock–like representations {#sect3}
===========================================================
For detailed computations and proofs of the results presented in this section see [@KaHa3].
Construction of ladder operators
--------------------------------
We now present the formulae describing explicitly the action of the generators (and hence of the whole algebra) of $P_{BF}^{(1,1)}$ on the carrier spaces $\bigoplus_{n=0}^{p} \bigoplus_{m=0}^{\infty}
\mathcal{V}_{m,n}$ for any positive integer $p$: [$$\label{equat3}
\begin{array}{l}
\blacktriangledown b^{-} \cdot | m, n, \alpha \rangle = \left\{%
\begin{array}{ll}
(-1)^{n}m | m-1, n, \alpha \rangle - 2(-1)^{n}nm | m-1, n, \beta \rangle, \ \underline{m:even} \\ \\
-(-1)^{n} \big(2n-m-(p-1) \big) | m-1, n, \alpha \rangle - 2(-1)^{n}n(m-1) | m-1, n, \beta \rangle, \ \underline{m:odd}
\end{array}
\right. \\
\blacktriangledown b^{-} \cdot | m, n, \beta \rangle = \left\{%
\begin{array}{ll}
-(-1)^{n} | m-1, n, \alpha \rangle + (-1)^{n}\big( 2n-m-p \big) | m-1, n, \beta \rangle, \ \underline{m:even} \\ \\
-(-1)^{n} | m-1, n, \alpha \rangle - (-1)^{n}(m-1) | m-1, n, \beta \rangle, \ \underline{m:odd}
\end{array}
\right. \\
\\
\blacktriangledown f^{-} \cdot | m, n, \alpha \rangle = n(p+1-n) | m, n-1, \alpha \rangle, \blacktriangledown b^{+} \cdot | m, n, \alpha \rangle = (-1)^{n} | m+1, n, \alpha \rangle - (-1)^{n}2n | m+1, n, \beta \rangle \\
\blacktriangledown f^{-} \cdot | m, n, \beta \rangle = | m, n-1, \alpha \rangle + (n-1)(p-n)| m, n-1, \beta \rangle, \quad \blacktriangledown b^{+} \cdot | m, n, \beta \rangle = -(-1)^{n} | m+1, n, \beta \rangle \\ \\
% \\
%b^{+} \cdot | m, n, \alpha \rangle = (-1)^{n} | m+1, n, \alpha \rangle + (-1)^{n-1}2n | m+1, n, \beta \rangle \\
%b^{+} \cdot | m, n, \beta \rangle = (-1)^{n-1} | m+1, n, \beta \rangle \\
% \\
\!\!
\begin{array}{lll}
\blacktriangledown f^{+} \! \cdot \! | m, n, \alpha \rangle = \left\{%
\begin{array}{cc}
| m, n+1, \alpha \rangle, \underline{if \ n \leq p-1} \\
0, \qquad \qquad \underline{if \ n \geq p}
\end{array}
\right.
& &
\blacktriangledown f^{+} \! \cdot \! | m, n, \beta \rangle = \left\{%
\begin{array}{cc}
| m, n+1, \beta \rangle, \underline{if \ n \leq p-1} \\
0, \qquad \qquad \underline{if \ n \geq p}
\end{array}
\right.
\end{array}
\end{array}$$ ]{} for all integers $0 \leq m$, $0 \leq n \leq p$. The direct proof [@KaHa3] of these formulae involves lengthy “normal–ordering" algebraic computations inside $P_{BF}^{(1,1)}$. We must take into account: [$(a).$]{} the relations , of $P_{BF}^{(1,1)}$, [$(b).$]{} the relations together with [$b^{-}|0\rangle\!=\!f^{-}|0\rangle\!=\!b^{-}
f^{+}|0\rangle\!=\!f^{-}b^{+}|0\rangle\!=\!0$]{} and [$(c).$]{} the structure and properties of the corresponding carrier space as described in Sect. \[sect2\].
Apart from the direct proof, eq. have also been checked and verified via the [Quantum](http://homepage.cem.itesm.mx/lgomez/quantum/) [@GMFD] add–on for Mathematica 7.0, which is an add-on for performing symbolic algebraic computations, including the use of generalized Dirac notation. What we have actually verified via the use of this package, is that the actions formulae preserve all of the relations , of $P_{BF}^{(1,1)}$.
Irreducibility
--------------
In the following diagram we provide a “visual" interpretation of relations i.e. of the action of the generators of $P_{BF}^{(1,1)}$ on the direct summand subspaces of the Fock-like carrier space $\bigoplus_{n=0}^{p} \bigoplus_{m=0}^{\infty}
\mathcal{V}_{m,n}$. We can easily figure out that we have a kind of *generalized creation–annihilation operators* acting on a two dimensional ladder of subspaces: [ ]{}\
Initiating from the remark that we are dealing with a cyclic module which moreover can be generated by any of its elements, we can prove [@KaHa3] that the Fock–like representations, explicitly given by and visually represented in the above diagram, are irreducible representations (or: simple $P_{BF}^{(1,1)}$-modules) for any $p \in \mathbb{N}^{*}$.
Klein–group Grading of the representation
-----------------------------------------
Defining $\verb"deg"|m,n,\alpha\rangle\!=\!\verb"deg"|m,n,\beta
\rangle\!=\!\big(m\ \textsf{mod}\ 2,\ n\ \textsf{mod}\ 2
\big)\!\in\!\mathbb{Z}_{2}\!\times\!\mathbb{Z}_{2}$ for the carrier spaces and $\verb"deg" b^{\pm}\!=\!(1,0)$, $\verb"deg" f^{\pm}\!=\!
(0,1)$ for the algebra, the Fock–like representations of $P_{BF}^{(1,1)}$ over $\bigoplus_{n=0}^{p} \bigoplus_{m=0}^{\infty}
\mathcal{V}_{m,n}$, become $(\mathbb{Z}_{2} \times
\mathbb{Z}_{2})$–graded modules, $\forall p\in\mathbb{N}^{*}$. (For proof and details see [@KaHa3]).
[KK would like to thank the whole staff of <span style="font-variant:small-caps;">Ifm</span>, <span style="font-variant:small-caps;">Umsnh</span> for providing a challenging and stimulating atmosphere while preparing this article. His work was supported by the research project <span style="font-variant:small-caps;">Conacyt</span>/No. J60060. AHA was supported by <span style="font-variant:small-caps;">Cic</span> 4.16 and <span style="font-variant:small-caps;">Conacyt</span>/No. J60060; he is also grateful to <span style="font-variant:small-caps;">Sni</span>. ]{}
References {#references .unnumbered}
==========
[9]{} Gómez-Muñoz J L and Delgado-Cepeda F http://homepage.cem.itesm.mx/lgomez/quantum/ Tec de Monterrey
Green H S 1953 *Phys. Rev.* **90** 2 270
Greenberg O W and Messiah A M L 1965 *Phys. Rev.* **138** 5B 1155-67
Kanakoglou K, Daskaloyannis C and Herrera–Aguilar A 2010 *AIP Conf. Proc.* **1256** 193-200 Kanakoglou K and Herrera–Aguilar A 2011 *International Journal of Algebra* **5** 9 [413-428](http://www.m-hikari.com/ija/forth/kanakoglouIJA9-12-2011.pdf) (*e-print* [arXiv:1006.4120v3 \[math.RT\]](http://arxiv.org/abs/1006.4120))
Lievens S, Stoilova N I and Van der Jeugt 2008 *Commun. Math. Phys.* **281** 805-26
Wigner E P 1950 *Physical Review* **77** 5 711-12
Yang W and Jing S 2001 *Science in China (Series A)* **44** 9 1167-73 (*e-print* [arXiv:math-ph/0212004v1](http://arxiv.org/PS_cache/math-ph/pdf/0212/0212004v1.pdf))
Yang W and Jing S 2001 *Mod. Phys. Letters* A **16** 15 963-71 (*e-print* [arXiv:math-ph/0212003v1](http://arxiv.org/PS_cache/math-ph/pdf/0212/0212003v1.pdf))
[^1]: in the sense that it is an inner product space, but not necessarily complete with respect to the inner product.
|
---
abstract: 'We introduce a new type of quadrature, known as approximate Gaussian quadrature (AGQ) rules using $\epsilon$-quasiorthogonality, for the approximation of integrals of the form $\int f(x) \, {{\mathrm d}}\alpha(x)$. The measure $\alpha(\cdot )$ can be arbitrary as long as it possesses finite moments $\mu_n$ for sufficiently large $n$. The weights and nodes associated with the quadrature can be computed in low complexity and their count is inferior to that required by classical quadratures at fixed accuracy on some families of integrands. Furthermore, we show how AGQ can be used to discretize the Fourier transform with few points in order to obtain short exponential representations of functions.'
author:
- |
Pierre-David Létourneau,$^a$ Eric Darve,$^b$\
$^a$ Institute of Computational and Mathematical Engineering (ICME), Stanford University, CA\
$^b$ Mechanical Engineering Department, Stanford University, CA
bibliography:
- 'biblio.bib'
title: 'Gaussian Quadrature Rule using $\epsilon$-Quasiorthogonality'
---
Introduction {#Intro}
============
In this paper, we present a new kind of quadrature rule for approximating integrals by sums of the form, $$\label{discrete}
\int f(x) \, {{\mathrm d}}\alpha(x) \approx \sum_{i=1}^n w_i f(x_i)$$ having the following characteristics:
1. The measure $\alpha( \cdot )$ can be *arbitrary* (positive, signed, complex, ...) as long as it satisfies some weak condition.
2. The nodes and weights associated with the quadrature rule can be obtained in low computational complexity through a simple numerical algorithm.
3. The quadrature is at least as accurate as the Gaussian quadrature rule, and in many cases is significantly more accurate.
4. Low-order rules are able to integrate high-order polynomials with high accuracy.
The scheme presented in the current work uses a strategy similar to classical Gaussian quadrature rules (of which a few examples can be found in Table \[classicalGaussQuad\]). The Gaussian quadrature rule is designed to integrate exactly polynomials of degree at most $2n-1$ using $n$ quadrature points and weights: $$\int x^k \, {{\mathrm d}}\alpha(x)$$ for various weight functions $\frac{{{\mathrm d}}\alpha}{{{\mathrm d}}x}$ (see Table \[classicalGaussQuad\]).
Name Interval Measure (${{\mathrm d}}\alpha / {{\mathrm d}}x$ )
---------------------------- --------------------- -------------------------------------------------------------
Gauss-Legendre $[-1,1]$ $1$
Gauss-Laguerre $[0, \infty)$ $e^{-x}$
Gauss-Hermite $(-\infty, \infty)$ $e^{-x^2} $
Gauss-Jacobi $(-1,1)$ $(1-x)^{\alpha}(1+x)^{\beta} \; , \;\; \alpha, \beta > -1 $
Chebyshev-Gauss (1st kind) $(-1,1)$ $1/\sqrt{1-x^2}$
Chebyshev-Gauss (2nd kind) $[-1,1]$ $\sqrt{1-x^2}$
: Examples of classical Gaussian quadratures
\[classicalGaussQuad\]
The paper is structured as follows. In Section \[GQ\], a brief overview of classical Gaussian quadratures will be presented. In Section \[sec:agq\], the concept of quasiorthogonal polynomial and approximate Gaussian quadrature will be introduced together with an error analysis. This will be followed in Section \[NS\] by numerical results. In the same section, we will discuss representations of functions by short sums of exponentials.
Gaussian quadrature {#GQ}
===================
Gaussian quadratures are schemes used to approximate definite integrals of the form, $$\int_a^b f(x) \, {{\mathrm d}}\alpha(x)$$ by a finite weighted sum of the form, $$\sum_{n=0}^N w_n \, f(x_n)$$ where $a<b \in \mathbb{R}$. The coefficients $\{ w_n \}$ are generally referred to as the *weights* of the quadrature, whereas the points $\{ x_n \}$ are referred to as the *nodes*. An $(N+1)$-node Gaussian quadrature can integrate polynomials up to degree $2N+1$ *exactly* and is generally well-suited for the integration of functions that are well-approximated by polynomials.
In what follows, we will briefly describe how the nodes and weights of classical Gaussian quadratures can be obtained based on the classical theory of orthogonal polynomials. For this purpose, we shall denote the real and complex numbers by $\mathbb{R}$ and $\mathbb{C}$ respectively. $\alpha( \cdot )$ will represent an arbitrary measure (possibly complex) on $(\mathbb{R}, \mathcal{B})$ or $(\mathbb{C}, \mathcal{B})$ unless otherwise stated. Vectors are represented by lower case letter e.g., $v$. The $i^{th}$ component of a vector $v$ will be written as $v_i$, and we shall use super-indices of the form $v^{(j)}$ when multiple vectors are under consideration.
We begin by introducing four key objects: the orthogonal polynomials, the Lagrange interpolants, the moments of a measure $\alpha(\cdot)$ and the Hankel matrix associated with such a measure.
[**(Orthogonal polynomial)**]{} A sequence $\{ p^{(k)} (x) \}_{k=0}^{\infty}$ of polynomials of degree $k$ is said to be a sequence of orthogonal polynomials with respect to a positive measure $\alpha(\cdot)$ if, $$\int p^{(k)}(x) p^{(l)} (x) \, {{\mathrm d}}\alpha(x) =
\left\{
\begin{array}{ll}
0 & \mbox{if } k \not = l \\
c_k & \mbox{if } k = l
\end{array}
\right.$$ If in addition $c_k = 1 \; \forall k \in \mathbb{N}$, then the sequence is called *orthonormal*.
We shall hereafter assume that all such polynomials are monic, i.e., that they can be written as, $$p^{(k)} (x) = x^k + \sum_{n=0}^{k-1} p^{(k)}_n x^n$$ where $\{ p^{(k)}_n \}_{n=0}^{k-1}$ are some (potentially complex) coefficients. We then introduce Lagrange interpolants,
[**(Lagrange interpolant)**]{} Given a set of $(d+1)$ data points $\{ (x_n, y_n) \}_{n=0}^d$, the Lagrange interpolant is the unique polynomial $L (x)$ of degree $d$ such that, $$L (x_n) = y_n , \; n = 0 ... d$$ It can be written explicitly as, $$L (x) = \sum_{n=0}^d y_n\, \ell_n (x)$$ where, $$\ell_n (x) = \prod_{\substack{{m=0}\\ {m \not = n}}}^d \frac{x-x_m}{x_n-x_m}$$ and $\ell_n (x)$ is referred to as the $n^{th}$ Lagrange basis polynomial.
Finally we introduce the moments as well as the Hankel matrix associated with a measure $\alpha(\cdot)$,
[**(Moment)**]{} Given an arbitrary measure $\alpha(\cdot)$ on $(\mathbb{R}, \mathcal{B})$, its $n^{th}$ moment $\mu_n$ is defined by the following Lebesgue integral, $$\mu_n = \int x^n \, {{\mathrm d}}\alpha(x)$$ whenever it exists.
[**(Hankel matrix)**]{} An $(N+1) \times (M+1)$ matrix $H$ is called the $(N+1) \times (M+1)$ Hankel matrix associated with the measure $\alpha( \cdot )$ if its entries take the form, $$\begin{pmatrix}
\mu_0 & \mu_1 & \cdots & \mu_{M} \\
\mu_1 & \mu_2 & \cdots & \mu_{M+1}\\
\vdots &\vdots &\vdots &\vdots \\
\mu_{N} & \mu_{N+1} & \cdots & \mu_{N+M} \end{pmatrix}
\label{eq:hankel}$$ i.e., $H_{ij} = \mu_{i+j}$, where $(\mu_0, \mu_1, \ldots, \mu_{N+M} )$ are the first $(N+M)$ moments of $\alpha(\cdot)$ whenever they exist.
With these quantities we can now present the main results associated with classical Gaussian quadratures,
[**(Gaussian quadrature)**]{} Consider a positive measure $\alpha(\cdot)$ on $([a,b], \mathcal{B})$ (with $a,b \in \mathbb{R}$ potentially infinity) and a sequence of orthonormal polynomials $\{ p^{(k)} (x) \}_{k=0}^{\infty}$ with respect to $\alpha( \cdot)$. Then, the quadrature rule with nodes $\{ x_n\}_{n=0}^k$ consisting in the zeros of $p^{(k+1)}(x)$ and weights $\{ w_n \}_{n=0}^k$ given by, $$w_n = \int \ell_n (x) \, {{\mathrm d}}\alpha(x)$$ integrates polynomials of degree $\leq 2k+1$ exactly.
This is a classical result which can be found in [@Meurant] for instance. Explicit expression for the error incurred in the case of smooth integrand also exist.
To close this section, we introduce a further result characterizing the coefficients of the orthogonal polynomials $\{ p^{(k)}(x) \} $. As we shall see in the next section, this characterization lies at the heart of our scheme,
Consider a positive measure $\alpha(\cdot)$ on $([a,b], \mathcal{B})$ (with $a,b \in \mathbb{R}$ potentially infinity) and a sequence of orthogonal polynomials $\{ p^{(k)} (x) \}_{k=0}^{\infty}$ with respect to $\alpha( \cdot)$. Then, the coefficients $\{ p^{(k+1)}_n \}_{n=0}^k$ of the $(k+1)^{th}$ orthogonal polynomial $p^{(k+1)}(x)$ satisfy the following Hankel system, $$Hp =
\begin{pmatrix}
\mu_0 & \mu_1 & \cdots & \mu_{k+1} \\
\mu_1 & \mu_2 & \cdots & \mu_{k+2}\\
\vdots &\vdots &\vdots &\vdots \\
\mu_{k} & \mu_{k+1} & \cdots & \mu_{2k+1}
\end{pmatrix}
\begin{pmatrix}
p_0^{(k+1)} \\
p_1^{(k+1)} \\
\cdots \\
p_{k}^{(k+1)}
\end{pmatrix} = 0$$ where $\{ \mu_n \}$ are the moments of the measure $\alpha ( \cdot ) $, whenever they exist. \[AGQ:characterization\]
First write, $$p^{(k+1)} (x) = \sum_{n=0}^{k+1} p^{(k+1)}_n x^n$$ Let $0 \leq j \leq k $. Then, from orthogonality we have, $$0 = \int p^{(k+1)} (x) \, x^j \, {{\mathrm d}}\alpha(x) = \sum_{n=0}^{k+1} p^{(k+1)}_n \int x^{n+j}\, {{\mathrm d}}\alpha(x) = \sum_{n=0}^{k+1} p^{(k+1)}_n \mu_{n+j}$$ Putting all these equations in matrix form provides the desired result.
The Hankel matrices associated with positive measures commonly encountered with classical Gaussian quadratures have been the subject of extensive study in the past (known as the moment problem). In some cases, they can be proved to be invertible although extremely ill-conditioned (see $\cite{Shohat}$ for details). On the other hand, less is known per regards to more general measures. In any case, in the event where the resulting Hankel matrix would be invertible, it can be expected to be ill-conditioned. Indeed, as an example it can be shown that for a large class of positive measures, the smallest eigenvalue of the $N \times N$ associated Hankel matrix scales like ${\mathcal{O}}\left (\frac{ \sqrt{N} }{\sigma^{2N}} \right ) $, where $\sigma$ depends only on the interval considered and is equal to $(1+\sqrt{2})$ for the interval $[-1,1]$ (see [@Widom:1966]).
The question we treat in the next section is whether such Hankel matrices arising from arbitrary measures can be used to derive Gaussian-like quadratures, and what this inherent ill-conditioning entails.
Approximate Gaussian quadrature (AGQ) {#sec:agq}
=====================================
In this section, we describe the concept of approximate Gaussian quadrature. For this purpose, we will need the concept of $\epsilon$-quasiorthogonal polynomial, which we introduce for the first time below. Before doing so however, we first point to the following key observation.
Let $H$ be a $N \times M$ with rank $0 < d < M $. Then, there exists $D \le d+1$ and a vector $a \not = 0$ such that $$Ha = 0, \quad \text{with $a_i = 0$ for all $i > D$}$$ \[AGQ:low\_rank\]
The rank of $H$ is $d$. Therefore if we consider the first $d+1$ columns for $H$ they are linearly dependent. Denote $D$ the smallest integer such that the first $D$ columns of $H$ are linearly dependent. We have $D \le d+1$ and, by definition, there is $a \neq 0$ such that $Ha = 0$ with $a_i = 0$, $i>D$.
We also have the following corollary,
\[AGQ:quasiortho\_cor\] Assume that the $N \times (N+1)$ Hankel matrix $H$ associated with the measure $\alpha(\cdot)$ exists. If $H$ has rank $d<N$ then there exists a nontrivial polynomial $p(x)$ with degree $(D-1)$ where $D \leq d+1$ such that, $$\int p(x) x^j \, {{\mathrm d}}\alpha(x) = 0$$ for all $j = 0,$ …, $N$.
Let $K$ be such that $$D = \inf \{ 0\leq n \leq N : \mathrm{rank}( H(:, 1:n) )= n \}$$ where $H(:, 1:n)$ is the matrix containing the first $n$ columns of $H$. By theorem \[AGQ:low\_rank\], there exists a vector $a \not = 0$ such that $Ha = 0$ and $a_i = 0$ for $i>D$.\
Let $p(x)$ be the polynomial with coefficients given by $a$, i.e. $$p(x) = \sum_{n=0}^D a_n x^n$$ Then, $$\begin{aligned}
\int p(x) x^j \, {{\mathrm d}}\alpha(x) &= \int \sum_{n=0}^{D} a_n x^{n+j} \, {{\mathrm d}}\alpha(x)\\
&= \sum_{n=0}^D a_n \mu_{n+j} \\
& = (H a)_j = 0\end{aligned}$$ since $a$ belongs to the null-space of $H$.
The consequences of this corollary are far-reaching and constitute the crux of the scheme presented here. Indeed, although we do not generally expect the Hankel matrix $H$ associated with some measure $\alpha( \cdot)$ to be *exactly* low-rank as in the case of Theorem \[AGQ:low\_rank\] (e.g., $H$ has full rank in the case of classical Gaussian quadratures) we can expect that in some cases $H$ will be *approximately* low rank. In other words, given $0 < \epsilon \ll 1 $ we expect, $$D \approx \max \{ 1 \leq i \leq N : \sigma_i > \epsilon \, \sigma_1 \}$$ where $\{ \sigma_i \}$ are the singular values of $H$, to be much smaller than $N$, i.e., $D \ll N$. We show for instance in Figure \[svd\] the first $50$ singular values of the Hankel matrix ($N=250$) associated with the Lebesgue measure in $[-1,1]$. The $y$-axis scales as a logarithm in base $10$, and it is seen that the singular values decay faster than exponentially.
In light of the above discussion, we might expect in these circumstances the existence of a polynomial $p(x)$ of degree $D \approx \max \{ 1 \leq i \leq N : \sigma_i > \epsilon \sigma_1 \}$ such that $$\left | \int p(x) x^j \, {{\mathrm d}}\alpha(x) \right | \lesssim \epsilon$$ for all $0 \leq j \leq N$, and this leads us to the introduction of the concept of $\epsilon$-quasiorthogonal polynomial which we now define,
A polynomial $p(x)$ is called $\epsilon$-quasiorthogonal of order $N$ with respect to the measure $\alpha(\cdot)$ and the basis $\{ L_n (x) \}$ if, $$\left | \int p(x) \, L_n(x) \, {{\mathrm d}}\alpha(x) \right | \leq \epsilon$$ for all $n=0,$ …, $N$.
Importantly, this definition imposes no restriction per regards to the measure $\alpha(\cdot)$, in opposition with orthogonal polynomials which demand the measure to be positive ([@Szego]). In this sense, the relation described is not one of orthogonality for it is not possible to define a nondegenerate inner-product unless $\alpha( \cdot )$ is positive. This is why we chose the name *quasi*-orthogonal. We also note that given $\epsilon \geq \sigma_N(H)$ such polynomial always exists for it suffices to pick $a$ aligned with the right singular vector associated with the smallest singular value $\sigma_N(H)$.
From a computational standpoint, there exists an efficient scheme to find such polynomials given a measure $\alpha( \cdot)$ and some $\epsilon>0$. This is the subject of Section \[AGQ:comp\]. For the remaining of this section, we will focus on demonstrating how such polynomials can be used to obtain efficient quadratures. As will be shown, the construction of the scheme shares a lot with that of classical Gaussian quadrature. This is what constitutes the origin of the denomination.
We will need the following technical lemma which proof is provided in appendix,
\[AGQ:tech\_lemma\] Let $\alpha(\cdot)$ be an arbitrary measure on $(\mathbb{R}, \mathcal{B})$, and $$p(x) =x^{d+1} + \sum_{n=0}^{d} p_n x^n$$ be a monic $\epsilon$-quasiorthogonal polynomial of degree $d+1$ and order $N$ associated with $\alpha(\cdot)$. Further, let, $$f(x) = \sum_{n=0}^{N+d} f_n \, L_n(x)$$ be some polynomial of degree $N+d$ and $ \tilde{q}(x) = \sum_{n=0}^{d} \tilde{q}_n x^n$ be the Lagrange interpolant of $q(x)$ associated with the zeros of $p(x)$. Finally, let $r(x) = \sum_{n=0}^{N} r_n x^n$ be the unique polynomial such that $ q(x) - \tilde{q}(x)= p(x) r(x) $. Then, $$\begin{aligned}
\sum_{n=0}^N |r_n| \leq \lVert\Gamma^{-1} \bar{q} \rVert_{1} \end{aligned}$$ where $\bar{q} =\left( q_{d+1}, q_{d+2}, \ldots, q_{N+d}\right )^T $ and $\Gamma$ is the $N \times N$ Toeplitz matrix such that $[ \Gamma ]_{i,j} = p_{j-i}$ if $0\leq j-i \leq d$ and $0$ otherwise. \[AGQ:errorlemma\]
We are now ready to prove our main theorem.
[**(Approximate Gaussian quadrature)**]{} \[AGQ:AGQ\_thm\] Consider an arbitrary measure $\alpha(\cdot)$ on $(\mathbb{R}, \mathcal{B})$. Let $p(x)$ be a monic $\epsilon$-quasiorthogonal polynomial of degree $d+1$ and order $N$ with respect to $\alpha( \cdot )$, where $0<\epsilon<1$. Then, the quadrature rule with nodes $\{ x_n \}_{n=0}^{d}$ consisting in the zeros of $p(x)$ and weights $\{ w_n \}_{n=0}^{d}$ given by, $$\label{AGQ:AGQ_thm:weight}
w_n = \int \ell_n (x) \, {{\mathrm d}}\alpha(x)$$ where $\ell_n (x)$ is the $n^{th}$Lagrange basis polynomial associated with the nodes, integrates polynomials $q(x)$ of degree $\leq N+d$ with an error bounded by, $$\left | \int q(x) \, {{\mathrm d}}\alpha(x) - \sum_{n=0}^d w_n \, q(x_n) \right | \leq \lVert\Gamma^{-1} \bar{q} \rVert_{1} \, \epsilon$$ where $\{ q_n \}_{n=0}^{N+d}$ are the coefficients of $q(x)$, $\bar{q} =\left( q_{d+1} , q_{d+2} , \ldots, q_{N+d}\right )^T $ and $\Gamma$ is the $N \times N$ Toeplitz matrix such that $[ \Gamma ]_{i,j} = p_{j-i}$ if $0\leq j-i \leq d$ and $0$ otherwise.
Let $q(x)$ be a polynomial of degree $ N + d$ and consider the Lagrange interpolant at the nodes $\{ x_n \}$, $$\tilde{q} (x) = \sum_{n=0}^d q(x_n) \ell_n (x)$$ Then consider, $$\begin{aligned}
I = \int \left [ q(x) - \tilde{q}(x) \right ] \, {{\mathrm d}}\alpha(x)\end{aligned}$$ The quantity $[q(x) - \tilde{q}(x)]$ is a polynomial of degree at most $(N+d)$ and has zeros located at each of the nodes $\{ x_n \}_{n=0}^d$. Therefore, by the factorization theorem for polynomials we can write, $$q(x) - \tilde{q}(x) = \prod_{n=0}^d (x - x_n) \, r(x)$$ where $r(x)$ is a polynomial of degree at most $N$. We further note that $\prod_{n=0}^d (x - x_n)$ is a monic polynomial of degree $d+1$ with zeros at $\{ x_n \}_{n=0}^d$ just as $p(x)$. Since monic polynomials are uniquely characterized by their roots we have, $$\prod_{n=0}^d (x - x_n) = p (x)$$ Therefore, $$\begin{aligned}
|I| &= \left | \int p(x) r(x) \, {{\mathrm d}}\alpha(x) \right | \leq \sum_{n=0}^N | r_n | \,\left | \int p(x) \, x^n \, {{\mathrm d}}\alpha(x) \right | \leq \sum_{n=0}^N | r_n | \, \epsilon \\\end{aligned}$$ where we used the $\epsilon$-quasiorthogonality of $p(x)$. Finally, thanks to Lemma \[AGQ:tech\_lemma\] we get, $$\left | \int q(x) \, {{\mathrm d}}\alpha(x) - \sum_{n=0}^d w_n \, q(x_n) \right | \leq \lVert \Gamma^{-1} \bar{q} \rVert_{1} \epsilon$$
Interestingly, the above analysis reveals that an AGQ of order $d$ is in fact *exact* for polynomials of degree $\leq d$.
Some advantages of AGQ is that there is no need for the measure $\alpha( \cdot )$ to have any specific properties beyond the existence of moments of high-enough order. Furthermore, the problem of the existence and uniqueness of the solution to the Hankel system is of no importance; in fact, the larger the null-space of $H$ the better it is.
Both characteristics are in sharp contrast with common wisdom regarding classical Gaussian quadratures. First, the positivity of the measure is key in proving the existence of a sequence of orthogonal polynomials necessary to build a classical quadrature (see [@Meurant], Theorem 2.7). Secondly, the notion of orthogonality is at the heart of modern numerical schemes used to obtain nodes and weights for it gives rise to a three-term recurrence relation that is thoroughly exploited computationally (see [@Golub:1969; @Meurant]).
Computational considerations {#AGQ:comp}
----------------------------
The first computational issue we describe here is that of finding an adequate $\epsilon$-quasiorthogonal polynomials of order $N$ given a measure $\alpha(\cdot)$ on $(\mathbb{R}, \mathcal{B})$, some $N \in \mathbb{N}$ and some value $ 0< \epsilon$. For this purpose, we note that a sufficient condition for a monic polynomial $p(x)$ of degree $(d+1)$ to fall within this category is to satisfy the following inequality, $$\lVert H(N,d) \, \bar{p} + h(d) \rVert_{\infty} \leq \epsilon$$ where $\bar{p} = [p_0, \, p_1 \, , \ldots, p_{d}]^T$, $p(x) = x^{d+1} + \sum_{n=0}^d p_n x^n$, $ h(d) = [\mu_{d+1}, \, \mu_{d+2} \, , \ldots, \mu_{d+N+1}]^T$ and $H(N,d)$ is the $(N+1)\times (d+1)$ Hankel matrix associated with the measure, i.e. $$H(N,d) =
\begin{pmatrix}
\mu_0 & \mu_1 & \cdots & \mu_{d} \\
\mu_1 & \mu_2 & \cdots & \mu_{d+1}\\
\vdots &\vdots &\vdots &\vdots \\
\mu_{N} & \mu_{N+1} & \cdots & \mu_{d+N}
\end{pmatrix}$$ The proof is analogous to that of Corollary \[AGQ:quasiortho\_cor\] and uses the definition of quasiorthogonal polynomials.
This inequality provides a constructive way for finding an $\epsilon$-quasiorthogonal polynomial of small degree. This is described in Algorithm \[poly\_alg\]; note that we replace the $\lVert \cdot \rVert_{\infty}$ norm by the more computationally-friendly $\lVert \cdot \rVert_2$ norm which is equivalent.
Let $(d+1) = \mathrm{rank}(H,\delta)$\
Solve $ \min_p \lVert H(N,d) \, p + h(d) \rVert_2 $\
[**Note:** ]{} The quadrature obtained from $p(x)$ integrates polynomials of degree $N+d$ with error prescribed by Theorem \[AGQ:AGQ\_thm\]. This error term involves the norm of the inverse of a matrix $\Gamma$ which is upper-triangular, Toeplitz with diagonal entries all equal to $1$ and remaining entries depending on the coefficients of the polynomial $p(x)$. In order to guarantee that an AGQ integrates polynomials of degree $\leq N+d$ with accuracy $\delta$ say, it is sufficient to set $\epsilon \leq \frac{\delta}{C}$ and constrain $p(x)$ to be such that $\lVert \Gamma^{-1} \rVert_{\infty} \leq C$ for some $C>0$. Upon obtaining some characterization of the set $\mathcal{S}_C := \{ p(x) : \lVert \Gamma^{-1} \rVert_{\infty} \leq C \}$, one could potentially carry out the steps described in Algorithm \[poly\_alg\] while restraining the solution to $\mathcal{S}_C$. One would thus guarantee the accuracy of the AGQ *a priori*. Unfortunately, such characterization is not readily available so one is left with the *a posteriori* estimates of Theorem \[AGQ:AGQ\_thm\]. On the other hand, numerical experiments point to the fact that the product $\lVert \Gamma^{-1} \rVert_{\infty} \, \epsilon$ does indeed decay in a fast manner as a function of the degree of $p(x)$, for $\bar{p}$ the solution of the least-squares problem having the smallest norm in Algorithm \[poly\_alg\]. In short, although AGQ in its current state performs well, some improvements are still possible. This constitutes a topic for future research.
Once such polynomial has been obtained, its roots constitute the nodes of the approximate Gaussian quadrature as per Theorem \[AGQ:AGQ\_thm\]. The cost of solving a thin $(N+1) \times (d+1)$ least-squares problem is ${\mathcal{O}}( [N+1) + (d+1)/3] (d+1)^2 ) $ (see [@Golub]). Since in general we expect $d \ll N$ the cost is *linear* in $N$. Also, each step of the while loop constitutes a rank-1 update of the system, so $p$ can be recomputed cheaply.
Another great computational aspect of the scheme is the availability of a simple analytical formula for the computation of the weights. Indeed, from Theorem \[AGQ:AGQ\_thm\] we have, $$w_n = \int \ell_n (x) \, {{\mathrm d}}\alpha(x) = \int \sum_{k=0}^d [\ell_n]_k x^k \, {{\mathrm d}}\alpha(x) = \sum_{k=0}^d [\ell_n]_k \, \mu_k$$ where $[\ell_n]_k $ is the $k^{th}$ coefficient of the $n^{th}$ Lagrange basis polynomial $ \ell_n (x)$, which can be obtained cheaply from the zeros of $ \ell_n (x)$, i.e., the nodes of the quadrature. We also noticed that it is generally possible to neglect nodes associated with small weights when such are present. This further reduces the cost of the method.
As a final comment, the accuracy of the scheme is highly dependent on the accuracy of the nodes. For this reason, we recommend performing the computations in extended arithmetic. In this paper, we used $Maple^\copyright$ in order to compute the nodes and weights of each approximate quadrature with high precision.
Numerical simulations {#NS}
=====================
In this section, we demonstrate the efficiency and the versatility of the scheme through a few numerical examples. In section \[NS:classical\], we compare fixed-order approximate Gaussian quadratures (AGQ) with two types of classical Gaussian quadratures (Gauss-Legendre and Gauss-Chebyshev) on monomials $x^n$ of increasing degree and show how it quickly becomes advantageous to use an approximate quadrature in those cases. Then in Section \[NS:sing\], we give examples related to functions with an integrable singularity at the origin.
In section \[NS:trig\], we show how the scheme can be applied to monomials on the complex circle, i.e., functions of the form $e^{{\imath}n x}$ where $0 \leq n$. The resulting quadratures are then used in Section \[NS:Beylkin\] to obtain approximations of functions through short exponential sums which is related to the method of Beylkin & Monzón [@Beylkin:2005; @Beylkin:2010].
Comparison with classical quadratures {#NS:classical}
-------------------------------------
In this section, we compare results between the approximate Gaussian quadrature scheme, the Gauss-Legendre $\left ( {{\mathrm d}}\alpha(x) = {{\mathrm d}}x \right ) $ and Gauss-Chebyshev $\left ( {{\mathrm d}}\alpha(x) = \frac{1}{\sqrt{1-x^2} }{{\mathrm d}}x \right ) $ quadrature.
### Integration of monomials
For this benchmark, we fix the order ($N$ in Section \[AGQ:comp\]) and study the error in approximating integrals of the form, $$\int_{-1}^1 x^n \, {{\mathrm d}}\alpha (x)$$ through quadratures involving different number of nodes ($d$ in Section \[AGQ:comp\]) where $n$ varies between $0$ to $700$.
Numerical results are shown in Figure \[GLcompare\] and \[GCcompare\]. They were obtained using $N=350$. The results need to be interpreted carefully. The choice of $N$ represents in effect the polynomial order that would be required to approximate a given function $f(x)$ to some accuracy $\epsilon$. A numerical quadrature will then be able to approximate the integral of $f(x)$ if it can integrate all monomials of degree less than $N$ with accuracy $\epsilon$. In Fig. \[GLcompare\] for example, we see that the Gauss-Legendre quadrature is exact to machine precision up to $n=39$. However the error increases rapidly to reach $10^{-3}$ near $n=350$. In contrast, although AGQ is not exact for $n \le 39$, the error up to $n \le 350$ remains lower than $10^{-4}$ with only 20 nodes. As we increase the number of nodes (middle and bottom plots) the gain below $n= 350$ is even more significant.
The behavior of AGQ in the top plot around $n \approx 40$ where Gauss-Legendre seems to outperform AGQ is not significant. Indeed if a polynomial of order $n \approx 40$ is sufficient to approximate $f$, we would reduce $N$. This would result in an AGQ quadrature much more accurate in the range $n \in [0,40]$.
On Figure \[bound:Legendre\] and \[bound:Chebyshev\], we also compare the theoretical bound obtained in Theorem \[AGQ:AGQ\_thm\] with the actual absolute error obtained through a 30-node AGQ for both the Lebesgue and Chebyshev measures respectively. In both cases, it is seen that the bound provides a reasonable estimate for the behavior of the error.
Finally, an interesting thing to be noted is that in both cases the nodes associated with the approximate Gaussian quadratures were *real* and the weights were *real and positive*; it is a known fact that this should be the case for classical Gaussian quadratures. However, this is by no means obvious for the case of approximate Gaussian quadratures, and we currently have no theory demonstrating that it is always the case for real positive measures.
### General integrands
An important difference between AGQ and Gaussian quadratures is that AGQ takes $N$ as a parameter. $N$ represents in effect the order of a polynomial that can approximate $f(x)$ to the desired accuracy. This is function-dependent and therefore may need to be adjusted in AGQ depending on the integrand, if one wishes to have a near optimal quadrature.
Generally speaking, AGQ should be able to outperform a classical Gaussian quadrature in all cases since a Gaussian quadrature is a special case of AGQ, by basically choosing $d=N-1$ where $d$ is the degree of the polynomial. Indeed, this is what we observed in our numerical tests. Whenever the classical Gaussian quadrature or CGQ performs well, no gain is obtained with AGQ. We note that, in this case, the usual numerical techniques to evaluate Gaussian quadrature nodes should be more effective than the numerical procedure we are advocating for AGQ (due to ill-conditioning for too stringent a tolerance as mentioned in the introduction).
Conversely, when the convergence of CGQ is slow, AGQ provides a significant improvement. This corresponds to situation where expanding $f$ using polynomials requires terms of high degree and then the approximation of AGQ for high order monomials makes a difference. This is illustrated in the examples below.
We used the following integrands to investigate the accuracy of AGQ: $$\begin{aligned}
\log \left ( 1- \frac{x}{1.05} \right ) &= - \sum_{n=0}^{\infty} \frac{1}{n (1.05)^n} \,x^n ,\;\;\; |x| \leq 1 \\
\frac{1}{ 1- \frac{x}{1.05} } &= - \sum_{n=0}^{\infty} \frac{1}{(1.05)^n} \,x^n ,\;\;\; |x| \leq 1 \\
e^{-10\,x} &= - \sum_{n=0}^{\infty} \frac{(-10)^n}{n!} \,x^n ,\;\;\; 0\leq x \leq 1\end{aligned}$$ The first two integrand have slowly-decaying coefficients and can be approximated in the interval $[-1,1]$ through a sum containing ${\mathcal{O}}( \log_{1.05}(1/\epsilon) ) $ terms for an accuracy of $\epsilon$. At $\epsilon$-machine ($\epsilon = 10^{-15}$) this implies approximately $700$ terms. The third integrand has very fast decay, and in this case only $50$ terms are sufficient.
For each case, we varied the number of nodes in the quadrature. Then for AGQ, we selected the integer $N$ that gave us the most accurate result. In practice, an algorithm would be required to estimate $N$ numerically but we will not address this question here. Results are show in Table \[general:log\]–\[general:exp\].
Number of nodes Optimal value for $N$ AGQ Gauss-Legendre
----------------- ----------------------- ----------------------- -----------------------
10 75 $2.16 \cdot 10^{-8}$ $1.39 \cdot 10^{-4}$
15 100 $1.08 \cdot 10^{-8}$ $3.94 \cdot 10^{-6}$
20 150 $2.05 \cdot 10^{-11}$ $1.26 \cdot 10^{-7}$
25 200 $3.99 \cdot 10^{-14}$ $4.31 \cdot 10^{-9}$
30 250 $1.61 \cdot 10^{-15}$ $1.54 \cdot 10^{-10}$
: Absolute error incurred by an AGQ and a Gauss-Legendre quadrature for the integration of $f(x) = \log \left ( 1- \frac{x}{1.05} \right ) $ over the interval $[-1,1]$ for various number of nodes.[]{data-label="general:log"}
Number of nodes Optimal value for $N$ AGQ Gauss-Legendre
----------------- ----------------------- ----------------------- ----------------------
10 75 $5.81 \cdot 10^{-5}$ $8.15 \cdot 10^{-3}$
15 100 $2.20 \cdot 10^{-6}$ $3.60 \cdot 10^{-4}$
20 150 $4.26 \cdot 10^{-9}$ $1.56 \cdot 10^{-5}$
25 200 $1.58 \cdot 10^{-11}$ $6.76 \cdot 10^{-7}$
30 250 $4.01 \cdot 10^{-13}$ $2.92 \cdot 10^{-8}$
35 300 $1.77 \cdot 10^{-15}$ $1.25 \cdot 10^{-9}$
: Absolute error incurred by an AGQ and a Gauss-Legendre quadrature for the integration of $f(x) = \frac{1}{ 1- \frac{x}{1.05} }$ over the interval $[-1,1]$ for various number of nodes.[]{data-label="general:geo"}
Number of nodes Optimal value for $N$ AGQ Gauss-Legendre
----------------- ----------------------- ----------------------- -----------------------
5 15 $1.09 \cdot 10^{-6}$ $8.82 \cdot 10^{-5}$
7 7 $1.29 \cdot 10^{-7}$ $1.29 \cdot 10^{-7}$
10 10 $1.02 \cdot 10^{-12}$ $1.02 \cdot 10^{-12}$
12 12 $4.44 \cdot 10^{-16}$ $4.44 \cdot 10^{-16}$
: Absolute error incurred for $e^{-10\,x} $ over the interval $[0,1]$. In that case, Gauss-Legendre converges very fast and AGQ simply provides a quadrature with the same accuracy. The two methods become essentially identical.[]{data-label="general:exp"}
We observe the superior accuracy of AGQ. The first two cases are challenging for CGQ and AGQ does significantly better. For the last case, CGQ converges extremely fast and then AGQ simply finds that the optimal choice is CGQ and provides an estimate with the same accuracy.
In summary, $N$ shoud be adjusted depending on the type of integrand. If the integrand is such that expansions in a polynomial basis possess slowy-decaying coefficients, AGQ will provide significantly greater accuracy. If on the contrary, a polynomial expansion converges very rapidly, both AGQ and CGQ will provide essentially identical (and fast) convergence.
We also stress that AGQ can be constructed for a wide range of measures whereas CGQ is restricted to positive measures (weight function) only.
Singular functions {#NS:sing}
------------------
We show how AGQ can be used to integrate functions with integrable singularities. For this purpose, we consider integrand of the form $x^n \log(x)$ for $x \in (0,1]$ and $0 \leq n \leq 700$. In this case, the integral of interest takes the form, $$\int_{0}^1 x^n \, \log(x) \, {{\mathrm d}}x$$ This quantity can either be seen as the integration of $x^n \, \log(x)$ with respect to Lebesgue measure or as the integration of the monomial $x^n$ with respect to the measure ${{\mathrm d}}\alpha(x) = \log(x) \, {{\mathrm d}}x$. Considering the latter, we build an AGQ of order $N=350$ with different number of nodes and display the absolute error as a function of the degree $n$ and the number of quadrature points. This is shown in Figure \[NS:log\]. Note that the bound is not plotted beyond $N=350$ for it is no more valid past this point.
We note that in this case we cannot perform a comparison with a classical Gaussian quadrature for no such quadrature exists as is the case with most measures but a few.
Quadrature for polynomials on the complex circle {#NS:trig}
------------------------------------------------
In this section, we are interested in integrands that take the form of trigonometric monomials, i.e., functions of the form, $$f(x) = e^{{\imath}n x}$$ where $0 \leq n$. As their name conveys, such functions are just homogeneous polynomials $z^n$ in the complex plane which have been restricted to the boundary of the unit circle, i.e., $z = e^{ix}$. Thanks to this close relationship with polynomials on the real axis, one can also develop approximate Gaussian quadratures for such functions as well. In fact it suffices to replace the moments $\mu_n$ by the trigonometric moments, $$\tau_n = \int (e^{{\imath}x})^n \, {{\mathrm d}}\alpha(x)
= \int z^n \, {{\mathrm d}}\alpha(z)$$ in all that has been presented above and similar results follow.
As an example, we built an AGQ of order $N=350$ for trigonometric polynomials with respect to the Lebesgue measure over the interval $[-1,1]$. The absolute error between our approximation and the exact value of the integral, $$\int_{-1}^1 e^{{\imath}n x} \, {{\mathrm d}}x = \frac{e^{{\imath}n} - e^{-{\imath}n}}{{\imath}n}$$ are presented in Figure \[NS:trigplot\]. There, it is seen that as little as $30$ quadrature points are necessary to integrate a complex exponential with frequency $n=500$ with $\approx 10^{-6}$ accuracy. We also plotted the theoretical bound of Theorem \[AGQ:AGQ\_thm\]. Again, it appears to be a good estimate.
It is interesting to look at the location of the nodes for such quadratures. An example is displayed in Figure \[NS:trignodes\]. The nodes are shown in the complex plane and appear to lie along a curve which rapidly moves upward from $-1$, slowly moves across, and rapidly moves back to $1$ on the real axis. This does not appear to be a coincidence given the fact that functions of the form $e^{{\imath}n x}$ decay exponentially and do not oscillate along the positive imaginary axis. Thus, the underlying curve could be some sort of path of *least oscillation* in an average sense over $0 \leq n \leq N$. At this point, this is a mere qualitative observation, but might be worth investigating in the future.
![Location in the complex plane of the nodes of a 20-node AGQ for trigonometric polynomials. The nodes appear to lie on a smooth curve with positive imaginary part.[]{data-label="NS:trignodes"}](trignodes.png){width="10cm"}
Approximation of functions through short exponential sums {#NS:Beylkin}
---------------------------------------------------------
In this section, we are interested in the approximation of functions by a short sum of exponentials. That is, given a function $f(x)$ defined over an interval $[a,b]$, we seek some approximation in the form, $$f(x) \approx \sum_{m=0}^d \alpha_m e^{ \beta_m x}$$ for $x \in [a,b]$, and where $d$ should be as small as possible. Such expansions can be viewed as more efficient representations of functions compared to Fourier transforms as they typically require fewer terms. They can form the starting point for various fast algorithms such as the fast multipole method, hierarchical matrices ($\mathcal H$-matrices), etc. Such techniques are particularly desirable when it comes to the solution of integral equations with translation-invariant kernels (see e.g., [@Letourneau:2012; @Beylkin:2005]). Very powerful techniques based on dynamical systems and recursion ideas were recently introduced by Beylkin & Monzón [@Beylkin:2005; @Beylkin:2010] in order to approach this problem. As was mentioned earlier, the latter inspired the current work.
We will show how AGQ can be used to derive similar approximations through the discretization of the Fourier transform. The final formulation shares some characteristics with the problem of Beylkin & Monzón that can be stated as follows: given the accuracy $\epsilon > 0$, for a smooth function $f(x)$ find the minimal number of complex weights $w_n$ and nodes $e^{t_m}$ such that, $$\left | f(x) - \sum_m w_m e^{t_m x} \right | < \epsilon$$ for $x \in I$, $I$ being some interval in $\mathbb{R}$.
Their scheme is based on an important result regarding Hankel matrices. Consider a Hankel matrix $H$ associated with a sequence $h_k$ where $h_k = f(x_k)$ are uniform samples of $f$. Assume that the null space of $H$ is non-trivial and consider the polynomial whose coefficients are given by a vector in the null space of $H$. The zeros of this polynomial, $\lambda_i$, satisfy the following property (see e.g., [@Boley:1998]), $$h_k = \sum_{i=1}^r \lambda_i^k \,d_i$$ for some $\{ d_i \}$, where $r$ is at most the number of columns of $H$. With our choice for $h_k$, one obtains, $$f(x_k) = \sum_{i=1}^r d_i \, e^{ \log( \lambda_i ) k }$$ which naturally extends to an interpolation formula for $f(\cdot)$.
In [@Beylkin:2005; @Beylkin:2010], the authors search for an approximate formula since in general the matrix $H$ is full rank and therefore no efficient representation, that would yield exactly $f(x_k)$, is possible. To achieve this, Beylkin et al. [@Beylkin:2005; @Beylkin:2010] show how $\lambda_i$ can be obtained as the roots of a polynomial whose coefficients are given as the entries of a con-eigenvector $u$, i.e., a vector such that, $$H u = \sigma \overline{u}$$ $\sigma$ being real and nonnegative. The error is then on the order of $\sigma$. They also show that the weights satisfy a well-conditioned Vandermonde system.
As will be seen, both our method and theirs involve a Hankel matrix with entries given by the uniform samples of the function to be approximated over the interval considered. However, the current approach avoids the solution of a con-eigenvalue problem altogether and allows for the direct computation of the weights rather than their computation through the solution of a Vandermonde system. Furthermore, since the quasi-orthogonal polynomial obtained through our scheme has small degree, the number of zeros that must be computed is also much smaller. This results in significant computational savings compared to the former method.
The resulting error estimates for both methods are different. Indeed, in the case of [@Beylkin:2005] one expects the error to be bounded *uniformly* by an expression on the order of the modulus of the small con-eigenvalue $\sigma$ (Theorem 2, [@Beylkin:2005]), and such value can be determined *a priori*. In our case however, the error in *not* uniform (as can be seen from the numerical examples). Furthermore, our current error estimate is *a posteriori*.
To begin with, consider a function $f(x) \in \mathcal{L}^2 ( \mathbb{R})$ uniformly sampled at $x_n = a + \frac{n (b-a)}{N}$, $n = 0... (N-1)$ for some $N \in \mathbb{N}$ and $a,b \in \mathbb{R}$, and use the Fourier transform to write, $$f(x_n) = \int_{-\infty}^{\infty} e^{2 \pi {\imath}x_n \xi} \hat{f} (\xi) \, {{\mathrm d}}\lambda( \xi)=\frac{N}{(b-a)}\, \int_{-\infty}^{\infty} e^{2 \pi {\imath}n \zeta} \, e^{2 \pi {\imath}a \zeta} \hat{f} \left (\frac{N}{b-a} \zeta \right ) \, {{\mathrm d}}\lambda( \zeta)$$ where $\hat{f} (\xi)$ denotes the Fourier transform of $f(x)$, and $\lambda( \cdot )$ is the Lebesgue measure. We note that $$\frac{N}{b-a} \, e^{2 \pi {\imath}a \zeta} \hat{f} \left (\frac{N}{b-a} \zeta \right )$$ can be seen as a Radon-Nykodym derivative of a certain measure $\alpha( \cdot)$ absolutely continuous with respect to Lebesgue measure (see [@Cohn]), i.e., $$\frac{{{\mathrm d}}\alpha }{ {{\mathrm d}}\lambda} (\zeta) = \frac{N}{b-a} \, e^{2 \pi {\imath}a \zeta} \hat{f} \left (\frac{N}{b-a} \zeta \right )$$ With this measure we have, $$f(x_n) = \int_{-\infty}^{\infty} e^{2 \pi {\imath}n \zeta} \, {{\mathrm d}}\alpha(\zeta) , \;\; n = 0... N$$ which is perfectly well-suited for discretization through an approximate Gaussian quadrature as described in the previous section. To find such quadrature, we first need the trigonometric moments of the measure. These moments turn out to have a very simple form. Indeed, a quick look at their definition shows that, $$\tau_n = \int_{\mathbb{T}} e^{{\imath}n \zeta} \, {{\mathrm d}}\alpha(\zeta) = \int_{\mathbb{T}} e^{{\imath}n \zeta} \, \left [ \frac{N}{b-a} \, e^{2 \pi {\imath}a \zeta} \hat{f} \left (\frac{N}{b-a} \zeta \right ) \right ] {{\mathrm d}}\lambda(\zeta) = f \left( a + n \frac{(b-a)}{N} \right )$$ At this point, we note that the Hankel matrix arising from such moments is exactly the same as the one described in [@Beylkin:2005] as previously mentioned.
Finally, the nodes $\{ w_n \}$ can be obtained through Eq.. In the end, we obtain $$f(x_n) \approx \sum_{m=0}^d w_m \, e^{ {\imath}n \zeta_m } , \;\;\; n=1,...,N$$ with error bounded by the expression provided in Theorem \[AGQ:AGQ\_thm\]. To obtain an approximation to $f(x)$ in all of $[a,b]$, we simply allow $\frac{n}{N}$ to vary continuously so that $$\frac{n}{N} = \frac{x-a}{b-a}$$ for $x \in [a,b]$ and write, $$\begin{aligned}
f(x) &\approx \sum_{m=0}^d \alpha_m \, e^{ {\imath}\beta_m x } \\
\alpha_m &= w_m \, e^{-{\imath}\frac{a}{b-a} N \xi_m } \\
\beta_m &= \frac{1}{b-a} N \xi_m\end{aligned}$$
When $x$ corresponds to a sample, i.e., $x = x_n$ for some $n$, this reduces to the previous expression. However, when $x$ lies between two samples this last formula should be seen as an interpolation. We do not currently have the complete theory describing the interpolation error. However, it was observed numerically that such error is generally of the same order as that associated with the closest sample whenever the function $f(x)$ is sufficiently oversampled. Numerical examples are provided below.
At this point, we describe an algorithm for the construction of such an approximation. The description can be found in pseudo-code in Algorithm \[approx\_alg\].
Pick $N \in \mathbb{N}$ sufficiently large (beyond the Nyquist rate)\
Compute $ \tau_n = f \left( a + n \frac{(b-a)}{N} \right )$\
Build the Hankel matrix $H_{i,j} = \tau_{i+j}$ for $i,j = 0 .. N$\
Proceed as described in Algorithm \[poly\_alg\] to find $p(x)$\
Compute $\{ x_n \}$, the nodes/zeros of $p(x)$\
Compute weights $w_n$ following Eq.\
Build approximation: $ \sum_{n} w_n \, e^{ {\imath}\frac{(x-a)}{(b-a)} N \, \frac{\log(x_n)}{{\imath}} }$
We now provide a few examples for the representation of some oscillatory functions: the Bessel functions of the first kind $J_{\nu} (100 \pi \, x )$ over the interval $[0,1]$ and for orders $ \nu \in \{ 0, 25 \}$. Such functions are relevant in problems involving the scattering of waves in two dimensions for instance. In both cases, the order of the AGQ is $N=400$ (note that the spectrum of both functions is bounded by about $400 \approx 100 \pi$) and a $40$-terms approximation is obtained using the scheme just introduced. The results are presented in Figure \[bessel0\] and \[bessel25\] respectively. Agreement within $10^{-10}$ and $10^{-7}$ absolute error is observed in each cases respectively. It should also be noted that the number of terms lies much below what should be expected with a standard Fourier series given the nature of the oscillations.
As a final example, we chose to represent the Dirichlet kernel, $$D_N (x) = \sum_{k=-N}^N e^{{\imath}k x} = \frac{\sin \left ( \pi( N + 1/2) \, x \right ) }{\sin \left ( \pi/2 \, x \right )}$$ over the interval $[-1,1]$. When applied through convolution, the Dirichlet kernel acts as a low-frequency filter. In this sense, a short exponential sum approximation can be used to speed up the filtering process.
We picked $N = 200$. To obtain the approximation, we proceeded as described in [@Beylkin:2005] and went on to first approximate, $$G_{200} (x) = \sum_{k \geq 0 } \frac{\sin(200 \pi (x+k))}{200 \pi (x+k)}$$ through a 40-term exponential sum and then built the Dirichlet kernel through the identity, $$D_{200} (x) = G_{200} (x) + G_{200} (1-x)$$ resulting in a 80-term approximation. It is shown in Figure \[dirichlet\]. The error is non-uniform as expected from Theorem \[AGQ:AGQ\_thm\] but still remains below $10^{-7}$ for all values in the interval.
Conclusion
==========
We have introduced a new type of quadrature closely related to Gaussian quadratures but which use the concept of $\epsilon$-quasiorthogonality to reduce the number of quadrature nodes and weights. Such quadratures have desirable computational properties and can be applied to a family much broader than that targeted by classical Gaussian quadratures. We have provided the theory for the existence of such quadratures and have provided error estimates together with practical ways of constructing them. We have also carried out various numerical examples displaying the versatility and performance of the method. Finally, we have described how AGQ can be used to approximate functions through short exponential sums and provided further numerical examples in these cases.
Acknowledgements
================
The authors would like to thank Professor Ying Wu from King Abdullah University of Science and Technology (KAUST) for supporting this research through her grant as well as the National Sciences and Engineering Research Council of Canada (NSERC) for their financial support.
[**Proof of Lemma \[AGQ:errorlemma\].**]{} First, thanks to the factorization theorem for polynomials (see e.g., [@Hungerford]) $$\label{error:1}
q(x) - \tilde{q} (x) = \sum_{i=0}^{d} (q_i - \tilde{q}_i ) x^i + \sum_{i=d+1}^{n+d} q_i x^i = \left( \sum_{i=0}^d p_i x^i \right ) \left ( \sum_{i=0}^n r_i x^i \right ) = p(x) r(x)$$ and from the Cauchy product, we have, $$\label{error:2}
\left( \sum_{i=0}^d p_i x^i \right ) \left ( \sum_{i=0}^N r_i x^i \right ) = \sum_{i=0}^{N+d} \left( \sum_{k=0}^i r_k p_{i-k} \right ) x^i$$ where it is understood that coefficients corresponding to indices outside the original range of definition of $p(x)$ and $r(x)$ are $0$. By matching coefficients of like powers in Eq. and and putting the linear system thus obtained in matrix form, one gets $$\Gamma r = \Gamma
\begin{pmatrix}
r_0 \\
r_1 \\
r_2 \\
\vdots \\
r_N
\end{pmatrix}
=
\begin{pmatrix}
q_0 - \tilde{q}_0 \\
\vdots \\
q_d - \tilde{q}_d \\
q_{d+1}\\
\vdots \\
q_{N+d}
\end{pmatrix}
= \kappa$$ where, $$\Gamma =
\begin{pmatrix}
p_0 & 0 & 0 & \cdots & 0 &0 \\
p_1 &p_0 & 0 & \cdots & 0 & 0\\
p_2 & p_1 & p_0 & \cdots & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 1 & p_{d}\\
0& 0 & 0 & \cdots & 0 & 1
\end{pmatrix}$$ $\Gamma$ is an $(N+d+1) \times (N+1)$ Toeplitz matrix characterized by the coefficients of the known quasi-orthogonal polynomial $p(x)$. We know form the existence and uniqueness theorem for the factorization of polynomials that there exists a unique solution to the above system. We further write, (assuming $N>d$) $$\Gamma =
\begin{pmatrix}
\Gamma_1 \\
\Gamma_2
\end{pmatrix}$$ where $\Gamma_1$ is a $d \times N$ matrix containing the first $d$ rows of $\Gamma$ and $\Gamma_2$ is a $N \times N$ matrix containing the last $N$ rows of $\Gamma$. It is to be noted that $\Gamma_2$ is an upper triangular matrix with diagonal entries all equal to $1$. Therefore, all eigenvalues of $\Gamma_2$ are equal to $1$. In particular, $\Gamma_2$ is invertible and we can write, $$\begin{pmatrix}
r_0 \\
r_1 \\
r_2 \\
\vdots \\
r_N
\end{pmatrix}
= \Gamma_2^{-1}
\begin{pmatrix}
q_{d+1}\\
\vdots \\
q_{N+d}
\end{pmatrix}$$ where $\Gamma_2^{-1} $ is also an upper triangular Toeplitz matrix with diagonal entries all equal to $1$. Therefore, $$\sum_{n=0}^N |r_n| = \lVert r \rVert_1 = \lVert \Gamma_2^{-1} \bar{q} \rVert_1$$
|
---
abstract: 'We present the first global, three-dimensional simulations of solar/stellar convection that take into account the influence of magnetic flux emergence by means of the Babcock-Leighton (BL) mechanism. We have shown that the inclusion of a BL poloidal source term in a convection simulation can promote cyclic activity in an otherwise steady dynamo. Some cycle properties are reminiscent of solar observations, such as the equatorward propagation of toroidal flux near the base of the convection zone. However, the cycle period in this young sun (rotating three times faster than the solar rate) is very short ($\sim$ 6 months) and it is unclear whether much longer cycles may be achieved within this modeling framework, given the high efficiency of field generation and transport by the convection. Even so, the incorporation of mean-field parameterizations in 3D convection simulations to account for elusive processes such as flux emergence may well prove useful in the future modeling of solar and stellar activity cycles.'
author:
- 'Mark S. Miesch$^1$ and Benjamin P. Brown$^2$'
title: 'Convective Babcock-Leighton Dynamo Models'
---
Introduction {#sec:intro}
============
The emergence of magnetic flux through the solar photosphere regulates solar variability and powers space weather. It is clear that this flux originates in the solar interior and is produced by the solar dynamo. However, it is currently unclear what role flux emergence plays in establishing the 22-year solar activity cycle. Is it an essential ingredient or merely a superficial by-product of deeper-seated dynamics?
One of the principle means by which flux emergence may act an an essential ingredient in the operation of the solar dynamo is through the so-called Babcock-Leighton (BL) mechanism [@babco61; @leigh64]. The BL mechanism arises through the dynamics of flux emergence. As a buoyant flux tube rises through the convection zone, the Coriolis force induces a twist in the axis of the tube that is manifested upon emergence as a poleward displacement between the trailing and leading edges of a bipolar active region. When the active region subsequently fragments and disperses due to surface convection and meridional flow, the redistribution of vertical flux induces a net electromotive force (emf) that converts mean toroidal field to poloidal field. Although doubts remain about the viability of the BL mechanism as the principle source of poloidal flux, its empirical foundation and robustness have made it an integral element in many recent mean-field dynamo models of the solar cycle [reviewed by @dikpa09; @charb10].
Current simulations of global solar and stellar convection do exhibit sustained dynamo action and, depending on the parameter regime, can produce magnetic cycles [@ghiza10; @racin11; @brown11]. Yet, these simulations do not have sufficient resolution to reliably capture flux emergence or the BL mechanism. [@nelso11] recently reported the first convective dynamo simulation to exhibit the spontaneous, self-consistent generation of buoyant magnetic flux structures generated by convectively-driven rotational shear. However, even these cannot realistically simulate the subtle, multi-scale dynamics of flux emergence and dispersal that underlies the BL mechanism. This would require (1) very low diffusion to form concentrated flux structures, (2) very high resolution to capture the destabilization and coherent rise of those structures, and (3) a realistic depiction of surface convection, meridional flow, and radiative transfer in order to capture the emergence, coalescence, fragmentation, and dispersal of bipolar active regions [@cheun10]. Each of these is a formidable computational challenge in its own right, stetching the limits of modern supercomputers. No global model is capable of unifying convective dynamos and the BL mechanism through direct numerical simulation.
Thus, the presence of magnetic cycles in global convection simulations demonstrates that the BL mechanism is not a necessary ingredient for cyclic activity. How this may or may not apply to the solar dynamo is a complex, unresolved issue and we make no attempt at a comprehensive discussion here. Our purpose is rather to investigate how flux emergence may alter the behavior of a convective dynamo by means of the BL mechanism. The BL mechanism is modeled using a mean-field parameterization intended to mimic dynamics that cannot be explicitly captured by the simulation itself for the reasons outlined above. Our approach is described in §\[model\] and simulation results are presented in §\[results\], along with interpretive discussion.
Before proceeding, it is worth emphasizing up front that the simulations we consider here are of a solar-like star rotating three times more rapidly than the Sun (3$\Omega_\odot$). This is done because it is a tidy numerical experiment; without BL forcing, this dynamo builds strong large-scale fields that do not undergo cycles. Other parameter regimes, including those at the solar rotation rate, will be considered in future work.
Model
=====
The starting point for our investigation is the convective dynamo simulation that we refer to as case D3 and that is described in detail by [@brown10]. We refer the reader to that paper for further information on the set up and results of the simulation as well as the ASH (Anelastic Spherical Harmonic) code that is used to solve the equations of magnetohydrodynamics (MHD) in a rotating spherical shell under the anelastic approximation. Solar values are used for the luminosity and the background stratification but the rotation rate is a factor of three faster than the Sun, as mentioned above. The spatial resolution of all simulations reported in this paper is 96 $\times$ 256 $\times$ 512 ($r$, $\theta$, $\phi$). The simulation domain spans from $r_1 = 0.718 R$ to $r_2 = 0.966 R$, where $R$ is the solar radius.
The salient feature of Case D3 that we are interested in here is the presence of persistent toroidal field structures that we term magnetic wreaths. The convection exhibits an intricate, evolving small-scale structure (Fig. \[fig:D3\]$a$) but produces a substantial differential rotation (Fig. \[fig:D3\]$b$) that in turn promotes the generation of prominent magnetic wreaths (Fig. \[fig:D3\]$c$). The mean toroidal field is approximately symmetric about the equator, with one wreath in each hemisphere of opposite polarity. Although continual buffeting by convective motions induces non-axisymmetric and temporal fluctuations, the mean field is remarkably persistent, retaining its essential structure indefinitely (Fig. \[fig:D3\]$d$). After they are established, the wreaths persist for at least 60 years (the duration of the simulation) with no sign of abating [@brown10]. This time interval is much longer than the rotation period (9.3 days), the convective turnover time scale (about 20 days) and the ohmic diffusion time (about 3.6 years).
The simulations described in this paper are all restarted from the same iteration of Case D3, defined as time $t = 0$. This includes the unmodified D3 run shown in Figure \[fig:D3\], which was continued beyond the restart iteration for comparison purposes. The absence of initial transients in Fig. \[fig:D3\] demonstrates that this simulation has reached a statistically steady state by the mutual reference time $t = 0$.
The anelastic MHD equations for the conservation of mass, momentum, and thermal energy are solved with no modification. The only difference between the simulations presented here and that presented in [@brown10] is in the magnetic induction equation where we add an additional poloidal source term $S(r,\theta)$ as follows: $$\label{eq:indy}
\frac{{\partial}{{\bf B}}}{{\partial}t} = {\mbox{\boldmath $\nabla \times$}}\left(
{{\bf v}}{\mbox{\boldmath $\times$}}{{\bf B}}- \eta {\mbox{\boldmath $\nabla \times$}}{{\bf B}}+ S {\mbox{\boldmath $\hat{\phi}$}}\right)$$ The additional term is intended to mimic the generation of mean poloidal field by the BL mechanism. Following the mean-field BL dynamo model of @rempe06, we choose $$S(r,\theta) = \alpha f(r) g(\theta) \hat{B}_\phi$$ where $f(r)$ and $g(\theta)$ are radial and latitudinal profiles $$\label{frad}
f(r) = \mbox{max}\left[ 0, 1 - \frac{(r-r_2)^2}{d^2}\right]$$ $$g(\theta) = \frac{3 \sqrt{3}}{2} ~ \sin^2\theta \cos \theta$$ and $\hat{B}_\phi$ is a measure of the mean toroidal flux at the base of the convection zone $$\label{Bhat}
\hat{B}_\phi = \int_{r_1}^{r_b} h(r) \left<B_\phi\right> dr ~~~.$$ Brackets $<>$ denote an average over longitude and $h$ is an averaging kernal given by $h(r) = h_0 \left(r - r_1\right)\left(r_b - r\right)$. The integration is confined to a region near the base of the convection zone, below $r_b = 0.79 R$ and the normalization $h_0$ is defined such that $\int_{r_1}^{r_b} h(r) dr = 1$.
Note that the radial profile in equation (\[frad\]) is nonzero only near the top of the convection zone, for $r > r_2 - d$. We choose $d = 20$ Mm so the poloidal source operates above $r = 0.937 R$. Thus, the BL term is nonlocal in the sense that the poloidal source near the surface is proportional to the mean toroidal flux near the base of the convection zone. This is typical of BL dynamo models [e.g. @rempe06].
The amplitude of the BL term, $\alpha$, includes an algebraic quenching of the form $\alpha = \alpha_0 (1 + B_t^2/B_q^2)^{-1}$ where $B_q = 1$ MG is the quenching field strength and $B_t^2 = (1/2) \int_0^\pi \hat{B}^2 \sin\theta d\theta$. In practice the fields generated are much less than $B_q$ so the quenching plays little role.
The magnetic diffusivity $\eta$ in all simulations is the same as in Case D3, varying from 1.56-7.69 $\times 10^{12}$ cm$^2$ s$^{-1}$ from the bottom to the top of the convection zone, $\propto
{\overline{\rho}}^{-1/2}$, where ${\overline{\rho}}$ is the background density.
The objective of this paper is to vary the amplitude of the BL term $\alpha_0$ in order to investigate how flux emergence may alter the nature of the dynamo. We emphasize again that the 3D MHD equations are unmodified apart from the $S(r,\theta)$ term in eq. (\[eq:indy\]) so setting $\alpha_0 = 0$ reproduces Case D3 as shown in Figure \[fig:D3\] and as described at length in @brown10.
Results
=======
Figure \[fig:bflys\] shows “butterfly” diagrams (latitude-time plots of the mean toroidal field near the base of the convection zone) for a series of simulations with progressively increasing values of the BL forcing amplitude $\alpha_0$. For $\alpha_0 = $ 1 m s$^{-1}$ the BL term is too weak to sigificantly influence the operation of the dynamo (Fig. \[fig:bflys\]$a$). The axisymmteric component of the poloidal field near the surface does increase but not enough to effect the maintenance of persistent wreaths in the lower convection zone.
For $\alpha_0 = $ 10 m s$^{-1}$ the results are very different. The wreaths essentially vanish within a few years of simulation time (Fig. \[fig:bflys\]$b$). This can be attributed to the differential rotation operating on the mean poloidal field generated by the BL mechanism via the $\Omega$-effect. The sense of the poloidal field produced by the BL term is such that the $\Omega$-effect generates toroidal flux in the upper convection zone that is of opposite polarity to the sense of the wreaths. Convection rapidly mixes this flux, bringing together opposite polarities that are annihilated through ohmic dissipation. The magnetic energy in the mean fields drops rapidly, decaying by a factor of 10$^6$ by $t \sim $ 30 yrs.
Yet, beyond about 30 years, the magnetic energy begins to rise again and the dynamo is reborn, saturating by about $t \sim $ 75 yrs (Fig. \[fig:bflys\]$c$) with a very different structure. Persistent toroidal wreaths are again present but they are symmetric about the equator, with two wreaths per hemisphere and a somewhat lower magnetic energy (about 27% less than in Case D3).
Increasing $\alpha_0$ by another order of magnitude produces prominent magnetic cycles (Fig. \[fig:bflys\]$d$). Toroidal wreaths form at mid latitudes in each hemisphere and propagate toward the equator, reminiscent of the solar butterfy diagram (Fig. \[fig:bflys\]$e$). However, the cycle period is much shorter than in the Sun; about 6 months compared to 22 years. Nonlinear modulation of the cycle is evident, with waxing and waning amplitudes and transient, weaker substructure in the butterfly diagram near the equator.
It is clear from Fig. \[fig:bflys\]$e$ that reversals in the two hemispheres are not precisely sychronized. A more careful analysis reveals that the phase difference shifts over time, suggesting that the two hemispheres are to some extent decoupled. For example, the northern hemisphere leads the southern hemisphere from $t \sim $ 4.5 – 11.5 yrs but the reverse is true for $t \sim $ 12 – 14.5 yrs. Similar shifts in the hemispheric phase difference (albeit less pronounced) have been reported in sunspot records; in particular, the southern hemisphere was apparently leading in cycles 18-20 while the northern was leading in cycles 21-23 [@mcint12].
The symmetry of the dynamo about the equator can be quantified by the parity ${\cal P} = (B_s^2 - B_a^2)/(B_s^2 + B_a^2)$, where $B_s$ and $B_a$ are the symmetric and antisymmtric components of $\left<B_\phi\right>$ respectively (sampled at $r = 0.84 R$). The parity of Case D3 and for $\alpha_0 = $ 1 m s$^{-1}$ ranges between -0.5 and -1 while that for $\alpha_0$ = 10 m s$^{-1}$ ranges between positive 0.5-1. By contrast, the cyclic dynamo ($\alpha_0 = $ 100 m s$^{-1}$) shifts between positive and negative parity as time proceeds. It does not exhibit the high synchronization of the solar cycle which exhibits a persistent negative parity.
As expected, the addition of a BL term has a substantial influence on the amplitude and structure of the mean poloidal field. Without it, the poloidal field has a roughly octupolar structure, such that $\left<B_r\right>$ is radially outward near the north pole, inward near the core of the northern wreath, and antisymmetric about the equator [@brown10]. As noted above, the BL term generates opposing poloidal flux at mid-latitudes near the surface, producing multi-polar structure and enhancing dissipation. This plays an essential role in establishing the cycles of Fig \[fig:bflys\]$d$-$e$ and is evident in movies of the mean-field evolution. The typical amplitude of the mean poloidal field near the surface for $\alpha_0 = 100$ m s$^{-1}$, 1-2 kG, is much larger than for $\alpha_0 = 0$ ($\sim$ 200 G in Case D3) but the overall (3D) magnetic energy is about a factor of two smaller.
The transition from steady to cyclic dynamos occurs when the BL term $S$ competes with the fluctuating emf $\left<{{\bf v}}^\prime {\mbox{\boldmath $\times$}}{{\bf B}}^\prime\right>$, where ${{\bf v}}^\prime = {{\bf v}}- \left<{{\bf v}}\right>$ and ${{\bf B}}^\prime = {{\bf B}}- \left<{{\bf B}}\right>$. This in turn occurs when $\alpha_0$ becomes comparable to the velocity of the convective motions, $V_c$. The relevant scale for $V_c$ is the rms value of the meridional components of ${{\bf v}}^\prime$, which is about 120 m s$^{-1}$ near the top of the convection zone in Case D3, decreasing with depth.
A legitimate question is whether the cyclic dynamo in Fig. \[fig:bflys\]$d$,$e$ is operating in an essentially mean-field, axisymmteric mode. In other words, if the BL term were to dominate the generation of mean poloidal field and the differential rotation were to dominate the generation of mean toroidal field via the $\Omega$-effect, then one might expect this 3D simulation to behave very similarly to an analogous, axisymmetric mean-field model. Under this mean-field scenario, coupling between the poloidal and toroidal source regions might occur through the nonlocality of the BL term, the mean meridional flow, and the turbulent diffusion $\eta$. The primary role of the resolved convective motions would then be to maintain the mean flows.
In order to assess whether this mean-field scenario is indeed a valid interpretation of the cyclic activity shown in Fig.\[fig:bflys\]$d$,$e$, we have initiated another simulation in which we have artificially suppressed the fluctuating emf. More specifically, we have replaced the ${{\bf v}}{\mbox{\boldmath $\times$}}{{\bf B}}$ term in equation (\[eq:indy\]) with only mean-field induction $\left<{{\bf v}}\right> {\mbox{\boldmath $\times$}}\left<{{\bf B}}\right>$. This simulation was restarted from that shown in Fig. \[fig:bflys\]$d$,$e$ at $t = 8.25$ yrs with the same parameters. The only difference is the absence of the fluctuating emf.
The dynamo mode changes dramatically, as demonstrated in Fig. \[fig:bfly\_mf\]. The non-axisymmetric magnetic field is quickly dissipated by ohmic diffusion, with the corresponding energy decreasing by six orders of magnitude within two years (shear promotes more rapid dissipation than the nominal 3.6 year diffusion time scale). The total magnetic energy is about a factor of three larger than in the progenitor simulation of Fig. \[fig:bflys\]$d$,$e$ and is dominated by a strong, axisymmtric toroidal field that is symmetric about the equator (positive parity) and reverses cyclically with a period of about 1.3 yrs. The butterfly diagram in Fig. \[fig:bfly\_mf\] suggests poleward propagation but closer inspection of the mean field evolution reveals a dynamo wave that propagates toward the rotation axis, with a cylindrical orientation for the wave front. This is consistent with the cylindrical nature of the differential rotation profile (Fig. \[fig:D3\]$b$) but is strikingly different from the progenitor simulation in Fig. \[fig:bflys\]$d$,$e$ which exhibits virtually no mean toroidal field at the equator even when the parity is positive. In short, cycles are longer, stronger, and less solar-like without a turbulent emf.
Thus, the cyclic dynamo in Fig. \[fig:bflys\]$d$,$e$ is not operating as a mean-field dynamo, or at least not in a naive sense. Convection contributes mean field generation and transport that plays an essential role in shaping the dynamo even for $\alpha_0$ as high as 100 m s$^{-1}$. Parity selection in cyclic dynamos is a subtle issue and space limitations preclude a thorough discussion here. We only remark that mean-field BL models have demonstrated that the relatively homogeneous nature of turbulent pumping [@guerr08], the antisymmetric structure of a supplementary $\alpha$-effect operating near the base of the convection zone [@dikpa01c], and efficient turbulent mixing that peaks in the upper convection zone [@hotta10] can all help promote negative (dipolar) parity. These mean-field results are loosely consistent with the 3D convection simulations reported here but warrant a more careful investigation which we reserve for future work.
In summary, we have shown that flux emergence can promote cyclic magnetic activity in a convective dynamo by means of the Babcock-Leighton mechanism. Although this result is to some extent anticipated by mean-field dynamo models, we have demonstrated it for the first time in a global convection simulation. The BL mechanism is not required to achieve cycles in convective dynamo simulations but, as we have shown here, it may help shape cycle properties such as the period, amplitude, and equatorward propagation of toroidal flux.
However, achieving cyclic activity through the BL mechanism is not easy. To make a significant impact on the dynamo, the amplitude of the BL $\alpha$-effect must be comparable to the convective velocity, $V_c$. In particular, we find that $\alpha_0 > $ 10 m s$^{-1}$ is required to induce cyclic activity when $V_c \sim$ 100 m s$^{-1}$ in the upper convection zone. By comparison, typical values of $\alpha$ used in mean-field BL dynamo models of the solar cycle are less than 1 m s$^{-1}$ [e.g. @dikpa09; @charb10].
The relatively large value of $\alpha_0$ used here, together with the strong shear $\vert {\mbox{\boldmath $\nabla$}}\Omega\vert$ and the efficiency of convective transport, can likely account for the very short cycle period in this young, rapidly-rotating Sun ($\Omega = 3 \Omega_\odot$). Might other parameters produce a 22-yr period comparable to the solar cycle? This remains to be seen. Whether the cycle period is regulated by a dynamo wave or flux transport, the large value of $\alpha_0$ needed to promote cyclic activity ($> 10$ m s$^{-1}$) and the short convective turnover time scale ($\sim $ 20 days) may favor relatively short cycle periods. Longer cycle periods can generally be achieved in mean-field BL dynamo models by reducing the efficiency of poloidal flux transport [e.g @dikpa09; @charb10] but we do not have that freedom here. Here the efficiency of transport is set by the convection, which in turn is an output of the simulation, regulated mainly by factors such as the stellar mass and luminosity that are set by observations. This implies either that the mean field generation and transport by solar convection is much less efficient than in the models considered here (due possibly to dynamical quenching of the turbulent $\alpha$ and $\beta$-effects or overestimation of the convective velocity), or that the solar dynamo does not adhere to a simple BL paradigm.
We thank Mausumi Dikpati and Matthias Rempel for insightful discussions and comments on the manuscript and we thank the anonymous referee for a prompt and constructive report. This work is supported by NASA grants NNH09AK14I (SR&T) and NNX08AI57G (HTP) and computing resources were provided by the NASA High-End Computing (HEC) Program and the NSF-sponsored Teragrid resources at NICS and TACC. B.P. Brown is supported in part by NSF Astronomy and Astrophysics postdoctoral fellowship AST 09-02004. NCAR is sponsored by NSF and CMSO is is supported by NSF grant PHY 08-21899.
[15]{} natexlab\#1[\#1]{}
Babcock, H. W. 1961, , 133, 572
Brown, B. P., Browning, M. K., Brun, A. S., Miesch, M. S., & Toomre, J. 2010, , 711, 424
Brown, B. P., Miesch, M. S., Browning, M. K., Brun, A. S., & Toomre, J. 2011, , 731, 69 (19pp)
Charbonneau, P. 2010, Living Reviews in Solar Physics, 7, http://www.livingreviews.org/lrsp-2010-3
Cheung, M., Rempel, M., Title, A. M., & Schüssler, M. 2010, , 720, 233
Dikpati, M. & Gilman, P. A. 2001, , 559, 428
—. 2009, Space Sci. Rev., 144, 67
Ghizaru, M., Charbonneau, P., & Smolarkiewicz, P. K. 2010, , 715, L133
Guerrero, G. & de Gouveia Dal Pino, E. 2008, Astron. Astrophys., 485, 267
Hotta, H. & Yokoyama, T. 2010, , 714, L308
Leighton, R. B. 1964, , 140, 1547
McIntosh, S. [et al.]{} 2012, in preparation
Nelson, N. J., Brown, B. P., Brun, A. S., Miesch, M. S., & Toomre, J. 2011, , 739, L38 (5pp)
Racine, E., Charbonneau, P., Ghizaru, M., Bouchat, A., & Smolarkiewicz, P. K. 2011, , 735, 46 (22pp)
Rempel, M. 2006, , 647, 662
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abstract: 'Two Kähler metrics on one complex manifold are said to be c-projectively equivalent if their $J$-planar curves, i.e., curves defined by the property that their acceleration is complex proportional to their velocity, coincide. The degree of mobility of a Kähler metric is the dimension of the space of metrics that are c-projectively equivalent to it. We give the list of all possible values of the degree of mobility of simply connected $2n$-dimensional Riemannian Kähler manifolds. We also describe all such values under the additional assumption that the metric is Einstein. As an application, we describe all possible dimensions of the space of essential c-projective vector fields of Kähler and Kähler-Einstein Riemannian metrics. We also show that two c-projectively equivalent Kähler Einstein metrics (of arbitrary signature) on a closed manifold have constant holomorphic curvature or are affinely equivalent.'
address:
- 'Institute of Mathematics, Friedrich-Schiller-Universität Jena, Jena, Germany.'
- 'Institute of Mathematics, Friedrich-Schiller-Universität Jena, Jena, Germany.'
author:
- 'Vladimir S. Matveev and Stefan Rosemann'
nocite: '[@*]'
title: 'Conification construction for Kähler manifolds and its application in c-projective geometry'
---
Introduction
============
The conification construction will be recalled in §\[sec:construction\]. It is a special case of a local construction from [@ACM] that given a $2n$-dimensional Kähler manifold $(M, g, J)$ (of arbitrary signature) produces a $(2n+2)$-dimensional Kähler manifold. If we apply this construction to the Fubini-Study metric $g_{FS}$, we obtain the flat metric.
There are many similar and more general constructions that were described and successfully applied in Kähler geometry before, for example the Calabi construction or the interplay between Sasakian and Kähler manifolds.
The main results of the paper are applications of the construction in the theory of c-projectively equivalent metrics, let us explain what this theory is about.
Let $(M,g,J)$ be a Kähler manifold of real dimension $2n\geq 4$ with Levi-Civita connection $\nabla$. A regular curve $\gamma:I\rightarrow M$ is called *$J$-planar* for $(g,J)$ if $$\begin{aligned}
\left(\nabla_{\dot{\gamma}}\dot{\gamma}\right)\wedge\dot{\gamma}\wedge J\dot{\gamma}=0\label{eq:hplanar}\end{aligned}$$ holds at each point of the curve. The condition can equivalently be rewritten as follows: there exist smooth functions $\alpha(t), \beta(t)$ such that $$\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)= \alpha(t) \cdot \dot \gamma(t) + \beta(t) \cdot J\dot\gamma(t) .$$ Every geodesic is evidently a $J$-planar curve. The set of $J$-planar curves is geometrically a much bigger set of curves than the set of geodesics. For example in every point and in every direction there exist infinitely many geometrically different $J$-planar curves, see figure \[manyhplanar\].
![Given $p\in M$, $X\in T_p M$, there are infinitely many $J$-planar curves $\gamma$ such that $\gamma(0)=p,\dot{\gamma}(0)=X$.[]{data-label="manyhplanar"}](manyhplanar){width=".3\textwidth"}
Two metrics $g,\tilde{g}$ on the complex manifold $(M,J)$ which are Kähler w.r.t. the complex structure $J$ are called *c-projectively equivalent* if their $J$-planar curves coincide. A trivial example of c-projectively equivalent metrics is when the metric $\tilde{g}$ is proportional to $g$ with a constant coefficient. Another trivial (in the sense it is relatively easy to treat it at least in the Riemannian, i.e., positively-definite case) example is when the metrics are *affinely equivalent*, that is when their Levi-Civita connections coincide. If these metrics are Kähler w.r.t. the same complex structure, they are of course c-projectively equivalent since the equation defining $J$-planar curves involves the connection and the complex structure only.
The theory of c-projectively equivalent Kähler metrics is a classical one. It was started in the 50th in Otsuki et al [@Otsuki1954] and for a certain period of time was one of the main research directions of the japanese (Obata, Yano) and soviet (Sinjukov, Mikes) differential geometric schools, see the survey [@Mikes] or the books [@Yanobook; @Sinjukov] for an overview of the classical results. In the recent time, the theory of c-projectively equivalent metrics has a revival. A number of new methods appeared within or were applied in the c-projective setting and classical conjectures were solved. Moreover, c-projectively equivalent metrics independently appeared under the name Hamiltonian $2$-forms, see [@ApostolovI; @ApostolovII; @ApostolovIII; @ApostolovIV], which are also closely related to conformal Killing or twistor $2$-forms studied in [@moruianusemmelmann; @semmelmann]. These relations will be explained in more detail in Section \[sec:reltoothers\]. The c-projectively equivalent metrics also play a role in the theory of (finitely-dimensional) integrable systems [@Kiyo1997] where they are closely related to the so called Kähler-Liouville metrics, see [@Kiyohara2010].
Most classical sources use the name “h-projective” or “holomorphically-projective” for what we call “c-projective” in our paper. We also used “h-projective” in our previous publications [@FKMR; @MatRos]. Recently a group of geometers studying c-projective geometry from different viewpoints decided to change the name from h-projective to c-projective, since a c-projective change of connections, though being complex in the natural sense, is generically not holomorphic. The prefix “c-” is chosen to be reminiscent of “complex-” but is not supposed to be pronounced nor regarded as such since “complex projective” is already used differently in the literature.
As we recall in Section \[sec:basics\], the set of metrics c-projectively equivalent to a given one (say, $g$) is in one-to-one correspondence with the set of nondegenerate hermitian symmetric $(0,2)$-tensors $A$ satisfying the equation $$\begin{aligned}
(\nabla_Z A)(X,Y)=g(Z,X)\lambda(Y)+g(Z,Y)\lambda(X)+\omega(Z,X)\lambda(JY)+\omega(Z,Y)\lambda(JX)\label{eq:mainA}\end{aligned}$$ for all $X,Y,Z\in TM$. Here $\lambda$ is a $1$-form which is easily seen to be equal to $\lambda=\tfrac{1}{4}d\mathrm{trace}\,A$ (here, $A$ is viewed as a $(1,1)$-tensor by “raising one index” with the metric). Since this equation is linear, the space of its solutions is a linear vector space. Its dimension is called the *degree of mobility* of $(g,J)$ and will be denoted by $D(g,J)$. Locally, $D(g,J)$ coincides with the dimension of the set (equipped with its natural topology) of metrics c-projectively equivalent to $g$.
Our main result is the list of all possible degrees of mobility of Riemannian Kähler metrics on simply connected manifolds (in what follows we always assume that simply connectedness implies connectedness). The degree of mobility is always $\ge 1$ since the metrics of the form $\textrm{const}\cdot \ g$ provide a one-parameter family of metrics c-projectively equivalent to $g$. One can show that for a generic metric the degree of mobility is precisely $1$. This statement, though known in folklore, is up to our knowledge nowhere published. Let us mention therefore that by [@Mikes] irreducible symmetric Riemannian Kähler spaces of nonconstant holomorphic curvature have degree of mobility precisely $1$. Using this result one could show, similar to [@hall §3.1], that for any (local) Kähler metric $g$ and every $\epsilon>0$ there exists a Kähler metric $g'$ that is $\varepsilon-$close to $g$ in the $C^\infty$ topology such that there exists $\epsilon'>0$ such that for every metric $g''$ that is $\varepsilon'-$close to $g'$ in the $C^\infty$ topology the degree of mobility of $g''$ is precisely $1$.
\[thm:degree\] Let $(M,g,J)$ be a simply connected Riemannian Kähler manifold of real dimension $2n\geq 4$. Suppose at least one metric c-projectively equivalent to $g$ is not affinely equivalent to it.
Then, the degree of mobility $D(g,J)$ belongs to the following list:
- $2$,
- $k^2+\ell$, where $k=0,...,n-1$ and $\ell=1,...,\big[\tfrac{n+1-k}{2}\big]$.
- $(n+1)^2$,
Moreover, every value from this list that is greater than or equal to $2$ is the degree of mobility of a certain simply connected $2n$-dimensional Kähler manifold $(M, g,J)$ such that there exists a c-projectively but not affinely equivalent metric $\tilde g$.
In the theorem above, $\big[\ .\ \big]$ denotes the integer part. The condition $D(g,J)\ge 2$ is due to the assumption that there exists a metric that is c-projectively but not affinely equivalent (and therefore not proportional) to $g$.
![The possible values for the degree of mobility $D$ for $2\leq\tfrac{1}{2}\mathrm{dim} M\leq 8$. []{data-label="1"}](degree){width=".7\textwidth"}
If all metrics that are c-projectively equivalent to $g$ are affinely equivalent to $g$, then it is not a big deal to obtain in the Riemannian situation the list of all possible degrees of mobility of such metrics on simply connected manifolds by using the same circle of ideas as in the proof of Theorem \[thm:degree\] (see also Section \[sec:ideas\] below): it is $$\{k^2 + \ell \mid k\le n-1, \ 1 \le \ell \le {n-k}\}\bigcup \{ n^2\}.$$
Special cases of Theorem \[thm:degree\] were known before. It is a classical result (see e.g. [@ApostolovI Proposition 4] or [@Sinjukov Chap. V, §3]) that the maximum value $D(g,J)=(n+1)^2$ implies that the metric has constant holomorphic curvature and is attained on simply connected manifolds of constant holomorphic curvature. It was also previously known ([@ApostolovII Proposition 10] and [@FKMR Lemma 6]) that in the case when the dimension is $2n=4$ the degree of mobility (on a simply connected manifold) takes the values $1,2,9$ only. We also see that the submaximal degree of mobility is $(n-1)^2+ 1 = n^2 - 2n +2$. This value was also known before, see [@Mikes §1.2], though we did not find a place where this statement was proved.
Under the additional assumption that the manifold $M$ is closed, the analog of Theorem \[thm:degree\] is also essentially known and the list of possible degrees of mobility is $\{1,2, (n+1)^2\}$. Indeed, as it was shown in [@FKMR], if $D(g,J)\geq 3$, $(M,g,J)$ must be equal to $(\mathbb{C}P(n),\mbox{const}\cdot g_{FS},J_{standard})$ and therefore its degree of mobility is $(n+1)^2$. On the other hand, there are many examples (constructed in [@ApostolovII; @ApostolovIII; @ApostolovIV] or [@Kiyohara2010]) of closed Kähler manifolds different from $(\mathbb{C}P(n),\mbox{const}\cdot g_{FS},J_{standard})$ admitting c-projectively equivalent but not affinely equivalent Riemannian Kähler metrics. Thus, on closed Kähler manifolds (and here the assumption that the manifold is simply connected is not important), $D(g,J)$ takes the values $1,2$ and $(n+1)^2$ only.
The dimension of the space of essential c-projective vector fields
------------------------------------------------------------------
A (possibly, local) diffeomorphism $f:M\rightarrow M$ of a Kähler manifold $(M,g,J)$ is called *c-projective transformation* if it is *holomorphic* (i.e., preserves $J$) and if it sends $J$-planar curves to $J$-planar curves. A clearly equivalent requirement is that the pullback $f^*g$ is c-projectively equivalent to $g$. A c-projective transformation is called *essential*, if it is not an isometry. A vector field is called a *c-projective vector field* if its (locally defined) flow consists of c-projective transformations; we call it essential if it is not a Killing vector field.
For a given Kähler structure $(g,J)$, let $\mathfrak{c}(g,J)$ and $\mathfrak{i}(g,J)$ denote the Lie algebras of c-projective and holomorphic Killing vector fields respectively. Both are linear vector spaces and $\mathfrak{c}(g,J) \supseteq \mathfrak{i}(g,J)$. The quotient vector space $\mathfrak{c}(g,J)/\mathfrak{i}(g,J)$ will be called the *space of essential c-projective vector fields*. Its dimension is $\dim(\mathfrak{c}(g,J))- \dim(\mathfrak{i}(g,J))$. In the proof of Theorem \[thm:hprotrafo\] it will be clear that under the assumption that the degree of mobility is $\ge 3$ this vector space could be (canonically) viewed as a subspace of $\mathfrak{c}(g,J)$. It is not a subalgebra though: typically the commutator of two vector fields from this space is a nontrivial Killing vector field. Let us also mention that the number $\dim(\mathfrak{c}(g,J))- \dim(\mathfrak{i}(g,J))$ remains the same if we replace $g$ by a c-projectively equivalent metric $\tilde g$.
\[thm:hprotrafo\] Let $(M,g,J)$ be a simply connected Riemannian Kähler manifold of real dimension $2n\geq 4$. Suppose at least one metric c-projectively equivalent to $g$ is not affinely equivalent to it.
Then, the dimension of the space $\mathfrak{c}(g,J)/\mathfrak{i}(g,J)$ is
- $0,1$, or
- $k^2+\ell-1$, where $k=0,...,n-1$ and $\ell=1,...,\big[\tfrac{n+1-k}{2}\big]$, or
- $(n+1)^2-1$.
Moreover, each of the values of the above list is equal to the number $\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))$ for a certain $2n$-dimensional simply connected Riemanian Kähler manifold $(M,g,J)$ such that there exists a metric that is c-projectively but not affinely equivalent to $g$.
Under the assumption that the manifold is closed, the analog of Theorem \[thm:hprotrafo\] is again known and is due to [@MatRos] where the classical Yano-Obata conjecture is proved; this conjecture implies that on any closed Riemannian Kähler manifold $\mathrm{dim}(\mathfrak{c}(g,J))=\dim(\mathfrak{i}(g,J))$ unless the manifold is $(\mathbb{C}P(n),\mbox{const}\cdot g_{FS},J_{standard})$ .
Einstein metrics
-----------------
Our next group of results concerns Kähler-Einstein metrics. Note that Einstein metrics play a special important role in the c-projective geometry since they are closely related to the normal sections of the so-called prolongation connection of the metrizability equation, see [@EMN].
The analog of Theorem \[thm:degree\] under the additional assumption that the metric is Einstein is
\[thm:degreeeinstein\] Let $(M,g,J)$ be a simply connected Riemannian Kähler-Einstein manifold of real dimension $2n\geq 4$. Assume at least one metric c-projectively equivalent to $g$ is not affinely equivalent to it. Then, the degree of mobility $D(g,J)$ is equal to one of the following numbers:
- $2$,
- $k^2+\ell$, where $k=0,...,n-2$ and $\ell=1,...,\big[\tfrac{n+1-k}{3}\big]$,
- $(n+1)^2$.
Conversely, each of the numbers of the above list that is greater than or equal to $2$ is the degree of mobility of a certain simply connected Riemannian Kähler-Einstein manifold $(M^{2n},g,J)$ admitting a metric $\tilde g$ that is c-projectively equivalent but not affinely equivalent to $g$.
The analog of Theorem \[thm:hprotrafo\] under the additional assumption that the metric is Einstein looks as follows:
\[thm:hprotrafoeinstein\] Let $(M,g,J)$ be a simply connected Riemannian Kähler-Einstein manifold of real dimension $2n\geq 4$. Suppose at least one metric c-projectively equivalent to $g$ is not affinely equivalent to it.
Then, the dimension of the space $\mathfrak{c}(g,J)/\mathfrak{i}(g,J)$ is
- $0,1$, or
- $k^2+\ell-1$, where $k=0,...,n-2$ and $\ell=1,...,\big[\tfrac{n+1-k}{3}\big]$, or
- $(n+1)^2-1$.
Moreover, each of the values of the above list is equal to $\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))$ for a certain $2n\ge 4$-dimensional simply connected Riemannian Kähler-Einstein manifold $(M,g,J)$ such that there exists a metric that is c-projectively but not affinely equivalent to $g$.
We see that the main difference in the lists of Theorems \[thm:degree\] and \[thm:degreeeinstein\] respectively Theorems \[thm:hprotrafo\] and \[thm:hprotrafoeinstein\] is that in the general case we divide ${n+1-k}$ by $2$ and in the Einstein case we divide ${n+1-k}$ by $3$. The additional difference is that $k$ goes up to $n-1$ in the general case and up to $n-2$ in the Einstein case. As a by-product we also obtain that in dimension $2n=4$, two c-projectively equivalent Riemannian Einstein metrics that are not affinely equivalent must be of constant holomorphic curvature; this result was known before and is in [@haddad1].
Under the assumption that the manifold is closed, the list of the degrees of mobility is again much more simple as the next theorem shows:
\[thm:einstein\] Suppose $g$ and $\tilde{g}$ are c-projectively equivalent Kähler-Einstein metrics of arbitrary signature on a closed connected complex manifold $(M,J)$ of real dimension $2n\geq 4$. Then, $g$ and $\tilde{g}$ are affinely equivalent unless $(M,g,J)$ is $(\mathbb{C}P(n),\mbox{const}\cdot g_{FS},J_{standard})$.
Note that as examples constructed in [@ApostolovII] show, the assumption that the second metric is also Einstein is important for Theorem \[thm:einstein\].
As a by-product, in the proof of Theorems \[thm:hprotrafoeinstein\] and \[thm:einstein\] we obtain the following
\[+2\] [Assume two nonproportional Kähler-Einstein metrics (of arbitrary signature) on a complex $2n\ge 4$-dimensional manifold $(M,J)$ are c-projectively equivalent. Then, any Kähler metric that is c-projectively equivalent to them is also Einstein.]{}
Modulo Theorem \[+2\] and under the additional assumption that $g$ is Riemannian, Theorem \[thm:einstein\] follows from known global results in Kähler-Einstein geometry[^1]. More precisely, if the first Chern class $c_1(M)$ is nonpositive, Theorem \[thm:einstein\] follows from [@Sinjukov2 Theorem 1]. The proof of [@Sinjukov2] is a standard application of the Weitzenböck formula which shows nonexistence of a Hamiltonian Killing vector field, see also [@besse page 89] or [@moroianu page 77] (it is known that nonaffine c-projective equivalence implies the existence of a Hamiltonian Killing vector field, see e.g. [@ApostolovI; @Kiyohara2010]).
If $c_1(M)>0$, then by Bando and Mabuchi [@bandomabuchi] (see also [@besse Addendum D]), for any two Kähler-Einstein metrics $g,\tilde{g}$ on a closed connected complex manifold $(M,J)$, there exists a bi-holomorphism $f:M\rightarrow M$, contained in the connected component of the group of bi-holomorphic transformations of $(M,J)$, such that $f^* g=\textrm{const} \cdot \tilde{g}$. This bi-holomorphism is of course a c-projective transformation of $g$. Now, by Theorem \[+2\], we have a one-parameter family of c-projectively equivalent Einstein metrics which gives us a one-parameter family of such bi-holomorphisms. By the standard rigidity argument, we obtain then the existence a nontrivial (i.e., containing not only isometries) connected Lie group of c-projective transformations and [@MatRos Theorem 1] implies that the manifold is $(\mathbb{C}P(n),\mbox{const}\cdot g_{FS},J_{standard})$.
As a direct corollary of Theorem \[thm:einstein\] we obtain
Let $(M,g,J)$ be a closed Kähler-Einstein manifold of arbitrary signature of real dimension $2n\geq 4$. Then, every c-projective vector field is an affine (i.e., connection-preserving) vector field unless $(M,g,J)=(\mathbb{C}P(n),\mbox{const}\cdot g_{FS},J_{standard})$.
Relation to hamiltonian and conformal Killing $2$-forms {#sec:reltoothers}
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Let $(M,g,J)$ be a Kähler manifold of real dimension $2n$ with Kähler form $\omega=g(.,J.)$. A hermitian $2$-form $\phi$ is called *hamiltonian $2$-form* if $$\begin{aligned}
\nabla_X \phi=X^\flat\wedge J\lambda+(JX)^\flat\wedge\lambda\label{eq:hamiltonian2form}\end{aligned}$$ for a certain $\lambda\in \Omega^1(M)$, where we define $J\lambda=\lambda\circ J$. It is straight-forward to see that $\lambda=\tfrac{1}{4}d\,\mathrm{trace}_\omega \phi$, where $\mathrm{trace}_\omega \phi=\sum_{i=1}^{2n}\phi(Je_i,e_i)$ for an orthonormal frame $e_1,...,e_{2n}$. Hamiltonian $2$-forms where introduced in [@Apostolov0] and have been studied further in [@ApostolovI; @ApostolovII; @ApostolovIII; @ApostolovIV]. Among other interesting results and applications in Kähler geometry [@ApostolovIII; @ApostolovIV], a complete local [@ApostolovI] and global [@ApostolovII] classification of Riemannian Kähler manifolds admitting hamiltonian $2$-forms have been obtained.
Let $\phi$ be a hamiltonian $2$-form and consider the corresponding symmetric hermitian $(0,2)$-tensor $A=\phi(J.,.)$. It is easy to see that this correspondence sends solutions of to solutions of and vice versa, thus hamiltonian $2$-forms and hermitian symmetric solutions of , i.e. c-projectively equivalent Kähler metrics, are essentially the same objects. We immediately obtain from Theorems \[thm:degree\] and \[thm:degreeeinstein\] the next two corollaries.
\[cor:hamilt\] Let $(M,g,J)$ be a simply connected Riemannian Kähler manifold of real dimension $2n\geq 4$. Suppose there exist at least one hamiltonian $2$-form on $M$ that is not parallel.
Then, the dimension of the space of hamiltonian $2$-forms belongs to the following list:
- $2$,
- $k^2+\ell$, where $k=0,...,n-1$ and $\ell=1,...,\big[\tfrac{n+1-k}{2}\big]$.
- $(n+1)^2$,
Moreover, every value from this list that is greater than or equal to $2$ is the dimension of the space of hamiltonian $2$-forms of a certain simply connected $2n$-dimensional Kähler manifold $(M, g,J)$ that admits a non-parallel hamiltonian $2$-form.
\[cor:hamilteinstein\] Let $(M,g,J)$ be a simply connected Riemannian Kähler-Einstein manifold of real dimension $2n\geq 4$. Assume that there exists at least one hamiltonian $2$-form on $M$ that is not parallel. Then, the dimension of the space of hamiltonian $2$-forms is equal to one of the following numbers:
- $2$,
- $k^2+\ell$, where $k=0,...,n-2$ and $\ell=1,...,\big[\tfrac{n+1-k}{3}\big]$,
- $(n+1)^2$.
Conversely, each of the numbers of the above list that is greater than or equal to $2$ is the dimension of the space of hamiltonian $2$-forms of a certain simply connected Riemannian Kähler-Einstein manifold $(M^{2n},g,J)$ that admits a nonparallel hamiltonian $2$-form.
Following [@ApostolovI Appendix A] and [@moruianusemmelmann], we briefly recall the relation between hamiltonian $2$-forms and conformal Killing or twistor $2$-forms studied in [@moruianusemmelmann; @semmelmann]. On a $m$-dimensional Riemannian manifold $(M,g)$ a *conformal Killing or twistor $2$-form* is a $2$-form $\psi$ satisfying
$$\begin{aligned}
\nabla_X \psi=X^\flat\wedge \alpha+i_X \beta\label{eq:defkilling2form}\end{aligned}$$
for certain $\alpha\in \Omega^1(M)$ and $\beta\in \Omega^3(M)$. It is straight-forward to see that $$\beta=\frac{1}{3}d\psi\mbox{ and }\alpha=-\frac{1}{n-1}\delta \psi,$$ where $(\delta \psi)(X)=\sum_{i=1}^m (\nabla_{e_i}\psi)(X,e_i)$ for an orthonormal frame $e_1,...,e_m$.
Now assume that $(M,g,J)$ is a Kähler manifold of real dimension $m=2n$, with Kähler form $\omega=g(.,J.)$. Then, as shown in [@ApostolovI Appendix A] and [@moruianusemmelmann Lemma 3.11], the defining equation for a conformal Killing $2$-form which in addition is assumed to be hermitian, equivalently reads $$\begin{aligned}
\nabla_X \psi=X^\flat\wedge \alpha-(JX)^\flat\wedge J\alpha+J\alpha(X)\omega.\label{eq:defkilling2formherm}\end{aligned}$$
On the other hand, setting $\alpha'=J\lambda$, becomes $$\begin{aligned}
\nabla_X \phi=X^\flat\wedge \alpha'-(JX)^\flat\wedge J\alpha'\label{eq:hamiltonian2form2}\end{aligned}$$ Thus, if $\phi$ is a hamiltonian $2$-form we set $f_\lambda=\tfrac{1}{4}\mathrm{trace}_\omega \phi$ and see that $$\begin{aligned}
\psi=\phi-f_\lambda\omega\label{eq:hamiltoconf}\end{aligned}$$ is a conformal Killing $2$-form with $\alpha=\alpha'$. Conversely, let $\psi$ be a conformal Killing $2$-form. In the case $2n>4$, we have $J\alpha=df_\alpha$, where $f_\alpha=\tfrac{1}{(2n-4)}\mathrm{trace}_\omega\psi$, see [@ApostolovI Appendix A] and [@moruianusemmelmann Lemma 3.8]. Then, $$\begin{aligned}
\phi=\psi-f_\alpha\omega\label{eq:conftohamil}\end{aligned}$$ is a hamiltonian $2$-form with $\alpha'=\alpha$, and the linear mappings in and are inverse to each other. Thus, when $2n>4$, hamiltonian $2$-forms and conformal (hermitian) Killing $2$-forms are essentially the same objects and Corollaries \[cor:hamilt\] and \[cor:hamilteinstein\] also descibe the possible dimensions of the space of hermitian conformal Killing $2$-forms for simply-connected Riemannian Kähler respectively Riemannian Kähler-Einstein manifolds of dimension $2n>4$ that admit a non-parallel conformal Killing $2$-form.
Relation to parallel $(0,2)$-tensors on the conification and circles of ideas used in the proof {#sec:ideas}
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As we mentioned above and will recall later, the system of equations (such that the dimension of the space of solutions is the degree of mobility) is linear and overdetermined. In theory, there exist algorithmic (sometimes called prolongation-projection or Cartan-Kähler) methods to understand the dimension of the space of solutions of such a system. In practice, these methods are effective in small dimensions only, or return the maximal and submaximal values for the possible dimension of the space of solutions. Actually, the previously known results are obtained by versions of these methods.
A key observation that allowed us to solve the problem in its full generality is that the problem can be reduced to the following geometric one:
[*Find all possible dimensions of the spaces of parallel symmetric hermitian (0,2)-tensors on conifications of Kähler manifolds.*]{}
The reduction goes as follows. First of all, at least for the proof of Theorem \[thm:degree\], we may assume that the degree of mobility of $(M^{2n},g,J)$ is $\ge 3$ (since otherwise it is $2$ and this case is clear).
Then, as we recall in Theorem \[thm:hprosystem\], the solutions of the system are in one-one correspondence with the solutions of the system .
Since the construction of the conification $(\hat M^{2n+2}, \hat g, \hat J)$ of $(M^{2n},g,J)$ in general only works locally (we explain this in detail in Section \[sec:construction\] and Theorem \[thm:coneconstruction1\]) we apply it assuming that our manifold is diffeomorphic to a disc. The extension of the results to simply connected manifolds is then a standard application of the theorem of Ambrose-Singer and will be explained in Section \[sec:proofthmdegreeglobal\].
Consider now the number $B$ from . If $B=-1$, then the solutions of are essentially parallel symmetric hermitian $(0,2)$-tensors on $(\hat M, \hat g, \hat J)$, see Theorem \[thm:coneconstruction2\]. If $B\ne -1$ but $B\ne 0$, one can always make $B=-1$ by an appropriate scaling of the metric $g$. If the initial constant $B$ was positive, then $\hat{g}$ has signature $(2,2n)$.
The case $B=0$ requires the following additional work: we show that on any simply connected neighborhood such that its closure is compact there exists a c-projectively equivalent Riemannian metric with $B\ne 0$, see Section \[sec:mainfeatures\]. For this new metric the above reduction works. Evidently, c-projectively equivalent metrics have the same degrees of mobility. Thus, also in this case, instead of solving the initial problem, we study the possible dimensions of the space of parallel symmetric hermitian $(0,2)$-tensors on the conification of a Kähler manifold.
To do this, let us first assume, for simplicity and since the ideas will be already visible in this setting, that $B$ is negative so the conification we will work with is a Riemannian manifold. In the final proof we will assume that the signature is $(2,2n)$ which poses additional difficulties. In order to calculate the dimension of the space of parallel symmetric hermitian $(0,2)$-tensors on the conification $(\hat M, \hat g, \hat J)$, let us consider the (maximal orthogonal) holonomy decomposition of the tangent space at a certain point of $\hat M$: $$\begin{aligned}
T_p\hat M=T_0\oplus T_1\oplus...\oplus T_\ell\label{eq:decomptangenttrivial-1}.\end{aligned}$$ where $T_0$ is flat (in the sense that the holonomy group acts trivially on it) and all other $T_i$ are irreducible. Clearly, each $T_i$ is $\hat J$-invariant so it has even dimension $2k_i$.
It is well known, at least since de Rham [@DR], that parallel symmetric $(0,2)$-tensor fields on a Riemannian manifold $\hat M$ are in one-one correspondence with the $(0,2)$-tensors on $T_p \hat M$ of the form $$\label{presentation}
\sum_{i,j=1}^{2k_0}c_{ij} \tau_i \otimes \tau_j + C_1 g_1 + ...+ C_\ell g_\ell,$$ where $\{\tau_i\}$ is a basis in $T_0^*$ and $g_i$ is the restriction of $\hat g$ to $T_i$. The assumption that the tensor is symmetric and hermitian implies that the $2k_0\times 2k_0$-matrix $c_{ij}$ is symmetric and is hermitian w.r.t. to the restriction of $\hat J$ to $T_0$. This gives us a $k_0^2 $-dimensional space of such tensors. If $\hat M$ is flat, i.e. if the initial metric $g$ has constant holomorphic curvature, we have $k_0= n+1$ (=half of the dimension of the conification of our $2n$-dimensional $M$) which gives us the number $(n+1)^2$. Suppose now that our manifold is not flat so $\ell\ge 1$. We show that the dimension of each $T_i$ for $i\ge 1$ is $\ge 4$ (i.e., that $k_i\ge 2$). The key observation that is used here is that each $T_i$ (or, more precisely, the restriction of $\hat g$ to the integral leafs of the integrable distribution $T_i$) is a cone manifold. Since it is not flat, it has dimension $>2$ and since the dimension is even it must be $\ge 4$. Then, $k=0,...,n-1$ and $\ell$ is at most $ \big[\tfrac{n+1-k}{2}\big]$, because the sum of the dimensions of all $T_i$ with $i\ge 1$ is $2(n+1)- 2k_0$ and each $T_i$ “takes” at least four dimensions, see .
Now, in the case our manifold is Einstein, we show that the dimension of each $T_i$ is $\ge 6$. This statement is due to the nonexistence of Ricci-flat but nonflat cones of dimension $4$. Then, $k=0,...,n-2$ and $\ell$ is at most $ \big[\tfrac{n+1-k}{3}\big]$, which gives us the list from Theorem \[thm:degreeeinstein\].
In the case the conification of the manifold has signature $(2,2n)$, we use recent results of [@FedMat Theorem 5], where an analog of was proven under the additional assumption that our manifold of signature $(2,2n)$ is a cone manifold (and conifications of Kähler manifolds of dimension $2n$ are indeed cone manifolds over $2n+1$-dimensional manifolds). The additional work is required though since the number $k$ in this case is not necessary $\tfrac{\dim(T_0)}{2}$ but could also be $\tfrac{\dim(T_0)}{2} + 1$. In the latter case we show that one of $T_i$ with $i\ge 1$ must necessary have dimension $\ge 6$ (resp. $\ge 8$ in the Einstein case) and the list of possible dimensions of the space of parallel symmetric hermitian $(0,2)$-tensors remains the same.
Let us now touch the proof of Theorems \[thm:hprotrafo\] and \[thm:hprotrafoeinstein\]. The restriction $\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))\le D(g,J)-1$ is straightforward. Now, in the case $D(g,J)\geq 3$, under the additional assumption that the constant $B\ne 0$, we actually have $$\label{-2} \mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))= D(g,J)-1.$$
Indeed, for every solution of we canonically construct a c-projective vector field. Two solutions of give the same c-projective vector field if and only if their difference is a multiple of $g$. The formula gives us the lists from Theorems \[thm:hprotrafo\] and \[thm:hprotrafoeinstein\]. The case when $B=0$ can be reduced to the case $B\ne 0$ by the same trick as in the proofs of Theorems \[thm:degree\] and \[thm:degreeeinstein\].
Organization of the paper
-------------------------
In Section \[sec:basics\] we recall basic statements in c-projective geometry that were proved before and that will be used in the proof.
In Section \[sec:construction\], we describe the construction of the conification $(\hat{M},\hat{g},\hat{J})$ of a Kähler manifold $(M,g,J)$ and will show that solutions to the system with $B=-1$ on $M$ correspond to parallel hermitian symmetric $(0,2)$-tensors on $\hat{M}$.
Section \[sec:mainfeatures\] is a technical one, its first goal is to explain that w.l.o.g. we can assume that $B$ in is equal to $-1$. The only nontrivial step here is in Section \[sec:proofoflemma\] and is as follows: if for the initial metric $B=0$, we change the metric to a c-projectively equivalent one such that $B\neq 0$ for the new metric. The second goal of Section \[sec:mainfeatures\] is to prove the additional statement concerning the case of an Einstein metric. Roughly speaking, one goal is to show that when we change the metric to make $B\ne 0$, the metric we obtain is still Einstein. Another goal is to show that the conification construction applied to Kähler-Einstein metrics gives a Ricci-flat metric.
In Section \[sec:proofs\] we prove the Theorems \[thm:degree\] and \[thm:degreeeinstein\]. In Section \[sec:proofthmdegree\] we will essentially calculate the possible dimensions of the space of parallel symmetric hermitian $(0,2)$-tensors on Kähler manifolds that arise from the conification construction. This will complete the proofs of the Theorems \[thm:degree\] and \[thm:degreeeinstein\] in the local situation. In Section \[sec:proofthmdegreeglobal\] we will extend our local results to the global situation. In Section \[sec:realization\] we complete the proof of Theorem \[thm:degree\] respectively Theorem \[thm:degreeeinstein\] and show that each of the values in these theorems is the degree of mobility of a certain Kähler metric respectively Kähler-Einstein metric.
In Section \[sec:thmhprotrafo\] we prove Theorems \[thm:hprotrafo\] and \[thm:hprotrafoeinstein\] and in the final Section \[sec:thmeinstein\] we prove Theorem \[thm:einstein\].
Basic facts in the theory of c-projectively equivalent metrics {#sec:basics}
==============================================================
C-projective equivalence of Kähler metrics as a system of PDE
-------------------------------------------------------------
Let $g$ and $\tilde{g}$ be two Kähler metrics on the complex manifold $(M,J)$ of real dimension $2n\geq 4$.
It is well-known, see [@Tashiro1956 equation (1.7)], that $g$ and $\tilde{g}$ are c-projectively equivalent if and only if for a certain $1$-form $\Phi$, the Levi-Civita connections $\nabla,\tilde{\nabla}$ of $g,\tilde{g}$ respectively satisfy $$\begin{aligned}
\tilde{\nabla}_X Y-\nabla_X Y=\Phi(X)Y+\Phi(Y)X-\Phi(JX)JY-\Phi(JY)JX\label{eq:lchpro}\end{aligned}$$ for all vector fields $X,Y$. In the tensor index notation, reads $$\label{M1}
\tilde \Gamma^i_{jk} - \Gamma^i_{jk}= \delta^i_j \Phi_k + \delta^i_k \Phi_j - J^i_{\ j} \Phi_sJ^s_{\ k} - J^i_{\ k} \Phi_s J^s_{\ j}.$$
Actually, the $1$-form $\Phi$ is exact, i.e., it is the differential of a function, and the function is explicitly given in terms of $g$ and $\tilde g$. Indeed, contracting w.r.t. $i$ and $k$, we obtain $$\tfrac{1}{2}\partial_i\ln \left(\frac{\det \tilde g}{ \det g}\right)= 2(n+1) \Phi_i$$ so $\Phi_i= \phi_{,i} = d\phi$ for the function $\phi$ given by $$\label{M2}
\phi:= \tfrac{1}{4(n+1)}\ln \left(\frac{\det \tilde g}{\det g}\right).$$
The equation allows us to reformulate the condition that the metrics $g$ and $\tilde g$ are c-projectively equivalent as a system of PDE on the components of $\tilde g$ whose coefficients depend on $g$ (and its derivatives). Indeed, in view of , the condition $\tilde{\nabla} \tilde{g}=0$ which is the defining equation for $\tilde \nabla$ reads $$\label{eisenhart}
\nabla_Z \tilde{g}(X,Y) - \Phi(X) \tilde{g}(Z,Y)- \Phi(Y) \tilde{g}(Z,X) - \Phi(JX) \tilde{\omega}(Z,Y) - \Phi(JY) \tilde{\omega}(Z,X)- 2 \Phi(Z) \tilde{g}(X,Y)=0,$$ where we denote by $\tilde{\omega}$ the Kähler $2$-form $\tilde{\omega}=\tilde{g}(.,J.)$
We will view this condition as a system of PDE on the entries of $\tilde g$ whose coefficients depend on the entries of $g$. This system of equations is nonlinear (since the entries of $\Phi$ depend algebraically on the entries $\tilde g$). A remarkable observation by Mikes et al [@DomMik1978] is that one can make this system linear by a clever substitution. For this, consider the symmetric hermitian $(0,2)$-tensor $A$ and the $1$-form $\lambda$ given by $$\begin{aligned}
A&=&A(g,\tilde{g})=\left(\frac{\mathrm{det}\,\tilde{g}}{\mathrm{det}\,g}\right)^{\tfrac{1}{2(n+1)}}g\tilde{g}^{-1}g.\label{eq:defA} \\
\lambda& =& -\Phi g^{-1} A. \label{eq:relationphilambda}\end{aligned}$$ Here $g^{-1}$ is viewed as an isomomorphism $g^{-1}:T^*M\to TM$ given by the condition $g(g^{-1}\psi, Y):=\psi(Y) $ for all $Y$ and $A$ is viewed as a mapping $A:TM\to T^*M$ given by the condition $AX(Y)= A(X,Y)$. If we view $(0,2)$-tensors as matrices and $(0,1)$-tensors as $2n$-tuples, the matrix of $A$ (up to multiplying it with the scalar expression involving the determinants of the metrics) is the product of the matrices $g$, $\tilde g^{-1}$, and again $g$. The 1-form $\lambda$ in the matrix-notation is minus the product $\Phi$, $g^{-1}$ and $A$. Using index notation, $$A_{ij} = \left(\frac{\mathrm{det}\,\tilde{g}}{\mathrm{det}\,g}\right)^{\tfrac{1}{2(n+1)}} g_{is} \tilde g^{sr} g_{rj} \textrm{ and $\lambda_i = -\Phi_s g^{sr} A_{r i}$ } ,$$ where $\tilde g^{ij}$ is the inverse $(2,0)$-tensor to $\tilde g_{ij}$.
Straightforward calculations show that the condition is equivalent to the formula . In index notation, the equation reads $$\label{Mbasic}
a_{ij,k} = \lambda_i g_{jk}+ \lambda_j g_{ik}+ J^s_{ \ j}\lambda_s J_{ik}+ J^s_{ \ i}\lambda_s J_{jk}.$$ Note that contracting the equation with $g^{ij}$ we obtain that the one-form $\lambda$ is the differential of the function $\tfrac{1}{4}\mathrm{trace}_g(A)$. In view of this, could be viewed as a linear system of PDE on the entries of $A$ only.
Note also that the formula is invertible: the tensor $A$ given by is nondegenerate, and the metric $\tilde g$ is given in the terms of $g$ and $A$ by $$\begin{aligned}
\tilde g=\tfrac{1}{\sqrt{\det(A)}} gA^{-1}g.\label{eq:inverse}\end{aligned}$$
\[rem:affine\] The metrics $g,\tilde{g}$ are affinely equivalent, if and only if the tensor $A$ in is parallel, i.e., if and only of $\lambda$ in is identically zero.
\[rem:metricsol\] Since is linear and the metric $g$ is always a solution of , for every solution $A$ of , the tensor $A+\mathrm{const}\cdot g$ is again a solution. Thus, if $A$ is degenerate, we can choose (at least locally) the constant such that $A+\mathrm{const}\cdot g$ is a non-degenerate solution and, hence, corresponds to a metric $\tilde{g}$ that is c-projectively equivalent to $g$.
Let us denote by $\mathcal{A}(g,J)$ the linear space of symmetric hermitian solutions $A$ of . The *degree of mobility $D(g,J)$* of a Kähler structure $(g,J)$ is the dimension of $\mathcal{A}(g,J)$.
\[rem:isomorphism\] If the metrics $g,\tilde{g}$ are c-projectively equivalent, the spaces $\mathcal{A}(g,J)$ and $\mathcal{A}(\tilde{g},J)$ are isomorphic and hence, $D(g,J)=D(\tilde{g},J)$. This statement is probably expected and evident, the proof can be found for example in [@MatRos Lemma 1].
The results recalled above are rather classical. The next result [@FKMR Theorem 3] is a recent one and plays a key role in our paper.
\[thm:hprosystem\] Let $(M,g,J)$ be a connected Kähler manifold of degree of mobility $D(g,J)\geq 3$ and of real dimension $2n\geq 4$. Then, there exists a unique constant $B$ such that for every $A\in \mathcal{A}(g,J)$ with the corresponding $1$-form $\lambda$ there exists the unique function $\mu$ such that $(A,\lambda,\mu)$ satisfies $$\begin{aligned}
\begin{array}{c}
(\nabla_Z) A (X,Y)=g(Z,X)\lambda(Y)+g(Z,Y)\lambda(X)+\omega(Z,X)\lambda(JY)+\omega(Z,Y)\lambda(JX)\vspace{1mm}\\
(\nabla_Z \lambda) (X)=\mu g(Z,X)+BA(Z,X)\vspace{1mm}\\
\nabla_Z \mu =2B\lambda(Z).
\end{array}\label{eq:hprosystem}\end{aligned}$$ for all vector fields $X,Y,Z$.
Conification construction and solutions of as parallel tensors on the conification {#sec:construction}
==================================================================================
Let $(M,g,J)$ be a Kähler manifold of arbitrary signature with Kähler $2$-form $\omega=g(.,J.)$. We explicitly allow $\dim M = 2n= 2$ here. We also suppose that the form $\omega$ is an exact form, $\omega=d\tau$. This is always true if $H^2(M,\mathbb{R})=0$ and in particular, it is always true if the manifold is diffeomorphic to the ball. This is sufficient for our purposes since we will apply the conification construction only to subsets of such kind.
We consider the manifold $P=\mathbb{R}\times M,$ where $t$ will denote the standard coordinate on $\mathbb{R}$ and the natural projection to $M$ is denoted by $\pi:P \to M$.
On $P$, we define the $1$-form $$\theta=dt-2\tau,$$ where for readability we omit the symbol for the pullback of $\tau$ to $P$ (actually, the formula above should be $\theta= dt - 2 \pi^*\tau$). We will also omit the symbols of the $\pi$-pullback in all formulas below so if in some formula we sum or compare a $(0,k)$-tensor defined on $M$ with a $(0,k)$-tensor defined on $P$, the tensor on $M$ should be pulled back to $P$ by $\pi$.
Clearly, $d\theta=-2d\tau=-2\omega$. Let us define the metric $h$ on $P$ by $$h=\theta^2+g \ \ (\textrm{where $\theta^2 =\theta\otimes\theta$}).$$
\[rem:changeoftau\] The freedom in the choice of $\tau$ is not essential for us since (as it is straightforward to check) the change $\tau\longmapsto \tilde{\tau}=\tau+df$ yields a metric $\tilde{h}$ that is isometric to $h$: the mapping $\phi(t,p)=(t-2f(p),p)$ satisfies $\phi^*\theta=\tilde{\theta}$ and hence, $\phi^* h=\tilde{h}$.
Further, let us denote by $\mathcal{H}=\mathrm{kern}(\theta)$ the “horizontal” distribution of $P$ defined by $\theta$. For any vector field $X$ on $M$ we define its *horizontal lift* $X^\theta$ on $P$ by the properties $$\begin{aligned}
\pi_{*}(X^\theta)= X \textrm{ and } \theta(X^\theta)=0.\label{eq:horizontallift}\end{aligned}$$
We consider now the *cone* over $(P,h)$, that is we consider the $2(n+1)$-dimensional manifold $(\hat{M}=\mathbb{R}_{>0}\times P,\hat{g}=dr^2+r^2h)$. Let us denote by $\hat \pi:\hat M\to M $ the natural projection $(r,t,p)\mapsto p \in M$. The kernel of the differential of this projection is spanned by $\xi=r\partial_r$, which will be called the *cone vector field*, and $\eta=\partial_t$. In the literature on Sasaki manifolds, $\eta$ is sometimes refered to as Reeb vector field.
Next, we introduce an almost complex structure $\hat{J}$ on $\hat M$ (which later appears to be a complex structure) by the formula $$\hat{J}\xi=\eta, \hat{J}\eta=-\xi,\ \mbox{ and } \hat{J}X^{\theta}=(JX)^{\theta}.$$
We will call the $2(n+1)$-dimensional manifold $(\hat{M},\hat{g},\hat{J})$ the *conification* of $(M,g,J)$.
We took the name “conification” from the recent paper [@ACM] of Alekseevsky et al where this construction has been obtained in a more general situation. The relation between the two constructions is explained in [@ACM Example 1].
We have
\[thm:coneconstruction1\] Let $(M,g,J)$ be a Kähler manifold of real dimension $2n\geq 2$ and suppose that the Kähler $2$-form $\omega=g(.,J.)$ satisfies $\omega=d\tau$ for a certain $1$-from $\tau$. Then the conification $(\hat{M},\hat{g},\hat{J})$ is a Kähler manifold.
Conversely, suppose $(\hat{M},\hat{g},\hat{J})$ is a Kähler manifold which locally, in a neighborhood of every point of a dense open subset, is a cone, i.e. $(\hat{M},\hat{g})$ is of the form $(\hat{M}=\mathbb{R}_{>0}\times P,\hat{g}=dr^2+r^2h)$ for a certain pseudo-Riemannian manifold $(P,h)$. Then, $(\hat{M},\hat{g},\hat{J})$ arises locally as the conification of its Kähler quotient $(M,g,J)$.
The Kähler quotient is taken with respect to the action of the hamiltonian Killing vector field $\eta=\hat{J}\xi$, where $\xi=r\partial_r$ is the cone vector field. This will be explained in more detail in the proof of Theorem \[thm:coneconstruction1\].
We could also assume the less restrictive condition that the Kähler class $[\omega]\in H^{2}(M,\mathbb{R})$ is integer, and hence, $\omega$, up to scale, is the curvature of a certain connection one-form $\theta$ on some $S^1$-bundle $P$ over $M$ as in [@ACM]. Then, the metrics $h$ on $P$ and $\hat{g}$ on the cone $\hat{M}$ over $P$ can be defined in the same way as above and the proof of Theorem \[thm:coneconstruction1\] will be literally the same as the proof that we will give below. In this more general situation, the role of $\eta=\partial_t$ is then played by the fundamental vector field $\eta$ of the $S^1$-action on $P$ that satisfies $\theta(\eta)=1$.
Actually from the construction it is immediately clear that $\hat J^2 = -{\mathrm{Id}}$. It is also straight-forward to check that $\hat g$ is hermitian w.r.t. $\hat J$ by checking the condition $\hat{g}(\hat{J}u,\hat{J}v)=\hat{g}(u,v)$ for all possible combinations of tangent vectors $u,v$ of the form $\xi,\eta$ and $X^\theta$. What remains is to show that $\hat J$ is parallel with respect to the Levi-Civita connection of $\hat g$. This will be done in Section \[sec:proof of coneconstruction1\].
Let us explain how the system on $M$ relates to the parallel $(0,2)$-tensors on $\hat{M}$.
\[thm:coneconstruction2\] Let $(M,g,J)$ be a Kähler manifold of real dimension $2n\geq 4$, suppose that the Kähler $2$-form $\omega=g(.,J.)$ satisfies $\omega=d\tau$ for a certain $1$-from $\tau$. Then, there exists an isomorphism between the space of solutions $(A,\lambda,\mu)$ of with $B=-1$ and the space of parallel symmetric hermitian $(0,2)$-tensors $\hat{A}$ on $(\hat{M},\hat{g},\hat{J})$. The isomorphism is explicit and is given by $$\begin{aligned}
(A,\lambda,\mu)\leftrightarrow \hat{A}=\mu dr^2-r dr\odot\lambda+r^2(\mu\theta^2+\theta\odot\lambda(J.)+A),\label{eq:Acccone}\end{aligned}$$ where we omit the symbol for the $\hat \pi$-pullback of $\mu,\lambda,\lambda(J.)$ and $A$ to $\hat{M}$. In the formula , $X\odot Y=X\otimes Y+Y\otimes X$ is the symmetric tensor product.
If we use $\tilde{\tau}=\tau+df$ instead of $\tau$ to construct the conification, the diffeomorphism $\phi:\hat{M}\rightarrow \hat{M}$ given by $\phi(r,t,p)=(r,t-2f(p),p)$ (compare also Remark \[rem:changeoftau\]) satisfies $\phi^* \theta=\tilde{\theta}$. Thus, $\phi$ sends $\hat{g}$, the Kähler $2$-form $\hat{\omega}=\hat{g}(.,\hat{J}.)=r\theta\wedge dr+r^2\omega$ and $\hat{A}$ given by to the corresponding objects constructed by using $\tilde{\tau}$.
The proof of theorems \[thm:coneconstruction1\] and \[thm:coneconstruction2\] is by direct calculations and will be done in Sections \[sec:proof of coneconstruction1\] and \[sec:isom\]. The proof of Theorem \[thm:coneconstruction1\] is contained in [@ACM] and will be given for self-containedness and because we will need all the formulas from the proof later on.
Proof of Theorem \[thm:coneconstruction1\]. {#sec:proof of coneconstruction1}
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Recall that for a vector field $X$ on $M$, the natural lift of $X$ to the horizontal distribution $\mathcal{H}=\mathrm{kern}(\theta)\subseteq TP$ will be denoted by $X^{\theta}$, see equation .
Let $X,Y$ denote vector fields on $M$. The Levi-Civita connection ${\tensor[^h]{\nabla}{}}$ of the metric $h$ on $P$ is given by the fomulas $$\begin{aligned}
{\tensor[^h]{\nabla}{}}_{\eta}\eta&=0,\,\,\,{\tensor[^h]{\nabla}{}}_{\eta}X^{\theta}={\tensor[^h]{\nabla}{}}_{X^{\theta}}\eta=(JX)^{\theta},\,\,\,{\tensor[^h]{\nabla}{}}_{X^{\theta}}Y^{\theta}=(\nabla_X Y)^{\theta}+\omega(X,Y)\eta
\label{eq:connsasaki}\end{aligned}$$
Let $u,v,w$ be vector fields on $P$. Using the Koszul formula $$2h({\tensor[^h]{\nabla}{}}_{u}v,w)=uh(v,w)+vh(w,u)-wh(u,v)-h(u,[v,w])+h(v,[w,u])+h(w,[u,v])$$ we calculate $$2h({\tensor[^h]{\nabla}{}}_{\eta}\eta,\eta)=0\mbox{ and }2h({\tensor[^h]{\nabla}{}}_{\eta}\eta,X^{\theta})=0,$$ thus, ${\tensor[^h]{\nabla}{}}_{\eta}\eta=0$ as we claimed. Now we calculate $2h({\tensor[^h]{\nabla}{}}_{\eta}X^{\theta},\eta)=0$ and $$2h({\tensor[^h]{\nabla}{}}_{\eta}X^{\theta},Y^{\theta})=-h(\eta,[X^{\theta},Y^{\theta}])=-\theta([X^{\theta},Y^{\theta}])=d\theta(X^{\theta},Y^{\theta})=-2\omega(X,Y).$$ This shows that ${\tensor[^h]{\nabla}{}}_{\eta}X^{\theta}=(JX)^{\theta}$.
To verify the last equation in , we calculate $$2h({\tensor[^h]{\nabla}{}}_{X^{\theta}}Y^{\theta},\eta)=h(\eta,[X^{\theta},Y^{\theta}])=2\omega(X,Y)\mbox{ and }2h({\tensor[^h]{\nabla}{}}_{X^{\theta}}Y^{\theta},Z^{\theta})=2g(\nabla_X Y,Z).$$ Combining these two equations gives us the third equation in .
The formulas for the Levi-Civita connection $\hat{\nabla}$ of the Riemannian cone $(\hat{M}=\mathbb{R}_{>0}\times P,\hat{g}=dr^2+r^2 h)$ over a pseudo-Riemannian manifold $(P,h)$ are given by $$\begin{aligned}
\hat{\nabla}\xi={\mathrm{Id}},\,\,\,\hat{\nabla}_{X}Y=\nabla_{X}Y-h(X,Y)\xi,\label{eq:LCcone}\end{aligned}$$ where $\xi=r\partial_r$ and $X,Y$ are vector fields on $P$. This is well-known, see for example [@Mounoud2010 Fact 3.2]. For $(P=\mathbb{R}\times M,h=\theta^2+g)$ as above we can combine these formulas with to obtain $$\begin{aligned}
\begin{array}{c}
\hat{\nabla}\xi={\mathrm{Id}},\,\,\,\hat{\nabla}\eta=\hat{J},\,\,\,\hat{\nabla}_{\xi}X^\theta=X^\theta,\,\,\,\hat{\nabla}_{\eta}X^\theta=\hat{J}X^\theta,\vspace{1mm}\\
\hat{\nabla}_{X^\theta}Y^\theta=(\nabla_{X}Y)^\theta+\omega(X,Y)\eta-g(X,Y)\xi.
\end{array}\label{eq:LCconncone}\end{aligned}$$ Using these equations, it is easy to check that $\hat{\nabla}\hat{J}=0$. A straight-forward way to do it, is to show that the equation $\hat{\nabla}_u (\hat{J}v)=\hat{J}\hat{\nabla}_u v$ is fulfilled for $u,v$ of the form $\xi,\eta$ and $X^\theta$.
Since $\hat g$ is evidently symmetric, nondegenerate and hermitian with respect to $\hat J$, the conification $(\hat{M},\hat{g},\hat{J})$ is a Kähler manifold as we claimed. This completes the proof of the first statement of Theorem \[thm:coneconstruction1\].
The other direction of Theorem \[thm:coneconstruction1\] immediately follows from
\[lem:coneconstruction1b\] Suppose $(\hat{M}^{2n+2},\hat{g},\hat{J})$ is a Kähler manifold which is locally, in a neighborhood of every point of a dense open subset, a cone, i.e. $(\hat{M},\hat{g})$ is of the form $(\hat{M}=\mathbb{R}_{>0}\times P,\hat{g}=dr^2+r^2h)$ for a certain $(2n+1)$-dimensional pseudo-Riemannian manifold $(P,h)$. Then, $(\hat{M},\hat{g},\hat{J})$ is locally the conification of its Kähler quotient $(M^{2n},g,J)$, where the quotient is taken w.r.t. the action of the hamiltonian Killing vector field $r\hat{J}\partial_r$.
We work on an open subset of $\hat{M}$ such that on this subset, $(\hat{M},\hat{g})$ is of the form $(\hat{M}=\mathbb{R}_{>0}\times P,\hat{g}=dr^2+r^2h)$. Consider the vector fields $\xi= r\partial r$ and $\eta=\hat{J}\xi$. Since $\eta $ is orthogonal to $\xi$, the derivative of $r$ in the direction of $\eta$ is zero and therefore $\partial_r$ commutes with $\eta$. Consequently, $\eta$ is essentially a vector field on the manifold $P$, i.e. in a coordinate system $(r,x^1,...,x^{2n+1})$, where $x^1,...,x^{2n+1}$ denote coordinates on $P$, the $\partial_r$-component of $\eta$ is zero and the $\partial_{x^i}$-components do not depend on $r$.
Inserting $\eta$ into the metric $\hat{g}$, we see that $h(\eta,\eta)=1$ and since $\eta$ is a Killing vector field for $\hat{g}$ it follows that $\eta$ is also Killing for $h$.
Let us take the quotient of $P$ by the action of the local flow of $\eta$, to obtain a (local quotient) bundle $\pi:P\rightarrow M$, where $M$ is a manifold of real dimension $2n$. Since we are working locally anyway, this bundle can be viewed as $P=\mathbb{R}\times M$ with coordinate $t$ on the $\mathbb{R}$-component such that $\eta=\partial_t$. Further, we introduce the $1$-form $\theta=h(\eta,.)$ on $P$ and denote by $\mathcal{H}=\mathrm{kern}\,\theta$, the horizontal distribution. Since $\eta$ is Killing, this distribution is invariant with respect to the action of the flow of $\eta$. For tangent vectors $X,Y\in T_p M$, we denote by $X^\theta,Y^\theta \in \mathcal{H}$ their horizontal lifts to a certain point in $\pi^{-1}(p)\subseteq P$. Defining $$g(X,Y)=h(X^\theta,Y^\theta),$$ we see that the right-hand side does not depend on the choice of the base point of $X^\theta,Y^\theta$ in $\pi^{-1}(p)$, hence $g$ defines a Riemannian metric on $M$.
Consider the endomorphism $J'={\tensor[^h]{\nabla}{}}\eta:TP\rightarrow TP$. From it immediately follows that $$J'\eta={\tensor[^h]{\nabla}{}}_\eta \eta=\hat{\nabla}_\eta \eta-h(\eta,\eta)\xi=\hat{J}\eta+\xi=0.$$ In the same way, we obtain $$J'X^\theta={\tensor[^h]{\nabla}{}}_{X^\theta}\eta=\hat{\nabla}_{X^\theta}\eta=\hat{J}X^\theta.$$
Thus, we have that $J':\mathcal{H}\rightarrow \mathcal{H}$ defines an almost complex structure and by setting $(JX)^\theta=J'X^\theta$, we obtain an almost complex structure $J:TM\rightarrow TM$ on $M$ which is indeed independent of the choice of base point of the lift $X^\theta$ of $X\in T_p M$, since the flow of $\eta$ preserves $\hat{J}$.
Using the definition of $(g,J)$ and the fact that $\eta$ is a Killing vector field for $h$, we obtain $$g(X,JY)=h(X^\theta,{\tensor[^h]{\nabla}{}}_{Y^\theta}\eta)=-\frac{1}{2}d\theta(X^\theta,Y^\theta).$$
On the one hand, this shows that $\omega=g(.,J.)$ is a $2$-form or equivalently, that $g$ is hermitian with respect to $J$, i.e., $g(J.,J.)=g$. On the other hand, since $d\theta$ is horizontal in the sense that it vanishes when $\eta$ is inserted, this shows that $d\theta=-2\pi^{*}\omega$. From this, it also follows that $\omega$ is closed.
Since $\theta([X^\theta,Y^\theta])=-d\theta(X^\theta,Y^\theta)=2\omega(X,Y)$, we obtain that $[X,Y]^\theta=[X^\theta,Y^\theta]-2\omega(X,Y)\eta$. Using this, it is straight-forward to see that the Nijenhuis torsion $$N_{J}(X,Y)=[X,Y]-[JX,JY]+J[JX,Y]+J[X,JY]$$ of $J$ lifts to the corresponding Nijenhuis torsion of $\hat{J}$ which is vanishing, more precisely $$(N_{J}(X,Y))^\theta=N_{\hat{J}}(X^\theta,Y^\theta)=0.$$ Thus, $J$ is integrable and $(M,g,J)$ is a Kähler manifold. From our construction, it is clear that $(\hat{M},\hat{g},\hat{J})$ coincides with the conification of $(M,g,J)$. This completes the proof of the lemma.
The construction of $g$ from $\hat{g}$ as presented in Lemma \[lem:coneconstruction1b\] is of course well-known and coincides with the Kähler quotient of $(\hat{M},\hat{g},\hat{J})$ w.r.t. the action of the hamiltonian Killing vector field $\eta$, see [@Hitchin] for a short explanation of Kähler quotients and symplectic reduction. The fact that the conification procedure can be reversed by taking the Kähler quotient was also mentioned in [@ACM].
Proof of Theorem \[thm:coneconstruction2\] {#sec:isom}
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Let us first recall the following fact proved before for example in [@FedMat Theorem 8], [@Matveev2010 Lemma 1] or [@Mounoud2010 Proposition 3.1]:
There is an isomorphism between the space of symmetric parallel $(0,2)$-tensors $\hat{A}$ on the Riemannian cone $(\hat{M}=\mathbb{R}_{>0}\times P,\hat{g}=dr^2+r^2h)$ and solutions $(L,\sigma,\rho)\in \Gamma(S^2T^* M\oplus T^*M\oplus \mathbb{R})$ of the linear PDE system $$\begin{aligned}
\begin{array}{c}
({\tensor[^h]{\nabla}{}}_{Z} L) (X,Y)=h(Z,X)\sigma(Y)+h(Z,Y)\sigma(X)\vspace{1mm}\\
({\tensor[^h]{\nabla}{}}_{Z} \sigma) (X)=\rho h(Z,X)-L(Z,X)\vspace{1mm}\\
{\tensor[^h]{\nabla}{}}_{Z} \rho =-2\sigma(Z)
\end{array}\label{eq:system}\end{aligned}$$ on $(P,h)$. The isomorphism is explicitly given by $$\begin{aligned}
(L,\sigma,\rho)\leftrightarrow \hat{A}=\rho dr^2-r dr\odot \sigma+r^2L,\label{eq:isomorphism}\end{aligned}$$ where we omitted the symbol for the pullback of objects from $P$ to $\hat{M}$.
Now let $(P=\mathbb{R}\times M,h=\theta^2+g)$ be defined as in the previous section, where $(M,g,J)$ is a $2n$-dimensional Kähler manifold with exact Kähler $2$-form. Let us prove a technical lemma that gives us a characterisation of solutions $(L,\sigma,\rho)$ of the system on $(P,h)$ which are invariant with respect to the action of the flow of $\eta$:
\[lem:dtinvariance\] The solution $(L,\sigma,\rho)$ of on $(P,h)$ is $\eta$-invariant if and only if $$\begin{aligned}
\sigma(\eta)=0,\,\sigma((JX)^{\theta})=L(\eta,X^{\theta}),L((JX)^{\theta},(JX)^{\theta})=L(X^{\theta},Y^{\theta})\mbox{ and }\rho=L(\eta,\eta).\label{eq:dtinvariance}\end{aligned}$$ for all $X,Y\in TM$.
Using and , we have that $\mathcal{L}_{\eta}L=0$ if and only if $$0=(\mathcal{L}_{\eta}L)(\eta,\eta)=2\sigma(\eta),\,\,\,0=(\mathcal{L}_{\eta}L)(\eta,X^{\theta})=\sigma(X^{\theta})+L(\eta,(JX)^{\theta}),$$ $$0=(\mathcal{L}_{\eta}L)(X^{\theta},Y^{\theta})=L((JX)^{\theta},Y^{\theta})+L(X^{\theta},(JY)^{\theta}).$$ These are the first three equations in . From the invariance of $\sigma$ it follows $$0=(\mathcal{L}_{\eta}\sigma)(\eta)=\rho-L(\eta,\eta),$$ which is the last equation in . The condition $$0=(\mathcal{L}_{\eta}\sigma)(X^{\theta})=-L(\eta,X^{\theta})+\sigma((JX)^{\theta})$$ is equivalent to the second equation in . The invariance of $\rho$ is satisfied automatically since $\mathcal{L}_\eta\rho=-2\sigma(\eta)=0$.
Next we show
\[lem:isom\] There is an isomorphism between the space of solutions $(A,\lambda,\mu)$ of on $(M,g,J)$ for $B=-1$ and $\eta$-invariant solutions $(L,\sigma,\rho)$ of on $(P,h)$.
With respect to the decomposition $TP=\mathbb{R}\,\eta\oplus\mathcal{H}$, the correspondence is given by $$\begin{aligned}
L=\mu\theta^2+\theta\otimes \lambda(J.)+\lambda(J.)\otimes\theta+A,\,\sigma=\lambda,\,\rho=\mu,\label{eq:A'sasaki}\end{aligned}$$ where we omit the symbol for the pullback of objects from $M$ to $P$.
First let us show, that $(L,\sigma,\rho)$ in defines a solution of . By direct calculation using the formulas for the Levi-Civita connection of $h$ we obtain $$({\tensor[^h]{\nabla}{}}_{\eta}L)(\eta,\eta)=\eta L(\eta,\eta)-2L({\tensor[^h]{\nabla}{}}_{\eta}\eta,,\eta)=0=h(\eta,\eta)\sigma(\eta)+h(\eta,\eta)\sigma(\eta),$$ $$({\tensor[^h]{\nabla}{}}_{\eta}L)(X^{\theta},\eta)=-L((JX)^{\theta},\eta)=\lambda(X)=h(\eta,X^{\theta})\sigma(\eta)+h(\eta,\eta)\sigma(X^{\theta}),$$ $$({\tensor[^h]{\nabla}{}}_{\eta}L)(X^{\theta},Y^{\theta})=-A(JX,Y)-A(X,JY)=0=h(\eta,X^{\theta})\sigma(Y^{\theta})+h(\eta,Y^{\theta})\sigma(X^{\theta}).$$ Further, using the equations in with $B=-1$, we calculate $$({\tensor[^h]{\nabla}{}}_{Z^{\theta}}L)(\eta,\eta)=Z(\mu)+2\lambda(Z)=0=h(Z^{\theta},\eta)\sigma(\eta)+h(Z^{\theta},\eta)\sigma(\eta),$$ $$({\tensor[^h]{\nabla}{}}_{Z^{\theta}}L)(X^{\theta},\eta)=(\nabla_Z \lambda)(JX)-\mu g(Z,JX)+A(Z,JX)=0=h(Z^{\theta},X^{\theta})\sigma(\eta)+h(Z^{\theta},\eta)\sigma(X^{\theta})$$ and finally $$({\tensor[^h]{\nabla}{}}_{Z^{\theta}}L)(X^{\theta},Y^{\theta})=g(Z,X)\lambda(Y)+g(Z,Y)\lambda(X)=h(Z^{\theta},X^{\theta})\sigma(Y^{\theta})+h(Z^{\theta},Y^{\theta})\sigma(X^{\theta}).$$ We have shown that $L$ defined in satisfies the first equation in . For $\sigma$ we obtain $$({\tensor[^h]{\nabla}{}}_{\eta}\sigma)(\eta)=0=\rho h(\eta,\eta)-L(\eta,\eta),$$ $$({\tensor[^h]{\nabla}{}}_{\eta}\sigma)(X^{\theta})=-\sigma((JX)^{\theta})=-\lambda(JX)=\rho h(\eta,X^{\theta})-L(\eta,X^{\theta})$$ and $$({\tensor[^h]{\nabla}{}}_{Z^{\theta}}\sigma)(X^{\theta})=\mu g(Z,X)-A(Z,X)=\rho h(Z^{\theta},X^{\theta})-L(Z^{\theta},X^{\theta}).$$ Thus, $\sigma$ satisfies the second equation in . Finally, for $\rho$ we have $${\tensor[^h]{\nabla}{}}_{\eta}\rho=0=-2\sigma(\eta)\mbox{ and }{\tensor[^h]{\nabla}{}}_{X^{\theta}}\rho=-2\lambda(X)=-2\sigma(X^{\theta}).$$
Now we show that under the correspondence , the $\eta$-invariant solution $(L,\sigma,\rho)$ of descends to a solution $(A,\lambda,\mu)$ of with $B=-1$. Using , we calculate $$(\nabla_{Z}A)(X,Y)=({\tensor[^h]{\nabla}{}}_{Z^{\theta}}L)(X^{\theta},Y^{\theta})+\omega(Z,X)\sigma((JY)^\theta)+\omega(Z,Y)\sigma((JX)^\theta)$$ $$=h(Z^{\theta},X^{\theta})\sigma(Y^{\theta})+h(Z^{\theta},Y^{\theta})\sigma(X^{\theta})+\omega(Z,X)\sigma((JY)^{\theta})+\omega(Z,Y)\sigma((JX)^{\theta})$$ $$=g(Z,X)\lambda(Y)+g(Z,Y)\lambda(X)+\omega(Z,X)\lambda(JY)+\omega(Z,Y)\lambda(JX),$$ which is the first equation in . To verify the second equation, we calculate $$(\nabla_{Z}\lambda)(X)=({\tensor[^h]{\nabla}{}}_{Z^{\theta}}\sigma)(X^{\theta})=\rho h(Z^{\theta},X^{\theta})-L(Z^{\theta},X^{\theta})=\mu g(Z,X)-A(Z,X).$$ Finally, $\nabla_Z \mu=-2\sigma(Z^{\theta})=-2\lambda(Z)$. This completes the proof of Lemma \[lem:isom\].
Recall that we already have an isomorphism between the space of symmetric parallel $(0,2)$-tensors $\hat{A}$ on $(\hat{M},\hat{g},\hat{J})$ and solutions $(L,\sigma,\rho)$ of on $(P,h)$. To prove Theorem \[thm:coneconstruction2\], it remains to show that hermitian symmetric parallel $(0,2)$-tensors $\hat{A}$ correspond to $\eta$-invariant solutions $(L,\sigma,\rho)$.
Using the definition of $\hat{J}$, it is easy to see that $\hat{A}$ in is hermitian, i.e., $\hat{A}(\hat{J}.,\hat{J}.)=\hat{A}$, if and only if $(L,\sigma,\rho)$ satisfies the equations . By Lemma \[lem:dtinvariance\], it follows that $\hat{A}$ is hermitian if and only if $(L,\sigma,\rho)$ is $\eta$-invariant. This completes the proof of Theorem \[thm:coneconstruction2\].
How to reduce the investigation of $\mathcal{A}(g,J)$ of dimension $\ge 3$ locally to the investigation of the space of parallel hermitian $(0,2)$-tensors on the conification {#sec:mainfeatures}
==============================================================================================================================================================================
If $B=-1$ in , the reduction was done in the previous section. For our purposes it is sufficiently to assume that the manifold is diffeomorphic to the $2n$-dimensional ball, the transition “any ball” $\longrightarrow$ “any simply-connected manifold” will be done in Section \[sec:proofthmdegreeglobal\]. The goal of the section is to show that on every neighborhood $U$ that is diffeomorphic to the ball and such that the closure of $U$ is compact one can achieve $B=-1$ by replacing the metric by its c-projectively equivalent. If $B\ne 0$, this could be done by a scaling of $g$, see the proof of Corollary \[cor:changeofmetric\]. If $B=0$, then we do need to change the metric by a essentially c-projectively equivalent one:
\[lem:changeofmetric\] Let $(M,g,J)$ be a connected Riemannian Kähler manifold of real dimension $2n\geq 4$. Assume there exists a solution $(A, \lambda, \mu)$ of with $B=0$ such that $\lambda \ne 0$.
Then for every open simply connected subset $U\subseteq M$ with compact closure, there exists a Riemannian Kähler metric $\tilde{g}$ on $U$ that is c-projectively equivalent to $g$, such that for any solution $(\tilde A,\tilde \lambda)$ of the system for the metric $\tilde g$ there exists a function $\tilde \mu$ such that $(\tilde A,\tilde \lambda, \tilde \mu)$ satisfies for the metric $\tilde{g}$ and such that the corresponding constant $\tilde{B}$ is different from $0$.
The proof of Lemma \[lem:changeofmetric\] will be given in Section \[sec:proofoflemma\].
As we already remarked, $D(g,J)$ is the same for all c-projectively equivalent metrics. As an direct application of Lemma \[lem:changeofmetric\] we obtain
\[cor:changeofmetric\] Let $(M,g,J)$ be a connected Riemannian Kähler manifold of real dimension $2n\geq 4$ and of degree of mobility $D(g,J)\geq 3$. Suppose there exists at least one metric c-projectively equivalent to $g$ and not affinely equivalent to it. Then, on each open simply connected subset $U\subseteq M$ with compact closure, the degree of mobility $D(g_{|U},J_{|U})$ is equal to the dimension of the space of solutions of with $B=-1$ for a certain positively or negatively definite Kähler metric $\tilde{g}$ on $U$ that is c-projectively equivalent to $g$.
By Lemma \[lem:changeofmetric\], on every open simply-connected subset $U$ with the required properties, we can find a Riemannian Kähler metric $g'$, c-projectively equivalent to $g$, such that $\mathcal{A}(g,J)$ is isomorphic to the space of solutions of the system for $g'$ with a certain constants $B'\neq 0$. For the rescaled metric $\tilde{g}= -B'g'$, the system holds now with a constant $\tilde{B}=-1$. Depending on the sign of $B'$, the new metric $\tilde{g}$ is either positively or negatively definite.
Proof of Lemma \[lem:changeofmetric\]: the constant $B$ in the system can be made non-zero by an arbitrary small change of the metric in the c-projective class {#sec:proofoflemma}
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The proof of Lemma \[lem:changeofmetric\] is divided into several steps. First we show how the constant $B$ changes if one chooses another metric in the c-projective class.
\[lem:changeofmetric1\] Let $(M,g,J)$ be a Riemannian Kähler manifold of real dimension $2n\geq 4$ and let $D(g,J)\geq 3$. Suppose $\tilde{g}$ is c-projectively equivalent to $g$ and let $A=A(g,\tilde{g})\in \mathcal{A}(g,J)$ be given by formula . Let $\lambda$ and $\mu$ be the $1$-form and function respectively such that $(A,\lambda,\mu)$ constitutes a solution of the system for $g$ with constant $B$ and let $\Lambda=g^{-1}\lambda$. Then the constant $\tilde{B}$ in the system for $\tilde{g}$ is given by $$\begin{aligned}
\tilde{B}=(\mathrm{det}A)^{\frac{1}{2}}(g(A^{-1}\Lambda,\Lambda)-\mu).\label{eq:trafoB}\end{aligned}$$
Let us view the tensor $A=A(g,\tilde{g})$ in equation equivalently as a $(1,1)$-tensor $g^{-1}A$ by raising the “left index” of $A$ by contraction with the inverse metric $g^{-1}$. To simplify notation, we will denote $g^{-1}A$ again by $A$ such that the equation now reads $$\begin{aligned}
\nabla_X A=g(.,X)\Lambda+g(.,\Lambda)X+g(.,JX)J\Lambda+g(.,J\Lambda)JX.\label{eq:mainAalternative}\end{aligned}$$ From the defining equation it follows that $A^{-1}=A(\tilde{g},g)$, thus $\tilde{A}=A^{-1}$ is a solution of written down in terms of $\tilde{g}$. The corresponding vector field $\tilde{\Lambda}$ can be expressed in terms of $A$ and $\Lambda$. Indeed, a straight-forward calculation, using , , and yields $$\tilde{\nabla}_{X}\tilde{A}=-(\mathrm{det}A)^{\frac{1}{2}}(\tilde{g}(.,X)A^{-1}\Lambda+\tilde{g}(.,A^{-1}\Lambda)X+\tilde{g}(.,JX)JA^{-1}\Lambda+\tilde{g}(.,JA^{-1}\Lambda)JX).$$ Comparing this with the expected form of equation for $\tilde{g}$, we see that $$\begin{aligned}
\tilde{\Lambda}=-(\mathrm{det}A)^{\frac{1}{2}}A^{-1}\Lambda.\label{eq:trafolambda}\end{aligned}$$ We will use this equation to calculate the second equation in the system for $\tilde{g}$. First we note that $$\nabla_X (\mathrm{det}A)^{\frac{1}{2}}=2(\mathrm{det}A)^{\frac{1}{2}}g(A^{-1}\Lambda,X).$$ Using this together with , , and , a straight-forward calculation yields $$\tilde{\nabla}_X \tilde{\Lambda}=(\mathrm{det}A)^{\frac{1}{2}}(g(A^{-1}\Lambda,A^{-1}\Lambda)-B)X+(\mathrm{det}A)^{\frac{1}{2}}(g(A^{-1}\Lambda,\Lambda)-\mu)\tilde{A}X.$$ Comparing this with the expected form of the second equation in for $\tilde{g}$, we see that $\tilde{B}$ is given by as we claimed.
We consider the case when for the metric $g$ the constant $B$ in is vanishing. The third equation in shows that the function $\mu$ is necessarily a constant. Next we show that we can always find a solution $(A,\lambda,\mu)$ of such that $\mu\neq 0$.
\[lem:changeofmetric2\] Let $(M,g,J)$ be a connected Kähler manifold of real dimension $2n\geq 4$ and of degree of mobility $D(g,J)\geq 3$. Suppose the system holds for $B=0$ and that at least one metric c-projectively equivalent to $g$ is not affinely equivalent to it. Then on every open simply connected subset $U\subseteq M$, we can find a solution $(A,\lambda,\mu)$ of such that $\mu\neq 0$.
Recall from Remark \[rem:affine\] that if $\tilde{g}$ is c-projectively equivalent to $g$ but not affinely equivalent to it, the $1$-form $\lambda$ corresponding to $A=A(g,\tilde{g})\in \mathcal{A}(g,J)$ is not identically zero. Let us work with this solution $A$. The equations in show that for the corresponding $1$-form $\lambda$ we have $$\nabla_X \lambda =\mu g(X,.)$$ for a certain constant $\mu$.
Suppose $\mu=0$, i.e. $\nabla_X \lambda=0$. Consider the $1$-form $Ag^{-1}\lambda$ (where as always both $g^{-1}:T^{*}M\rightarrow TM$ and $A:TM\rightarrow T^{*}M$ are viewed as bundle morphisms). Calculating its covariant derivative using , we obtain $$\begin{aligned}
\nabla_X(Ag^{-1}\lambda)=(\nabla_X A)g^{-1}\lambda=\lambda(g^{-1}\lambda)g(X,.)+\lambda(X)\lambda-\lambda(JX)\lambda(J.).\label{eq:oneformderivative}\end{aligned}$$ Recall that $\lambda$ is the differential of a function, i.e. $\lambda=\nabla f$ for a certain function $f$. On the other hand, on the open neighborhood $U$ also the $1$-form $\lambda(J.)$ is the differential of a function $f':U\rightarrow \mathbb{R}$. This follows from the fact that $\lambda(J.)$ is parallel (and hence closed) and $U$ is simply connected. Let us set $c=\lambda(g^{-1}\lambda)$ (which is a non-zero constant) and define the $1$-form $\sigma=Ag^{-1}\lambda-f\lambda+f'\lambda(J.).$ It follows from , that $$\nabla_X\sigma=c g(X,.).$$ On the other hand, it is straight-forward to check that the symmetric hermitian $(0,2)$-tensor $$\tilde{A}=\sigma\otimes \sigma+ \sigma(J.)\otimes \sigma(J.)$$ satisfies . The corresponding $1$-form $\tilde{\lambda}$ is given by $c\sigma$ and satisfies $\nabla_X \tilde{\lambda}=c^2 g(X,.).$ Thus $(\tilde{A},\tilde{\lambda},\tilde{\mu}=c^2)$ is the desired solution of with $B=0$ but $\tilde{\mu}\neq 0$.
Now we are able to give the proof of Lemma \[lem:changeofmetric\]. Let us suppose that $B=0$ and let $U$ be an open simply connected subset with compact closure. By Lemma \[lem:changeofmetric2\], we can find a solution $(A,\lambda,\mu)$ of on $U$ such that $\mu\neq 0$ and after rescaling we can suppose that $\mu=1$.
For arbitrary real numbers $t$, we define the triple $$A(t)=t(\lambda\otimes\lambda+\lambda(J.)\otimes \lambda(J.))+g,\,\,\,\lambda(t)=t\lambda,\,\,\,\mu(t)=t\mu=t.$$ Obviously the triple $(A(t),\lambda(t),\mu(t))$ is a solution of for $g$ with $B=0$. Moreover, since $U$ has compact closure, we find $\epsilon>0$ such that for all $t\in (-\epsilon,\epsilon)$ the solution $A(t)$ is non-degenerate on $U$ and the metric $\tilde{g}_t$ in the c-projective class of $g$ which corresponds to $A(t)$ (that is, $\tilde{g}_t$ is defined by $A(g,\tilde{g}_t)=A(t)$ in equation ) is positively definite.
Using Lemma \[lem:changeofmetric1\], we see that the constant $\tilde{B}_t$ in the system for $\tilde{g}_t$ is given by $$\tilde{B}(t)=(\mathrm{det}A(t))^{\frac{1}{2}}(g(A(t)^{-1}\Lambda(t),\Lambda(t))-\mu(t))=(\mathrm{det}A(t))^{\frac{1}{2}}(t^2g(A(t)^{-1}\Lambda,\Lambda)-t),$$ where as usual $\Lambda=g^{-1}\lambda$. We want to show that $\tilde{B}(t)$ is non-zero for some $t_0\in (-\epsilon,\epsilon)$. Since $(\mathrm{det}A(t))^{\frac{1}{2}}$ is positive anyway it suffices to show that $$f(t)=t^2g(A(t)^{-1}\Lambda,\Lambda)-t$$ is non-zero for some $t_0\in (-\epsilon,\epsilon)$. Taking the derivative of $f$ when $t=0$, we obtain $\tfrac{d f}{dt}(0)=-1,$ thus, there is some $t_0\in (-\epsilon,\epsilon)$ such that $\tilde{B}(t_0)$ is non-zero. The metric $\tilde{g}_{t_0}$ has the properties as required in Lemma \[lem:changeofmetric\]. This completes the proof of Lemma \[lem:changeofmetric\].
Though we explicitly used in the proof of Lemma \[lem:changeofmetric2\] that the metric $g$ is Riemannian, Lemma \[lem:changeofmetric2\] remains true for metrics of arbitrary signature (but the proof in arbitrary signature is longer and uses nontrivial results of [@EMN]).
Conification of Einstein manifolds {#-4}
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Suppose $(M^{2n\ge 4}, g, J)$ is a Kähler-Einstein manifold of arbitrary signature. We assume that the symplectic form $\omega= g(.,J.)$ is exact so that we can consider the conification $(\hat M^{2n+2}, \hat g, \hat J)$ introduced in Section \[sec:construction\]. Our goal is to show that the investigation of solutions of on $(M,g,J)$ reduces to the investigation of parallel tensors on the conification $(\hat{M},\hat{g},\hat{J})$ for a Ricci flat metric $\hat{g}$. We start with the following technical statement.
\[lem:scalarB\] Let $(M,g,J)$ be a Kähler-Einstein manifold of real dimension $2n\geq 4$. Suppose there exists a solution $(A, \lambda, \mu) $ of such that $\lambda\ne 0$. Then, $$\begin{aligned}
B=-\frac{\mathrm{Scal}(g)}{4n(n+1)},\label{eq:scalarB}\end{aligned}$$ where $\mathrm{Scal}(g)$ is the scalar curvature of $g$.
Take a solution $A$ of such that $\lambda\ne 0$. As usual we denote $\Lambda=g^{-1}\lambda$. It is known (see [@ApostolovI Proposition 3], [@FKMR Corollary 3] or equation (13) and the sentence below in [@DomMik1978]) that $J\Lambda$ is a Killing vector field which in particular implies that $\lambda$ is non-zero in every point of an open and everywhere dense subset. We define the function $\sigma=\mathrm{trace}\nabla\Lambda$. From the second equation in , we see that $\sigma$ and $\mu$ are related by $\sigma=2n\mu+B\mathrm{trace}(A).$ Thus, taking the covariant derivative of this equation and inserting the third equation of , we obtain $$\begin{aligned}
\nabla\sigma=2n\nabla\mu+4B\lambda=4B(n+1)\lambda.\label{eq:musigma}\end{aligned}$$
On the other hand, since $J\Lambda$ is Killing, we have the identity $\nabla_X\nabla\Lambda=-JR(X,J\Lambda)$. Using the symmetries of the curvature tensor $R$ of the Kähler metric $g$, we have $\mathrm{trace}(JR(X,JY))=2\mathrm{Ric}(g)(X,Y)$. Combining the last two equations yields $\nabla\sigma=-2\mathrm{Ric}(g)(.,\Lambda)$ and inserting this into , we have $$-\mathrm{Ric}(g)(.,\Lambda)=2B(n+1)\lambda.$$ Since $g$ is Kähler-Einstein, that is $\mathrm{Ric}(g)=\frac{\mathrm{Scal}(g)}{2n}g$, we evidently have .
\[lem:eins\] Let $(M,g,J)$ be a Kähler-Einstein manifold of real dimension $2n\geq 4$ and let the symplectic form $\omega= g(., J.)$ be exact. Assume $ {\mathrm{Scal}(g)}= 4n(n+1)$. Then, the conification of $(M,g,J)$ is Ricci flat. Moreover, if a conification of a certain Kähler-Einstein manifold $(M^{\ge 4},g,J)$ is Ricci-flat, then $g$ is Einstein with scalar curvature $ {\mathrm{Scal}(g)}= 4n(n+1)$.
By direct calculation, using the formulas , we obtain that the curvature tensor $\hat{R}$ of $(\hat{M},\hat{g},\hat{J})$ is given by the formulas $$\begin{aligned}
\hat{R}(.,.)\xi=\hat{R}(.,.)J\xi=0,\label{eq:curvcon0}\end{aligned}$$ where $\xi=r\partial_r$ is the cone vector field on $(\hat{M},\hat{g},\hat{J})$, and $$\begin{aligned}
\hat{R}(X^\theta,Y^\theta)Z^\theta=(R(X,Y)Z-4H(X,Y)Z)^\theta,\label{eq:curvcon}\end{aligned}$$ where $R$ denotes the curvature tensor of $(M,g,J)$, $$H(X,Y)Z=\frac{1}{4}(g(Z,Y)X-g(Z,X)Y+\omega(Z,Y)JX-\omega(Z,X)JY+2\omega(X,Y)JZ)$$ is the algebraic curvature tensor of constant holomorphic curvature equal to one and $X^\theta$ denotes the horizontal lift of tangent vectors $X\in TM$ to the distribution $\mathcal{H}=\mathrm{span}\{\xi,J\xi\}^\perp\subseteq T\hat{M}$ (see also Section \[sec:construction\] for the notation). Having these formulas, Lemma \[lem:eins\] follows from simple linear algebra:
From it is clear that $\mathrm{Ric}(\hat{g})(\xi,.)=\mathrm{Ric}(\hat{g})(J\xi,.)=0$, where $\xi$ denotes the cone vector field on $\hat{M}$. A straight-forward calculation using yields $$\mathrm{Ric}(\hat{g})(X^\theta,Y^\theta)=r^2\left(\mathrm{Ric}(g)(X,Y)-2(n+1)g(X,Y)\right).$$ implying that if $ {\mathrm{Scal}(g)}= 4n(n+1)$ then $$\mathrm{Ric}(\hat{g})(X^\theta,Y^\theta)=r^2\left(\mathrm{Ric}(g)(X,Y)-\frac{\mathrm{Scal}(g)}{2n}g(X,Y)\right)=0$$ so $\hat{g}$ is Ricci flat, and if $\hat{g}$ is Ricci flat then $g$ is Einstein with $ {\mathrm{Scal}(g)}= 4n(n+1)$.
\[lem:einsteincproeinstein\] Let $(M,g,J)$ be a Kähler-Einstein manifold of real dimension $2n\geq 4$. Suppose there exists a solution $(A, \lambda, \mu) $ of such that $\lambda\ne 0$. Then, every metric $\tilde{g}$, c-projectively equivalent to $g$, is also Kähler-Einstein.
Let $A=A(g,\tilde{g})$ be the solution of given by . In the second equation $\nabla\lambda=\mu g+B A$ of , we express $\lambda$ in terms of the $1$-form $\Phi$ by using the relation . A straight-forward calculation yields $$(\mu-g(A^{-1}\Lambda,\Lambda))gA^{-1}+Bg=-\nabla \Phi+\Phi\otimes\Phi-\Phi(J.)\otimes\Phi(J.),$$ where $A$ is equivalently viewed as a $(1,1)$-tensor and $\Lambda=g^{-1}\lambda$ as usual. Using $\tilde{g}=(\mathrm{det}\,A)^{-\frac{1}{2}}gA^{-1}$ and the transformation rule for $B$, we can rewrite this into the form $$\begin{aligned}
Bg-\tilde{B}\tilde{g}=-\nabla \Phi+\Phi\otimes\Phi-\Phi(J.)\otimes\Phi(J.),\label{eq:kaehlereinstein}\end{aligned}$$
On the other hand, using the transformation rule for the Levi-Civita connections of the two metrics, it is straight-forward to show and well-known (see [@Tashiro1956 equation (1.11)]) that the Ricci-tensors corresponding to $g$ and $\tilde{g}$ are related by $$\begin{aligned}
\mathrm{Ric}(\tilde{g})=\mathrm{Ric}(g)-2(n+1)(\nabla\Phi-\Phi\otimes\Phi+\Phi(J.)\otimes\Phi(J.)).\label{eq:riccitafo}\end{aligned}$$
Combining this equation with , we obtain $$\mathrm{Ric}(\tilde{g})+2(n+1)\tilde{B}\tilde{g}=\mathrm{Ric}(g)+2(n+1)Bg.$$ Since the assumptions of Lemma \[lem:scalarB\] are satisfied and $g$ is Einstein, we see from formula that the right-hand side of the last equation is vanishing identically. Then, $\tilde{g}$ is Einstein.
Combining Lemma \[lem:changeofmetric\] with Lemmas \[lem:scalarB\] and \[lem:einsteincproeinstein\], we obtain
\[cor:changeofmetriceinstein\] Let $(M,g,J)$ be a Riemannian Kähler-Einstein manifold of real dimension $2n\geq 4$ and of degree of mobility $D(g,J)\geq 3$. Suppose there exists a solution $(A, \lambda, \mu) $ of such that $\lambda\ne 0$.
Then, on each open simply connected subset $U\subseteq M$ with compact closure, the degree of mobility $D(g_{|U},J_{|U})$ is equal to the dimension of the space of solutions of with $B= -\frac{\mathrm{Scal}(\tilde{g})}{4n(n+1)}= -1$ for a certain positively or negatively definite Kähler-Einstein metric $\tilde{g}$ on $U$ that is c-projectively equivalent to $g$.
It follows from Corollary \[cor:changeofmetriceinstein\], Theorem \[thm:coneconstruction2\] and Lemma \[lem:eins\] that (at least in the local setting) we reduced the study of the degrees of mobility of $2n$-dimensional Kähler-Einstein Riemannian metrics to the study of the possible dimensions of the space of parallel hermitian symmetric $(0,2)$-tensors of Ricci-flat cone Kähler manifolds $(\hat{M},\hat{g},\hat{J})$ of dimension $2(n+1)$ where $\hat{g}$ is positively definite or has signature $(2,2n)$.
In the proof of Theorem \[thm:degreeeinstein\] we will need one more observation.
\[lem:ricciflatcone\] Let $(\hat{M}=\mathbb{R}_{>0}\times P,\hat{g}=dr^2+r^2 h,\hat{J})$ be a Kähler manifold which is the cone over an $(2n+1)$-dimensional pseudo-Riemannian manifold $(P,h)$.
1. If $\mathrm{dim}\,\hat{M}<6$ and $\hat{g}$ is Ricci flat, then $\hat{g}$ is flat.
2. Let $\hat{g}$ have signature $(2,2n)$. If $\mathrm{dim}\,\hat{M}<8$, $\hat{g}$ is Ricci flat and $X$ is a non-zero parallel null-vector field on $\hat{M}$, then $\hat{g}$ is flat.
\(1) Using it is straight-forward to calculate that the curvature tensor $\hat{R}$ of $\hat{g}$ is given by the formulas $$\begin{aligned}
\hat{R}(.,.)\partial_r=0\mbox{ and }\hat{R}(X,Y)Z=R(h)(X,Y)Z-(h(Z,Y)X-h(Z,X)Y),\label{eq:curvaturecone}\end{aligned}$$ where $X,Y,Z\in TP$ and $R(h)$ is the curvature tensor of $h$. Then, calculating the Ricci tensor of $\hat{g}$, it is straight-forward to see that if $\hat{g}$ is Ricci flat, $h$ is Einstein with scalar curvature $\mathrm{Scal}(h)=(\mathrm{dim}\,P)(\mathrm{dim}\,P-1)$. If in addition $\mathrm{dim}\,\hat{M}<6$, we have that $\mathrm{dim}\,P<4$ and therefore $h$ has constant curvature equal to $1$. Inserting this back into , we obtain that $\hat{R}=0$ as we claimed.
\(2) Let $\xi=r\partial_r$ denote the cone vector field on $\hat{M}$ and let $X$ be the parallel non-zero null vector field. Since $\hat{\nabla}\xi={\mathrm{Id}}$, we obtain that $\hat{g}(X,\xi)\neq 0$ on a dense and open subset of $\hat{M}$. Indeed, if $\hat{g}(X,\xi)=0$ on an open subset $U$ we can take the covariant derivative of this equation in the direction of $Y\in T\hat{M}$ to obtain that $\hat{g}(X,Y)=0$ in every point $p\in U$ for all $Y\in T_p \hat{M}$. This implies $X=0$ on $\hat{M}$ which contradicts our assumption. By similar arguments one also obtains that $\hat{g}(JX,\xi)\neq 0$ on a dense and open subset of $\hat{M}$. Let us work in a point $p\in \hat{M}$ such that $\hat{g}(X,\xi)\neq 0$ and $\hat{g}(JX,\xi)\neq 0$ at $p$. We suppose that $\mathrm{dim}\,\hat{M}=6$ and show that $\hat{g}$ is flat.
It is an easy exercise to show that there exist a basis $X,Y,Z,JX,JY,JZ$ of $T_p \hat{M}$ in which $\hat{g}$ takes the form $$\begin{aligned}
\hat{g}=\left(\begin{array}{cccccc}0&1&0&0&0&0\\1&0&0&0&0&0\\0&0&1&0&0&0\\0&0&0&0&1&0\\0&0&0&1&0&0\\0&0&0&0&0&1\end{array}\right)\nonumber\end{aligned}$$ and such that $\mathrm{span}\{X,Y,\hat{J}X,\hat{J}Y\}=\mathrm{span}\{X,\xi,\hat{J}X,\hat{J}\xi\}$. Hence any endomorphism of $T_p \hat{M}$ that commutes with $\hat{J}$ and vanishes on $\xi$ and $X$ has to vanish on $Y$ as well. This holds in particular for the curvature endomorphisms $\hat{R}=\hat{R}(u,v):T_p\hat{M}\rightarrow T_p\hat{M}$ for every pair of vectors $u,v\in T_p \hat{M}$. Since $\hat{R}$ commutes with $\hat{J}$ and is skew-symmetric with respect to $\hat{g}$ it takes the form $$\begin{aligned}
\hat{R}=\left(\begin{array}{cccccc}a&0&b&-A&-B&-C\\0&-a&c&-D&-A&-E\\-c&-b&0&-E&-C&-F\\A&B&C&a&0&b\\D&A&E&0&-a&c\\E&C&F&-c&-b&0\end{array}\right).\nonumber\end{aligned}$$ in the basis $X,Y,Z,JX,JY,JZ$ from above. Since $\hat{R}$ vanishes on $X,Y,JX,JY$ it implies that $a=b=c=A=B=C=D=E=0$. Furthermore, using the condition that $\hat{g}$ is Ricci flat, i.e., $\mathrm{trace}(\hat{J}\hat{R})=0$ yields $F=0$. Thus the curvature tensor $\hat{R}$ is vanishing in every point of a dense and open subset, implying that $\hat{g}$ is flat as we claimed.
Proof of Theorems \[thm:degree\] and \[thm:degreeeinstein\] {#sec:proofs}
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Proof of the first statement of the Theorems \[thm:degree\] and \[thm:degreeeinstein\] in the local situation {#sec:proofthmdegree}
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Let $(M,g,J)$ be a Kähler manifold of real dimension $2n\geq 4$. Our goal is to show that for every open simply connected subset $U\subseteq M$ with compact closure and the property that the Kähler form $\omega$ is exact on $U$, the degree of mobility $D(g_{|U},J_{|U})$ is given by one of the values either in the list of Theorem \[thm:degree\] for a general metric or in the list of Theorem \[thm:degreeeinstein\] under the additional assumption that the metric is Einstein. We will prove this simultaneously.
By Corollary \[cor:changeofmetric\] and Theorem \[thm:coneconstruction2\], the number $D(g_{|U},J_{|U})$ is precisely the dimension of the space of parallel hermitian symmetric $(0,2)$-tensors on the conification $(\hat{U},\hat{\tilde{g}},\hat{J})$ of $(U,\tilde{g}_{|U},J_{|U})$, where $\tilde{g}$ is a certain metric on $U$ that is c-projectively equivalent to $g$. Moreover, since $\tilde{g}$ is either positively or negatively definite, the metric $\hat{\tilde{g}}$ will be either positively definite or has signature $(2,2n)$. We also know in view of Lemma \[lem:einsteincproeinstein\] that if the metric $g$ is Einstein, then the metric $ {\tilde{g}}$ is also Einstein so the metric $\hat{\tilde{g}}$ is Ricci-flat.
To avoid cumbersome notations, we will drop the “hat” and the “tilde” in the notation for the conification. The local version of the Theorems \[thm:degree\] and \[thm:degreeeinstein\] that we are going to prove in this section reads
\[thm:degreelocal\] Let $(M,g,J)$ be a simply connected Kähler manifold of real dimension $2n+2\geq 6$ which is a cone over a $(2n+1)$ dimensional manifold. Further, let $g$ be either positively definite or have signature $(2,2n)$. Then, the dimension $D$ of the space of parallel hermitian symmetric $(0,2)$-tensors is given by one of the values in the list of Theorem \[thm:degree\]. Moreover, if the metric $g$ is Ricci-flat, then the dimension $D$ of the space of parallel hermitian symmetric $(0,2)$-tensors is given by one of the values in the list of Theorem \[thm:degreeeinstein\].
The proof of Theorem \[thm:degreelocal\] in the case when the metric $g$ is positively definite is more simple than in the case when the signature is $(2,2n)$. Moreover, the arguments for the proof when $g$ is positively definite are implicitly contained in the proof when the signature is $(2,2n)$. We therefore restrict to the latter case and assume that $g$ has signature $(2,2n)$ in what follows; the Riemannian signature is explained in Section \[sec:ideas\] and we leave it as an easy exercise.
Let $p$ be an arbitrary point in $M$. Consider a maximal orthogonal holonomy decomposition of $T_{p}M$. $$\begin{aligned}
T_pM=T_0\otimes T_1\otimes...\otimes T_\ell\label{eq:decomptangenttrivial}. \end{aligned}$$ Here $T_0$ is a nondegenerate subspace of $T_pM$ such that the holonomy group acts trivially and such that it is $J$-invariant, and $T_i$ for $i\ge 1$ are $J$-invariant nondegenerate subspace invariant w.r.t. the action of the holonomy group. We assume that the decomposition is maximal in the sense that no $T_i$, $i\ge 1$ has a holonomy-invariant nontrivial nondegenerate subspace and therefore can not be decomposed further. The existence of such a decomposition is standard and follows for example from [@Wu].
If in addition the initial manifold is Ricci-flat, then the restriction of the curvature tensor to each $T_i$ is also Ricci-flat.
It is well known that symmetric hermitian [*parallel*]{} $(0,2)$-tensor fields on $M$ are in one-one correspondence with symmetric hermitian $(0,2)$-tensors on $T_pM$ that are invariant w.r.t. the action of the holonomy group. As it was shown in [@FedMat Theorem 5], every symmetric [*holonomy-invariant*]{} $(0,2)$-tensor on $T_pM$ has the form $$\label{presentation1}
\sum_{\alpha, \beta =1}^{2k}c_{\alpha \beta } \tau_\alpha \otimes \tau_\beta + C_1 g_1 + ...+ C_\ell g_\ell.$$ Here $\{\tau_i\}_{i=1,...,2k}$ is a basis in the subspace of $T^*M$ consisting of those elements that are invariant w.r.t. the holonomy group, and $g_i$, $i=1,..., \ell $ is the restriction of $g$ to $T_i$ viewed as $(0,2)$-tensors on $T_pM$. Note that in the case of indefinite signature the number $k$ must not coincide with $k_0:= \tfrac{1}{2}\dim T_0$, since it might exist a light-like holonomy-invariant vector such that it is orthogonal to all vectors from $T_0$. In the signature $(2,2n)$ we have $k=k_0$ or $k=k_0+1$. The coefficients $c_{\alpha\beta}$ satisfy $c_{\alpha\beta}=c_{\beta\alpha}$ so $(c_{\alpha\beta})$ is a symmetric matrix. Our assumption that the parallel tensor is hermitian implies that the matrix $c_{\alpha\beta}$ is hermitian.
Clearly, the dimension of the space of the tensors of the form is the number of free parameters $c_{\alpha\beta}$, $C_i$. It is well known that the space of symmetric hermitian $2k\times 2k$ matrices has dimension $k^2$ so the first term of gives us $k^2$ dimensions and we obtain $ k^2 + \ell$ in total which is as we claimed. Our goal is to show that $k$ and $\ell$ satisfy the restrictions in the Theorems \[thm:degree\] and \[thm:degreeeinstein\].
Suppose $\ell =0$ (that is $g$ is flat and hence, the initial $2n$-dimensional metric has constant holomorphic curvature). Then, $k_0=n+1$ and we obtain that the dimension of the space of the parallel hermitian tensors is $(n+1)^2$ as we want.
Suppose $\ell\ge 1 $ and take $i\ge 1$. By [@FedMat Lemma 2], the dimension of $T_i$ is $\ge 3$ and since it is even, we have $\dim(T_i)\ge 4$. Moreover, under the additional assumption that $T_i$ is Ricci-flat, we have $\dim(T_i)\ge 6$ by Lemma \[lem:ricciflatcone\].
Suppose now $T_i$ with $i\ge 1$ contains a nonzero holonomy-invariant vector. Let us show that then the dimension of this $T_i$ is $\ge 6$. Let us denote this vector by $v$. Note that the vector $Jv$ is also holonomy-invariant and any linear combination of $v$ and $Jv$ is light-like since otherwise there would exist a nontrivial holonomy-invariant nondegenerate (two-dimensional) subspace.
We extend $T_i$ and also $v,Jv\in T_i$ to the whole manifold by parallel translating these objects along all possible ways starting at $p$. The extension of $T_i$ is well-defined and gives us an integrable distribution on $M$. The extensions of $v$ and $Jv$ are also well-defined and are parallel vector fields.
It is sufficient to show that under the assumption $\dim(T_i)=4$ the restriction of the curvature to this distribution vanishes, since this will imply in view of the theorem of Ambrose-Singer that the holonomy group acts trivially on $T_i$ which contradicts the assumption that $T_i$ has no nontrivial holonomy-invariant nondegenerate subspaces. We choose a generic point $q$. Since the point is generic, then by [@FedMat Lemma 5 and Lemma 2] there exists a vector $u\in T_i(q)$ such that $g(u,u)\ne 0$ and such that $R(u,.).= 0$. We consider the basis $\{u, Ju, v, Jv\}$ of $T_i(q)$. This is indeed a basis since the vectors $u$ and $Ju$ (resp. $v$ and $Jv$) are nonproportional and therefore linearly independent and no nontrivial linear combination of $u$ and $Ju$ can be equal to a nontrivial linear combination of $v$ abd $Jv$ since any linear combination of $v$ and $Jv$ is light-like and any linear combination of $u$ and $Ju$ is not light-like. Now, the vectors $u,v, Ju, Jv$ satisfy the condition $R(u,.).= R(Ju,.).= R(v,.).= R(Jv,.)= 0$. Indeed, $R(u,.).=0$ is essentially the choice of our vector, $R(Ju,.).=0$ is because the Riemanian curvature of a Kähler metric is $J$-invariant, $ R(v,.).= R(Jv,.)= 0$ is fulfilled because $v$ and $Jv$ are parallel. Thus, $\dim(T_i)\ge 6$.
Let us now suppose that $T_i$ is Ricci flat and contains a nonzero (and therefore light-like) holonomy-invariant vector. Then, it follows from Lemma \[lem:ricciflatcone\] that $\dim(T_i)\ge 8$.
We obtain that the number $k$ is at most $n-1$ and the number $\ell$ is at most $\left[\frac{n-k-1}{2}\right]$. Indeed, suppose there exists a nonzero holonomy-invariant vector contained in one $T_i$ with $i\ge 1$. Without loss of generality we may assume $i=1$. As we explained above, the dimension of $T_1$ is $\ge 6$ and the dimension of all other $T_j$ for $j\ge 2$ is at least $4$. The dimension of $T_0$ is $2k- 2$. Then, $$\underbrace{\dim(T_0)}_{2k-2} + \underbrace{\dim(T_1)}_{\ge 6}+ \underbrace{\dim(T_2)}_{\ge 4}+ ... +\underbrace{\dim(T_\ell)}_{\ge 4}=2(n+1)\label{-1}$$ implying $k\le n-1$ and $\ell\le \left[\frac{n-k-1}{2}\right]$ as we want.
Suppose now there exists no nonzero holonomy-invariant vector contained in one $T_i$ with $i\ge 1$. Then, $\dim(T_0)=2k$ and the dimension of all $T_j$ for $j\ge 1$ is at least $4$. Here we obtain $\ell\le \left[\frac{n-k-1}{2}\right]$ by the same argument. Indeed, in this case $$\label{+1}
\underbrace{\dim(T_0)}_{2k } + \underbrace{\dim(T_1)}_{\ge 4}+ ... +\underbrace{\dim(T_\ell)}_{\ge 4}=2(n+1)$$ implying $\ell\le \left[\frac{n-k-1}{2}\right]$ as we want.
Assume now the metric $\hat g$ is Ricci-flat. Then, each $T_i$ is Ricci flat, so its dimension is $\ge 6$. As we have shown above, if $T_i$ (with $i\ge 1$) contains a nonzero holonomy-invariant vector, then $\dim(T_i)\ge 8$, and the analog of looks $$\underbrace{\dim(T_0)}_{2k-2} + \underbrace{\dim(T_1)}_{\ge 8}+ \underbrace{\dim(T_2)}_{\ge 6}+ ... +\underbrace{\dim(T_\ell)}_{\ge 6}=2(n+1).$$ implying $k\le n-2$ and $\ell\le \left[\frac{n-k-1}{3}\right]$ as we want. If there exists no nonzero holonomy-invariant vector contained in one $T_i$ with $i\ge 1$. Then, $\dim(T_0)=2k$, the dimension of all $T_j$ for $j\ge 1$ is at least $6$ and we obtain $k\le n-2$ and $\ell\le \left[\frac{n-k-1}{3}\right]$ by the same argument. Indeed, in this case $$\underbrace{\dim(T_0)}_{2k } + \underbrace{\dim(T_1)}_{\ge 6}+ ... +\underbrace{\dim(T_\ell)}_{\ge 6}=2(n+1)$$ implying $k\le n-2$ and $\ell\le \left[\frac{n-k-1}{3}\right]$ as we want. Theorem \[thm:degreelocal\] is proved.
Proof of the first parts of Theorems \[thm:degree\] and \[thm:degreeeinstein\] in the global situation {#sec:proofthmdegreeglobal}
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In this section, we complete the proof of the first parts of the Theorems \[thm:degree\] and \[thm:degreeeinstein\]. Let $(M,g,J)$ be a simply connected Kähler manifold of real dimension $2n\geq 4$.
We call a subset $U\subseteq M$ a *ball* if $U$ is open, homeomorphic to an open $2n$-ball in $\mathbb{R}^{2n}$ and has compact closure $\bar{U}$.
Since a ball $U$ satisfies all the assumptions in Corollary \[cor:changeofmetric\], Theorem \[thm:coneconstruction2\] and Theorem \[thm:degreelocal\], we obtain that the degree of mobility $D(g_{|U},J_{|U})$ of the restriction of the Kähler structure is given by one of the values in the list of Theorem \[thm:degree\].
If in addition $g$ is Einstein, it follows from Corollary \[cor:changeofmetriceinstein\], Theorem \[thm:coneconstruction2\] and Theorem \[thm:degreelocal\] that $D(g_{|U},J_{|U})$ is given by one of the values in the list of Theorem \[thm:degreeeinstein\].
Recall from [@ApostolovI Proposition 4] or [@Mikes equation (1.3)] that the space $\mathcal{A}(g,J)$ of hermitian symmetric solutions of equation is isomorphic to the subspace $\mathrm{Par}(E,\nabla^E)$ of the space of sections of a certain vector bundle $\pi:E\rightarrow M$ whose elements are parallel with respect to a certain connection $\nabla^E$ on $E$. In particular, we have $D(g,J)=\mathrm{dim}\,\mathrm{Par}(E,\nabla^E)$.
The next statement will complete the proof of the first parts of the Theorems \[thm:degree\] and \[thm:degreeeinstein\].
\[lem:extensiontoglobal\] Let $M$ be a simply connected manifold and let $\pi:E\rightarrow M$ be a vector bundle over $M$ with a connection $\nabla^E$. Let $I\subseteq \mathbb{N}$ be a set of nonnegative integers and suppose that for every ball $U\subseteq M$, we have $\mathrm{dim}\,\mathrm{Par}(E_{|U},\nabla^E)\in I$. Then, $\mathrm{dim}\,\mathrm{Par}(E,\nabla^E)\in I$.
![We can choose a tubular neighborhood $U$ of the union $\bigcup_{i=1}^N c_i([0,1])$ of the curves $c_1,...,c_N:[0,1]\rightarrow M$ such that $U$ is a ball.[]{data-label="2"}](parallel){width=".6\textwidth"}
First let us introduce some notions. Let $p\in U$, where $U$ is any simply connected open subset of $M$, and let $H(U,p)$ be the holonomy group of the restriction $\nabla^E:\Gamma(E_{|U})\rightarrow \Gamma(T^* U\otimes E_{|U})$ in the point $p$. The space $\mathrm{Par}(E_{|U},\nabla^E)$ is isomorphic to the holonomy-invariant elements $P(U,p)=\{u\in E_p:hu=u\,\forall h\in H(U,p)\}$ in the fiber $E_p$, the isomorphism is given by parallel extension of $u\in P(U,p)$ to a parallel section on $U$. Since $U$ is simply connected, the group $H(U,p)$ is connected. Then, $P(U,p)$ coincides with $$\{u\in E_p:hu=0\,\forall h\in\mathfrak{h}(U,p)\},$$ where $\mathfrak{h}(U,p)$ is the Lie algebra of $H(U,p)$. Let $R^E\in \Gamma(\Lambda^2 T^*M\otimes\mathrm{End}(E))$ be the curvature of $\nabla^E$. By the theorem of Ambrose-Singer (see e.g. [@Kob]), $\mathfrak{h}(U,p)$ as a vector space is generated by elements of the form $$\begin{aligned}
(\tau_c)^{-1}R^E(X,Y)\tau_c:E_p\rightarrow E_p,\label{eq:generators}\end{aligned}$$ where $X,Y\in T_q M$, $q\in M$, and $\tau_c:E_{c(0)}\rightarrow E_{c(1)}$ is the parallel displacement along a certain piece-wise smooth curve $c:[0,1]\rightarrow U$ with $c(0)=p$ and $c(1)=q$.
Since $\mathfrak{h}(M,p)$ is a finite-dimensional vector space there exist finitely many curves $c_1,...,c_N:[0,1]\rightarrow M$ starting at $p$ such that $\mathfrak{h}(M,p)$ as a vector space is generated by finitely many elements of the form $$\begin{aligned}
(\tau_{c_i})^{-1}R^E(X,Y)\tau_{c_i}:E_p\rightarrow E_p.\label{eq:generators1}\end{aligned}$$ If we sligtly perturbe these curves, the corresponding elements will still generate $\mathfrak{h}(M,p)$ so we may assume that the curves have no intersections and self-intersections. Then, a sufficiently thin tubular neighborhood $U$ of the union of the curves $c_i$ is a ball, see figure \[2\].
The degree of mobility of the restriction of the Kähler structure to the ball $U$ clearly coincides with the degree of mobility of the Kähler structure on the whole manifold since the holonomy groups have the same algebras and therefore coincide.
Proof of the second “realization” part of Theorems \[thm:degree\] and \[thm:degreeeinstein\] {#sec:realization}
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Let $2n\geq 4$, $k\in \{0,...,n-1\}$ and $\ell\in\{1,...,[\frac{n+1-l}{2}]\}$. We need to construct a $2n$-dimensional simply-connected Riemannian Kähler manifold $(M,g,J)$ such that $D(g,J)=k^2+\ell$ and such that in the case $k^2+\ell\ge 2$ there exists a metric $\tilde g$ that is c-projectively but not affinely equivalent to $g$.
The construction is as follows: we consider the direct product $$\begin{aligned}
(\hat{M},\hat{g},\hat{J})=(M_0,g_0,J_0)\times(M_1,g_1,J_1)\times...\times(M_\ell,g_\ell,J_\ell)\label{eq:decompmhat}\end{aligned}$$ of Riemannian Kähler manifolds. The manifold $(M_0,g_0,J_0)$ is the standard $R^{2k}$ with the standard flat metric and the standard complex structure. The Riemannian Kähler manifolds $(M_i,g_i,J_i)$ for $i\ge 1$ satisfy the following conditions: they have dimension $\ge 4$, admit no nontrivial parallel hermitian symmetric $(0,2)$-tensor field, are cone manifolds, and the sum of their dimensions is $2(n+1- k)$. The existence of such $(M_i,g_i,J_i)$ is trivial: because of the condition $\ell\in\{1,...,[\frac{n+1-l}{2}]\}$ there exists a decomposition of $2(n+1- k)$ in the sum of the integer numbers $2k_1+...+2k_\ell$ such that every $k_i\ge 2$. Now, as the manifold $(M_i,g_i,J_i)$ we take the conification of the standard $(R^{2k_i-2}, g_{flat}, J_{standard})$. They are cone manifolds and they admit no nontrivial parallel hermitian symmetric $(0,2)$-tensor fields since for example by Theorem \[thm:coneconstruction2\] the existence of such a tensor field will imply that the constant $B$ of the standard $(R^{2k_i-2}, g_{flat}, J_{standard})$ is $B=-1$ though it is equal to zero.
Let us also note, in view of the proof of Theorem \[thm:coneconstruction1\], that $(M_0,g_0,J_0)=(\mathbb{R}^{2k}\setminus \{0\},g_{flat},J_{0})$ coincides (at least locally) with the conification of $(\mathbb{C}P(k-1),g_{FS},J_{standard})$ via the Hopf fibration $S^{2k-1}\rightarrow \mathbb{C}P(n)$.
The direct product is clearly a Riemannian Kähler manifold. By [@FedMat Lemma 5], it is a cone manifold, so by Theorem \[thm:coneconstruction1\] it is (at least in a neighborhood of almost every point) the conificiation of a certain $2n$-dimensional Kähler manifold. This manifold has degree of mobility $k^2+\ell$ since the dimension of parallel symmetric hermitian $(0,2)$ tensors on its conification (which is ) is $k^2+\ell$ as we want. This completes the proof of Theorem \[thm:degree\].
Now, in order to construct a Kähler-Einstein metric with degree of mobility $D(g,J)=k^2+\ell$ where $k\in \{0,...,n-2\}$ and $\ell\in\{1,...,[\frac{n+1-l}{3}]\}$ we proceed along the same lines of ideas used above but assume in addition that the manifolds $(M_i,g_i,J_i)$ are Ricci-flat (and as such manifolds we can take conifications of Kähler-Einstein manifolds with scalar curvature chosen in correspondence with section \[-4\]) and irreducible. The restrictions $k\in \{0,...,n-2\}$ and $\ell\in\{1,...,[\frac{n+1-l}{3}]\}$ imply that this is possible. This completes the proof of Theorem \[thm:degreeeinstein\].
Proof of Theorems \[thm:hprotrafo\] and \[thm:hprotrafoeinstein\] {#sec:thmhprotrafo}
=================================================================
We need to show that $\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))$ is given by one of the values in the list of Theorem \[thm:hprotrafo\] for a generic metric or by one of the values in the list of Theorem \[thm:hprotrafoeinstein\] if the metric is Einstein. We assume that the manifold is simply connected and that the Kähler metric is Riemannian.
Denote by $\mathfrak{h}(g,J)$ the Lie algebra of homothetic vector fields of $(M,g,J)$, i.e., vector fields $v$ satisfying $\mathcal{L}_v g=\mbox{const}\cdot g$, where $\mathcal{L}_v$ denotes the Lie derivative with respect to $v$. Consider the following sequence $$\begin{aligned}
0\rightarrow \mathfrak{h}(g,J)\hookrightarrow \mathfrak{c}(g,J)\overset{f}{\longrightarrow} \mathcal{A}(g,J)/\mathbb{R}g\rightarrow 0,\label{eq:sequence}\end{aligned}$$ where the mapping $f$ is given by $$f(v)=-\frac{1}{2}\left(\mathcal{L}_v g-\frac{\mathrm{trace}(g^{-1}\mathcal{L}_v g)}{2(n+1)}g\right)\mbox{ mod }\mathbb{R}g.$$ It is straight-forward to check that the tensor contained in the brackets on the right-hand side is indeed a solution of , for a proof see [@MatRos Lemma 2]. From the formula for $f$, it is straight-forward to see that $f(v)=0 \mbox{ mod }\mathbb{R}g$ if and only if $v$ is a homothetic vector field. Thus, the kernel of $f$ coincides with the image of the inclusion map from $\mathfrak{h}(g,J)$ to $\mathfrak{c}(g,J)$ and the sequence is exact at the first two stages. In particular, it follows that $$\begin{aligned}
\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{h}(g,J))\leq D(g,J)-1.\label{eq:inequality}\end{aligned}$$
From this inequality we see that in the case $D(g,J)=2$ the codimension of $\mathfrak{i}(g,J)$ in $\mathfrak{c}(g,J)$ is at most equal to one so $\dim(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))$ is $1$ or $0$. These two values are equal to $\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))$ for certain $2n\ge 4$-dimensional Riemannian Kähler-Einstein manifolds $(M, g,J)$ admitting a c-projectively equivalent metric which is not affinely equivalent to it. Indeed, most of the closed Riemannian Kähler-Einstein manifolds $(M,g,J)$ constructed in [@ApostolovII] that admit c-projectively equivalent metrics which are not affinely equivalent to $g$, are of non-constant holomorphic curvature and therefore admit no non-killing c-projective vector field by the Yano-Obata conjecture [@MatRos]. These examples have therefore $\dim(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))=0$. In order to consider the case $D(g, J)\ge 3$, and also to construct examples with $\dim(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))=1$, we need the following
\[-5\] Let $(M,g,J)$ be a connected Kähler manifold of real dimension $2n\ge 4$ such that the equation admits a solution $(A, \lambda, \mu)$ with $\lambda\ne 0$. Assume $B\ne 0$. Then, $$\begin{aligned}
\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))= D(g,J)-1.\label{eq:Dhprovec}\end{aligned}$$
Without loss of generality we can assume that $B=-1$. Let us first show that $g$ admits no (local) homothety which is not an isometry. Suppose $F:M\rightarrow M$ is a homothety for $g$, i.e. $F^* g=cg$ for a certain constant $c$. For the new metric $\tilde g= cg$, the system holds for the constant $\tilde B= B(cg)=\tfrac{1}{c}B$. But the constant $B$ is unique: since $cg$ and $g$ are isometric via $F$, the constants $B$ and $B(cg)$ (i.e., the constant $B$ corresponding to the metric $cg$) must coincide. It follows that $c=1$ and consequently every homothety is an isometry.
In view of this, the sequence reads $$\begin{aligned}
0\rightarrow \mathfrak{i}(g,J)\hookrightarrow \mathfrak{c}(g,J)\overset{f}{\longrightarrow} \mathcal{A}(g,J)/\mathbb{R}g\rightarrow 0,\label{eq:sequence1}\end{aligned}$$ Let us now show that the sequence is exact which of course immediately implies the equality .
In order to do it, we show the existence of a splitting $$h:\mathcal{A}(g,J)/\mathbb{R}g\rightarrow\mathfrak{c}(g,J)$$ of the sequence . The mapping $h$ is explicit and sends a solution $A$ to the corresponding vector field $\Lambda=g^{-1}\lambda$. Using the system , it is straight-forward to check that $\Lambda$ is a c-projective vector field (for an explicit proof see [@Tanno1978 Proposition 10.3]). Moreover, the vector field $\Lambda$ is the same for $A$ and $A+\mbox{const}\cdot g$, hence, $h$ is well-defined and linear. For the composition of $f$ and $h$, we calculate $$f( h (A\mbox{ mod }\mathbb{R}g))=f(\Lambda)=-\frac{1}{2}\left(2\nabla\lambda-\frac{\mathrm{trace}(\nabla\Lambda)}{n+1}g\right)\mbox{ mod }\mathbb{R}g$$ $$\overset{\eqref{eq:hprosystem}}{=}-\frac{1}{2}\left(\frac{2\mu g-2(n+1)A+\mathrm{trace}(A)g}{n+1}\right)\mbox{ mod }\mathbb{R}g.$$ The third equation in implies $d\mu=-2\lambda=-\tfrac{1}{2}d\mathrm{trace}A$. Thus, the functions $2\mu$ and $-\mathrm{trace}A$ coincide up to adding a constant. This shows that $$f( h (A\mbox{ mod }\mathbb{R}g))=A \mbox{ mod }\mathbb{R}g,$$ implying that is a splitting exact sequence and holds.
The image of the map $h$ is precisely the “canonical” space of essential c-projective vector fields whose existence we announced in the introduction.
Combining Lemma \[-5\] with Theorem \[thm:degree\], we obtain the list from Theorem \[thm:hprotrafo\]. Combining Lemma \[-5\] with Theorem \[thm:degreeeinstein\], we obtain the list from Theorem \[thm:hprotrafoeinstein\] (under the additional assumption that $B\neq0$).
Note that by using Lemma \[-5\], also the values from the lists of Theorem \[thm:hprotrafo\] or Theorem \[thm:degreeeinstein\] can be obtained as the number $\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))$ since in Section \[sec:realization\] we also constructed metrics in all considered dimensions admitting solutions $(A, \lambda, \mu)$ of with $\lambda\ne 0$ such that their degree of mobility is two. The only case that cannot be constructed in this way is a $4$-dimensional Kähler-Einstein structure $(g,J)$ with $\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))=1$ since we can only produce examples of constant holomorphic curvature by using the procedure from Section \[sec:realization\]. We therefore construct an explicit example: consider the local $4$-dimensional Riemannian Kähler structure $(g,\omega,J=-g^{-1}\omega)$ given in coordinates $x,y,s,t$ by $$\begin{aligned}
\begin{array}{c}
g=(x-y)( dx^2+ dy^2)+\frac{1}{x-y}\left[\left(ds+x dt\right)^2+\left(ds+ y dt\right)^2\right],\vspace{1mm}\\
\hat{\omega}=dx\wedge(ds+y dt) +dy\wedge( ds+x dt).
\end{array}\nonumber\end{aligned}$$
This Kähler structure is a special case of those obtained in [@ApostolovI; @BMMR]. It is straight-forward to check that $g$ is Ricci-flat but non-flat and that the $(1,1)$-tensor $$\begin{aligned}
A=x\,\partial_x\otimes dx+y\,\partial_y\otimes dy+(x+y)\partial_s \otimes ds+xy\,\partial_s\otimes dt-\partial_t\otimes ds
\nonumber\end{aligned}$$ is contained in $\mathcal{A}(g,J)$ (when viewed as $(0,2)$-tensor) and is non-parallel (and thus, corresponds to a Kähler metric $\tilde{g}$, that is c-projectively equivalent to $g$ and not affinely equivalent). Moreover, the vector field $v=x\,\partial_{x}+y\,\partial_{y}+2s\,\partial_s+t\,\partial_t$ is a c-projective vector field for $g$ and it is not Killing (thought, it is in fact an infinitesimal homothety, i.e. we have $\mathcal{L}_v g=3g$). For this metric we have $\dim(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))=1$.
Let us now consider the case $B=0$. In this situation, as in the proof of Theorem \[thm:degree\], we change the metric in the c-projective class to make $B$ non-zero. This can be done on every open connected subset with compact closure, see Corollary \[cor:changeofmetric\] and Corollary \[cor:changeofmetriceinstein\]. The next lemma shows that the number $\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))$ remains the same.
Let $(M,g,J)$ be a simply connected Kähler manifold of real dimension $2n\geq 4$. Then, $\mathrm{dim}\,\mathfrak{i}(g,J)=\mathrm{dim}\,\mathfrak{i}(\tilde{g},J)$ for any metric $\tilde{g}$ that is c-projectively equivalent to $g$.
We give a shorter version of the proof from [@EMN]. Let $K$ be a Killing vector field for $(g,J)$. It follows that $K$ is also symplectic for the Kähler $2$-form $\omega=g(.,J.)$. Since we are working on a simply connected space, every symplectic vector field arises from a hamiltonian function $f$, i.e. $K=X_f$, where $X_f$ is defined by $g(X_f,J.)=df$. The condition that $K$ is Killing is equivalent to the condition that $\nabla\nabla f$ is hermitian.
Let $\phi$ be the function given by and consider the function $\mathrm{e}^{2\phi} f$. Let us show that this function is the hamiltonian function for a Killing vector field for $\tilde{g}$. The geometry behind this statement is explained in [@EMN].
We need to show that the symmetric $(0,2)$-tensor field $\tilde{\nabla}\tilde{\nabla}(e^{2\phi}f)$ is hermitian.
First of all, it is well-known, see for example [@ApostolovI Proposition 3] or [@Kiyohara2010 Lemma 3.2], that the function $\mathrm{e}^{-2\phi}$ is the hamiltonian for a Killing vector field for $g$ and, swapping the metrics $g$ and $\tilde g$, that $\mathrm{e}^{2\phi}$ is the hamiltonian function for a Killing vector field for $\tilde{g}$. Consequently, its hessian $\tilde{\nabla}\tilde{\nabla}(e^{2\phi})$ is hermitian.
Using the transformation law , we calculate $$\tilde{\nabla}\tilde{\nabla}(e^{2\phi}f)=f(\tilde{\nabla}\tilde{\nabla}e^{2\phi})+e^{2\phi}(\tilde{\nabla} df+2\Phi\odot df)$$ $$=f(\tilde{\nabla}\tilde{\nabla}e^{2\phi})+e^{2\phi}(\nabla df+\Phi\odot df+\Phi(J.)\odot df(J.)).$$
We see that the right-hand side of the above equation is hermitian, thus, the Hamiltonian vector field of $e^{2\phi}f$ is a Killing vector field for $\tilde g$. If we choose another hamiltonian function $f+\mbox{const}$ for $K$, the mapping $$K\longmapsto \tilde{K}=\tilde{X}_{e^{2\phi}f},$$ where $\tilde{X}_f$ is defined by $\tilde{g}(\tilde{X}_f,J.)=df$, is only defined up to adding constant multiples of $X_{e^{2\phi}}$. Thus, $\mathrm{dim}\,\mathfrak{i}(g,J)$ coincides with $\mathrm{dim}\,\mathfrak{i}(\tilde{g},J)$ as we claimed.
Since obviously $\mathfrak{c}(g,J)=\mathfrak{c}(\tilde{g},J)$, it follows from the lemma that on each open simply connected neighborhood $U$, the number $\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))$ does not depend on the choice of the metric in the c-projective class. Suppose in addition that $U$ has compact closure. Then, by Corollary \[cor:changeofmetric\] there exists a Riemannian metric $\tilde{g}$ on $U$ which is c-projectively equivalent to $g$ and such that the system for $\tilde{g}$ holds with a constant $\tilde{B}=-1$. Thus, by the already proven part, when restricted to a simply connected open subset $U$ with compact closure, the number $\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))$ is given by one of the values from the list of Theorem \[thm:hprotrafo\] and, in the Einstein situation, it is given by one of the values from the list of Theorem \[thm:hprotrafoeinstein\].
In order to prove Theorem \[thm:hprotrafo\] and \[thm:hprotrafoeinstein\] on the whole manifold, we again use Lemma \[lem:extensiontoglobal\]. It is known that Killing vector fields could be viewed as parallel sections of a certain vector bundle. The same is true for c-projective vector fields, for example because c-projective geometry is a parabolic geometry, see for example [@CapGoverHammerl Section 3.3], [@cap Section 4.6] or [@hrdina], and infinitesimal symmetries of parabolic geometries are sections of a certain vector bundle. Actually, in [@EMN] the vector bundle and also the connection on it are explicitly constructed. By Lemma \[lem:extensiontoglobal\], the number $\mathrm{dim}(\mathfrak{c}(g,J)/\mathfrak{i}(g,J))$ on the whole manifold is the same as this number for the restriction of the Kähler structure to a certain ball and above we have shown that this value is contained in the list of Theorem \[thm:hprotrafo\] or, in the Einstein situation, in the list of Theorem \[thm:hprotrafoeinstein\].
Proof of Theorem \[thm:einstein\] {#sec:thmeinstein}
=================================
Let us first recall the following statement from [@haddad2] (and give a full proof since the publication is not easy to find)
\[lem:einsteincproeinstein2\] Let $g,\tilde{g}$ be c-projectively equivalent Kähler-Einstein metrics on the connected complex manifold $(M,J)$ of real dimension $2n\geq 4$. Then, for every $A\in \mathcal{A}(g,J)$ with corresponding $1$-form $\lambda$, there exists a function $\mu$ such that $(A,\lambda,\mu)$ satisfies with $B=-\frac{\mathrm{Scal}(g)}{4n(n+1)}$.
Denote by $A=A(g,\tilde{g})$ the solution of given by . We can insert the relation between the $1$-forms $\Phi$ in and $\lambda$ in into to obtain the change of the Ricci-tensors in terms of $A$ and $\lambda$. Denoting by $\Lambda=g^{-1}\lambda$ the vector field corresponding to $\lambda$, a straight-forward calculation shows $$\mathrm{Ric}(\tilde{g})=\mathrm{Ric}(g)+2(n+1)(g(A^{-1}\nabla\Lambda.,.)-g(A^{-1}\Lambda,\Lambda)g(A^{-1}.,.)).$$ Now suppose both metrics are Einstein, that is $\mathrm{Ric}(\tilde{g})=\tilde{c}\tilde{g}$ and $\mathrm{Ric}(g)=cg$ for constants $c=\tfrac{\mathrm{Scal}(g)}{2n},\tilde{c}=\tfrac{\mathrm{Scal}(\tilde{g})}{2n}$. Inserting this into the last equation and multiplying with $g^{-1}$ from the left yields $$\tilde{c}g^{-1}\tilde{g}=c\mathrm{Id}+2(n+1)(A^{-1}\nabla\Lambda-g(A^{-1}\Lambda,\Lambda)A^{-1}).$$ By , $\tilde{g}$ can be written as $\tilde{g}=(\mathrm{det}A)^{-\tfrac{1}{2}}gA^{-1}$. Inserting this into the last equation and multiplying with $A$ from the left, we obtain $$\tilde{c}(\mathrm{det}A)^{-\tfrac{1}{2}}\mathrm{Id}=cA+2(n+1)(\nabla\Lambda-g(A^{-1}\Lambda,\Lambda)\mathrm{Id}).$$ Rearranging terms yields $$\begin{aligned}
\nabla\Lambda=\mu \mathrm{Id}+BA,\label{eq:secondeqsystem}\end{aligned}$$ where we defined $$\begin{aligned}
\mu=\frac{\bar{c}(\mathrm{det}A)^{-\tfrac{1}{2}}}{2(n+1)}+g(A^{-1}\Lambda,\Lambda)\mbox{ and }B=-\frac{c}{2(n+1)}.\label{eq:functionandconstant}\end{aligned}$$ Equation is exactly the second equation in . It remains to show that the third equation on the function $\mu$ is satisfied as well. In [@FKMR Remark 5] it was noted that if the second equation in the system holds for $B$ equal to a constant, the third equation in is satisfied automatically. This is sufficient for our purposes, however, we show that the third equation can be obtained directly by taking the covariant derivative of . We obtain $$\nabla_X \nabla\Lambda=(\nabla_X\mu) \mathrm{Id}+B\nabla_X A\overset{\eqref{eq:mainA}}{=}(\nabla_X\mu) \mathrm{Id}+B(g(.,X)\Lambda+g(.,\Lambda)X+g(.,JX)J\Lambda+g(.,J\Lambda)JX).$$ Taking the trace of this equation yields $$\begin{aligned}
\mathrm{trace}(\nabla_X \nabla\Lambda)=2n\nabla_X\mu+4Bg(X,\Lambda).\label{eq:thirdeqderiv}\end{aligned}$$ As in the proof of Lemma \[lem:scalarB\], we can use that $J\Lambda$ is Killing to obtain the identity $\nabla_X\nabla \Lambda=-JR(X,J\Lambda)$. Together with the usual identities for the Ricci-tensor of a Kähler metric, this yields $$\mathrm{trace}(\nabla_X \nabla\Lambda)=-\mathrm{trace}(J R(X,J\Lambda))=-2\mathrm{Ric}(X,\Lambda)=-2cg(X,\Lambda),$$ where we used the Einstein condition in the last step. Inserting the above formula and $B$ from into , we obtain the third equation in .
We have shown that when $g,\tilde{g}$ are c-projectively equivalent Kähler-Einstein metrics, there exists a function $\mu$ and a constant $B$ such that the triple $(A=A(g,\tilde{g}),\lambda,\mu)$ satisfies for the metric $g$ (of course, by interchanging the roles of $g,\tilde{g}$ this can be obtained also in terms of $\tilde{g}$). Since in the case $D(g,J)=2$ every $A'\in \mathcal{A}(g,J)$ is a linear combination of $\mathrm{Id}$ and the solution $A$ from above, we obtain the proof of Lemma \[lem:einsteincproeinstein2\] for $D(g,J)=2$. On the other hand, in the case $D(g,J)\geq 3$, Lemma \[lem:einsteincproeinstein2\] follows as a direct application of Theorem \[thm:hprosystem\] above.
Combining Lemmas \[lem:einsteincproeinstein\] and \[lem:einsteincproeinstein2\], we obtain Theorem \[+2\].
Let us now prove Theorem \[thm:einstein\], that is, let us show that two c-projectively equivalent Kähler-Einstein metrics on a closed connected complex manifold have constant holomorphic curvature unless they are affinely equivalent.
We have shown that the triple $(A,\lambda,\mu)$, where $A$ is the tensor from constructed by the two metrics and $\lambda$ is the corresponding $1$-form from , satisfies the system for a certain constant $B$. If this constant is zero, we see from that the function $\mu$ is constant and $\nabla\lambda$ is parallel. Since $\lambda$ is the differential of a function and the manifold is closed, there are points where $\nabla\lambda$ is positively and negatively definite respectively (corresponding to the minimum and maximum value respectively of the function). Since $\nabla\lambda$ is parallel, it actually has to vanish identically, thus, $\lambda$ is parallel. Since it vanishes at points where the corresponding function is maximal, $\lambda$ has to be identically zero. Using Remark \[rem:affine\], this implies that the two Einstein metrics are affinely equivalent.
Now suppose that the metrics are not affinely equivalent. In particular, $B$ is not zero and the function $\mu$ is not constant. Using the equations from the system , we can succesively replace the covariant derivatives of $A$ and $\lambda$ to obtain that $\mu$ satisfies the third order system $$\begin{aligned}
\begin{array}{c}
(\nabla\nabla\nabla \mu)(X,Y,Z)=B[2(\nabla_{X}\mu)g(Y,Z)+(\nabla_{Z}\mu)g(X,Y)+(\nabla_{Y}\mu)g(X,Z)\vspace{2mm}\\-(\nabla_{JZ}\mu)g(JX,Y)-(\nabla_{JY}\mu)g(JX,Z)]
\end{array}.\label{eq:tanno}\end{aligned}$$ of partial differential equations. This equation was studied in [@HiramatuK; @Tanno1978]. There it was shown that the existence of non-constant solutions of this equation on a closed connected Kähler manifold implies that $B<0$ and the metric $g$ has constant holomorphic curvature equal to $-4B$. By interchanging the roles of $g$ and $\tilde{g}$, this statement holds for $\tilde{g}$ as well. This completes the proof of Theorem \[thm:einstein\].
**Acknowledgements.** {#acknowledgements. .unnumbered}
---------------------
We are grateful to D. V. Alekseevsky, D. Calderbank, M. Eastwood, A. Ghigi, V. Kiosak and C. Tønnesen-Friedman for discussions and useful comments to this paper. Also, we thank Deutsche Forschungsgemeinschaft (Research training group 1523 — Quantum and Gravitational Fields) and FSU Jena for partial financial support.
[99]{}
D. V. Alekseevsky, V. Cortes, T. Mohaupt, *Conification of Kähler and hyper-Kähler manifolds*, accepted to Comm. Math. Phys., arXiv:1205.2964, 2012
V. Apostolov, D. Calderbank, P. Gauduchon, *The geometry of weakly self-dual Kähler surfaces*, Compositio Math., [**135**]{} , no. 3, 279–322, 2003, MR1956815
V. Apostolov, D. Calderbank, P. Gauduchon, *Hamiltonian 2-forms in Kähler geometry. I. General theory*, J. Differential Geom. [**73**]{}, no. 3, 359–412, 2006
V. Apostolov, D. Calderbank, P. Gauduchon, C. Tønnesen-Friedman, *Hamiltonian 2-forms in Kähler geometry. II. Global classification*, J. Differential Geom. [**68**]{}, no. 2, 277–345, 2004
V. Apostolov, D. Calderbank, P. Gauduchon, C. Tønnesen-Friedman *Hamiltonian 2-forms in Kähler geometry. III. Extremal metrics and stability*, Invent. Math. [**173**]{}, no. 3, 547–601, 2008
V. Apostolov, D. Calderbank, P. Gauduchon, C. Tønnesen-Friedman, *Hamiltonian 2-forms in Kähler geometry. IV. Weakly Bochner-flat Kähler manifolds*, Comm. Anal. Geom. [**16**]{}, no. 1, 91–126, 2008
S. Bando, T. Mabuchi, *Uniqueness of Einstein Kähler metrics modulo connected group actions*, Algebraic geometry, Sendai, 1985, 11–40, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987, MR0946233
A. Besse, *Einstein manifolds*, Springer, 1987
A. V. Bolsinov, V. S. Matveev, T. Mettler, S. Rosemann, *Four-dimensional Kähler metrics admitting essential c-projective vector fields,* in preparation.
A. Čap, *Correspondence spaces and twistor spaces for parabolic geometries*, J. Reine Angew. Math. [**582**]{}, 143–172, 2005, MR2139714
A. Čap, A. R. Gover, M. Hammerl, *Holonomy reductions of Cartan geometries and curved orbit decompositions*, arXiv:1103.4497 \[math.DG\], 2011
G. De Rham, [*Sur la reductibilité d’un espace de Riemann,* ]{} Comment. Math. Helv. [**26**]{}, 328–344, 1952.
M. Eastwood, V. Matveev, K. Neusser, *C-projective geometry: background and open problems* , in preparation. A. Fedorova, V. Kiosak, V. Matveev, S. Rosemann, [*The only Kähler manifold with degree of mobility at least 3 is $(CP(n), g_{Fubini-Study})$*]{}, Proc. London Math. Soc., [**105**]{}, no. 1, 153–188, 2012, doi: 10.1112/plms/pdr053, 2012
A. Fedorova, V. Matveev, *Degree of mobility for metrics of lorentzian signature and parallel (0,2)-tensor fields on cone manifolds*, arXiv:1212.5807 \[math.DG\], 2012
H. Hiramatu, [*Integral inequalities in Kählerian manifolds and their applications*]{}, Period. Math. Hungar. [**12**]{}, no. 1, 37–47, 1981, MR0607627, Zbl 0427.53032.
N. J. Hitchin, A. Karlhede, U. Lindström, M. Rocek, *Hyper-Kähler metrics and supersymmetry*, Com. Math. Phys. 108 (4) 535–589, 1987, MR877637
J. Hrdina, *Almost complex projective structures and their morphisms*, Arch. Math. (Brno) [**45**]{}, no. 4, 255–264, 2009, MR2591680
K. Kiyohara, *Two classes of Riemannian manifolds whose geodesic flows are integrable*, Mem. Amer. Math. Soc. [**130**]{}, no. 619, viii+143 pp., 1997
K. Kiyohara, P. J. Topalov, *On Liouville integrability of h-projectively equivalent Kähler metrics*, Proc. Amer. Math. Soc. [**139**]{}, 231–242, 2011.
V. Kiosak, M. Haddad, *On A-harmonic Kähler spaces,* Geometry of generalized spaces, 41–45, Penz. Gos. Ped. Inst., Penza, 1992.
V. Kiosak, M. Haddad, *On holomorphic-projective transformations of A-harmonic Kähler spaces,* preprinted in Ukr. NIINTI 20.08.1991 no. 1217-UK91.
S. Kobayashi, K. Nomizu, *Foundations of Differential Geometry II*, John Wiley and Sons, Inc., 1996.
V. S. Matveev, *Gallot-Tanno theorem for pseudo-Riemannian manifolds and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist*, J. Diff. Geom. Appl. [**28**]{}, no. 2, 236–240, 2010
V. S. Matveev, P. Mounoud, *Gallot-Tanno Theorem for closed incomplete pseudo-Riemannian manifolds and applications*, Ann. Glob. Anal. Geom. [**38**]{}, 259–271, 2010
V. S. Matveev, [*Geodesically equivalent metrics in general relativity,*]{} J. Geom. Phys. [**62**]{}, no. 3, 675–691, 2012.
V. S. Matveev, S. Rosemann, *Proof of the Yano-Obata conjecture for h-projective transformations*, J. Differential Geom. [**92**]{}, no. 1, 221–261, 2012
J. Mikes, V. V. Domashev, *On The Theory Of Holomorphically Projective Mappings Of Kaehlerian Spaces*, Math. Zametki [**23**]{}, no. 2, 297–303, 1978
J. Mikes, *Holomorphically projective mappings and their generalizations.*, J. Math. Sci. (New York) [**89**]{}, no. 3, 1334–1353, 1998
A. Moroianu, *Lecture Notes on Kähler geometry*, http://www.math.polytechnique.fr/$\sim$moroianu/tex/kg.pdf
A. Moroianu, U. Semmelmann, *Twistor forms on Kähler manifolds*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) [**2**]{}, no. 4, 823–845, 2003, MR2040645
T. Otsuki, Y. Tashiro, *On curves in Kaehlerian spaces*, Math. Journal of Okayama University [**4**]{}, 57–78, 1954
U. Semmelmann, *Conformal Killing forms on Riemannian manifolds*, Math. Z. [**245**]{}, no. 3, 503–527, 2003, MR2021568
N. S. Sinjukov, [*Geodesic mappings of Riemannian spaces.*]{} (in Russian) “Nauka”, Moscow, 1979, MR0552022, Zbl 0637.53020.
N. S. Sinyukov, E. N. Sinyukova, *Holomorphically projective mappings of special Kählerian spaces*, (Russian) Mat. Zametki [**36**]{}, no. 3, 417–423, 1984, MR0767221
S. Tanno, *Some Differential Equations On Riemannian Manifolds*, J. Math. Soc. Japan [**30**]{}, no. 3, 509–531, 1978
Y. Tashiro, *On A Holomorphically Projective Correspondence In An Almost Complex Space*, Math. Journal of Okayama University [**6**]{}, 147–152, 1956
H. Wu, *On the de Rham decomposition theorem,* Illinois J. Math. [**8**]{}, 291–311, 1964.
K. Yano, [*Differential geometry on complex and almost complex spaces.*]{} International Series of Monographs in Pure and Applied Mathematics, Vol. [**49**]{}, A Pergamon Press Book. The Macmillan Co., New York 1965 xii+326 pp.
[^1]: We are grateful to D. Calderbank and C. Tønnesen-Friedman for pointing this out to us.
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abstract: |
The description of spontaneous symmetry breaking that underlies the connection between classically ordered objects in the thermodynamic limit and their individual quantum mechanical building blocks is one of the cornerstones of modern condensed matter theory and has found applications in many different areas of physics. The theory of spontaneous symmetry breaking however, is inherently an [*equilibrium*]{} theory, which does not address the [*dynamics*]{} of quantum systems in the thermodynamic limit. Here, we will use the example of a particular antiferromagnetic model system to show that the presence of a so-called thin spectrum of collective excitations with vanishing energy –one of the well-known characteristic properties shared by all symmetry-breaking objects– can allow these objects to also spontaneously break time-translation symmetry in the thermodynamic limit. As a result, that limit is found to be able, not only to reduce quantum mechanical equilibrium averages to their classical counterparts, but also to turn individual-state quantum dynamics into classical physics. In the process, we find that the dynamical description of spontaneous symmetry breaking can also be used to shed some light on the possible origins of Born’s rule.
We conclude by describing an experiment on a condensate of exciton polaritons which could potentially be used to experimentally test the proposed mechanism.
author:
- Jasper van Wezel
title: 'Quantum Dynamics in the Thermodynamic Limit.'
---
1: Introduction
---------------
Combining many elementary particles into a single interacting system may result in collective behaviour that qualitatively differs from the properties allowed by the physical theory governing the individual building blocks. This realisation –immortalised by P.W. Anderson in his famous phrase ’More is Different’ [@Anderson72]– not only forms the basis of much of the research being done in condensed matter physics today, but has also found applications in areas ranging from string theory to cosmology. The theory of Spontaneous Symmetry Breaking which formalises these ideas first took shape over fifty years ago [@Landau37; @Goldstone62; @Anderson63:book; @Anderson52; @Anderson58; @Nambu60], and was completed in the context of quantum magnetism only two decades ago by the detailed description of the classical state as a combination of thin spectrum states, emerging as $N \to \infty$ because of the singular nature of the thermodynamic limit [@Lieb62; @Kaiser89; @Kaplan90]. The same description of the classical state emerging from the thin spectrum has since been shown to also directly apply to the cases of quantum crystals, antiferromagnets, Bose-Einstein condensates and superconductors [@vanWezel05; @vanWezel06; @vanWezel07; @vanWezel07:SC; @Birol07].
The connection between the quantum mechanical properties of microscopic particles and the classical behaviour of symmetry broken macroscopic objects has now again come to the forefront of modern science because of our technological capability to create ever larger and heavier quantum superpositions in the laboratory. Superconducting flux qubits harbour counterrotating streams of supercurrent consisting of up to $10^{11}$ Cooper pairs [@Wal00; @Chiorescu00; @Mooij:flux03], while Bose Einstein condensates of the order of $10^5$ Rubidium atoms can be routinely brought into superpositions of different momentum states [@Anderson95; @Davis95; @Stenger99; @Kozuma99]; Young’s double slit experiment has now been done using $C_{60}$ molecules instead of single photons or electrons [@Zeilinger99]; and an experiment has even been proposed to create a Schrödinger cat-like state of a mesoscopic mirror superposed over a macroscopically discernible distance [@Marshall03].
Almost all of these experiments employ the rigidity associated with a spontaneously broken symmetry to create and manipulate their ’macroscopic’ superpositions. Roughly speaking, the typical setup consists of a well defined, symmetry broken object in isolation (a superconductor, Bose Einstein condensate or crystal) which is brought into superposition by coupling it to a carefully selected quantum state. Although the theory of spontaneous symmetry breaking can be used to understand the stability and rigidity of macroscopic classical states such as superconductors or crystals, it says nothing about the quantum dynamics of such objects interacting with microscopic quantum states. The reason is that the standard description of spontaneous symmetry breaking is an inherently [*equilibrium*]{} description: it explains how macroscopic operators (such as the order parameter) can acquire finite expectation values and still be in stable equilibrium, but it does not say anything about the [*dynamics*]{} of these objects away from equilibrium.
A theoretical framework which does addresses the interaction of a macroscopic object with its microscopic quantum mechanical environment, is the study of decoherence [@Zurek81; @Joos85; @CaldeiraLeggett]. The basic idea of decoherence is that the entanglement of a certain quantum state with the many states of its environment can lead that state to behave effectively classically as long as the environmental states remain unobservable. This phenomenon has many practical implications, not in the least in the field of quantum information technology, where decoherence forms the main hurdle to be overcome in the race towards a working quantum computer. In the description of the interaction of a single macroscopic object with a single quantum state however, the theory of decoherence cannot be applied. The problem is that decoherence has to always refer to the properties of an [*ensemble average*]{}: after deciding which of the environmental degrees of freedom cannot be measured, one has to trace them out of the full density matrix describing the combined system of object and environment. Doing this (partial) trace is exactly equivalent to taking the quantum mechanical expectation value of the operators describing the unobserved states, and as such is only defined within an ensemble and cannot be used to say anything about the outcomes of [*single-shot*]{} experiments [@Adler; @Bassi]. In this paper, we will develop a description of dynamical spontaneous symmetry breaking that is meant to augment the earlier theories of equilibrium spontaneous symmetry breaking and decoherence in the areas where these theories do not apply. It will describe the quantum dynamics of individual experiments in which macroscopic and microscopic systems are allowed to interact. We will find that the presence of thin spectrum states in symmetry-broken objects allows these systems to also spontaneously break the unitarity of quantum mechanical time evolution. This result explains why truly macroscopic objects do not dynamically delocalise even if they are allowed to interact and entangle with an observable quantum mechanical environment. At the same time it also sheds light on what happens if the classical state is forced into a superposition state by an interaction with a carefully chosen quantum state.
In section 2 we start out with a short review of the equilibrium theory of spontaneous symmetry breaking. The role of the thin spectrum and the singular nature of the thermodynamic limit will be highlighted. In section 3 we then review the theory of decoherence and point out why it refers only to ensemble averages. We then turn to dynamic spontaneous symmetry breaking in section 4, using a model antiferromagnetic system as an example. It is argued there that the thin spectrum states and the thermodynamic limit can cooperate to allow the spontaneous breakdown of quantum mechanical unitarity. The resulting dynamics of a single quantum state in the thermodynamic limit is studied. We then continue in section 5 by describing the fate of a macroscopic object that is forced into a quantum superposition through the interaction with a microscopic quantum state. The results are again clarified using the example of the model antiferromagnet, and are shown to shed new light on the emergence of Born’s rule. Finally, in section 6, we describe a possible experimental test of the ideas of sections 4 and 5 using a condensate of exciton polaritons. We end in section 7 with a summary and conclusions.
2: Equilibrium Spontaneous Symmetry Breaking
--------------------------------------------
Classically, spontaneous symmetry breaking just corresponds to the evolution from a high symmetry metastable state into a ground state with lower symmetry. Quantum mechanically however, the situation becomes a bit more involved. First of all, there are in general nonzero tunneling matrix elements between different symmetry broken states, so that strictly speaking time evolution should cause any symmetry broken state to spread out and restore its symmetry. In practise though, this finite lifetime of a symmetry broken state can be easily shown to be long compared to the age of the universe for any realistic macroscopic system. Secondly, the symmetry broken states of a finite size system do not have to be ground states. In fact, they usually are not even eigenstates of the system. To establish how the system can end up in a state that is not an eigenstate of the underlying Hamiltonian, we will here use the specific example of the Lieb-Mattis model [@Lieb62; @Kaiser89; @Kaplan90; @vanWezel06; @vanWezel07]. This model is defined by the Hamiltonian: $$\begin{aligned}
H_{\text{LM}}
&= \frac{2 J}{N} {\bf S}_A \cdot {\bf S}_B \nonumber \\
&= \frac{J}{N} \left[ S^2 - S_A^2 - S_B^2 \right].
\label{Hlm}\end{aligned}$$ Here $N$ spin-$\frac{1}{2}$s are distributed over a bipartite lattice, with ${\bf S}_{A/B}$ the total spin of the $A/B$ sublattice and $S^z_{A/B}$ its $z$-projection. Each spin on the $A$ sublattice thus has an interaction with every spin on the $B$ sublattice and vice versa. The positive interaction strength $J$ is divided by $N$ to make the model extensive. $S$ is the total spin of the combined sublattices: $S=S_A+S_B$. The reason for considering specifically the Lieb-Mattis model with its infinitely long ranged interactions, is that it captures the relevant physics of a broad class of Heisenberg models with short ranged interactions. To say that a particular model for an antiferromagnet is invariant under SU(2) spin rotations is equivalent to stating that its Hamiltonian commutes with the total spin operator: $\left[H,S^2 \right]=0$. It is thus immediately obvious that total spin is a good quantum number for any isotropic antiferromagnet and that all their eigenstates can be labelled by such a total spin quantum number. For the description of the collective properties of the system as a whole (i.e. strictly infinite wavelength), the total spin is the only relevant part of the Hamiltonian. As far as the total spin is concerned, the Lieb-Mattis model coincides exactly with all other antiferromagnetic models. That is to say, if one takes [*any*]{} model for an antiferromagnet with short ranged interactions (such as for example the nearest neighbour Heisenberg model) and looks at the model in Fourier space, then the $k=0$ and $k=\pi$ modes together form [*exactly*]{} the Lieb Mattis-Hamiltonian [@vanWezel06; @vanWezel07]. At the same time, the finite wavelength, $k\neq 0,\pi$ modes are gapped and dispersionless in the Lieb-Mattis model due to the infinite ranged interactions, which makes it ideally suited for studying just the collective behaviour of antiferromagnets. The discussion of this model can also be easily adapted to describe spontaneous symmetry breaking in quantum crystals, superconductors and Bose-Einstein condensates [@vanWezel06; @vanWezel07; @vanWezel07:SC; @Birol07].
From the second expression in equation it is immediately clear that the ground state of the Lieb-Mattis system is a singlet state with zero total spin. This non-degenerate ground state is isotropic in spin space and thus fully respects the symmetry of its Hamiltonian. The heart of the workings of spontaneous symmetry breaking lies in the realisation that every many-particle Hamiltonian which possesses a continuous symmetry that is unbroken in its ground state (such as the Lieb-Mattis Hamiltonian), gives rise to a tower of low-energy states called the *thin spectrum* [@vanWezel05; @vanWezel07]. The states in this thin spectrum represent global (infinite wavelength) excitations that can be seen as the centre of mass properties of the collective system [@vanWezel07; @Birol07]. In the present model of equation the thin spectrum consists of total spin states, which only cost an energy of order $J/N$ to excite. These states thus become degenerate with the symmetric ground state in the thermodynamic limit. They are called the [*thin*]{} spectrum of the model because of the vanishing weight that these states have in the partition function. Excitations that change the size of the sublattice spins are separated from the ground state by an energy gap of size $J$, and can thus be ignored in the present (low energy) discussion. Without loss of generality we also set the $z$ projection of the total spin to be zero from here on.
The crucial observation is now that the strength of the field needed to give rise to a fully ordered ground state depends on the total number of particles, $N$, in the system. Because the energy separation between two consecutive thin spectrum states scales as $1/N$, the field strength necessary to explicitly break the symmetry decreases with system size. In the thermodynamic limit (where $N \to \infty$) all of the thin spectrum states collapse onto the ground state to form a degenerate continuum of states. Within this continuum even an *infinitesimally* small symmetry breaking field is enough stabilise a fully ordered, symmetry broken ground state. The system is thus said to spontaneously break its symmetry in that limit. To make this explicit in the present model, we add a symmetry breaking staggered magnetic field to the Hamiltonian: $$\begin{aligned}
H_{\text{LM}}=\frac{2 J}{N} {\bf S}_A \cdot {\bf S}_B - B \left( S^z_A - S^z_B \right).
\label{HlmSB}\end{aligned}$$ The staggered magnetisation only has non-zero matrix elements between consecutive thin spectrum levels [@vanWezel07]: $$\begin{aligned}
\left< S' \left| S^z_A - S^z_B \right| S \right> & = \delta_{S'+1,S} f_S + \delta_{S'-1,S} f_{S'} \nonumber \\
& \simeq \frac{N}{4} \left( \delta_{S'+1,S} + \delta_{S'-1,S} \right),
\label{Bmatrix}\end{aligned}$$ where $f_S \equiv \sqrt{\{ S^2 [ \left(S_A+S_B+1\right)^2 -S^2 ] \} /\{4S^2-1\}}$, and the approximation in the last line holds if $S_A=S_B=N/4$ and $1 \ll S \ll N$ [@Kaiser89]. The Schrödinger equation for the Lieb-Mattis model, $H_{\text{LM}} |n\rangle = E^n |n\rangle$, can be expanded in the total spin basis using $|n\rangle \equiv \sum_S u_S^n |S\rangle$. Upon taking the continuum limit it then reads $$\begin{aligned}
-\frac{1}{2}\frac{\partial^2}{\partial S^2} u_S^n + \frac{1}{2} \omega^2 S^2 u_S^n = \nu^n u_S^n,
\label{un}\end{aligned}$$ with $\omega=2/N \sqrt{J/B}$ and $\nu^n = 2 E^n / (B N) + 1$. This equation describes a harmonic oscillator and its eigenfunctions are given in terms of the well known Hermite polynomials. The expansion of these harmonic wavefuntions in the total spin basis brings to the fore the crucial role played by the thin spectrum in the mechanism of spontaneous symmetry breaking: because the total spin states all become degenerate in the limit $N \to \infty$, it then becomes arbitrarily easy to create the antiferromagnetic Néel state $|n=0\rangle = \sum_S u_S^0 |S\rangle$. Mathematically this translates into the non-commuting limits for the equilibrium expectation values of the order parameter [@vanWezel07] $$\begin{aligned}
\lim_{N \to \infty} \lim_{B \to 0} \left<\frac{S_A^z-S_B^z}{N/2}\right> &= 0 \nonumber \\
\lim_{B \to 0} \lim_{N \to \infty} \left<\frac{S_A^z-S_B^z}{N/2}\right> &= 1.
\label{limits}\end{aligned}$$ The same instability can also been seen by looking at the energy of the ground state in the presence of the symmetry breaking field. That energy is proportional to $-NB$ and thus an infinite amount of energy could be gained in the thermodynamic limit by aligning with an infinitesimally small symmetry breaking field.
An alternative, equivalent way of phrasing this singular property of the thermodynamic limit is to say that the limits of equation imply that even in the absence of $B$, quantum fluctuations of the order parameter which tend to disorder the symmetry broken state take an infinitely long time to have any measurable effect on a truly macroscopic system. Under equilibrium conditions, the system will thus be stable in a symmetry broken state that is not an eigenstate of its Hamiltonian.
Strictly speaking equation only allows truly infinite-size systems to spontaneously select a direction for their sublattice magnetisation. A large, but not infinitely large, system requires a finite symmetry breaking field to stabilise one of the symmetry broken states over the exact ground state. A true staggered magnetic field that points up on each site of the $A$ sublattice and down on the $B$ sublattice does not exist in nature. Because the strength of the required field becomes increasingly weaker as the size of the antiferromagnet grows, it can be argued however that [*any*]{} field which has a component that resembles a staggered magnetic field will be enough to stabilise the symmetry broken state in a large enough antiferromagnet. Such a weak staggered field could be provided in practise by magnetic impurities, local fields or even by a second antiferromagnet at an ever increasing distance from the first.
3: Decoherence
--------------
We have seen in the last section how spontaneous symmetry breaking enables a macroscopic collection of quantum mechanical particles to occur in an effectively classical symmetry broken state under equilibrium conditions. A different route from quantum mechanics to effectively classical behaviour is provided by the process of decoherence. Decoherence happens on all length scales (i.e. it does not require the object of interest to be macroscopic), and is a direct consequence of the inability of observers to monitor each and every degree of freedom of a typical quantum environment. At the heart, decoherence is the process in which a carefully prepared quantum state gets entangled with different states in its environment. Because the observer cannot measure all states of the environment, he can see only part of the final entangled state, and this partial state looks effectively classical. In this section we will use the Lieb-Mattis model as an example to highlight the different conceptual steps involved in the decoherence process.
Consider the Hamiltonian of equation . Its eigenstates can be written as $|m,S\rangle \equiv |S_A=S_B=N/4-m/2,S\rangle$ (where we have assumed $S^z=0$ and $S_A=S_B$ without loss of generality). The excitations $m$ represent magnons or spin waves while the excitations $S$ form the thin spectrum of this model. Because the thin spectrum excitations make only a vanishingly small contribution to the free energy of the Lieb-Mattis antiferromagnet if $N$ is large, they will be very hard to observe experimentally (for relatively small $N$ the thin spectrum states of molecular antiferromagnets can and have been experimentally observed [@Waldmann03; @Waldmann05]). For large systems we can thus regard the thin spectrum as a ’quantum environment’ for the magnon excitations. To study decoherence in this system we will first prepare a superposition state in the magnon sector, then we will let the magnon and the thin spectrum excitations interact and become entangled, and finally we will disregard the thin spectrum states and find that magnon states on their own have become an effectively classical mixture.
To prepare the initial magnon superposition, let us assume that we can access the exact ground state of the $N$-spin system and subsequently let it interact with a separate two-spin singlet state $\sqrt{1/2} \left[ |\uparrow \downarrow \rangle - |\downarrow \uparrow\rangle \right]$ through the instantaneous interaction defined by: $$\begin{aligned}
H = \left\{ \begin{array}{ll}
\frac{2 J}{N} {\bf S}_A \cdot {\bf S}_B + J {\bf S}_1 \cdot {\bf S}_2 & \text{for}~t<0 \\
\frac{2 J}{N+2} \left({\bf S}_A + {\bf S}_1 \right) \cdot \left( {\bf S}_B + {\bf S}_2 \right) & \text{for}~t>0.
\end{array} \right.
\label{Hint}\end{aligned}$$ Here ${\bf S}_{1/2}$ refer to the two initially separated spins, and the interaction is turned on at time $t=0$. In terms of the eigenstates of the Hamiltonian at positive times, the initial state can easily be shown to correspond to the state $\sqrt{1/2} \left[|m=0,S=0\rangle - |m=2,S=0\rangle \right]$ for large $N$ (where now $m$ and $S$ refer to the $N+2$-spin system). That is, for large $N$ the initial state of the two-spin system is encoded in the number and relative phase of the magnon excitations in the final state [@masterthesis].
Next, we would like to entangle the magnons with the thin spectrum so that the quantum information initially encoded in the magnon states is spread out over the environment. One way of achieving this is to instantaneously introduce a symmetry breaking field $B \left( S^z_A + S_1^z - S^z_B - S_2^z \right)$ into the Hamiltonian at some positive time $t_0$. After some straightforward algebra the state of our systems at times $\tau=t-t_0$ is then found to be $$\begin{aligned}
\hspace{-6pt} \left| \psi \right> = \sqrt{\frac{1}{2}} \sum_{n,S} u_S^n u_0^n \left[ e^{-\frac{i}{\hbar}E^n_0 \tau} \left| 0,S \right> - e^{-\frac{i}{\hbar}E^n_2 \tau} \left| 2,S \right> \right]
\label{state}\end{aligned}$$ where $u_S^n$ are the harmonic wavefunctions defined in equation and $E^n_m$ is the energy of the $n^{\text{\tiny{th}}}$ harmonic wavefunction in the presence of $m$ magnons. We can write this final entangled state in the form of a density matrix through the definition $\rho(\tau) = | \psi \rangle \langle \psi |$. Notice that all the quantum information encoded in the initial two-spin singlet state is still present in the final density matrix $\rho(\tau)$. Because purely quantum mechanical time evolution is always strictly unitary, time inversion symmetry is automatically preserved and there is always a way (at least in principle) to evolve the system back to its original state. If we now decide that the thin spectrum states are unobservable, and trace them out of our density matrix [@Neumann55], we end up with a reduced density matrix describing the dynamics of the magnons only. In doing so however, the time inversion symmetry is lost along with some of the quantum information. To be specific, the reduced density matrix $\rho_{\text{red}}$ will be given by: $$\begin{aligned}
\rho_{\text{red}}(\tau) &= \text{Tr}_{\text{thin}} \phantom{.} \rho(\tau) \nonumber \\
&= \sum_{S} \left<S \left| \right. \psi \right> \left< \psi \left| \right. S\right> \nonumber \\
&= \sum_{m,m'} \left|m\right> \left\{ \sum_S \psi^{\phantom{*}}_{m,S} \psi^{*}_{m',S} \right\} \left<m'\right|.
\label{reduced}\end{aligned}$$ In the last line we have written the entangled wavefunction as $|\psi\rangle=\sum_{m,S} \psi(m,S) |m\rangle|S\rangle$ to show explicitly that taking the partial trace over the thin spectrum states is equivalent to calculating the usual quantum mechanical ensemble-averaged expectation value with respect to these states.
To complete the analysis of our model interaction, we should explicitly calculate the reduced density matrix elements of equation . The diagonal elements of the resulting $2$x$2$ matrix are easily seen to be $1/2$. For the off-diagonal elements the calculation involves a summation over terms which differ only by the phase factor $e^{-\frac{i}{\hbar}(E^n_0-E^n_2) \tau}$. After some algebra one finds that these phases interfere destructively [@masterthesis], so that after a time $\tau_{\text{coh}}\sim \hbar / \sqrt{J B}$ the reduced density matrix becomes effectively diagonal. We thus find that the initial, pure density matrix loses its coherence and becomes a diagonal, mixed reduced density matrix within a time $\tau_{\text{coh}}$. Because for large enough $N$ the environmental states are unobservable this constitutes a ’for all practical purposes’ reduction from quantum to classical physics within the ensemble average. In any one single, individual experimental realisation of the above procedure however, one ends up with the full density matrix defined by equation , and one cannot use the expectation values of equation to conclude anything about that one specific experiment. In particular, in the classic Young’s double slit experiment, the observation that each single electron produces only a single dot on the photographic plate, can [*not*]{} be explained by invoking decoherence and averaging over the many degrees of freedom of the plate [@Tonomura89; @Adler; @Bassi].
Although the presence of the thin spectrum can lead to decoherence in real qubits [@vanWezel05], the interaction of the Lieb-Mattis antiferromagnet and the two-spin state considered in this section is of course a highly pathological example. In reality there will never be infinite ranged interactions, instantaneous changes to the Hamiltonian or full experimental control over the prepared states. Moreover, experiments typically involve finite temperatures and external environments that do not resemble the thin spectrum states of our model. However, the general idea of constructing a meaningful quantum superposition, letting it interact and entangle with its environment, and then looking only at the result averaged over the environmental degrees of freedom to find decoherence, remains essentially unaltered in more realistic situations [@CaldeiraLeggett]. In particular the conclusion that the the theory of decoherence is applicable only within the realm of ensemble averages remains intact throughout.
4: Dynamic Spontaneous Symmetry Breaking
----------------------------------------
As we have seen, both the theory of spontaneous symmetry breaking and the theory of decoherence have only a limited domain of applicability. Because macroscopic states typically have a lot of interaction with their environments, decoherence explains the reduction of pure macroscopic states to mixed states in situations where not all degrees of freedom can be explicitly monitored, but only in an (ensemble) averaged sense. Spontaneous symmetry breaking on the other hand can be used to demonstrate the stability of macroscopically ordered, classical states using the singular nature of the thermodynamic limit and the properties of the thin spectrum, but only under equilibrium conditions. The most general situation involving macroscopic objects –that of individual-state quantum dynamics in the thermodynamic limit– cannot be addressed within either of these frameworks.
In this section we will show that the presence of a thin spectrum in objects that can undergo spontaneous symmetry breaking also allows these objects to spontaneously break the (unitary) time translation symmetry of quantum mechanical time evolution. The resulting dynamical version of the process of spontaneous symmetry breaking naturally leads to the observed stability of macroscopic objects even in the presence of interactions with a quantum environment.
The approach to spontaneously breaking time translation symmetry is exactly analogous to the spontaneous breaking of more usual symmetries: we will introduce a vanishingly small non-unitary perturbation to the free Hamiltonian and demonstrate that this results in a qualitative change to the dynamics of a macroscopic object, even in the limit of taking the field strength to zero. The conclusion must thus be that the quantum dynamics of these macroscopic objects is infinitely sensitive to any non-unitary perturbation of the type considered. In other words: purely unitary quantum dynamics is unstable in the thermodynamic limit in the same way that the total spin singlet state of a macroscopic antiferromagnet is an unstable state under equilibrium conditions. As a result the unitary time translation symmetry of macroscopic quantum objects will be spontaneously broken and give rise instead to classical dynamics.
At this point one may wonder about the physical origin of the symmetry breaking field. As with the usual equilibrium symmetry breaking, large but finite sized systems will require a very small but nonetheless finite symmetry breaking field. Non-unitary fields however are strictly forbidden in quantum theory. The origin of a non-unitary symmetry breaking field must therefore lie outside of quantum mechanics. There are many possible candidates that could in principle insert a vanishingly small non-unitary correction into quantum mechanics. A notable example is the theory of general relativity, in which general covariance rather than unitarity is the guiding principle. Because of this, gravity has (in a different setting) been considered before as a possible non-unitary influence on mesoscopic systems [@Diosi89; @Penrose:96; @vanWezel:penrose]. In this paper we will not speculate about the possible origins of the non-unitary field, but merely recognise that there are non-unitary physical theories outside of the realm of quantum mechanics, and that only an infinitesimally small contribution from one of these sources would be enough to spontaneously break the unitarity of quantum dynamics in the thermodynamic limit.
We thus consider once again the Lieb-Mattis model for an antiferromagnet, but now in the presence of a non-unitary symmetry breaking field: $$\begin{aligned}
H=\frac{2 J}{N} {\bf S}_A \cdot {\bf S}_B + i b \left( S^z_A - S^z_B \right).
\label{Hdssb}\end{aligned}$$ The rationale of which specific form of non-unitary field is to be included in this equation is again exactly analogous to the case of equilibrium spontaneous symmetry breaking: one should in principle consider every conceivable field. The system will of course be stable with respect to the vast majority of them, but as long as there is one that has an effect in the limit in which its strength is sent to zero, the system will be unstable. In the equilibrium case considered before, we have seen that the symmetric singlet state is unstable with respect to a staggered magnetic field along the $z$-axis. We could have also considered other symmetry breaking fields, such as a uniform magnetic field along the $z$-axis. It is easy to show however that such a field would not lead to the non-commuting limits of equation . The Lieb-Mattis system is thus shown to be unstable under equilibrium conditions with respect to antiferromagnetic ordering, but not with respect to ferromagnetic ordering. The situation in the dynamical case is analogous: most fields have no effect on the quantum dynamics of the system if their strength is sent to zero; But as soon as there is one field that does influence the dynamics even if it is infinitesimally weak, the dynamics is found to be unstable. Notice also that in the equilibrium case, the antiferromagnet is in fact unstable towards staggered magnetic fields along any axis. In practise, the resulting orientation of the order parameter is therefore randomly chosen, just as in the case of classical symmetry breaking. In equation we have chosen a non-unitary version of the staggered magnetic field to break time translational symmetry, because to be able to have an effect in the thermodynamic limit, the symmetry breaking field must couple to the order parameter of the system. The orientation along the $z$-axis rather than any other axis is chosen for convenience only.
The time evolution operator $U(t) \equiv \exp (-iHt/\hbar)$ implied by equation has a non-unitary component, and thus no longer automatically conserves the total energy of the system (defined as $\langle H \rangle$ with $b \to 0$). This problem is automatically solved in the thermodynamic limit though. The staggered magnetisation only has non-zero matrix elements between consecutive states in the thin spectrum (see equation ). Since all thin spectrum states become degenerate with the ground state in the limit $N \to \infty$, the time evolution defined through $H$ cannot alter the total energy of the system in that limit. Other problems that are usually associated with non-unitary quantum dynamics (conservation of normalisability, commutativity, and so on) are likewise automatically solved in the limit of vanishing $b$ and large $N$.
To visualise the time evolution defined by $U(t)$, consider a general initial state $| \psi(t=0) \rangle = \sum_S \psi_S(t=0) |S\rangle$ (we again take $S_A$ and $S_B$ maximal and $S^z=0$). Using $| \psi(t) \rangle = U(t) | \psi(0) \rangle$ we then find the generalised (non-unitary) Schrödinger equation to be: $$\begin{aligned}
\hspace{-6pt} \dot{\psi}_S = \frac{-i}{\hbar}\frac{J}{N} S(S+1) \psi_S + \frac{b}{\hbar} \left( f_{S+1} \psi_{S+1} + f_S \psi_{S-1} \right)
\label{psidot}\end{aligned}$$ with $f_S$ the matrix elements defined in equation . This differential equation for the time evolution of a general initial wavefunction cannot easily be solved analytically (taking the limit in which $S$ becomes a continuous variable and $1 \ll S \ll N$, there is a solution in terms of Whittaker functions, but this explicit solution is not very enlightening for our present purposes). One can however integrate equation forward in time numerically, and we can study the effect of the unitarity breaking field by comparing the resulting time evolutions of different initial states. Two initial states of particular interest are the completely symmetric singlet state and the symmetry broken antiferromagnetic Néel state.
![(Color online) Left: The staggered magnetisation as a function of time. To make the plot the values $J=10$ and $b=1$ were used, and time was measured in units of $\hbar s$. The curves range from $N=20$ (rightmost curve) to $N=400$ (leftmost curve) and represent the evolution starting from the completely symmetric singlet state.\
Right: The dependence of the halftime on the parameters of the model. The top plot shows that $t_{1/2} \propto 1/b$, the middle plot that $t_{1/2} \propto 1/N$, and the bottom plot that $t_{1/2}$ is independent of $J$.[]{data-label="symplot"}](Fig1){width="\columnwidth"}
In the case of the symmetric initial state the time evolution of equation leads the unitarity breaking field to amplify the weight of states with a finite order parameter (i.e. its component in the wavefunction becomes a monotonously increasing exponential function), so that a fully ordered state is quickly formed. In figure \[symplot\] the time evolution of the order parameter is shown for different values of $b$, $J$ and $N$. It is immediately clear that the half-time associated with the reduction towards an ordered state must be proportional to $1/(Nb)$, so that the thermodynamic limit in this case is found to be a singular limit: if we let $b$ go to zero before sending $N$ to infinity, the symmetric singlet state remains an eigenstate of $H$ and under time evolution it can only pick up a total phase; if on the other hand even just an infinitesimally small field $b$ is present while the thermodynamic limit is taken, the time evolution governed by $H$ gives rise to an instantaneous reduction of the symmetric state to the fully ordered state with the order parameter pointing in the direction of $b$. Analogous to the equilibrium description, this non-commuting order of limits signals the sensitivity of the system to even infinitesimally small perturbations. In this case it is the unitary time translational symmetry of quantum dynamics itself that is spontaneously broken, and as a result the symmetric singlet state will be spontaneously and instantaneously reduced to an ordered Néel state.
![(Color online) Left: The staggered magnetisation along the $z$ axis as a function of time. To make the plot the values $J=10$ and $N=200$ were used, and time was measured in units of $\hbar s$. The curves range from $b=0.1$ (rightmost curve) to $b=2$ (leftmost curve) and represent the evolution starting from the state with full antiferromagnetic order along the $x$ axis.\
Right: The dependence of the halftime on the parameters of the model. The top plot shows that $t_{1/2} \propto \sqrt{1/b}$, the middle plot that $t_{1/2}$ is independent of $N$, and the bottom plot that $t_{1/2} \propto \sqrt{1/J}$.[]{data-label="xtozplot"}](Fig2){width="\columnwidth"}
Starting from a fully ordered initial state, the picture changes drastically. The state which has antiferromagnetic order aligned with the field $b$ to start with, will not be influenced at all. That state is just a stable state with respect to the generator of time evolution $U(t)$. The evolution of the initial state with full Néel order along the $x$ axis (at a $90$ degree angle with the field $b$) is shown in figure \[xtozplot\]. The effect of the presence of the unitarity breaking term is clearly to align the initial order parameter with the field $b$. The timescale on which this process takes place however is proportional to $\sqrt{1/(Jb)}$. This time is just the ergodic time of the Lieb-Mattis system and it becomes infinitely long in the thermodynamic limit with a vanishing symmetry breaking field. The difference between this ’turning time’ and the ’ordering time’ of the symmetric state considered before is due to the fact that for large objects any fully ordered state becomes exactly orthogonal to all differently ordered states, while the symmetric state always keeps a finite overlap with all of them [@Anderson:SolidState]. The lifetime of the symmetric state is therefore determined simply by the strength of the amplification due to the unitarity breaking field, while the turning time of a fully ordered initial state is set by the ergodic time of the system. Starting from the ordered state, the limit $N \to \infty$ is thus no longer singular: regardless of the size of $N$, the limit $b \to 0$ will reduce any dynamics to just the standard quantum mechanical time evolution. The dynamics of the ordered state, in other words, is stable with respect to the unitarity breaking field $b$.
Summarising, it has become clear that even an infinitesimally small unitarity breaking field is enough in the thermodynamic limit to instantaneously convert a symmetric initial state into a fully ordered state. Once such an ordered state has been formed however, it is stable with respect to any differently aligned unitarity breaking field. The former instability explains why the interaction with its environment cannot cause the wavefunction of a macroscopically ordered state to spread. After all, the more symmetric, spread-out wavepacket would be an unstable state, and it would spontaneously and instantaneously be brought back to the ordered state. At the same time the stability of the macroscopically ordered state itself ensures that such a state cannot spontaneously change the direction of its order parameter.
In the above analyses we have only considered symmetry breaking fields that are constant in time. Because of the dynamical nature of the spontaneous symmetry breaking process, it would actually be more natural to also include time dependent non-unitary fields. Since the strength of the field is taken to be infinitesimal, the time dependence of such a field must lie in its spatial orientation. As we have seen, ordered states are stable with respect to any orientation of the symmetry breaking field, and will thus also be stable with respect to a fluctuating field. The symmetric state on the other hand is sensitive to the direction of the field $b$: it is along this direction that the ordered state is formed. A fluctuating symmetry breaking field will thus cause the quantum dynamics of a symmetric state to amplify different orientations of the order parameter at different times. As a result both the direction and the size of the overall staggered magnetisation will undergo a random walk. As soon as the size of the magnetisation is large enough however, the dynamics again reduces to that of the ordered state, and the influence of the symmetry breaking field will no longer be felt. Because the symmetric state reacts infinitely fast to an infinitesimal perturbation in the thermodynamic limit, the whole process of undergoing a random walk and picking out an orientation for the order parameter will still be effectively instantaneous, and the earlier conclusions about the stability of quantum dynamics in the thermodynamic limit remain unaltered even in the presence of a fluctuating field.
5: Macroscopic Superpositions and Born’s Rule
---------------------------------------------
Having established that a macroscopically ordered state is stable and will not be driven into a quantum superposition of differently ordered states by its environment, the question arises what would happen to a macroscopic system that is forced into a superposition by some strong external force. Instead of a gentle and continuous spreading of the wavepacket (such as the one caused by the environment, which is subject to the instability discussed before), consider a quantum mechanical operation which quickly drives a macroscopic system into a superposition of ordered states with well separated orientations of their order parameters (the instantaneous coupling of the order parameter to a quantum superposition would in general do the trick). To be specific, consider the initial state $$\begin{aligned}
\left| \psi(0) \right> = \alpha \left| AFM \right>_x + \beta \left| AFM \right>_z.
\label{psi0}\end{aligned}$$ Here $|AFM\rangle_x$ signifies an antiferromagnetic Néel state ordered along the $x$ axis. The time evolution of the order parameter measured along the $z$ axis, starting from the initial state with $\alpha=\beta=\sqrt{1/2}$ is shown in figure \[theta\]. Here we again consider a constant symmetry breaking field $b$ and the time evolution defined by equation .
![(Color online) The evolution of the order parameter as a function of time (in units of $\hbar s$) for different constant orientations of the unitarity breaking field. Each set of three curves consists of different numbers of spins which are initially prepared in an equal-weight superposition of being ordered along the $z$ axis and along the $x$ axis. The angle $\theta$ between the unitarity breaking field and the $z$ axis $0.2~\pi$ for the upper set, $0.25~\pi$ in the middle and $0.3~\pi$ for the lowest set. The inset shows the fate of the order parameter in the thermodynamic limit, as a function of $\theta$.[]{data-label="theta"}](Fig3){width="0.8\columnwidth"}
The evolution of this initial state can be seen as a a combination of the two processes encountered before. First there is a fast reduction of the initial state to a single ordered state within a timescale $\propto 1/(Nb)$. The choice of which ordered state results from this fast initial evolution depends only on the chosen direction of the unitarity breaking field, and not on the weights of the different ordered states in $|\psi(0)\rangle$ (as can be seen in figure \[bornplot\]). After the fast reduction to a single ordered state, the slow process of rotating the order parameter towards alignment with the field $b$ takes over. This secondary process happens in a time which scales as $\propto \sqrt{1/(Jb)}$. In the limit that the number of particles goes to infinity before the unitarity breaking field is sent to zero, the result is thus a spontaneous, instantaneous reduction of the initial state to just a single one of the ordered states present in the original superposition.
The observation that the selection of the ordered state to be singled out by the spontaneous dynamics depends on the chosen (constant) orientation of $b$ signifies the fact that the initial state is unstable with respect to two different and competing perturbations: one for each orientation of the order parameter present in the initial superposition. The two possible stable final states are mutually exclusive since for any choice of unitarity breaking field, only one orientation of the staggered magnetisation results.
As mentioned before, the dynamical nature of the symmetry breaking process implies that we should really consider a time-dependent, fluctuating symmetry breaking field rather than only a constant field. In the presence of such a fluctuating field, it is clear that there must be a competition between the two instabilities of the initial state. In general, this gives rise to a statistical outcome of the reduction process (just like the instabilities of the singlet state gave rise to a statistical, random selection of the orientation of its order parameter under equilibrium conditions). The resulting dynamic process could be somewhat reminiscent of the evolutions considered in the GRW and CSL models for quantum state reduction [@Pearle89; @Ghirardi90], and consist of a random sequence of amplifying one or the other ordered state until one of them completely dominates.
![(Color online) The evolution of the order parameter as a function of time (in units of $\hbar s$) starting from the superposition state $\sqrt{ {1/5}} \left| AFM \right>_x + \sqrt{ {4/5}} \left| AFM \right>_z$. Each set of three curves represents the time evolution with three different values for $N$ in the presence of a single, constant orientation of the unitarity breaking field $b$. From top to bottom the angle between $b$ and the $z$ axis for the different sets is $0.1~\pi$, $0.2~\pi$, $0.25~\pi$, $0.3~\pi$ and $0.4~\pi$. The point at which the initial fast reduction process starts favouring the $x$ orientation over the $z$ orientation is seen to be at $0.25~\pi$.[]{data-label="bornplot"}](Fig4){width="0.8\columnwidth"}
It was shown recently by Zurek, using the concept of ENVariance [@Zurek03], that one can obtain conclusions about the statistics of the final results of a dynamic competition between instabilities such as the one considered here, without knowing the exact dynamics governing the competition process [@Zurek05]. It is shown in the appendix that Zurek’s proof is applicable here without the need for any assumptions regarding our system. Following his derivation one finds that the only possible result of the dynamic competition between different instabilities of the initial state of equation is the emergence of Born’s rule: the probability of a certain direction of the order parameter emerging from the process is given by the square of its weight in the initial wavefunction [@Born26]. Notice that this result is not an expectation value: it is valid even for the quantum dynamics of a single macroscopic object that is forced into a superposition state.
6: Experimental Predictions
---------------------------
The dynamic spontaneous symmetry breaking process described in the previous sections results in unaltered, purely unitary quantum dynamics for microscopic particles, but also gives rise to spontaneous and non-unitary effects in the thermodynamic limit. For truly macroscopic objects the non-unitarity will be effectively instantaneous, and the quantum dynamics of such objects correspondingly reduces to classical physics. Somewhere in between the micro and macro scales however, there must be a class of mesoscopic objects which are just sensitive enough to the presence of a small (but finite) time translation symmetry breaking field to undergo non-unitary dynamics on timescales that are measurable by human standards. The scale at which this happens should in fact be the same scale at which collections of interacting quantum particles become large enough to be meaningfully ascribed a (stable) orderparameter and considered classical, symmetry broken objects under equilibrium conditions. This prediction can in principle be exploited to experimentally test the ideas which are put forward in this paper.
The greatest obstacle in realising such an experimental test will be decoherence. Quite apart from the issue of its applicability to only ensemble averages, decoherence is of course a real physical phenomenon which severely complicates the observation of quantum effects in systems coupled to a reservoir. To observe the breakdown of unitary quantum dynamics, one will thus have to find a way to experimentally distinguish its effects from those of the usual environmental decoherence. The most obvious way of doing that is to look at single-shot experiments only. If the famous experiment of Zeilinger et al. [@Zeilinger99], interfering C$_{60}$ molecules one at a time, could be scaled up to truly macroscopic proportions, it would form the ideal testing ground for observing the transition from quantum to classical behaviour. The crossover scale could then be directly compared with the scale at which ordering and rigidity appear under equilibrium conditions, and this could be used to examine the role played by dynamic spontaneous symmetry breaking. Such macroscopic interference experiments however, seem to be very far from what can presently be experimentally realised.
We thus have to look for a different Schrödinger-cat like state of a mesoscopic system which is large enough to feel the effects of non-unitarity, but small enough to still have a measurably long reduction time. Creating such mesoscopic superpositions in the lab surely is not an easy task, but significant experimental progress towards its realisation is already being made in setups in for example quantum computation (superconducting flux qubits and Cooper pair boxes) or cold atom physics (Bose Einstein condensates in optical traps). Note however that the superposition must be a combination of states with different orientations of the order parameter itself. Superpositions of elementary excitations (such as magnons, phonons or supercurrents) which do not affect the order parameter, are not subject to the spontaneous reduction process described in the previous sections, even in a truly infinite system. Although distinguishing any non-unitary dynamics from the effects of decoherence is known to be very hard in most systems [@vanWezel:penrose], there is at least one experimental arena in which there seems to be, at least in principle, an opportunity for doing so: the Bose-Einstein condensation of exciton polaritons in semiconductor microcavities [@Kasprzak06; @Keeling07; @Wouters07].
Exciton polaritons are composite particles built partly from particle-hole pairs (excitons) and partly from photons. This unique combination of light and matter allows the particles to have strong interactions (due to their excitonic nature) while also being susceptible to direct experimental manipulation (due to their coupling to light). Although the short lifetime of the excitons implies that the condensate formed from polaritons in semiconductor microcavities is necessarily in a dynamical rather than a thermal equilibrium, it has been shown that the condensed phase shares many properties of the usual atomic Bose-Einstein condensate: it is a coherent state of spontaneously broken symmetry with an associated Goldstone mode [@Wouters07]. Recently it has been proposed that the dynamical nature of the polariton condensation can be used to explicitly break the U(1) phase symmetry present in a continuously, resonantly pumped experiment using an additional continuous probing laser [@Wouters07; @Amo07]. The coherence of the condensate can be independently tested by looking at the coherence and polarisation of the light emitted by recombining excitons [@Ciuti01]. If the pumping power is large enough to create a polariton condensate in a truly classical, symmetry broken state, then the condensate should retain its coherence even after the probing laser has been turned off. At lower power the condensate wavefunction will instead spread out over phase space and look symmetric again. Building on these results, the following experiment comes to mind. One can use the lack of number conservation in the condensate’s dynamical equilibrium [@Amo07], to create a superposition of different order parameters by subjecting the polaritons to a superposition of two different probing laser beams. The resulting macroscopic superposition is then expected to spontaneously collapse into just one ordered state for high enough pumping power due to dynamical spontaneous symmetry breaking, while lower pumping power (and the absence of symmetry breaking) should lead only to quantum beatings between the states of the initial superposition. If the transition from collapse behaviour to quantum beatings occurs at the same pumping power at which a single condensate has been seen to remain stable after turning off the probing laser, that would form a strong experimental indication of the involvement of dynamic spontaneous symmetry breaking.
7: Conclusions
--------------
In summary, we have shown here that macroscopic objects which spontaneously break a continuous symmetry under equilibrium conditions are also subject to a spontaneous breakdown of quantum mechanics’ unitary time translation symmetry. The coincidence of objects liable to dynamic spontaneous symmetry breaking with those liable to equilibrium spontaneous symmetry breaking is ensured by the crucial role played by the thin spectrum which is known to characterise the latter objects. Dynamic spontaneous symmetry breaking augments the well known theories of equilibrium spontaneous symmetry breaking and decoherence in the domains where these theories do not apply, and so leads to the symmetry broken state being not just the only stable ground state under equilibrium conditions, but also the only stable state dynamically. The quantum dynamics of any symmetric state, and more generally any superposition of differently ordered states, is almost infinitely sensitive to non-unitary perturbations in the thermodynamic limit, and such states must thus spontaneously and instantaneously be reduced to a state with only a single order parameter.
Applying this description of dynamic spontaneous symmetry breaking to the ordered states in our classical world, it becomes clear why these ordered classical states do not seem to be bothered by the interaction with their quantum environments: any buildup of quantum uncertainty is immediately reduced by the dynamical symmetry breaking process. Using the description instead to study the fate of a superposition of different classical states, one finds that only a single state can survive the spontaneous breakdown of quantum dynamics, and that the probability for finding any one particular outcome must be given by Born’s rule.
The predicted spontaneous breaking of unitary quantum time evolution can in principle be tested experimentally if one has a controlled way of constructing superpositions of differently ordered mesoscopic states. One type of system in which this may possibly be achieved is given by the polariton condensates in which the phase of the order parameter can be selected using the coherence of an incident laser beam.
Acknowledgements
----------------
I would like to gratefully acknowledge countless stimulating and insightful discussions with Jeroen van den Brink and Jan Zaanen.
Appendix A: Quantum Measurement
-------------------------------
In the main text we investigated the stability of a macroscopic state created by a quantum mechanical operation which quickly drives an ordered system into a superposition of differently ordered states with well separated orientations of their order parameters. One instance in which such a process is believed to occur is quantum measurement. By its very nature a quantum measurement is defined to be a process in which some property of a microscopic quantum state is translated into a specific pointer state of a macroscopic measurement machine [@Zurek81; @Joos85]. The different pointer states of such a machine must be easily distinguishable, classical states. In practise this always implies that they are symmetry broken states with different values or orientations for their order parameters. If we take these properties of the measurement machine at face value then it is clear that the measurement of a superposed quantum state must also lead to a superposition of pointer states in the measurement device because of the unitarity of quantum mechanical time evolution [@Bassi]. This simple observation already lead John von Neumann to postulate a collapse process which takes place after the usual quantum mechanical time evolution, and acts only on macroscopic superpositions [@Neumann55]. The explanation of why the collapse process exists, why it only acts on pointer states and not on microscopic states and why it gives rise to Born’s rule (dictating the probability of a certain outcome) is known as the quantum measurement problem. Many attempts have been made to either introduce a specific collapse process into quantum mechanics or to avoid the problem altogether by interpreting the mathematics of quantum mechanics in a different way. However, neither of these approaches has yet lead to a satisfactory resolution of all of the questions posed by the measurement problem.
Our analysis of the quantum dynamics of a superposition of differently ordered states in the thermodynamic limit suggests the following description of quantum measurement: a quantum measurement machine is any system with a well developed order parameter that can be coupled to a microscopic quantum system in such a way that the orientation of the order parameter after the coupling process has been completed, represents the property of the microscopic state that is to be measured. In general such a coupling should give rise to macroscopic superpositions of the order parameter, but the dynamical, spontaneous breakdown of quantum mechanics’ unitary time evolution ensures the spontaneous reduction of such superpositions into a state with just a single well defined order parameter. Because the macroscopic superposition state is subject to multiple competing instabilities, the outcome of the reduction process is probabilistic. The probability for obtaining a specific outcome is automatically guaranteed to agree with Born’s rule due to the properties of the process of dynamic spontaneous symmetry breaking (see also Appendix B).
Using dynamic spontaneous symmetry breaking, we have arrived at a clear-cut definition of what a measurement machine is; why it is subject to a collapse process; why this collapse does not influence microscopic quantum states; and we have recovered Born’s rule. The quantum measurement problem is thus reduced to the problem of identifying possible sources of non-unitary perturbations to the theory of quantum mechanics, which could drive the dynamic spontaneous symmetry breaking process. Regardless of its source, any non-unitary influence which can couple to a suitable order parameter will be amplified by the symmetry breaking process, and yield the expected macroscopic dynamics.
Appendix B: Detailed Derivation of Born’s Rule
----------------------------------------------
In this appendix we will give the detailed derivation of the emergence of Born’s rule from the dynamic spontaneous breaking of quantum mechanical time translation symmetry as applied to the case of the Lieb-Mattis antiferromagnet. There are three main requirements that need to be satisfied in order for the following derivation to be applicable. These requirements are: (1) The spontaneous evolution must yield a final state with only a single orientation of the order parameter, and the selection of the specific order parameter to be realised must be a probabilistic process; (2) The probability of obtaining a certain outcome may only depend on its weight in the initial superposition; (3) If the initial superposed state is entangled with some other, external quantum mechanical object with which the antiferromagnet has no further interaction, then the probability for finding a certain final orientation of the antiferromagnetic order parameter should not be affected by the precise state of the external quantum mechanical object.
To see that these requirements are all satisfied by the process of dynamic spontaneous symmetry breaking described before, consider the initial state $$\begin{aligned}
\left| \psi(0) \right> = \alpha \left| e1 \right> \otimes \left| AFM \right>_x + \beta \left| e2 \right> \otimes \left| AFM \right>_z,
\label{alphabeta}\end{aligned}$$ where $|\alpha|^2 + |\beta|^2=1$, $|AFM\rangle_x$ is the state with full antiferromagnetic order along the $x$ axis, and the states $|e1\rangle$ and $|e2\rangle$ are some external states which have no further interaction with the antiferromagnet whatsoever. The Hilbert space of the combined system of antiferromagnet and external states can be written as a product of the space of states of the antiferromagnet and the space of external states. Following the discussion of the quantum dynamics of a superposed macroscopic state in the main text, it is clear that the dynamics of the initial state $\left| \psi(0) \right>$ is unstable with respect to two orientations of the symmetry breaking field. Since the two instabilities of $\left| \psi(0) \right>$ must compete with each other, only one of the two available stable states can be realised, and the selection of which state is realised in the presence of a fluctuating symmetry breaking field is a probabilistic process, as stated in requirement one. Furthermore, since the competition between instabilities takes place on an infinitesimally short timescale, it cannot be influenced by the finite energy scale $J$. The [*fluctuating*]{} field $b$ is guaranteed by symmetry not to favour either one of the two possible final states. The only thing left to determine the probability of finding a certain final state is then the choice of the initial state itself: i.e. only the weights $\alpha$ and $\beta$ can determine the probability distribution of final states, in agreement with requirement two. That these weights in fact do influence the probability distribution is obvious from the fact that the initial states with $\alpha$ or $\beta$ equal to zero are stable states. The external states $|e1\rangle$ and $|e2\rangle$ cannot influence the spontaneous dynamics because all of the competition between the instabilities is governed by the unitarity breaking field $b$. This field acts only on the states of the antiferromagnet, and not on any other part of the Hilbert space (requirement three). The initial state $\left| \psi(0) \right>$ will thus be spontaneously and instantaneously reduced to either the state $\left| e1 \right> \otimes \left| AFM \right>_x$ or the state $\left| e2 \right> \otimes \left| AFM \right>_z$, while the probabilities $P_x(\psi)$ and $P_z(\psi)$ for finding either final state depend only on the values of $\alpha$ and $\beta$.
Building on these known properties of the final probabilities, let’s now follow Zurek’s arguments for obtaining the exact final probability distribution [@Zurek05]. First consider two different initial states: $$\begin{aligned}
\left| \psi \right> = \alpha \left| e1 \right> \otimes \left| AFM \right>_x + \beta \left| e2 \right> \otimes \left| AFM \right>_z \phantom{.} \nonumber \\
\left| \phi \right> = \alpha \left| e3 \right> \otimes \left| AFM \right>_x + \beta \left| e4 \right> \otimes \left| AFM \right>_z.
\label{statementA}\end{aligned}$$ Since the final probabilities can only depend on the weights of the classical states in the initial wavefunction (req. 2), it is immediately clear that $P_x(\psi)=P_x(\phi)$. This must hold independent of the external states $|e1\rangle$ through $|e4\rangle$ (req. 3), and thus it must also hold in the special case $|e1\rangle=e^{i \theta} |e3\rangle,~ |e2\rangle=|e4\rangle$, showing that the probability distribution cannot depend on the phases of the weights in the initial wavefunction.
Next, consider the initial states $$\begin{aligned}
\left| \psi \right> = \alpha \left| e1 \right> \otimes \left| AFM \right>_x + \beta \left| e2 \right> \otimes \left| AFM \right>_z \phantom{.} \nonumber \\
\left| \chi \right> = \alpha \left| e2 \right> \otimes \left| AFM \right>_z + \beta \left| e1 \right> \otimes \left| AFM \right>_x.
\label{statementB}\end{aligned}$$ Clearly, we must have $P_x(\psi)=P_z(\chi)$ for any choice of $\alpha$ and $\beta$. In the special case $|\alpha|=|\beta|$ we also know $P_z(\psi)=P_z(\chi)$, and thus we find that in that case $P_x(\psi)=P_z(\psi)$. In other words, if the sizes of the weights corresponding to two final states are equal, then so are the probabilities for finding these states. This statement can be trivially extended to yield the rule that a set of possible final states with equal weights in the initial wavefunction leads to equal probability for finding any one of the final states within that set. Continuing that line of thought, consider $$\begin{aligned}
\left| \psi \right> = \alpha \left| AFM \right>_i + \alpha \left| AFM \right>_j+ \alpha \left| AFM \right>_k + ...
\label{statementD}\end{aligned}$$ where $i$, $j$ and $k$ are different directions in real space. The combined probability $P_{i~\text{or}~j}(\psi)$ must then be equal to $P_i(\psi)+P_j(\psi)=2P_k(\psi)$, which follows directly from the additivity of probabilities and the mutual exclusivity of the three possible final states. That the final states are in fact mutually exclusive is guaranteed by requirement 1: in the thermodynamic limit $\left| AFM \right>_i$ and $\left| AFM \right>_j$ correspond to states with different directions of their order parameters, which can have no overlap and only one of which can be the result of the spontaneous dynamics. Extending this result, it is now clear that within a set of possible final states with equal weights in the initial wavefunction, a subset has a combined probability equal to the relative size of the subset times the total probability of the entire set.
Finally, consider the initial state $$\begin{aligned}
\hspace{-5pt} \left| \psi \right> = \sqrt{\frac{m}{N}} \left| e1 \right> \otimes \left| AFM \right>_x + \sqrt{\frac{n}{N}} \left| e2 \right> \otimes \left| AFM \right>_z.
\label{statementE}\end{aligned}$$ The probability $P_x(\psi)$ is independent of the external states (req. 3). We are therefore free to write $|e1\rangle$ and $|e2\rangle$ in a basis in which they are a sum of states with equal weights (such a basis can be shown to always exist [@Zurek05]):$$\begin{aligned}
\left| e1 \right> &= \sqrt{\frac{1}{m}} \left[ \left| E1_1 \right> + \left| E1_2 \right> + ... + \left| E1_m \right> \right] \phantom{.} \nonumber \\
\left| e2 \right> &= \sqrt{\frac{1}{n}} \left[ \left| E2_1 \right> + \left| E2_2 \right> + ... + \left| E2_n \right> \right] .
\label{statementE2}\end{aligned}$$ Reinserting these definitions into equation yields $$\begin{aligned}
\left| \psi \right> = \sqrt{\frac{1}{N}} \left[ \sum_{i=1}^m \left| E1_i \right> \otimes \left| AFM \right>_x + \right. \nonumber \\
\left. \sum_{j=1}^n \left| E2_j \right> \otimes \left| AFM \right>_z \right] .
\label{statementE3}\end{aligned}$$ In this expression all weights are equal, and using the previously found rules we must thus conclude that $P_x(\psi)=\frac{n}{m}P_z(\psi)$. In the case that the total probability for finding any outcome at all is one, this result precisely corresponds to Born’s rule: the probability for finding any specific final orientation of the order parameter is equal to the square of the weight of the corresponding state in the initial wavefunction [@Born26]. The extension of this result to include also weights which are square roots of non-rational numbers is trivial because the rational numbers are dense on the real line [@Zurek05].
[10]{}
P. Anderson, Science [**177**]{}, 393 (1972).
L. Landau, Phys. Z. Sowjetunion [**11**]{}, 542 (1937).
J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. [**127**]{}, 965 (1962).
P. Anderson, , Benjamin, New York, 1963.
P. Anderson, Phys. Rev. [**86**]{}, 694 (1952).
P. Anderson, Phys. Rev. [**112**]{}, 1900 (1958).
Y. Nambu, Phys. Rev. [**117**]{}, 648 (1960).
E. Lieb and D. Mattis, J. Math. Phys. [**3**]{}, 749 (1962).
C. Kaiser and I. Peschel, J. Phys. A [**22**]{}, 4257 (1989).
T. Kaplan, W. von der Linden, and P. Horsch, Phys. Rev. B [**42**]{}, 4663 (1990).
J. van Wezel, J. van den Brink, and J. Zaanen, Phys. Rev. Lett. [**94**]{}, 230401 (2005).
J. van Wezel, J. Zaanen, and J. van den Brink, Phys. Rev. B [**74**]{}, 094430 (2006).
J. van Wezel and J. van den Brink, Am. J. Phys. [**75**]{}, 635 (2007).
J. van Wezel and J. van den Brink, ArXiv: Cond-mat , 07061922 (2007).
T. Birol, T. Dereli, O. Müstecaplioglu, and L. You, Phys. Rev. A [**76**]{}, 043616 (2007).
C. van der Wal, A. ter Haar, F. Wilhelm, R. Schouten, C. Harmans, T. Orlando, S. Lloyd, and J. Mooij, Science [**290**]{}, 773 (2000).
I. Chiorescu, Y. Nakamura, C. Harmans, and J. Mooij, Science [**299**]{}, 1869 (2003).
I. Chiorescu, Y. Nakamura, C. Harmans, and J. Mooij, Science [**299**]{}, 1869 (2003).
M. Anderson, J. Ensher, M. Matthews, C.W., and E. Cornell, Science [**269**]{}, 198 (1995).
K. Davis, M. Mewes, M. Andrews, N. van Druten, D. Durfee, D. Kurn, and W. Ketterle, Phys. Rev. Lett. [**75**]{}, 3969 (1995).
J. Stenger, S. Inouye, A. Chikkatur, D. Stamper-Kurn, D. Pritchard, and W. Ketterle, Phys. Rev. Lett. [**82**]{}, 4569 (1999).
M. Kozuma, L. Deng, E. Hagley, J. Wen, R. Lutwak, K. Helmerson, S. Rolston, and W. Phillips, Science [**286**]{}, 2309 (1999).
M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, Nature [**401**]{}, 680 (1999).
W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, Phys. Rev. Lett. [**91**]{}, 130401 (2003).
W. Zurek, Phys. Rev. D [**24**]{}, 1516 (1981).
E. Joos and H. Zeh, Z. Phys. B [**59**]{}, 223 (1985).
A. Caldeira and A. Leggett, Ann. Phys. [**149**]{}, 374 (1983).
S. L. Adler, Stud. Hist. Phil. Mod. Phys. [**34**]{}, 135 (2003).
A. Bassi and G. Ghirardi, Phys. Lett. A [**275**]{}, 373 (2000).
O. Waldmann, T. Guidi, S. Carretta, C. Mondelli, and A. Dearden, Phys. Rev. Lett. [**91**]{}, 237202 (2003).
O. Waldmann, C. Dobe, H. Güdel, and H. Mutka, Phys. Rev. B [**74**]{}, 054429 (2005).
J. van Wezel, A divine game of dice, Master’s thesis, Leiden University, Leiden, The Netherlands, 2003.
J. von Neumann, , Princeton University Press, 1955.
A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa, Am. J. Phys. [**57**]{}, 117 (1989).
L. Diósi, Phys. Rev. A [**40**]{}, 1165 (1989).
R. Penrose, Gen. Rel. Grav. [**28**]{}, 581 (1996).
J. van Wezel T. Oosterkamp and J. Zaanen, ArXiv: Cond-mat , 0706.3976 (2007).
P. Anderson, , Perseus Books, 1997.
P. Pearle, Phys. Rev. A [**39**]{}, 2277 (1989).
G. Ghirardi, R. Grassi, and A. Rimini, Phys. Rev. A [**42**]{}, 1057 (1990).
W. Zurek, Phys. Rev. Lett. [**90**]{}, 120404 (2003).
W. Zurek, Phys. Rev. A [**71**]{}, 052105 (2005).
M. Born, Z. Phys. [**40**]{}, 167 (1926).
J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J.M.J. Keeling, F.M. Marchetti, M.H. Szymaśka, R. André, J.L. Staehli, V. Savona, P.B. Littlewood, B. Deceaud, and L.S. Dang Nature [**443**]{}, 409 (2006).
J. Keeling, F. Marchetti, M. Szymańska, and P. Littlewood, Semicond. Sci. Technol. [**22**]{}, R1 (2007).
M. Wouters and I. Carusotto, Phys. Rev. A [**76**]{}, 043807 (2007).
A. Amo, D. Sanvitto, D. Ballarini, F.P. Laussy, E. del Valle, M.D. Martin, A. Lemaitre, J. Bloch, D.N. Krizhanovskii, M.S. Skolnick, C. Tejedor, and L. Vina, ArXiv: Cond-mat , 0711.1539 (2007).
C. Ciuti, P. Schwendimann, and A. Quattropani, Phys. Rev. B [**63**]{}, 041303 (2001).
|
---
abstract: 'Closed essential surfaces in a 3–manifold can be detected by ideal points of the character variety or by algebraic non-integral representations. We give examples of closed essential surfaces not detected in either of these ways. For ideal points, we use Chesebro’s module-theoretic interpretation of Culler-Shalen theory. As a corollary, we construct an infinite family of closed hyperbolic Haken 3–manifolds with no algebraic non-integral representation into $\operatorname{PSL}_2 ( \CC)$, resolving a question of Schanuel and Zhang.'
address:
- |
Alex Casella,\
School of Mathematics and Statistics F07,\
The University of Sydney,\
NSW 2006 Australia\
(casella@maths.usyd.edu.au)\
–
- |
Charles Katerba,\
Department of Mathematical Sciences,\
Montana State University\
MT 59717 USA\
(charles.katerba@montana.edu)\
–
- |
Stephan Tillmann,\
School of Mathematics and Statistics F07,\
The University of Sydney,\
NSW 2006 Australia\
(tillmann@maths.usyd.edu.au)
author:
- 'Alex Casella, Charles Katerba and Stephan Tillmann'
bibliography:
- 'ces.bib'
title: |
Ideal points of character varieties, algebraic non-integral\
representations, and undetected closed essential surfaces in 3–manifolds
---
Introduction
============
A *knot manifold* is a compact, irreducible, orientable 3–manifold whose boundary consists of a single torus. We say that a knot manifold is *large* if it contains a closed essential surface. Here, an *essential surface* $\Sigma$ in a 3–manifold is an orientable, properly embedded surface with no sphere or boundary parallel components such that the homomorphism on fundamental groups induced by inclusion is injective for each connected component of $\Sigma$.
Let $N$ be a knot manifold and $\Gamma$ denote its fundamental group. Culler and Shalen [@CS1] construct essential surfaces in 3–manifolds from representations of $\Gamma$ into $\operatorname{SL}_2 ( \CC)$. This combines algebraic geometry, valuations and actions on trees, and has seen broad applications in 3–manifold topology (see for example [@BZ2; @BZ1; @CCGLS; @CGLS]). In this introduction, we assume some familiarity with Culler-Shalen theory—basic definitions and facts are collected in §\[sec:charvar\] and §\[sec:preliminaries\].
The set $\X_{\operatorname{SL}} ( N )$ of characters of representations of $\Gamma$ into $\operatorname{SL}_2 ( \CC)$ admits the structure of a complex affine algebraic set called the [character variety of $N$]{}. Essential surfaces can be associated to certain representations $\Gamma \to \operatorname{SL}_2 ( F)$, where $F$ is a field with a valuation $v \co F^\times \to \ZZ.$ The character variety provides two ways to find such representations: by passing to ideal points and by carrying algebraic non-integral, or ANI, representations. If the essential surface $\Sigma$ in $N$ is associated to an ideal point of a curve in $\X_{\operatorname{SL}}(N)$ it is *detected by (an ideal point of) the character variety*. Similarly, if $\Sigma$ is associated to an ANI-representation, we say $\Sigma$ is *ANI-detected by the character variety*. Similar terminology is adopted for the *boundary slopes* of essential surfaces; that is, unoriented isotopy classes of simple closed curves on the boundary of $N$ that can be represented by the boundary components of essential surfaces in $N$. The theory and its applications were extended to representations into $\operatorname{PSL}_2 ( \CC)$ by Boyer and Zhang [@BZ2].
This paper addresses the general question of which essential surfaces in a 3–manifold are detected by ideal points or ANI-representations. To this end, Chesebro and the third author [@CT] showed that there are boundary slopes of knot manifolds which are not detected by the character variety. Motegi showed that there are closed graph manifolds that contain essential tori [@MOT] not detected by the character variety. Boyer and Zhang [@BZ2 Theorem 1.8] showed that there are infinitely many closed hyperbolic 3–manifolds whose character varieties do not detect closed essential surfaces contained in these manifolds. Schanuel and Zhang [@SZ Example 17] gave an example of a closed hyperbolic 3–manifold with a closed essential surface that cannot be detected by an ideal point but is ANI-detected. We first turn our attention to knot manifolds, and as an application answer an open question of Schanuel and Zhang. The knots $10_{152}$, $10_{153}$ and $10_{154}$ were shown to be large by Burton, Coward and the third author [@BCT]. This paper shows:
Let $N$ be the complement in $S^3$ of the large hyperbolic knot $10_{152}$, $10_{153}$ or $10_{154}$. No closed essential surface in $N$ is detected by an ideal point of the character variety of $N$. \[thm:main\]
The proof of Theorem \[thm:main\] uses a module-theoretic approach to Culler-Shalen theory developed by Chesebro [@C]. As will be explained further in §\[sec:main\], this approach transforms the problem into a computation in commutative algebra. Thus, we are able to answer this question using algorithmic techniques implemented in the software package *Macaulay2* [@M2]. It also follows from [@C Proposition 5.2] that no closed essential surface in the complement of $10_{152}$, $10_{153}$ or $10_{154}$ is ANI-detected.
Joshua Howie informed the authors that $10_{152}$, $10_{153}$, and $10_{154}$ are the first non-alternating adequate knots in Rolfsen’s table. It would be interesting to know the extent to which Howie’s observation may give a geometric or topological obstruction to detecting closed essential surfaces by ideal points.
As a corollary to Theorem \[thm:main\], we answer Question 9 of Schanuel and Zhang [@SZ]. They ask whether there are large closed hyperbolic 3–manifolds which have no ANI-representations into $\operatorname{PSL}_2 ( \CC).$ We answer this question affirmatively.
There are infinitely many large closed hyperbolic 3–manifolds with no ANI-representations into $\operatorname{PSL}_2(\CC)$. \[cor:noani\]
Corollary \[cor:noani\] follows from Theorem \[thm:main\] by considering sufficiently large Dehn fillings of any of the knots given by the theorem. Under such a Dehn filling, a closed essential surface in the knot exterior will remain essential in the filled manifold [@CGLS Theorem 2.0.3] and the filled manifold will be hyperbolic by Thurston’s Hyperbolic Dehn Surgery Theorem. Using a result of Culler about lifting representations and a result of Chesebro connecting ANI-detected to detected closed essential surfaces, we show that the filled manifold cannot have any ANI-representations into $\operatorname{PSL}_2( \CC)$.
The remainder of the paper is structured as follows. In the next section, we review some basics concerning character varieties and describe an algorithm for their computation. Section \[sec:preliminaries\] outlines the essentials of Culler-Shalen theory and summarizes Chesebro’s module-theoretic approach. Section \[sec:main\] begins with a description of our computational techniques and heuristic for finding knots whose character varieties may not detect closed essential surfaces. We then prove the main results of this paper.
**Acknowledgements** The first author acknowledges support by the Commonwealth of Australia. The second author acknowledges support by an NSF-EAPSI Fellowship (project number 1713920). Research of the third author was supported by an Australian Research Council Future Fellowship (project number FT170100316). The authors thank Robert Loewe and Benjamin Lorenz for running the computation for $10_{154}$ on a cluster at TU Berlin.
Character varieties {#sec:charvar}
===================
We describe the construction of character varieties in the $\operatorname{SL}_2( \CC )$ case. The $\operatorname{PSL}_2 ( \CC)$ case is similar with only a few additional technicalities; see [@BZ2] for a detailed account of Culler-Shalen theory for $\operatorname{PSL}_2 ( \CC )$.
Let $\Gamma$ be a finitely presented group with presentation $\langle \gamma_1, \cdots, \gamma_n \mid r_1 , \ldots, r_m \rangle$. A function $\rho \co \{\gamma_{j} \}_{j=1}^n \to \operatorname{SL}_2 (\CC)$ extends to a representation if and only if $\rho ( r_i)$ is the $2 \times 2$ identity matrix for each $1 \leq i \leq m$. The set $\R_{\operatorname{SL}}(\Gamma) := \operatorname{Hom}( \Gamma, \operatorname{SL}_2 ( \CC))$ is therefore in natural 1-1 correspondence with an algebraic set in $\CC^{4n}$ and hence called the *representation variety of $\Gamma$*.
For each $\gamma \in \Gamma$ there is a regular function $I_\gamma \co \R_{\operatorname{SL}}( \Gamma ) \to \CC$ given by $I_\gamma ( \rho ) = \operatorname{tr}\rho ( \gamma )$. Let $T( \Gamma)$ denote the ring with unity generated the set $\{I_\gamma \}_{\gamma \in \Gamma}$.
[(cf. [@CS1] and [@GMA])]{} The ring $T(\Gamma)$ is generated by the set $$\mathcal{G} = \{ I_{\gamma_{i_1} \cdots \gamma_{i_k}} \mid \text{for } 1 \leq i_1 < \cdots < i_k \leq n \text{ and } k \leq 3 \}.$$ \[prop:tgfg\]
We include a sketch of a proof of this proposition to remind the reader that there is an algorithm to write each $I_\gamma$ for $\gamma \in \Gamma$ as a polynomial in the elements of $\mathcal{G}$. This algorithm is based on the following trace identities [@GMA Lemmas 4.1 and 4.1.1]. If $A, B, C, D \in \operatorname{SL}_2 ( \CC)$, then
$$\begin{aligned}
\operatorname{tr}A & = \operatorname{tr}A^{-1} \label{eq:1} \\
\operatorname{tr}AB & = \operatorname{tr}A \operatorname{tr}B - \operatorname{tr}A {{B}^{-1}} \\
\operatorname{tr}ACB & = \operatorname{tr}A \operatorname{tr}BC + \operatorname{tr}B \operatorname{tr}AC + \operatorname{tr}C \operatorname{tr}AB - \operatorname{tr}A \operatorname{tr}B \operatorname{tr}C - \operatorname{tr}ABC \\
2 \cdot \operatorname{tr}ABCD & = \operatorname{tr}A \operatorname{tr}BCD + \operatorname{tr}B \operatorname{tr}ACD + \operatorname{tr}C \operatorname{tr}ABD - \operatorname{tr}D \operatorname{tr}ACB + \operatorname{tr}BC \operatorname{tr}AD \nonumber \\
& \qquad + \operatorname{tr}AB \operatorname{tr}CD - \operatorname{tr}AC \operatorname{tr}BD + \operatorname{tr}B \operatorname{tr}D \operatorname{tr}AC - \operatorname{tr}A \operatorname{tr}B \operatorname{tr}CD \nonumber \\
& \qquad - \operatorname{tr}B \operatorname{tr}C \operatorname{tr}AD
\end{aligned}$$
Given $\gamma \in \Gamma$, write $\gamma$ as a word in the generators $\{\gamma_i\}_{i = 1}^n$. Use the first and second trace identities to write $I_\gamma$ as a polynomial in trace functions of words with no inverses or exponents on letters higher than one. With the fourth trace identity one can reduce the word length to at most 3. Finally, the third identity allows one to write the trace functions in lexicographic order.
Order the $N = n(n^2 +5)/6$ elements of $\mathcal{G}$ lexicographically and use this ordering to define a map $t \co \R_{\operatorname{SL}} ( \Gamma ) \to \CC^N$ by $t( \rho ) = ( I_g ( \rho ) )_{ g \in \mathcal{G}}$. Culler and Shalen proved that the image of $t$ is a closed algebraic set that is in 1-1 correspondence with the set of $\operatorname{SL}_2( \CC)$-characters of $\Gamma$ [@CS1]. Thus, $\X_{\operatorname{SL}}(\Gamma) := t ( \R_{\operatorname{SL}}( \Gamma))$ is called the *character variety of $\Gamma$*. The elements of $\mathcal{G}$ serve as coordinate functions on $\X_{\operatorname{SL}}( \Gamma)$.
González-Acuña and Montesinos-Amilibia [@GMA] exhibited specific defining equations for the character variety of a finitely presented group. First, they exhibit a finite collection of polynomials which cut out the character variety for the free group $F_n$ on $n$ letters. Next, they prove that $\X_{\operatorname{SL}} ( \Gamma )$ is the algebraic subset of $\X_{\operatorname{SL}} ( F_n )$ cut out by the $m(n+1)$ polynomials $$\begin{aligned}
\{ I_{r_1} - 2, I_{ \gamma_1 r_1} - I_{\gamma_1}, \ldots, I_{ \gamma_n r_1} - I_{\gamma_n}, I_{r_2} -2, \ldots, I_{\gamma_n r_2} - I_{\gamma_n}, \ldots, I_{\gamma_n r_m } - I_{\gamma_n} \}. \end{aligned}$$ Each of the above trace functions can be computed algorithmically, as can defining equations for the character variety of the free group on $n$ letters. By the fourth trace identity, we see that $\X_{\operatorname{SL}}(\Gamma)$ is cut out of $\CC^N$ by polynomials with rational coefficients. This gives the following proposition:
Given a presentation $\langle \gamma_1, \ldots, \gamma_n | r_1, \ldots, r_m \rangle $ for a group $\Gamma$, defining equations for $\X_{\operatorname{SL}}( \Gamma)$ can be computed algorithmically. Moreover, $\X_{\operatorname{SL}} ( \Gamma)$ is defined over $\QQ$. \[prop:dfneqns\]
Elements of Culler-Shalen theory {#sec:preliminaries}
================================
Fix a knot manifold $N$, let $\Gamma$ denote the fundamental group of $N$, and set $\X_{\operatorname{SL}} ( N ) = \X_{\operatorname{SL}} (\Gamma) $.
If $F$ is a field with a valuation $v \co F^\times \to \ZZ$, then Bass-Serre theory gives a simplicial tree $T_v$ associated to $v$ and an action of $\operatorname{SL}_2 ( F)$ on $T$ [@JPS]. A representation $\rho \co \Gamma \to \operatorname{SL}_2 ( F)$ induces an action of $\Gamma$ on $T_v$. When this action is nontrivial (that is, when no vertex of $T_v$ is fixed by $\Gamma$) a construction due to Stallings gives essential surfaces in $N$. Essential surfaces that can be built from the above procedure are *associated to the tree $T_v$*. The next theorem demonstrates a connection between the topology of $N$, the representation $\rho$, and the valuation $v$.
[[@CS1]]{} Suppose there is a representation $\rho \co \Gamma \to \operatorname{SL}_2 ( F ) $ where $F$ is a field with a valuation $v$ such that the induced action of $\Gamma$ on the tree $T_v$ is nontrivial.
1. If $v ( \operatorname{tr}\rho ( \gamma) ) \geq 0$ for each $\gamma \in \pi_1 ( \bound N)$, then there is a closed essential surface associated to $T_v$.
2. Otherwise there is a unique element $\gamma \in \pi_1 ( \bound N) $ (up to inversion and conjugation) such that $v ( \operatorname{tr}\rho ( \gamma )) \geq 0$. In this case, every essential surface associated to $T_v$ has non-empty boundary and $\gamma$ represents the boundary slope of these surfaces.
\[thm:cs1\]
The character variety $\X_{\operatorname{SL}} ( N )$ provides two ways of finding fields equipped with valuations and hence essential surfaces in $N$: by passing to ideal points and by carrying algebraic non-integral representations.
*Ideal points:* The dimension of $\X_{\operatorname{SL}}(N)$ is at least one [@CCGLS], so take an irreducible curve $X \subset \X_{\operatorname{SL}}( N )$ with normalization $\phi \co \overline{X} \to X$ and a smooth projective model ${\widetilde{X}}$ for $X$. Then there is a birational isomorphism $\iota \co {\widetilde{X}} \to \overline{X}$ whose inverse is defined on all of $\overline{X}$. The elements of the set $${\widetilde{X}} - \iota^{-1} ( \overline{X} )$$ are the *ideal points of $X$*. Up to isomorphism $X$ has a unique smooth projective model, so the set of ideal points is well-defined. Each rational function on $X$ extends to a function ${\widetilde{X}} \to \CC P^1 = \CC \cup \{\infty \}$ and to each ideal point $\hat{x}$ of $X$ there is a natural valuation $v_{\hat{x}}$ on the function field $\CC(X) = \CC( {\widetilde{X}})$ given by $$v_{\hat{x}} (f) = \begin{cases}
-(\text{order of the pole of $f$ at $\hat{x}$}) & \text{if } f(\hat{x}) = \infty \\
0 &\text{it } f(\hat{x}) \in \CC - \{ 0 \} \\
\text{order of the zero of $f$ at $\hat{x}$} & \text{if } f(\hat{x}) = 0 \end{cases}.$$
There is a representation, often referred to as the *tautological representation*, $\P \co \Gamma \to \operatorname{SL}_2 ( \CC ( {\widetilde{X}}))$. That $X$ is a curve implies that the action of $\Gamma$ on the tree $T_{v_{\hat{x}}}$ for each ideal point $\hat{x}$ of $X$ is nontrivial. Hence Theorem \[thm:cs1\] applies.
*ANI-representations:* Suppose $F$ is a number field (i.e.a finite extension of $\QQ$) and $\rho \co \Gamma \to \operatorname{SL}_2 ( F)$ is a representation. We say $\rho$ is *algebraic non-integral*, or *ANI* if there is some element $\gamma \in \Gamma$ such that $\operatorname{tr}\rho ( \gamma)$ is not integral over $\ZZ$. Recall that the integral closure of $\ZZ$ in $F$ is the intersection of all the valuation rings of $F$. Since $\operatorname{tr}\rho ( \gamma)$ is not integral over $\ZZ$, there is some valuation $v$ on $F$ such that $v(\operatorname{tr}\rho (\gamma)) < 0$. Hence, there is an essential surface in $N$ associated to the tree $T_v$ by Theorem \[thm:cs1\]
Chesebro [@C] noticed a connection between the detection of essential surfaces by ideal points and an infinite collection of modules associated to the coordinate ring $\CC[X]$ of $X$ which we now describe.
For any unital subring $R$ of $\CC$, let $T_R ( X)$ denote the $R$-subalgebra of $\CC[X]$ generated by $\mathcal{G}$, noting that $T_\CC ( X) = \CC[X]$. If $\gamma \in \pi_1 ( N )$, then $I_\gamma$ is an element of $T_{\ZZ}(X)$ and hence $R[I_\gamma] \subseteq T_R(X)$ for each unital subring $R\subseteq \CC$. In particular, we may view $T_R(X)$ as a $R[I_\gamma]$-module. The following theorem relates the detection of essential surfaces by ideal points to these modules.
[(Chesebro [@C Theorem 1.2])]{} Let $X$ be an irreducible component of $\X_{\operatorname{SL}}(N)$ and take $R \in \{ \ZZ, \QQ, \CC \}$. Then
1. $T_R(X)$ is not finitely generated as an $R[I_\gamma]$-module for each $\gamma \in \pi_1 ( \bound N)$ if and only if $X$ detects a closed essential surface.
2. Otherwise, $T_R(X)$ is not finitely generated as a $R[I_\gamma]$-module for some $\gamma \in \pi_1 (\bound N)$ if and only if $\gamma$ represents a boundary slope detected by $X$.
\[thm:cheese\]
Main Results {#sec:main}
============
We aim to prove that there are knots in $S^3$ whose complement contains closed essential surfaces, none of which are detected by the character variety. A corollary to Theorem \[thm:cheese\] which replaces $X$ with an arbitrary union of irreducible components of $\X_{\operatorname{SL}} ( N)$ will be our main tool.
Suppose $Y = X_1 \cup \cdots \cup X_n$ is a union of irreducible components of $\X_{\operatorname{SL}}(N)$. Then $Y$ does not detect a closed essential surface if and only if $\CC[Y]$ is finitely generated as a $\CC[I_\gamma]$-module for some $\gamma \in \pi_1 ( \bound N)$ which is not a boundary slope of $N$.
Moreover, when $\CC[Y]$ is finitely generated, it is a free $\CC[I_\gamma]$-module. \[cor:union\]
Take an element $\gamma \in \pi_1 ( \bound N)$ which does not represent a boundary slope of $N$. If $Y$ does not detect a closed essential surface, then, for each $i$, $\CC[X_i]$ is a finitely generated $\CC[I_\gamma]$-module by Theorem \[thm:cheese\]. Since $X_i$ is irreducible, $\CC[X_i]$ is an integral domain, and so $\CC[X_i]$ is torsion free over $\CC[I_\gamma]$. In particular, $\CC[X_i]$ is free over $\CC[I_\gamma]$ since $\CC[I_\gamma]$ is a principle ideal domain.
Suppose $\X_{\operatorname{SL}}(N) \subseteq \CC^k$. Let $\Phi \co \CC[z_1, \ldots, z_k] \to \CC[Y]$ and $\phi_i \co \CC[z_1, \ldots, z_k] \to \CC[X_i]$ denote the natural epimorphisms induced by the inclusions $Y \hookrightarrow \CC^k$ and $X_i \hookrightarrow \CC^k$. Define a function $\Psi \co \CC[Y] \to \oplus_1^n \CC[X_i]$ by $$\Psi ( \Phi( f )) = ( \phi_1 ( f ) , \ldots, \phi_n ( f )) \quad \text{for} \quad f\in \CC[z_1, \ldots, z_k].$$ If we regard $\oplus_1^n \CC[X_i]$ as a $\CC[I_\gamma]$-module with the diagonal action, then $\Psi$ is $\CC[I_\gamma]$-linear.
We claim that $\Psi$ is injective. Take a polynomial $f \in \CC[z_1, \ldots, z_n]$ such that $\phi_i ( f) $ is the zero function on $X_i$ for each $i$. Then $f$ is an element of the ideal $I(X_i)$ of $X_i$ and hence $$f \in \bigcap_1^n I(X_i) = I(Y).$$ Thus $\Psi$ is a monomorphism and $\CC[Y]$ is isomorphic to a submodule of a finitely generated free module. Finally, since $\CC[I_\gamma]$ is a PID, $\CC[Y]$ must be a finitely generated free $\CC[I_\gamma]$-module.
Now suppose toward a contradiction that $\CC[Y]$ is finitely generated over $\CC[I_\gamma]$ and that $Y$ detects a closed essential surface. Then, by Theorem \[thm:cheese\], $\CC[X_i]$ is not finitely generated over $\CC[I_\gamma]$ for some $1 \leq i \leq n$. But the inclusion $X_i \hookrightarrow Y$ induces a surjection $\CC[Y] \to \CC[X_i]$. This gives a contradiction since the image of a generating set for $\CC[Y]$ would generate $\CC[X_i]$.
Corollary \[cor:union\] demonstrates how we transform the question of whether or not $\X_{\operatorname{SL}}(N)$ detects a closed essential surface into a computational commutative algebra problem. The program *Macaulay2* uses inexact numbers when working over $\CC$, so we must extend Corollary \[cor:union\] so that the coefficient field is $\QQ$ where the program performs exact calculations. The proof of the following corollary is essentially [@K Proposition 16].
Take $Y = X_1 \cup \cdots \cup X_n$ to be a union of irreducible components of $\X_{\operatorname{SL}}(N)$ and fix $\gamma \in \pi_1 ( \bound N)$. Suppose $Y$ is defined over $\QQ$. Then $T_\QQ ( Y)$ is finitely generated over $\QQ[I_\gamma]$ if and only if $\CC[Y]$ is finitely generated over $\CC[I_\gamma]$.
In particular $Y$ does not detect a closed essential surface if and only if $T_\QQ (Y)$ is finitely generated over $\QQ[I_\gamma]$ for some $\gamma \in \pi_1 ( \bound N)$ that does not represent a boundary slope of $N$. \[cor:overqq\]
First, observe that any generating set for $T_\QQ ( Y )$ over $\QQ[ I_\gamma]$ will automatically generate $\CC[Y]$ over $\CC[I_\gamma]$.
Now suppose $\CC[Y]$ is finitely generated over $\CC[I_\gamma]$. Then $\CC[Y]$ is a free module and we may take a free basis $\mathcal{B} = \{b_1, \ldots, b_m\}$ lying in $T(X)$ (see the remarks following Corollary 2.5 in [@C]).
We claim $\mathcal{B}$ spans $T_\QQ ( X)$ over $\QQ[I_\gamma]$. Take an element $f \in T_\QQ(X)$ and write $ f = \sum_1^m p_j b_j $ for some $p_j \in \CC[I_\gamma]$. The field automorphism group ${\mathrm{Aut}}( \CC / \QQ )$ acts on $\CC[Y]$ since $Y$ is defined over $\QQ$. Moreover, since $f \in T_\QQ (X)$, if $\sigma \in {\mathrm{Aut}}( \CC / \QQ)$, $$\sigma\cdot f = f, \quad \text{so} \quad \sum_1^m (p_j - \sigma \cdot p_j) b_j = 0.$$ $\mathcal{B}$ is a free basis, so $p_j = \sigma \cdot p_j$ for every $\sigma \in {\mathrm{Aut}}( \CC / \QQ)$. The fixed field of ${\mathrm{Aut}}( \CC / \QQ )$ is $\QQ$ [@MFT Theorem 9.29], so each $p_j \in \QQ[I_\gamma]$.
Corollaries \[cor:union\] and \[cor:overqq\] provide a way to prove our main result, Theorem \[thm:main\]. For a given knot manifold $N$, to determine whether $\X_{\operatorname{SL}}(N)$ detects any closed essential surface, one must first find a slope $\alpha$ which is not a boundary slope of $N$, then decide if $T_\QQ ( \X_{\operatorname{SL}}( N ))$ is finitely generated as a $\QQ[I_\alpha]$-module. Fortunately, the `basis` command in *Macaulay2* finds generating sets for modules over specified rings [@M2].
To begin our search, we needed to know which knots in $S^3$ have large complements. Burton, Coward, and the third author [@BCT] developed an algorithm to check precisely this and [@BCT Appendix F] lists all 1019 large knots in $S^3$ that can be represented with diagrams with at most 12 crossings.
Chesebro proved that if an irreducible curve in $\X_{\operatorname{SL}}(N)$ ANI-detects a closed essential surface, then it detects a closed essential surface with ideal points [@C Proposition 5.2]. Goodman, Heard, and Hodgson compiled a partial list of all hyperbolic knots and links with up to 12 crossings whose discrete and faithful representations are ANI. Using this table, we created a list of all hyperbolic large knots with 12 or fewer crossings whose discrete faithful representation is either not ANI or its integrality is unknown. For each knot on our list, we recorded the number of tetrahedra in the triangulation given by the program *SnapPy* [@SnapPy]. Using the notation from Rolfsen’s table [@ROL], the knots in our table with the fewest tetrahedra in their *SnapPy* triangulation are $10_{152}$ with $9$ tetraheadra, $10_{153}$ with $8$, and $10_{154}$ with $10$.
![The knots $10_{152}$, $10_{153}$, and $10_{154}$. []{data-label="fig:knots"}](10152.pdf "fig:"){width=".3\textwidth"}![The knots $10_{152}$, $10_{153}$, and $10_{154}$. []{data-label="fig:knots"}](10153.pdf "fig:"){width=".3\textwidth"}![The knots $10_{152}$, $10_{153}$, and $10_{154}$. []{data-label="fig:knots"}](10154.pdf "fig:"){width=".3\textwidth"}
We need to show that no essential surfaces is detected by an ideal point of the character variety. We give the details for $N$ the complement of the knot $10_{153}$; the calculations for $10_{152}$ and $10_{154}$ follow along the same lines. The package *HIKMOT* [@HIKMOT] certifies that the interior of $N$ admits a finite volume hyperbolic metric. Now *SnapPy* [@SnapPy] gives the following presentation for the fundamental group of $N$: $$\pi_1 ( N ) \cong \langle a, b, c\mid abAbCaabAbcB, abCBcAc \rangle$$ where capital letters denote inverses. A basis for $\pi_1 ( \bound N )$ is $$\{\; \mu = BAABa, \; \lambda = BAACaabCAbCBa \;\}.$$ Note that ${{\mu}^{-1}} \lambda$ is conjugate to the element $CaabCAbC$ in $\pi_1(N)$, so $I_{{{\mu}^{-1}} \lambda} = I_{CaabCAbC}$.
Now we use Propositions \[prop:tgfg\] and \[prop:dfneqns\] to compute defining equations for $\X_{\operatorname{SL}} (N)$. Using coordinates $$( x = I_a , y = I_b, z = I_c, w = I_{ab}, t = I_{ac}, u = I_{bc}, v = I_{abc} )$$ we find that $\X_{\operatorname{SL}}(N)$ is cut out of $\CC^7$ by 9 polynomials $\{p_0, \ldots, p_8\}$: $p_0$ being the polynomial defining the character variety of the free group on 3 letters and the other 8 coming from our presentation for $\pi_1 ( N)$. This collection of polynomials is long and unwieldy, so we elect not to display these polynomials here; the interested reader can find the polynomials in the ancillary files [@CKT-anc].
Define a new ideal $I$ generated by $\{p_0, \ldots, p_8, s - I_{{{\mu}^{-1}}\lambda}(x,y,z,w,t,u,v) \}$. Then the zero set of $I$ is an embedding of $\X_{\operatorname{SL}}(N)$ into $\CC^8$ with coordinates $(x,y,z,w,t,u,v,s)$ such that $I_{{{\mu}^{-1}}\lambda}$ is the coordinate function $s$.
Set $S = \QQ[s]$ and $ R = S[x,y,z,w,t,u,v]$. Then $R/I \cong T_{\QQ} ( \X_{\operatorname{SL}} ( N ))$ since $\X_{\operatorname{SL}}(N)$ is defined over $\QQ$. To investigate whether or not $T_{\QQ} ( \X_{\operatorname{SL}} (N) )$ is finitely generated as an $S$-module, we first compute a Gröbner basis for $I$, then execute the command $$\text{ \texttt{ basis( R / I , SourceRing => S ) }}$$ in *Macaulay2*. The command gives either an error if the module is not finitely generated over $S$ or a list of generators. In our case, we get a list $\mathcal{L}$ of 48 monomials which can be found in the ancillary files [@CKT-anc].
While $\mathcal{L}$ may not be a free basis for $T_\QQ( \X_{\operatorname{SL}} (N))$ over $\QQ[I_{{{\mu}^{-1}} \lambda}]$, it is guaranteed to be at least a generating set. In particular, by Corollary \[cor:overqq\], ${{\mu}^{-1}} \lambda$ is not a boundary slope of $N$ and $\X_{\operatorname{SL}}(N)$ does not detect any closed essential surfaces even though $N$ is a large knot manifold.
The defining equations for $\X_{\operatorname{SL}}(N)$ were computed using a *Mathematica* notebook written by Ashley, Burelle, and Lawton [@ABL] that is based on the Free Group Toolbox Version 2.0 *Mathematica* notebook written by William Goldman [@GOLD]. We independently verified the equations with a computation from first principles.
We first performed the above calculation for the knots $10_{152}$ and $10_{153}$ on a 2014 MacBook Air with a 1.4GHz processor and only 8GB of memory. The calculation for $10_{154}$ took 62 hours on a cluster with a 2.6GHz processor and used 33GB of memory at TU Berlin. This was done by Robert Loewe and Benjamin Lorenz. Both machines used version 1.11 of *Macaulay2*.
We end this note with an application of Theorem \[thm:main\] answering Question 9 of Schanuel and Zhang [@SZ] concerning algebraic non-integral representations into $\operatorname{PSL}_2 ( \CC)$. Even though the trace of an element of $\operatorname{PSL}_2 ( \CC)$ is not well-defined, it is up to sign. An element of a number field $F$ is integral over $\ZZ$ if and only if its negative is, so we say a representation $\rho \co \Gamma \to \operatorname{PSL}_2 (F)$ is *algebraic non-integral* if there is some $\gamma \in \Gamma$ such that the trace of a lift of $\rho ( \gamma)$ to $\operatorname{SL}_2 (\CC)$ is algebraic non-integral.
There are infinitely many large closed hyperbolic 3–manifolds with no ANI-representations into $\operatorname{PSL}_2 ( \CC)$.
Let $N$ denote the complement of any knot with the property that the character variety of its complement has no ideal point detecting a closed essential surface (such as the knot $10_{152}$, $10_{153}$ or $10_{154}$). Fix a basis $\{ \mu, \lambda \}$ for $\pi_1( \bound N) \subseteq \pi_1 ( N)$ such that $\mu$ is a meridian for $N$ and $\lambda$ may be represented by a simple closed curve on $\bound N$. Let $N(p/q)$ denote the closed 3–manifold obtained by performing $p/q$ Dehn filling on $N$; that is, attach a solid torus to $\bound N$ such that the meridian of the solid torus is glued to $\bound N$ along a primitive curve representing $\mu^p \lambda^q$. We obtain a presentation for $\pi_1 ( N ( p/q))$ by simply adding the relation $\mu^p \lambda^q$ to a presentation for $\pi_1(N)$. In particular, there is an epimorphism $\pi_1 (N ) \to \pi_1 ( N ( p/q))$ which induces inclusions $$\X_{\operatorname{SL}} ( N(p/q)) \hookrightarrow \X_{\operatorname{SL}} ( N ) \quad \text{and} \quad \X_{\operatorname{PSL}}(N(p/q)) \to \X_{\operatorname{PSL}}(N).$$
There is a sequence $\{p_i/q_i \}_{i=1}^\infty$ of slopes on $\bound N$ such that
1. $N(p_i / q_i)$ is a large hyperbolic closed 3–manifold;
2. $p_i / q_i$ is not a boundary slope of $N$.
That $N(p_i/q_i)$ can be taken to be hyperbolic follows from Thurston’s Hyperbolic Dehn Surgery Theorem [@THUR]. We may assume $N(p_i/q_i)$ is large by [@CGLS Theorem 2.0.3]. Finally, $N$ has only finitely many boundary slopes [@HAT], so we may choose $p_i/q_i$ that are not boundary slopes of $N$.
Fix $p_i / q_i$ and let $r \co \pi_1 ( N) \to \pi_1 ( N( p_i / q_i ))$ be the quotient map. Suppose towards a contradiction that $N( p_i / q_i )$ has an algebraic non-integral representation $\rho \co \pi_1 ( N(p_i/q_i)) \to \operatorname{PSL}_2 ( \CC)$. Then $\rho \circ r$ is an ANI representation of $\pi_1(N)$ into $\operatorname{PSL}_2 ( \CC)$. By [@CU], this representation lifts to an ANI-representation $ {\widetilde{\rho}} \co \pi_1 (N ) \to \operatorname{SL}_2 ( \CC).$
We claim that ${\widetilde{\rho}}$ must ANI-detect a closed essential surface in $N$. Since $\rho\circ r$ factors through $\pi_1( N ( p_i / q_i))$, we must have ${\widetilde{\rho}}( \mu^{p_i}\lambda^{q_i}) = I \text{ or } -I$, where $I$ is the $2 \times 2$ identity matrix. In particular, the character $\chi_{{\widetilde{\rho}}}$ evaluates to $\pm 2$ at $\mu^{p_i}\lambda^{q_i}$, both of which are integral over $\ZZ$. Thus, by Theorem \[thm:cs1\], $\chi_{{\widetilde{\rho}}}$ either ANI-detects a closed essential surface or $\mu^{p_i}\lambda^{q_i}$ is a boundary slope of $N$. This is not the case by construction, proving our claim.
Finally, Chesebro showed that if $\X_{\operatorname{SL}}(N)$ ANI-detects a closed essential surface, then it detects one by ideal points [@C Proposition 5.2]. This contradicts our hypothesis, so $N(p_i / q_i)$ has no ANI-representations into $\operatorname{PSL}_2 ( \CC)$.
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---
abstract: 'We demonstrate that massive simulated galaxies assemble in two phases, with the initial growth dominated by compact in situ star formation, whereas the late growth is dominated by accretion of old stars formed in subunits outside the main galaxy. We also show that 1) gravitational feedback strongly suppresses late star formation in massive galaxies contributing to the observed galaxy colour bimodality that 2) the observed galaxy downsizing can be explained naturally in the two-phased model and finally that 3) the details of the assembly histories of massive galaxies are directly connected to their observed kinematic properties.'
---
The two phased formation of early-type galaxies
===============================================
High-resolution numerical zoom-in simulations of massive early-type galaxies have shown that there are two distinct phases in their formation histories ([@2007ApJ...658..710N Naab et al. 2007]; [@2010ApJ...725.2312O Oser et al. 2010]; [@2012MNRAS.425..641L Lackner et al. 2012]). At high redshifts of $z\sim 3-6$ the galaxies assemble rapidly through compact $(r<r_{\rm eff})$ in situ star formation. The later growth of the galaxies proceeds predominantly through the accretion of stars formed in subunits outside the main galaxy. The majority of the accreted stars are added to the outskirts of the galaxies at larger radii $(r>r_{\rm eff})$ providing a satisfactory explanation for the observed size growth of massive early-type galaxies (ETGs) since $z\sim 3$ until the present-day ([@2009ApJ...699L.178N Naab et al. 2009]; [@2009ApJ...697.1290B Bezanson et al. 2009]; [@2012ApJ...744...63O Oser et al. 2012]).
Here we show that in addition to the size growth of ETGs, the two phased formation picture can also shed light on other observational results of ETGs, such as the observed galaxy bimodality, the downsizing of massive galaxies and the observed dichotomy of the kinematic properties of ETGs. The simulation sample we use to address these questions contains 9 galaxies simulated at high-resolution ([@2012ApJ...754..115J Johansson et al. 2012]) using the Gadget-2 code ([@2005MNRAS.364.1105S Springel 2005]). The simulations include cooling for a primordial composition, star formation and feedback from type II supernovae, but exclude supernova driven winds and AGN feedback. We run three models (A2,C2,E2) at very high spatial ($\epsilon_{\star}=0.125 \ \rm kpc$) and mass resolution ($m_{\star}\sim 10^{5} M_{\odot}$) with the remaining six simulations (U,Y,T,Q,M,L) simulated at a somewhat lower resolution of ($\epsilon_{\star}=0.25 \ \rm kpc$, $m_{\star}\sim 10^{6} M_{\odot}$). We organize the galaxies in our simulation sample into three groups of three galaxies each depending on whether their late assembly history $(z\lesssim 2)$ is dominated primarily by dissipationless minor merging (mostly accreted stars: galaxies C2,U,Y), a mixed dissipationless/dissipational (mostly accreted and some in situ stars: galaxies A2,Q,T) or a primarily dissipational formation history (significant amount of in situ stars: galaxies E2,M,L). We find that this classification of the assembly histories of ETGs is useful as it broadly separates more massive ETGs forming dissipationlessly from less massive ETGs with a more dissipational formation history.
![The net heating (solid line) and net cooling rates (dashed lines) for virial $(r<r_{\rm vir})$ non-starforming diffuse gas, for which the density is below the star formation threshold of $n_{\rm th}<0.205 \ \rm cm^{-3}$ shown at redshifts of $z=0$ (left) and $z=3$ (right). The fraction $f_{\rm{sf}}$ of dense starforming gas $(n>n_{\rm th})$ is also given. Typically, the heating rate dominates over the cooling rate at all redshifts for the low-density non-starforming gas.[]{data-label="fig1"}](johansson_ph_fig1.eps){width="14.0cm"}
The bimodality of the local galaxy population
=============================================
Recent large-field sky surveys have unequivocally demonstrated that local galaxies show a bimodal distribution. Local galaxies with stellar masses above a critical mass of $M_{\rm crit}\simeq 3\times 10^{10} M_{\odot}$ are typically red spheroidal galaxies with old stellar populations, whereas galaxies below this critical mass are typically blue, star-forming galaxies with somewhat younger stellar populations. The observed bimodality is usually explained theoretically using models in which the star formation is efficiently quenched in massive haloes above $M\sim 10^{12} M_{\odot}$ ([@2006MNRAS.368....2D Dekel & Birnboim 2006]; [@2011MNRAS.417.2676G Gabor et al. 2011]).
The late assembly of our simulated galaxies is dominated by dry minor merging building up the accreted stellar component. The infalling satellite galaxies captured through dynamical friction will cause gaseous wakes from which energy is transferred to the surrounding gas. Collectively together with the heating caused by supersonic collisions and shocks caused by infalling cold gas this process is deemed gravitational heating. In Fig. \[fig1\] we compare the shock-induced heating rates of the diffuse non-starforming gas with the corresponding cooling rates. The heating rates dominate over the cooling rates at all redshifts with the more massive galaxies (U,A2) showing higher heating rates than the slightly lower mass galaxies (C2,E2), as expected by the scaling of the gravitational feedback energy, $(\Delta E)_{\rm grav}\propto v_{c}^{2}$, where $v_{c}$ is the circular velocity of the galaxy ([@2009ApJ...697L..38J Johansson et al. 2009b]). The inclusion of gravitational heating helps in maintaining a hot gaseous halo and thus inhibits star formation contributing to the observed galaxy bimodality. However, gravitational heating is predominantly important in the outer parts of galaxies and some form of additional feedback, most probably AGN feedback (e.g. [@2009ApJ...690..802J Johansson et al. 2009a]), is required to stop late central star formation in very massive galaxies (e.g. galaxy U at $z=0$).
Downsizing of massive galaxies
==============================
Several recent observations have shown that old, massive red metal-rich galaxies were already in place at high redshifts of $z\sim 2-3$. This observational result can be seen as a manifestation of cosmic downsizing, in which galaxies seem to form anti-hierarchically in the sense that the most massive galaxies formed a significant proportion of their stars at high redshifts, compared to lower mass systems that exhibit a more continuous star formation history throughout the cosmic epoch (e.g. [@2004Natur.430..181G Glazebrook et al. 2004]).
![The stellar mass assembly histories of our simulated galaxy sample, with the solid colour bars showing the contribution of in situ formed stellar mass and the dashed colour bars representing the contribution of accreted stellar mass shown at redshifts of $z=0$ (left) and $z=3$ (right). At high redshifts ($z\gtrsim 3$) the galaxies assemble rapidly through in situ star formation, whereas the late ($z\lesssim 3$) assembly history is dominated by accreted stars, with the more massive galaxies ending up with a proportionally larger fraction of accreted stars.[]{data-label="fig2"}](johansson_ph_fig2.eps){width="14.0cm"}
In Fig. \[fig2\] we show the assembly histories of all our simulated galaxies depicting separately the masses in the in-situ formed and accreted stellar components. At $z=3$ the stellar components in all galaxies have been assembled rapidly through mainly in situ star formation, fueled by cold gas flows and hierarchical mergers of multiple star-bursting subunits. At lower redshifts $(z\lesssim 3)$ the subsequent growth of the stellar component proceeds predominantly through the accretion of existing stellar clumps. The galaxies in Fig. \[fig2\] are ordered in decreasing final stellar mass from top to bottom and we can immediately see that the fraction of accreted stars at $z=0$ increases as a function of galaxy mass. The accreted stellar component forms on average in low mass galaxies very early in the Universe. However, the accreted stellar component is added to the massive galaxies much later and at lower redshifts than the in situ formed stars. In addition, the metallicity of the central in situ formed stars is on average higher than the metallicity of the accreted stars that are formed in lower mass galaxies and added later to the outskirts of galaxies, resulting in a negative metallicity gradient, in agreement with the observations. Thus, the counter-intuitive concept of downsizing can be explained in the two phased formation mechanism. Massive galaxies form their central stellar mass in situ and then accrete substantial amounts of stars that were formed even earlier in smaller subsystems. Hence by $z\sim 2-3$ the most massive galaxies have the oldest stellar populations compared to lower mass galaxies that are still forming in situ stars.
Kinematic properties of massive galaxies
========================================
![The effective ellipticity $(\epsilon_{\rm eff})$ is plotted against the ratio of the major axis rotation and central velocity dispersion $(v_{\rm maj}/\sigma_{0})$. The contours show the 95% probability location of the galaxies in the $(\epsilon_{\rm eff}-v_{\rm maj}/\sigma_{0})$-plane derived from 500 random viewing angles of our simulation data. The overplotted symbols are observational data from ([@1994MNRAS.269..785B Bender et al. 1994]) demonstrating that our simulated galaxies seem to be largely consistent with the observations. The dashed line shows the theoretical value for an oblate isotropic rotator.[]{data-label="fig3"}](johansson_ph_fig3.eps){width="14.5cm"}
Observations in the late 1980s using slit spectroscopy found that the ETG population can be broadly separated into a population of massive slowly-rotating systems $(v/\sigma<0.1)$ with boxy isophotes and a population of fast-rotating $(v/\sigma\sim 1)$ ETGs with more disky isophotes found typically at somewhat lower masses. Recent results from the volume limited ATLAS$^{\rm 3D}$ survey utilizing a modern integral-field-unit (IFU) confirmed this dichotomy showing that about $\sim 15\%$ of local ETGs rotate slowly with no indications of an embedded disk component, whereas the majority ($\sim 85\%$) of the local ETGs show significant disk-like rotation ([@2011MNRAS.413..813C Cappellari et al. 2011]).
We derive the kinematic properties of our simulated galaxies using 500 random projections in order to assess the mean properties of the simulated galaxies averaged over all sightlines. In Fig. \[fig3\] we plot the 95% probability of finding a simulated galaxy in the $\epsilon_{\rm eff}-(v_{\rm maj}/\sigma_{0})$ plane, where $\epsilon_{\rm eff}$ measures the effective ellipticity of the galaxies and the ratio of the major axis rotation and velocity dispersion $(v_{\rm maj}/\sigma_{0})$ is a measure of the rotational support of the galaxies. Galaxies U,C,E2 show projections with very low rotational support, whereas the other galaxies (Y,A2,Q,T,L,M) show significantly more rotational support in very good agreement with the theoretical prediction for an oblate isotropic rotator shown by the dashed line in Fig. \[fig3\]. The most massive galaxy U shows mostly round projections with very low $\epsilon_{\rm eff}$, whereas all the other galaxies show projections extending in $\epsilon_{\rm eff}$ from zero up to 0.4. We see a correlation between the rotational support and the in situ/accreted fraction, both galaxies U,C that have a large accreted fraction are slowly rotating and galaxy Y would most probably also been slowly rotating if it had not experienced a late $(z<0.5)$ major merger. Thus, within our rather narrow mass range the majority of the simulated galaxies are consistent with being rotationally supported disky ellipticals, whereas the most massive galaxy in our sample is consistent with being a roundish slow-rotator.
, R., [Saglia]{}, R. P., & [Gerhard]{}, O. E. 1994, *MNRAS*, 269, 785
, R., [van Dokkum]{}, P. G., [Tal]{}, T., et al. 2009, *ApJ*, 697, 1290
, M., [Emsellem]{}, E., [Krajnovi[ć]{}]{}, D., et al., 2011, *MNRAS*, 413, 813
, A. & [Birnboim]{}, Y. 2006, *MNRAS*, 368, 2
, J. M., [Dav[é]{}]{}, R., [Oppenheimer]{}, B. D., [Finlator]{}, K., 2011, *MNRAS*, 417, 2676
, K., [Abraham]{}, R. G., [McCarthy]{}, P. J., et al. 2004, *Nature*, 430, 181
, P. H., [Naab]{}, T., & [Burkert]{}, A. 2009a, *ApJ*, 690, 802
, P. H., [Naab]{}, T., & [Ostriker]{}, J. P. 2009b, *ApJL*, 697, L38
, P. H., [Naab]{}, T., & [Ostriker]{}, J. P. 2012, *ApJ*, 754, 115
, C. N., [Cen]{}, R., [Ostriker]{}, J. P., & [Joung]{}, M. R., 2012 *MNRAS*, 425, 641
, T., [Johansson]{}, P. H., [Ostriker]{}, J. P., & [Efstathiou]{}, G. 2007, *ApJ*, 658, 710
, T., [Johansson]{}, P. H., & [Ostriker]{}, J. P. 2009, *ApJL*, 699, L178
, L., [Ostriker]{}, J. P., [Naab]{}, T., [Johansson]{}, P. H., & [Burkert]{}, A. 2010, *ApJ*, 725, 2312
, L., [Naab]{}, T., [Ostriker]{}, J. P., & [Johansson]{}, P. H. 2012, *ApJ*, 744, 63
, V. 2005, *MNRAS*, 364, 1105
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---
abstract: 'The paper investigates the second-order blameworthiness or duty to warn modality “one coalition knew how another coalition could have prevented an outcome”. The main technical result is a sound and complete logical system that describes the interplay between the distributed knowledge and the duty to warn modalities.'
author:
- Pavel Naumov
- Jia Tao
bibliography:
- 'sp.bib'
title: Duty to Warn in Strategic Games
---
Introduction
============
On October 27, 1969, Prosenjit Poddar, an Indian graduate student from the University of California, Berkeley, came to the parents’ house of Tatiana Tarasoff, an undergraduate student who recently immigrated from Russia. After a brief conversation, he pulled out a gun and unloaded it into her torso, then stabbed her eight times with a 13-inch butcher knife, walked into the house and called the police. Tarasoff was pronounced dead on arrival at the hospital [@a17timeline].
In this paper we study the notion of blameworthiness. This notion is usually defined through the principle of alternative possibilities: an agent (or a coalition of agents) is blamable for ${\varphi}$ if ${\varphi}$ is true and the agent had a strategy to prevent it [@f69tjop; @w17]. This definition is also referred to as the counterfactual definition of blameworthiness [@c15cop]. In our case, Poddar is blamable for the death of Tatiana because he could have taken actions (to refrain from shooting and stabbing her) that would have prevented her death. He was found guilty of second-degree murder and sentenced to five years [@a17timeline]. The principle of alternative possibilities, sometimes referred to as “counterfactual possibility” [@c15cop], is also used to define causality [@lewis13; @h16; @bs18aaai]. A sound and complete axiomatization of modality “statement ${\varphi}$ is true and coalition $C$ had a strategy to prevent ${\varphi}$” is proposed in [@nt19aaai]. In related works, Xu [@x98jpl] and Broersen, Herzig, and Troquard [@bht09jancl] axiomatized modality “took actions that unavoidably resulted in ${\varphi}$” in the cases of single agents and coalitions respectively.
According to the principle of alternative possibilities, Poddar is not the only one who is blamable for Tatiana’s death. Indeed, Tatiana’s parents could have asked for a temporary police protection, hired a private bodyguard, or taken Tatiana on a long vacation outside of California. Each of these actions is likely to prevent Tatiana’s death. Thus, by applying the principle of alternative possibilities directly, we have to conclude that her parents should be blamed for Tatiana’s death. However, the police is unlikely to provide life-time protection; the parents’ resources can only be used to hire a bodyguard for a limited period time; and any vacation will have to end. These measures would only work if they knew an approximate time of a likely attack on their daughter. Without this crucial information, they had a strategy to prevent her death, but they did not know what this strategy was. If an agent has a strategy to achieve a certain outcome, knows that it has a strategy, and knows what this strategy is, then we say that the agent has a [*know-how strategy*]{}. Axiomatic systems for know-how strategies have been studied before [@aa16jlc; @fhlw17ijcai; @nt17aamas; @nt18ai; @nt18aaai; @nt19ai]. In a setting with imperfect information, it is natural to modify the principle of alternative possibilities to require an agent or a coalition to have a know-how strategy to prevent. In our case, parents had many different strategies that included taking vacations in different months. They did not know that a vacation in October would have prevented Tatiana’s death. Thus, they cannot be blamed for her death according to the modified version of the principle of alternative possibilities. We write this as $
\neg{{\sf B}}_{\mbox{\scriptsize parents}}(\mbox{``Tatiana is killed''}).
$ Although Tatiana’s parents did not know how to prevent her death, Dr. Lawrence Moore did. He was a psychiatrist who treated Poddar at the University of California mental clinic. Poddar told Moore how he met Tatiana at the University international student house, how they started to date and how depressed Poddar became when Tatiana lost romantic interest in him. Less than two months before the tragedy, Poddar shared with the doctor his intention to buy a gun and to murder Tatiana. Dr. Moore reported this information to the University campus police. Since the University knew that Poddar was at the peak of his depression, they could estimate the possible timing of the attack. Thus, the University knew what actions the parents could take to prevent the tragedy. In general, if a coalition $C$ knows how a coalition $D$ can achieve a certain outcome, then coalition $D$ has a [*second-order know-how*]{} strategy to achieve the outcome. This class of strategies and a complete logical system that describes its properties were proposed in [@nt18aamas]. We write ${{\sf B}}_C^D{\varphi}$ if [*${\varphi}$ is true and coalition $C$ knew how coalition $D$ could have prevented ${\varphi}$*]{}. In our case, $
{{\sf B}}_{\mbox{\scriptsize university}}^{\mbox{\scriptsize parents}}(\mbox{``Tatiana is killed''}).
$
After Tatiana’s death, her parents sued the University. In 1976 the California Supreme Court ruled that “When a therapist determines, or pursuant to the standards of his profession should determine, that his patient presents a serious danger of violence to another, he incurs an obligation to use reasonable care to protect the intended victim against such danger. The discharge of this duty may require the therapist to take one or more of various steps, depending upon the nature of the case. Thus it may call for him to warn the intended victim or others likely to apprise the victim of the danger, to notify the police, or to take whatever other steps are reasonably necessary under the circumstances.” [@t76opinion]. In other words, the California Supreme Court ruled that in this case the duty to warn is not only a moral obligation but a legal one as well. In this paper we propose a sound and complete logical system that describes the interplay between the distributed knowledge modality ${{\sf K}}_C$ and the second-order blameworthiness or [*duty to warn*]{} modality ${{\sf B}}^D_C$. The (first-order) blameworthiness modality ${{\sf B}}_C{\varphi}$ mentioned earlier could be viewed as an abbreviation for ${{\sf B}}_C^C{\varphi}$. For example, $
{{\sf B}}_{\mbox{\scriptsize Poddar}}(\mbox{``Tatiana is killed''})
$ because Poddar knew how he himself could prevent Tatiana’s death. The paper is organized as follows. In the next section we introduce and discuss the formal syntax and semantics of our logical system. In Section \[axioms section\] we list axioms and compare them to those in the related logical systems. Section \[examples section\] gives examples of formal proofs in our system. Section \[soundness section\] and Section \[completeness section\] contain the proofs of the soundness and the completeness, respectively. Section \[conclusion section\] concludes.
Syntax and Semantics {#syntax and semantics section}
====================
In this section we introduce the formal syntax and semantics of our logical system. We assume a fixed set of propositional variables and a fixed set of agents $\mathcal{A}$. By a coalition we mean any subset of $\mathcal{A}$. The language $\Phi$ of our logical system is defined by grammar: $${\varphi}:= p\;|\;\neg{\varphi}\;|\;{\varphi}\to{\varphi}\;|\;{{\sf K}}_C{\varphi}\;|\;{{\sf B}}^D_C{\varphi},$$ where $C$ and $D$ are arbitrary coalitions. Boolean connectives $\bot$, $\wedge$, and $\vee$ are defined through $\neg$ and $\to$ in the usual way. By ${{\sf \overline{K}}}_C{\varphi}$ we denote the formula $\neg{{\sf K}}_C\neg{\varphi}$ and by $X^Y$ the set of all functions from set $Y$ to set $X$.
\[game definition\] A game is a tuple $\left(I, \{\sim_a\}_{a\in\mathcal{A}},\Delta,\Omega,P,\pi\right)$, where
1. $I$ is a set of “initial states”,
2. $\sim_a$ is an “indistinguishability” equivalence relation on the set of initial states $I$, for each agent $a\in\mathcal{A}$,
3. $\Delta$ is a set of “actions”,
4. $\Omega$ is a set of “outcomes”,
5. a set of “plays” $P$ is an arbitrary set of tuples $(\alpha,\delta,\omega)$ such that $\alpha\in I$, $\delta\in\Delta^\mathcal{A}$, and $\omega\in\Omega$. Furthermore, we assume that for each initial state $\alpha\in I$ and each function $\delta\in\Delta^\mathcal{A}$, there is at least one outcome $\omega\in \Omega$ such that $(\alpha,\delta,\omega)\in P$,
6. $\pi(p)\subseteq P$ for each propositional variable $p$.
By a complete (action) profile we mean any function $\delta\in\Delta^\mathcal{A}$ that maps agents in $\mathcal{A}$ into actions in $\Delta$. By an (action) profile of a coalition $C$ we mean any function from set $\Delta^C$.
Figure \[example figure\] depicts a diagram of the game for the Tarasoff case. It shows two possible initial states: October and November that represent two possible months with the peak of Poddar’s depression. The actual initial state was October, which was known to the University, but not to Tatiana’s parents. In other words, the University could distinguish these two states, but the parents could not. We show the indistinguishability relation by dashed lines. At the peak of his depression, agent Poddar might decide not to attack Tatiana (action 0) or to attack her (action 1). Parents, whom we represent by a single agent for the sake of simplicity, might decide to take vacation in October (action 0) or November (action 1). Thus, in our example, $\Delta=\{0,1\}$. Set $\Omega$ consists of outcomes $dead$ and $alive$. Recall that a complete action profile is a function from agents into actions. Since in our case there are only two agents (Poddar and parents), we write action profiles as $xy$ where $x\in \{0,1\}$ is an action of Poddar and $y\in\{0,1\}$ is an action of the parents. The plays of the game are all possible valid combinations of an initial state, a complete action profile, and an outcome. The plays are represented in the diagram by directed edges. For example, the directed edge from initial state October to outcome $dead$ is labeled with action profile $11$. This means that $(\mbox{October},11,dead)\in P$. In other words, if the peak of depression is in October, Poddar decides to attack (1), and the parents take vacation in November (1), then Tatiana is dead. Multiple labels on the same edge of the diagram represent multiple plays with the same initial state and the same outcome. Function $\pi$ specifies the meaning of propositional variables. Namely, $\pi(p)$ is the set of all plays for which proposition $p$ is true.
Next is the core definition of this paper. Its item 5 formally defines the semantics of modality ${{\sf B}}_C^D$. Traditionally, in modal logic the satisfiability $\Vdash$ is defined as a relation between a state and a formula. This approach is problematic in the case of the blameworthiness modality because this modality refers to two different states: ${{\sf B}}^D_C{\varphi}$ if statement ${\varphi}$ is true in [*the current*]{} state and coalition $C$ knew how coalition $D$ could have prevented ${\varphi}$ in [*the previous*]{} state. In other words, the meaning of formula ${{\sf B}}^D_C{\varphi}$ depends not only on the current state, but on the previous one as well. We resolve this issue by defining the satisfiability as a relation between a [*play*]{} and a formula, where a play is a triple consisting of the previous state $\alpha$, the complete action profile $\delta$, and an outcome (state) $\omega$. We distinguish initial states from outcomes to make the presentation more elegant. Otherwise, this distinction is not significant.
We write $\omega\sim_C\omega'$ if $\omega\sim_a\omega'$ for each agent $a\in C$. We also write $f=_X g$ if $f(x)=g(x)$ for each element $x\in X$.
\[sat\] For any game $\left(I, \{\sim_a\}_{a\in\mathcal{A}},\Delta,\Omega,P,\pi\right)$, any formula ${\varphi}\in\Phi$, and any play $(\alpha,\delta,\omega)\in P$, the satisfiability relation $(\alpha,\delta,\omega)\Vdash{\varphi}$ is defined recursively as follows:
1. $(\alpha,\delta,\omega)\Vdash p$ if $(\alpha,\delta,\omega)\in \pi(p)$,
2. $(\alpha,\delta,\omega)\Vdash \neg{\varphi}$ if $(\alpha,\delta,\omega)\nVdash {\varphi}$,
3. $(\alpha,\delta,\omega)\Vdash{\varphi}\to\psi$ if $(\alpha,\delta,\omega)\nVdash{\varphi}$ or $(\alpha,\delta,\omega)\Vdash\psi$,
4. $(\alpha,\delta,\omega)\Vdash{{\sf K}}_C{\varphi}$ if $(\alpha',\delta',\omega')\Vdash{\varphi}$ for each $(\alpha',\delta',\omega')\in P$ such that $\alpha\sim_C\alpha'$,
5. $(\alpha,\delta,\omega)\Vdash{{\sf B}}_C^D{\varphi}$ if $(\alpha,\delta,\omega)\Vdash{\varphi}$ and there is a profile $s\in \Delta^D$ such that for each play $(\alpha',\delta',\omega')\in P$, if $\alpha\sim_C\alpha'$ and $s=_D\delta'$, then $(\alpha',\delta',\omega')\nVdash{\varphi}$.
Going back to our running example, $$(\mbox{October},11,dead)\Vdash{{\sf B}}_{\mbox{\scriptsize university}}^{\mbox{\scriptsize parents}}(\mbox{``Tatiana is killed''})$$ because $
(\mbox{October},11,dead)\Vdash\mbox{``Tatiana is killed''}
$ and $$(\alpha',\delta',\omega')\nVdash(\mbox{``Tatiana is killed''})$$ for each play $(\alpha',\delta',\omega')\in P$ such that $\alpha'\sim_{\mbox{\scriptsize university}}\mbox{October}$ and $\delta'(\mbox{parents})=0$.
Because the satisfiability is defined as a relation between plays and formulae, one can potentially talk about two forms of knowledge [*about a play*]{} in our system: [*a priori*]{} knowledge in the initial state and [*a posteriori*]{} knowledge in the outcome. The knowledge captured by the modality ${{\sf K}}$ as well as the knowledge implicitly referred to by the modality ${{\sf B}}$, see item (5) of Definition \[sat\], is [*a priori*]{} knowledge about a play. In order to define a posteriori knowledge in our setting, one would need to add an indistinguishability relation on outcomes to Definition \[game definition\]. We do not consider a posteriori knowledge because one should not be blamed for something that the person only knows how to prevent [*post-factum*]{}.
Since we define the second-order blameworthiness using distributed knowledge, if a coalition $C$ is blamable for not warning coalition $D$, then any superset $C'\supseteq C$ could be blamed for not warning $D$. One might argue that the definition of blameworthiness modality ${{\sf B}}^D_C$ should include a minimality condition on the coalition $C$. We do not include this condition in item (5) of Definition \[sat\], because there are several different ways to phrase the minimality, all of which could be expressed through our basic modality ${{\sf B}}^D_C$.
First of all, we can say that $C$ is the minimal coalition among those coalitions that knew how $D$ could have prevented ${\varphi}$. Let us denote this modality by $[1]^D_C{\varphi}$. It can be expressed through ${{\sf B}}^D_C$ as: $$[1]^D_C{\varphi}\equiv{{\sf B}}^D_C{\varphi}\wedge \neg\bigvee_{E\subsetneq C}{{\sf B}}^D_E{\varphi}.$$
Second, we can say that $C$ is the minimal coalition that knew how [*somebody*]{} could have prevented ${\varphi}$: $$[2]^D_C{\varphi}\equiv{{\sf B}}^D_C{\varphi}\wedge \neg\bigvee_{E\subsetneq C}\bigvee_{F\subseteq\mathcal{A}}{{\sf B}}^F_E{\varphi}.$$
Third, we can say that $C$ is the minimal coalition that knew how [*the smallest*]{} coalition $D$ could have prevented ${\varphi}$: $$[3]^D_C{\varphi}\equiv{{\sf B}}^D_C{\varphi}\wedge \neg\bigvee_{E\subseteq \mathcal{A}}\bigvee_{F\subsetneq D}{{\sf B}}^F_E{\varphi}\wedge \neg\bigvee_{E\subsetneq C}{{\sf B}}^D_E{\varphi}.$$
Finally, we can say that $C$ is the minimal coalition that knew how [*some smallest*]{} coalition could have prevented ${\varphi}$: $$[4]_C{\varphi}\equiv \bigvee_{D\subseteq\mathcal{A}}\left({{\sf B}}^D_C{\varphi}\wedge \neg\bigvee_{E\subseteq \mathcal{A}}\bigvee_{F\subsetneq D}{{\sf B}}^F_E{\varphi}\wedge \neg\bigvee_{E\subsetneq C}{{\sf B}}^D_E{\varphi}\right).$$ The choice of the minimality condition depends on the specific situation. Instead of making a choice between several possible alternatives, in this paper we study the basic blameworthiness modality without a minimality condition through which modalities $[1]^D_C{\varphi}$, $[2]^D_C{\varphi}$, $[3]^D_C{\varphi}$, $[4]_C{\varphi}$, and possibly others could be defined.
Axioms {#axioms section}
======
In addition to the propositional tautologies in language $\Phi$, our logical system contains the following axioms:
1. Truth: ${{\sf K}}_C{\varphi}\to{\varphi}$ and ${{\sf B}}^D_C{\varphi}\to{\varphi}$,
2. Distributivity: ${{\sf K}}_C({\varphi}\to\psi)\to({{\sf K}}_C{\varphi}\to {{\sf K}}_C\psi)$,
3. Negative Introspection: $\neg{{\sf K}}_C{\varphi}\to{{\sf K}}_C\neg{{\sf K}}_C{\varphi}$,
4. Monotonicity: ${{\sf K}}_C{\varphi}\to{{\sf K}}_E{\varphi}$ and ${{\sf B}}^D_C{\varphi}\to{{\sf B}}^F_E{\varphi}$,\
where $C\subseteq E$ and $D\subseteq F$,
5. None to Act: $\neg{{\sf B}}^\varnothing_C{\varphi}$,
6. Joint Responsibility: if $D\cap F=\varnothing$, then\
${{\sf \overline{K}}}_C{{\sf B}}^D_C{\varphi}\wedge{{\sf \overline{K}}}_E{{\sf B}}_E^F\psi\to ({\varphi}\vee\psi\to{{\sf B}}_{C\cup E}^{D\cup F}({\varphi}\vee\psi))$,
7. Strict Conditional: ${{\sf K}}_C({\varphi}\to\psi)\to({{\sf B}}^D_C\psi\to({\varphi}\to {{\sf B}}^D_C{\varphi}))$,
8. Introspection of Blameworthiness: ${{\sf B}}^D_C{\varphi}\to{{\sf K}}_C({\varphi}\to{{\sf B}}^D_C{\varphi})$.
The Truth, the Distributivity, the Negative Introspection, and the Monotonicity axioms for modality ${{\sf K}}$ are the standard axioms from the epistemic logic S5 for distributed knowledge [@fhmv95]. The Truth axiom for modality ${{\sf B}}$ states that a coalition can only be blamed for something that has actually happened. The Monotonicity axiom for modality ${{\sf B}}$ captures the fact that both distributed knowledge and coalition power are monotonic.
The None to Act axiom is true because the empty coalition has only one action profile. Thus, if the empty coalition can prevent ${\varphi}$, then ${\varphi}$ would have to be false on the current play. This axiom is similar to the None to Blame axiom $\neg{{\sf B}}_\varnothing{\varphi}$ in [@nt19aaai].
The Joint Responsibility axiom shows how the blame of two separate coalitions can be combined into the blame of their union. This axiom is closely related to Marc Pauly [@p02] Cooperation axiom, which is also used in coalitional modal logics of know-how [@aa16jlc; @nt17aamas; @nt18ai; @nt18aaai] and second-order know-how [@nt18aamas]. We formally prove the soundness of this axiom in Lemma \[joint responsibility soundness\].
Strict conditional ${{\sf K}}_C({\varphi}\to\psi)$ states that formula ${\varphi}$ is known to $C$ to imply $\psi$. By contraposition, coalition $C$ knows that if $\psi$ is prevented, then ${\varphi}$ is also prevented. The Strict Conditional axiom states that if $C$ could be second-order blamed for $\psi$, then it should also be second-order blamed for ${\varphi}$ as long as ${\varphi}$ is true. A similar axiom is present in [@nt19aaai].
Finally, the Introspection of Blameworthiness axiom says that if coalition $C$ is second-order blamed for ${\varphi}$, then $C$ knows that it is second-order blamed for ${\varphi}$ as long as ${\varphi}$ is true. A similar Strategic Introspection axiom for second-order know-how modality is present in [@nt18aamas].
We write $\vdash{\varphi}$ if formula ${\varphi}$ is provable from the axioms of our system using the Modus Ponens and the Necessitation inference rules: $$\dfrac{{\varphi},{\varphi}\to\psi}{\psi},
\hspace{20mm}
\dfrac{{\varphi}}{{{\sf K}}_C{\varphi}}.$$ We write $X\vdash{\varphi}$ if formula ${\varphi}$ is provable from the theorems of our logical system and an additional set of axioms $X$ using only the Modus Ponens inference rule.
\[super distributivity\] If ${\varphi}_1,\dots,{\varphi}_n\!\vdash\!\psi$, then ${{\sf K}}_C{\varphi}_1,\dots,{{\sf K}}_C{\varphi}_n\!\vdash\!{{\sf K}}_C\psi$.
By the deduction lemma applied $n$ times, assumption ${\varphi}_1,\dots,{\varphi}_n\vdash\psi$ implies that $
\vdash{\varphi}_1\to({\varphi}_2\to\dots({\varphi}_n\to\psi)\dots).
$ Thus, by the Necessitation inference rule, $$\vdash{{\sf K}}_C({\varphi}_1\to({\varphi}_2\to\dots({\varphi}_n\to\psi)\dots)).$$ Hence, by the Distributivity axiom and the Modus Ponens rule, $$\vdash{{\sf K}}_C{\varphi}_1\to{{\sf K}}_C({\varphi}_2\to\dots({\varphi}_n\to\psi)\dots).$$ Then, again by the Modus Ponens rule, $${{\sf K}}_C{\varphi}_1\vdash{{\sf K}}_C({\varphi}_2\to\dots({\varphi}_n\to\psi)\dots).$$ Therefore, ${{\sf K}}_C{\varphi}_1,\dots,{{\sf K}}_C{\varphi}_n\vdash{{\sf K}}_C\psi$ by applying the previous steps $(n-1)$ more times.
The next lemma capture a well-known property of S5 modality. Its proof could be found, for example, in [@nt18aamas].
\[positive introspection lemma\] $\vdash {{\sf K}}_C{\varphi}\to{{\sf K}}_C{{\sf K}}_C{\varphi}$.
Examples of Derivations {#examples section}
=======================
The soundness of our logical system is established in the next section. Here we prove several lemmas about our formal system that will be used later in the proof of the completeness.
\[alt fairness lemma\] $\vdash{{\sf \overline{K}}}_C{{\sf B}}^D_C{\varphi}\to({\varphi}\to{{\sf B}}^D_C{\varphi})$.
Note that $\vdash {{\sf B}}_C^D{\varphi}\to{{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi})$ by the Introspection of Blameworthiness axiom. Thus, $\vdash \neg{{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi})\to \neg{{\sf B}}_C^D{\varphi}$, by the law of contrapositive. Then, $\vdash {{\sf K}}_C(\neg{{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi})\to \neg{{\sf B}}_C^D{\varphi})$ by the Necessitation inference rule. Hence, by the Distributivity axiom and the Modus Ponens inference rule, $$\vdash {{\sf K}}_C\neg{{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi})\to {{\sf K}}_C\neg{{\sf B}}_C^D{\varphi}.$$ At the same time, by the Negative Introspection axiom: $$\vdash \neg{{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi})\to{{\sf K}}_C\neg{{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi}).$$ Then, by the laws of propositional reasoning, $$\vdash \neg{{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi})\to {{\sf K}}_C\neg{{\sf B}}_C^D{\varphi}.$$ Thus, by the law of contrapositive, $$\vdash \neg{{\sf K}}_C\neg{{\sf B}}_C^D{\varphi}\to {{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi}).$$ Since ${{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi})\to({\varphi}\to{{\sf B}}_C^D{\varphi})$ is an instance of the Truth axiom, by propositional reasoning, $$\vdash \neg{{\sf K}}_C\neg{{\sf B}}_C^D{\varphi}\to ({\varphi}\to{{\sf B}}_C^D{\varphi}).$$ Therefore, $\vdash {{\sf \overline{K}}}_C{{\sf B}}_C^D{\varphi}\to ({\varphi}\to{{\sf B}}_C^D{\varphi})$ by the definition of ${{\sf \overline{K}}}_C$.
\[alt cause lemma\] If $\vdash {\varphi}\leftrightarrow \psi$, then $\vdash {{\sf B}}^D_C{\varphi}\to{{\sf B}}^D_C\psi$.
By the Strict Conditional axiom, $$\vdash {{\sf K}}_C(\psi\to{\varphi})\to({{\sf B}}_C^D{\varphi}\to(\psi\to {{\sf B}}_C^D\psi)).$$ Assumption $\vdash {\varphi}\leftrightarrow \psi$ implies $\vdash \psi\to {\varphi}$ by the laws of propositional reasoning. Hence, $\vdash {{\sf K}}_C(\psi\to {\varphi})$ by the Necessitation inference rule. Thus, by the Modus Ponens rule, $
\vdash {{\sf B}}_C^D{\varphi}\to(\psi\to {{\sf B}}_C^D\psi).
$ Then, by the laws of propositional reasoning, $$\label{sofia}
\vdash ({{\sf B}}_C^D{\varphi}\to\psi)\to ({{\sf B}}_C^D{\varphi}\to {{\sf B}}_C^D\psi).$$ Observe that $\vdash {{\sf B}}_C^D{\varphi}\to{\varphi}$ by the Truth axiom. Also, $\vdash {\varphi}\leftrightarrow \psi$ by the assumption of the lemma. Then, by the laws of propositional reasoning, $\vdash {{\sf B}}_C^D{\varphi}\to\psi$. Therefore, $
\vdash {{\sf B}}_C^D{\varphi}\to {{\sf B}}_C^D\psi
$ by the Modus Ponens inference rule from statement (\[sofia\]).
\[add cK lemma\] ${\varphi}\vdash {{\sf \overline{K}}}_C{\varphi}$.
By the Truth axioms, $\vdash{{\sf K}}_C\neg{\varphi}\to\neg{\varphi}$. Hence, by the law of contrapositive, $\vdash{\varphi}\to \neg{{\sf K}}_C\neg{\varphi}$. Thus, $\vdash{\varphi}\to {{\sf \overline{K}}}_C{\varphi}$ by the definition of the modality ${{\sf \overline{K}}}_C$. Therefore, ${\varphi}\vdash {{\sf \overline{K}}}_C{\varphi}$ by the Modus Ponens inference rule.
The next lemma generalizes the Joint Responsibility axiom from two coalitions to multiple coalitions.
\[super joint responsibility lemma\] For any integer $n\ge 0$, $$\{{{\sf \overline{K}}}_{E_i}{{\sf B}}^{F_i}_{E_i}\chi_i\}_{i=1}^n,\chi_1\vee\dots\vee\chi_n
\vdash {{\sf B}}^{F_1\cup\dots\cup F_n}_{E_1\cup\dots\cup E_n}(\chi_1\vee \dots\vee\chi_n),$$ where sets $F_1,\dots,F_n$ are pairwise disjoint.
We prove the lemma by induction on $n$. If $n=0$, then disjunction $\chi_1\vee\dots\vee \chi_n$ is Boolean constant false $\bot$. Hence, the statement of the lemma, $\bot\vdash{{\sf B}}_\varnothing^\varnothing\bot$, is provable in the propositional logic.
Next, assume that $n=1$. Then, from Lemma \[alt fairness lemma\] using Modus Ponens rule twice, we get ${{\sf \overline{K}}}_{E_1}{{\sf B}}_{E_1}^{F_1}\chi_1,\chi_1\vdash{{\sf B}}_{E_1}^{F_1}\chi_1$.
Assume now that $n\ge 2$. By the Joint Responsibility axiom and the Modus Ponens inference rule, $$\begin{aligned}
&&\hspace{-8mm}{{\sf \overline{K}}}_{E_1\cup \dots \cup E_{n-1}}{{\sf B}}_{E_1\cup \dots \cup E_{n-1}}^{F_1\cup \dots \cup F_{n-1}}(\chi_1\vee\dots\vee\chi_{n-1}),
{{\sf \overline{K}}}_{E_n}{{\sf B}}_{E_n}^{F_n}\chi_n,\\
&&\hspace{-8mm}\chi_1\vee\dots\vee\chi_{n-1}\vee\chi_n\vdash {{\sf B}}_{E_1\cup \dots \cup E_{n-1}\cup E_{n}}^{F_1\cup \dots \cup F_{n-1}\cup F_{n}}(\chi_1\vee\dots\vee\chi_{n-1}\vee \chi_n).\end{aligned}$$ Hence, by Lemma \[add cK lemma\], $$\begin{aligned}
&&\hspace{-8mm}{{\sf B}}_{E_1\cup \dots \cup E_{n-1}}^{F_1\cup \dots \cup F_{n-1}}(\chi_1\vee\dots\vee\chi_{n-1}),
{{\sf \overline{K}}}_{E_n}{{\sf B}}_{E_n}^{F_n}\chi_n,\chi_1\vee\dots\vee\chi_{n-1}\vee\chi_n\\
&&\vdash {{\sf B}}_{E_1\cup \dots \cup E_{n-1}\cup E_{n}}^{F_1\cup \dots \cup F_{n-1}\cup F_{n}}(\chi_1\vee\dots\vee\chi_{n-1}\vee \chi_n).\end{aligned}$$ At the same time, by the induction hypothesis, $$\{{{\sf \overline{K}}}_{E_i}{{\sf B}}_{E_i}^{F_i}\chi_i\}_{i=1}^{n-1},\chi_1\vee\dots\vee\chi_{n-1}
\vdash {{\sf B}}_{E_1\cup \dots \cup E_{n-1}}^{F_1\cup \dots \cup F_{n-1}}(\chi_1\vee \dots\vee\chi_{n-1}).$$ Thus, $$\begin{aligned}
&&\hspace{-5mm}\{{{\sf \overline{K}}}_{E_i}{{\sf B}}_{E_i}^{F_i}\chi_i\}_{i=1}^n,\chi_1\vee\dots\vee\chi_{n-1},\chi_1\vee\dots\vee\chi_{n-1}\vee\chi_n\\
&&\vdash {{\sf B}}_{E_1\cup \dots \cup E_{n-1}\cup E_{n}}^{F_1\cup \dots \cup F_{n-1}\cup F_{n}}(\chi_1\vee \dots\vee\chi_{n-1}\vee\chi_n).\end{aligned}$$ Note that $\chi_1\vee\dots\vee\chi_{n-1}\vdash\chi_1\vee\dots\vee\chi_{n-1}\vee\chi_n$ is provable in the propositional logic. Thus, $$\begin{aligned}
&&\hspace{-10mm}\{{{\sf \overline{K}}}_{E_i}{{\sf B}}_{E_i}^{F_i}\chi_i\}_{i=1}^n,\chi_1\vee\dots\vee\chi_{n-1}\nonumber\\
&&\hspace{0mm} \vdash {{\sf B}}_{E_1\cup \dots \cup E_{n-1}\cup E_{n}}^{F_1\cup \dots \cup F_{n-1}\cup F_{n}}(\chi_1\vee \dots\vee\chi_{n-1}\vee\chi_n).\label{part 1}\end{aligned}$$ Similarly, by the Joint Responsibility axiom and the Modus Ponens inference rule, $$\begin{aligned}
&&\hspace{-8mm}{{\sf \overline{K}}}_{E_1}{{\sf B}}_{E_1}^{F_1}\chi_1,{{\sf \overline{K}}}_{E_2\cup \dots \cup E_n}{{\sf B}}_{E_2\cup \dots \cup E_n}^{F_2\cup \dots \cup F_n}(\chi_2\vee\dots\vee\chi_n),\\
&&\hspace{-8mm}\chi_1\vee(\chi_2\vee\dots\vee\chi_n)\vdash {{\sf B}}_{E_1\cup \dots \cup E_{n-1}\cup E_n}^{F_1\cup \dots \cup F_{n-1}\cup F_n}(\chi_1\vee(\chi_2\vee\dots\vee \chi_n)).\end{aligned}$$ Because formula $\chi_1\vee(\chi_2\vee\dots\vee \chi_n)\leftrightarrow \chi_1\vee\chi_2\vee\dots\vee \chi_n$ is provable in the propositional logic, by Lemma \[alt cause lemma\], $$\begin{aligned}
&&\hspace{-8mm}{{\sf \overline{K}}}_{E_1}{{\sf B}}_{E_1}^{F_1}\chi_1,{{\sf \overline{K}}}_{E_2\cup \dots \cup E_n}{{\sf B}}_{E_2\cup \dots \cup E_n}^{F_2\cup \dots \cup F_n}(\chi_2\vee\dots\vee\chi_n),\\
&&\hspace{-8mm}\chi_1\vee\chi_2\vee\dots\vee\chi_n\vdash {{\sf B}}_{E_1\cup \dots \cup E_{n-1}\cup E_n}^{F_1\cup \dots \cup F_{n-1}\cup F_n}(\chi_1\vee\chi_2\vee\dots\vee \chi_n).\end{aligned}$$ Hence, by Lemma \[add cK lemma\], $$\begin{aligned}
&&\hspace{-7mm}{{\sf \overline{K}}}_{E_1}{{\sf B}}_{E_1}^{F_1}\chi_1,{{\sf B}}_{E_2\cup \dots \cup E_n}^{F_2\cup \dots \cup F_n}(\chi_2\vee\dots\vee\chi_n),\chi_1\vee\chi_2\vee\dots\vee\chi_n\\
&&\vdash {{\sf B}}_{E_1\cup \dots \cup E_{n-1}\cup E_n}^{F_1\cup \dots \cup F_{n-1}\cup F_n}(\chi_1\vee\chi_2\vee\dots\vee \chi_n).\end{aligned}$$ At the same time, by the induction hypothesis, $$\{{{\sf \overline{K}}}_{E_i}{{\sf B}}_{E_i}^{F_i}\chi_i\}_{i=2}^n,\chi_2\vee\dots\vee\chi_n
\vdash {{\sf B}}_{E_2\cup\dots\cup E_n}^{F_2\cup\dots\cup F_n}(\chi_2\vee \dots\vee\chi_n).$$ Thus, $$\begin{aligned}
&&\hspace{-8mm}\{{{\sf \overline{K}}}_{E_i}{{\sf B}}_{E_i}^{F_i}\chi_i\}_{i=1}^n,\chi_2\vee\dots\vee\chi_n,\chi_1\vee\chi_2\vee\dots\vee\chi_n\\
&&\vdash {{\sf B}}_{E_1\cup\dots\cup E_{n-1}\cup E_n}^{F_1\cup\dots\cup F_{n-1}\cup F_n}(\chi_1\vee\chi_2\vee\dots\vee\chi_n).\end{aligned}$$ Note that $\chi_2\vee\dots\vee\chi_{n}\vdash\chi_1\vee\dots\vee\chi_{n-1}\vee\chi_n$ is provable in the propositional logic. Thus, $$\begin{aligned}
&&\hspace{-10mm}\{{{\sf \overline{K}}}_{E_i}{{\sf B}}_{E_i}^{F_i}\chi_i\}_{i=1}^n,\chi_2\vee\dots\vee\chi_n\nonumber\\
&&\hspace{0mm} \vdash {{\sf B}}_{E_1\cup\dots\cup E_{n-1}\cup E_n}^{F_1\cup\dots\cup F_{n-1}\cup F_n}(\chi_1\vee\chi_2\vee\dots\vee\chi_n).\label{part 2}\end{aligned}$$ Finally, note that the following statement is provable in the propositional logic for $n\ge 2$, $$\vdash\chi_1\vee\dots\vee\chi_n\to(\chi_1\vee\dots\vee\chi_{n-1})\vee
(\chi_2\vee\dots\vee\chi_n).$$ Therefore, from statement (\[part 1\]) and statement (\[part 2\]), $$\{{{\sf \overline{K}}}_{E_i}{{\sf B}}_{E_i}^{F_i}\chi_i\}_{i=1}^n,\chi_1\vee\dots\vee\chi_n
\vdash {{\sf B}}_{E_1\cup\dots\cup E_n}^{F_1\cup\dots\cup F_n}(\chi_1\vee \dots\vee\chi_n).$$ by the laws of propositional reasoning.
Our last example rephrases Lemma \[super joint responsibility lemma\] into the form which is used in the proof of the completeness.
\[five plus plus\] For any $n\ge 0$, any sets $E_1,\dots,E_n\subseteq C$, and any pairwise disjoint sets $F_1,\dots,F_n\subseteq D$, $$\{{{\sf \overline{K}}}_{E_i}{{\sf B}}^{F_i}_{E_i}\chi_i\}_{i=1}^n,{{\sf K}}_C({\varphi}\to\chi_1\vee\dots\vee\chi_n)\vdash{{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi}).$$
Let $X=\{{{\sf \overline{K}}}_{E_i}{{\sf B}}^{F_i}_{E_i}\chi_i\}_{i=1}^n$. Then, by Lemma \[super joint responsibility lemma\], $$X,\chi_1\vee\dots\vee\chi_n\vdash {{\sf B}}_{E_1\cup\dots\cup E_n}^{F_1\cup\dots\cup F_n}(\chi_1\vee\dots\vee\chi_n).$$ Hence, by the Monotonicity axiom, $$X,\chi_1\vee\dots\vee\chi_n\vdash {{\sf B}}_{C}^D(\chi_1\vee\dots\vee\chi_n).$$ $$\hspace{-15mm}\mbox{Thus, }\hspace{5mm}X,{\varphi}, {\varphi}\to\chi_1\vee\dots\vee\chi_n\vdash {{\sf B}}_{C}^D(\chi_1\vee\dots\vee\chi_n)$$ by the Modus Ponens inference rule. Hence, by the Truth axiom, $$X,{\varphi}, {{\sf K}}_C({\varphi}\to\chi_1\vee\dots\vee\chi_n)\vdash {{\sf B}}_{C}^D(\chi_1\vee\dots\vee\chi_n).$$
The following formula is an instance of the Strict Conditional axiom ${{\sf K}}_C({\varphi}\to\chi_1\vee\dots\vee\chi_n)\to({{\sf B}}_C^D(\chi_1\vee\dots\vee\chi_n)\to({\varphi}\to{{\sf B}}_C^D{\varphi}))$. Thus, by the Modus Ponens applied twice, $$X,{\varphi}, {{\sf K}}_C({\varphi}\to\chi_1\vee\dots\vee\chi_n)\vdash {\varphi}\to{{\sf B}}_C^D{\varphi}.$$ Then, $
X,{\varphi}, {{\sf K}}_C({\varphi}\to\chi_1\vee\dots\vee\chi_n)\vdash {{\sf B}}_C^D{\varphi}$ by the Modus Ponens.\
Thus, $
X, {{\sf K}}_C({\varphi}\to\chi_1\vee\dots\vee\chi_n)\vdash {\varphi}\to{{\sf B}}_C^D{\varphi}$ by the deduction lemma. Hence, $$\{{{\sf K}}_C{{\sf \overline{K}}}_{E_i}{{\sf B}}_{E_i}^{F_i}\chi_i\}_{i=1}^n, {{\sf K}}_C{{\sf K}}_C({\varphi}\to\chi_1\vee\dots\vee\chi_n)\vdash {{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi})$$ by Lemma \[super distributivity\] and the definition of set $X$. Then, $$\{{{\sf K}}_{E_i}{{\sf \overline{K}}}_{E_i}{{\sf B}}_{E_i}^{F_i}\chi_i\}_{i=1}^n, {{\sf K}}_C{{\sf K}}_C({\varphi}\to\chi_1\vee\dots\vee\chi_n)\vdash {{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi})$$ by the Monotonicity axiom, the Modus Ponens inference rule, and the assumption $E_1,\dots,E_n\subseteq C$. Thus, $$\{{{\sf \overline{K}}}_{E_i}{{\sf B}}_{E_i}^{F_i}\chi_i\}_{i=1}^n, {{\sf K}}_C{{\sf K}}_C({\varphi}\to\chi_1\vee\dots\vee\chi_n)\vdash {{\sf K}}_C({\varphi}\to{{\sf B}}_C^D{\varphi})$$ by the definition of modality ${{\sf \overline{K}}}$, the Negative Introspection axiom, and the Modus Ponens rule. Therefore, by Lemma \[positive introspection lemma\] and the Modus Ponens inference rule, the statement of the lemma is true.
Soundness {#soundness section}
=========
The soundness of the Truth, the Distributivity, the Negative Introspection, the Monotonicity, and the None to Blame axioms is straightforward. Below we prove the soundness of the Joint Responsibility, the Strict Conditional, and the Introspection of Blameworthiness axioms as separate lemmas.
\[joint responsibility soundness\] If $D\cap F=\varnothing$, $(\alpha,\delta,\omega)\Vdash {{\sf \overline{K}}}_C{{\sf B}}_C^D{\varphi}$, $(\alpha,\delta,\omega)\Vdash {{\sf \overline{K}}}_E{{\sf B}}_E^F\psi$, and $(\alpha,\delta,\omega)\Vdash {\varphi}\vee\psi$, then $(\alpha,\delta,\omega)\Vdash {{\sf B}}_{C\cup E}^{D\cup F}({\varphi}\vee\psi)$.
By Definition \[sat\] and the definition of modality ${{\sf \overline{K}}}$, assumption $(\alpha,\delta,\omega)\Vdash{{\sf \overline{K}}}_C{{\sf B}}^D_C{\varphi}$ implies that there is a play $(\alpha_1,\delta_1,\omega_1)$ such that $\alpha\sim_C\alpha_1$ and $(\alpha_1,\delta_1,\omega_1)\Vdash{{\sf B}}^D_C{\varphi}$. Thus, again by Definition \[sat\], there is an action profile $s_1\in\Delta^D$ such that for each play $(\alpha',\delta',\omega')\in P$, if $\alpha_1\sim_C\alpha'$ and $s_1=_D\delta'$, then $(\alpha',\delta',\omega')\nVdash{\varphi}$. Recall that $\alpha\sim_C\alpha_1$. Thus, for each play $(\alpha',\delta',\omega')\in P$, $$\label{november}
\alpha\sim_C\alpha'\wedge s_1=_D\delta'\to (\alpha',\delta',\omega')\nVdash{\varphi}.$$ Similarly, assumption $(\alpha,\delta,\omega)\Vdash{{\sf \overline{K}}}_E{{\sf B}}^F_E\psi$ implies that there is a profile $s_2\in\Delta^F$ such that for each play $(\alpha',\delta',\omega')\in P$, $$\label{december}
\alpha\sim_E\alpha'\wedge s_2=_F\delta'\to (\alpha',\delta',\omega')\nVdash\psi.$$ Let $s\in\Delta^{D\cup F}$ be the action profile: $$\label{january}
s(a)=
\begin{cases}
s_1(a), & \mbox{if } a\in D,\\
s_2(a), & \mbox{if } a\in F.
\end{cases}$$ Action profile $s$ is well-defined because $D\cap F=\varnothing$. Statements (\[november\]), (\[december\]), and (\[january\]) by Definition \[sat\] imply that for each play $(\alpha',\delta',\omega')\in P$ if $\alpha\sim_{C\cup E}\alpha'$ and $s=_{D\cup F}\delta'$, then $(\alpha',\delta',\omega')\nVdash{\varphi}\vee\psi$. Recall that $(\alpha,\delta,\omega)\Vdash{\varphi}\vee\psi$. Therefore, $(\alpha,\delta,\omega)\Vdash{{\sf B}}^{D\cup F}_{C\cup E}({\varphi}\vee\psi)$ by Definition \[sat\].
If $(\alpha,\delta,\omega)\Vdash {{\sf K}}_C({\varphi}\to\psi)$, $(\alpha,\delta,\omega)\Vdash {{\sf B}}_C^D\psi$, and $(\alpha,\delta,\omega)\Vdash {\varphi}$, then $(\alpha,\delta,\omega)\Vdash {{\sf B}}_C^D{\varphi}$.
By Definition \[sat\], assumption $(\alpha,\delta,\omega)\Vdash {{\sf K}}_C({\varphi}\to\psi)$ implies that for each play $(\alpha',\delta',\omega')\in P$ of the game if $\alpha\sim_C\alpha'$, then $(\alpha',\delta',\omega')\Vdash{\varphi}\to\psi$.
By Definition \[sat\], assumption $(\alpha,\delta,\omega)\Vdash {{\sf B}}^D_C\psi$ implies that there is an action profile $s\in \Delta^D$ such that for each play $(\alpha',\delta',\omega')\in P$, if $\alpha\sim_C\alpha'$ and $s=_D\delta'$, then $(\alpha',\delta',\omega')\nVdash\psi$.
Hence, for each play $(\alpha',\delta',\omega')\in P$, if $\alpha\sim_C\alpha'$ and $s=_D\delta'$, then $(\alpha',\delta',\omega')\nVdash{\varphi}$. Therefore, $(\alpha,\delta,\omega)\Vdash {{\sf B}}^D_C{\varphi}$ by Definition \[sat\] and the assumption $(\alpha,\delta,\omega)\Vdash {\varphi}$ of the lemma.
If $(\alpha,\delta,\omega)\Vdash {{\sf B}}^D_C{\varphi}$, then $(\alpha,\delta,\omega)\Vdash {{\sf K}}_C({\varphi}\to{{\sf B}}^D_C{\varphi})$.
By Definition \[sat\], assumption $(\alpha,\delta,\omega)\Vdash {{\sf B}}^D_C{\varphi}$ implies that there is an action profile $s\in \Delta^D$ such that for each play $(\alpha',\delta',\omega')\in P$, if $\alpha\sim_C\alpha'$ and $s=_D\delta'$, then $(\alpha',\delta',\omega')\nVdash{\varphi}$.
Let $(\alpha',\delta',\omega')\in P$ be a play where $\alpha\sim_C\alpha'$ and $(\alpha',\delta',\omega')\Vdash {\varphi}$. By Definition \[sat\], it suffices to show that $(\alpha',\delta',\omega')\Vdash {{\sf B}}^D_C{\varphi}$.
Consider any play $(\alpha'',\delta'',\omega'')\in P$ such that $\alpha'\sim_C\alpha''$ and $s=_D\delta''$. Then, since $\sim_C$ is an equivalence relation, assumptions $\alpha\sim_C\alpha'$ and $\alpha'\sim_C\alpha''$ imply $\alpha\sim_C\alpha''$. Thus, $(\alpha'',\delta'',\omega'')\nVdash{\varphi}$ by the choice of action profile $s$. Therefore, $(\alpha',\delta',\omega')\Vdash {{\sf B}}^D_C{\varphi}$ by Definition \[sat\] and the assumption $(\alpha',\delta',\omega')\Vdash {\varphi}$.
Completeness {#completeness section}
============
The standard proof of the completeness for individual knowledge modality ${{\sf K}}_a$ defines states as maximal consistent sets [@fhmv95]. Two such sets are indistinguishable to an agent $a$ if these sets have the same ${{\sf K}}_a$-formulae. This construction does [*not*]{} work for distributed knowledge because if two sets share ${{\sf K}}_a$-formulae and ${{\sf K}}_b$-formulae, they do not necessarily have to share ${{\sf K}}_{a,b}$-formulae. To overcome this issue, we use the Tree of Knowledge construction, similar to the one in [@nt18ai]. An important change to this construction proposed in the current paper is placing elements of a set $\mathcal{B}$ on the edges of the tree. This change is significant for the proof of Lemma \[B child exists lemma\].
Let $\mathcal{B}$ be an arbitrary set of cardinality larger than that of the set $\mathcal{A}$. Next, for each maximal consistent set of formulae $X_0$, we define the canonical game $G(X_0)=\left(I,\{\sim_a\}_{a\in\mathcal{A}},\Delta,\Omega,P,\pi\right)$.
\[canonical outcome\] The set of outcomes $\Omega$ consists of all sequences $X_0,(C_1,b_1),X_1,(C_2,b_2),\dots,(C_n,b_n),X_n$, where $n\ge 0$ and for each $i\ge 1$, $X_i$ is a maximal consistent subset of $\Phi$, (i) $C_i\subseteq\mathcal{A}$, (ii) $b_i\in\mathcal{B}$, and (iii) $\{{\varphi}\;|\;{{\sf K}}_{C_i}{\varphi}\in X_{i-1}\}\subseteq X_i$.
If $x$ is a nonempty sequence $x_1,\dots,x_n$ and $y$ is an element, then by $x::y$ and $hd(x)$ we mean sequence $x_1,\dots,x_n,y$ and element $x_n$ respectively.
We say that outcomes $w,u\in\Omega$ are [*adjacent*]{} if there are coalition $C$, element $b\in\mathcal{B}$, and maximal consistent set $X$ such that $w=u::(C,b)::X$. The adjacency relation forms a tree structure on set $\Omega$, see Figure \[tree figure\]. We call it [*the Tree of Knowledge*]{}. We say that edge $(w,u)$ is [*labeled*]{} with each agent in coalition $C$ and is [*marked*]{} with element $b$. Although vertices of the tree are sequences, it is convenient to think about the maximal consistent set $hd(\omega)$, not a sequence $\omega$, being a vertex of the tree.
For any outcome $\omega\in\Omega$, let $Tree(\omega)$ be the set of all $\omega'\in\Omega$ such that sequence $\omega$ is a prefix of sequence $\omega'$.
Note that $Tree(\omega)$ is a subtree of the Tree of Knowledge rooted at vertex $\omega$, see Figure \[tree figure\].
\[canonical sim\] For any two outcomes $\omega,\omega'\in\Omega$ and any agent $a\in\mathcal{A}$, let $\omega\sim_a\omega'$ if all edges along the unique path between nodes $\omega$ and $\omega'$ are labeled with agent $a$.
\[canonical sim is equivalence relation\] Relation $\sim_a$ is an equivalence relation on $\Omega$.
\[transport lemma\] ${{\sf K}}_C{\varphi}\in hd(\omega)$ iff ${{\sf K}}_C{\varphi}\in hd(\omega')$, if $\omega\sim_C \omega'$.
By Definition \[canonical sim\], assumption $\omega\sim_C \omega'$ implies that all edges along the unique path between nodes $\omega$ and $\omega'$ are labeled with all agents of coalition $C$. Thus, it suffices to prove the statement of the lemma for any two adjacent vertices along this path. Let $\omega'=\omega::(D,b)::X$. Note that $C\subseteq D$ because edge $(\omega,\omega')$ is labeled with all agents in coalition $C$. We start by proving the first part of the lemma.
$(\Rightarrow)$ Suppose ${{\sf K}}_C{\varphi}\in hd(\omega)$. Thus, $hd(\omega)\vdash {{\sf K}}_C{{\sf K}}_C{\varphi}$ by Lemma \[positive introspection lemma\]. Hence, $hd(\omega)\vdash {{\sf K}}_D{{\sf K}}_C{\varphi}$ by the Monotonicity axiom. Thus, ${{\sf K}}_D{{\sf K}}_C{\varphi}\in hd(\omega)$ because set $hd(\omega)$ is maximal. Therefore, ${{\sf K}}_C{\varphi}\in X=hd(\omega')$ by Definition \[canonical outcome\].
$(\Leftarrow)$ Assume ${{\sf K}}_C{\varphi}\notin hd(\omega)$. Thus, $\neg{{\sf K}}_C{\varphi}\in hd(\omega)$ by the maximality of the set $hd(\omega)$. Hence, $hd(\omega)\vdash {{\sf K}}_C\neg{{\sf K}}_C{\varphi}$ by the Negative Introspection axiom. Then, $hd(\omega)\vdash {{\sf K}}_D\neg{{\sf K}}_C{\varphi}$ by the Monotonicity axiom. Thus, ${{\sf K}}_D\neg{{\sf K}}_C{\varphi}\in hd(\omega)$ by the maximality of set $hd(\omega)$. Then, $\neg{{\sf K}}_C{\varphi}\in X=hd(\omega')$ by Definition \[canonical outcome\]. Therefore, ${{\sf K}}_C{\varphi}\notin hd(\omega')$ because set $hd(\omega')$ is consistent.
\[diamond corollary\] If $\omega\sim_C \omega'$, then ${{\sf \overline{K}}}_C{\varphi}\in hd(\omega)$ iff ${{\sf \overline{K}}}_C{\varphi}\in hd(\omega')$.
The set of the initial states $I$ of the canonical game is the set of all equivalence classes of $\Omega$ with respect to relation $\sim_\mathcal{A}$.
\[canonical initial state\] $I=\Omega/\sim_\mathcal{A}$.
\[well-defined lemma\] Relation $\sim_C$ is well-defined on set $I$.
Consider outcomes $\omega_1,\omega_2,\omega'_1$, and $\omega'_2$ where $\omega_1\sim_C\omega_2$, $\omega_1\sim_\mathcal{A}\omega'_1$, and $\omega_2\sim_\mathcal{A}\omega'_2$. It suffices to show $\omega'_1\sim_C\omega'_2$. Indeed, the assumptions $\omega_1\sim_\mathcal{A}\omega'_1$ and $\omega_2\sim_\mathcal{A}\omega'_2$ imply $\omega_1\sim_C\omega'_1$ and $\omega_2\sim_C\omega'_2$. Thus, $\omega'_1\sim_C\omega'_2$ because $\sim_C$ is an equivalence relation.
\[alpha iff omega\] $\alpha\sim_C\alpha'$ iff $\omega\sim_C\omega'$, for any states $\alpha,\alpha'\in I$, any outcomes $\omega\in\alpha$ and $\omega'\in\alpha'$, and any $C\subseteq\mathcal{A}$.
In [@nt19aaai], the domain of actions $\Delta$ of the canonical game is the set $\Phi$ of all formulae. Informally, if an agent employs action ${\varphi}$, then she [*vetoes*]{} formula ${\varphi}$. The set $P$ specifies under which conditions the veto takes place. Here, we modify this construction by requiring the agent, while vetoing formula ${\varphi}$, to specify a coalition $C$ and an outcome $\omega$. The veto will take effect only if coalition $C$ cannot distinguish the outcome $\omega$ from the current outcome. One can think about this construction as requiring the veto ballot to be signed by a key only known, distributively, to coalition $C$. This way only coalition $C$ knows how the agent must vote.
$\Delta =\{({\varphi},C,\omega)\;|\;{\varphi}\in\Phi, C\subseteq\mathcal{A}, \omega\in \Omega\}.$
\[canonical play\] The set $P\subseteq I\times \Delta^\mathcal{A}\times \Omega$ consists of all triples $(\alpha,\delta,u)$ such that (i) $u\in\alpha$, and (ii) for any outcome $v$ and any formula ${{\sf \overline{K}}}_C{{\sf B}}^D_C\psi\in hd(v)$, if $\delta(a)=(\psi,C,v)$ for each agent $a\in D$ and $u\sim_C v$, then $\neg\psi\in hd(u)$.
\[canonical pi\] $\pi(p)=\{(\alpha,\delta,\omega)\in P\;|\; p\in hd(\omega)\}$.
This concludes the definition of the canonical game $G(X_0)$. In Lemma \[termination lemma\], we show that this game satisfies the requirement of item (5) from Definition \[game definition\]. Namely, for each $\alpha\in I$ and each complete action profile $\delta\in\Delta^\mathcal{A}$, there is at least one $\omega\in \Omega$ such that $(\alpha,\delta,\omega)\in P$.
As usual, the completeness follows from the induction (or “truth”) Lemma \[induction lemma\]. To prove this lemma we first need to establish a few auxiliary properties of game $G(X_0)$.
\[B child exists lemma\] For any play $(\alpha,\delta,\omega)\in P$ of game $G(X_0)$, any formula $\neg({\varphi}\to {{\sf B}}_C^D{\varphi})\in hd(\omega)$, and any profile $s\in\Delta^D$, there is a play $(\alpha',\delta',\omega')\in P$ such that $\alpha\sim_C\alpha'$, $s =_D\delta'$, and ${\varphi}\in hd(\omega')$.
Let the complete action profile $\delta'$ be defined as: $$\label{choice of delta'}
\delta'(a)=
\begin{cases}
s(a), & \mbox{ if } a\in D,\\
(\bot,\varnothing,\omega), & \mbox{ otherwise}.
\end{cases}$$ Then, $s=_D\delta'$. Consider the following set of formulae: $$\begin{aligned}
X&\!\!=\!\!&\!\{{\varphi}\}\;\cup\;\{\psi\;|\;{{\sf K}}_C\psi\in hd(\omega)\}\\
&&\!\cup\;\{\neg\chi\;|\;{{\sf \overline{K}}}_E{{\sf B}}_E^F\chi\in hd(v), E\subseteq C, F\subseteq D, \\
&&\hspace{12mm} \forall a\in F(\delta'(a)=(\chi,E,v)), \omega\sim_E v\}.\end{aligned}$$
Set $X$ is consistent.
Suppose the opposite. Thus, there are formulae ${{\sf \overline{K}}}_{E_1}{{\sf B}}_{E_1}^{F_1}\chi_1,\dots,{{\sf \overline{K}}}_{E_n}{{\sf B}}_{E_n}^{F_n}\chi_n$, outcomes $v_1,\dots,v_n\in\Omega$, $$\begin{aligned}
\mbox{ and formulae }&&\hspace{-5mm}{{\sf K}}_C\psi_1,\dots,{{\sf K}}_C\psi_m\in hd(\omega),\label{choice of psi-s}\\
\mbox{such that }&&\hspace{-5mm}{{\sf \overline{K}}}_{E_i}{{\sf B}}_{E_i}^{F_i}\chi_i\in hd(v_i)\;\forall i\le n,\label{choice of chi2-s}\\
&&\hspace{-5mm}E_1,\dots,E_n\subseteq C,\hspace{2mm}F_1,\dots,F_n\subseteq D,\label{choice of Es}\\
&&\hspace{-5mm}\delta'(a)=(\chi_i,E_i,v_i)\;\forall i\le n\;\forall a\in F_i,\label{choice of votes}\\
&&\hspace{-5mm}\omega\sim_{E_i} v_i\;\forall i\le n,\label{choice of omega'2}\\
\mbox{ and }&&\hspace{-5mm}\psi_1,\dots,\psi_m,\neg\chi_1,\dots,\neg\chi_n\vdash\neg{\varphi}.\label{choice of cons}\end{aligned}$$ Without loss of generality, we assume that formulae $\chi_1,\dots,\chi_n$ are distinct. Thus, assumption (\[choice of votes\]) implies that $F_1,\dots,F_n$ are pairwise disjoint. Assumption (\[choice of cons\]) implies $$\psi_1,\dots,\psi_m\vdash{\varphi}\to\chi_1\vee\dots\vee\chi_n$$ by the propositional reasoning. Then, $${{\sf K}}_C\psi_1,\dots,{{\sf K}}_C\psi_m\vdash{{\sf K}}_C({\varphi}\to\chi_1\vee\dots\vee\chi_n)$$ by Lemma \[super distributivity\]. Hence, by assumption (\[choice of psi-s\]), $$hd(\omega)\vdash{{\sf K}}_C({\varphi}\to\chi_1\vee\dots\vee\chi_n).$$
At the same time, ${{\sf \overline{K}}}_{E_1}{{\sf B}}^{F_1}_{E_1}\chi_1,\dots, {{\sf \overline{K}}}_{E_n}{{\sf B}}^{F_n}_{E_n}\chi_n\in hd(\omega)$ by assumption (\[choice of chi2-s\]), assumption (\[choice of omega’2\]), and Corollary \[diamond corollary\]. Thus, $hd(\omega)\vdash {{\sf K}}_C({\varphi}\to{{\sf B}}^D_C{\varphi})$ by Lemma \[five plus plus\], assumption (\[choice of Es\]), and the assumption that sets $F_1,\dots,F_n$ are pairwise disjoint. Hence, by the Truth axiom, $hd(\omega)\vdash {\varphi}\to{{\sf B}}^D_C{\varphi}$, which contradicts the assumption $\neg({\varphi}\to{{\sf B}}^D_C{\varphi})\in hd(\omega)$ of the lemma because set $hd(\omega)$ is consistent. Thus, $X$ is consistent.
Let $X'$ be any maximal consistent extension of set $X$ and $\omega'_b$ be the sequence $\omega::(C, b)::X'$ for each element $b\in\mathcal{B}$. Then, $\omega'_b\in\Omega$ for each element $b\in\mathcal{B}$ by Definition \[canonical outcome\] and the choice of sets $X$ and $X'$. Also ${\varphi}\in X\subseteq hd(\omega'_b)$ for each $b\in\mathcal{B}$ by the choice of sets $X$ and $X'$.
Note that family $\{Tree(\omega'_b)\}_{b\in \mathcal{B}}$ consists of pair-wise disjoint sets. This family has the same cardinality as set $\mathcal{B}$. Let $$V=\{v\in \Omega\;|\; \delta'(a)=(\psi,E,v), a\in \mathcal{A}, \psi\in\Phi,E\subseteq\mathcal{A}\}.$$ The cardinality of $V$ is at most the cardinality of set $\mathcal{A}$. By the choice of set $\mathcal{B}$, its cardinality is larger than the cardinality of set $\mathcal{A}$. Thus, there exists a set $Tree(\omega'_{b_0})$ in family $\{Tree(\omega'_b)\}_{b\in \mathcal{B}}$ disjoint with set $V$: $$\label{disjoint equation}
Tree(\omega'_{b_0}) \cap V=\varnothing.$$ Let $\omega'$ be the outcome $\omega'_{b_0}$.
\[E subseteq C claim\] If $\omega' \sim_E v$ for some $v\in V$, then $E\subseteq C$.
Consider any agent $a\in E$. By Definition \[canonical sim\], assumption $\omega' \sim_E v$ implies that each edge along the unique path connecting vertex $\omega$ with vertex $v$ is labeled with agent $a$. At the same time, $v\notin Tree(\omega')$ by statement (\[disjoint equation\]) and because $\omega'=\omega'_{b_0}$. Thus, the path between vertex $\omega'$ and vertex $v$ must go through vertex $\omega$, see Figure \[signature figure\]. Hence, this path must contain edge $(\omega',\omega)$. Since all edges along this path are labeled with agent $a$ and edge $(\omega',\omega)$ is labeled with agents from set $C$, it follows that $a\in C$.
Let initial state $\alpha'$ be the equivalence class of outcome $\omega'$ with respect to the equivalence relation $\sim_{\mathcal{A}}$. Note that $\omega\sim_C\omega'$ by Definition \[canonical outcome\] because $\omega'=\omega::(C,b_0)::X'$. Therefore, $\alpha\sim_C\alpha'$ by Corollary \[alpha iff omega\].
$(\alpha',\delta',\omega')\in P$.
First, note that $\omega'\in\alpha'$ because initial state $\alpha'$ is the equivalence class of outcome $\omega'$. Next, consider an outcome $v\in\Omega$ $$\begin{aligned}
\mbox{and a formula }&&\hspace{-5mm}{{\sf \overline{K}}}_E{{\sf B}}_E^F\chi\in hd(v), \label{K bar}\\
\mbox{such that }&&\hspace{-5mm}\omega'\sim_E v,\hspace{3mm}\label{omega' E v}\\
\mbox{and }&&\hspace{-5mm}\forall a\in F\, (\delta'(a)=(\chi,E,v)).\label{delta'}
\end{aligned}$$ By Definition \[canonical play\], it suffices to show that $\neg\chi\in hd(\omega')$.
[**Case I:**]{} $F=\varnothing$. Then, $\neg{{\sf B}}^F_E\chi$ is an instance of the None to Act axiom. Thus, $\vdash{{\sf K}}_E\neg{{\sf B}}^F_E\chi$ by the Necessitation inference rule. Hence, $\neg{{\sf K}}_E\neg{{\sf B}}^F_E\chi\notin hd(v)$ by the consistency of the set $hd(v)$, which contradicts the assumption (\[K bar\]) and the definition of modality ${{\sf \overline{K}}}$.
[**Case II:**]{} $\varnothing\neq F\subseteq D$. Thus, there exists an agent $a\in F$. Note that $\delta'(a) = (\chi, E, v)$ by assumption (\[delta’\]). Hence, $v\in V$ by the definition of set $V$. Thus, $E\subseteq C$ by Claim \[E subseteq C claim\] and assumption (\[omega’ E v\]). Then, $\neg\chi\in X$ by the definition of set $X$, the assumption of the case that $F\subseteq D$, assumption (\[K bar\]), assumption (\[omega’ E v\]), and assumption (\[delta’\]). Therefore, $\neg\chi\in hd(\omega')$ because $X\subseteq X'=hd(\omega'_{b_0})=hd(\omega')$ by the choice of set $X'$, set of sequences $\{\omega'_{b}\}_{b\in\mathcal{B}}$, and sequence $\omega'$.
[**Case III:**]{} $F\nsubseteq D$. Consider any agent $a\in F\setminus D$. Thus, $\delta'(a)=(\bot,\varnothing,\omega)$ by equation (\[choice of delta’\]). Thus, $\chi\equiv\bot$ by statement (\[delta’\]) and the assumption $a\in F$. Hence, formula $\neg\chi$ is a tautology. Therefore, $\neg\chi\in hd(\omega')$ by the maximality of set $hd(\omega')$.
This concludes the proof of the lemma.
\[delta exists lemma\] For any outcome $\omega\in\Omega$, there is a state $\alpha\in I$ and a complete profile $\delta\in \Delta^\mathcal{A}$ such that $(\alpha,\delta,\omega)\in P$.
Let initial state $\alpha$ be the equivalence class of outcome $\omega$ with respect to the equivalence relation $\sim_{\mathcal{A}}$. Thus, $\omega\in\alpha$. Let $\delta$ be the complete profile such that $\delta(a)=(\bot,\varnothing,\omega)$ for each $a\in \mathcal{A}$. To prove $(\alpha,\delta,\omega)\in P$, consider any outcome $v\in\Omega$, any formula ${{\sf \overline{K}}}_C{{\sf B}}_C^D\chi\in hd(v)$ such that $$\label{delta = chi}
\forall a\in D\,(\delta(a)=(\chi, C,v)).$$ By Definition \[canonical play\], it suffices to show that $\neg\chi\in hd(\omega)$.
[**Case I**]{}: $D=\varnothing$. Thus, $\vdash\neg{{\sf B}}_C^D\chi$ by the None to Act axiom. Hence, $\vdash{{\sf K}}_C\neg{{\sf B}}_C^D\chi$ by the Necessitation rule. Then, $\neg{{\sf K}}_C\neg{{\sf B}}_C^D\chi\notin hd(v)$ because set $hd(v)$ is consistent. Therefore, ${{\sf \overline{K}}}_C{{\sf B}}_C^D\chi\notin hd(v)$ by the definition of modality ${{\sf \overline{K}}}$, which contradicts the choice of ${{\sf \overline{K}}}_C{{\sf B}}_C^D\chi$.
[**Case II**]{}: $D\neq\varnothing$. Then, there is an agent $a\in D$. Thus, $\delta(a)=(\chi, C, v)$ by statement (\[delta = chi\]). Hence, $\chi\equiv\bot$ by the definition of action profile $\delta$. Then, $\neg\chi$ is a tautology. Therefore, $\neg\chi\in hd(\omega)$ by the maximality of set $hd(\omega)$.
\[termination lemma\] For each $\alpha\in I$ and each complete action profile $\delta\in\Delta^\mathcal{A}$, there is at least one outcome $\omega\in \Omega$ such that $(\alpha,\delta,\omega)\in P$.
By Definition \[canonical initial state\], initial state $\alpha$ is an equivalence class. Since each equivalence class is not empty, there must exist an outcome $\omega_0\in \Omega$ such that $\omega_0\in \alpha$. By Lemma \[delta exists lemma\], there is an initial state $\alpha_0\in I$ and a complete action profile $\delta_0\in \Delta^\mathcal{A}$ such that $(\alpha_0,\delta_0,\omega_0)\in P$. Then, $\omega_0\in \alpha_0$ by Definition \[canonical play\]. Hence, $\omega_0$ belongs to both equivalence classes $\alpha$ and $\alpha_0$. Thus, $\alpha=\alpha_0$. Therefore, $(\alpha,\delta_0,\omega_0)\in P$.
\[N child exists lemma\] For any play $(\alpha,\delta,\omega)\in P$ and any $\neg{{\sf K}}_C{\varphi}\in hd(\omega)$, there is a play $(\alpha',\delta',\omega')\in P$ such that $\alpha\sim_C\alpha'$ and $\neg{\varphi}\in hd(\omega')$.
Consider the set $X=\{\neg{\varphi}\}\;\cup\;\{\psi\;|\;{{\sf K}}_C\psi\in hd(\omega)\}$. First, we show that set $X$ is consistent. Suppose the opposite. Then, there are formulae ${{\sf K}}_C\psi_1,\dots,{{\sf K}}_C\psi_n\in hd(\omega)$ such that $
\psi_1,\dots,\psi_n\vdash{\varphi}.
$ Hence, $
{{\sf K}}_C\psi_1,\dots,{{\sf K}}_C\psi_n\vdash{{\sf K}}_C{\varphi}$ by Lemma \[super distributivity\]. Thus, $hd(\omega)\vdash{{\sf K}}_C{\varphi}$ because ${{\sf K}}_C\psi_1,\dots,{{\sf K}}_C\psi_n\in hd(\omega)$. Hence, $\neg{{\sf K}}_C{\varphi}\notin hd(\omega)$ because set $hd(\omega)$ is consistent, which contradicts the assumption of the lemma. Therefore, set $X$ is consistent.
Recall that set $\mathcal{B}$ has larger cardinality than set $\mathcal{A}$. Thus, there is at least one $b\in\mathcal{B}$. Let set $X'$ be any maximal consistent extension of set $X$ and $\omega'$ be the sequence $\omega::(C,b)::X'$. Note that $\omega'\in\Omega$ by Definition \[canonical outcome\] and the choice of sets $X$ and $X'$. Also, $\neg{\varphi}\in X\subseteq X'=hd(\omega')$ by the choice of sets $X$ and $X'$.
By Lemma \[delta exists lemma\], there is an initial state $\alpha'\in I$ and a profile $\delta'\in \Delta^\mathcal{A}$ such that $(\alpha',\delta',\omega')\in P$. Note that $\omega\sim_C\omega'$ by Definition \[canonical sim\] and the choice of $\omega'$. Thus, $\alpha\sim_C\alpha'$ by Corollary \[alpha iff omega\].
\[induction lemma\] $(\alpha,\delta,\omega)\Vdash{\varphi}$ iff ${\varphi}\in hd(\omega)$.
We prove the lemma by induction on the complexity of formula ${\varphi}$. If ${\varphi}$ is a propositional variable, then the lemma follows from Definition \[sat\] and Definition \[canonical pi\]. If formula ${\varphi}$ is an implication or a negation, then the required follows from the induction hypothesis and the maximality and the consistency of set $hd(\omega)$ by Definition \[sat\]. Assume that formula ${\varphi}$ has the form ${{\sf K}}_C\psi$.
$(\Rightarrow):$ Let ${{\sf K}}_C\psi\notin hd(\omega)$. Thus, $\neg{{\sf K}}_C\psi\in hd(\omega)$ by the maximality of set $hd(\omega)$. Hence, by Lemma \[N child exists lemma\], there is a play $(\alpha',\delta',\omega')\in P$ such that $\alpha\sim_C\alpha'$ and $\neg\psi\in hd(\omega')$. Then, $\psi\notin hd(\omega')$ by the consistency of set $hd(\omega')$. Thus, $(\alpha',\delta',\omega')\nVdash\psi$ by the induction hypothesis. Therefore, $(\alpha,\delta,\omega)\nVdash{{\sf K}}_C\psi$ by Definition \[sat\].
$(\Leftarrow):$ Let ${{\sf K}}_C\psi\in hd(\omega)$. Thus, $\psi\in hd(\omega')$ for any $\omega'\in\Omega$ such that $\omega\sim_C\omega'$, by Lemma \[transport lemma\]. Hence, by the induction hypothesis, $(\alpha',\delta',\omega')\Vdash\psi$ for each play $(\alpha',\delta',\omega')\in P$ such that $\omega\sim_C\omega'$. Thus, $(\alpha',\delta',\omega')\Vdash\psi$ for each $(\alpha',\delta',\omega')\in P$ such that $\alpha\sim_C\alpha'$, by Lemma \[alpha iff omega\]. Therefore, $(\alpha,\delta,\omega)\Vdash{{\sf K}}_C\psi$ by Definition \[sat\].
Assume formula ${\varphi}$ has the form ${{\sf B}}^D_C\psi$.
$(\Rightarrow):$ Suppose ${{\sf B}}^D_C\psi\notin hd(\omega)$.
[**Case I**]{}: $\psi\notin hd(\omega)$. Then, $(\alpha,\delta,\omega)\nVdash\psi$ by the induction hypothesis. Thus, $(\alpha,\delta,\omega)\nVdash{{\sf B}}^D_C\psi$ by Definition \[sat\].
[**Case II**]{}: $\psi\in hd(\omega)$. Let us show that $\psi\to{{\sf B}}^D_C\psi\notin hd(\omega)$. Indeed, if $\psi\to{{\sf B}}^D_C\psi\in hd(\omega)$, then $hd(\omega)\vdash {{\sf B}}^D_C\psi$ by the Modus Ponens rule. Thus, ${{\sf B}}^D_C\psi\in hd(\omega)$ by the maximality of set $hd(\omega)$, which contradicts the assumption above.
Since set $hd(\omega)$ is maximal, statement $\psi\to{{\sf B}}^D_C\psi\notin hd(\omega)$ implies that $\neg(\psi\to{{\sf B}}^D_C\psi)\in hd(\omega)$. Hence, by Lemma \[B child exists lemma\], for any action profile $s\in \Delta^D$, there is a play $(\alpha',\delta',\omega')$ such that $\alpha\sim_C\alpha'$, $s=_D\delta'$, and $\psi\in hd(\omega')$. Thus, by the induction hypothesis, for any action profile $s\in \Delta^D$, there is a play $(\alpha',\delta',\omega')$ such that $\alpha\sim_C\alpha'$, $s=_D\delta'$, and $(\alpha',\delta',\omega')\Vdash \psi$. Therefore, $(\alpha,\delta,\omega)\nVdash{{\sf B}}^D_C\psi$ by Definition \[sat\].
$(\Leftarrow):$ Let ${{\sf B}}^D_C\psi\in hd(\omega)$. Hence, $hd(\omega)\vdash\psi$ by the Truth axiom. Thus, $\psi\in hd(\omega)$ by the maximality of the set $hd(\omega)$. Then, $(\alpha,\delta,\omega)\Vdash\psi$ by the induction hypothesis.
Next, let $s\in \Delta^D$ be the action profile of coalition $D$ such that $s(a)=(\psi,C,\omega)$ for each agent $a\in D$. Consider any play $(\alpha',\delta',\omega')\in P$ such that $\alpha\sim_C\alpha'$ and $s=_D\delta'$. By Definition \[sat\], it suffices to show that $(\alpha',\delta',\omega')\nVdash \psi$.
Assumption ${{\sf B}}^D_C\psi\in hd(\omega)$ implies $hd(\omega)\nvdash \neg{{\sf B}}^D_C\psi$ because set $hd(\omega)$ is consistent. Thus, $hd(\omega)\nvdash {{\sf K}}_C\neg{{\sf B}}^D_C\psi$ by the contraposition of the Truth axiom. Hence, $\neg{{\sf K}}_C\neg{{\sf B}}^D_C\psi\in hd(\omega)$ by the maximality of $hd(\omega)$. Then, ${{\sf \overline{K}}}_C{{\sf B}}^D_C\psi\in hd(\omega)$ by the definition of modality ${{\sf \overline{K}}}$. Recall that $s(a)=(\psi,C,\omega)$ for each agent $a\in D$ by the choice of the action profile $s$. Also, $s=_D\delta'$ by the choice of the play $(\alpha',\delta',\omega')$. Hence, $\delta'(a)=(\psi,C,\omega)$ for each agent $a\in D$. Thus, $\neg\psi\in hd(\omega')$ by Definition \[canonical play\] and because ${{\sf \overline{K}}}_C{{\sf B}}^D_C\psi\in hd(\omega)$ and $(\alpha',\delta',\omega')\in P$. Then, $\psi\notin hd(\omega')$ by the consistency of set $hd(\omega')$. Therefore, $(\alpha',\delta',\omega')\nVdash \psi$ by the induction hypothesis.
Next is the strong completeness theorem for our system.
\[completeness theorem\] If $X\nvdash{\varphi}$, then there is a game, and a play $(\alpha,\delta,\omega)$ of this game such that $(\alpha,\delta,\omega)\Vdash\chi$ for each $\chi\in X$ and $(\alpha,\delta,\omega)\nVdash{\varphi}$.
Assume that $X\nvdash{\varphi}$. Hence, set $X\cup\{\neg{\varphi}\}$ is consistent. Let $X_0$ be any maximal consistent extension of set $X\cup\{\neg{\varphi}\}$ and let game $\left(I,\{\sim_a\}_{a\in\mathcal{A}},\Delta,\Omega,P,\pi\right)$ be the canonical game $G(X_0)$. Also, let $\omega_0$ be the single-element sequence $X_0$. Note that $\omega_0\in \Omega$ by Definition \[canonical outcome\]. By Lemma \[delta exists lemma\], there is an initial state $\alpha\in I$ and a complete action profile $\delta\in \Delta^\mathcal{A}$ such that $(\alpha,\delta,\omega_0)\in P$. Hence, $(\alpha,\delta,\omega_0)\Vdash\chi$ for each $\chi\in X$ and $(\alpha,\delta,\omega_0)\Vdash\neg{\varphi}$ by Lemma \[induction lemma\] and the choice of set $X_0$. Thus, $(\alpha,\delta,\omega_0)\nVdash{\varphi}$ by Definition \[sat\].
Conclusion {#conclusion section}
==========
In this paper, we proposed a formal definition of the second-order blameworthiness or duty to warn in the setting of strategic games. Our main technical result is a sound and complete logical system that describes the interplay between the second-order blameworthiness and the distributed knowledge modalities. \[end of paper\]
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abstract: 'Using first-principles electronic structure calculations, we studied the electronic and thermoelectric properties of [SrTiO$_3$]{} based oxide materials and their nanostructures identifying those nanostructures which possess highly anisotropic electronic bands. We showed recently that highly anisotropic flat-and-dispersive bands can maximize the thermoelectric power factor, and at the same time they can produce low dimensional electronic transport in bulk semiconductors. Although most of the considered nanostructures show such highly anisotropic bands, their predicted thermoelectric performance is not improved over that of [SrTiO$_3$]{}. Besides highly anisotropic character, we emphasize the importance of the large weights of electronic states participating in transport and the small effective mass of charge carriers along the transport direction. These requirements may be better achieved in binary transition metal oxides than in ABO$_3$ perovskite oxide materials.'
author:
- 'Daniel I. Bilc, Calin G. Floare, Liviu P. Zârbo, Sorina Garabagiu'
- 'Sebastien Lemal, and Philippe Ghosez'
title: 'First-principles Modelling of [SrTiO$_3$]{} based Oxides for Thermoelectric Applications'
---
INTRODUCTION
============
Thermoelectric (TE) technology exploits the ability of certain materials for direct and reversible conversion of thermal energy into electricity. This double edge ability gives TE technology a strong appeal in almost all energy-related applications. Nevertheless, many fundamental problems have to be solved and in particular the intrinsic efficiency of TE materials still needs to be significantly improved before TE technology becomes a competitive alternative. The efficiency of a TE material depends on the dimensionless figure of merit, $ZT=(S^2\sigma$T$)/\kappa_{th}$, where $\sigma$ is the electrical conductivity, $S$ is the thermopower or Seebeck coefficient, $T$ is the absolute temperature, $\kappa_{th}$ is the total thermal conductivity including electronic and lattice contributions, and $S^2\sigma$ is the power factor ($PF$). Improving TE efficiency is not obvious because the parameters entering $ZT$ are interlinked, and cannot be optimized independently. Moreover, $ZT$ has to be optimized in a $T$ range at which the TE devices will operate. Many interesting TE applications function at high $T$, and for practical applications other aspects are also very important such as: the cost of materials, their stability at high $T$, their toxicity, and availability.
Oxide materials are not known to exhibit the highest TE performance, but they offer stability in oxidizing and corrosive environments at high $T$ (T$ > $800 - 1000 K). In this context, oxides appear very appealing for high $T$ applications. Consequently, efforts have been devoted over the last 15 years to the optimization of TE properties of both n-type and p-type oxide materials. Good candidates for n-type materials include Nb-, W-, La-, Ce-, Pr-, Nd-, Sm-, Gd-, Dy-, and Y-doped SrTiO$_3$ [@Ohta2005; @Okuda; @Zhu2011; @Mei2011; @Mei2013; @Hyeon2014; @Weidenkaff2014; @Buscaglia2014; @Koumoto2015; @Wu2015; @Tritt2015; @Reaney2016; @Chen2016], Nb-, La-, Nd-, Sm-, and Gd-doped Sr$_2$TiO$_4$, and Sr$_3$Ti$_2$O$_7$ [@Koumoto2009], La-doped CaMnO$_3$ [@Matsubara] or Al-, Ge-, Ni- and Co-doped ZnO [@Ohtaki; @Arai1997; @Zeng2013; @Ichinose2015; @Zhai2015], Ce-doped In$_2$O$_3$ [@Yang2015], Er-doped CdO [@Fu2013; @Wang2015], TiO$_2$ [@Brown2014], Nb$_2$O$_5$ [@Brown2014], WO$_3$ [@Brown2014], while for p-type materials the most promising compounds are Ca$_3$Co$_4$O$_9$ [@Masset] with $ZT$$\sim$ 0.3 at 1000 K [@Ohta2007], and BiCuSeO with $ZT$$\sim$ 1.4 at 923 K.[@Zhao2013] Many studies have concerned doped [SrTiO$_3$]{}, demonstrating the largest $ZT$$\sim$ 0.4 in SrTi$_{0.8}$Nb$_{0.2}$O$_{3}$ films at 1000 K [@Okuda], and $ZT$$\sim$ 0.41 in bulk Sr$_{1 - 3x/2}$La$_x$TiO$_{3}$ at 973 K.[@Reaney2016] Different strategies have been proposed to further increase TE efficiency. Attempts to decrease the lattice thermal conductivity $\kappa_{l}$ by atomic substitution of Sr by Ba have been envisaged but seem to negatively affect the TE performance.[@Ohta2008] A more promising approach is the reduction of $\kappa_{l}$ from scattering of phonons at interfaces, highlighted in layered Ruddlesden-Popper (RP) compounds. [@Koumoto] Also, Ohta [*et. al.*]{} demonstrated significant enhancement of $S$ arising from electron confinement and the formation of two dimensional electron gas (2-DEG) in [SrTiO$_3$]{}/Nb-[SrTiO$_3$]{} superlattices.[@Ohta2007] Although very promising, the inactive [SrTiO$_3$]{} interlayer must be sufficiently large to avoid electron tunneling and to get significant enhancement of $S$ [@Mune], but this decreases the effective TE performance. In order to maximize TE performance, we showed that the 2-DEG has to be achieved in doped semiconducting nanostructures rather than those with metallic character.[@Garcia] Few first-principles studies have investigated the electronic properties of Nb-doped [SrTiO$_3$]{} [@Astala; @Guo; @Zhang] and Sr$_2$TiO$_4$ [@Yun] and a few studies have addressed TE properties of [SrTiO$_3$]{} [@Usui; @Zhang2; @Kinaci; @Ricinschi2013; @Zou2013; @Yamanaka2013; @Kahaly2014; @Singsoog2014; @Zhang3], BaTiO$_3$ [@Zhang3], PbTiO$_3$ [@Roy2016], CaTiO$_3$ [@Zhang3], KTaO$_3$ [@Janotti2016], HoMnO$_3$ [@Ahmad2015], CaMnO$_3$ [@Zhang2011; @Zhang4; @Molinari2014; @Zhang2015], Ca$_3$Co$_4$O$_9$ [@Maensiri2016], ZnO [@Ong; @Jia2011; @Jantrasee2014; @Chen2015; @Zheng2015], Cu$_2$O [@Chen], CdO [@Parker2015], TiO$_2$ [@Kioupakis2015], and V$_2$O$_5$ [@Chumakov]. At this stage, a complete understanding of the transport properties and moreover of the band structure engineering in oxide materials is still missing.
Employing the concept of electronic band structure engineering and our guidance ideas, we showed recently that very anisotropic flat-and-dispersive electronic bands are able to maximze the $PF$ and carrier concentration $n$ of bulk semiconductors, and at the same time to produce low-dimensional electronic transport (low-DET).[@Bilc2015] In practice this is typically achieved from the highly directionsl character of some orbitals like the $d$ states. Transition metal (TM) oxides with $d$-type conduction states appear as a well suited playground to explore this concept. Also the rich crystal chemistry of oxides encourages strategies of multiscale nanostructuring [@Biswas; @Heremans] by considering hybrid crystal structures that contain discrete structural blocks or layers. The nanostructuring of such hybrid materials is possible in order to engineer their electronic band structures and to lower their $\kappa_{l}$. Therefore, in this theoretical work, we studied TE properties of [SrTiO$_3$]{} based materials and their nanostructures. We have considered the ([SrTiO$_3$]{})$_m$-([LaVO$_3$]{})$_1$ and ([SrTiO$_3$]{})$_m$-([KNbO$_3$]{})$_1$ superlattices (SL). [LaVO$_3$]{} is a Mott insulator with an electronic band gap $E_{g}\sim$1.1 eV [@Inaba], which shows a $PF$$\sim$ 0.06 $m W/mK^2$ at 1000 K. [@Wang] For ([SrTiO$_3$]{})$_m$-[LaVO$_3$]{} nanostructures we expect an electronic transport achieved through [LaVO$_3$]{} layers since it has a smaller band gap that [SrTiO$_3$]{} ($E_{g}\sim$3.3 eV), which may give rise to low-DET and enhanced $PF$. We consider ([SrTiO$_3$]{})$_m$-[KNbO$_3$]{} SL in order to compare their results with those of Nb-doped [SrTiO$_3$]{}. We also studied TE properties of Sr and Co based naturally-ordered Ruddlesden-Popper compounds (AO\[ABO$_3$\]$_m$), which are more easy to control experimentally and more realistic for practical applications than artificial nanostructures.
TECHNICAL DETAILS
=================
The structural, electronic and TE properties of considered oxide structures were studied within density functional theory (DFT) formalism using the hybrid functional B1-WC.[@B1-WC] B1-WC hybrid functional describes the electronic properties (band gaps) and the structural properties with a better accuracy than the usual simple functionals, being more appropriate for correlated materials such as oxides.[@B1-WC; @Goffinet; @Prikockyte] For comparison of bulk [SrTiO$_3$]{} results, we also used approximations based on LDA [@LDA], GGA(PBE [@PBE] and GGA-WC [@GGA-WC]) usual simple functionals. The electronic structure calculations have been performed using the linear combination of atomic orbitals method as implemented in CRYSTAL first-principles code.[@Crystal] We used localized Gaussian-type basis sets including polarization orbitals and considered all the electrons for Ti [@Bredow], O [@Piskunov2004], V [@Mackrodt], K and Nb [@Dovesi1991], F and Co [@Peintinger]. The Hartree-Fock pseudopotential for Sr [@Piskunov2004], and the Stuttgart energy-consistent pseudopotential for La [@Cao] were used.
In order to go beyond the rigid band approximation, we considered 3$\times$3$\times$3 [SrTiO$_3$]{} perovskite supercells with $P \overline{\it 1}$ symmetry, which incorporate explicitly two Nb and two La doping elements per supercell. For ([SrTiO$_3$]{})$_m$-([LaVO$_3$]{})$_1$ and ([SrTiO$_3$]{})$_m$-([KNbO$_3$]{})$_1$ nanostructures, we have considered $a \times a \times c$ SL with $P4mm$ symmetry to treat the nonmagnetic and ferromagnetic (FM) order, and $\sqrt{2} a \times \sqrt{2} a \times c$ SL with $P4bm$ symmetry for the FM and antiferromagnetic (AFM) orders, where $a$ is the lattice constant of cubic perovskite structure. For AO\[ABO$_3$\]$_m$ SL with $I4/mmm$ symmetry the body centered primitive cells were used in calculations. According to the position of F at the apical site, the symmetry of Sr$_2$CoO$_3$F has been reduced to $Cmcm$ and $Cmmm$ space groups for type I and type II ordered structures, respectively. For these F ordered structures, we used the face centered primitive cells in our calculations.
Brillouin zone integrations were performed using the following meshes of $k$-points: 6$\times$6$\times$6 for bulk [SrTiO$_3$]{}, Sr$_2$TiO$_4$, and Sr$_2$CoO$_3$F, 3$\times$3$\times$3 for La and Nb doped 3$\times$3$\times$3 [SrTiO$_3$]{} supercells, 6$\times$6$\times$4 for ([SrTiO$_3$]{})$_1$-([LaVO$_3$]{})$_1$ and ([SrTiO$_3$]{})$_1$-([KNbO$_3$]{})$_1$ SL, 6$\times$6$\times$1 for ([SrTiO$_3$]{})$_5$-([LaVO$_3$]{})$_1$ and ([SrTiO$_3$]{})$_5$-([KNbO$_3$]{})$_1$ SL, and 4$\times$4$\times$4 for Sr$_3$Ti$_2$O$_7$. The self-consistent-field calculations were considered to be converged when the energy changes between interactions were smaller than 10$^{-8}$ Hartree. An extra-large predefined pruned grid consisting of 75 radial points and 974 angular points was used for the numerical integration of charge density. Full optimizations of the lattice constants and atomic positions have been performed with the optimization convergence of 5$\times$10$^{-5}$ Hartree/Bohr in the root-mean square values of forces and 1.2$\times$10$^{-3}$ Bohr in the root-mean square values of atomic displacements. The level of accuracy in evaluating the Coulomb and exchange series is controlled by five parameters.[@Crystal] The values used in our calculations are 7, 7, 7, 7, and 14.
The transport coefficients were estimated in the Boltzmann transport formalism and the constant relaxation time approximation using BoltzTraP transport code.[@Boltztrap] The electronic band structures (energies), used in the transport calculations, were calculated with electronic charge densities converged for denser $k$-point meshes (doubling the $k$-point meshes used in optimization calculations). The transport coefficients were very well converged for the energies calculated on $k$-point meshes of 59$\times$59$\times$59 for bulk [SrTiO$_3$]{}, Sr$_2$TiO$_4$, Sr$_3$Ti$_2$O$_7$, and Sr$_2$CoO$_3$F, 27$\times$27$\times$27 for La doped 3$\times$3$\times$3 [SrTiO$_3$]{} supercell, 59$\times$59$\times$41 for ([SrTiO$_3$]{})$_1$-([LaVO$_3$]{})$_1$ and ([SrTiO$_3$]{})$_1$-([KNbO$_3$]{})$_1$ SL, and 59$\times$59$\times$23 for ([SrTiO$_3$]{})$_5$-([LaVO$_3$]{})$_1$ and ([SrTiO$_3$]{})$_5$-([KNbO$_3$]{})$_1$ SL.
The effective masses were obtained by calculating values of energy close to conduction band (CB) minimum and valence band (VB) maximum while moving from the extremum points along the three directions of the orthogonal reciprocal lattice vectors $k_{i}$ (i=x, y, z). The energy values, $\epsilon_{\vec{k}}$, were fitted up to 10-th order polynomials in $k_{i}$. In general $\epsilon_{\vec{k}}$ can be expanded about an extremum point as: $$\frac{2m_{e}}{\hbar^{2}} \epsilon_{\vec{k}} = \sum_{i,j} \frac{m_{e}}{m_{ij}} k_{i}k_{j}
$$ where $m_{ij}$ are the components of the effective mass tensor (i, j=x, y, z) and $m_{e}$ is the free electron mass. In the present expansion, we have only the diagonal components of the effective mass tensor $m_{ii}$, since all the interaxial angles are 90$^\circ$.
RESULTS
=======
The oxide compounds under study are well known to exhibit various types of structural phase transitions with temperature. Since we are interested in TE at high T, in first approximation we restricted our investigation to the high symmetry phases of $P m\overline{\it 3}m$ for bulk perovskites and $P4mm$, $P4bm$, and $I4/mmm$ for superlattices. Some of them are also expected to show magnetic order at low T and be in a paramagnetic configuration at high T. For the later we compare different magnetic orders to find the systems with lower total energy and more structurally stable.
Bulk [SrTiO$_3$]{} and its alloys
---------------------------------
The structural (lattice constant) and electronic (band gap) properties of bulk [SrTiO$_3$]{} optimized within the different functionals are given in Table \[Table1\]. LDA underestimates the lattice constant and the atomic volume, whereas PBE overestimates these properties, which is the typical behaviour of LDA and PBE functionals. GGA-WC describes very well the structural properties of [SrTiO$_3$]{}, being a functional developed for solids. All the simple functionals underestimate [SrTiO$_3$]{} band gap, which is an inherent problem of DFT. B1-WC hybrid, which mixes the GGA-WC with a small percentage of exact exchange (16$\%$) describes simultaneously both the structural and electronic properties with good accuracy.
[0.49]{} [@cccccc]{} & LDA & PBE & GGA-WC & B1-WC & Exp.\
$a$(Å) & 3.864 & 3.943 & 3.898 & 3.880 & 3.890$^a$\
$E_g (eV) $ & 2.24 & 2.25 & 2.25 & 3.57 & 3.25$^b$\
\
\
For a more complete theoretical characterization, we have studied also TE properties ($\sigma$, $S$, and $PF$) of bulk [SrTiO$_3$]{} within B1-WC hybrid, and the other used simple functionals. In the constant relaxation time approximation, the relaxation time $\tau$ is considered as a constant $\tau=\tau_0$ independent of energy and temperature $T$, and is estimated from fitting of the experimental electrical conductivity $\sigma_{exp}$ at a given doping carrier concentration $n$ and $T$. $\tau$ within the different functionals was estimated by fitting the room temperature experimental values $\sigma_{exp}=1.4\times10^5$ S/m at $n=8\times10^{20}$ cm$^{-3}$ [@Muta], and $\sigma_{exp}=1.667\times10^5$ S/m at $n=1\times10^{21}$ cm$^{-3}$ [@Okuda] (Table 2). The resulting value of $\tau$ is $\sim0.43\times10^{-14}$ s, which very similar in all the used functionals (see Table \[Table2\]). The estimated values of $\sigma$, $S$, and $PF$ at 300 K as a function of chemical potential within B1-WC hybrid are given in Fig. \[TranSTO\]. For n-type doping, $PF$ is $\sim$1 mW/mK$^2$, being underestimated with respect to experiment ($PF_{exp} \sim$3 mW/mK$^2$).[@Muta; @Okuda] This underestimation of $PF$ is due to a low value of thermopower $S\sim$ -77 $\mu$V/K in comparison with the experimental value of $S_{exp} \sim$ -147 $\mu$V/K at $n=1\times10^{21}$ cm$^{-3}$ and 300 K. $PF$ estimated within the usual simple functionals (LDA, GGA-WC) has the same value $\sim$1 mW/mK$^2$ (Fig. \[PFAllFunc\]).
[0.49]{} [@cccc]{} &$\quad$ LDA $\quad$ & GGA-WC & B1-WC\
&\
$\tau$( 10$^{-14}$ s) & 0.44 & 0.46 & 0.43\
&\
$\tau$( 10$^{-14}$ s) & 0.43 & 0.45 & 0.42\
![\[TranSTO\] (Color online) Electrical conductivity $\sigma$, Seebeck coefficient $S$, and power factor $PF=S^2 \sigma$ dependence on chemical potential for [SrTiO$_3$]{} estimated at 300 K within B1-WC using the relaxation time $\tau=0.43\times10^{-14}$ s. The n-type doping carrier concentration $n$ of $1\times10^{21}$ cm$^{-3}$ is shown in dashed vertical line.](STO_TranProp "fig:")\
![\[PFAllFunc\] (Color online) Power factor $PF=S^2 \sigma$ dependence on chemical potential for [SrTiO$_3$]{} estimated at 300 K within different functionals using the relaxation time $\tau=0.43\times10^{-14}$ s. The different $PF$ peak positions within GGA-WC and LDA are due to lower band gap $E_g$ values. $PF$ for 3$\times$3$\times$3 [SrTiO$_3$]{}, and La:[SrTiO$_3$]{} supercells, estimated at 300 K within B1-WC, are also shown.](PF_AllFunc "fig:")\
The promising TE properties of [SrTiO$_3$]{} based oxides were found for strong n-type doping ($n \sim10^{21}$ cm$^{-3}$, carrier concentrations which are more than one order of magnitude higher than those of typical TE materials such as PbTe, Bi$_2$Te$_3$). At these high concentrations, the underestimation of $PF$ within all considered functionals may be generated by the incomplete validity of rigid band structure approximation (Fig. \[PFAllFunc\]) . In order to check this, we have considered 3$\times$3$\times$3 [SrTiO$_3$]{} supercells which explicitly incorporate La and Nb doping elements, and studied their structural, electronic and transport properties. These doping elements introduce one electron to the systems, and change also the electronic states close to the Fermi level (chemical potential) with respect to that of bulk [SrTiO$_3$]{} (see the density of states DOS from Fig. \[DOS333STO\](a) and (b)). Indeed at these high electronic concentrations the rigid band structure approximation is not completely valid.
![\[DOS333STO\] (Color online) Density of states (DOS) of: (a) Nb doped [SrTiO$_3$]{} (Nb:[SrTiO$_3$]{}), and (b) La doped [SrTiO$_3$]{} (La:[SrTiO$_3$]{}) at electronic concentrations $n \sim 1.2\times10^{21} cm^{-3}$. The Fermi energy E$_F$ is shown in red dashed line. DOS of bulk [SrTiO$_3$]{} is shown in background brown color.](SrTiO3_Nb2_333 "fig:")\
![\[DOS333STO\] (Color online) Density of states (DOS) of: (a) Nb doped [SrTiO$_3$]{} (Nb:[SrTiO$_3$]{}), and (b) La doped [SrTiO$_3$]{} (La:[SrTiO$_3$]{}) at electronic concentrations $n \sim 1.2\times10^{21} cm^{-3}$. The Fermi energy E$_F$ is shown in red dashed line. DOS of bulk [SrTiO$_3$]{} is shown in background brown color.](SrTiO3_La2_333 "fig:")\
![\[StrucSTOLVO\] (Color online) (a)-(b) ([SrTiO$_3$]{})$_m$([KNbO$_3$]{})$_1$ nanostructures with nonmagnetic ground state (m=1,5), and (c)-(d) ([SrTiO$_3$]{})$_m$([LaVO$_3$]{})$_1$ nanostructures with AFM ground state (m=1,5). The AFM spin order on V atoms is shown by arrows.](SrTiO3_LaVO3_Struct "fig:")\
We studied TE properties of 3$\times$3$\times$3 [SrTiO$_3$]{} supercells, which include explicitly La doping elements (3$\times$3$\times$3 La:SrTiO3). The relaxation time determined by fitting $\sigma_{exp}$ at $n=1\times10^{21}$ cm$^{-3}$ and 300 K is the same as that of bulk [SrTiO$_3$]{} ($\tau=0.45\times10^{-14}$ s). Although the electronic states near the Fermi level of 3$\times$3$\times$3 La:[SrTiO$_3$]{} supercell are slightly different than those of bulk [SrTiO$_3$]{} at high values of $n=1.2\times10^{21}$ cm$^{-3}$ (Fig. \[DOS333STO\](b)), $PF$ is comparable with that of bulk [SrTiO$_3$]{} (Fig. \[PFAllFunc\]). Therefore, the underestimation of experimental power factors $PF_{exp}$ of $\sim$2-3 mW/mK$^2$ [@Okuda; @Muta; @Ohta2005b] is not due to the change of electronic states close to Fermi level generated by doping. Kinaci [*et. al.*]{} also showed that La, Nb and Ta doping do not change significantly the electronic states close to the Fermi level, and TE properties of [SrTiO$_3$]{} alloys are comparable with those of bulk [SrTiO$_3$]{}.[@Kinaci] $S$ values for [SrTiO$_3$]{} alloys are slightly lower, whereas $\sigma$ values are slightly larger than those of bulk [SrTiO$_3$]{}.[@Kinaci] These results suggest that $PF$ of [SrTiO$_3$]{} alloys is not expected to increase significantly with respect to that of bulk [SrTiO$_3$]{}.
We assign the underestimation of $PF_{exp}$ to the enhancement of carrier effective mass due to the electron-phonon coupling interaction, which is compatible with the fact that the electronic transport in n-type [SrTiO$_3$]{} has a polaronic nature.[@Mazin] A factor of 3 larger inertial effective mass $m_{i}^*$ was obtained from experimental optical conductivity relative to the theoretical $m_{i}^*$ value of $\sim$0.63$m_e$ estimated within LDA.[@Mazin] At a given carrier concentration, larger experimental effective masses generate larger $S_{exp}$ by lowering the chemical potential relative to CB bottom. We have estimated $m_{i}^*$ according to the relation [@Snyder]: $$\frac{1}{m_{i}^*}=\frac{1}{3}(\frac{2}{m_l} + \frac{1}{m_h})$$ 0.40$m_e$(0.39$m_e$), 6.09$m_e$(6.1$m_e$), and 0.58$m_e$(0.57$m_e$) where $m_l$ and $m_h$ are the light and heavy effective masses
![\[BndSTOLVO\] Electronic band structure of: (a) ([SrTiO$_3$]{})$_1$([KNbO$_3$]{})$_1$, (b) ([SrTiO$_3$]{})$_5$([KNbO$_3$]{})$_1$, (c) ([SrTiO$_3$]{})$_1$([LaVO$_3$]{})$_1$, (d) ([SrTiO$_3$]{})$_5$([LaVO$_3$]{})$_1$ superlattices, and (e) bulk [SrTiO$_3$]{} estimated within B1-WC.](SrTiO3KNbO3_11_band06 "fig:") ![\[BndSTOLVO\] Electronic band structure of: (a) ([SrTiO$_3$]{})$_1$([KNbO$_3$]{})$_1$, (b) ([SrTiO$_3$]{})$_5$([KNbO$_3$]{})$_1$, (c) ([SrTiO$_3$]{})$_1$([LaVO$_3$]{})$_1$, (d) ([SrTiO$_3$]{})$_5$([LaVO$_3$]{})$_1$ superlattices, and (e) bulk [SrTiO$_3$]{} estimated within B1-WC.](SrTiO3KNbO3_51_band06 "fig:") ![\[BndSTOLVO\] Electronic band structure of: (a) ([SrTiO$_3$]{})$_1$([KNbO$_3$]{})$_1$, (b) ([SrTiO$_3$]{})$_5$([KNbO$_3$]{})$_1$, (c) ([SrTiO$_3$]{})$_1$([LaVO$_3$]{})$_1$, (d) ([SrTiO$_3$]{})$_5$([LaVO$_3$]{})$_1$ superlattices, and (e) bulk [SrTiO$_3$]{} estimated within B1-WC.](SrTiO3LaVO3_11_P4bm_AFM_band06 "fig:") ![\[BndSTOLVO\] Electronic band structure of: (a) ([SrTiO$_3$]{})$_1$([KNbO$_3$]{})$_1$, (b) ([SrTiO$_3$]{})$_5$([KNbO$_3$]{})$_1$, (c) ([SrTiO$_3$]{})$_1$([LaVO$_3$]{})$_1$, (d) ([SrTiO$_3$]{})$_5$([LaVO$_3$]{})$_1$ superlattices, and (e) bulk [SrTiO$_3$]{} estimated within B1-WC.](SrTiO3LaVO3_51_P4bm_AFM_band06 "fig:") ![\[BndSTOLVO\] Electronic band structure of: (a) ([SrTiO$_3$]{})$_1$([KNbO$_3$]{})$_1$, (b) ([SrTiO$_3$]{})$_5$([KNbO$_3$]{})$_1$, (c) ([SrTiO$_3$]{})$_1$([LaVO$_3$]{})$_1$, (d) ([SrTiO$_3$]{})$_5$([LaVO$_3$]{})$_1$ superlattices, and (e) bulk [SrTiO$_3$]{} estimated within B1-WC.](SrTiO3_tran_Cryst_band06 "fig:")\
of the three fold degenerate Ti $t_{2g}$ bands which form the CB bottom. The estimated $m_l$, $m_h$, and $m_{i}^*$ values are within B1-WC(LDA), respectively. Since in the transport calculations we do not account for the polaronic nature of [SrTiO$_3$]{} conductivity, this translate into small values of the estimated relaxation time (see Table \[Table2\]).
Band structure engineering in [SrTiO$_3$]{} based nanostructures
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We considered ([SrTiO$_3$]{})$_m$([KNbO$_3$]{})$_1$ and ([SrTiO$_3$]{})$_m$([LaVO$_3$]{})$_1$ SL nanostructures with m=1, 5 for which we studied the electronic and transport properties, and described the relation between size of nanostructures and their TE properties by looking at the effect of quantum confinement on $PF$. ([SrTiO$_3$]{})$_m$([KNbO$_3$]{})$_1$ SL with m=1, 5 have a nonmagnetic ground state. Their structures and electronic band structures are shown in Figs. \[StrucSTOLVO\](a),(b), and \[BndSTOLVO\](a),(b). In comparison to bulk [SrTiO$_3$]{}, ([SrTiO$_3$]{})$_1$([KNbO$_3$]{})$_1$ SL possess smaller $E_g$, and an electronic band which is very flat along $\Gamma$Z direction and dispersive in the other orthogonal directions of the Brillouin zone (see Fig. \[BrillouinZone\](b)). This very flat-and-dispersive band forms the CB bottom and has a Nb $d_{xy}$ orbital character. The electronic states associated to this very anisotropic band, which participate in the electronic transport, have a reduced weight (short $\Gamma$Z distance in the Brillouin zone). The weight is proportional with the density of states DOS and the carrier pocket volumes inside of the Brillouin zone. The reduced weight can be seen more easily from DOS scaled to ABO$_3$ formula unit (f.u.) (see Fig. \[dosSTOLVO\](a)). The electronic states inside of [SrTiO$_3$]{} band gap have a small weight, which generate in the inplane direction power factors $PF_{xx}$ smaller than that of bulk [SrTiO$_3$]{} (Fig. \[PFSTOLVO\](a)). In the cross plane direction, ([SrTiO$_3$]{})$_1$([KNbO$_3$]{})$_1$ SL show large power factors $PF_{zz}$ but at very high $n$ values (chemical potential $\mu$ $\sim$2.75 eV) which can not be achieved in experiment. Increasing the quantum confinement in the case of ([SrTiO$_3$]{})$_5$([KNbO$_3$]{})$_1$ SL, decreases the weight of very anisotropic flat-and-dispersive Nb $d$ band. The decrease in weight of this anisotropic band can be seen from the electronic band structure and DOS (see Figs. \[BndSTOLVO\](b), \[dosSTOLVO\](a)), and produces a $PF$ drop relative to ([SrTiO$_3$]{})$_1$([KNbO$_3$]{})$_1$ SL (Fig. \[PFSTOLVO\](a)).
![\[BrillouinZone\] (Color online) Brillouin zone of: (a) simple cubic [SrTiO$_3$]{}, (b) tetragonal ([SrTiO$_3$]{})$_m$([KNbO$_3$]{})$_1$ and ([SrTiO$_3$]{})$_m$([LaVO$_3$]{})$_1$ SL (m=1,5), (c) body centered tetragonal SrO\[SrTiO$_3$\]$_m$ (m=1, 2), and (d) one face centered tetragonal Sr$_2$CoO$_3$F (ground state structure from Fig. \[StrucAOABO3\](c)). The high symmetry points along the directions used in electronic band structures, and the orthogonal reciprocal $k_i$ vectors (i=x,y,z) are also shown.](STO_TE_BrillouinZone "fig:")\
![\[dosSTOLVO\] (Color online) Total density of states (DOS) of: (a) ([SrTiO$_3$]{})$_m$([KNbO$_3$]{})$_1$, and (b) ([SrTiO$_3$]{})$_m$([LaVO$_3$]{})$_1$ superlattices (m=1,5) scaled to ABO$_3$ formula unit. DOS of bulk [SrTiO$_3$]{} is also shown in background brown color.](SrTiO3KNbO3_m1_dos "fig:")\
![\[dosSTOLVO\] (Color online) Total density of states (DOS) of: (a) ([SrTiO$_3$]{})$_m$([KNbO$_3$]{})$_1$, and (b) ([SrTiO$_3$]{})$_m$([LaVO$_3$]{})$_1$ superlattices (m=1,5) scaled to ABO$_3$ formula unit. DOS of bulk [SrTiO$_3$]{} is also shown in background brown color.](SrTiO3LaVO3_m1_P4bm_AFM_up_dos "fig:")\
SL formed by ([SrTiO$_3$]{})$_m$([LaVO$_3$]{})$_1$ with m=1, 5 have an antiferromagnetic (AFM) ground state. Their structures and electronic band structures are shown in Figs. \[StrucSTOLVO\](c),(d), and \[BndSTOLVO\](c),(d). These SL possess two electronic bands laying inside of [SrTiO$_3$]{} band gap, which are very flat in $\Gamma$A and AZ directions, and weakly dispersive in the other directions of Brillouin zone (Fig. \[BrillouinZone\](b)). These flat bands create a narrow energy distribution with a very large weight on the top of valence band, being generated by V $d_{xz}$ and $d_{yz}$ orbitals. The very large weight of this narrow energy distribution can be seen from DOS, and this distribution generates $PF$s smaller than those of [SrTiO$_3$]{} (see Figs. \[dosSTOLVO\](b), and \[PFSTOLVO\](b)). This shows that a single or multiple very flat bands having large effective masses in all directions of Brillouin zone are not able to enhance TE performance, since the charge carriers associated to such flat bands are very localized and unable to participate in electronic transport. Increasing the quantum confinement in ([SrTiO$_3$]{})$_5$([LaVO$_3$]{})$_1$ SL, also lowers the weight of narrow energy distribution and $PF$ of these SL (see Figs. \[dosSTOLVO\](b), and \[PFSTOLVO\](b)).
![\[PFSTOLVO\] (Color online) Power factor $PF=S^2 \sigma$ dependence on chemical potential $\mu$ of: (a) ([SrTiO$_3$]{})$_m$([KNbO$_3$]{})$_1$, and (b) ([SrTiO$_3$]{})$_m$([LaVO$_3$]{})$_1$ superlattices (m=1,5) estimated at 300 K within B1-WC using the relaxation time $\tau=0.43\times10^{-14}$ s. ](STOKNO_m1_PF "fig:")\
![\[PFSTOLVO\] (Color online) Power factor $PF=S^2 \sigma$ dependence on chemical potential $\mu$ of: (a) ([SrTiO$_3$]{})$_m$([KNbO$_3$]{})$_1$, and (b) ([SrTiO$_3$]{})$_m$([LaVO$_3$]{})$_1$ superlattices (m=1,5) estimated at 300 K within B1-WC using the relaxation time $\tau=0.43\times10^{-14}$ s. ](STOLVO_m1_PF "fig:")\
![\[StrucAOABO3\] (Color online) Structures of (a) Sr$_2$TiO$_4$, and (b) Sr$_3$Ti$_2$O$_7$. Model structures of Sr$_2$CoO$_3$F with: (c) AFM order on Co and one F atom in apical position, (d) FM order on Co and one F atom in apical position, (e) AFM order on Co and two F atoms in apical position, and (f) FM order on Co and two F atoms in apical position. The spin order on Co atoms is shown by arrows.](SrOSrTiO3_Struct "fig:")\
Band structure engineering in AO\[ABO$_3$\]$_m$ naturally-ordered Ruddlesden-Popper phases
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Highly anisotropic flat-and-dispersive bands can be found also in AO\[ABO$_3$\]$_m$ Ruddlesden-Popper naturally-ordered compounds. These compounds are formed from ABO$_3$ perovskite layers separated by an AO atomic layer and can nowadays be grown epitaxially with atomic-scale control.[@Schlom] To search for highly anisotropic bands, we have considered SrO\[SrTiO$_3$\]$_m$ (m=1 and 2) and SrO\[SrCoO$_2$F\]$_1$ compounds (Fig. \[StrucAOABO3\]). The insertion of SrO atomic layer in the crystallographic direction $Oz$ creates the quantum confinement of electronic states in $\Gamma$Z direction from the band structure of Sr$_2$TiO$_4$ and Sr$_3$Ti$_2$O$_7$ (see Fig. \[BndAOABO3\](a),(b)). It can be seen that CB bottom is formed by such very anisotropic bands, which generate narrow energy distributions with small weights (small length of $\Gamma$Z direction). The small weights of these distributions close to CB minimum (energy $\sim$1.75 - 2 eV) can be seen more easily from scaled DOS per f.u. of
![\[BndAOABO3\] Electronic band structure of: (a) Sr$_2$TiO$_4$, (b) Sr$_3$Ti$_2$O$_7$, (c) Sr$_2$CoO$_3$F (ground state structure from Fig. \[StrucAOABO3\](c)), and (d) bulk [SrTiO$_3$]{} estimated within B1-WC.](Sr2TiO4_band06 "fig:") ![\[BndAOABO3\] Electronic band structure of: (a) Sr$_2$TiO$_4$, (b) Sr$_3$Ti$_2$O$_7$, (c) Sr$_2$CoO$_3$F (ground state structure from Fig. \[StrucAOABO3\](c)), and (d) bulk [SrTiO$_3$]{} estimated within B1-WC.](Sr3Ti2O7_band06 "fig:") ![\[BndAOABO3\] Electronic band structure of: (a) Sr$_2$TiO$_4$, (b) Sr$_3$Ti$_2$O$_7$, (c) Sr$_2$CoO$_3$F (ground state structure from Fig. \[StrucAOABO3\](c)), and (d) bulk [SrTiO$_3$]{} estimated within B1-WC.](Sr2CoO3F_2_AFM_band06 "fig:") ![\[BndAOABO3\] Electronic band structure of: (a) Sr$_2$TiO$_4$, (b) Sr$_3$Ti$_2$O$_7$, (c) Sr$_2$CoO$_3$F (ground state structure from Fig. \[StrucAOABO3\](c)), and (d) bulk [SrTiO$_3$]{} estimated within B1-WC.](SrTiO3_tran_Cryst_band06_1 "fig:")\
Sr$_2$TiO$_4$ and Sr$_3$Ti$_2$O$_7$, (Fig. \[dosAOABO3\](a)).
TE properties have been estimated using the same value of $\tau$ as that of bulk [SrTiO$_3$]{}, because we want to compare the electronic contribution given by the electronic band structure of these naturally-ordered compounds to that of bulk [SrTiO$_3$]{}. Due to the small weight of narrow energy distribution, the n-type $PF$ corresponding to the chemical potential $\sim$1.75 - 2 eV in $Ox$ direction ($PF_{xx}$) is smaller than that of bulk [SrTiO$_3$]{} (Fig. \[PFAOABO3\](a)). In the approximation that $\tau$ of SrO\[SrTiO$_3$\]$_m$ compounds is similar to that of bulk [SrTiO$_3$]{}, the n-type $PF$s is not improved.
From Co based AO\[ABO$_3$\]$_m$ compounds, we have explored the cobalt oxyfluoride Sr$_2$CoO$_3$F, in which F substitute O form apical position of CoO$_6$ octahedron.[@Tsujimoto] In Figure \[StrucAOABO3\](c-f) are shown the model structures in which F substitute one O atom (Fig. \[StrucAOABO3\](c),(d)) or two O atoms (Fig. \[StrucAOABO3\](e),(f)) form apical positions and Co atoms have AFM/FM order. The analysis of structural properties shows that the ground state structure is the structure in which F substitute one O atom form apical positions and Co atoms have AFM order (Fig. \[StrucAOABO3\](c)). These results are in agreement with the experimental study, which finds G-type antiferromagnetic order of Co in Sr$_2$CoO$_3$F.[@Tsujimoto] For the ground state structure, we studied the electronic and transport properties. The electronic band structure of Sr$_2$CoO$_3$F contains two electronic bands with Co $d$ orbital character in the (0.75eV, 1.25eV) energy interval (Fig. \[BndAOABO3\](c)). These bands do not have a very anisotropic character, requirement identified to maximize $PF$.[@Bilc2015] As a result the energy distribution of the two Co bands is narrow and has large weight, which can be seen from DOS (Fig. \[dosAOABO3\](b)). In the approximation that $\tau$ of Sr$_2$CoO$_3$F is comparable to that of [SrTiO$_3$]{}, the transport calculations show that $PF$ of these compounds is smaller than that of [SrTiO$_3$]{} (Fig. \[PFAOABO3\](b)).
![\[dosAOABO3\] (Color online) Total density of states (DOS) of: (a) Sr$_2$TiO$_4$ and Sr$_3$Ti$_2$O$_7$, and (b) Sr$_2$CoO$_3$F (ground state structure from Fig. \[StrucAOABO3\](c)) scaled to formula unit (f.u.= Sr$_2$TiO$_4$, Sr$_{1.5}$TiO$_{3.5}$, and Sr$_2$CoO$_3$F, respectively). ](SrOSrTiO3_1m_dos "fig:")\
![\[dosAOABO3\] (Color online) Total density of states (DOS) of: (a) Sr$_2$TiO$_4$ and Sr$_3$Ti$_2$O$_7$, and (b) Sr$_2$CoO$_3$F (ground state structure from Fig. \[StrucAOABO3\](c)) scaled to formula unit (f.u.= Sr$_2$TiO$_4$, Sr$_{1.5}$TiO$_{3.5}$, and Sr$_2$CoO$_3$F, respectively). ](SrOCoO2F_11_dos "fig:")\
![\[PFAOABO3\] (Color online) Power factor $PF=S^2 \sigma$ dependence on chemical potential $\mu$ of: (a) Sr$_2$TiO$_4$ and Sr$_3$Ti$_2$O$_7$, and (c) Sr$_2$CoO$_3$F naturally-ordered superlattices estimated at 300 K within B1-WC using the relaxation time $\tau=0.43\times10^{-14}$ s. ](SrOSTO_1m_PF "fig:")\
![\[PFAOABO3\] (Color online) Power factor $PF=S^2 \sigma$ dependence on chemical potential $\mu$ of: (a) Sr$_2$TiO$_4$ and Sr$_3$Ti$_2$O$_7$, and (c) Sr$_2$CoO$_3$F naturally-ordered superlattices estimated at 300 K within B1-WC using the relaxation time $\tau=0.43\times10^{-14}$ s. ](SrOSrCoO2F_11_PF "fig:")\
Comparison of the different nanostructures with bulk [SrTiO$_3$]{}
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In the approximation that $\tau$ of the considered nanostructures is comparable to that of bulk [SrTiO$_3$]{}, none of the nanostructures shows higher TE performance than bulk [SrTiO$_3$]{}, in spite of the fact that some of them possess highly anisotropic flat-and-dispersive TM $d$ electronic bands. In addition, the electronic states associated to these bands which participate in transport must have significant weights in order to maximize $PF$ and $n$. The weights are proportional to DOS which depends on the density of states effective mass $m_{d}^* = \gamma^{2/3} (m_{l}^{2}m_{h})^{1/3} $, where $\gamma$ is the carrier pocket degeneracy or band multiplicity.[@Bilc2006] Therefore in Table \[Table3\], we show the estimated values of DOS($\mu$)/f.u. at the chemical potential $\mu$ which optimizes n- and p-type $PF$s, $m_{d}^*$, and band anisotropy ratio $R =m_{h}/m_{l}$. The usual anisotropic behavior is for heavy masses $m_h$ across SL direction ($Oz$), and light masses $m_l$ in the inplane SL direction ($Ox$, $Oy$). Although ([SrTiO$_3$]{})$_1$([KNbO$_3$]{})$_1$ SL show very large $R$ and large $m_{d}^*$ values, their DOS($\mu$)/f.u. corresponding to the maximum n-type $PF$ is about one order of magnitude smaller than that of [SrTiO$_3$]{}. The confinement of Nb $d_{xy}$ states achieved in these SL is able to create very large anisotropic electronic bands, but with small weights of the electronic states participating in transport due to their small carrier pocket volume in the Brillouin zone. ([SrTiO$_3$]{})$_1$([LaVO$_3$]{})$_1$ SL at Z point of VB maximum, show unusual band anisotropic behavior with $m_l$ along $z$ direction and $m_h$ along $x$ and $y$ directions, which gives large values for $R$, $m_{d}^*$, DOS($\mu$)/f.u. at the p-type $PF$ maximum, and larger p-type power factor across SL direction ($PF_{zz}$) than along SL direction ($PF_{xx}$). Similar to ([SrTiO$_3$]{})$_1$([KNbO$_3$]{})$_1$ SL, the confinement of Ti $t_{2g}$ ($d_{xy}$) states participating in transport of Sr$_2$TiO$_4$ and Sr$_3$Ti$_2$O$_7$ naturally-ordered SL creates large anisotropy ratios, but at the same time detrimental reduced weights. On the other hand, Co $t_{2g}$ ($d_{xz}$, and $d_{yz}$) states involved in the n-type transport of Sr$_2$CoO$_3$F have large weights, but small anisotropy ratio. For comparison, we show in Table \[Table3\] the corresponding values for the full Heusler Fe$_2$TiSi which shows very large n-type $PF$s.[@Bilc2015] These very large $PF$s are achieved for concomitant large anisotropy ratio and weights, and small $m_l$ effective mass which gives large carrier mobilities along the transport direction.
[0.5]{} [@cccccccc]{} & & $m_h$ & $m_l$ & $R$ & $\gamma$ & $m_{d}^*$&DOS($\mu$)\
[SrTiO$_3$]{}&CB($\Gamma$)& 6.1 & 0.4 & 15.3 & 3 & 2.06 & 14.6\
([SrTiO$_3$]{})$_1$([KNbO$_3$]{})$_1$&CB($\Gamma$)& 219.4 & 0.27 & 812.6 & 1 & 2.52 & 1.5\
([SrTiO$_3$]{})$_1$([LaVO$_3$]{})$_1$&CB($\Gamma$)& 4.66 & 0.41 & 11.4 & 1 & 0.92 & 2.4\
&VB(Z) & 15.62 & 0.8 & 19.5 & 1 & 5.8 & 90.9\
&VB(X) & 13.92 & 1.73 & 8.1 & 1 & 3.47 &\
Sr$_2$TiO$_4$ &CB($\Gamma$)& 74 & 0.42 & 176.2 & 1 & 2.36 & 4.7\
Sr$_3$Ti$_2$O$_7$ &CB($\Gamma$)& 50 & 0.4 & 125 & 1 & 2.0 & 4.9\
Sr$_2$CoO$_3$F &CB(Z) & 5.7 & 1.54 & 3.7 & 1 & 2.91 & 32\
& & & 2.8 & 2.0 & & &\
&VB(T) & 244.6 & 0.36 & 679.4 & 1 & 3.65 & 5\
& & & 0.55 & 444.7 & & &\
Fe$_2$TiSi$^a$ &CB($\Gamma$)& 90 & 0.2 & 450 & 3 & 3.19 & 31.2\
\
Conclusions
===========
Using the concept of electronic band structure engineering we tried to design materials possessing highly anisotropic electronic bands in ([SrTiO$_3$]{})$_m$([KNbO$_3$]{})$_1$ and ([SrTiO$_3$]{})$_m$([LaVO$_3$]{})$_1$ (m=1 and 5) artificial superlattices, and in SrO\[SrTiO$_3$\]$_m$ (m=1 and 2) and SrO\[SrCoO$_2$F\]$_1$ naturally-ordered superlattices. In spite of the fact that almost all superlattices possess such highly anisotropic electronic bands created by the confinement of TM $d$ states, which is a signature of low-DET, their $PF$s are not better than that of [SrTiO$_3$]{}. The origin of this TE performance is the small weights of electronic states participating in transport, which are associated to the highly anisotropic electronic bands. The experimental evidences for the decreased effective TE performance of quantum wells and two-dimensional electron gas systems, caused by the contribution of barrier layers used to create the confinement, support our conclusion.[@Hicks; @Ohta2007] Another detrimental effect on TE performance is the large $m_l$ values along the transport direction of [SrTiO$_3$]{} and related oxide materials, which are a factor $\sim$2 larger than those of Fe based Heusler compounds [@Bilc2015], and a factor of $\sim$16 larger than those of usual thermoelectrics such as PbTe.[@Bilc2006] If we account for the polaronic conductivity of [SrTiO$_3$]{}, these factors are $\sim$ 3 times larger. Although [SrTiO$_3$]{} possesses highly directional TM $d$ electronic states active in transport, these states do not generate very large PF’s. The origin of this TE performance is the low electron mobility as a result of the polaronic nature of electrical conductivity. In [SrTiO$_3$]{} and related perovskite oxides, there is an important TM $d$ - O $p$ hybridization with covalent character, which appears to favour the polaronic conductivity. Binary TM oxides possessing high structural symmetries with stronger TM $d$ - $d$ atomic interactions, may show high anisotropy, large weights and high mobilities of the charge carriers giving improved thermoelectric performance over ABO$_3$ perovskite oxides.
The authors acknowledge financial support from the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project number PN-II-PT-PCCA-2013-4-1119. Ph. G. acknowledges the ARC project AIMED and the F.R.S.-FNRS project HiT4FiT.
[99]{}
S. Ohta et al., Appl. Phys. Lett. 2005, 87, 092108. T. Okuda, K. Nakanishi, S. Miyasaka and Y. Tokura, Phys. Rev. B 2001, 63, 113104. P.-P. Shang, B.-P. Zhang , Y. Liu, J.-F. Li, H.-M. Zhu, J. Electr. Mater. 2011, 40, 926. H. C. Wang, C. l. Wang, W. B Su, J. Liu, Y. Sun, H. Peng, L. M. Mei, JACS 2011, 94, 838. J. Liu, C. L. Wang, Y. Li, W. B. Su, Y. H. Zhu, J. C. Li, and L. M. Mei, J. Appl. Phys. 2013, 114, 223714. K. Park, J. S. Son, S. Ill Woo, K. Shin, M.-W. Oh, S.-D. Parkc, and T. Hyeon, J. Mater. Chem. A, 2014, 2, 4217. A. V. Kovalevsky, A. A. Yaremchenko, S. Populoh, P. Thiel, D. P. Fagg, A. Weidenkaff, and J. R. Frade, Phys. Chem. Chem. Phys. 2014, 16, 26946. M. T. Buscaglia et al., J. Eur. Ceram. Soc. 2014, 34, 307. N. Wang, H. Chen, H. He, W. Norimatsu, M. Kusunoki, and K. Koumoto, Sci. Rep. 2013, 3, 3449. B. Zhang, J. Wang, T. Zou, S. Zhang, X. Yaer, N. Ding, C. Liu, L. Miao, Y. Lia, and Y. Wu, J. Mater. Chem. C 2015, 3, 11406. A. M. Dehkordi, S. Bhattacharya, T. Darroudi, X. Zeng, H. N. Alshareef, T. M. Tritt, J. Vis. Exp. 2015, 102, e52869. Z. Lu, H. Zhang, W. Lei, D. C. Sinclair, and I. M. Reaney, Chem. Mater. 2016, 28, 925. E. Li, N. Wang, H. He, and H. Chen, Nanosc. Res. Lett. 2016, 11, 188. Y. Wang, K. H. Lee, H. Ohta, and K. Koumoto, J. Appl. Phys. 2009, 105, 103701. I. Matsubara et al. Appl. Phys. Lett. 2001, 78, 3627. M. Ohtaki, T. Tsubota and K. Egushi, J. Appl. Phys. 1996, 79, 1816. T. Tsubota, M. Ohtaki, K. Eguchi, and H. Arai, J. Mater. Chem. 1997, 7, 85. Z.-H. Wu, H.-Q. Xie, and Q.-F. Zeng, J. Inorg. Mater. 2013, 28, 921. S. Saini , P. Mele, H. Honda, K. Matsumoto, K. Miyazaki, and A. Ichinose, J. Electr. Mater. 2014, 43, 2145. Z. H. Wu, H. Q. Xie, and Y. B. Zhai, J. Nanosci. Nanotechnol. 2015, 15, 3147. J.-L. Lan, Y. Liu, Y.-H. Lin, C.-W. Nan, Q. Cai, and X. Yang, Sci. Rep. 2015, 5, 7783. S. F. Wang, F. Q. Liu, Q. Lu, S. Y. Dai, J. L. Wang, W. Yu, G. S. Fu, J. Eur. Ceram. Soc. 2013, 33, 1763. X. R. Zhang, H. L. Li, and J. L. Wang, J. Adv. Ceram. 2015, 4, 226. M. Backhaus-Ricoult, J. Rustad, L. Moore, C. Smith, and J. Brown, Appl. Phys. A 2014, 116, 433.
A. C. Masset et al., Phys. Rev. B 2000, 62, 166. H. Ohta et al., Nature Materials 2007, 6, 129. J. Sui, J. Li, J. He, Y.-L. Pei, D. Berardan, H. Wu, N. Dragoe, W. Caia, and L.-D. Zhao, Energy Environ. Sci. 2013, 6, 2916. H. Ohta, K. Sugiura and K. Koumoto, Inorg. Chem. 2008, 47, 8429. K. Koumoto, I. Terasaki and R. Funahashi, MRS Bulletin 2006, 31, 206. Y. Mune et al. Appl. Phys. Lett. 2007, 91, 192105. P. Garcia-Fernandez, M. Verissimo-Alves, D. I. Bilc, P. Ghosez, and J. Junquera, Phys. Rev. B 2012, 86, 085305. R. Astala and P. D. Bristowe, J. Phys.: Condens. Matter 2002, 14, L149. X. G. Guo et al., Phys. Lett. A 2003, 317, 501. C. Zhang et al., Mat. Chem. Phys. 2008, 107, 215. J. N. Yun and Z. Y. Zhang, Chin. Phys. B 2009, 18, 2945. H. Usui et al., Phys. Rev. B 2010, 81, 205121. R. Zhang et al., J. Am. Ceram. Soc. 2010, 93, 1677. A. Kinaci, C. Sevik, and T. Çağin, Phys. Rev. B 2010, 82, 155114. J. D. Baniecki1, M. Ishii, H. Aso, K. Kurihara, and D. Ricinschi, J. Appl. Phys. 2013, 113, 013701. D. F. Zou, Y. Y. Liu, S. H. Xie, J. G. Lin, J. Y. Li, Chem. Phys. Lett. 2013, 586, 159. K. Shirai and K. Yamanaka, J. Appl. Phys. 2013, 113, 053705. M. U. Kahaly, and U. Schwingenschlögl, J. Mater. Chem. A 2014, 2, 10379. K. Singsoog, T. Seetawan, A. Vora-Ud, and C. Thanachayanont, Integr. Ferroelectr. 2014, 155, 111. R.-Z. Zhang, X.-Y. Hu, P. Guo, C.-l. Wang, Phys. B - Cond. Matt. 2012, 407, 1114. A. Roy, Phys. Rev. B 2016, 93, 100101. B. Himmetoglu and A. Janotti, J. Phys.: Condens. Matter 2016, 28, 065502. B. Khan, H. A. R. Aliabad, N. Razghandi, M. Maqbool, S. J. Asadabadi, and I. Ahmad, Comput. Phys. Commun. 2015, 187, 1. F. P. Zhang et al., Physica B: Condens. Matter 2011, 406, 1258. F. P. Zhang et al., J. Phys. Chem. Sol. 2013, 74, 1859. M. Molinari, D. A. Tompsett, S. C. Parker, F. Azough, and R. Freer, J. Mater. Chem. A 2014, 2, 14109. X. H. Zhang et al., J. Alloy. Compd. 2015, 634, 1. P. Srepusharawoot, S. Pinitsoontorn, and S. Maensiri, Comput. Mater. Sci. 2016, 114, 64. K. P. Ong, D. J. Singh, P. Wu, Phys. Rev. B 2011, 83, 115110. X. Qua, W. Wanga, S. Lv, D. Jia, Solid State Commun. 2011, 151, 332. S. Jantrasee, S. Pinitsoontorn, and P. Moontragoon, J. Electr. Mater. 2014, 43, 1689. A. Alvarado, J. Attapattu, Y. Zhang, C. F. Chen, J. Appl. Phys. 2015, 118, 165101. Z. Huang, T. Y. Lu, H. Q. Wang, J. C. Zheng, AIP Adv. 2015, 5, 097204. X. Chen, D. Parker, M. H. Du, and D. J. Singh, New J. Phys. 2013, 15, 043029. L. Lindsay and D. S. Parker, Phys. Rev. B 2015, 92, 144301. D. Bayerl and E. Kioupakis, Phys. Rev. B 2015, 91, 165104. Y. Chumakov et al., J. Electr. Matter. 2013, 42, 1597.
D. I. Bilc, G. Hautier, D. Waroquiers, G.-M. Rignanese, and Ph. Ghosez, Phys. Rev. Lett. 2015, 114, 136601. K. Biswas et al., Nature 2012, 489, 414. J. P. Heremans, M. S. Dresselhaus, L. E. Bell, and D. T. Morelli, Nature Nanotech. 2013, 8, 471. F. Inaba et al., Phys. Rev. B 1995, 52, 2221. Q. Wang et al., Sci. Technol. Adv. Mat. 2004, 5, 543. D. I. Bilc, R. Orlando, R. Shaltaf, G.-M. Rignanese, Jorge Iniguez and Ph. Ghosez, Phys. Rev. B 2008, 77, 165107. M. Goffinet, P. Hermet, D. I. Bilc, and Ph. Ghosez, Phys. Rev. B 2009, 79, 014403. A. Prikockyte, D. Bilc, P. Hermet, C. Dubourdieu, and Ph. Ghosez, Phys. Rev. B 2011, 84, 214301. J. P. Perdew and A. Zunger, Phys. Rev. B 1981, 23, 5048. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 1996, 77, 3865. Z. Wu and R. E. Cohen, Phys. Rev. B 2006, 73, 235116. R. Dovesi, R. Orlando, B. Civalleri, C. Roetti, V. R. Saunders, and C. M. Zicovich-Wilson, Z. Kristallogr. 2005, 220, 571. T. Bredow , P. Heitjans , M. Wilkening, Phys. Rev. B 2004, 70, 115111. S. Piskunov, E. Heifets, R. I. Eglitis, and G. Borstel, Comp. Mat. Sci. 2004, 29, 165. R. Dovesi, C. Roetti, C. Freyria Fava, M. Prencipe, and V.R. Saunders, Chem. Phys. 1991, 156, 11. M. F. Peintinger, D. Vilela Oliveira, and T. Bredow, J. Comput. Chem. 2013, 34, 451. W. C. Mackrodt, N. M. Harrison, V. R. Saunders, N. L. Allan, M. D. Towler, E. Apra, and R. Dovesi, Phil. Mag. A 1993, 68, 653. X. Cao, M. Dolg, J. Molec. Struct. (Theochem) 2002, 581, 139. G. K. H. Madsen and D. J. Singh, Comput. Phys. Commun. 175, 67 (2006). H. Muta, K. Kurosaki, S. Yamanaka, J. Alloys Comp. 2005, 392, 306. K. H. Hellwege, and A. M. Hellwege (Eds.), Ferroelectrics and Related Substances, New Series, vol. 3, Landolt-Bornstein, Springer Verlag, Berlin, 1969, group III. K. van Benthem, C. Elsasser, and R. H. French, J. Appl. Phys. 2001, 90, 6156. S. Ohta, T. Nomura, H. Ohta and K. Koumoto, J. Appl. Phys. 2005, 97, 034106. J. L. M. van Mechelen, D. van der Marel, C. Grimaldi, A. B. Kuzmenko, N. P. Armitage, N. Reyren, H. Hagemann, and I. I. Mazin, Phys. Rev. Lett. 2008, 100, 226403. Y. Pei, A. D. LaLonde, H. Wang, and G. J. Snyder, Energy Environ. Sci. 2012, 5, 7963. Y. F. Nie, Y. Zhu, C.-H. Lee, L. F. Kourkoutis, J. A. Mundy, J. Junquera, Ph. Ghosez, D. J. Baek, S. Sung, X. X. Xi, K. M. Shen, D. A. Muller, and D. G. Schlom, Nature Commun. 2014, 5, 4530. Y. Tsujimoto et al., Inorg. Chem. 2012, 51, 4802. D. I. Bilc, S. D. Mahanti, and M. G. Kanatzidis, Phys. Rev. B 2006, 74, 125202. L. D. Hicks, T. C. Harman, X. Sun, and M. S. Dresselhaus, Phys. Rev. B 1996, 53, R10493.
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abstract: 'In this paper we develop a geometric analysis and a numerical algorithm, based on indirect methods, to solve optimal guidance of endo-atmospheric launch vehicle systems under mixed control-state constraints. Two main difficulties are addressed. First, we tackle the presence of Euler singularities by introducing a representation of the configuration manifold in appropriate local charts. In these local coordinates, not only the problem is free from Euler singularities but also it can be recast as an optimal control problem with only pure control constraints. The second issue concerns the initialization of the shooting method. We introduce a strategy which combines indirect methods with homotopies, thus providing high accuracy. We illustrate the efficiency of our approach by numerical simulations on missile interception problems under challenging scenarios.'
author:
- 'Riccardo Bonalli, Bruno Hérissé and Emmanuel Trélat[^1][^2][^3]'
bibliography:
- 'references.bib'
title: 'Optimal Control of Endo-Atmospheric Launch Vehicle Systems: Geometric and Computational Issues'
---
Geometric optimal control, Indirect methods, Numerical homotopy methods, Guidance of vehicle systems.
Introduction {#secIntro}
============
Optimal Guidance of Launch Vehicle Systems
------------------------------------------
of autonomous launch vehicle systems towards rendezvous regions is a complex task, often considered in aerospace applications. It can be modeled as an optimal control problem with the objective of finding a control law enabling the vehicle to join some target region considering prescribed constraints as well as performance criteria. The rendezvous region may be static as well as a moving point if, for example, the mission consists of reaching a maneuvering goal. Then, an important challenge consists of developing analysis and algorithms able to provide *high numerical precision* for optimal trajectories, considering rough onboard processors, i.e., reduced computational capabilities.
In the engineering community, one of the most widespread approaches to solve such kind of task relies on *explicit guidance laws* (see, e.g., [@LinTsai; @Lin; @shinar1991optimal; @morgan2011minimum; @Nahshon]). They correct errors coming from perturbations and misreading of the system. Nonetheless, trajectories induced by guidance laws are usually not optimal because of some approximations that are required to develop a closed-form expression. On the other hand, computation of trajectories is often achieved by adopting *direct methods* (see, e.g., [@Paris; @RossOld; @Ross; @petersen2013model; @weiss2015model]). These techniques consist of discretizing each component of the optimal control problem (the state, the control, etc.) to reduce it to a nonlinear constrained optimization problem. A high degree of robustness is provided while, in general, no deep knowledge of properties related to the structure of the dynamical system is needed, making these methods particularly easy to use in practice. However, their efficiency is proportional to the computational load which often obliges to use them offline.
Good candidates to deal with onboard processing of optimal trajectories are *indirect methods* (see, e.g., [@calise1979singular; @calise1998design; @Lu1; @Lu3; @Pontani]). They leverage necessary conditions for optimality coming from the *Pontryagin Maximum Principle* (PMP) (see, e.g., [@pontryagin1987mathematical; @lee1967foundations]) to wrap the optimal guidance problem into a two-point boundary value problem, leading to accurate and fast algorithms (see, e.g. [@betts1998survey]). The advantages of indirect methods, whose more basic version is known as *shooting method*, are their extremely good numerical accuracy and the fact that, when they converge, convergence is very quick. However, *initializing* indirect methods is a challenging task. Moreover, further methodological difficulties arise in designing algorithms that are based on indirect methods.
Additional Methodological Issues: Euler Coordinates Singularities coming from Mixed Control-State Constraints
-------------------------------------------------------------------------------------------------------------
Obtaining efficient solutions for optimal guidance may oblige to consider both demanding performance criteria and possible onerous missions to accomplish. Since, in this situation, the vehicle is subject to several strong mechanical strains, some stability constraints must be imposed, which are modeled as *mixed control-state constraints*. These optimal control problems are more difficult to tackle by the PMP (see, e.g., [@bryson1975applied; @hartl1995survey; @de2001maximum; @clarke2010optimal]). Indeed, further Lagrange multipliers appear, for which obtaining useful information may be arduous and has been the object of many studies in the existing literature (see, e.g., [@jacobson1971new; @maurer1977optimal; @bonnard2003optimal; @bonnans2007well; @arutyunov2011maximum]).
A widespread approach in aeronautics to avoid to deal with these particular mixed control-state constraints consists of reformulating the original guidance problem using some *local Euler coordinates*, under which the structural constraints become pure control constraints (see, e.g., [@bonnard2003optimal]; we discuss this change of coordinates in Section \[localPMP\]). The transformation allows to consider the usual PMP, and then, classical shooting methods. However, Euler coordinates are not global and their singularities prevent from solving all reachable configurations, reducing the number of feasible missions.
Statement of Contributions
--------------------------
The main objective of this paper consists of designing a numerical strategy based on indirect methods to solve optimal guidance of endo-atmospheric launch vehicle systems. This strategy is able to provide global solutions lying in the configuration manifold by tackling the presence of Euler coordinates singularities introduced by mixed control-state constraints. The contribution is twofold: we first provide a geometric analysis of necessary conditions for optimality from which we derive a numerical scheme ensuring convergence of indirect methods when mixed control-state constraints are considered. The advantage of this strategy is that solutions satisfying mixed control-state constraints can be found by merely employing usual shooting methods.
Specifically, our contributions go as follows:\
1) *Geometric analysis of necessary conditions for optimality:* The solution that we propose to bypass the problem of Euler coordinates singularities consists of reformulating the optimal guidance problem within an intrinsic viewpoint, using geometric control (it does not seem that this framework has been investigated in the optimal guidance context so far).
We build additional local coordinates that cover the singularities of the previous ones (see Section \[secIntro\].B) and in which the mixed control-state constraints can be expressed as pure control constraints (see Section \[secIntro\].B) as well. Moreover, these two sets of local coordinates form an atlas of the configuration manifold and we prove, by using geometric control techniques, that the local PMP formulations in these charts, which have only pure control constraints, are (locally) equivalent to the global PMP formulation with mixed constraints. This justifies the implementation of indirect methods to solve the original problem by employing classical shooting algorithms on the two local problems (with pure control constraints).
We stress the fact that the introduction of these particular local coordinates provides, in turn, two main benefits. On one hand, there is no limit on the feasible missions that can be simulated, and, on the other hand, the optimal guidance problem is not conditioned by multipliers depending on mixed constraints, then, standard shooting or multi-shooting methods can be easily put in practice. This is at the price of changing chart (that is, local coordinates), which slightly complicates the implementation of indirect method, but, importantly, does not affect their efficiency.\
2) *Indirect method based on numerical homotopy procedures:* Our second aim consists of providing a numerical algorithm based on indirect methods. The main advantage of indirect methods is their extremely good numerical accuracy. Indeed, they inherit of the very quick convergence properties of the Newton method. Nevertheless, it is known that their main drawback is related to their initialization. We address this issue by adopting *homotopy methods* (see, e.g., [@allgower2003introduction]).
The basic idea of homotopy methods is to solve a difficult problem step by step starting from a simpler problem (that we call *problem of order zero*) by parameter deformation. Combined with the shooting problem derived from the PMP, homotopies consist of deforming the problem into a simpler one (i.e., on which a shooting method can be easily initialized) and then of solving a series of shooting problems step by step to come back to the original problem. One of the main issues is then being able to design an appropriate problem of order zero, which should “resemble” some extent of the initial problem but at the same time should be “easy to solve”.
Homotopy procedures have proved to be reliable and robust for problems like orbit transfer, atmospheric reentry or planar tilting maneuvers (see, e.g., [@EmmanuelH; @Petit; @zhu2016minimum; @zhu2016planar]). Here, we propose a numerical homotopy scheme to solve the shooting problem coming from the optimal guidance framework, ensuring high numerical accuracy of optimal trajectories.
To practically show the efficiency of this homotopy algorithm, we give numerical solutions of the *endo-atmospheric missile interception* problem (presented, for example, in [@cottrell1971optimal]). We design an appropriate problem of order zero which is a good candidate to initialize the first homotopic iterations. Then, we solve the original problem by a *linear continuation method* (i.e., the simplest homotopy scheme, see, e.g., [@allgower2003introduction]).
Structure of the Paper
----------------------
The paper is organized as follows. Section \[SectionProblem\] contains details on the model under consideration and the optimal guidance problem. Section \[secTheoretical\] is devoted to the PMP formulation of our problem, its intrinsic geometric behavior analysis and the computations of the optimal controls as functions of the state and the costate (which represents a crucial step to correctly define numerical indirect methods. Singular controls are analyzed too). In Sections \[sectHomotopy\] and \[sectAppl\] we provide the numerical homotopy scheme, giving global numerical solutions for the endo-atmospheric missile interception problem. Finally, Section \[conclSect\] contains conclusions and perspectives.
Optimal Guidance Problem {#SectionProblem}
========================
Model Dynamics for Guidance Systems {#SectionForces}
-----------------------------------
We focus on a class of launch vehicles modeled as a three-dimensional axial symmetric cylinder, where $\bm{u}$ denotes its principal body axis, steered by a control system (for example, based on steering fins or a reaction control system). We denote by $Q$ the point of the vehicle where this system is placed. Let $O$ be the center of the Earth, $\bm{K}$ be the northsouth axis of the planet and consider an orthonormal inertial frame $(\bm{I},\bm{J},\bm{K})$ centered at $O$. For the applications presented, the effect of the rotation of the Earth can be neglected. Denoting by $G$ the center of mass of the vehicle which is assumed to lie on $\bm{u}$, the motion is described by the variables $(\bm{r}(t),\bm{v}(t),\bm{u}(t))$, where $\bm{r}(t) = x(t) \bm{I} + y(t) \bm{J} + z(t) \bm{K}$ is the trajectory of $G$ while the vector $\bm{v}(t) = \dot{x}(t) \bm{I} + \dot{y}(t) \bm{J} + \dot{z}(t) \bm{K}$ denotes its velocity.
We denote by $m$ the mass of the vehicle, whose evolution is given as function of the mass flow rate, denoted by $q$. The air density is denoted by $\rho(\bm{r})$ (a standard exponential law of type $\rho_0 \exp(-(\|\bm{r}\|-r_T) / h_r)$ is considered, where $\rho_0 > 0$, $r_T$ is the radius of the Earth and $h_r$ is a reference altitude) while $S$ denotes a constant reference surface for aerodynamical forces. The forces that act on the vehicle are (see, e.g., [@pucci2015nonlinear; @carlucci2018ballistics]):
- gravity $\bm{g} = -g(\bm{r}) m \frac{\bm{r}}{\| \bm{r} \|}$;
- drag $\bm{D} = -\frac{S}{2} \rho(\bm{r}) C_D \| \bm{v} \| \bm{v}$, where $C_D = C_{D_0} + C_{D_1} \left( \frac{\| \bm{u} \times \bm{v} \|}{\| \bm{v} \|} \right)^2$ is a quadratic approximation of the drag coefficient ($C_{D_0}$, $C_{D_1}$ are constant;
- lift $\bm{L} = \frac{S}{2} \rho(\bm{r}) C_{L_{\alpha}} \big( \bm{v} \times (\bm{u} \times \bm{v}) \big)$, where $C_{L_{\alpha}}$ is constant;
- thrust $\bm{T} = f_T(t) \bm{u}$, where $f_T$ is a given nonnegative function which is proportional to the mass flow $q$.
Structural optimization ensures that torques do not affect the dynamics of the momentum. As a standard result (see, e.g. [@pucci2015nonlinear; @carlucci2018ballistics]), the following dynamics is obtained
$$\begin{aligned}
\label{firstDyn}
\begin{cases}
\displaystyle \dot{\bm{r}}(t) = \bm{v}(t) , \ \dot{\bm{v}}(t) = \bm{f}(t,\bm{r}(t),\bm{v}(t),\bm{u}(t)) := \frac{\bm{T}(t,\bm{u}(t))}{m(t)} + \medskip \\
\displaystyle \hspace{10pt} \frac{\bm{g}(\bm{r}(t))}{m(t)} + \frac{\bm{D}(\bm{r}(t),\bm{v}(t),\bm{u}(t))}{m(t)} + \frac{\bm{L}(\bm{r}(t),\bm{v}(t),\bm{u}(t))}{m(t)} \ .
\end{cases}\end{aligned}$$
General Optimal Guidance Problem
--------------------------------
System must be closed with some stability constraints. In particular, the velocity must be always positively oriented w.r.t. the principal body axis and, for controllability reasons, the velocity $\bm{v}$ must lies inside a cone whose symmetry axis is the body axis $\bm{u}$, and that has amplitude $0 < \alpha_{\max} \le \pi/6$, where $\alpha_{\max}$ is the *maximal angle of attack*. From this and , the full dynamics of our system becomes $$\begin{aligned}
\label{guidanceDyn}
\begin{cases}
\dot{\bm{r}}(t) = \bm{v}(t) \quad , \quad \dot{\bm{v}}(t) = \bm{f}(t,\bm{r}(t),\bm{v}(t),\bm{u}(t)) \medskip \\
\displaystyle (\bm{r}(t),\bm{v}(t)) \in N \quad , \quad \bm{u}(t) \in S^2 \medskip \\
\bm{r}(0) = \bm{r}_0 \ , \ \bm{v}(0) = \bm{v}_0 \quad , \quad (\bm{r}(T),\bm{v}(T)) \in M \subseteq N \medskip \\
\displaystyle c_1(\bm{v}(t),\bm{u}(t)) := -\bm{v}(t) \cdot \bm{u}(t) \le 0 \medskip \\
\displaystyle c_2(\bm{v}(t),\bm{u}(t)) := \bigg( \frac{\| \bm{u}(t) \times \bm{v}(t) \|}{\| \bm{v}(t) \| \sin \alpha_{\max}} \bigg)^2 - 1 \le 0
\end{cases}\end{aligned}$$ where $N$ is an open subset of $\mathbb{R}^6 \setminus \{ 0 \}$ consisting of all possible scenarios (see Remark \[remN\] in Section \[localPMP\]), $S^2 = \{ \bm{u} \in \mathbb{R}^3 : \| \bm{u} \|^2 = 1 \}$ is the unit sphere in $\mathbb{R}^3$, $(\bm{r}_0,\bm{v}_0) \in N$ are given initial values, $T$ is the final time and $M$ is a subset of $N$ representing given final conditions. The control variable on which we act is represented by the principal body axis $\bm{u}$.
In this general context, a mission depends on which specific task the launch vehicle has to accomplish, which in turn depends on the cost that has to be minimized and on the set $M$ of final conditions. Then, given any function $g : \mathbb{R} \times \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}$ of class $C^1$, we define the General Optimal Guidance Problem (**GOGP**) to be the optimal control problem that consists of minimizing the generic cost $$C(T,\bm{r}(\cdot),\bm{v}(\cdot),\bm{u}(\cdot)) = g(T,\bm{r}(T),\bm{v}(T))$$ under the dynamical control system (\[guidanceDyn\]). The final time $T$ may be free or not. The generality of this cost allows one to consider various launch vehicle missions: for instance, in the case of endo-atmospheric landing problem one wants to minimize the error between the final position and some desired target point, or, in the case of missile interception one may want to maximize the final velocity.
In what follows, to apply indirect methods it will be needed to compute optimal controls using the PMP (see also Section \[controlSection\]). This may become difficult to accomplish unless one considers further (merely technical) assumptions on $g$ and $M$. More specifically, we assume the following:
\[assM\] The set $M$ is a submanifold of $N$. Moreover, at least one between the following two conditions is satisfied:
1. The final time $T$ is free and $\displaystyle \frac{\partial g}{\partial t}(T,\bm{r},\bm{v}) \neq 0$;
2. It holds $M = \Big\{ (\bm{r},\bm{v}) \in N : F(\bm{r},\bm{v}) = 0 \Big\}$, where $F$ is a smooth submersion. Moreover, for every local chart $(x_1,\dots,x_6)$ (local coordinates) of $(\bm{r},\bm{v}) \in M$ in $N$, there exists a variable $x_i$ such that $\frac{\partial g}{\partial x_i}(T,\bm{r},\bm{v}) \neq 0$.
Pontryagin Maximum Principle Analysis and Optimal Controls in Two Local Charts {#secTheoretical}
==============================================================================
PMP for Problems with Mixed Control-State Constraints {#maxSect}
-----------------------------------------------------
The main objective of this paper consists of providing a numerical strategy to solve (**GOGP**) via indirect methods. They are based on necessary conditions for optimality that arise by applying the PMP to (**GOGP**) (see, e.g., [@betts1998survey]): in this section, we recall such necessary conditions for optimality.
The formulation of (**GOGP**) contains two mixed control-state constraints: $c_1$ and $c_2$. In the presence of such kind of constraints, the PMP can be efficiently employed only under further regularity assumptions on $c_1$ and $c_2$ (see, e.g., [@dmitruk2009development]). Indeed, it is required that the *rank condition* $$\label{rankCond}
\textnormal{rank} \left( \begin{array}{cc}
\partial_{u_1} c_2 & u _1 \\
\partial_{u_2} c_2 & u_2 \\
\partial_{u_3} c_2 & u_3
\end{array} \right)(\bm{v},\bm{u}) = 2$$ holds when $c_2(\bm{v},\bm{u}) = 0$ and $\| \bm{u} \|^2 = 1$ (see, e.g., [@dmitruk1993maximum; @dmitruk2009development]). Straightforward computations show that is always satisfied, therefore, the PMP can be applied to (**GOGP**), leading to the following necessary conditions for optimality as follows.
Denote $\bm{p} = (\bm{p}_1,\bm{p}_2) \in \mathbb{R}^3 \times \mathbb{R}^3$, $\bm{\mu} = (\mu_0,\mu_1,\mu_2) \in \mathbb{R}^3$ and, as usual in the framework of the PMP, define $$\begin{gathered}
\label{ham}
H(t,\bm{r},\bm{v},\bm{p},\bm{\mu},\bm{u}) := H^0(t,\bm{r},\bm{v},\bm{p},\bm{u}) + \mu_0 ( \| \bm{u} \|^2 - 1 ) \\
+ \mu_1 c_1(\bm{v},\bm{u}) + \mu_2 c_2(\bm{v},\bm{u}) := \Big( \bm{p}_1 \cdot \bm{v} + \bm{p}_2 \cdot \bm{f}(t,\bm{r},\bm{v},\bm{u}) \Big) \\
+ \mu_0 ( \| \bm{u} \|^2 - 1 ) + \mu_1 c_1(\bm{v},\bm{u}) + \mu_2 c_2(\bm{v},\bm{u})\end{gathered}$$ to be the *Hamiltonian* of (**GOGP**) (see, e.g., [@dmitruk1993maximum; @dmitruk2009development]). According to the PMP with mixed control-state constraints (see, e.g. [@pontryagin1987mathematical; @hestenes1965calculus; @dmitruk2009development]), if $(\bm{r}(\cdot),\bm{v}(\cdot),\bm{u}(\cdot))$ is optimal for (**GOGP**) with final time $T$, there exist a non-positive scalar $p^0$, an absolutely continuous curve $\bm{p} : [0,T] \rightarrow \mathbb{R}^6$ called *adjoint vector*, and functions $\mu_0(\cdot)$, $\mu_1(\cdot)$, $\mu_2(\cdot) \in L^{\infty}([0,T],\mathbb{R})$, with $(\bm{p}(\cdot),p^0) \neq 0$, such that the so-called *extremal* $(\bm{r}(\cdot),\bm{v}(\cdot),\bm{p}(\cdot),p^0,\mu_0(\cdot),\mu_1(\cdot),\mu_2(\cdot),\bm{u}(\cdot))$ satisfies almost everywhere in the time-interval $[0,T]$:
- **Adjoint Equations** $$\begin{aligned}
\label{adjointSystem}
\begin{cases}
\displaystyle \left(\begin{array}{c} \dot{\bm{r}}(t) \\ \dot{\bm{v}}(t) \end{array}\right) = \frac{\partial H}{\partial \bm{p}}(t,\bm{r}(t),\bm{v}(t),\bm{p}(t),\bm{\mu}(t),\bm{u}(t)) \medskip \\
\displaystyle \dot{\bm{p}}(t) = -\frac{\partial H}{\partial (\bm{r},\bm{v})}(t,\bm{r}(t),\bm{v}(t),\bm{p}(t),\bm{\mu}(t),\bm{u}(t))
\end{cases}\end{aligned}$$
- **Maximality Conditions** $$\label{maxCond}
\displaystyle H^0(t,\bm{r}(t),\bm{v}(t),\bm{p}(t),\bm{u}(t)) \ge H^0(t,\bm{r}(t),\bm{v}(t),\bm{p}(t),\bm{u})$$ for every vector $\bm{u} \in S^2$ that satisfies $c_1(\bm{v}(t),\bm{u}) \le 0$ and $c_2(\bm{v}(t),\bm{u}) \le 0$. Moreover, it holds $$\label{maxCondDer}
\displaystyle \frac{\partial H}{\partial \bm{u}}(t,\bm{r}(t),\bm{v}(t),\bm{p}(t),\bm{\mu}(t),\bm{u}(t)) = 0$$
- **Complementarity Slackness Conditions** $$\label{slack}
\begin{cases}
\mu_1(t) c_1(\bm{v}(t),\bm{u}(t)) = 0 \\
\mu_2(t) c_2(\bm{v}(t),\bm{u}(t)) = 0
\end{cases} \ , \ \mu_1(t) \le 0 \ , \ \mu_2(t) \le 0$$
- **Transversality Conditions** $$\label{transvCond2}
\displaystyle \bm{p}(T) - p^0 \frac{\partial g}{\partial (\bm{r},\bm{v})}(T,\bm{r}(T),\bm{v}(T)) \perp T_{(\bm{r}(T),\bm{v}(T))} M$$ where $T_{(\bm{r}(T),\bm{v}(T))} M$ is the tangent space of $M$ at $(\bm{r}(T),\bm{v}(T)) \in M$. If the final time $T$ is free, then $$\label{transvCond1}
\displaystyle \max_{\bm{u}} H^0(T,\bm{r}(T),\bm{v}(T),\bm{p}(T),\bm{u}) = - p^0 \frac{\partial g}{\partial t}(T,\bm{r}(T),\bm{v}(T))$$ where the maximum is taken over vectors $\bm{u} \in S^2$ that satisfy $c_1(\bm{v}(T),\bm{u}) \le 0$ and $c_2(\bm{v}(T),\bm{u}) \le 0$.
The extremal is said *normal* if $p^0 \neq 0$ and, in this case, we set $p^0 = -1$. Otherwise, the extremal is said *abnormal*.
As pointed out in the introduction, obtaining rigorous and useful information on the multipliers $\mu_1(\cdot)$, $\mu_2(\cdot)$ may be difficult, which consequently makes challenging applying indirect methods, as stated by the previous conditions, to (**GOGP**).
Local Model with Respect to Two Local Charts {#localPMP}
--------------------------------------------
A change of coordinates can be used to transform the mixed control-state constraints $c_1$ and $c_2$ into pure control constraints, allowing to use standard indirect methods. This is commonly used in aerospace (see, e.g., [@bonnard2003optimal]), though without the global (geometric) insight that we propose in this paper. However, this transformation acts only locally, preventing one from representing the whole configuration manifold $N$. For sake of clarity, we first recall this standard transformation, and then, we show how to fix the problem of Euler singularities by introducing further coordinates, in which, $c_1$ and $c_2$ remain pure control constraints. In turn, this allows us to locally apply standard indirect methods to solve (**GOGP**), by employing a simplified version of -.\
### Reduction to Pure Control Constraints via Local Coordinates {#firstChart}
We denote by $(r,L,\ell)$ the spherical coordinates of the center of mass $G$ of the vehicle w.r.t. $(\bm{I},\bm{J},\bm{K})$, where $r$ is the distance between $O$ and $G$ (Section \[SectionForces\]), $L$ the latitude and $\ell$ the longitude. We denote $(\bm{e}_L,\bm{e}_{\ell},\bm{e}_r)$ the *North-East-Down* (NED) frame, a moving frame centered at $G$, where $-\bm{e}_r$ is the local vertical direction, $(\bm{e}_L,\bm{e}_{\ell})$ is the local horizontal plane and $\bm{e}_L$ is pointing to the North. By definition, we have $$\begin{aligned}
\begin{cases}
\bm{e}_L = -\sin(L) \cos({\ell}) \bm{I} - \sin(L) \sin({\ell}) \bm{J} + \cos(L) \bm{K} \\
\bm{e}_{\ell} = -\sin({\ell}) \bm{I} + \cos({\ell}) \bm{J} \\
\bm{e}_r = -\cos(L) \cos({\ell}) \bm{I} - \cos(L) \sin({\ell}) \bm{J} - \sin(L) \bm{K}
\end{cases}\end{aligned}$$ for which $\bm{r} = -r \bm{e}_r$ and we have $$\label{derNED}
\begin{split}
&\dot{\bm{e}}_L = - \dot{\ell} \sin(L) \bm{e}_{\ell} + \dot{L} \bm{e}_r \ , \ \dot{\bm{e}}_{\ell} = \dot{\ell} \sin(L) \bm{e}_L + \dot{\ell} \cos(L) \bm{e}_r \\
&\dot{\bm{e}}_r = -\dot{L} \bm{e}_L - \dot{\ell} \cos(L) \bm{e}_{\ell} \ .
\end{split}$$ Then, the transformation from the frame $(\bm{I},\bm{J},\bm{K})$ to the frame $(\bm{e}_L,\bm{e}_{\ell},\bm{e}_r)$ is the following rotation (i.e., a mapping in $SO(3) = \{ R \in GL_3(\mathbb{R}) : R^{\top} R = I \ , \ \textnormal{det}(R) = 1 \})$ $$R(L,{\ell}) := \left( \begin{array}{ccc}
-\sin(L) \cos({\ell}) & -\sin(L) \sin({\ell}) & \cos(L) \\
-\sin({\ell}) & \cos({\ell}) & 0 \\
-\cos(L) \cos({\ell}) & -\cos(L) \sin({\ell}) & -\sin(L) \end{array} \right) .$$
[r]{}[0.2]{} {width="18.00000%"}
To obtain $c_1$ and $c_2$ as pure control constraints, further coordinates for the velocity of the vehicle must be introduced. Using the classical formulation in the azimuth/path angle coordinates (see, e.g., [@bonnard2003optimal]), the *first velocity frame* $(\bm{i}_1,\bm{j}_1,\bm{k}_1)$ is
$$\begin{aligned}
\label{frame1}
\begin{cases}
\displaystyle \bm{i}_1 := \frac{\bm{v}}{v} = \cos(\gamma) \cos(\chi) \bm{e}_L + \cos(\gamma) \sin(\chi) \bm{e}_{\ell} - \sin(\gamma) \bm{e}_r \medskip \\
\bm{j}_1 := -\sin(\gamma) \cos(\chi) \bm{e}_L - \sin(\gamma) \sin(\chi) \bm{e}_{\ell} - \cos(\gamma) \bm{e}_r \\
\bm{k}_1 := -\sin(\chi) \bm{e}_L + \cos(\chi) \bm{e}_{\ell}
\end{cases}\end{aligned}$$ where we denote $v = \| \bm{v} \|$. Therefore, the rotation from the frame $(\bm{e}_L,\bm{e}_{\ell},\bm{e}_r)$ to the frame $(\bm{i}_1,\bm{j}_1,\bm{k}_1)$ is $$R_a(\gamma,\chi) = \left( \begin{array}{ccc}
\cos(\gamma) \cos(\chi) & \cos(\gamma) \sin(\chi) & -\sin(\gamma) \\
-\sin(\gamma) \cos(\chi) & -\sin(\gamma) \sin(\chi) & -\cos(\gamma) \\
-\sin(\chi) & \cos(\chi) & 0 \end{array} \right) .$$
It is important to note that $(r,L,{\ell},v,\gamma,\chi)$ represent local coordinates for the dynamics of (**GOGP**). In the context of differential geometry, this means that there exists a local chart of $\mathbb{R}^6 \setminus \{ 0 \}$ whose coordinates are exactly $(r,L,{\ell},v,\gamma,\chi)$. Indeed, denote $U = \Big[ (0,\infty) \times \left( -\frac{\pi}{2} , \frac{\pi}{2} \right) \times ( -\pi , \pi ) \Big]^2$ and define the mapping $\varphi^{-1}_a : U \longrightarrow \mathbb{R}^6 \setminus \{ 0 \}$ such that $$\begin{gathered}
\label{map1}
\varphi^{-1}_a(r,L,{\ell},v,\gamma,\chi) = \bigg( r \cos(L) \cos({\ell}) , r \cos(L) \sin({\ell}) , \\
r \sin(L) , R^{\top}(L,{\ell}) R^{\top}_a(\gamma,\chi) \Bigg( \begin{array}{c} v \\ 0 \\ 0 \end{array} \Bigg) \bigg) .\end{gathered}$$ This mapping is an injective immersion and its inverse is a local chart of $\mathbb{R}^6 \setminus \{ 0 \}$ (in the sense of differential geometry) when restricted to $U_a := \varphi^{-1}_a(U)$, which is an open subset of $\mathbb{R}^6 \setminus \{ 0 \}$. Exploiting (\[derNED\]) and the definition of $(\bm{i}_1,\bm{j}_1,\bm{k}_1)$, in the coordinates provided by (\[map1\]), the derivative of $\bm{v}$ is $$\begin{gathered}
\label{velocity1}
\dot{\bm{v}} = \dot{v} \bm{i}_1 + \left( v \dot{\gamma} - \frac{v^2}{r} \cos(\gamma) \right) \bm{j}_1 + \\
\left( v \cos(\gamma) \dot{\chi} - \frac{v^2}{r} \cos^2(\gamma) \sin(\chi) \tan(L) \right) \bm{k}_1 \ .\end{gathered}$$
Finally, we introduce new control variables (which are functions of the original control $\bm{u}$), under which, $c_1$ and $c_2$ can be reformulated as pure control constraints. For this, define the new control $\bm{w} = R_a(\gamma,\chi) R(L,{\ell}) \bm{u}$. Then, the constraint functions become (by using the fact that $v > 0$ by definition) $$\label{localConstraint1}
c_1(\bm{w}) = -w_1 \ , \ c_2(\bm{w}) = \frac{w^2_2 + w^2_3}{\sin^2(\alpha_{\max})} - 1 \ , \ \bm{w} \in S^2$$ which are pure control constraints. Denote the normalized drag and lift coefficients respectively by $d = \frac{1}{2 m} \rho S C_{D_0}$, $c_m = \frac{1}{2 m} \rho S C_{L_{\alpha}}$ and the efficiency factor by $\eta > 0$ (see, e.g., [@pucci2015nonlinear; @carlucci2018ballistics; @pepy2014indirect]). By introducing $\omega(t) = \frac{f_T(t)}{m(t) v(t)} + v(t) c_m(t) > 0$, with the help of (\[velocity1\]), the local evaluation of the dynamics in (\[guidanceDyn\]) using the local chart $\varphi_a$ immediately gives $$\begin{aligned}
\label{dynFirst}
\begin{cases}
\dot{r} = v\sin(\gamma) \ , \ \dot{L} = \displaystyle \frac{v}{r} \cos(\gamma) \cos(\chi) \ , \ \dot{\ell} = \displaystyle \frac{v}{r} \frac{\cos(\gamma) \sin(\chi)}{\cos(L)} \medskip \\
\dot{v} = \displaystyle \frac{f_T}{m} w_1 -\left(d + \eta c_m (w^2_2 + w^2_3) \right) v^2 - g \sin(\gamma) \medskip \\
\dot{\gamma} = \displaystyle \omega w_2 + \left(\frac{v}{r} - \frac{g}{v}\right) \cos(\gamma) \medskip \\
\dot{\chi} = \displaystyle \frac{\omega}{\cos(\gamma)} w_3 + \frac{v}{r} \cos(\gamma) \sin(\chi) \tan(L) \ .
\end{cases}\end{aligned}$$ It is crucial to note that $\gamma = \pm\pi/2$ are singularities for .
The previous computations allow us to reformulate (**GOGP**) by introducing a new optimal control problem, named (**GOGP**)$_a$, which si locally equivalent to (**GOGP**) and has only pure control constraints: this represents one of the two sought optimal control problems on which we run classical indirect methods. It consists of minimizing the cost $$C_a(T,r,L,{\ell},v,\gamma,\chi,\bm{w}) = g(T,\varphi^{-1}_a(r,L,{\ell},v,\gamma,\chi)(T))$$ subject to the dynamics (\[dynFirst\]) and the control constraints (\[localConstraint1\]).\
### Additional Coordinates to Manage Eulerian Singularities {#secondChart}
Even if formulation (**GOGP**)$_a$ is widely used in the aerospace community, it prevents one from completely describing the original problem (**GOGP**) because of its local nature. Indeed, in several situations, demanding performance criteria (costs $C$) and onerous missions (final conditions $M$) force optimal trajectories to pass through points that do not lie within the domain of the local chart $\varphi_a$ (i.e., $U_a$), and then, by exploiting merely (**GOGP**)$_a$ either the optimality could be lost or, in the worst case, the numerical computations may fail.
[r]{}[0.24]{} {width="25.00000%"}
Here, the novelty consists of introducing another set of coordinates that covers the singularities (with respect to the path angle $\gamma$) of chart $(U_a,\varphi_a)$ in which the constraints $c_1$ and $c_2$ are pure control constraints, as provided by expressions (\[localConstraint1\]).
For this, by mimicking the previous case, we introduce a new, *second velocity frame* $(\bm{i}_2,\bm{j}_2,\bm{k}_2)$, defined as
$$\begin{aligned}
\label{frame2}
\begin{cases}
\displaystyle \bm{i}_2 = \frac{\bm{v}}{v} = \cos(\theta) \sin(\phi) \bm{e}_L + \sin(\theta) \bm{e}_{\ell} + \cos(\theta) \cos(\phi) \bm{e}_r \medskip \\
\bm{j}_2 = -\sin(\theta) \sin(\phi) \bm{e}_L + \cos(\theta) \bm{e}_{\ell} - \sin(\theta) \cos(\phi) \bm{e}_r \\
\bm{k}_2 = -\cos(\phi) \bm{e}_L + \sin(\phi) \bm{e}_r
\end{cases}\end{aligned}$$ and the transformation (rotation) from the frame $(\bm{e}_L,\bm{e}_{\ell},\bm{e}_r)$ to the frame $(\bm{i}_2,\bm{j}_2,\bm{k}_2)$ is given by $$R_b(\theta,\phi) = \left( \begin{array}{ccc}
\cos(\theta) \sin(\phi) & \sin(\theta) & \cos(\theta) \cos(\phi) \\
-\sin(\theta) \sin(\phi) & \cos(\theta) & -\sin(\theta) \cos(\phi) \\
-\cos(\phi) & 0 & \sin(\phi) \end{array} \right) .$$
The new local chart $(U_b,\varphi_b)$ is given by its domain $U_b = \varphi^{-1}_b(U)$ (see Section \[firstChart\] for the definition of $U$) and $$\begin{gathered}
\varphi^{-1}_b(r,L,{\ell},v,\theta,\phi) = \bigg( r \cos(L) \cos({\ell}) , r \cos(L) \sin({\ell}) , \\
r \sin(L) , R^{\top}(L,{\ell}) R^{\top}_b(\theta,\phi) \Bigg( \begin{array}{c} v \\ 0 \\ 0 \end{array} \Bigg) \bigg) .\end{gathered}$$ This new local chart covers the singularities with respect to the path angle $\gamma$ of the local chart $(U_a,\varphi_a)$. In these new coordinates, the derivative of the velocity is $$\begin{gathered}
\label{velocity2}
\dot{\bm{v}} = \dot{v} \bm{i}_2 + \bigg( v \dot{\theta} - \frac{v^2}{r} \sin(\theta) \big( \cos(\phi) + \sin(\phi) \tan(L) \big) \bigg) \bm{j}_2 \\
+ \bigg( \frac{v^2}{r} \cos^2(\theta) \Big( \sin(\phi) + \tan^2(\theta) \big( \sin(\phi) - \tan(L) \cos(\phi) \big) \Big) \\
+ v \dot{\phi} \cos(\theta) \bigg) \bm{k}_2 \ .\end{gathered}$$
As in the previous case, we now introduce new control variables (which are complementary to the local control $\bm{w}$), by defining $\bm{z} = R_b(\theta,\phi) R(L,{\ell}) \bm{u}$. Simple computations show that the constraints $c_1$ and $c_2$ are given in this local chart by $$\label{localConstraint2}
c_1(\bm{z}) = -z_1 \ , \ c_2(\bm{z}) = \frac{z^2_2 + z^2_3}{\sin^2(\alpha_{\max})} - 1 \ , \ \bm{z} \in S^2 \ .$$ Using the same notations as in the previous case, with the help of expression (\[velocity2\]), the local evaluation of the dynamics in (\[guidanceDyn\]) by using the local chart $\varphi_b$ immediately gives $$\begin{aligned}
\label{dynSecond}
\begin{cases}
\dot{r} = -v\cos(\theta)\cos(\phi) \ , \ \dot{L} = \displaystyle \frac{v}{r} \cos(\theta) \sin(\phi) \ , \ \dot{\ell} = \displaystyle \frac{v}{r} \frac{\sin(\theta)}{\cos(L)} \medskip \\
\dot{v} = \displaystyle \frac{f_T}{m} z_1 -\left(d + \eta c_m (z^2_2 + z^2_3) \right) v^2 + g \cos(\theta) \cos(\phi) \medskip \\
\dot{\theta} = \displaystyle \omega z_2 + \frac{v}{r} \sin(\theta) \Big( \cos(\phi) + \sin(\phi) \tan(L) \Big) - \frac{g}{v} \sin(\theta) \cos(\phi) \medskip \\
\dot{\phi} = \displaystyle -\frac{\omega}{\cos(\theta)} z_3 + \frac{v}{r} \cos(\theta) \bigg( \sin(\phi) + \tan^2(\theta) \Big( \sin(\phi) \medskip \\
\displaystyle \hspace{5ex} - \tan(L) \cos(\phi) \Big) \bigg) - \frac{g}{v} \frac{\sin(\phi)}{\cos(\theta)} \ .
\end{cases}\end{aligned}$$
We define a second optimal control problem, named (**GOGP**)$_b$, which is locally equivalent to (**GOGP**) and has only pure control constraints: this represents the second sought optimal control problem on which we run classical indirect methods. It consists of minimizing the cost $$C_b(T,r,L,{\ell},v,\theta,\phi,\bm{z}) = g(T,\varphi^{-1}_b(r,L,{\ell},v,\theta,\phi)(T))$$ subject to the dynamics (\[dynSecond\]) and the control constraints (\[localConstraint2\]).
\[remN\] The mappings $\varphi^{-1}_a : U \rightarrow \mathbb{R}^6 \setminus \{ 0 \}$, $\varphi^{-1}_b : U \rightarrow \mathbb{R}^6 \setminus \{ 0 \}$ are not defined respectively for the values $\chi = \pi$, $\phi = \pi$: these singularities can be covered by extending $\varphi^{-1}_a$ and $\varphi^{-1}_b$ also on $\Big[ (0,\infty) \times \left( -\frac{\pi}{2} , \frac{\pi}{2} \right) \times ( 0 , 2 \pi ) \Big]^2$. Nevertheless, the framework of this paper concerns launch vehicles able to cover bounded distances (in the region of one hundred kilometers). From these remarks, without loss of generality, we define the configuration manifold of (**GOGP**) to be $N := U_a \cup U_b$.
Equivalence between Global and Local Formulations {#secChange}
-------------------------------------------------
From the previous sections, it is clear that, within the open set $U_a \subseteq \mathbb{R}^6 \setminus \{ 0 \}$, (**GOGP**) is equivalent to (**GOGP**)$_a$ while, within the open set $U_b \subseteq \mathbb{R}^6 \setminus \{ 0 \}$, (**GOGP**) is equivalent to (**GOGP**)$_b$. However, it is not clear whether the PMP formulation related to (**GOGP**), which is a problem with mixed control-state constraints, is equivalent respectively to the dual formulation of (**GOGP**)$_a$ locally within $U_a$, and with the dual formulation of (**GOGP**)$_b$ locally within $U_b$, which are problems with pure control constraints. More precisely, we have a priori three different tuples of multipliers, namely:
- $(\bm{p}(\cdot),p^0,\mu_1(\cdot),\mu_2(\cdot))$ related to (**GOGP**);
- $(p_a(\cdot),p^0_a)$ related to (**GOGP**)$_a$;
- $(p_b(\cdot),p^0_b)$ related to (**GOGP**)$_b$.
Nothing ensures that $p_a(\cdot)$, $p_b(\cdot)$ are related to $\bm{p}(\cdot)$ within $U_a$, $U_b$, respectively. In such situation, recasting the analysis of necessary conditions for optimality and related indirect methods from (**GOGP**) to (**GOGP**)$_a$, (**GOGP**)$_b$ may cause inconsistencies because, a priori, the adjoint vectors $p_a(\cdot)$, $p_b(\cdot)$, coming from the local formulations, evolve independently.
We fix this gap by showing that $p_a(\cdot)$, $p_b(\cdot)$ can be consistently related to $\bm{p}(\cdot)$, which will justify the study and the development of indirect methods for (**GOGP**)$_a$, (**GOGP**)$_b$ to solve (**GOGP**). In particular, we prove that it is always possible to choose the previous multipliers so that the local projections of $(\bm{p}(\cdot),p^0)$ onto charts $(U_a,\varphi_a)$ and $(U_b,\varphi_b)$ are equivalent respectively to $(p_a(\cdot),p^0_a)$ and to $(p_b(\cdot),p^0_b)$.
\[mainTheo\] *Consider the manifold $N = U_a \cup U_b \subseteq \mathbb{R}^6 \setminus \{ 0 \}$ of all possible scenarios for (**GOGP**). Suppose that $(\bm{r}(\cdot),\bm{v}(\cdot),\bm{u}(\cdot))$ is an optimal solution for (**GOGP**) in $[0,T]$. There exist multipliers $(\bm{p}(\cdot),p^0,\mu_1(\cdot),\mu_2(\cdot))$ satisfying the PMP formulation (\[adjointSystem\])-(\[transvCond1\]) and multipliers $(p_a(\cdot),p^0_a)$, $(p_b(\cdot),p^0_b)$ related to the classical PMP formulations with pure control constraints respectively of problem (**GOGP**)$_a$ and of problem (**GOGP**)$_b$, such that $p^0_a = p^0_b = p^0$ and* $$\label{adjPull}
\bm{p}(t) = \begin{cases} (\varphi_a)^*_{\varphi_a(\bm{r}(t),\bm{v}(t))} \cdot p_a(t) \quad , \quad (\bm{r}(t),\bm{v}(t)) \in U_a \\
(\varphi_b)^*_{\varphi_b(\bm{r}(t),\bm{v}(t))} \cdot p_b(t) \quad , \quad (\bm{r}(t),\bm{v}(t)) \in U_b \end{cases}$$ *where $(\cdot)^*$ is the pullback (see [@agrachev2013control] for such definitions).*
The proof of Theorem \[mainTheo\] is done in Appendix \[proofApp\]. The main idea is the following. By the PMP for problems with mixed control-state constraints, there exists a global adjoint vector $\bm{p}(\cdot)$ for (**GOGP**) which we restrict it to the domain of one of the two local charts built previously, for instance, $(U_a,\varphi_a)$. Then, via the local maximality condition (\[maxCondDer\]) and the transformation between $\bm{u}$ and $\bm{w}$ (see Sections \[firstChart\], \[secondChart\]), one shows that the covector $(\varphi^{-1}_a)^* \cdot \bm{p}(\cdot)$ satisfies the PMP formulation with pure control constraints related to (**GOGP**)$_a$.
Let us clarify how one can exploit this result to solve (**GOGP**) by indirect methods. Assume to have an optimal solution $(\bm{r}(\cdot),$ $\bm{v}(\cdot),$ $\bm{u}(\cdot))$ for (**GOGP**) in $[0,T]$. Without loss of generality, we can assume that $(\bm{r},\bm{v})(0) \in U_a$. If a guess for the optimal value of $p_a(0)$ (or equivalently of $\bm{p}(0)$, see ) is known, we can solve (**GOGP**) by running an indirect method on (**GOGP**)$_a$ starting from $p_a(0)$. Suppose that, at a given time $\tau_1 \in (0,T)$, the optimal trajectory is such that $(\bm{r},\bm{v})(\tau_1) \in U_b \setminus U_a$, i.e., our solution crosses a *singular region* of the first local chart. Then, Theorem \[mainTheo\] allows us to stop the numerical computations at a time $\tau_2 < \tau_1$ such that $(\bm{r},\bm{v})(\tau_2) \in U_a \cap U_b$ and then run an indirect method on (**GOGP**)$_b$ starting from $p_b(\tau_2) = (\varphi_a \circ \varphi^{-1}_b)^*_{\varphi_a(\bm{r}(\tau_2),\bm{v}(\tau_2))} p_a(\tau_2)$ (see ), therefore avoiding the singularity related to $\varphi_a$ when reaching the point $(\bm{r},\bm{v})(\tau_1) \in U_b \setminus U_a$ (see Figure \[fig:charts\] below). This procedure can be iterated every time a jump from $U_a$ to $U_b$ (as well as a jump from $U_b$ to $U_a$) occurs in the optimal trajectory. The adjoint vector related to (**GOGP**) is recovered thanks to (\[adjPull\]). This methodology allows one to describe global optimal solutions for any feasible mission related to problem (**GOGP**).
It is worth noting that, even if modifying indirect methods by implementing the change of coordinates introduces further computations, the numerical transformation between local adjoint vectors takes a negligible part of the total computational time an indirect method needs to converge (as simulations show, see Section \[secNumerical\]), justifying our approach.
![Optimal trajectory crossing the domains of the two local charts.[]{data-label="fig:charts"}](mappings.jpg){width="25.00000%"}
Optimal Control as Functions of the Adjoint Vectors {#controlSection}
---------------------------------------------------
In the previous section, we showed (by Theorem \[mainTheo\]) that implementing indirect methods on (**GOGP**) is equivalent to developing indirect methods for the local PMP formulations related to (**GOGP**)$_a$ and to (**GOGP**)$_b$. In this section, we provide optimal controls as functions of the states and the adjoint vectors, by adopting the previous formalism based on (**GOGP**)$_a$, (**GOGP**)$_b$. This provides formulations needed to run indirect methods on problems (**GOGP**)$_a$, (**GOGP**)$_b$.\
Let $(\bm{r}(\cdot),\bm{v}(\cdot),\bm{u}(\cdot))$ be an optimal solution for (**GOGP**) in $[0,T]$, and $\bm{p}(\cdot)$, $\bm{p}_a(\cdot) = (p^a_r,p^a_L,p^a_{\ell},p^a_v,p_{\gamma},p_{\chi})(\cdot)$ and $\bm{p}_b(\cdot) = (p^b_r,p^b_L,p^b_{\ell},p^b_v,p_{\theta},p_{\phi})(\cdot)$ be the related adjoint vectors respectively for (**GOGP**), (**GOGP**)$_a$ and for (**GOGP**)$_b$ as in Theorem \[mainTheo\] (see also Section \[localPMP\] for the definition of the local problems). As pointed out previously, thanks to Theorem \[mainTheo\], the computation of the optimal control $\bm{u}$ can be achieved by focusing on the optimal values of the local controls $\bm{w}$, $\bm{z}$, which are the projections of $\bm{u}$ onto $U_a$, $U_b$, respectively (see also Section \[localPMP\]). Hereafter, when clear from the context, we skip the dependence on $t$ to keep better readability. For sake of clarity, we denote $C_a := p^a_v \frac{f_T}{m}$, $C_b := p^b_v \frac{f_T}{m}$, $D_a := p^a_v \eta c_m v^2$ and $D_b := p^b_v \eta c_m v^2$ (see Section \[localPMP\] for notations).
Expressions for optimal controls $\bm{w}$, $\bm{z}$ as functions of the local states and the local adjoint vectors can be achieved by studying the local versions of the Maximality Condition . From the PMP for pure control constraints applied to (**GOGP**)$_a$, (**GOGP**)$_b$ (resulting as a special case of conditions -, see also [@pontryagin1987mathematical]), locally almost everywhere where they are defined, respectively related to (**GOGP**)$_a$ and to (**GOGP**)$_b$, these maximality conditions are given by
$$\begin{gathered}
\label{firstHam}
\displaystyle \bm{w}(t) = {\textnormal{argmax}}\Bigg\{ C_a w_1 - D_a (w^2_2 + w^2_3) + p_{\gamma} \omega w_2 + p_{\chi} \frac{\omega}{\cos(\gamma)} w_3 \\
w^2_1 + w^2_2 + w^2_3 = 1 \ , \ w_1 \ge 0 \ , \ w^2_2 + w^2_3 \le \sin^2(\alpha_{\max}) \Bigg\}\end{gathered}$$
$$\begin{gathered}
\label{secondHam}
\displaystyle \bm{z}(t) = {\textnormal{argmax}}\Bigg\{ C_b z_1 - D_b (z^2_2 + z^2_3) + p_{\theta} \omega z_2 - p_{\phi} \frac{\omega}{\cos(\theta)} z_3 \\
z^2_1 + z^2_2 + z^2_3 = 1 \ , \ z_1 \ge 0 \ , \ z^2_2 + z^2_3 \le \sin^2(\alpha_{\max}) \Bigg\} \ .\end{gathered}$$
Solving these maximization conditions may lead to either *regular* or *nonregular controls*, depending on the value of $(p_{\gamma}(\cdot),p_{\chi}(\cdot))$, $(p_{\theta}(\cdot),p_{\phi}(\cdot))$ on non-zero measure subsets of $[0,T]$. Indeed, by definition, regular controls are the regular points of *the end-point mapping* while nonregular controls are its critical points (see, e.g., [@trelat2012optimal]). Then, for (**GOGP**), regular controls consist of controls whose extremal satisfies either $p_{\gamma}|_J(\cdot) \neq 0$ or $p_{\chi}|_J(\cdot) \neq 0$, within a non-zero measure subset $J \subseteq [0,T]$, if the system travels along the first local chart $(U_a,\varphi_a)$ within $J$. On the other hand, regular controls satisfy $p_{\theta}|_J(\cdot) \neq 0$ or $p_{\phi}|_J(\cdot) \neq 0$ if the system covers the second local chart $(U_b,\varphi_b)$ within $J$. Conversely, nonregular controls consist of controls for which there exists a non-zero measure subset $J \subseteq [0,T]$ such that $p_{\gamma}|_J(\cdot) = p_{\chi}|_J(\cdot) = 0$ in the first chart, and $p_{\theta}|_J(\cdot) = p_{\phi}|_J(\cdot) = 0$ in the second chart.
We analyze separately regular and nonregular controls.\
### Regular Controls
Suppose that, locally within a non-zero measure subset $J \subseteq [0,T]$, either $p_{\gamma}|_J(\cdot) \neq 0$ or $p_{\chi}|_J(\cdot) \neq 0$ if the system travels along the first chart $(U_a,\varphi_a)$ within $J$. Otherwise, $p_{\theta}|_J(\cdot) \neq 0$ or $p_{\phi}|_J(\cdot) \neq 0$. In this case, regular controls appear. Explicit expressions for these are easily derived from (\[firstHam\]), (\[secondHam\]) by using the Karush-Kuhn-Tucker conditions (see, e.g., [@NoceWrig06]), if we assume the following:
\[assumpSimply\] *For points $(\varepsilon,x) \in \mathbb{R}_+ \times \mathbb{R}$ such that $(1 + \varepsilon) x^2 \le \sin^2(\alpha_{\max})$, where $0 < \alpha_{\max} \le \pi / 6$ is constant, the following holds: $\sqrt{1 - (1 + \varepsilon) x^2} \cong \Big( 1 - (1 + \varepsilon) x^2 / 2 \Big)$.*
This assumption is not limiting. Indeed, for most of the applications that are based on the dynamical model developed for (**GOGP**), the maximal angle of attack $\alpha_{\max}$ is actually lower than $\pi / 6$ (because of controllability issues). Moreover, this assumption has already implicitly been used to recover the explicit expressions of the drag and the lift listed in Section \[SectionForces\]. Under Assumption \[assumpSimply\], we provide the computations for the explicit expressions of regular controls in Appendix \[regularApp\]. Note that regular controls are well defined in each of the two charts $(U_a,\varphi_a)$, $(U_b,\varphi_b)$, but their local expressions reach singular values as soon as the optimal trajectory gets close to the boundary of $U_a$, $U_b$, respectively.\
### Nonregular Controls {#secSingular}
In some cases, locally within a non-zero measure subset $J \subseteq [0,T]$, it may happen that $p_{\gamma}|_J(\cdot) = p_{\chi}|_J(\cdot) = 0$ in the first local chart, or $p_{\theta}|_J(\cdot) = p_{\phi}|_J(\cdot) = 0$ in the second local chart. The control is then nonregular and the evaluation of optimal controls is harder to achieve than in the regular case. Here, (\[firstHam\]) and (\[secondHam\]) are
$$\begin{gathered}
\label{singularHam}
\displaystyle \bm{w}(t) = {\textnormal{argmax}}\Big\{ C_a w_1 - D_a (w^2_2 + w^2_3) \, \mid \,
w^2_1 + w^2_2 + w^2_3 = 1 , \\
w_1 \ge 0 , w^2_2 + w^2_3 \le \sin^2(\alpha_{\max}) \Big\}\end{gathered}$$
$$\begin{gathered}
\label{singularHam2}
\displaystyle \bm{z}(t) = {\textnormal{argmax}}\Big\{ C_b z_1 - D_b (z^2_2 + z^2_3) \ \mid \ z^2_1 + z^2_2 + z^2_3 = 1 , \\
z_1 \ge 0 \ , \ z^2_2 + z^2_3 \le \sin^2(\alpha_{\max}) \Big\} \ .\end{gathered}$$
The Karush-Kuhn-Tucker conditions are no more helpful because, depending on the value of $C_a$ or $C_b$, many uncountable values of $(w_2,w_3)$ or $(z_2,z_3)$ are optimal. Instead, a geometric study is required. It is in the case of nonregular controls that Assumption \[assM\] becomes particularly useful to manage hard computations, as well as the following one:
\[assVel\] *Let $J \subseteq [0,T]$ be a non-zero measure subset. Along $J$, any optimal trajectory associated with a nonregular control in $J$ satisfies (see Section \[localPMP\] for notations)* $$\| \bm{v} \|^2 > \frac{3}{2} g(\bm{r}) h_r \bigg( \sqrt{1 + \frac{4}{9} \frac{1}{g(\bm{r}) h_r} \left( \frac{f_T}{m d} \right) } - 1 \bigg) \ .$$
It is important to note that, for our applications, the magnitude of the velocities of the vehicles is in general large enough when $f_T > 0$, so that Assumption \[assVel\] is always satisfied, as numerical simulations confirm. In particular, this assumption is required only for nonregular arcs, i.e., in the case of regular optimal controls no boundaries on the velocities are imposed. Under Assumption \[assVel\], we provide the expressions of nonregular optimal controls in Appendix \[singularApp\], which, together with the results above, lead straightforwardly to the following:
\[regSing\] *Under Assumption \[assumpSimply\], regular optimal controls for (**GOGP**) are well-defined and have univocal explicit expressions. On the other hand, under Assumption \[assM\] and Assumption \[assVel\], any nonregular optimal control for (**GOGP**) is well-defined and has a univocal explicit expression.*
It is worth noting that nonregular controls are not very common: the absence of nonregular controls has been widely studied in the context of optimal control and it has been shown that, under appropriate assumptions, regular controls appear *almost always* (see, e.g., [@bonnard1997generic; @chitour2008singular]). Running Monte-Carlo simulations on (**GOGP**) for many different realistic scenarios in the context of missile interception problems, we have never found nonregular controls (see Section \[secBatch\]). However, for sake of completeness, we have provided full descriptions of both regular and nonregular controls so that indirect methods for (**GOGP**) are always correctly defined (see Proposition \[regSing\]).
Numerical Indirect Method for General Optimal Guidance Problems {#sectHomotopy}
===============================================================
In the previous sections, we showed that (**GOGP**) can be locally converted into two optimal control problems with pure control constraints, and that the PMP formulations of such problems locally match with the original global one. This allows one to run indirect methods on these local problems to provide solutions for (**GOGP**) with transmission conditions. In this section, we describe a general numerical scheme to solve (**GOGP**) which combines shooting methods with homotopy procedures (see, e.g., [@allgower2003introduction]). The main idea consists of introducing a family of problems parametrized by some quantity $\bm{\lambda}$, so that: 1) the problem related to $\bm{\lambda} = 0$, named *problem of order zero*, is simple to solve by shooting methods; 2) we solve (**GOGP**) by combining shooting methods with an iterative procedure that makes $\bm{\lambda}$ vary with continuity, starting from the solution obtained for $\bm{\lambda} = 0$. We begin the section by discussing the formulation of the problem of order zero.
Designing the Problem of Order Zero {#subSect0}
-----------------------------------
The problem of order zero, from which the iterative shooting path starts, should be, on one hand, handy to solve via basic shooting methods and, on the other hand, as close as possible to (**GOGP**) to efficiently recover a solution for the original problem by some homotopy. It is worth noting that two difficulties prevent shooting methods to easily converge: 1) the presence of the gravity, the thrust and Earth’s curvature that considerably complexify the dynamics; 2) the more demanding the mission is (which is represented by the cost $g$ and the target $M$), the less intuition one has on the structure of optimal solutions. These facts led us to derive the following heuristic problem of order zero for (**GOGP**): denoted by (**GOGP**)$_0$, it consists of minimizing the simplified cost $$C_0(T,\bm{r}(\cdot),\bm{v}(\cdot),\bm{u}(\cdot)) = g_0(T,\bm{r}(T),\bm{v}(T))$$ subject to the simplified dynamics $$\begin{aligned}
\begin{cases}
\dot{\bm{r}}(t) = \bm{v}(t) \quad , \quad \dot{\bm{v}}(t) = \bm{f}_0(t,\bm{r}(t),\bm{v}(t),\bm{u}(t)) \medskip \\
(\bm{r}(t),\bm{v}(t)) \in N \quad, \quad \bm{u}(t) \in S^2 \medskip \\
\displaystyle \bm{r}(0) = \bm{r}_0 \ , \ \bm{v}(0) = \bm{v}_0 \quad , \quad (\bm{r}(T),\bm{v}(T)) \in M_0 \subseteq N \medskip \\
\displaystyle c_1(\bm{v}(t),\bm{u}(t)) \le 0 \quad , \quad c_2(\bm{v}(t),\bm{u}(t)) \le 0 \ .
\end{cases}\end{aligned}$$ Here, $g_0$, $\bm{f}_0$ and $M_0$ represent simplified versions for the original cost $g$, the original dynamics $f$, and the original target $M$, respectively. Consistently with our remarks above, we define the simplified dynamics $\bm{f}_0$ such that the contributions of the gravity, the thrust and Earth’s curvature are removed from the original dynamics, that is $$\label{simplyDyn}
\bm{f}_0(t,\bm{r},\bm{v},\bm{u}) := \bm{f}(t,\bm{r},\bm{v},\bm{u}) - \bigg( \frac{\bm{T}(t,\bm{u})}{m} + \frac{\bm{g}(\bm{r})}{m} - \bm{\omega}_{\textnormal{NED}}(\bm{r},\bm{v}) \times \bm{v} \bigg)$$ where $\bm{\omega}_{\textnormal{NED}}(\bm{r},\bm{v})$ represents the angular velocity of the NED frame $(\bm{e}_L,\bm{e}_l,\bm{e}_r)$ w.r.t. the inertial frame $(\bm{I},\bm{J},\bm{K})$ (the sign minus in front of it is consistent with our convention, see Section \[firstChart\]). It is important to evaluate (\[simplyDyn\]) only by using charts $(U_a,\varphi_a)$, $(U_b,\varphi_b)$, otherwise its explicit expression could appear more complex than the original dynamics, especially because of the presence of the term $\bm{\omega}_{\textnormal{NED}}\times \bm{v}$.
From what we pointed out above, the simplified cost $g_0$ and target $M_0$ should be designed such that it is easy to make a shooting method converge for (**GOGP**)$_0$. This may require that, for instance, we choose $g_0$, $M_0$ so that optimal trajectories for (**GOGP**)$_0$ do not meet Euler singularities, i.e., they lie entirely within the chart domain $U_a$ (or $U_b$), or also, that optimal strategies are not of bang-bang type. In a very general context, it may be not evident to provide appropriate $g_0$, $M_0$ such that (**GOGP**)$_0$ is easy to solve by shooting methods, and this strongly depends on the nature of the original mission. The engineer intuition is often crucial at this step. Also, exploiting techniques from geometric control or dynamical system theory applied to mission design (see, e.g., [@trelat2012optimal]) may lead to design relevant formulations for $g_0$, $M_0$. Hereafter, we show how to define (**GOGP**)$_0$ for missile interception (see Section \[sectAppl\]).
Solving (**GOGP**)$_0$ by standard shooting methods leads to a solution $(\bm{r}_0(\cdot),\bm{v}_0(\cdot),\bm{u}_0(\cdot))$ for (**GOGP**)$_0$ with adjoint variables $(\bm{p}_0(\cdot),p^0_0)$. Thanks to Theorem \[mainTheo\], from now on, we do not report the multipliers related to the mixed constraints.
Homotopies Initialized by the Problem of Order Zero {#sectHomotopy2}
---------------------------------------------------
Now that the problem of order zero has been defined, we provide optimal strategies for (**GOGP**) by iteratively solving a sequence of shootings problems indexed by some parameter $\bm{\lambda}$, using the adjoint variables related to (**GOGP**)$_0$.
We first introduce the family of problems, denoted by (**GOGP**)$_{\bm{\lambda}}$, depending on the parameter $\bm{\lambda}$. For every $\bm{\lambda} = (\lambda_1,\lambda_2) \in [0,1]^2$, the optimal control problem (**GOGP**)$_{\bm{\lambda}}$ consists of minimizing the parametrized cost $$C_{\bm{\lambda}}(T,\bm{r}(\cdot),\bm{v}(\cdot),\bm{u}(\cdot)) = g_{\bm{\lambda}}(T,\bm{r}(T),\bm{v}(T))$$ subject to the parametrized dynamics $$\begin{aligned}
\begin{cases}
\dot{\bm{r}}(t) = \bm{v}(t) \quad , \quad \dot{\bm{v}}(t) = \bm{f}_{\bm{\lambda}}(t,\bm{r}(t),\bm{v}(t),\bm{u}(t)) \medskip \\
(\bm{r}(t),\bm{v}(t)) \in N \quad, \quad \bm{u}(t) \in S^2 \medskip \\
\displaystyle \bm{r}(0) = \bm{r}_0 \ , \ \bm{v}(0) = \bm{v}_0 \quad , \quad (\bm{r}(T),\bm{v}(T)) \in M_{\bm{\lambda}} \subseteq N \medskip \\
\displaystyle c_1(\bm{v}(t),\bm{u}(t)) \le 0 \quad , \quad c_2(\bm{v}(t),\bm{u}(t)) \le 0 \ .
\end{cases}\end{aligned}$$ Here, the parametrized cost $g_{\bm{\lambda}}$ and dynamics $\bm{f}_{\bm{\lambda}}$ are $$\begin{gathered}
g_{\bm{\lambda}}(T,\bm{r},\bm{v}) := g_0(T,\bm{r},\bm{v}) + \lambda_1 \Big( g(T,\bm{r},\bm{v}) - g_0(T,\bm{r},\bm{v}) \Big) \\
\bm{f}_{\bm{\lambda}}(t,\bm{r},\bm{v},\bm{u}) := \bm{f}_0(t,\bm{r},\bm{v},\bm{u}) + \lambda_1 \Big( \bm{f}(t,\bm{r},\bm{v},\bm{u}) - \bm{f}_0(t,\bm{r},\bm{v},\bm{u}) \Big)\end{gathered}$$ involving only the first component of $\bm{\lambda}$, and the parametrized target $M_{\bm{\lambda}}$ is chosen such that it depends only on the second component of $\bm{\lambda}$ and satisfies $M_{\lambda_2 = 0} \equiv M_0$, $M_{\lambda_2 = 1} \equiv M$. Remark that the problem of order zero (**GOGP**)$_0$ corresponds to $\bm{\lambda} = (0,0)$ while the original (target) problem (**GOGP**) corresponds to $\bm{\lambda} = (1,1)$. In this case, the homotopy procedure consists of solving a series of shooting problems by making $\bm{\lambda}$ continuously pass from $(0,0)$ to $(1,1)$.
We propose to solve (**GOGP**) by Algorithm \[ref:algoCont\] below, continuation scheme which operates on the family of problems (**GOGP**)$_{\bm{\lambda}}$ introduced above, starting from problem (**GOGP**)$_0$.
Algorithm \[ref:algoCont\] operates a continuation via bisection method on the coordinates of parameter $\bm{\lambda}$ and is considered successful if it ends with $\bm{\lambda} = (1,1)$. If line **10** of Algorithm \[ref:algoCont\] is called frequently, the convergence rate may become slow. To prevent such behavior, acceleration steps may be considered (see, e.g., [@allgower2003introduction]). Numerical simulations show that splitting the continuation on the hard terms of the dynamics and on the mission helps obtaining better performance. The theoretical convergence of Algorithm \[ref:algoCont\] is established under appropriate assumptions. Indeed, it is known that homotopy methods may fail whenever bifurcation points, singularities or different connected components are found (see, e.g., [@allgower2003introduction; @trelat2012optimal]). However, the absence of *conjugate points* and of *abnormal minimizers* (remark that these may be different from nonregular controls) are sufficient conditions for homotopies to converge (see [@trelat2012optimal]). Algorithm \[ref:algoCont\] may be combined with numerical procedures computing conjugate points (see [@bonnard2007second]). For what concerns (**GOGP**), we have solved many different realistic scenarios via Monte-Carlo simulations for missile interception problems, showing an empirical efficiency for Algorithm \[ref:algoCont\] in such context (in particular, see Section \[secIPOPT\]).
Launch Vehicle Application: Endo-Atmospheric Missile Interception {#sectAppl}
=================================================================
In this section, we apply Algorithm \[ref:algoCont\] to solve (**GOGP**) in the context of *endo-atmospheric interception* (see, e.g., [@cottrell1971optimal]). The problem consists of steering a missile towards a given target, optimizing some criterion. We are interested in the *mid-course phase* which starts when the vehicle reaches a given threshold of the magnitude of the velocity. The target consists of a predicted interception point and, since this point may change over time, fast and accurate computations are needed.
The Optimal Interception Problem (**OIP**) consists of the specific (**GOGP**) for which the cost $g$ and the target $M$ are $$\label{interception}
\displaystyle g(T,\bm{r}(T),\bm{v}(T)) = C_1 T - \| \bm{v}(T) \|^2$$ where $C_1 \ge 0$ and the final time $T$ is either fixed or free. This cost is set up to maximize the chances to complete the mission with reasonable delays. Moreover, $M$ fixes the final position and orientation of the vehicle. Assumption \[assM\] is satisfied.
For numerical simulations, a *solid-fuel propelled missile* is employed, with the following numerical values:
- $c_m(0) = 7.5 \cdot 10^{-4} \textnormal{m}^{-1}$, $d(0) = 5 \cdot 10^{-5} \textnormal{m}^{-1}$, $\eta = 0.442$, $h_r = 7500 \textnormal{m}$, $\alpha_{\max} = \pi/6$, $q_0 = 0.025 \textnormal{s}^{-1}$, $f^0_T = 37.5 \textnormal{m} \cdot \textnormal{s}^{-2}$
- $\displaystyle \frac{q}{m(0)}(t) = \begin{cases}
q_0 \ , \ t \le 20 \\
0 \ , \ t > 20
\end{cases} , \frac{f_T}{m(0)}(t) = \begin{cases}
f^0_T \ , \ t \le 20 \\
0 \ , \ t > 20 \ .
\end{cases}$
The shooting problems in Algorithm \[ref:algoCont\] are solved using a C++ environment and *hybrd.c* [@minPACK] while a fixed time-step explicit fourth-order Runge-Kutta method is used to integrate differential equations (whose number of integration steps varies between 250 and 350). Computations are done on a system Ubuntu 12.04 (32-bit), with 7.00 Gb of RAM.
Choices for the Problem of Order Zero {#secOrderZero}
-------------------------------------
Without loss of generality, a problem of order zero (**OIP**)$_0$ for (**OIP**) can be chosen such that its optimal trajectory lies in the domain of the local chart $(U_a,\varphi_a)$. The following problem of order zero is considered (see [@bonalli2017analytical]) $$\begin{aligned}
\begin{cases}
\qquad \min \quad -v^2(T) \quad , \quad (w_2,w_3) \in \mathbb{R}^2 \bigskip \\
\dot{r} = v \sin(\gamma) \ , \ \dot{L} = \displaystyle \frac{v}{r} \cos(\gamma) \cos(\chi) \ , \ \dot{l} = \displaystyle \frac{v}{r} \frac{\cos(\gamma) \sin(\chi)}{\cos(L)} \medskip \\
\dot{v} = -(d + \eta c_m (w^2_2 + w^2_3)) v^2 \ , \ \dot{\gamma} = v c_m w_2 \ , \ \dot{\chi} = \displaystyle \frac{v c_m}{\cos(\gamma)} w_3
\end{cases}\end{aligned}$$ where, consistently with the arguments in Section \[subSect0\], the contribution of the thrust, the gravity and of $\bm{\omega}_{\textnormal{NED}} \times \bm{v}$ are removed, no constraints on the controls are imposed and $C_1 = 0$. More specifically, the target set $M_0$ can be chosen such that (**OIP**)$_0$ is feasible and the PMP applied to (**OIP**)$_0$ allows one to recover an approximated explicit guidance law which successfully initializes shooting methods for (**OIP**)$_0$. For the sake of conciseness, we do not report all details for such computations and the interested reader is referred to [@bonalli2017analytical].
![Optimal trajectories $\bm{r}(\cdot)$ and constraints $c_2(\bm{r}(\cdot),\bm{v}(\cdot))$ (in the chart $(U_a,\varphi_a)$) for the first mission, i.e., cost $T - \| \bm{v}(T) \|^2$ and target $M^1$. The dashed-yellow curves represent the explicit guidance law used to initialize a shooting on (**OIP**)$_0$ (see Section \[secOrderZero\]), whose solution is dashed-red. The dashed-brown curves represent optimal quantities for $\bm{\lambda} = (1,0)$ and the solid-blue curves for $\bm{\lambda} = (1,1)$, i.e., for the original problem.[]{data-label="figMission1"}](firstFig.png){width="65.00000%"}
![Optimal trajectories $\bm{r}(\cdot)$ and constraints $c_2(\bm{r}(\cdot),\bm{v}(\cdot))$ (in the chart $(U_a,\varphi_a)$) for the second mission, i.e., cost $T - \| \bm{v}(T) \|^2$ and target $M^2$. The dashed-yellow curves represent the explicit guidance law used to initialize a shooting on (**OIP**)$_0$ (see Section \[secOrderZero\]), whose solution is dashed-red. The dashed-brown curves represent optimal quantities for $\bm{\lambda} = (1,0)$ and the solid-blue curves for $\bm{\lambda} = (1,1)$, i.e., for the original problem. In the big black box, we show the two-dimensional projection of the trajectory for (**OIP**) onto the plane $(L \cdot r_T,r-r_T)$. The small boxes show changes of local chart.[]{data-label="figMission2"}](secondFig.png){width="65.00000%"}
Numerical Simulations on Two Realistic Missions {#secNumerical}
-----------------------------------------------
In this section, we apply Algorithm \[ref:algoCont\] to solve two realistic interception missions for (**OIP**). We consider free final time problems for which $C_1 = 1$ in . Details for the choice of the family of problems have been provided in Section \[sectHomotopy2\]. In particular, the initial conditions and the target for the problem of order zero in standard units are (in the local chart $(U_a,\varphi_a)$) $$(r-r_T,L \cdot r_T,l \cdot r_T,v,\gamma,\chi)(0) = (1000,0,0,500,0,0)$$ $$M_0 = \Big\{ (r-r_T,L \cdot r_T,l \cdot r_T) = (5000,14000,0) \ , \ (\gamma,\chi) = (-\pi/6,0) \Big\}$$ where $r_T$ is Earth’s radius. Since we fix , the two missions are unambiguously defined by , i.e., respectively by $$\begin{gathered}
M^1 = \Big\{ (r-r_T,L \cdot r_T,l \cdot r_T) = (5000,14000,-2000) \ , \\
(\gamma,\chi) = (-\pi/6,\pi/6) \Big\}\end{gathered}$$ $$\begin{gathered}
M^2 = \Big\{ (r-r_T,L \cdot r_T,l \cdot r_T) = (7900,7500,2000) \ , \\
(\gamma,\chi) = (-\pi/4,-\pi/4) \Big\}\end{gathered}$$ that are provided in standard units. The second mission is more challenging because abrupter maneuvers will be required. The parametrized target sets $M_{\lambda_2}$ for (**OIP**)$_{\bm{\lambda}}$ are convex combinations in $\lambda_2$ of $M_0$ and $M^1$, $M^2$, respectively.\
### First Mission
When solving the first mission, Algorithm \[ref:algoCont\] provides $(T,\| \bm{v}(T) \|) = (21.4,753.7)$ as optimal values (in standard units). Optimal trajectories and values for constraints are given in Figure \[figMission1\]. The computations take around 1.6 s, for which 14 iterations on $\lambda_1$ and 11 iterations on $\lambda_2$ are required. This is due to the minimal time in the cost which makes the structure of the solutions more complex. In Figure \[figMission1\] b), we see that, even if constraint $c_2$ is not satisfied by the explicit guidance law of Section \[secOrderZero\], the latter correctly initializes Algorithm \[ref:algoCont\], so that saturations on constraint $c_2$ are satisfied by the solution of the original problem.\
### Second Mission
For the second mission, considered to be more challenging, Algorithm \[ref:algoCont\] provides $(T,\| \bm{v}(T) \|) = (29.03,475.2)$ as optimal values (in standard units). Optimal trajectories and values for constraints are given in Figure \[figMission2\]. Here, the computations take around 2.3 s, where 14 iterations on $\lambda_1$ and 26 iterations on $\lambda_2$ are required. A higher number of iterations on $\lambda_2$ occurs because, when proposing to intercept a target quite close to the initial point, the vehicle is led to perform abrupt maneuvers to recover an optimal solution. Moreover, two changes of local chart as designed in Section \[secChange\] are involved in this mission. Indeed, from Figure \[figMission2\] a), we see that trajectories are close both to $\gamma = \pi/2$, singular value for $(U_a,\varphi_a)$, and to $\theta = \pi/2$, critical value for $(U_b,\varphi_b)$ (see the small black boxes in the two-dimensional projection onto $(L \cdot r_T,r-r_T)$ in Figure \[figMission2\] a)). Even if the change of coordinates is not compulsory to solve this mission, it considerably increases the performances of the algorithm. Indeed, without it, simulations on this mission would take around 8.2 s with 19 iterations on $\lambda_1$ and 121 iterations on $\lambda_2$. Other tests show that some scenarios cannot be solved without the change of local chart. This shows a glimpse of the benefits in performance that one may achieve when adopting the change of local chart provided in this paper.
Performance Test via Batch Simulations {#secBatch}
--------------------------------------
In this section, we test the efficiency of Algorithm \[ref:algoCont\] on a batch of simulations. We consider free final time problems (**OIP**) without minimal final time, i.e., cost is chosen such that $C_1 = 0$. Monte-Carlo simulations are run on missions with the same initial conditions used in the previous simulations and values for the final target uniformly, randomly chosen in $$r - r_T \in [4000,8000] \ , \ L \cdot r_T \in [14000,18000] \ ,$$ $$l \cdot r_T \in [-4000,4000] \ , \ \gamma , \chi \in [-\pi/3,\pi/3]$$ which gather realistic interception mission scenarios. We report in Table 1 the results obtained on 1000 missions.
Since the parametrized costs $C_{\lambda}$ do not change, the number of iterations on $\lambda_1$ is constant and equal to 7. The absence of minimal final time in the cost considerably improves computational times. Moreover, as mentioned at the end of Section \[secSingular\], every solution provided by Algorithm \[ref:algoCont\] on this batch consists of regular controls, i.e., nonregular controls do not appear. From the fact that 99.7% of the missions are solved, these tests empirically show that the structure of the problem of order zero that we described in Sections \[subSect0\], \[secOrderZero\] for missile interception applications preserves enough information so that shooting methods combined with homotopies efficiently converge to a solution for the original problem. Remark that, without considering the change of local chart, an additional 9.5% of missions would have failed.
[| C | C | C | C |]{} Successful missions & Average time & Average nb. of iterations on $\lambda_2$ & Changes of local chart\
99.7% & 0.85 seconds & 6 & 9.5%\
Table 1 : Average results obtained from solving (**OIP**) by Algorithm \[ref:algoCont\] on 1000 missions. The averages are computed only on successful cases.
Comparisons with State-of-the-Art Direct Methods {#secIPOPT}
------------------------------------------------
In this section, we compare results and performance of Algorithm \[ref:algoCont\] with state-of-the-art direct methods for optimization problem. We discretize (**OIP**) in time and solve the related nonlinear optimization problem in AMPL [@fourer1993ampl] combined with IpOpt [@wachter2006implementation] as nonlinear solver (see, e.g., [@gollmann2014theory; @rodrigues2017optimal; @ma2018direct]).
To have specific and fair results to compare, we run Algorithm \[ref:algoCont\] and AMPL/IpOpt on three missions, by considering fixed final time problems (**OIP**) (and with $C_1 = 0$ in ). The initial conditions are the same as in Sections \[secNumerical\], \[secBatch\], so that the missions are given by the following scenarios:
- (**SC**)$_1$ : $T = 20$, $r - r_T = 7030$, $L \cdot r_T = 9450$, $l \cdot r_T = 1400$, $\gamma, \chi = -0.55$
- (**SC**)$_2$ : $T = 23$, $r - r_T = 7465$, $L \cdot r_T = 8475$, $l \cdot r_T = 1700$, $\gamma, \chi = -0.67$
- (**SC**)$_3$ : $T = 29$, $r - r_T = 7900$, $L \cdot r_T = 7500$, $l \cdot r_T = 2000$, $\gamma, \chi = -0.79$
in standard units. These missions are listed in ascending order by difficulty, so that the last mission requires a change of chart to be solved by Algorithm \[ref:algoCont\]. The AMPL/IpOpt code is run considering the local chart $(U_a,\varphi_a)$, which is the formulation for (**OIP**) that is found in the literature. Moreover, to obtain comparable computational times, a second-order Runge-Kutta method is used to integrate ODEs both in Algorithm \[ref:algoCont\] and in AMPL/IpOpt. We test various time-steps for AMPL/IpOpt, while setting 80 time-steps for Algorithm \[ref:algoCont\].
Results are given in Table 2. The finer the step-size of the time-discretization scheme for IpOpt is, the better optimal solutions are obtained, but this is at the price of additional computational time, which is particularly higher for the last, more challenging mission. For comparable computational times, we obtain slightly better solutions than AMPL/IpOpt.
Conclusions and Perspectives {#conclSect}
============================
In this paper we have developed a geometric analysis that we have used to design a numerical algorithm, based on indirect methods, to solve optimal control problems for endo-atmospheric launch vehicle systems. Considering the original problem with mixed control-state constraints in an intrinsic geometric framework, we have recast it into an optimal control problem with pure control constraints by restriction to two sets of local coordinates (local charts). We have solved the original problem by combining classical shooting methods and homotopies, bypassing singularities of Euler coordinates. We have provided numerical simulations for optimal interception missions, showing similar (sometimes better) performance than state-of-the-art methods in numerical optimal control.
Additional contributions may be considered to refine the dynamics. In particular, state and control delays may be important to take into account further dynamical strains and phenomena like the *non-minimum phase*, a classical issue for launch vehicles applications (see, e.g., [@balas2012adaptive]). Motivated by the convergence result established in [@bonalli2017delay; @bonalli2018delay], we could add one further homotopic step on the delay. For computational times, even if many simulations on different missions for (**OIP**) show that Algorithm \[ref:algoCont\] can run between 0.5 Hz and 1 Hz, we cannot ensure a real-time processing yet. This may be achieved by combining Algorithm \[ref:algoCont\] with offline computations: we evaluate offline optimal strategies for several possible missions that will initialize online spatial continuations (i.e. on the parameter $\lambda_2$) to solve any new feasible mission.
[D | D | D | D | D | D | D |]{}
& & &\
& $\| v(T) \|$ & time & $\| v(T) \|$ & time & $\| v(T) \|$ & time\
& 763.1 & 0.66 & 609.8 & 0.92 & 480.0 & 2.3\
& 846.7 & 0.42 & 663.3 & 0.29 & 568.6 & 0.76\
& 780.3 & 1.1 & 635.7 & 1.6 & 531.2 & 1.9\
& 760.5 & 5.3 & 618.1 & 5.8 & 494.9 & 6.7\
& 767.5 & 10 & 614.5 & 10 & 478.3 & 12\
Table 2 : Optimal results obtained from solving (**OIP**) by Algorithm \[ref:algoCont\] and AMPL/IpOpt. Quantities are reported in standard units.
Proof of the Consistency for Local Adjoint Vectors {#proofApp}
--------------------------------------------------
Here, we provide a proof of Theorem \[mainTheo\]. By similarity between the local charts $(U_a,\varphi_a)$, $(U_b,\varphi_b)$, without loss of generality, we prove the assert considering the chart $(U_a,\varphi_a)$.\
Denote $\bm{q} = (\bm{r},\bm{v})$ and, for the sake of clarity in the notation, let us denote an optimal solution for (**GOGP**) in $[0,T]$ by $(\bar{\bm{q}}(\cdot),\bar{\bm{u}}(\cdot))$. Select times $s_1, s_2 \in (0,T)$ such that $s_1 < s_2$ and $\bar{\bm{q}}([s_1,s_2]) \subseteq U_a$, and consider the notation $x = \varphi_a(\bm{q})$, $\bm{q} \in U_a$. Since $\bar{\bm{q}}([s_1,s_2]) \subseteq U_a$, in what follows, we merely need to consider (**GOGP**) when restricted to $U_a$. Therefore, when restricted to $[s_1,s_2]$, the curve $(\bar{\bm{q}}(\cdot),\bar{\bm{u}}(\cdot))$ is a solution for the following local version of (**GOGP**) in $U_a$ (which is correctly well-defined up to multiplications by appropriate smooth cut-off functions): $$\begin{aligned}
(\textbf{GOGP})_{\textnormal{loc}} \
\begin{cases}
\displaystyle \ \min \ g\big(s_2,\bm{q}(t),\bm{u}(t)\big) \medskip \\
\dot{\bm{q}}(t) = \bm{h}\big(t,\bm{q}(t),\bm{u}(t)\big) \ , \ \bm{q}(t) \in U_a \medskip \\
\bm{q}(s_1) = \bar{\bm{q}}(s_1) \ , \ \bm{q}(s_2) = \bar{\bm{q}}(s_2) \medskip \\
\bm{c}\big(\bm{q}(t),\bm{u}(t)\big) \le 0 \ , \ \textnormal{a.e.} \ [0,T]
\end{cases}\end{aligned}$$ where $s_1$, $s_2$ are fixed and, for sake of brevity, we denoted $\bm{h} = (\bm{v},\bm{f})$, $c_0(\bm{q},\bm{u}) := \| \bm{u} \|^2 - 1$ (even if $c_0$ does not depend on $\bm{q}$, this notation will be useful hereafter) and $\bm{c} = (c_0,c_1,c_2)$. On the other hand, with the definitions in Section \[localPMP\] (see , ), the local version of (**GOGP**)$_{\textnormal{loc}}$ w.r.t. $(U_a,\varphi_a)$ writes as $$\begin{aligned}
(\textbf{GOGP})_a
\begin{cases}
\displaystyle \ \min \ g\big(s_2,\varphi_a^{-1}\big(x(t)\big),\Phi\big(x(t),\bm{w}(t)\big)\big) \medskip \\
\dot{x}(t) = d\varphi_a \cdot \bm{h} \big(t,\varphi_a^{-1}\big(x(t)\big),\Phi\big(x(t),\bm{w}(t)\big)\big) \medskip \\
x(s_1) = \varphi_a^{-1}(\bar{\bm{q}}(s_1)) \ , \ x(s_2) = \varphi_a^{-1}(\bar{\bm{q}}(s_2)) \medskip \\
\bm{c}(\bm{w}(t)) = \bm{c}\big(\varphi_a^{-1}\big(x(t)\big),\Phi\big(x(t),\bm{w}(t)\big)\big) \le 0
\end{cases}\end{aligned}$$ where $d\varphi_a$ is the differential of $\varphi_a$ and (recall Section \[localPMP\]) $$\Phi : U \times \mathbb{R}^3 \rightarrow \mathbb{R}^3 : (x,\bm{w}) \mapsto R^{\top}(x) R^{\top}_a(x) \bm{w} \ .$$ Then, by denoting $\bar{\bm{w}}(\cdot) = R_a(\bar{x}(\cdot)) R(\bar{x}(\cdot)) \bar{\bm{u}}|_{[s_1,s_2]}(\cdot)$ with $\bar{x}(\cdot) = \varphi_a(\bar{\bm{q}}|_{[s_1,s_2]}(\cdot))$, $(\bar{x}(\cdot),\bar{\bm{w}}(\cdot))$ is optimal for (**GOGP**)$_a$.\
For what follows, recall definitions and notations in Section \[maxSect\]. By applying the PMP to (**GOGP**) (i.e., relations -), we obtain the existence of a non-positive scalar $p^0$, an absolutely continuous mapping $\bm{p} : [0,T] \rightarrow \mathbb{R}^6$ and a vector function $\bm{\mu}(\cdot) \in L^{\infty}([0,T],\mathbb{R}^3)$, with $(\bm{p}(\cdot),p^0) \neq 0$, such that, almost everywhere in $[0,T]$, the following holds $$\label{adjApp}
\displaystyle \dot{\bm{p}}(t) = -\frac{\partial H^0}{\partial \bm{q}}(t,\bar{\bm{q}}(t),\bm{p}(t),p^0,\bar{\bm{u}}(t)) - \bm{\mu}(t) \cdot \frac{\partial \bm{c}}{\partial \bm{q}}(\bar{\bm{q}}(t),\bar{\bm{u}}(t))$$ $$\label{maxCondApp}
\displaystyle H^0(t,\bar{\bm{q}}(t),\bm{p}(t),p^0,\bar{\bm{u}}(t)) = \underset{\bm{c}(\bar{\bm{q}}(t),\bm{u}) \le 0}{\max} H^0(t,\bar{\bm{q}}(t),\bm{p}(t),p^0,\bm{u})$$ $$\label{maxDerApp}
\displaystyle \frac{\partial H^0}{\partial \bm{u}}(t,\bar{\bm{q}}(t),\bm{p}(t),p^0,\bar{\bm{u}}(t)) + \bm{\mu}(t) \cdot \frac{\partial \bm{c}}{\partial \bm{u}}(\bar{\bm{q}}(t),\bar{\bm{u}}(t)) = 0$$ and, in addition, conditions (\[slack\])-(\[transvCond1\]) hold. Since the quantity $\bm{c}\big(\bm{q},\Phi\big(\varphi_a(\bm{q}),\bm{w}\big)\big)$ does not depend on the state variable $\bm{q}$ (see , ), by differentiating it w.r.t. $\bm{q}$, one obtains $$\frac{\partial \bm{c}}{\partial \bm{q}}(\bar{\bm{q}}(t),\bar{\bm{u}}(t)) + \frac{\partial \bm{c}}{\partial \bm{u}}(\bar{\bm{q}}(t),\bar{\bm{u}}(t)) \cdot \frac{\partial \Phi}{\partial \bm{q}}(\bar{x}(\cdot),\bar{\bm{w}}(\cdot)) = 0 \ .$$ Moreover, by multiplying the previous expression by $\bm{\mu}(t)$ and plugging it into (\[maxDerApp\]), we straightforwardly have $$\bm{\mu}(t) \cdot \frac{\partial \bm{c}}{\partial \bm{q}}(\bar{\bm{q}}(t),\bar{\bm{u}}(t)) = \frac{\partial H^0}{\partial \bm{u}}(t,\bar{\bm{q}}(t),\bm{p}(t),p^0,\bar{\bm{u}}(t)) \cdot \frac{\partial \Phi}{\partial \bm{q}}(\bar{x}(\cdot),\bar{\bm{w}}(\cdot))$$ such that, a.e. in $[s_1,s_2]$, the adjoint equation (\[adjApp\]) becomes $$\begin{gathered}
\label{localAdjApp}
\displaystyle \dot{\bm{p}}(t) = -\frac{\partial H^0}{\partial \bm{q}}(t,\bar{\bm{q}}(t),\bm{p}(t),p^0,\bar{\bm{u}}(t)) \\
\displaystyle -\frac{\partial H^0}{\partial \bm{u}}(t,\bar{\bm{q}}(t),\bm{p}(t),p^0,\bar{\bm{u}}(t)) \cdot \frac{\partial \Phi}{\partial \bm{q}}(\bar{x}(\cdot),\bar{\bm{w}}(\cdot)) \ .\end{gathered}$$ Then, by defining $p(t) = (\varphi_a^{-1})^*_{\bar{\bm{q}}(t)} \cdot \bm{p}(t)$ for every $t \in [s_1,s_2]$, it is straightforward to obtain from (\[localAdjApp\]) and standard symplectic geometry computations (see, e.g., [@agrachev2013control]) that $$\begin{gathered}
\label{localAdjFinalApp}
\displaystyle \dot{p}(t) = -p(t) \cdot \frac{\partial}{\partial x}\Big(d\varphi_a \cdot \bm{h} \big(t,\varphi_a^{-1}(x),\Phi(x,\bm{w}))\Big)(t,\bar{x}(t),\bar{\bm{w}}(t)) \ .\end{gathered}$$ Moreover, from the properties of $\Phi$, we immediately see that the maximality condition (\[maxCondApp\]) reads as $$\label{maxCondFinalApp}
\displaystyle H^0_a(t,\bar{x}(t),p(t),p^0,\bar{\bm{w}}(t)) \ge H^0_a(t,\bar{x}(t),p(t),p^0,\bm{w})$$ for $\bm{w}$ such that $\bm{c}(\bm{w}) = \bm{c}\big(\varphi_a^{-1}\big(\bar{x}(t)\big),\Phi\big(\bar{x}(t),\bm{w}\big)\big) \le 0$ where $$H^0_a(t,x,p,p^0,\bm{w}) := p \cdot \Big(d\varphi_a \cdot \bm{h} \big(t,\varphi_a^{-1}(x),\Phi(x,\bm{w}))\Big)$$ From conditions (\[localAdjFinalApp\]), (\[maxCondFinalApp\]), it is easily deduced that $(p(\cdot),p^0)$ is the sought multiplier for the PMP formulation related to problem (**GOGP**)$_a$ (see, e.g., [@pontryagin1987mathematical]). Theorem \[mainTheo\] is proved.
Computation of Regular Controls {#regularApp}
-------------------------------
In this section we compute regular controls for (**GOGP**) under Assumption \[assumpSimply\]. We first assume that the system evolves in $(U_a,\varphi_a)$, within a non-zero measure subset $J \subseteq [0,T]$. Then, it holds $p_{\gamma}|_J(\cdot) \neq 0$ or $p_{\chi}|_J(\cdot) \neq 0$ (see Section \[controlSection\]).\
If $p^a_v|_J(\cdot) = 0$, by definition $C_a|_J(\cdot) = D_a|_J(\cdot) = 0$ and then, from (\[firstHam\]) and the Cauchy-Schwarz inequality, we obtain $$\displaystyle w_2 = \frac{\sin(\alpha_{\max}) p_{\gamma}}{\sqrt{p^2_{\gamma} + \frac{p^2_{\chi}}{\cos^2(\gamma)}}} \quad , \quad w_3 = \frac{\sin(\alpha_{\max}) p_{\chi}}{\cos(\gamma) \sqrt{p^2_{\gamma} + \frac{p^2_{\chi}}{\cos^2(\gamma)}}} \ .$$ Therefore, $w_1 = \sqrt{1 - (w^2_2 + w^2_3)}$ thanks to constraint $c_1$.\
We analyze now the harder case $p^a_v|_J(\cdot) \neq 0$. Denote $\lambda = p_{\gamma} \omega$, $\rho = p_{\chi} \frac{\omega}{\cos(\gamma)}$. In the following, we apply the Karush-Kuhn-Tucker conditions to (\[firstHam\]). For this, we first remark that any optimum for (\[firstHam\]) satisfies $w_1 > 0$. Moreover, if the constraints in (\[firstHam\]) were active at the optimum, then this point would satisfy $\bm{w} \in S^2$, $w^2_2 + w^2_3 = \sin^2(\alpha_{\max})$, and consequently, the gradients of these constraints evaluated at the optimum would satisfy the linear independence constraint qualification (see, e.g., [@NoceWrig06]). By applying the Karush-Kuhn-Tucker conditions to (\[firstHam\]) (without considering $w_1 \ge 0$, thanks to what we said above), we infer the existence of a non-zero multiplier $(\eta_1,\eta_2) \in \mathbb{R} \times \mathbb{R}_+$ which satisfies $$\begin{aligned}
\begin{cases}
C_a - 2 \eta_1 w_1 = 0 \quad , \quad 2 (\eta_1 + \eta_2 + D_a) w_2 - \lambda = 0 \medskip \\
2 (\eta_1 + \eta_2 + D_a) w_3 - \rho = 0 \ , \ \eta_2 (w^2_2 + w^2_3 - \sin^2(\alpha_{\max})) = 0 \ .
\end{cases}\end{aligned}$$ Since either $\lambda \neq 0$ or $\rho \neq 0$, one necessarily has $\eta_1 + \eta_2 + D_a \neq 0$ so that the optimal control satisfies $\rho w_2 = \lambda w_3$. We proceed considering $\lambda \neq 0$, i.e., $w_3 = (\rho/\lambda) w_2$. The problem is reduced to the study of the following optimization $$\begin{gathered}
\max \Bigg\{ C_a w_1 - \bigg(1 + \frac{\rho^2}{\lambda^2}\bigg) (D_a w^2_2 - \lambda w_2) \ \mid \\
w^2_1 + \bigg(1 + \frac{\rho^2}{\lambda^2}\bigg) w^2_2 = 1 \ , \ \bigg(1 + \frac{\rho^2}{\lambda^2}\bigg) w^2_2 \le \sin^2(\alpha_{\max}) \Bigg\} \ .\end{gathered}$$ In other words, we seek points $(w_1,w_2)$ such that the relations $$\begin{gathered}
\label{parabola}
w_1 = \frac{1}{C_a} \bigg(1 + \frac{\rho^2}{\lambda^2}\bigg) (D_a w^2_2 - \lambda w_2) + \frac{C}{C_a} \ , \\
\ w^2_1 + \bigg(1 + \frac{\rho^2}{\lambda^2}\bigg) w^2_2 = 1 \ , \ \bigg(1 + \frac{\rho^2}{\lambda^2}\bigg) w^2_2 \le \sin^2(\alpha_{\max})\end{gathered}$$ hold with the largest possible $C \in \mathbb{R}$. Several cases occur:
- $C_a = 0 \ $:
Since $D_a \neq 0$, this case results in the maximization of a parabola under box constraints. By denoting $A = - \Big(1 + \frac{\rho^2}{\lambda^2}\Big) D_a$, $B = \Big(1 + \frac{\rho^2}{\lambda^2}\Big) \lambda$ and $D = \frac{|\lambda| \sin(\alpha_{\max})}{\sqrt{\lambda^2 + \rho^2}}$, we maximize $A w^2_2 + B w_2$ such that $-D \le w_2 \le D$. Then, one has $w_1 = \sqrt{1 - \Big(1 + \frac{\rho^2}{\lambda^2}\Big) w^2_2}$, where:
- $w_2 = -D$ if $A > 0$, $B < 0$ or $A > 0$, $B < -2 |A| D$;
- $w_2 = -\frac{B}{2 A}$ if $A > 0$, $-2 |A| D \le B \le 2 |A| D$;
- $w_2 = D$ if $A > 0$, $B > 0$ or $A < 0$, $B > 2 |A| D$.
- $C_a > 0 \ $:
The optimum is given by the contact point between the parabola and the ellipse given in (\[parabola\]) that lies in the positive half-plane $w_1 > 0$. Under Assumption \[assumpSimply\], this is given by matching the first derivatives of these curves. More specifically, this provides $w_1 = \sqrt{1 - \frac{\lambda^2 + \rho^2}{(C_a + 2 D_a)^2}}$, $w_2 = \frac{\lambda}{C_a + 2 D_a}$ if $\frac{\lambda^2 + \rho^2}{(C_a + 2 D_a)^2} \le \sin^2(\alpha_{\max})$. However, saturations may arise, i.e., $w_1 = \cos(\alpha_{\max})$ and $w_2 = -\frac{|\lambda| \sin(\alpha_{\max})}{\displaystyle \sqrt{\lambda^2 + \rho^2}}$ if $\frac{\lambda}{C_a + 2 D_a} < -\frac{|\lambda| \sin(\alpha_{\max})}{\sqrt{\lambda^2 + \rho^2}}$, or $w_2 = \frac{|\lambda| \sin(\alpha_{\max})}{\sqrt{\lambda^2 + \rho^2}}$ if $\frac{\lambda}{C_a + 2 D_a} > \frac{|\lambda| \sin(\alpha_{\max})}{\sqrt{\lambda^2 + \rho^2}}$.
- $C_a < 0 \ $:
In this case, since $w_1 > 0$, the optimum becomes the point of intersection beetwen the parabola and the upper part of the ellipse given in (\[parabola\]) for which $C$ takes the maximum value. Only saturations are allowed. Indeed, by studying the position of the minimum of the parabola, we obtain $w_1 = \cos(\alpha_{\max})$ and $w_2 = -\frac{|\lambda| \sin(\alpha_{\max})}{\sqrt{\lambda^2 + \rho^2}}$ if $\frac{\lambda}{D_a} > 0$, or $w_2 = \frac{|\lambda| \sin(\alpha_{\max})}{\sqrt{\lambda^2 + \rho^2}}$ if $\frac{\lambda}{D_a} < 0$.
Clearly, a similar procedure holds when $\rho \neq 0$, $w_2 = (\lambda/\rho) w_3$.\
At this step, we have found the optimal strategy in the regular case for the first local chart $(U_a,\varphi_a)$. By the similarity of (\[firstHam\]) and (\[secondHam\]), similar results hold true for the local control $\bm{z}$ using instead the second local chart $(U_b,\varphi_b)$, for which $\lambda$ and $\rho$ are replaced respectively by $p_{\theta} \omega$ and by $-p_{\phi} \frac{\omega}{\cos(\theta)}$.\
We have found the behavior of any regular controls.
Computation of Nonregular Controls {#singularApp}
----------------------------------
In this section we compute nonregular optimal controls for (**GOGP**), under Assumption \[assM\] and Assumption \[assVel\], within a non-zero measure subset $J \subseteq [0,T]$. In what follows, we will need the adjoint equations related to (**GOGP**)$_a$. These come from applying the PMP for problems with pure control constraints (see, e.g., [@pontryagin1987mathematical]) to (**GOGP**)$_a$ and are listed below: $$\begin{gathered}
\displaystyle \dot{p}^a_r = p^a_L \frac{v}{r^2} \cos(\gamma) \cos(\chi) + p^a_l \frac{v}{r^2} \frac{\cos(\gamma) \sin(\chi)}{\cos(L)} + p_{\gamma} \bigg( \frac{v c_m}{h_r} w_2 \\
+ \frac{v}{r^2} \cos(\gamma) + \frac{\partial g}{\partial r} \frac{\cos(\gamma)}{v} \bigg) + p_{\chi} \bigg( \frac{v c_m}{h_r \cos(\gamma)} w_3 + \frac{v}{r^2} \cos(\gamma) \sin(\chi) \tan(L) \bigg) \medskip \\
+ p^a_v \bigg( \frac{\partial g}{\partial r} \sin(\gamma) - \frac{v^2}{h_r} \big( d + \eta c_m (w^2_2 + w^2_3) \big) \bigg)\end{gathered}$$ $$\displaystyle \dot{p^a}_L = -p^a_l \frac{v}{r} \frac{\cos(\gamma) \sin(\chi) \tan(L)}{\cos(L)} - p_{\chi} \frac{v}{r} \frac{\cos(\gamma) \sin(\chi)}{\cos^2(L)} \ , \ \dot{p}^a_l = 0$$ $$\begin{gathered}
\displaystyle \dot{p}^a_v = -p^a_r \sin(\gamma) - p^a_L \frac{\cos(\gamma) \cos(\chi)}{r} - p^a_l \frac{\cos(\gamma) \sin(\chi)}{r \cos(L)} \\
\displaystyle + 2 p^a_v v \big( d + \eta c_m (w^2_2 + w^2_3) \big) + p_{\gamma} \bigg( \frac{\omega}{v} w_2 - \frac{\cos(\gamma)}{r} - \frac{g}{v^2} \cos(\gamma) \bigg) \\
\displaystyle + p_{\chi} \bigg( \frac{\omega}{v} \frac{w_3}{\cos(\gamma)} - \frac{\cos(\gamma) \sin(\chi) \tan(L)}{r} \bigg)\end{gathered}$$ $$\begin{gathered}
\displaystyle \dot{p}_{\gamma} = -p^a_r v \cos(\gamma) + p^a_L \frac{v}{r} \sin(\gamma) \cos(\chi) + p^a_l \frac{v}{r} \frac{\sin(\gamma) \sin(\chi)}{\cos(L)} + p_{\gamma} \bigg( \frac{v}{r} \\
\displaystyle - \frac{g}{v} \bigg) \sin(\gamma) + p^a_v g \cos(\gamma) + p_{\chi} \bigg( \frac{v}{r} \sin(\gamma) \sin(\chi) \tan(L) - \frac{\omega \sin(\gamma)}{\cos^2(\gamma)} w_3 \bigg)\end{gathered}$$ $$\displaystyle \dot{p}_{\chi} = p^a_L \frac{v}{r} \cos(\gamma) \sin(\chi) - p^a_l \frac{v}{r} \frac{\cos(\gamma) \cos(\chi)}{\cos(L)} - p_{\chi} \frac{v}{r} \cos(\gamma) \cos(\chi) \tan(L) .$$
The important result that allows us to work out explicit expressions for nonregular controls consists of showing that, under Assumption \[assM\], it holds $p^a_v|_J(\cdot) \neq 0$, $p^b_v|_J(\cdot) \neq 0$, i.e., nonregular controls are not degenerate. This arises as follows.
\[propSingCont\] Suppose $p_{\gamma}|_J(\cdot) = p_{\chi}|_J(\cdot) = 0$ (as well as $p_{\theta}|_J(\cdot) = p_{\phi}|_J(\cdot) = 0$), i.e., nonregular controls appear. Then, under Assumption \[assM\], $p^a_v|_J(\cdot) \neq 0$ (as well as $p^b_v|_J(\cdot) \neq 0$).
*Proof:* We prove the statement considering the first local chart $(U_a,\varphi_a)$. For the second local chart, similar computations hold. By contradiction, suppose that $p_{\gamma}|_J(\cdot) = p_{\chi}|_J(\cdot) = p^a_v|_J(\cdot) = 0$. From the adjoint equations of coordinates $p^a_v$, $p_{\gamma}$ and $p_{\chi}$ (given above) restricted to $J$, we obtain $$\left( \arraycolsep=1.5pt \begin{array}{ccc}
-v \cos(\gamma) & \displaystyle \frac{v}{r} \sin(\gamma) \cos(\chi) & \displaystyle \frac{v}{r} \frac{\sin(\gamma) \sin(\chi)}{\cos(L)} \medskip \\
0 & \displaystyle \frac{v}{r} \cos(\gamma) \sin(\chi) & \displaystyle -\frac{v}{r} \frac{\cos(\gamma) \cos(\chi)}{\cos(L)} \medskip \\
-\sin(\gamma) & \displaystyle \frac{\cos(\gamma) \cos(\chi)}{r} & \displaystyle \frac{\cos(\gamma) \sin(\chi)}{r \cos(L)} \medskip \\
\end{array} \right) \left( \begin{array}{c}
p^a_r \medskip \\
p^a_L \medskip \\
p^a_l
\end{array} \right) =
\left( \begin{array}{c}
0 \medskip \\
0 \medskip \\
0
\end{array} \right) \ .$$ The determinant of the matrix is $\frac{v^2 \cos(\gamma)}{r^2 \cos(L)} \neq 0$ and then $(p^a_r,p^a_L,p^a_l)|_J(\cdot) = 0$. This implies that the adjoint vector is zero everywhere in $[0,T]$. Assumption \[assM\], the transversality conditions and $\bm{p}(\cdot) \equiv 0$ (from Theorem \[mainTheo\]) give $p^0 = 0$, raising thus a contradiction because it holds $(\bm{p}(\cdot),p^0) \neq 0$. $_{\Box}$\
### First Local Chart Representation
We start assuming that the system evolves in the first local chart $(U_a,\varphi_a)$, within a non-zero mesure subset $J \subseteq [0,T]$. Our objective consists in studying (\[singularHam\]). Thanks to Lemma \[propSingCont\], from now on, we assume $p_{\gamma}|_J(\cdot) = p_{\chi}|_J(\cdot) = 0$, $p^a_v|_J(\cdot) \neq 0$ and, when clear from the context, we skip the dependence on $t$ to keep better readability. Moreover, we introduce the following local forms for the dynamics of (**GOGP**)$_a$ (recall Section \[firstChart\]): $$\begin{gathered}
\displaystyle X(t,\bm{r},\bm{v}) := v\sin(\gamma) \frac{\partial}{\partial r} + \frac{v}{r} \cos(\gamma) \cos(\chi) \frac{\partial}{\partial L} + \frac{v}{r} \frac{\cos(\gamma) \sin(\chi)}{\cos(L)} \frac{\partial}{\partial l} \\
\displaystyle - \left(d v^2 + g \sin(\gamma) \right) \frac{\partial}{\partial v} + \left(\frac{v}{r} - \frac{g}{v}\right) \cos(\gamma) \frac{\partial}{\partial \gamma} + \frac{v}{r} \cos(\gamma) \sin(\chi) \tan(L) \frac{\partial}{\partial \chi}\end{gathered}$$ $$\displaystyle Y_1(t,\bm{r},\bm{v}) := \frac{f_T}{m} \frac{\partial}{\partial v} \quad , \quad Y_Q(t,\bm{r},\bm{v}) := -\eta c_m v^2 \frac{\partial}{\partial v}$$ $$\displaystyle Y_2(t,\bm{r},\bm{v}) := \omega \frac{\partial}{\partial \gamma} \quad , \quad Y_3(t,\bm{r},\bm{v}) := \frac{\omega}{\cos(\gamma)} \frac{\partial}{\partial \chi} \ .$$ The Lie bracket of two vector fields $X$, $Y$ is defined as the derivation $[X,Y](f) := X(Yf) - Y(Xf)$, $f \in C^{\infty}$ (see, e.g., [@agrachev2013control]). The following classical result holds (see, e.g., [@bonnard2003optimal]).
\[lemmaLie\] Using the first local chart $(U_a,\varphi_a)$, for times $t \in J$ such that $(\bm{r},\bm{v})(t)$ lies within $U_a$, the following holds: $$\begin{gathered}
\label{lie1}
\displaystyle \frac{d}{d t} \big\langle \bm{p} , Y_2 \big\rangle = \big\langle \bm{p} , \frac{\partial}{\partial t} Y_2 \big\rangle + \big\langle \bm{p} , [X,Y_2] \big\rangle + w_1 \big\langle \bm{p} , [Y_1,Y_2] \big\rangle \\
\displaystyle + w_3 \big\langle \bm{p} , [Y_3,Y_2] \big\rangle + (w^2_2 + w^2_3) \big\langle \bm{p} , [Y_Q,Y_2] \big\rangle\end{gathered}$$ $$\begin{gathered}
\label{lie2}
\displaystyle \frac{d}{d t} \big\langle \bm{p} , Y_3 \big\rangle = \big\langle \bm{p} , \frac{\partial}{\partial t} Y_3 \big\rangle + \big\langle \bm{p} , [X,Y_3] \big\rangle + w_1 \big\langle \bm{p} , [Y_1,Y_3] \big\rangle \\
\displaystyle + w_2 \big\langle \bm{p} , [Y_2,Y_3] \big\rangle + (w^2_2 + w^2_3) \big\langle \bm{p} , [Y_Q,Y_3] \big\rangle\end{gathered}$$ $$\begin{gathered}
\label{lie3}
\displaystyle \frac{d}{d t} \big\langle \bm{p} , [X,Y_2] \big\rangle = \big\langle \bm{p} , \frac{\partial}{\partial t} [X,Y_2] \big\rangle + \big\langle \bm{p} , [X,[X,Y_2]] \big\rangle \\
\displaystyle + w_1 \big\langle \bm{p} , [Y_1,[X,Y_2]] \big\rangle + w_2 \big\langle \bm{p} , [Y_2,[X,Y_2]] \big\rangle \\
\displaystyle + w_3 \big\langle \bm{p} , [Y_3,[X,Y_2]] \big\rangle + (w^2_2 + w^2_3) \big\langle \bm{p} , [Y_Q,[X,Y_2]] \big\rangle\end{gathered}$$ $$\begin{gathered}
\label{lie4}
\displaystyle \frac{d}{d t} \big\langle \bm{p} , [X,Y_3] \big\rangle = \big\langle \bm{p} , \frac{\partial}{\partial t} [X,Y_3] \big\rangle + \big\langle \bm{p} , [X,[X,Y_3]] \big\rangle \\
\displaystyle + w_1 \big\langle \bm{p} , [Y_1,[X,Y_3]] \big\rangle + w_2 \big\langle \bm{p} , [Y_2,[X,Y_3]] \big\rangle \\
\displaystyle + w_3 \big\langle \bm{p} , [Y_3,[X,Y_3]] \big\rangle + (w^2_2 + w^2_3) \big\langle \bm{p} , [Y_Q,[X,Y_3]] \big\rangle \ .\end{gathered}$$
The idea that we develop here seeks explicit expressions for the optimal controls $\bm{w}(\cdot)$ by analyzing expressions (\[lie1\])-(\[lie4\]). Our strategy is based on the following remarks, which come from symbolic Lie bracket computations on the local fields:
1. $[Y_1,Y_2]$, $[Y_Q,Y_2]$ are proportional to $\frac{\partial}{\partial \gamma}$;
2. $[Y_1,Y_3]$, $[Y_2,Y_3]$, $[Y_Q,Y_3]$, $[Y_2,[X,Y_3]]$ lie along $\frac{\partial}{\partial \chi}$;
3. When $p_{\gamma}|_J(\cdot) = p_{\chi}|_J(\cdot) = 0$, then $\big\langle \bm{p} , [X,[X,Y_3]] \big\rangle$, $\big\langle \bm{p} , [Y_1,[X,Y_3]] \big\rangle$, $\big\langle \bm{p} , [Y_Q,[X,Y_3]] \big\rangle$ lie along $\dot{p}_{\chi}$;
4. When $p_{\gamma}|_J(\cdot) = p_{\chi}|_J(\cdot) = 0$, $\big\langle \bm{p} , \frac{\partial}{\partial t} [X,Y_2] \big\rangle$ lies along $\big\langle \bm{p} , [X,Y_2] \big\rangle$ and $\big\langle \bm{p} , \frac{\partial}{\partial t} [X,Y_3] \big\rangle$ lies along $\big\langle \bm{p} , [X,Y_3] \big\rangle$.
From $p_{\gamma}|_J(\cdot) = p_{\chi}|_J(\cdot) = 0$, (A) and (B) applied to (\[lie1\]) and (\[lie2\]) give $\big\langle \bm{p} , [X,Y_2] \big\rangle \big|_J = \big\langle \bm{p} , [X,Y_3] \big\rangle \big|_J = 0$. These expressions, plugged into (\[lie4\]) using (B), (C) and (D), lead to $$\label{exprSing}
w_3 \big\langle \bm{p} , [Y_3,[X,Y_3]] \big\rangle = 0 \ , \ \textnormal{in } J \ .$$ Seeking explicit expressions for the nonregular controls from (\[exprSing\]) becomes a hard and tedious task when $\big\langle \bm{p} , [Y_3,[X,Y_3]] \big\rangle = 0$. This because more many time derivatives are required, which provide complex expressions of Lie brackets. In this situation, the environmental conditions concerning the feasibility of (**GOGP**) (represented by Assumption \[assVel\]) play an important role in making these further time derivatives of Lie brackets not necessary for our purpose. Indeed, we have the following:
\[lemmaDuality\] Assume that Assumption \[assVel\] holds. Then, one has $\big\langle \bm{p} , [Y_3,[X,Y_3]] \big\rangle \neq 0$ almost everywhere in $J$.
*Proof:* By contradiction, suppose that $\big\langle \bm{p} , [Y_3,[X,Y_3]] \big\rangle = 0$ a.e. within $J$. This implies that $\cos(\chi) p^a_L + \frac{\sin(\chi)}{\cos(L)} p^a_l = 0$ a.e. within $J$. The previous expression, combined with the adjoint equation for $p_{\chi}$ (given above), gives $p^a_L|_J(\cdot) = p^a_l|_J(\cdot) = 0$. On the other hand, from the adjoint equation of $p_{\gamma}$ (see above), we have $(v p^a_r - g p^a_v)|_J(\cdot) = 0$. Combining this expression with its derivative w.r.t. time and imposing $p^a_v|_J(\cdot) \neq 0$ lead to $$v^4 + 3 g(\bm{r}) h_r v^2 - g(\bm{r}) h_r \left( \frac{f_T w_1}{m (d + \eta c_m (w^2_2 + w^2_3))} \right) = 0 \ .$$ First of all, if $f_T = 0$ a contradiction arises immediately. The only physically meaningful solution for this equation is $$v = \sqrt{\frac{3}{2} g(\bm{r}) h_r} \sqrt{ \sqrt{1 + \frac{4}{9} \frac{1}{g(\bm{r}) h_r} \left( \frac{f_T w_1}{m (d + \eta c_m (w^2_2 + w^2_3))} \right) } - 1 }$$ and, from $w_1 \in [0,1]$, Assumption \[assVel\] gives a contradiction. $_{\Box}$
The previous results allow us to reformulate (\[singularHam\]) as $$\displaystyle (w_1,w_2) = {\textnormal{argmax}}\Big\{ C_a w_1 - D_a w^2_2 \mid w^2_1 + w^2_2 = 1, w^2_2 \le \sin^2(\alpha_{\max}) \Big\}$$ that we can solve. Remark that $D_a \neq 0$, $C_a \neq 0$ iff $f_T \neq 0$.
Suppose first that $C_a = 0$. In this case, it is clear that the maximization problem above is solved by $w_1 = 1$, $w_2 = 0$ if $D_a > 0$ and $w_1 = \cos(\alpha_{\max})$, $w^2_2 = \sin^2(\alpha_{\max})$ if $D_a < 0$. Let now $C_a \neq 0$. Exploiting a graphical study, it is not difficult to see that the solutions are now given by $w_1 = 1$, $w_2 = 0$ if $C_a > 0$ and $w_1 = \cos(\alpha_{\max})$, $w^2_2 = \sin^2(\alpha_{\max})$ if $C_a < 0$.
To conclude, it remains to establish the value of the coordinate $w_2$ when $w_1 = \cos(\alpha_{\max})$ and $w^2_2 = \sin^2(\alpha_{\max})$. For this, we may use expression (\[lie3\]). Indeed, it is clear that, when $\big\langle \bm{p} , [Y_2,[X,Y_2]] \big\rangle \neq 0$, it holds (recall statements (A)-(D)) $$\begin{gathered}
w_2 \displaystyle = -\frac{\big\langle \bm{p} , [X,[X,Y_2]] \big\rangle}{\big\langle \bm{p} , [Y_2,[X,Y_2]] \big\rangle} - w_1 \frac{\big\langle \bm{p} , [Y_1,[X,Y_2]] \big\rangle}{\big\langle \bm{p} , [Y_2,[X,Y_2]] \big\rangle} \\
- w^2_2 \frac{\big\langle \bm{p} , [Y_Q,[X,Y_2]] \big\rangle}{\big\langle \bm{p} , [Y_2,[X,Y_2]] \big\rangle} \ .\end{gathered}$$ If instead $\big\langle \bm{p} , [Y_2,[X,Y_2]] \big\rangle = 0$ a.e. in $J$, then, suppose that $\big\langle \bm{p} , [Y_2,[Y_2,[X,Y_2]]] \big\rangle \neq 0$. Differentiating as done in (\[lie3\]), (\[lie4\]), by using the same arguments as above we have $$\begin{gathered}
w_2 \displaystyle = -\frac{\Big\langle \bm{p} , [Y_2,[X,[X,Y_2]]] \Big\rangle}{\Big\langle \bm{p} , [Y_2,[Y_2,[X,Y_2]]] \Big\rangle} - w_1 \frac{\Big\langle \bm{p} , [Y_2,[Y_1,[X,Y_2]]] \Big\rangle}{\Big\langle \bm{p} , [Y_2,[Y_2,[X,Y_2]]] \Big\rangle} \\
- w^2_2 \frac{\Big\langle \bm{p} , [Y_2,[Y_Q,[X,Y_2]]] \Big\rangle}{\Big\langle \bm{p} , [Y_2,[Y_2,[X,Y_2]]] \Big\rangle} \ .\end{gathered}$$ We can prove that actually one between the two previous formulas always holds, giving then the sought conclusion.
Under Assumption \[assVel\], almost everywhere in $J$: $$\big\langle \bm{p} , [Y_2,[X,Y_2]] \big\rangle \neq 0 \quad \textnormal{or} \quad \big\langle \bm{p} , [Y_2,[Y_2,[X,Y_2]]] \big\rangle \neq 0 \ .$$
*Proof:* By contradiction, suppose that $\big\langle \bm{p} , [Y_2,[X,Y_2]] \big\rangle = 0$ and $\big\langle \bm{p} , [Y_2,[Y_2,[X,Y_2]]] \big\rangle = 0$ a.e. in $J$. From these, one recovers respectively the following two expressions $$\bigg( \sin(\gamma) p^a_r + \frac{\cos(\gamma) \cos(\chi)}{r} p^a_L + \frac{\cos(\gamma) \sin(\chi)}{r \cos(L)} p^a_l - \frac{g \sin(\gamma)}{v} p^a_v \bigg) \bigg|_J(\cdot) = 0$$ $$\bigg( \cos(\gamma) p^a_r - \frac{\sin(\gamma) \cos(\chi)}{r} p^a_L - \frac{\sin(\gamma) \sin(\chi)}{r \cos(L)} p^a_l - \frac{g \cos(\gamma)}{v} p^a_v \bigg) \bigg|_J(\cdot) = 0$$ which lead to $\cos(\chi) p^a_L + \frac{\sin(\chi)}{\cos(L)} p^a_l = 0$ a.e. within $J$. This relation, combined with the adjoint equation of $p_{\chi}$ (see above), gives $p^a_L|_J(\cdot) = p^a_l|_J(\cdot) = 0$. On the other hand, the adjoint equation of $p_{\gamma}$ (above) provides $(v p^a_r - g p^a_v)|_J(\cdot) = 0$. Then, as in the proof of Lemma \[lemmaDuality\], a contradiction arises. $_{\Box}$\
### Second Local Chart Representation
The approach proposed in the previous section is no more exploitable when using the second local chart $(U_b,\varphi_b)$ and problem (\[singularHam2\]). Indeed, the terms of the gravity and the curvature of the Earth contained in (\[dynSecond\]) make the computations on the Lie algebra generated by the local fields hard to treat. However, we can still recover nonregular arcs by proceeding as follows.
Thanks to the previous computation, we know the explicit expressions for nonregular controls for every point in $U_a$. Then, it is enough to compute possible nonregular controls for trajectories in $U_b \setminus U_a$. From (\[frame1\]), (\[frame2\]), one sees that these trajectories lie in the following submanifold of $\mathbb{R}^6 \setminus \{ 0 \}$ $$S_b := \left\{ (\bm{r},\bm{v}) \in \mathbb{R}^6 \setminus \{ 0 \} \ \mid \ \bm{v} \ \mathbin{\!/\mkern-5mu/\!} \ \bm{r} \right\}$$ which corresponds, by forcing the coordinates of the chart $(U_b,\varphi_b)$, to points such that $\theta = 0$, $\phi = 0$ or $\theta = 0$, $\phi = \pi$. Following the previous argument, suppose that there exists a non-zero measure subset $J \subseteq [0,T]$ for which the optimal trajectory $(\bm{r},\bm{v})(\cdot)$ arisen from a nonregular control $\bm{u}(\cdot)$ is such that $(\bm{r},\bm{v})(t) \in S_b$ for every $t \in J$. In particular, suppose that $\theta|_J(\cdot) = 0 \ , \ \phi|_J(\cdot) = 0$ or $\phi|_J(\cdot) = \pi$. Then, almost everywhere in $J$, the trajectory $(\bm{r},\bm{v})(\cdot)$ satisfies $$\begin{aligned}
\begin{cases}
\dot{r} = \pm v \ , \ \dot{L} = 0 \ , \ \dot{l} = 0 \ , \ \dot{\theta} = \displaystyle \omega z_2 \ , \ \dot{\phi} = \displaystyle -\omega z_3 \medskip \\
\dot{v} = \displaystyle \frac{f_T}{m} z_1 -\left(d + \eta c_m (z^2_2 + z^2_3) \right) v^2 \pm g \ .
\end{cases}\end{aligned}$$ Since the values of $\theta$ and $\phi$ remain the same along $J$, their derivative w.r.t. the time must be zero. Therefore, almost everywhere in $J$, any nonregular control satisfies $z_1|_J(\cdot) = 1$, $z_2|_J(\cdot) = 0$ and $z_3|_J(\cdot) = 0$. This concludes our analysis.
[Riccardo Bonalli]{} obtained his MSc in Mathematical Engineering from Politecnico di Milano, Italy, in 2014, and his PhD in applied mathematics from Sorbonne Université, France, in 2018, in collaboration with ONERA - The French Aerospace Lab, France. He is now postdoctoral researcher at the Department of Aeronautics and Astronautics, at Stanford University, California. His main research interests concern the theoretical and numerical optimal control with applications in aerospace engineering and robotics.
[Bruno Hérissé]{} received the Engineering degree and the Master degree from the École Supérieure d’Électricité (SUPELEC), Paris, France, in 2007. After three years of research with CEA List, he received the Ph.D. degree in robotics from the University of Nice Sophia Antipolis, Sophia Antipolis, France, in 2010. Since 2011, he has been a Research Engineer with ONERA, the French Aerospace Lab, Palaiseau, France. His research interests include optimal control and vision-based control with applications in aerospace sytems and aerial robotics.
[Emmanuel Trélat]{} was born in 1974. He is currently full professor at Sorbonne Université (Paris 6). He is the director of the Fondation Sciences Mathématiques de Paris. He is editor in chief of the journal ESAIM: Control, Optimization and Calculus of Variations, and is associated editor of many other journals. He has been awarded the SIAM Outstanding Paper Prize (2006), Maurice Audin Prize (2010), Felix Klein Prize (European Math. Society, 2012), Blaise Pascal Prize (french Academy of Science, 2014), Big Prize Victor Noury (french Academy of Science, 2016). His research interests range over control theory in finite and infinite dimension, optimal control, stabilization, geometry, numerical analysis, with a special interest to optimal control applied to aerospace.
[^1]: R. Bonalli is with Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions, équipe CAGE, F-75005 Paris, France and ONERA, DTIS, Université Paris Saclay, F-91123 Palaiseau, France, e-mail: rbonalli@stanford.edu, riccardo.bonalli@etu.upmc.fr, riccardo.bonalli@onera.fr.
[^2]: B. Hérissé is with ONERA, DTIS, Université Paris Saclay, F-91123 Palaiseau, France, e-mail: bruno.herisse@onera.fr.
[^3]: E. Trélat is with Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions, équipe CAGE, F-75005 Paris, France, e-mail: emmanuel.trelat@sorbonne-universite.fr.
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abstract: 'We reanalyze the formation and evolution of galaxies in the hierarchical clustering scenario. Using a semi-analytic model (SAM) of galaxy formation described in this paper, which we hereafter call the Mitaka model, we extensively investigate the observed scaling relations of galaxies among photometric, kinematic, structural and chemical characteristics. In such a scenario, spheroidal galaxies are assumed to be formed by major merger and subsequent starburst, in contrast to the traditional scenario of monolithic cloud collapse. As a new ingredient of SAMs, we introduce the effects of dynamical response to supernova-induced gas removal on size and velocity dispersion, which play an important role on dwarf galaxy formation. In previous theoretical studies of dwarf galaxies based on the monolithic cloud collapse given by Yoshii & Arimoto and Dekel & Silk, the dynamical response was treated in the extremes of a purely baryonic cloud and a baryonic cloud fully supported by surrounding dark matter. To improve this simple treatment, in our previous paper, we formulated the dynamical response in more realistic, intermediate situations between the above extremes. While the effects of dynamical response depend on the mass fraction of removed gas from a galaxy, how much amount of the gas remains just after major merger depends on the star formation history. A variety of star formation histories are generated through the Monte Carlo realization of merging histories of dark halos, and it is found that our SAM naturally makes a wide variety of dwarf galaxies and their dispersed characteristics as observed. It is also found that our result strongly depends on the adopted redshift dependence of star formation timescale, because it determines the gas fraction in high-redshift galaxies for which major mergers frequently occur. We test four star formation models. The first model has a constant timescale of star formation independent of redshift. The last model has a timescale proportional to the dynamical timescale of the galactic disk. The other models have timescales intermediates of these two. The last model fails to reproduce observations, because it predicts only little amount of the leftover gas at major mergers, therefore giving too weak dynamical response on size and velocity dispersion of dwarf spheroidals. The models, having a constant timescale of star formation or a timescale very weakly dependent on redshift, associated with our SAM, succeed to reproduce most observations from giant to dwarf galaxies, except that the adopted strong supernova feedback in this paper does not fully explain the color–magnitude relation under the cluster environment and the Tully-Fisher relation. A direction of overcoming this remaining problem is also discussed.'
author:
- Masahiro Nagashima
- Yuzuru Yoshii
title: 'Hierarchical Formation of Galaxies with Dynamical Response to Supernova-Induced Gas removal '
---
Introduction
============
Dwarf galaxies give us many useful insights on the galaxy formation. They occupy a dominant fraction among galaxies in number and have a wide variety of structures and formation histories. Furthermore it would be easier to understand how physical processes such as heating by supernova explosions affect their evolution, because of shallowness of their gravitational potential well, compared to massive galaxies that have more complex formation histories and physical processes. Equally important is that massive galaxies have evolved via continuous mergers and accretion of less massive galaxies, according to the recent standard scenario of large-scale structure formation in the universe with the cold dark matter (CDM). This indicates that it is essential to understand the formation of dwarf galaxies even in understanding the formation of massive galaxies, because they would be formed by mergers of their [*building blocks*]{} like dwarf galaxies.
Before the CDM model gained its popularity, traditional models of monolithic cloud collapse such as the galactic wind model for elliptical galaxies [@els62; @l69; @i77; @s79; @ys79; @ay86; @ay87; @ka97] and the infall model for spiral galaxies [@ayt91] had been widely used in analyses of galaxy evolution. These are simple but strong tools to explain various observations and then have contributed to construction of a basic picture of galaxy evolution. Based on such traditional models, @ds86 systematically investigated many aspects of dwarf galaxies. Especially they focused on the role of supernova (SN) feedback, that is, heating up and sweeping out of the galactic gas by multiple SN explosions, in the processes of formation and evolution of dwarf galaxies. Since a large amount of heated gas is expelled by the galactic wind, the self-gravitating system expands as a result of dynamical response to the gas removal [@h80; @m83; @v86]. They considered two limiting cases for the gas removal. One is a purely self-gravitating gas cloud, and another is a gas cloud embedded in a dominant dark halo. They showed that many observed scaling relations are well explained by their model. Combined with the evolutionary population synthesis code, @ya87 extended their analysis to estimating directly observable photometric properties, while only considering the purely self-gravitating gas cloud.
Since the hierarchical clustering of dark halos predicted by the CDM model becomes a standard structure formation scenario, the galaxy formation scenario must be modified so as to be consistent with the hierarchical merging of dark halos. Explicitly taking into account the Monte Carlo realization of merging histories of dark halos based on the distribution function of initial density fluctuations, the so-called semi-analytic models (SAMs) of galaxy formation have been developed [e.g., @kwg93; @c94; @bcf96; @ngs99; @sp99; @spf01; @ntgy01; @nytg02]. SAMs include several important physical processes such as star formation, supernova feedback, galaxy merger, population synthesis and so on. While SAMs well reproduce many observed properties of galaxies, most of analyses have been limited to those of massive galaxies. Thus we focus on the formation of dwarf galaxies in the framework of the SAMs in this paper. In the analyses of dwarf galaxies, the dynamical response to gas removal, which is known to play an important role as shown by @ds86 and @ya87, must be taken into account. While self-consistent equilibrium models taking into account baryon and dark matter have been analyzed by @ys87 and @cp92, we focus on the dynamical response of baryons within a dark matter halo. Since we have derived the mathematical formula of dynamical response for the galaxies consisting of baryon and dark matter in @ny03, it is possible to incorporate them into our SAM.
The main purpose of this paper is a reanalysis of @ds86 and @ya87 in the framework of SAMs. Thus we mainly focus on the formation of elliptical galaxies, which are assumed to be formed by major merger and subsequent starburst. In order to do this, we construct the Mitaka model, which is a SAM including the effects of dynamical response. So far many scaling relations among photometric, structural, and kinematical parameters of elliptical galaxies have been observed, such as the color-magnitude relation [e.g., @b59], the velocity dispersion-magnitude relation [@fj76], and the surface brightness-size relation [@k77; @kow83]. To understand how these relations are originated, especially on scales of dwarf galaxies, helps to clarify the processes of galaxy formation and evolution in the context of the cosmological structure formation scenario. Although some authors found by principal component analysis that elliptical galaxies are distributed over the so-called fundamental plane in the three-dimensional space among photometric, structural and kinematical parameters [@wko85; @d87; @dd87], we focus on the direct observables in this paper rather than their principal component projection.
This paper is outlined as follows. In §2 we describe our SAM. In §3 we constrain model parameters in our SAM using local observations. In §4 we compare the theoretical predictions of SAM galaxies with various observations. In §5 we examine consistency check of our SAM with other observations. In §6 we show the cosmic star formation history. In §7 we discuss the nature of galaxies on the cooling diagram that has been traditionally used for understanding the formation of galaxies. In §8 we provide summary and conclusion.
Model
=====
The galaxy formation scenario that we use is as follows. In the CDM universe, dark matter halos cluster gravitationally and merge in a manner that depends on the adopted power spectrum of the initial density fluctuations. In each of the merged dark halos, radiative gas cooling, star formation, and gas reheating by supernovae occur. The cooled dense gas and stars constitute [*galaxies*]{}. These galaxies sometimes merge together in a common dark halo, and then more massive galaxies form. Repeating these processes, galaxies form and evolve to the present epoch.
Some ingredients of our SAM are revised. Modifications include the shape of mass function of dark halos, the star formation (SF) timescale, the merger timescale of galaxies taking into account the tidal stripping of subhalos, and the dynamical response to gas removal caused by starburst during major merger. The details are described below.
Merging Histories of Dark Halos {#sec:mh}
-------------------------------
The merging histories of dark halos are realized by a Monte Carlo method proposed by @sk99, based on the extended Press-Schechter (PS) formalism [@bcek91; @b91; @lc93]. This formalism is an extension of the Press-Schechter formalism [@ps74], which gives the mass function of dark halos, $n(M)$, to estimate the mass function of progenitor halos with mass $M_{1}$ at a redshift $z_{0}+\Delta z$ of a single dark halo with mass $M_{0}$ collapsing at a redshift $z_{0}$, $n(M_{1}; z_{0}+\Delta z|M_{0}; z_{0})dM_{1}$. According to this mass function, a set of progenitors is realized. By repeating this, we obtain a [*merger tree*]{}. Realized trees are summed with a weight given by a mass function at output redshift. Dark halos with circular velocity $V_{\rm circ}\geq V_{\rm low}=30$km s$^{-1}$ are regarded as isolated halos, otherwise as diffuse accreted matter.
In our previous papers, we adopted the PS mass function to provide the weight for summing merger trees. Recent high-resolution $N$-body simulations, however, suggest that the PS mass function should be slightly corrected [e.g., @j01]. According to the notation in @j01, the original PS mass function is written by $$f(\sigma;{\rm PS})=\sqrt{\frac{2}{\pi}}\frac{\delta_{c}}{\sigma}
\exp\left(-\frac{\delta_{c}^{2}}{2\sigma^{2}}\right),$$ and the cumulative mass function $n(M)$ is related with the above function by $$f(\sigma;X)=\frac{M}{\rho_{0}}\frac{dn(M)}{d\ln\sigma^{-1}},$$ where $X$ specifies a model such as PS, $\rho_{0}$ is the mean density of the universe, $\sigma$ denotes the standard deviation of the density fluctuation field, and $\delta_{c}$ is the critical density contrast for collapse assuming spherically symmetric collapse [@t69; @gg72]. In this paper we use the following mass function given by @yny03 [hereafter YNY] instead of the PS mass function, $$f(\sigma;{\rm YNY})=A(1+x^{C})x^{D}\exp(-x^{2}),$$ where $x=B\delta_{c}/\sqrt{2}\sigma$, $A=2/[\Gamma(D/2)+\Gamma(\{C+D\}/2)], B=0.893, C=1.39$ and $D=0.408$. This is a fitting function that satisfies the normalization condition, that is, the integration over all rage of $\nu$ is unity. These functions are plotted in Figure \[fig:massfn\] by the solid line (YNY) and the dot-dashed line (PS). We also show other often used formula given by @st99, $$f(\sigma;{\rm ST})=A\sqrt{\frac{2a}{\pi}}\left[1+\left(\frac{\sigma^{2}}{a\delta_{c}^{2}}\right)^{p}\right]\frac{\delta_{c}}
{\sigma}\exp\left(-\frac{a\delta_{c}^{2}}{2\sigma^{2}}\right),$$ and by @j01, $$f(\sigma;{\rm J})=0.315\exp(-|\ln\sigma^{-1}+0.61|^{3.8}),$$ where $A=0.3222, a=0.707$ and $p=0.3$ for the former (ST) and the latter (J) formula is valid over the range $-1.2\leq\ln\sigma^{-1}\leq 1.05$.
The number density of dark halos with large mass given by the above three functions (YNY, ST and J) and by the $N$-body simulation are similar to those given by analytic estimation [e.g., @yng96; @monaco98; @n01] and are therefore trustworthy. With $N$-body simulations, it is difficult to estimate the mass function at low mass because of the limited resolution and the uncertainty in the identification of dark halos. The very high resolution $N$-body simulations, made using the adaptive mesh refinement method by @yy01 and @y02, predict a mass function similar to that of ST. This mass function extends to a mass 10 times smaller than that of @j01 and has a slightly different slope. Thus we adopt the YNY mass function as above. A comparison between the YNY mass function and the $N$-body simulation is discussed in a separate paper [@yny03]. In §\[sec:paramset\] we show how the mass function of dark halos affects the luminosity function of galaxies.
In short, we realize merger trees by using the extended PS formalism for dark halos whose mass function is given by the YNY at each output redshift.
In this paper, we only consider a recent standard $\Lambda$CDM model, that is, $\Omega_{0}=0.3, \Omega_{\Lambda}=0.7, h=0.7$ and $\sigma_{8}=0.9$, where these parameters denote the mean density of the universe, the cosmological constant, the Hubble parameter and the normalization of the power spectrum of the initial density fluctuation field. The shape of the power spectrum given by @s95, which is a modified one of @bbks [hereafter BBKS] taking into account the baryonic effects, is adopted.
Tidal Stripping of Subhalos {#sec:tidal}
---------------------------
Most of recent high resolution $N$-body simulations suggest that swallowed dark halos survive in their host halo as [*subhalos*]{}. Envelopes of those subhalos are stripped by tidal force from the host halo. We assume the radius of a tidally stripped subhalo $r_{\rm t}$ by $$\frac{r_{\rm t}}{r_{\rm s}}=\frac{r_{\rm peri}}{r_{\rm apo}}\frac{M_{\rm h}}{M_{\rm s}}\left(\frac{V_{\rm circ,s}}{V_{\rm
circ,h}}\right)^{3},$$ where $r_{\rm peri}$ and $r_{\rm apo}$ are the pericenter and apocenter for the orbit of the subhalo, respectively, and subscripts “h” and “s” indicate the host halo and subhalo, respectively. In this paper, a ratio of $r_{\rm peri}/r_{\rm apo}=0.2$ is assumed [@gmglqs98; @oh99; @oh00]. Because a singular isothermal profile for subhalos is assumed, their mass decreases proportional to $r_{\rm t}/r_{\rm s}$. The mass of stripped subhalos is used in the estimation of dynamical response to gas removal on the size and velocity dispersion when satellite galaxies merge together (see §§\[sec:response\]).
Gas Cooling, Star Formation and Supernova Feedback
--------------------------------------------------
The mean mass density in dark halos is assumed to be proportional to the cosmic mean density at the epoch of collapse using a spherically symmetric collapse model [@t69; @gg72]. Each collapsing dark halo contains baryonic matter with a mass fraction $\Omega_{\rm
b}/\Omega_{0}$, where $\Omega_{\rm b}$ is the baryon density parameter. We adopt a value of $\Omega_{\rm b}h^{2}=0.02$ that is recently suggested by the BOOMERANG Project measuring the anisotropy of the cosmic microwave background [@boomerang]. This value is an intermediate one between $(0.64-1.4)\times 10^{-2}$ given by analysis of light element abundance produced by big bang nucleosynthesis [@syb00] and $(2.24\pm 0.09)\times 10^{-2}$ given by analysis of cosmic microwave background observed by $WMAP$ [@s03]. The baryonic matter consists of diffuse hot gas, dense cold gas, and stars.
When a halo collapses, the hot gas is shock-heated to the virial temperature of the halo with an isothermal density profile. A part of the hot gas cools and accretes to the disk of a galaxy until subsequent collapse of dark halos containing this halo. The amount of the cold gas involved is calculated by using metallicity-dependent cooling functions provided by @sd93. The difference of cooling rates between the primordial and metal-polluted gases is prominent at $T\sim 10^{6}$K due to line-cooling of metals. Chemical enrichment in hot gas is consistently solved with star formation and SN feedback. The cooling is, however, very efficient in dark halos with a virial temperature of $T\sim 10^{6}$K even in the case of the primordial gas, so the metallicity dependence of cooling rate only slightly affects our results. In order to avoid the formation of unphysically large galaxies, the cooling process is applied only to halos with $V_{\rm
circ}\leq V_{\rm cut}=$250 km s$^{-1}$. This manipulation would be needed, because the simple isothermal distribution forms so-called “monster galaxies” due to too efficient cooling at the center of halos. While @c00 adopted another isothermal distribution with central core instead of such a simple cutoff of the cooling and @bbflbc03 considered some additional mechanisms such as the heating of hot gas by SNe and by heat conduction from outside as well as its removal by superwinds from halos, we take the above simple approach. The value of $V_{\rm cut}$ is rather small compared with our previous paper and other SAMs. This is caused by our assumption that invisible stars have negligible fraction, which is introduced to darken luminosity of galaxies (§§\[sec:photo\]). This smaller value of $V_{\rm cut}$ makes the color of large galaxies less red, which shows up on a bright portion of the color-magnitude relation of elliptical galaxies (§§\[sec:cmr\]).
Stars in disks are formed from the cold gas. The SF rate (SFR) $\dot{M}_{*}$ is given by the cold gas mass $M_{\rm cold}$ and a SF timescale $\tau_{*}$ as $\dot{M}_{*}=M_{\rm cold}/\tau_{*}$. Now we consider two SF models. One is a constant star formation (CSF), in which $\tau_{*}$ is constant against redshift. Another is a dynamical star formation (DSF), in which $\tau_{*}$ is proportional to the dynamical timescale of the halo, which allows for the possibility that the SF efficiency is variable with redshift. We then express these SF timescales as $$\begin{aligned}
\tau_{*}=\left\{
\begin{array}{ll}
\displaystyle{\tau_{*}^{0}[1+\beta(V_{\rm circ})]} & \mbox{(CSF)},\\
\displaystyle{\tau_{*}^{0}[1+\beta(V_{\rm circ})]
\left[\frac{\tau_{\rm dyn}(z)}{\tau_{\rm dyn}(0)}\right]
(1+z)^{\sigma}} & \mbox{(DSF)},
\end{array}
\right.
\label{eqn:sft}\end{aligned}$$ where $\tau_{*}^{0}$ and $\sigma$ are free parameters, and $\beta$ indicates the ratio of the SF timescale to the reheating timescale by the SN feedback defined by equation (\[eqn:beta\]) (see below). Pure DSF occurs when $\sigma=0$. Because the timescale of cold gas consumption is equal to $\tau_{*}/(1+\beta-R)$, where $R$ is the returned mass fraction from evolved stars ($R=0.25$ in this paper), the mass fraction of cold gas in galaxies that is nearly constant against their magnitude is automatically adjusted by multiplying $(1+\beta)$. Hence the parameter $\alpha_{*}$ originally introduced by @c94 is eliminated by introducing the factor $(1+\beta)$. The parameter $\tau_{*}^{0}$ is so chosen as to match the mass fraction of cold gas with the observed fraction (see §3). Thereby the SF-related parameters are constrained according to @c00. Since not all of cold gas might be observed, the observed data give a lower limit to mass fraction of cold gas.
Figure \[fig:sfr\] shows the redshift dependence of SF timescale for the four SF models of CSF (dot-dashed line), DSF0 ($\sigma=0$; solid line), DSF1 ($\sigma=0.5$; dotted line) and DSF2 ($\sigma=1$; dashed line). Objects, which collapse at higher redshift, have higher density and therefore shorter dynamical timescale. Evidently, the DSF with smaller $\sigma$ gives more rapid conversion of cold gas into stars, compared with the CSF. The four SF models predict different mass fraction of cold gas at high redshift, leading to quite different characteristics of dwarf galaxies. This indicates, as will be clarified later, that the redshift dependence of SF timescale can be constrained particularly from observed structures and photometric properties of dwarf galaxies.
Massive stars explode as Type II SNe and heat up the surrounding cold gas. This SN feedback reheats the cold gas at a rate of $\dot{M}_{\rm
reheat}={M_{\rm cold}}/{\tau_{\rm reheat}}$, where the timescale of reheating is given by $$\tau_{\rm reheat}=\frac{ \tau_{*}}{\beta(V_{\rm circ})},$$ where $$\beta(V_{\rm circ})\equiv\left(\frac{V_{\rm circ}}{V_{\rm hot}}
\right)^{\alpha_{\rm hot}}.
\label{eqn:beta}$$ The free parameters $V_{\rm hot}$ and $\alpha_{\rm hot}$ are determined by matching the local luminosity function of galaxies with observations.
With the above equations and parameters, we obtain the masses of hot gas, cold gas, and disk stars as a function of time or redshift. Chemical enrichment is also taken into account, adopting the [*heavy-element yield*]{} of $y=2Z_{\odot}$, assuming the instantaneous recycling approximation with the returned mass fraction from evolved stars $R=0.25$. All of newly produced metals are released into cold gas, then by SN feedback, a part of them is expelled into hot gas. Some metals in hot gas are brought back to cold gas by subsequent cooling, and are accumulated in stars by their formation.
Mergers of dark halos and galaxies
----------------------------------
When two or more progenitor halos have merged, the newly formed larger halo should contain at least two or more galaxies which had originally resided in the individual progenitor halos. By definition, we identify the central galaxy in the new common halo with the central galaxy contained in the most massive one of the progenitor halos. Other galaxies are regarded as satellite galaxies. These satellites merge by either dynamical friction or random collision. The timescale of merging by dynamical friction is given by $\tau_{\rm mrg}=f_{\rm mrg}\tau_{\rm
fric}$, where $\tau_{\rm fric}$ is given by @bt87, which is estimated from masses of the new common halo and the tidally truncated subhalo. The parameter $f_{\rm mrg}$ is set to 0.7 in this paper. When the time elapsed after merging of a progenitor halo exceeds $\tau_{\rm
mrg}$, the satellite galaxy is accreted to the central galaxy. On the other hand, the mean free timescale of random collision of satellite galaxies $\tau_{\rm coll}$ is given by @mh97. With a probability $\Delta t/\tau_{\rm coll}$, where $\Delta t$ is the time step corresponding to the redshift interval $\Delta z$ of merger tree of dark halos, a satellite galaxy merges with another satellite picked out randomly [@sp99].
Consider the case when two galaxies of masses $m_1$ and $m_2 (>m_1)$ merge together. If the mass ratio $f=m_1/m_2$ is larger than a certain critical value of $f_{\rm bulge}$, we assume that a starburst occurs and that all of the cold gas turns into stars and hot gas, which fills the resulting halo, and all of the stars populate the bulge of a new galaxy. On the other hand, if $f<f_{\rm bulge}$, no starburst occurs, and a smaller galaxy is simply absorbed into the disk of a larger galaxy. Throughout this paper we use $f_{\rm bulge}=0.5$, which gives a consistent morphological fraction in galaxy number counts.
Size of Galaxies and Dynamical Response to Starburst-induced Gas Removal {#sec:response}
------------------------------------------------------------------------
We assume that the size of spiral galaxies is determined by a radius at which the gas is supported by rotation, under the conservation of specific angular momentum of hot gas that cools and contracts. We also assume that the initial specific angular momentum of the gas is the same as that of the host dark halo. Acquisition of the angular momentum of dark halos is determined by tidal torques in the initial density fluctuation field [@w84; @ct96a; @ct96b; @ng98]. The distribution of the dimensionless spin parameter $\lambda_{H}$, which is defined by $\lambda_{H}\equiv L|E|^{1/2}/GM^{5/2}$ where $L$ is the angular momentum and $E$ is the binding energy, is well approximated by a log-normal distribution [@mmw98], $$p(\lambda_{H})d\lambda_{H}=
\frac{1}{\sqrt{2\pi}\sigma_{\lambda}}
\exp\left[-\frac{(\ln\lambda_{H}-\ln\bar{\lambda})^2}
{2\sigma_{\lambda}^{2}}\right] d\ln\lambda_{H},
\label{eqn:spin}$$ where $\bar{\lambda}$ is the mean value of spin parameter and $\sigma_{\lambda}$ is its logarithmic variance. We adopt $\bar{\lambda}=0.03$ and $\sigma_{\lambda}=0.5$. When the specific angular momentum is conserved, the effective radius $r_{e}$ of a presently observed galaxy at $z=0$ is related to the initial radius $R_{i}$ of the progenitor gas sphere via $r_{e}=(1.68/\sqrt{2})\lambda_{H}R_{i}$ [@f79; @fe80; @f83]. The initial radius $R_{i}$ is set to be the smaller one between the virial radius of the host halo and the cooling radius. A disk of a galaxy grows due to cooling and accretion of hot gas from more distant envelope of its host halo. In our model, when the estimated radius by the above equation becomes larger than that in the previous time-step, the radius grows to the new larger value in the next step. At that time, the disk rotation velocity $V_{d}$ is set to be the circular velocity of its host dark halo.
Size estimation of high-redshift spiral galaxies, however, carries uncertainties because of the large dispersion in their observed size distribution. For example, @s99 suggests only mild evolution of disk size against redshift, taking into account the selection effects arising from the detection threshold of surface brightness, although the above simple model predicts disk size proportional to virial radius $R_{\rm vir}$ of host dark halos evolving as $R_{\rm vir}\propto 1/(1+z)$ for fixed mass. Allowing for the possibility that the conservation of angular momentum is not complete, we generalize this size estimation by introducing a simple redshift dependence, $$r_{e}=\frac{1.68}{\sqrt{2}}\lambda_{H}R_{i}(1+z)^{\rho},
\label{eqn:sizerho}$$ where $\rho$ is a free parameter. We simply use $\rho=1$ as a reference value in this paper. The effect of changing $\rho$ emerges in the selection effects due to the cosmological dimming of surface brightness and in the dust extinction, because the dust column density also changes with galaxy size. This is discussed in §\[sec:consistency\].
Size of early-type galaxies, which likely form from galaxy mergers, are primarily determined by the virial radius of the baryonic component. When a major merger of galaxies occurs, assuming the energy conservation, we estimate the velocity dispersion of the merged system. Now we assign the subscript 0 to the merged galaxy, and subscripts 1 and 2 to the central and satellite galaxies, respectively, in the case of central-satellite merger, or to larger and smaller galaxies, respectively, in the case of satellite-satellite merger. Using the virial theorem, the total energy for each galaxy is $$E_{i}=-\frac{1}{2}[M_{b}V_{b}^{2}+(M_{d}+M_{\rm cold})V_{d}^{2}],$$ where $M_{b}$ and $M_{d}$ are the masses of bulge and disk, respectively, and $V_{b}$ and $V_{d}$ are the velocity dispersion of bulge and the rotation velocity of disk, respectively. Assuming the virial equilibrium, the binding energy $E_{b}$ between the progenitors just before the merger is given by $$E_{b}=-\frac{E_{1}E_{2}}{(M_{2}/M_{1})E_{1}+(M_{1}/M_{2})E_{2}}.$$ Then we obtain $$E_{1}+E_{2}+E_{b}=E_{0}.$$ Just after the merger there is only the bulge component consisting of cold gas and stars in the merger remnant, whose velocity dispersion is directly estimated from the above equation. This procedure is similar to @c00, although they argued it in terms of size estimation. Then the size of the system just after the merger is defined by $$r_{i}=\frac{GM_{i}}{2V_{b}^{2}},$$ where $M_{i}=M_{*}+M_{\rm cold}$ is the total baryonic mass of the merged system.
Next, the cold gas turns into stars and hot gas. Newly formed stellar mass is nearly equal to $M_{\rm cold}/(1+\beta)$ and the rest of the cold gas is expelled from the merged system to the halo by SN feedback. The final mass after the mass loss, $M_{f}$, can be estimated from the known $\beta$. Assuming the density distributions of baryonic and dark matters, the dynamical response on the structural parameters to the mass loss can be estimated. In this paper we adopt the Jaffe model [@jaffe] for baryonic matter and the singular isothermal sphere for dark matter, and assume slow (adiabatic) gas removal compared with dynamical timescale of the system. Defining the ratios of mass, size, density and velocity dispersion at final state relative to those at initial state by ${\cal M}, R, Y$ and $U$, the response under the above assumption is approximately given by $$\begin{aligned}
R&\equiv&\frac{r_{f}}{r_{i}}=\frac{1+D/2}{{\cal M}+D/2},\\
U&\equiv&\frac{V_{b,f}}{V_{b,i}}=\sqrt{\frac{YR^{2}+Df(z_{f})/2}{1+Df(z_{i})/2}},\end{aligned}$$ where ${\cal M}=YR^{3}$, $D=1/y_{i}z_{i}^{2}$, $y$ and $z$ are the ratios of density and size of baryonic matter to those of dark matter and, $f(z)$ is a function defined in Appendix. The subscripts $i$ and $f$ stand for the initial and final states in the mass loss process. The details are shown in Appendix and @ny03. The parameter $D$ indicates the contribution of dark matter to the gravitational potential felt by baryonic matter in the central region of a halo. In actual calculation, we use a circular velocity at the center of dark halos, $V_{\rm cent}$ defined below, to estimate $D$ as $2V_{\rm
cent}^{2}/V_{b,i}^{2}$, instead of the ratios of size $y_{i}$ and density $z_{i}$ which characterize the global property of dark halos. If there is negligible dark matter ($D\to 0$), the well-known result of adiabatic invariant $(M_{*}+M_{\rm cold})r$ emerges [e.g., @ya87]. In contrast, if there is negligible baryonic matter ($D\to\infty$), $R$ and $U$ become unity, that is, size and velocity dispersion do not change during mass loss. The effect of the dynamical response is the most prominent for dwarf galaxies of low circular velocity.
Considering realistic situations, the baryonic matter often condenses in the central region and becomes denser than the average density in the halo. It is likely that the depth of gravitational potential well is changed when a part of baryonic mass is removed due to SN feedback. To take into account this process, we define a central circular velocity of dark halo $V_{\rm cent}$. When a dark halo collapses without any progenitors, $V_{\rm cent}$ is set to $V_{\rm circ}$. After that, although the mass of the dark halo grows up by subsequent accretion and/or mergers, $V_{\rm cent}$ remains constant or decreases by the dynamical response. When the mass is doubled, $V_{\rm cent}$ is set to $V_{\rm circ}$ at that time again. The dynamical response to mass loss from a central galaxy of a dark halo by SN feedback lowers $V_{\rm
cent}$ of the dark halo as follows: $$\frac{V_{{\rm cent},f}}{V_{{\rm cent},i}}=
\frac{M_{f}/2+M_{d}(r_{i}/r_{d})}{M_{i}/2+M_{d}(r_{i}/r_{d})}.$$ The change of $V_{\rm cent}$ in each time step is only a few per cent. Under these conditions the approximation of static gravitational potential of dark matter is valid during starburst.
Once a dark halo falls into its host dark halo, it is treated as a subhalo. Because we assume that subhalos do not grow up in mass, the central circular velocity of the subhalos monotonically decreases. Thus this affects the dynamical response later when mergers between satellite galaxies occur. We approximate that the resultant density distribution remains to be isothermal with $V_{\rm cent}$ at least within the galaxy size.
Photometric Properties and Morphological Identification {#sec:photo}
-------------------------------------------------------
The above processes are repeated until the output redshift and then the SF history of each galaxy is obtained. For the purpose of comparison with observations, we use a stellar population synthesis approach, from which the luminosities and colors of model galaxies are calculated. Given the SFR as a function of time or redshift, the absolute luminosity and colors of individual galaxies are calculated using a population synthesis code by @ka97. The stellar metallicity grids in the code cover a range from $Z_{*}=$0.0001 to 0.05. Note that we now define the metallicity as the mass fraction of metals. The initial stellar mass function (IMF) that we adopt is the power-law IMF of Salpeter form, with lower and upper mass limits of $0.1M_{\odot}$ and $60M_{\odot}$, respectively.
In most of SAM analyses, it has been assumed that there is a substantial fraction of invisible stars such as brown dwarfs. @c94 introduced a parameter defined as $\Upsilon=(M_{\rm lum}+M_{\rm
BD})/M_{\rm lum}$, where $M_{\rm lum}$ is the total mass of luminous stars with mass larger than $0.1M_\odot$ and $M_{\rm BD}$ is that of invisible brown dwarfs. A range of $\Upsilon\sim 1-3$ has been assumed depending on ingredients of SAMs. For example, @c00 assumed $\Upsilon=3.07$ in the case of $\Omega_{\rm b}=0.04$. In this paper, however, we do not assume the existence of substantial fraction of invisible stars. If a large value of $\Upsilon$ is adopted, the mass-to-light ratio of galaxies is too high to agree with observations. Thus we fix $\Upsilon=1$. @sp99 also adopted a small value of $\Upsilon=1.25$ in their $\Lambda$CDM.3 model for $\Omega_{\rm b}=0.037$ (in their notation $f^{*}_{\rm lum}=1/\Upsilon=0.8$).
The optical depth of internal dust is consistently estimated by our SAM. We take the usual assumption that the abundance of dust is proportional to the metallicity of cold gas, and then the optical depth is proportional to the column density of metals. Then the optical depth $\tau$ is given by $$\tau\propto\frac{M_{\rm cold}Z_{\rm cold}}{r_{e}^{2}},
\label{eqn:dust}$$ where $r_{e}$ is the effective radius of the galactic disk. There are large uncertainties in estimating the proportionality constant, but we adopt about a factor of two smaller value compared with that in @c00, otherwise it predicts too strong extinction to reproduce galaxy number counts, presumably because our chemical yield is higher than theirs. Wavelength dependence of optical depth is assumed to be the same as the Galactic extinction curve given by @seaton79. Dust distribution is simply assumed to be the slab dust [@ddp89], according to our previous papers. We found that the resultant mean extinction for spiral galaxies is close to a model by @c00.
We classify galaxies into different morphological types according to the $B$-band bulge-to-disk luminosity ratio $B/D$. In this paper, following @sdv86, galaxies with $B/D\geq 1.52$, $0.68\leq
B/D<1.52$, and $B/D<0.68$ are classified as elliptical, lenticular, and spiral galaxies, respectively. @kwg93 and @bcf96 showed that this method of type classification well reproduces the observed type mix.
Parameter Settings {#sec:paramset}
==================
As already mentioned, we adopt a standard $\Lambda$CDM model. The cosmological parameters are $\Omega_{0}=0.3, \Omega_{\Lambda}=0.7,
h=0.7$ and $\sigma_{8}=0.9$. The baryon density parameter $\Omega_{\rm
b}=0.02h^{-2}$ is used.
The astrophysical parameters are constrained from local observations, according to the procedure discussed in @ntgy01 [@nytg02]. The adopted values are slightly different from those in our previous papers. This is mainly caused by adopting the different mass function of dark halos (§§\[sec:mh\]) and by fixing $\Upsilon=1$ (§§\[sec:photo\]). Values of these parameters are tabulated in Tables \[tab:astro0\] and \[tab:astro\].
[llll]{} $\begin{array}{ll}
{V_{\rm hot}}\\
{\alpha_{\rm hot}}
\end{array}$ & $\left. \begin{array}{ll}
150 \mbox{km~s}^{-1} \\
4
\end{array}\right\}$ & supernova feedback-related (§§2.3) & luminosity functions (Figure 3)\
$V_{\rm cut}$ & 250 km s$^{-1}$ & cooling cut-off (§§2.3) & luminosity functions (Figure 3)\
$V_{\rm low}$ & 30 km s$^{-1}$ & minimum circular velocity of dark halos (§§2.1) & —\
[$y$]{} & 2 $Z_{\odot}$ & heavy-element yield (§§2.3) & metallicity distribution (Figure 18)\
[$f_{\rm bulge}$]{} & 0.5 & major/minor merger criterion (§§2.4) & morphological counts\
[$f_{\rm mrg}$]{} & 0.7 & coefficient of dynamical friction timescale (§§2.4) & luminosity functions (Figure 3)\
$\begin{array}{ll}
{\bar{\lambda}}\\
{\sigma_{\lambda}}
\end{array}$ & $\left. \begin{array}{ll}
{0.03}\\
{0.5}
\end{array}\right\}$ & spin parameter distribution (§§2.5) & disk size (Figure 5)\
$\rho$ & 1 & redshift dependence of disk size (§§2.5) & faint galaxy number counts (Figures 20 and 21)\
[$\Upsilon$]{} & 1 & fraction of invisible stellar mass (§§2.6) & mass-to-light ratio (Figure 17)\
[ccc]{} CSF & 1.3 & –\
DSF0 & 1.7 & 0\
DSF1 & 1.3 & 0.5\
DSF2 & 1.5 & 1\
First, the SN feedback-related parameters ($V_{\rm hot}, \alpha_{\rm
hot}$) and merger-related parameter ($f_{\rm mrg}$) are almost uniquely determined if their values are so chosen as to reproduce the local luminosity function. Figure \[fig:lf\] shows theoretical results for CSF (solid line), DSF2 (dashed line), DSF1 (dot-dashed line) and DSF0 (dotted line). As in our previous papers, the SF timescale affects the local luminosity function only slightly. Symbols with errorbars represent observational results from the $B$-band redshift surveys, such as Automatic Plate Machine [APM; @l92], ESO Slice Project [ESP; @z97], Durham/United Kingdom Schmidt Telescope [UKST; @r98] and Two-Degree Field [2dF; @f99], and from the $K$-band redshift surveys given by @s98, Two Micron All Sky Survey [2MASS; @k01] and 2dF combined with 2MASS [@c01].
In Figure \[fig:lf\] we see how the mass function of dark halos affects the luminosity function. In the same figure we show the same model as CSF but for the PS mass function represented by the thin solid lines. As the PS mass function predicts more dark halos and hence more galaxies than the YNY mass function except for the largest mass scale. Thus, we need $V_{\rm hot}=150$ km s$^{-1}$, less than 280km s$^{-1}$ in the previous model, to weaken SN feedback which mainly determines the scale of the exponential cut-off of the luminosity function [@ntgy01; @nytg02]. We also investigated the effects of power spectrum of density fluctuations and confirmed that neither a $\sigma_{8}=1$ model nor a model with BBKS power spectrum without the baryonic effects is significantly different from our reference model. In §\[sec:consistency\], we discuss the slight effect that $\sigma_{8}$ has on high-redshift galaxies.
The effects of photoionization on the luminosity function have been discussed [@cn94; @ngs99; @bkw00; @ng01; @s02; @tstv02; @blbcf02a; @bflbc02b; @bfbcl03a]. There are two ways to suppress the gas cooling by photoionization. One effect is that the ultraviolet (UV) background from quasars and/or young stars prevents hot gas at the outer envelope from cooling. Another is that the Jeans mass becomes larger after cosmic reionization, which means larger $V_{\rm low}$. It has been found that both processes lower the faint-end of the luminosity function. For example, if we use a large value of $V_{\rm low}\simeq
70$ km s$^{-1}$ for our model, which corresponds to increasing the Jeans mass, the number of galaxies at the faint-end of the resultant luminosity function decreases by about a factor of two, less than that given by the APM survey at $-20\la M_{B}-5\log(h)\la-17$ (see Figure \[fig:lf\]). In that case we need weaker SN feedback, that is, smaller $V_{\rm hot}$ and/or smaller $\alpha_{\rm hot}$. We found, however, that such a large value of $V_{\rm low}$ does not make dwarf spheroidals at $M_{B}-5\log(h)\ga -12$ for $V_{\rm low}=70$ km s$^{-1}$ and at $M_{B}-5\log(h)\ga -10$ for $V_{\rm low}=50$ km s$^{-1}$. In addition, giant galaxies obtain more gas that has not cooled in smaller halos $V_{\rm circ}\leq V_{\rm low}$, causing the bright-end of the luminosity function to shift brighter. Note that this is not effective for $V_{\rm low}\la 40$ km s$^{-1}$. Keeping in mind that such effects might affect our analysis through the determination of SN feedback-related parameters, we use $V_{\rm low}=30$ km s$^{-1}$ in this paper.
Next, the SFR-related parameter ($\tau_*^0$) is determined by using the mass fraction of cold gas in spiral galaxies. The gas fraction depends on both the SN feedback-related and SFR-related parameters. The former parameters determine the gas fraction expelled from galaxies and the latter the gas fraction that is converted into stars. Therefore, in advance of determining the SFR-related parameters, the SN feedback-related parameters must be determined by matching the local luminosity function.
Figure \[fig:gas\] shows the ratio of cold gas mass relative to $B$-band luminosity of spiral galaxies as a function of their luminosity. Theoretical result is shown only for the CSF model by the solid line. Other SF models of DSF2, DSF1 and DSF0 provide almost the same results, and their differences from the CSF are only about 0.1 dex. We here assume that 75% of the cold gas in the models is comprised of hydrogen, i.e., $M_{\rm HI}=0.75M_{\rm cold}$. data, taken from @hr88, are shown by open squares with errorbars. Since their data do not include the fraction of H$_{2}$ molecules, they should be regarded as providing a lower limit to the mass fraction of cold gas. Adopted values of the parameters in the SF models are tabulated in Table \[tab:astro\]. Slight difference in the values of $\tau_{*}^{0}$ stems from the different redshift-dependence of SF timescale.
Figure \[fig:r\] shows the effective disk radii of local spiral galaxies as a function of their luminosity only for the CSF model. Other SF models also provide almost the same results as CSF and are not shown. Thus, with the use of ($\bar{\lambda}, \sigma_{\lambda}$)=(0.03, 0.5), all the SF models well reproduce the observed disk size-magnitude relation (thin solid line) compiled by @ty00 based on the data taken from @isib96, while showing a slightly steeper slope than the observed one.
Results
=======
Relationships between Size and Magnitude
----------------------------------------
First, we examine the size distribution of elliptical galaxies, because it is directly affected by the dynamical response to gas removal. Figure \[fig:rad1\] shows the contour distribution in the effective radius versus absolute $B$-magnitude diagram. The levels of contours in order from outside to inside are 0.02, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9 times the largest number of galaxies in grids. This way of description applies to all the contour distributions shown below. The four panels show the results for the models of CSF, DSF2, DSF1 and DSF0, respectively, as indicated. Effective radius $r_{e}$ of model galaxies are defined by 0.744$r_{b}$, where $r_{b}$ is the three-dimensional half-mass radius, assuming the de Vaucouleurs profile [@ny03]. The filled triangles and squares represent the data taken from @bbf92 [@bbf93] and @m98, respectively. When some of dwarf spheroidals are listed in both their tables, we use the data in @bbf92 [@bbf93]. Observed elliptical galaxies have two sequences of dispersed dwarf galaxies with low surface brightness and compact dwarf galaxies with high surface brightness.
As shown in Figure \[fig:sfr\], the CSF model has the longest SF timescale at high redshift among the four SF models. This means that when a major merger occurs in the CSF model, a significant fraction of gas is removed, so that the dynamical response of dwarf galaxies is expected to be the largest. Figure \[fig:rad1\] clearly shows that the CSF model ([*panel a*]{}) predict many dwarf galaxies with very large size of $r_{e}\sim 10-10^{2}h^{-1}$kpc for $M_{B}-5\log h\ga -10$. In the range of $-10\ga M_{B}-5\log h\ga-15$, the peak of distribution is on the sequence of dispersed dwarf ellipticals. On the other hand, the DSF0 model ([*panel d*]{}) has the shortest SF timescale at high redshift, so that the effects of gas removal are not significant. Most of galaxies are distributed along a single power-law sequence, corresponding to the compact dwarf galaxies. The slope is determined by $\alpha_{\rm hot}$ as shown in @ntgy01, in which they simply determined the size of elliptical galaxies by $GM_{b}/2V_{b}^{2}$. The behavior of SF timescale indicates that the models of DSF1 and DSF2 are intermediate between CSF and DSF0. This suggests that the SF timescale should be constant independent of redshift or very weakly dependent on redshift, but not proportional to the dynamical timescale of galaxies.
It should be noted that the gas removal is not a unique mechanism to make large ellipticals. If a galaxy just after major merger keeps a large fraction of gas, its size becomes larger than that of gas-poor system with the same stellar mass because the size is proportional to the total mass of stars and cold gas. This can be seen directly if we do not take into account the effects of dynamical response to gas removal. Figure \[fig:rad2\] shows the same result as Figure \[fig:rad1\] except without the effects of dynamical response. The overall shapes are similar, independent of SF timescale, because the effects of dynamical response are not taken into account. However the peak locations in Figure \[fig:rad2\] are different from those in Figure \[fig:rad1\]. This is because the mass of cold gas just after major merger determines its size as mentioned above and therefore the size depends on the SF timescale. In other words, the CSF model has a larger fraction of cold gas and hence a larger galactic mass at high redshift, compared with the DSF models. This situation without the effects of dynamical response corresponds to the case of [*dominant dark halo*]{} in @ds86, in which the size and velocity dispersion do not change during the gas removal. We therefore conclude that in realistic situations the dark matter is not always dominant in the gravitational potential. Note that while the $r_e - M_B$ relation is reproduced even without the effects of the dynamical response, as will shown in the next subsection, the Faber-Jackson relation is not reproduced unless the effects are properly taken into account (see Figures \[fig:fj1\] and \[fig:fj2\]).
In addition to size and magnitude, surface brightness is also an important observable quantity. Two sequences of dispersed and compact dwarf galaxies are prominent in the surface brightness versus absolute $B$-magnitude diagram, as shown in Figure \[fig:sb1\]. Surface brightness of model galaxies is defined as the average brightness in the area encircled by effective radius. For giant ellipticals, surface brightness is predicted to become brighter towards brighter magnitude with shallow slope, converging to $\mu_{e,B}\simeq 22$, as observed. For $M_{B}-5\log h\ga -18$, surface brightness widely spreads. This magnitude corresponds to $V_{\rm hot}$, that is, the magnitude at which $\beta$ becomes larger than unity. The CSF and DSF2 models make the widely spread distribution which reproduces two sequences simultaneously and thus are likely, while the DSF0 makes one sequence of compact ellipticals only.
In all the SF models there are many galaxies with very low surface brightness for $M_{B}-5\log h\ga-15$. Actually such galaxies cannot be detected unless the detection threshold of surface brightness is faint enough in galaxy survey observations. With a cutoff of $\mu_{e,B}=26.5$ taken into account in both the model and observed data, we show the surface brightness versus absolute $B$-magnitude diagram in Figure \[fig:sb3\] and the effective radius versus absolute $B$-magnitude diagram in Figure \[fig:rad3\]. For the models of CSF and DSF2 the resultant distributions in both of these diagrams are consistent with the data.
As another combination of observable quantities, the surface brightness versus size relation is often referred to as the Kormendy relation [@k77]. Figure \[fig:kor\] shows the theoretical distribution of elliptical galaxies with $\mu_{e,B}\leq 26.5$ in the surface brightness versus effective radius diagram. Extended distribution toward low surface brightness is clearly seen for the models of CSF and DSF2.
Faber-Jackson Relation
----------------------
Velocity dispersion of elliptical galaxies is also an independent dynamical observable. If the dark matter dominates the baryonic matter, the velocity dispersion of galaxies reflects that of their host dark halos. On the other hand, in the case of negligible dark matter, the velocity dispersion is substantially affected by dynamical response to gas removal in proportion to the mass fraction of removed gas. Thus the velocity dispersion versus absolute magnitude relation for elliptical galaxies, often called Faber-Jackson (FJ) relation, provides another strong constraint on galaxy formation [@fj76].
Figure \[fig:fj1\] shows the velocity dispersion versus absolute $B$-magnitude diagram for the models of CSF, DSF2, DSF1 and DSF0. Velocity dispersion of model galaxies is assumed to be isotropic and is converted to one-dimensional central dispersion by $\sigma_{0}(\rm
1D)=V_{\rm b}/\sqrt{3}$ after increased to the central value by a factor of $\sqrt{2}$ according to the de Vaucouleurs-like profile. For the DSF0 model, galaxies with velocity dispersion less than 10km s$^{-1}$ is scarcely distributed. Note that the cutoff circular velocity $V_{\rm
low}$, above which dark halos are identified as isolated objects, is 30km s$^{-1}$ nearly corresponding to the Jeans scale for collapse after cosmic reionization [@og96]. This makes the sequence that converges to $\sigma_{0}(\rm 1D)=30/\sqrt{3}\simeq 17$ km s$^{-1}$. For the DSF0 model, we see only such sequence because the smallest fraction of gas at major merger in this model gives very weak effects of dynamical response to gas removal. It should be noted that this tendency hardly depends on $V_{\rm low}$ because the SN feedback is very efficient for dwarf galaxies. In contrast, other SF models can reproduce the low velocity dispersion observed for local dwarf spheroidals for $M_{B}-5\log h\ga-15$.
It is possible that dwarf galaxies with very low velocity dispersion would remain undetected from actual observations, because such galaxies are expected to have very low surface brightness as a result of dynamical expansion by gas removal. Figure \[fig:fj3\] therefore shows the FJ relation after excluding galaxies of lower surface brightness with $\mu_{e,B}\geq 26.5$. Most of dwarf galaxies of low circular velocity are excluded from the CSF model and their distribution is in apparent disagreement with the data. This situation is a little improved for the models of DSF1 and DSF2. Figure \[fig:fj3\] clearly indicates the difficulty of making dwarf galaxies of low circular velocity and high surface brightness. In other words, the FJ relation strongly constrains the SF timescale requiring the mild or negligible redshift dependence, when compared with the dynamical timescale.
Here we examine the effects of the dynamical response on velocity dispersion. Figure \[fig:fj2\] shows the velocity dispersion versus absolute $B$-magnitude diagram without the effects of dynamical response as in Figure \[fig:rad2\]. Evidently, all four SF models cannot reproduce the dwarf galaxies of low circular velocity and give almost the same distribution on this diagram independent of SF timescale. This hilights the importance of dynamical response for the formation of dwarf galaxies. @kc98 predicted the FJ relation in their SAM with no dynamical response taken into account and claimed to find good agreement with observations. Their result is, however, limited only to $M_{B}\la -18$, where such dynamical response has no significant effect. Thus our SAM analysis is the first that has reproduced the observed velocity dispersion even for dwarf galaxies.
Mass-to-Light Ratio
-------------------
The mass, which is used to estimate the mass-to-light ratio, is the [*dynamical mass*]{} of $M_{\rm dyn}\propto r_{b}\sigma^{2}/G$. In our SAM, the size $r_{b}$ and the velocity dispersion $\sigma$ are determined by the amount of baryonic matter that escapes out of the system against the underlying dark matter potential. In this paper we define the dynamical mass by $M_{\rm dyn}=2r_{b}V_{b}^{2}/G$. Figure \[fig:ml1\] shows the mass-to-light ratio of elliptical galaxies as a function of absolute $B$-magnitude for the four SF models.
Our SAM well reproduces the observed mass-to-light ratio for giant ellipticals, and at least qualitatively the mass-to-light ratio for dwarf spheroidals that increases towards faint magnitude. As for giant ellipticals, observed data still have a large scatter. This might be caused by an uncertainty in estimating dynamical mass owing to anisotropic kinematics and rotation.
Such trend of the mass-to-light ratio for dwarf spheroidals is not caused by the dynamical response alone. Figure \[fig:ml2\] shows the mass-to-light ratio with no dynamical response taken into account, which turns out to be very similar to or slightly higher than that with dynamical response in Figure \[fig:ml1\]. Therefore, considerably high values up to $M/L_B\sim 10^{3}$ are caused by the dominant dark halo in which the size and velocity dispersion are kept unchanged during the starburst. Since the luminosity $L_{B}$ is proportional to the final mass $M_{f}$, it follows that $M/L_{B}\propto
M_{i}/M_{f}\propto\beta$ in the limit of $\beta\gg 1$, where $\beta$ measures the strength of the SN feedback in equation (\[eqn:beta\]). Using a relation $M\propto V_{\rm circ}^{3}$ from the spherical collapse model [@t69; @gg72], we obtain $$\frac{M}{L_{B}}\propto\beta\propto \sigma_{i}^{-\alpha_{\rm hot}}\propto
M_{f}^{-\alpha_{\rm hot}/(3+\alpha_{\rm hot})}.$$ If we adopt $\alpha_{\rm hot}=4$, then $M/L_{B}\propto M_{f}^{-4/7}
\propto 10^{8M_{B}/35}$. The slope of this relation explains the result in Figure \[fig:ml2\]. Therefore, the smaller mass fraction of dark matter lowers the mass-to-light ratio [@ds86; @ya87]. In apparent contrast to this, the DSF0 model scarcely forms galaxies of such high mass-to-light ratio. This is because the size is determined by the baryonic mass, $r_{e}\propto GM_{b}/V_{b}^{2}$, and because the removed gas is negligible due to the short SF timescale at high redshift.
By excluding galaxies of lower surface brightness with $\mu_{e,B}\geq 26.5$, we obtain gross agreement with the observed mass-to-light ratio in Figure \[fig:ml3\].
Metallicity-Magnitude Relation {#sec:metallicity}
------------------------------
Figure \[fig:metal3\] shows the mean stellar metallicity as a function of absolute $B$-magnitude for elliptical galaxies. The vertical axis represents the logarithmic iron abundance \[Fe/H\]. Shown by symbols are the data compiled by @m98 and those translated from Mg$_{2}$ index for those taken by @bbf93, whereas by contours the $L_B$-weighted average of logarithmic metal abundances of stars $\langle\log(Z_{*}/Z_{\odot})\rangle_{L_{B}}$ for the theoretical prediction. Since the theoretical results are almost independent of whether galaxies of low surface brightness are excluded, we only show the metallicity distribution excluding such galaxies with $\mu_{e,B}\geq
26.5$. We do not show the metallicity distribution for spirals, which is similar to that for ellipticals.
All the SF models well reproduce the observed tight relation between metallicity and absolute magnitude. This means that such relation is not affected by the redshift dependence of SF timescale, because we adopt the SN feedback in the same way irrespective of either continuous or burst formation of stars in galaxies. With strong SN feedback with $\alpha_{\rm hot}\simeq 4$, our SAM succeeds to explain the observed low metallicity \[Fe/H\]$\sim -2$ at $M_{B}-5\log h\simeq -10$, in spite of using a rather high value of metallicity yield $y=2Z_{\odot}$. Note that if $\alpha_{\rm hot}=2$, we obtain about a factor of three higher metallicity at that magnitude. This is consistent with the conclusion by @c00 that with $\alpha_{\rm hot}=2$ the metallicities of their dwarf galaxies are systematically higher than the data, while such predicted values reside in a range of large observed scatter.
Consistency with Other Observations {#sec:consistency}
===================================
In this section, we show the whole aspects of our SAM to check the consistency with other local and high-redshift observations. This will clarify the limitations of present SAM analyses and physical processes for which further investigation is required. In the first three subsections (§§5.1-5.3), high-redshift properties of model galaxies are examined by comparing the observed galaxies in the Hubble Deep Field [HDF, @w96] and the Subaru Deep Field [SDF, @m01]. Then, the Tully-Fisher relation (TFR) for local spiral galaxies (§5.4) and the color-magnitude relation (CMR) for cluster elliptical galaxies (§5.5) are discussed.
Faint Galaxy Number Counts {#sec:counts}
--------------------------
Counting galaxies as a function of apparent magnitude is one of the most important observable quantities for constraining the geometry of the universe and the evolution of galaxies [e.g., @yt88]. We already showed that our SAM can simultaneously reproduce the UV/optical galaxy counts in the HDF [@ntgy01] and the near-infrared galaxy counts in the SDF [@nytg02]. These previous works demonstrated that inclusion of the selection effects arising from the cosmological dimming of surface brightness of high-redshift galaxies derived by @y93 is essential, because the number count of galaxies is obtained by summing up the product of luminosity function and cosmological volume element to the accessible maximal redshift above which galaxies have the surface brightness fainter than the threshold and thus are not detected. Details of the method to estimate such selection effects are found in @y93, @ty00, @t01 and @ntgy01 [@nytg02].
In this paper the YNY mass function of dark halos (§§\[sec:mh\]) is used instead of the PS, and the dynamical response to gas removal is a novel ingredient in the analysis. With these modifications we reexamine the galaxy counts again. Figure \[fig:hdf\] shows the UV/optical galaxy counts in the HDF ($U_{300}, B_{450}, V_{606}$ and $I_{814}$ in the AB system). For reference, other ground-based observed counts are also plotted. The thick solid and dashed lines show the predictions of CSF and DSF2, respectively, taking into account the absorption by internal dust [@ty00] and by intervening clouds [@yp94], and the selection effects due to cosmological dimming of surface brightness [@y93]. We adopt $\rho=1$ in equation (\[eqn:sizerho\]), which determines the disk size at high redshift. The thin lines represent the same models, except without the selection effects. We do not show the models of DSF1 and DSF0, because they are much the same as CSF and DSF2 with only small difference comparable to observational errors. These models only slightly underpredict the $U_{300}$ count and overpredict the $I_{814}$ count, as pointed out by @ntgy01.
Figure \[fig:sdf\] shows the near-infrared galaxy counts in the SDF ($K'$). Like the HDF counts, the models of CSF and DSF2 well reproduce the SDF counts. For the purpose of comparison, the DSF0 model is also shown. Because of the highest SFR at high redshift, the stellar mass in this model increases rapidly, making too many faint galaxies to reproduce the observation. Therefore, the models of CSF and DSF2 give the predictions that agree with the HDF and SDF counts simultaneously.
Redshift Distribution
---------------------
Figure \[fig:hdfz\] shows the redshift distributions of $I_{814}$-selected galaxies in the HDF with $22\leq I_{814}\leq 24$ ([*panel a*]{}), $24\leq I_{814}\leq 26$ ([*panel b*]{}) and $26\leq
I_{814}\leq 28$ ([*panel c*]{}). The number of model galaxies in each panel is calculated over the same celestial area as the HDF. The thick solid, dashed and dot-dashed lines show the results of CSF, CSF with $\sigma_{8}=1$ and DSF0, respectively. The histogram in each panel is the observed photometric redshift distribution by @f00, in which they improved the redshift estimation over @fly99. The difference between the results of CSF and DSF0 is very small for bright galaxies with $I_{814}\leq 26$ ([*panels a*]{} and [*b*]{}). While the CSF model reproduces the observed redshift distribution over an entire range of apparent $I_{814}$-magnitudes considered, the DSF0 model predicts too many high-redshift galaxies for $I_{814}\geq 26$ ([*panel c*]{}), especially at $z\ga 3$. This is because the shorter SF timescale in the DSF0 model induces earlier formation of galaxies compared with CSF. These results are consistent with our previous results. We show the same model as CSF but for $\sigma_{8}=1$. Since the formation epoch of dark halos shifts to higher redshift, a little more galaxies are formed at higher redshifts. We also find that the DSF2 model reproduces the observed redshift distribution with negligible difference from the CSF model with $\sigma_{8}=1$.
Isophotal Area-Magnitude Relation {#sec:angular}
---------------------------------
Figure \[fig:angsize\] shows the isophotal area of galaxies plotted against their $K'$-magnitude, for which the same observational condition employed in the SDF survey is used to calculate the isophoto in the SF models. The solid line indicates the the mean relation with errorbars of 1$\sigma$ scatter, predicted by the CSF model. Results of other SF models are almost the same, and are not shown. The data plotted by the crosses are those for the SDF galaxies that are detected in both the $K'$- and $J$-bands. As stressed in our previous papers, the selection effects from the cosmological dimming of surface brightness of galaxies cannot be ignored in the SAM analysis of galaxy number counts. This indicates that the size of high-redshift galaxies must be modeled properly.
With the selection effects correctly taken into account, the predicted size should converge towards the limiting magnitude, because faint galaxies with larger area have surface brightnesses below the detection threshold and then remain undetected. We find from this figure that our SAM galaxies well reproduce the observed area-magnitude relation, and are consistent with the SDF galaxies, only when the selection bias against faint galaxies with high redshift and/or low surface brightness is taken into account in the analysis.
Tully-Fisher Relation (TFR)
---------------------------
Figure \[fig:tf\] shows the $I$-band TFR. The thick solid line indicates the predicted TFR with errorbars of 1$\sigma$ scatter for gas-rich spirals in the CSF model, having more than 10% mass fraction of cold gas in the total galactic mass, that is, $M_{\rm cold}/(M_{*}+M_{\rm cold})\geq 0.1$. This criterion is the same as that used by @c00. The thick dashed line is for all spirals. The tendency that gas-rich spirals for a given magnitude are always brighter than gas-poor spirals has already been shown in @c00. Results of other SF models are not shown because of negligible difference.
The thin dashed, dot-dashed and dotted lines are the best-fit results to the observed TFRs by @mfb92, @pt92, and @g97, respectively. We assume that the line-width $W$ is simply twice the disk rotation velocity $V_{\rm rot}$ as usual. For model galaxies, we set $V_{d}$ to be equal to $V_{\rm rot}$, as mentioned in §§\[sec:response\].
The predicted TFR slope for galaxies with small rotation velocity $V_{\rm rot}\la 80$km is steeper than that observed, while the predicted slope and magnitude for galaxies with larger velocity agree well with the observations. The TFR slope is determined by $\alpha_{\rm hot}$, because this parameter relates the mass fraction of stars in each galaxy, which gives luminosity, to the rotation velocity. We use $\alpha_{\rm hot}=4$ throughout as a standard, but smaller $\alpha_{\rm
hot}$ gives shallower slope. For the purpose of comparison, we also plot the CSF model with $\alpha_{\rm hot}=2$ by the dot-dashed line. In this case the slope becomes closer to that observed. Although @c00 adopted $\alpha_{\rm hot}=2$ and claimed agreement of their model with the observed TFRs, we found that this model predicts too many high-redshift galaxies to be reconciled with the observed number counts. Moreover, mean metallicity of stars in dwarf spheroidals is too high to be consistent with their observed metallicities, as shown in §§\[sec:metallicity\]. In order to reproduce both observations, it might be worth relaxing our assumptions in estimating the disk rotation velocity and/or other physical processes such as SN feedback. For example, the SN feedback parameter $\beta$, which is assumed to be constant, might depend on whether star formation is either continuous or burst-like. Furthermore, it might evolve with redshift. Dynamical response to gas removal, of course, provides a promising effect on disk rotation velocity. These possibilities should therefore be investigated in more detail.
Color-Magnitude Relation (CMR) {#sec:cmr}
------------------------------
Figure \[fig:cmr\] shows the $V-K$ color versus magnitude relation of cluster elliptical galaxies embedded in extended dark halo with $V_{\rm circ}=10^{3}$km s$^{-1}$. The thick solid, dashed, dot-dashed, and dotted lines represent the predicted CMRs in the SF models of CSF, DSF2, DSF1, and DSF0, respectively. The CMR for each model is obtained by averaging 50 realizations. Errorbars to the CMR denote the $1\sigma$ uncertainties in the DSF2 model, which are comparable to CSF and DSF1 but is slightly larger than DSF0. The thin dashed line is the observed CMR for galaxies in the Coma cluster by @ble92 and the thin solid line is the same but for the aperture-corrected CMR by @kaba98.
All the SF models are not reconciled with the observed CMR, although the colors at $M_{V}-5\log h\simeq -20$ are consistent with the observation. This is a consequence of adopting the SN feedback with $(\alpha_{\rm hot}, V_{\rm hot})=(4, 180{\rm km~s}^{-1})$, together with the satellite-satellite merger. As discussed by @kc98 and @ng01, a combination of smaller $\alpha_{\rm hot}
(\simeq 2)$ and larger $V_{\rm hot}$ is known to give a better fit to the observed CMR.
Like the TFR, the faint-end slope of the CMR is determined by $\alpha_{\rm hot}$, because the CMR is primarily a metallicity sequence [@ka97] and chemical enrichment of dwarf galaxies is determined by $\alpha_{\rm hot}$. The bright-end slope of the CMR is nearly flat, because the SN feedback becomes negligible. If we adopt a large value of $V_{\rm hot}\simeq 280$ km s$^{-1}$, this flat region moves brightwards. Thereby, an expected slope from $\alpha_{\rm hot}$ is realized over an almost entire range of magnitudes. Additionally, the satellite-satellite merger makes dwarf spheroidal galaxies brighter, while keeping their color unchanged. This shifts the CMR bluewards when seen at faint magnitudes.
With these considerations, we can improve the fit to the observed CMR by adjusting the SN feedback-related parameters ($\alpha_{\rm hot}$, $V_{\rm hot}$), and the merger-related parameter $f_{\rm mrg}$. However, appropriate choice of their values that could explain the observed CMR seems to invalidate other successes of our SAM. This difficulty is clearly the area of future investigation.
Furthermore, we find that a bright portion of the CMR is also affected by the cooling cutoff $V_{\rm cut}$. Figure \[fig:cmr2\] shows the $V_{\rm cut}$-dependence of CMR. The solid and dashed lines represent the CSF model with $V_{\rm cut}=250$ and 300 km s$^{-1}$, respectively. The dot-dashed and dotted lines are the same but for the DSF0 model. If we adopt larger $V_{\rm cut}$, the metallicity becomes larger because chemical enrichment continues until lower redshift. Note that the difference between the models of CSF and DSF0 is caused mainly by the difference of mean stellar age, that is, stars in the DSF0 model were born earlier and thus older than CSF. Since the CMR is sensitive to $V_{\rm cut}$, more knowledge of the gas cooling is needed to fix the cutoff on cluster scales.
This situation might be improved by taking the following procedures. First is to use a lower value of $\sigma_{8}$ below unity as recent observations suggest $\sigma_{8}\simeq 0.8$ [e.g., @s03]. This value provides a statistically late epoch for density fluctuations to collapse. Therefore, a large value of $V_{\rm cut}$ does not result in the formation of monster galaxies. Second is the aperture effects. @kg03 showed the importance of aperture effects by using their chemo-dynamical simulations. The predicted $V-K$ colors of their giant elliptical galaxies are on the average 3.2 at $M_{V}-5\log h\simeq -22$ and agree well with the observed colors within 5 kpc aperture. However, the observed $V-K$ color within 99kpc aperture is nearly equal to 3.0, which is even bluer than our predicted color at the same magnitude. Third is the effects of UV background. @ng01 showed that the photoionization by the UV background has a similar effect to the SN feedback. Therefore, introduction of the photoionization is equivalent to adopting larger $V_{\rm hot}$. Further investigation on the CMR as well as the TFR are needed along these lines.
It should be noted that @on03 have successfully reproduced the CMR slope by using a SAM combined with an $N$-body simulation. Their SF model corresponds to CSF with $\alpha_{\rm hot}=2$ and $V_{\rm hot}=200$ km s$^{-1}$. As explained above, however, adopting this small value of $\alpha_{\rm hot}$ fails to explain other observations unless the merger strength is adjusted. In this sense, high resolution $N$-body simulations is highly awaited to follow a full trace of merging histories of progenitor halos of clusters.
Cosmic Star Formation History {#sec:Madau}
=============================
Recently the cosmic SF history, which is a plot of SFR in a comoving volume against redshift, is widely used to examine the global SF history [@m96] and also in SAM analyses [@bcfl98; @spf01]. Since the SFR is very sensitive to the SF timescale, in Figure \[fig:Madau\] we plot the redshift evolution of cosmic SF rate for the four models with different SF timescales. Thick lines denote the total SFR and thin lines the SFR only for starburst. Symbols with errorbars indicate observational SFRs compiled by @aygm02. While there are large scatters between individual data points, the CSF and DSF2 models broadly agree with the observations.
As expected from the redshift dependence of SF timescale, the CSF model predicts a maximum SFR at relatively low redshift, $z\la 2$, and the redshift at which this maximum occurs becomes larger, in order from DSF2 (dashed line) via DSF1 (dot-dashed line) to DSF0 (dotted line) as the SF timescale becomes shorter at high redshift (see Figure \[fig:sfr\]).
The fraction of cold gas available for starburst becomes larger for longer SF timescale at high redshift. Because of the same reason as above, the CSF model by the thin solid line gives the largest fraction of cold gas at major merger and therefore the highest SFR for starburst among the four models under consideration. Accordingly, the dynamical response is the most effective in the CSF model.
Cooling Diagram Revisited
=========================
@s77 and @ro77 proposed the so-called cooling diagram, that is, the distribution of galaxies on the density versus temperature diagram. Since then, a number of authors used this diagram as crucial constraints on the formation of galaxies in the framework of monolithic cloud collapse scenario [e.g., @f82; @bfpr84]. For example, characteristic mass of galaxies can be evaluated, because the evolutionary path of gas clouds in this diagram gives an initial condition of density and temperature expected from density fluctuation spectrum in the early universe.
Figure \[fig:cd1\] shows the contours of galaxy distribution in the cooling diagram for ellipticals ([*top panel*]{}), lenticulars ([ *middle panel*]{}), and spirals ([*bottom panel*]{}). The velocity on the horizontal axis indicates the velocity dispersion of bulge component for elliptical and lenticular galaxies, and the disk rotation velocity for spiral galaxies. The baryon density on the vertical axis is estimated by dividing the baryonic mass in individual galaxies by their volume. We simply assume a sphere of effective radius for elliptical and lenticular galaxies, and a cylinder of scale height being one tenth of the effective radius for spiral galaxies. The contours are plotted excluding low surface brightness galaxies with $\mu_{e,B}\geq 26.5$.
The dashed and dot-dashed lines represent the cooling curves that the cooling timescale is equal to the gravitational free-fall timescale $\tau_{\rm cool}=\tau_{\rm grav}$ for the cases of primordial and solar compositions, respectively. These timescales are $$\begin{aligned}
\tau_{\rm cool}&=&\frac{3}{2}\frac{\rho_{\rm gas}}{\mu m_{\rm p}}\frac{kT}{n_{\rm e}^{2}\Lambda(T,Z_{\rm gas})},\\
\tau_{\rm grav}&=&\left(24\pi G\rho_{\rm tot}\right)^{-1/2},\end{aligned}$$ where $n_{\rm e}$ is the electron number density, $\rho_{\rm gas}$ is the gas density, $\rho_{\rm tot}$ is the total mass density including dark matter, and $\Lambda(T,Z_{\rm gas})$ is the cooling function depending on both the gas temperature $T$ and metallicity $Z_{\rm gas}$. Wherever necessary below, we transform the gas temperature into velocity dispersion based on the virial theorem, and identify the gas density with baryon density. The thin solid curves represent the sequences of galaxies originated from the CDM density fluctuations with density contrast of $1\sigma$ and 3$\sigma$, respectively. It is evident from this figure that our SAM galaxies are distributed within a region of $\tau_{\rm cool}<\tau_{\rm grav}$ and their morphologies are distinctly segregated from each other, as observed in this diagram.
Figure \[fig:cd2\] shows the distribution for all galaxies without imposing any selection bias against low surface brightness. Comparison of Figures \[fig:cd1\] and \[fig:cd1\] indicates that many dwarf galaxies have low surface brightness and are distributed towards a region of $\tau_{\rm cool}>\tau_{\rm grav}$ characterized by low baryon density and low circular velocity. This extended distribution is prominent only for elliptical galaxies for which the starburst-induced gas removal followed by dynamical response has the maximal effect. Such galaxies of low surface brightness would have been detected in recent ultradeep surveys where the detection threshold is set below $\mu_{e,B}=26.5$.
Summary and Conclusion
======================
We have investigated the formation and evolution of galaxies in the context of the hierarchical clustering scenario by using the Mitaka model, or our SAM in which the effects of dynamical response on size and velocity dispersion of galaxies are explicitly taken into account, according to the formula by @ny03 for galaxies consisting of baryon and dark matter. This paper is therefore an extension of previous analyses in the context of monolithic cloud collapse scenario [@ds86; @ya87].
A $\Lambda$-dominated flat universe, which is recently recognized as a standard, is exclusively used here. The investigation mainly focuses on elliptical galaxies which are assumed to be formed by major merger and starburst-induced gas removal followed by dynamical response of the systems. While a mass fraction of removed cold gas, after heated, is determined by the circular velocity of galaxies similar to the traditional collapse models, the total amount of cold gas depends on the formation history of galaxies which is realized by the Monte Carlo method based on the power spectrum of density fluctuation predicted by the CDM model.
The Mitaka model, which we have constructed in this paper, is found to reproduce a wide variety of observed characteristics of galaxies, particularly their scaling relations among various observables such as magnitude, surface brightness, size, velocity dispersion, mass-to-light ratio and metallicity. This strongly supports the CDM cosmology and merger hypothesis of elliptical galaxy formation even on scales of dwarf galaxies.
Most of model parameters related to star formation, SN feedback and galaxy merger are constrained by the local luminosity function and mass fraction of cold gas in spiral galaxies. As an extra parameter to be furthermore constrained, we have examined redshift dependence of SF timescale through comparison among the four SF models denoted by CSF, DSF0, DSF1, and DSF2. The CSF refers to a constant SF timescale against redshift and has been suggested to be consistent with observations of galaxy number counts [@ntgy01; @nytg02], quasar luminosity function [@kh00; @eng03], and number and metallicity evolution of damped Ly-$\alpha$ systems [@spf01; @ongy02]. The DSF0 refers to a SF timescale simply proportional to dynamical timescale. The DSF1 and DSF2 are intermediate between the CSF and DSF0 (see Figure \[fig:sfr\]). It is found that the DSF0 model fails to reproduce observed properties of local dwarf spheroidals and galaxy counts. Among the above four SF models, the CSF model has the longest SF timescale at high redshift and therefore the largest amount of cold gas at that epoch. Accordingly, the fraction of removed gas during starburst is the largest in the CSF model, giving many extended galaxies in agreement with observation. In this sense, the model of constant star formation (CFS) or at most mildly evolving star formation (DSF2) is favorable. Note that [*dominant dark halo*]{}, in which size and velocity dispersion do not change during gas removal as considered by @ds86, apparently explains the observed scatter of size. However, arguments from physical ground indicate that the dynamical response should play a significant role in forming galaxies with low velocity dispersion of $\sigma_{0}(\rm 1D)\la
10$km s$^{-1}$, as actually observed in the Local Group.
There are some areas of further improvements in the current Mitaka model. Because of strong dependence of SN feedback on circular velocity assuming $\alpha_{\rm hot}=4$, dwarf galaxies in our SF models are too faint to agree with expected magnitude from observed TFRs. Furthermore, because of the same reason, dwarf galaxies are too blue to agree with expected colors from observed CMRs for cluster elliptical galaxies. On the other hand, such large value of $\alpha_{\rm hot}$ stops chemical and photometric evolution of elliptical galaxies at early epochs and is required to explain their observed low stellar metallicity as well as the galaxy number counts in the HDF and SDF. In order to settle these contradictions, some new ingredients need to be introduced in the SAM analysis. At least, for example, we must know how efficient the gas cooling is in massive dark halos and whether SN feedback at starburst works in the same way as that in disks.
We adopted a fitting mass function of dark halos by @yny03 for given redshift instead of the often used PS mass function. This YNY mass function is slightly different from those by @st99 and @j01, but is confirmed to provide a better fit to recent $N$-body results given by @yy01 and @y02, which is discussed in more detail in a separate paper [@yny03]. Since the number density of dark halos affects SN feedback-related parameters to be chosen, it is very important to establish the mass function of dark halos, particularly on small mass scales, by high resolution $N$-body simulation. Although such SAM analyses have just begun [e.g., @bpfbj01; @h03a; @h03b], further investigation will obviously be required. We are upgrading the Mitaka model accommodated with full high resolution $N$-body simulations, and the results will be given elsewhere in the near future.
We thank Takashi Okamoto and Naoteru Gouda for useful suggestions. The authors are grateful to the anonymous referee for helpful comments to improve this paper. This work has been supported in part by the Grant-in-Aid for the Center-of-Excellence research (07CE2002) of the Ministry of Education, Science, Sports and Culture of Japan. MN acknowledges support from a PPARC rolling grant for extragalactic astronomy and cosmology.
Dynamical Response to Gas Removal
=================================
We show the dynamical response of size and velocity dispersion to supernova-induced gas removal in the two-component galaxies consisting of baryon and dark matter. The details are given in @ny03.
All elliptical galaxies in this paper have a density profile similar to the Jaffe model [@jaffe], $$\rho(r)=\frac{4\rho_{b}r_{b}^{4}}{r^{2}(r+r_{b})^{2}},$$ where $\rho_{b}$ and $r_{b}$ are the characteristic density and radius, respectively. This profile well approximate the de Vaucouleurs $r^{1/4}$ profile for stars. The effective (half-light) radius $r_{e}$ defined in the projected surface is related to the half-mass radius $r_{b}$ in three dimensional space as $r_{e}=0.744r_{b}$ [@ny03]. For the distribution of dark matter, the singular isothermal distribution ($\rho_{d}\propto r^{-2}$) is considered. We found that the Navarro-Frenk-White profile [@nfw97] provides almost the same results as those given by singular isothermal sphere. Note that according to recent high resolution $N$-body simulations, which have reveals that many substructures survive even in virialized halos, we assume that subhalos exist as underlying gravitational potential for satellite galaxies. As mentioned in §§\[sec:tidal\], these subhalos are tidally truncated.
Because the derivation is long and complicated, we describe it only briefly. Just after the merger, the system is virialized immediately. Then the starburst occurs and a part of gas is gradually removed. The size and velocity dispersion change following the gas removal adiabatically. During the gas removal, the dark matter distribution is assumed to be not affected.
In the followings we derive useful simplified formulae for the dynamical response. As shown in Appendix B of @ny03, the relationship between density and size for the adiabatic gas removal is given by $$Y=\frac{h(y_{i},z_{i})p(z_{f})+q(z_{f})}{y_{i}},\label{eqn:response}$$ where $Y\equiv y_{f}/y_{i}$, $y_{i}=\rho_{i}/\rho_{d}$, $z_{i,f}=r_{i,f}/r_{d}$, and the function $h, p$ and $q$ are $$\begin{aligned}
h(y,z)&=&yz^{4}+\frac{1}{2}[-z+(1+z)\ln (1+z)],\\
p(z)&=&\frac{1}{z^{4}},\\
q(z)&=&-\frac{1}{2z^{4}}[-z+(1+z)\ln (1+z)].\end{aligned}$$ In order to obtain the size after gas removal, we need to know the inverted function of the above and it will be very complicated. Therefore, we approximate the above function. The equation can be reduced to the following expression, $$Y=\frac{1}{R^{4}}+\frac{U_{\rm iso}(R,z_{i})}{2y_{i}z_{i}^{4}R^{4}},\label{eqn:full}$$ where $R=z_{f}/z_{i}$. Expanding $U_{\rm iso}(R,z_{i})$ around $R=1$ and $z_{i}=0$ and picking out the lowest order term, we obtain $$Y=\frac{1}{R^{4}}-\frac{D}{2}\frac{R-1}{R^{4}},$$ where $D=1/y_{i}z_{i}^{2}$. Defining the ratio of final to initial masses, ${\cal M}=M_{f}/M_{i}=YR^{3}$, the above equation is transformed to $${\cal M}=\frac{1}{R}-\frac{D}{2}\frac{R-1}{R}.$$ Then, inverting this, we obtain $$R=\frac{1+D/2}{{\cal M}+D/2}.$$ By using the virial theorem, the velocity dispersion is $$\frac{\sigma_{f}}{\sigma_{i}}=\sqrt{\frac{YR^{2}+Df(z_{f})/2}{1+Df(z_{i})/2}},$$ where $$f(z)=\frac{\ln(1+z)}{z}+\ln\left(1+\frac{1}{z}\right).$$
Figure \[fig:approx\] shows the relation between ${\cal M}$ and $R$ based on this approximate formula with $D=0.05$ and 2 (solid lines). For each value of $D$, the exact solution is also shown for two cases of $z_{i}=0.05$ (dot-dashed line) and 0.5 (dashed line). Clearly our approximate relation follows the exact solution quite well.
Ascasibar Y., Yepes G., Gottlöber S., Müller V., 2002, A&A, 387, 396 Arimoto, N., & Yoshii, Y. 1986, , 164, 260 Arimoto, N., & Yoshii, Y. 1987, , 173, 23 Arimoto, N., Yoshii, Y., & Takahara, F. 1991, , 253, 21 Bardeen, J. M., Bond, J. R., Kaiser, N., & Szalay, A. S. 1986, , 304, 15 Baum, W. A. 1959, PASP, 71, 106 Baugh, C. M., Cole, S., & Frenk, C. S. 1996, , 283, 1361 Baugh, C. M., Cole, S., Frenk, C. S., & Lacey, C. G. 1998, , 498, 504 Bender, R., Burstein, D., & Faber, S. M. 1992, , 399, 462 Bender, R., Burstein, D., & Faber, S. M. 1993, , 411, 153 Bender, R., Paquet, A., & Nieto, J.-L. 1991, , 246, 349 Benson, A. J., Pearce, F. R., Frenk, C. S., Baugh, C. M., & Jenkins, A. 2001, , 320, 261 Benson, A. J., Lacey, C. G., Baugh, C. M., Cole, S., & Frenk, C. S. 2002a, , 333, 156 Benson, A. J., Frenk, C. S., Lacey, C. G., Baugh, C. M., & Cole, S. 2002b, , 333, 177 Benson, A. J., Frenk, C. S., Baugh, C. M., Cole, S., & Lacey, C. G. 2003a, , 343, 679 Benson, A. J., Bower, R. G., Frenk, C. S., Lacey, C. G., Baugh, C. M., & Cole, S. 2003b, preprint (astro-ph/0302450) Binney, J., & Tremaine, S. 1987, Galactic Dynamics, Princeton Univ. Press, Princeton, NJ Blumenthal, G. R., Faber, S. M., Primack, J. R., & Rees, M. J. 1984, Nature, 311, 517 Bond, J. R., Cole, S., Efstathiou, G., & Kaiser, N. 1991, , 379, 440 Bower, R. 1991, , 248, 332 Bower, R., Lucey, J.R., & Ellis, R.S. 1992, , 254, 601 Bullock, J. S., Kravtsov, A. V., & Weinberg, D. H. 2000, , 517, 521 Catelan, P., & Theuns, T. 1996a, , 282, 436 Catelan, P., & Theuns, T. 1996b, , 282, 455 Chiba, M., & Nath, B.B. 1994, , 436, 618 Ciotti, L., Pellegrini, S. 1992, MNRAS, 255, 561 Cole, S., Aragon-Salamanca, A., Frenk, C. S., Navarro, J. F., & Zepf, S. E. 1994, , 271, 781 Cole, S., Lacey, C. G., Baugh, C. M., & Frenk, C. S. 2000, , 319, 168 Cole, S. et al. 2001, , 326, 255 Dekel A., Silk J., 1986, ApJ, 303, 39 Disney, M., Davies, J., & Phillipps, S. 1989, , 239, 939 Djorgovski, S., & Davis, M. 1987, , 313, 59 Dressler, A., Lynden-Bell, D., Burstein, D., Davies, R. L., Faber, S. M., Terlevich, R. J., & Wegner, G. 1987, , 313, 42 Eggen, O.J., Lynden-Bell, D., & Sandage, A.R. 1962, , 136, 748 Enoki, M., Nagashima, M., & Gouda, N. 2003, , 55, 133 Faber S.M., 1982, in Brück H. A., Coyne G. V., Longair M. S., eds, Astrophysical Cosmology. Pontificia Academia Scientiarum, p. 191 Faber, S. M., & Jackson, R. E. 1976, ApJ, 204, 668 Fall, S. M. 1979, , 281, 200 Fall, S. M., & Efstathiou, G. 1980, , 193, 189 Fall, S. M. 1983, in ‘Internal kinematics and dynamics of galaxies’, proceedings of the IAU symposium 100, Besancon, France, Dordrecht, D. Reidel, p.391 Fern[á]{}ndez-Soto, A., Lanzetta, K. M., & Yahil, A. 1999, , 513, 34 Folkes, S. et al. 1999, , 308, 459 Furusawa, H., Shimasaku, K., Doi, M., & Okamura, S. 2000, , 534, 624 Gardner, J. P., Sharples, R. M., Carrasco, B. E., & Frenk, C. S. 1996, , 282, L1 Ghigna, S., Moore, B. Governato, F., Lake, G., Quinn, T., & Stadel, J. 1998, , 300, 146 Giovanelli, R., Haynes, M.P., da Costa, L.N., Freudling, W., Salzer, J.J., & Wegner, G. 1997, , 477, L1 Glazebrook, K., Peacock, J.A., Miller, L., & Collins, C.A. 1994, , 266, 65 Gunn, J.E., & Gott, J.R. 1972, ApJ, 176, 1 Hall, P., & Mackay, C. B. 1984, , 210, 979 Held, E. V., de Zeeuw, T., Mould, J., & Picard, A. 1992, , 103, 851 Helly, J. C., Cole, S., Frenk, C. S., Baugh, C. M., Benson, A. J., & Lacey, C. 2003, , 338, 903 Helly, J. C., Cole, S., Frenk, C. S., Baugh, C. M., Benson, A. J., Lacey, C., & Pearce, F. R. 2003, , 338, 913 Hills, J. G. 1980, , 225, 986 Huchtmeier, W. K., & Richter, O. -G. 1988, , 203, 237 Ikeuchi, S. 1977, PTP, 58, 1742 Impey, C.D., Sprayberry, D., Irwin, M. J., & Bothun, G. D. 1996, , 105, 209 Jaffe, W. 1983, , 202, 995 Jenkins, A., Frenk, C. S., White, S. D. M., Colberg, J. M., Cole, S., Evrard, A. E., Couchman, H. M. P., & Yoshida, N. 2001, , 321, 372 Jones, L.R., Fong, R., Shanks, T., Ellis, R. S., & Peterson, B. A. 1991, , 249, 481 Kauffmann, G., & Charlot, S. 1998, , 294, 705 Kauffmann, G., & Haehnelt, M. 2000, , 311, 576 Kauffmann, G., White, S. D. M., & Guiderdoni, B. 1993, , 264, 201 Kawata, D., & Gibson, B.K. 2003, , 340, 908 Kochanek, C.S. et al. 2001, , 560, 566 Kodaira, K., Okamura, S., & Watanabe, M. 1983, , 274, L49 Kodama, T., & Arimoto, N. 1997, , 320, 41 Kodama, T., Arimoto, N., Barger, A.J., & Arag[ó]{}n-Salamanca A.1998, , 334, 99 Koo, D.C. 1986, , 311, 651 Kormendy, J. 1977, , 218, 333 Lacey, C.G., & Cole, S. 1993, , 262, 627 Larson, R. B. 1969, , 169, 229 Loveday, J., Peterson, B. A., Efstathiou, G., & Maddox, S. J. 1992, , 90, 338 Madau, P., Ferguson, H., Dickinson, M., Giavalisco, M., Steidel, C., & Fruchter, A. 1996, , 283, 1388 Maddox, S. J., Sutherland, W. J., Efstathiou, G., Loveday, J., & Peterson, B. A. 1990, , 247, 1p Maihara, T. et al. 2001, , 53, 25 Makino, J., & Hut, P. 1997, , 481, 83 Mateo, M. L. 1998, , 36, 435 Mathewson, D.S., Ford, V.L., & Buchhorn, M. 1992, , 81, 413 Mathieu, R. D. 1983, , 267, L97 McLeod, B.A., Bernstein, G.M., Rieke, M.J., Tollestrup, E.V., & Fazio, G.G. 1995, , 96, 117 Metcalfe, N., Shanks, T., Fong, R., & Jones, L. R. 1991, , 249, 498 Minezaki, T., Kobayashi, Y., Yoshii, Y., & Peterson, B.A. 1998, , 494, 111 Mo, H.J., Mao, S., & White, S.D.M. 1998, , 295, 319 Monaco, P. 1998, Fundam. Cosmic Phys., 19, 157 Nagashima, M. 2001, , 562, 7 Nagashima, M., & Gouda, N. 1998, , 301, 849 Nagashima, M., Gouda, N., & Sugiura, N. 1999, , 305, 449 Nagashima, M., & Gouda, N. 2001, MNRAS, 325, L13 Nagashima, M., Totani, T., Gouda, N., & Yoshii, Y. 2001, , 557, 505 Nagashima, M., Yoshii, Y., Totani, T., & Gouda, N. 2002, ApJ, 578, 675 Nagashima, M., & Yoshii, Y. 2003, , 340, 509 Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, , 490, 493 Netterfield, C. B. et al. 2002, , 571, 604 Okamoto, T., & Habe, A. 1999, , 516, 591 Okamoto, T., & Habe, A. 2000, , 52, 457 Okamoto, T., & Nagashima, M. 2003, , 587, 500 Okoshi, K., Nagashima, M., Gouda, N., & Yoshioka, S. 2004, , in press Ostriker, J.P., & Gnedin, N.Y. 1996, , 472, L63 Pierce, M., & Tully, R.B. 1992, , 387, 47 Press, W., & Schechter, P. 1974, , 187, 425 Ratcliffe, A., Shanks, T., Parker, Q., & Fong, R. 1998, , 293, 197 Rees, M.J., & Ostriker, J.P. 1977, , 179, 541 Saito, M. 1979, PASJ, 31, 193 Seaton, M. J. 1979, , 187, 73P Sheth, R., & Tormen, G. 1999, , 308, 119 Silk, J. 1977, , 211,638 Simard, L., Koo, D. C., Faber, S. M., Sarajedini, V. L., Vogt, N. P., Phillips, A. C., Gebhardt, K., Illingworth, G. D., & Wu, K. L. 1999, , 519, 563 Simien, F., & de Vaucouleurs, G. 1986, , 302, 564 Somerville, R. S. 2002, , 572, 23 Somerville, R.S., & Kolatt, T. 1999, , 305, 1 Somerville, R.S., & Primack, J. R. 1999, , 310, 1087 Somerville, R.S., Primack, J. R., & Faber, S. M. 2001, , 320, 504 Spergel, D.N. et al. 2003, preprint (astro-ph/0302209) Sugiyama, N. 1995, , 100,281 Sutherland, R., & Dopita, M. A. 1993, , 88, 253 Suzuki, T.K., Yoshii, Y., & Beers, T.C. 2000, , 540, 99 Szokoly, G.P., Subbarao, M.U., Connolly, A.J., & Mobasher, B. 1998, , 492, 452 Tomita, K. 1969, Prog. Theor. Phys., 42, 9 Totani, T., & Yoshii, Y. 2000, , 540, 81 Totani, T., Yoshii, Y., Maihara, T., Iwamuro, F., & Motohara, K. 2001, , 559, 592 Tully, R. B., Somerville, R. S., Trentham, N., & Verheijen, M. A. W. 2002, , 569, 573 Tyson, J.A. 1988, , 96, 1 Vader, J. P. 1986, , 305, 669 White, S.D.M. 1984, , 286, 38 Williams, R. T. et al. 1996, AJ, 112, 1335 Yahagi, H., & Yoshii, Y. 2001, , 558, 463 Yahagi, H. 2002, D. Thesis, University of Tokyo Yahagi, H., Nagashima, M., & Yoshii, Y. 2003, submitted Yano, T., Nagashima, M., & Gouda, N. 1996, , 466, 1 Yoshii, Y. 1993, , 403, 552 Yoshii, Y., & Arimoto, N. 1987, , 188, 13 Yoshii, Y., & Peterson, B. A. 1994, , 436, 551 Yoshii, Y., & Saio, H. 1979, PASJ, 31, 339 Yoshii, Y., & Saio, H. 1987, , 227, 677 Yoshii, Y., & Takahara, F. 1988, , 326, 1 Watanabe, M., Kodaira, K., & Okamura, S. 1985, , 292, 72 Zucca, E. et al. 1997, , 326, 477
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abstract: 'Greedy-GQ is an off-policy two timescale algorithm for optimal control in reinforcement learning [@maei2010toward]. This paper develops the first finite-sample analysis for the Greedy-GQ algorithm with linear function approximation under Markovian noise. Our finite-sample analysis provides theoretical justification for choosing stepsizes for this two timescale algorithm for faster convergence in practice, and suggests a trade-off between the convergence rate and the quality of the obtained policy. Our paper extends the finite-sample analyses of two timescale reinforcement learning algorithms from policy evaluation to optimal control, which is of more practical interest. Specifically, in contrast to existing finite-sample analyses for two timescale methods, e.g., GTD, GTD2 and TDC, where their objective functions are convex, the objective function of the Greedy-GQ algorithm is non-convex. Moreover, the Greedy-GQ algorithm is also not a linear two-timescale stochastic approximation algorithm. Our techniques in this paper provide a general framework for finite-sample analysis of non-convex value-based reinforcement learning algorithms for optimal control.'
author:
- |
[**Yue Wang**]{}\
Electrical Engineering\
University at Buffalo\
ywang294@buffalo.edu\
\
Electrical Engineering\
University at Buffalo\
szou3@buffalo.edu
bibliography:
- 'RL.bib'
title: 'Finite-sample Analysis of Greedy-GQ with Linear Function Approximation under Markovian Noise'
---
Introduction
============
Reinforcement learning (RL) is to find an optimal control policy to interact with a (stochastic) environment so that the accumulated reward is maximized [@sutton2018reinforcement]. It finds a wide range of applications in practice, e.g., robotics, computer games and recommendation systems [@minh2015; @Minh2016; @silver2016mastering; @kober2013reinforcement].
When the state and action spaces of the RL problem are finite and small, RL algorithms based on the tabular approach, which stores the action-values for each state-action pair, can be applied and usually have convergence guarantee, e.g., Q-learning [@watkins1992q] and SARSA [@Rummery1994]. However, in many RL applications, the state and action spaces are very large or even continuous. Then, the approach of function approximation can be used. Nevertheless, with function approximation in off-policy training, classical RL algorithms may diverge to infinity, e.g., Q-learning, SARSA and TD learning [@baird1995residual; @gordon1996chattering].
To address the non-convergence issue in off-policy training, a class of gradient temporal difference (GTD) learning algorithms were developed in [@maei2010toward; @maei2011gradient; @sutton2009fast; @Sutton2009b], including GTD, GTD2, TD with correction term (TDC), and Greedy-GQ. The basic idea is to construct squared objective functions, e.g., mean squared projected Bellman error, and then to perform stochastic gradient descent. To address the double sampling problem in gradient estimation, a weight doubling trick was proposed in [@sutton2009fast], which leads to a two timescale update rule. One great advantage of this class of algorithms is that they can be implemented in an online and incremental fashion, which is memory and computationally efficient.
The asymptotic convergence of these two timescale algorithms has been well studied under both i.i.d. and non-i.i.d. settings [@sutton2009fast; @Sutton2009b; @maei2010toward; @yu2017convergence; @borkar2009stochastic; @borkar2018concentration; @karmakar2018two]. Furthermore, the finite-sample analyses of these algorithms are of great practical interest for algorithmic parameter tuning and design of new sample-efficient algorithms. However, these problems remain unsolved until very recently [@dalal2018finite; @wang2017finite; @liu2015finite; @gupta2019finite; @xu2019two]. But, existing finite-sample analyses are only for the GTD, GTD2 and TDC algorithms, which are designed for evaluation of a given policy. The finite-sample analysis for the Greedy-GQ algorithm, which is to directly learn an optimal control policy, is still not understood and will be the focus of this paper. In this paper, we will develop the finite-sample analysis for the Greedy-GQ algorithm with linear function approximation under Markovian noise. More specifically, we focus on the general case with a single sample trajectory and non-i.i.d. data. We will develop explicit bounds on the convergence of the Greedy-GQ algorithm and understand its sample complexity as a function of various parameters of the algorithm.
Summary of Major Challenges and Contributions
---------------------------------------------
The major challenges and our main contributions are summarized as follows.
The objective function of the Greedy-GQ algorithm is the mean squared projected Bellman error (MSPBE). Unlike the objective functions of GTD, GTD2 and TDC, which are convex, the objective function of Greedy-GQ is non-convex since the target policy is also a function of the action-value function approximation (see for the objective function). In this case, the Greedy-GQ algorithm may not be able to converge to the global optimum, and existing analyses for GTD, GTD2 and TDC based on convex optimization theory cannot be directly applied. Moreover, the Greedy-GQ algorithm cannot be viewed as a linear two timescale stochastic approximation due to its non-convexity, and thus existing analyses for linear two timescale stochastic approximation are not applicable. Due to the non-convexity of the objective function, convergence to the global optimum may not be guaranteed. Therefore, we study the convergence of the gradient norm to zero (in an on-average sense, i.e., randomized stochastic gradient method [@ghadimi2013stochastic]), and we focus on convergence to stationary points. In this paper, we develop a novel methodology for finite-sample analysis of the Greedy-GQ algorithm, which solves reinforcement learning problems from a non-convex optimization perspective. This may be of independent interest for a wide range of reinforcement learning problems with non-convex objective functions.
In this paper, we focus on the most general scenario where there is a single sample trajectory and the data are non-i.i.d.. This non-i.i.d. setting will invalidate the martingale noise assumption commonly used in stochastic approximation (SA) analysis [@maei2010toward; @dalal2018finite; @borkar2018concentration]. Our approach is to analyze RL algorithms from a non-convex optimization perspective, and does not require the martingale noise assumption. Thus, our approach has a much broader applicability.
Moreover, the propagation of the stochastic bias in the gradient estimate caused by the Markovian noise in the two timescale updates makes the analysis even more challenging. We develop a comprehensive characterization of the stochastic bias and establish the convergence rate of the Greedy-GQ algorithm under constant stepsizes. More importantly, we develop a novel recursive approach of bounding the bias caused by the tracking error, i.e., the error in the fast timescale update. Specifically, our approach is to recursively plug the obtained bound back into the analysis to tighten the final bound on the bias.
We show that under constant stepsizes, i.e., $\alpha_t=\frac{1}{T^a}$ and $\beta_t=\frac{1}{T^b}$ for $0\leq t\leq T$, the Greedy-GQ algorithm converges as fast as $\mathcal O\left(\frac{1}{T^{1-a}}+\frac{\log T}{T^{\min\{b,a-b\}}}\right)$. We also derive the best choice of $a$ and $b$ so that the above rate is the fastest. Specifically, when $a=\frac{2}{3}$ and $b=\frac{1}{3}$, the Greedy-GQ algorithm converges as fast as $\mathcal O\left(\frac{\log T}{T^{\frac{1}{3}}}\right)$. We further characterize the trade-off between the convergence speed and the quality of the obtained policy. Specifically, the algorithm needs more samples to converge if the target policy is more “greedy", e.g., a larger parameter $\sigma$ in softmax makes the policy more “greedy", and will require more samples to converge. Our experiments also validate this theoretical observation.
Related Work
------------
In this subsection, we provide an overview of closely related work. Specifically, we here focus on value-based RL algorithms with function approximation. We note that there are many other types of approaches, e.g., policy gradient and fitted value/policy iteration, which are not discussed in this paper.
**TD, Q-learning and SARSA with function approximation.** TD with linear function approximation was shown to converge asymptotically in [@Tsitsiklis1997], and its finite-sample analysis was established in [@Dalal2018a; @Laksh2018; @bhandari2018finite; @srikant2019] under both i.i.d. and non-i.i.d. settings. Moreover, the finite-sample analysis of TD with over–parameterized neural function approximation was developed in [@cai2019neural]. Q-learning and SARSA with linear function approximation were shown to converge asymptotically under certain conditions [@melo2008analysis; @perkins2003convergent] and their finite-sample analyses were developed in [@zou2019finite; @chen2019performance]. However, these algorithms may diverge under off-policy training. Different from TD, Q-learning and SARSA, the Greedy-GQ algorithm follows a stochastic gradient descent type update. However, the updates of TD, Q-learning and SARSA do not exactly follow a gradient descent type, since the “gradient" therein is not gradient of any function [@maei2010toward]. Moreover, the Greedy-GQ algorithm is a two timescale one, and thus requires more involved analysis than these one timescale methods.
**GTD algorithms.** The GTD, GTD2 and TDC algorithms were shown to converge asymptotically in [@Sutton2009b; @sutton2009fast; @yu2017convergence]. Their finite-sample analyses were further developed recently in [@dalal2018finite; @wang2017finite; @liu2015finite; @gupta2019finite; @xu2019two] under i.i.d. and non-i.i.d. settings. The Greedy-GQ algorithm studied in this paper is fundamentally different from the above three algorithms. This is due to the fact that the Greedy-GQ algorithm is for optimal control and its objective function is non-convex; whereas the GTD, GTD2 and TDC algorithms are for policy evaluation, and their objective functions are convex. Therefore, new techniques need to be developed to tackle the non-convexity for the finite-sample analysis for Greedy-GQ. Moreover, general linear two timescale stochastic approximation has also been studied. Although the Greedy-GQ algorithm follows a two timescale update rule, but it is not linear. Furthermore, the general non-linear two timescale stochastic approximation was studied in [@borkar2018concentration]. However, the Greedy-GQ algorithm under Markovian noise does not satisfy the martingale noise assumption therein. Moreover, our paper uses a non-convex optimization based approach to develop the finite-sample analysis, which is different from the approach used in [@borkar2018concentration].
Preliminaries {#sec:pre}
=============
Markov Decision Process
-----------------------
In RL problems, a Markov Decision Process (MDP) is usually used to model the interaction between an agent and a stochastic environment. Specifically, an MDP consists of $(\mathcal{S},\mathcal{A}, \mathsf{P}, r, \gamma)$, where $\mathcal{S}\subset \mathbb R^d$ is the state space, $\mathcal{A}$ is a finite set of actions, and $\gamma\in(0,1)$ is the discount factor. Denote the state at time $t$ by $S_t$, and the action taken at time $t$ by $A_t$. Then the measure $\mathsf P$ denotes the action-dependent transition kernel of the MDP:
(S\_[t+1]{}U|S\_t=s,A\_t=a)=\_U(dx|s,a),
for any measurable set $U\subseteq \mathcal S$. The reward at time $t$ is given by $r_t=r(S_t,A_t,S_{t+1})$, which is the reward of taking action $A_t$ at state $S_t$ and transitioning to a new state $S_{t+1}$. Here $r:{\mathcal{S}}\times{\mathcal{A}}\times{\mathcal{S}}\to\mathbb R$ is the reward function, and is assumed to be uniformly bounded, i.e., $$\begin{aligned}
0\leq r(s,a,s')\leq r_{\max}, \forall (s,a,s')\in {\mathcal{S}}\times{\mathcal{A}}\times{\mathcal{S}}.\end{aligned}$$
A stationary policy maps a state $s\in{\mathcal{S}}$ to a probability distribution $\pi(\cdot|s)$ over ${\mathcal{A}}$, which does not depend on time. For a policy $\pi$, its value function $V^\pi: {\mathcal{S}}\to\mathbb R$ is defined as the expected accumulated discounted reward by executing the policy $\pi$ to obtain actions: $$\begin{aligned}
V^\pi\left(s_0\right)={\mathbb{E}}\left[\sum_{t=0}^{\infty}\gamma^t r(S_t,A_t,S_{t+1})|S_0=s_0\right].\end{aligned}$$ The action-value function $Q^\pi:{\mathcal{S}}\times{\mathcal{A}}\rightarrow\mathbb R$ of policy $\pi$ is defined as $$\begin{aligned}
Q^\pi(s,a)={\mathbb{E}}_{S'\sim \mathsf P(\cdot|s,a)}\left[r(s,a,S')+\gamma V^\pi(S')\right].\end{aligned}$$ The goal of optimal control in RL is to find the optimal policy $\pi^*$ that maximizes the value function for any initial state, i.e., to solve the following problem:
V\^\*(s)=\_V\^(s), s.
We can also define the optimal action-value function as
Q\^\*(s,a)=\_Q\^(s,a), (s,a).
Then, the optimal policy $\pi^*$ is greedy w.r.t. $Q^*$. The Bellman operator $\mathbf T$ is defined as $$\begin{aligned}
(\mathbf TQ)(s,a)=&\int_{\mathcal{S}}(r(s,a,s'){\nonumber}\\
&+\gamma \max_{b\in{\mathcal{A}}}Q(s',b))\mathsf{P}(ds'|s,a).\end{aligned}$$ It is clear that $\mathbf T$ is contraction in the sup norm defined as $\|Q\|_{\sup}=\sup_{(s,a)\in{\mathcal{S}}\times{\mathcal{A}}}|Q(s,a)|$, and the optimal action-value function $Q^*$ is the fixed point of $\mathbf T$ [@bertsekas2011dynamic].
Linear Function Approximation
-----------------------------
In many modern RL applications, the state space is usually very large or even continuous. Therefore, classical tabular approach cannot be directly applied due to memory and computational constraint [@sutton2018reinforcement]. In this case, the approach of function approximation can be applied, which uses a family of parameterized function to approximate the action-value function. In this paper, we focus on linear function approximation.
Consider a set of $N$ fixed base functions $\phi^{(i)}$: ${\mathcal{S}}\times{\mathcal{A}}\rightarrow \mathbb R,\, i=1,\ldots,N$. Further consider a family of real-valued functions $\mathcal Q=\{Q_\theta:\theta\in\mathbb R^N\}$ defined on ${\mathcal{S}}\times{\mathcal{A}}$, which consists of linear combinations of $\phi^{(i)}$, $i=1,\ldots,N$. Specifically,
Q\_(s,a)=\_[i=1]{}\^N (i)\^[(i)]{}\_[s,a]{}=\_[s,a]{}\^.
The goal is to find a $Q_\theta$ with a compact representation in $\theta$ to approximate the optimal action-value function $Q^*$.
Greedy-GQ Algorithm
-------------------
In this subsection, we introduce the Greedy-GQ algorithm, which was originally proposed in [@maei2010toward] to solve the problem of optimal control in RL under off-policy training.
For the Greedy-GQ algorithm, a fixed behavior policy $\pi_b$ is used to collect samples. It is assumed that the Markov chain $\{X_t,A_t\}_{t=0}^\infty$ induced by the behavior policy $\pi_b$ and the Markov transition kernel $\mathsf P$ is uniformly ergodic with the invariant measure denoted by $\mu$.
The main idea of the Greedy-GQ algorithm is to design an objective function, and further to employ a stochastic gradient descent optimization approach together with a weight doubling trick (a two timescale update) [@Sutton2009b] to minimize the objective function. Specifically, the goal is to minimize the following mean squared projected Bellman error (MSPBE):
\[eq:objective\] J()||T\^[\_]{}Q\_-Q\_||\_.
Here $\|Q(\cdot,\cdot)\|_\mu\triangleq\int_{s\in{\mathcal{S}},a\in{\mathcal{A}}}d\mu_{s,a}Q(s,a)$; $\mathbf T^{\pi}$ is the Bellman operator: $$\begin{aligned}
\mathbf T^{\pi}Q(s,a)\triangleq{\mathbb{E}}_{S',A'}[r(s,a,S')+\gamma Q(S',A'))],
$$ where $S'\sim \mathsf P(\cdot|s,a)$, and $A'\sim \pi(\cdot|S')$; $\mathbf \Pi$ is a projection operator which projects an action-value function to the function space $\mathcal{Q}$ with respect to $||\cdot||_{\mu}$, i.e., $\mathbf \Pi \hat Q=\arg\min_{Q\in \mathcal Q}\|Q-\hat Q\|_\mu$; and $\pi_\theta$ is a stationary policy, which is a function of $\theta$.
We note that the objective function in is non-convex since the parameter $\theta$ is also in the Bellman operator, i.e., $\pi_\theta$. Moreover, unlike GTD, GTD2 and TDC, the objective function of the Greedy-GQ algorithm is not a quadratic function of $\theta$. Thus, the Greedy-GQ algorithm is not a linear two timescale stochastic approximation algorithm. Define $\delta_{s,a,s'}(\theta)=r({s,a,s'})+\gamma\Bar{V}_{s'}(\theta)-\theta^\top \phi_{s,a}$, and $\Bar{V}_{s'}(\theta)=\sum_{a'} \pi_{\theta}(a'|s')\theta^\top \phi_{s',a'}$. In this way, the objective function in can be rewritten equivalently as follows $$\begin{aligned}
J({\theta})=&\mathbb{E}_{\mu}[\delta_{S,A,S'}({\theta})\phi_{S,A}]^\top \mathbb{E}_{\mu}[\phi_{S,A}\phi_{S,A}^\top ]^{-1}{\nonumber}\\
&\times\mathbb{E}_{\mu}[\delta_{S,A,S'}({\theta})\phi_{S,A}],\end{aligned}$$ where $(S,A)\sim\mu$, and $S'\sim \mathsf P(\cdot|S,A)$ is the subsequent state.
To compute a gradient to $J(\theta)$, we will need to compute the gradient to $\delta_{S,A,S'}(\theta)$, and thus the gradient to $\Bar{V}_{S'}(\theta)$. Suppose $\hat{\phi}_{S'}(\theta)$ is an unbiased estimate of the gradient to $\Bar{V}_{S'}(\theta)$ given $S'$, then $\psi_{S,A,S'}(\theta)=\gamma\hat\phi_{S'}(\theta)-\phi_{S,A}$ is a gradient of $\delta_{S,A,S'}(\theta)$. Then, the gradient to $J(\theta)/2$ can be computed as follows: $$\begin{aligned}
\label{eq:5}
&\mathbb{E}_{\mu}[\psi_{S,A,S'}(\theta)\phi_{S,A}^\top ]\mathbb{E}_{\mu}[\phi_{S,A}\phi_{S,A}^\top ]^{-1}\mathbb{E}_{\mu}[\delta_{S,A,S'}(\theta)\phi_{S,A}] \nonumber\\
&=-\mathbb{E}_{\mu}[\delta_{S,A,S'}(\theta)\phi_{S,A}]+\gamma\mathbb{E}_{\mu}[\hat{\phi}_{S'}(\theta)\phi_{S,A}^\top ]\omega^*(\theta),\end{aligned}$$ where $\omega^*(\theta)=\mathbb{E}_{\mu}[\phi_{S,A}\phi_{S,A}^\top ]^{-1} \mathbb{E}_{\mu}[\delta_{S,A,S'}({\theta})\phi_{S,A}].$ To get an unbiased estimate of , two independent samples of $(S,A,S')$ are needed, which is not applicable when there is a single sample trajectory. Then, a weight doubling trick [@Sutton2009b] was used in [@maei2010toward] to construct the Greedy-GQ algorithm with the following updates (see Algorithm \[al:1\] for more details): $$\begin{aligned}
&\theta_{t+1}=\theta_t+\alpha_t(\delta_{t+1}(\theta_t)\phi_t-\gamma(\omega_t^\top \phi_t)\hat{\phi}_{t+1}(\theta_t)),\\
&\omega_{t+1}=\omega_t+\beta_t(\delta_{t+1}(\theta_t)-\phi_t^\top \omega_t)\phi_t,\label{eq:omegaupdate}\end{aligned}$$ where $\alpha_t>0$ and $\beta_t>0$ are non-increasing stepsizes, $\delta_{t+1}(\theta)\triangleq\delta_{s_t,a_t,s_{t+1}}(\theta)$ and $\phi_t\triangleq\phi_{s_t,a_t}$. For more details of the derivation of the Greedy-GQ algorithm, we refer the readers to [@maei2010toward].
**Initialization:** $\theta_0$, $\omega_0$, $s_0$, $\phi^{(i)}$, for $i=1,2,...,N$ **Method:** $\pi_{\theta_0}\leftarrow{\mathrm \Gamma}(\phi^\top \theta_0)$ Observe $s_{t+1}$ and $r_{t}$ $\Bar{V}_{s_{t+1}}(\theta_{t}) \leftarrow \sum_{a'\in \mathcal{A}} \pi_{\theta_{t}}(a' |s_{t+1})\theta_{t}^\top \phi_{s_{t+1},a'}$ $\delta_{t+1}(\theta_{t})\leftarrow r_{t}+\gamma\Bar{V}_{s_{t+1}}(\theta_{t})-\theta_{t}^\top \phi_{t} $ $\hat{\phi}_{t+1}(\theta_{t})\leftarrow$ gradient of $\Bar{V}_{s_{t+1}}(\theta_{t})$ $\theta_{t+1} \leftarrow \theta_{t}+\alpha_{t}(\delta_{t+1}(\theta_{t})\phi_{t}-\gamma(\omega_{t}^\top \phi_{t})\hat{\phi}_{t+1}(\theta_{t}))$ $\omega_{t+1} \leftarrow \omega_{t}+\beta_{t}(\delta_{t+1}(\theta_{t})-\phi_{t}^\top \omega_{t})\phi_{t}$ **Policy improvement**: $\pi_{\theta_{t+1}}\leftarrow{\mathrm \Gamma}(\phi^\top \theta_{t+1})$
In Algorithm \[al:1\], ${\mathrm \Gamma}$ is a policy improvement operator, which maps an action-value function to a policy, e.g., greedy, $\epsilon$-greedy, and softmax and mellowmax [@Asadi2016].
Finite-Sample Analysis for Greedy-GQ
====================================
In this section, we will first introduce some technical assumptions, and then present our main results. We make the following standard assumptions.
The matrix $C=\mathbb{E}_{\mu}[\phi_t\phi_t^\top ]$ is non-singular.
$\|\phi_{s,a}\|_2\leq 1, \forall (s,a)\in{\mathcal{S}}\times{\mathcal{A}}$.
\[ass:1\] There exists some constants $m>0$ and $\rho \in (0,1)$ such that $$\begin{aligned}
\sup_{s\in\mathcal{S}} d_{TV}(\mathbb{P}(s_t |s_0=s), \mu) \leq m\rho^t ,\end{aligned}$$ for any $t>0$, where $d_{TV}$ is the total-variation distance between the probability measures.
In this paper, we focus on policies that are smooth. Specifically, $\pi_{\theta}(a|s)$ and $\nabla \pi_{\theta} (a|s)$ are Lipschitz functions of $\theta$.
\[assump:policy\] The policy $\pi_\theta(a|s)$ is $k_1$-Lipschitz and $k_2$-smooth, i.e., for any $(s,a) \in {\mathcal{S}}\times{\mathcal{A}}$, $$\begin{aligned}
\|\nabla \pi_{\theta}(a|s)\|\leq k_1, \forall \theta,\end{aligned}$$and, $$\begin{aligned}
\|\nabla\pi_{\theta_1}(a|s)-\nabla\pi_{\theta_2}(a|s)\| \leq k_2 \| \theta_1-\theta_2\|, \forall \theta_1,\theta_2
.\end{aligned}$$
We note that the smaller the $k_1$ and $k_2$ are, the smoother the policy is. This family contains many policies as special cases, e.g., softmax and mellowmax [@Asadi2016]. We also note that the greedy policy is not smooth, since it is not differentiable.
To justify the feasibility of Assumption \[assump:policy\] in practice, in the following, we first provide an example of the softmax policy, and show that it is Lipschitz and smooth in $\theta$. Consider the softmax operator, where for any $(a,s)\in {\mathcal{A}}\times{\mathcal{S}}$ and $\theta\in \mathbb R^N$, $$\begin{aligned}
\label{eq:softmax}
\pi_{\theta}(a|s)=\frac{e^{\sigma {\theta}^\top \phi_{s,a}}}{\sum_{a' \in \mathcal{A}}e^{\sigma {\theta}^\top \phi_{s,a'}}},\end{aligned}$$ for some $\sigma>0$.
\[lemma:softmax\_smooth\] The softmax policy $\pi_{\theta}(a|s)$ is $2\sigma$-Lipschitz and $8\sigma^2$-smooth, i.e., for any $(s,a)\in{\mathcal{S}}\times{\mathcal{A}}$, and for any $\theta_1,\theta_2\in\mathbb R^N$, $$\begin{aligned}
|\pi_{\theta_1}(a|s)-\pi_{\theta_2}(a|s)| &\leq 2\sigma \|\theta_1-\theta_2 \|,\\
\|\nabla\pi_{\theta_1}(a|s)-\nabla \pi_{\theta_2}(a|s) \|&\leq 8\sigma^2 \|\theta_1-\theta_2 \|.\end{aligned}$$
As $\sigma\rightarrow\infty$, the softmax policy approximates the greedy policy asymptotically, however its Lipschitz and smoothness constants also go to infinity.
It can be seen from that the objective function of the Greedy-GQ algorithm is non-convex. It may not be possible to guarantee the convergence of the algorithm to the global optimum. Therefore, to measure the convergence rate, we consider the convergence rate of the gradient norm to zero. Furthermore, motivated by the randomized stochastic gradient method in [@ghadimi2013stochastic], which is designed to analyze non-convex optimization problems, in this paper, we also consider a randomized version of the Greedy-GQ algorithm in Algorithm \[al:1\]. Specifically, let $M$ be an independent random variable with probability mass function ${\mathbb{P}}_M$. For steps from 1 to $M$, call the Greedy-GQ algorithm in Algorithm \[al:1\]. The final output is then $\theta_M$.
In the following theorem, we provide the convergence rate bound for $\mathbb{E}[\|\nabla J(\theta_M)\|^2]$ when constant stepsizes are used. Specifically, let $M \in \left\{1,2,...T\right\}$ and
\[eq:M\] (M=k)=.
\[thm:main\] Consider the following stepsizes: $\beta=\beta_t=\frac{1}{T^b}$, and $\alpha=\alpha_t=\frac{1}{T^a}$, where $\frac{1}{2}< a\leq 1$ and $0<b\leq a$. Then we have that for $T>0$, $$\begin{aligned}
\label{eq:theorembound}
\mathbb{E}[\|\nabla J(\theta_M)\|^2]=\mathcal{O}\left(\frac{1}{T^{1-a}}+\frac{\log T}{T^{\min\{b,a-b\}}}\right).\end{aligned}$$
Here we only provide the order of the bound in terms of $T$. An explicit bound can also be derived, which however is cumbersome and tedious. To understand how different parameters, e.g., $L, C, m,\rho$, affect the convergence speed, we refer the readers to equation in the appendix.
Although it is not explicitly characterized in , we note that as $k_1$ and $k_2$ increases, the bound will become looser and thus the algorithm will need more samples to converge. For a more “greedy" target policy with larger $k_1$ and $k_2$, it will require more samples to converge. This suggests a practical trade-off between the quality of the obtained policy and the sample complexity.
Theorem \[thm:main\] characterizes the relationship between the convergence rate and the choice of the stepsizes $\alpha_t$ and $\beta_t$. We further optimize over the choice of the stepsizes and obtain the best bound as in the following corollary.
\[col:1\] If we choose $a=\frac{2}{3}$ and $b=\frac{1}{3}$, then the best rate of the bound in is obtained as follows:
\[J(\_M)\^2\]=().
For the general non-convex optimization problem with a Lipschitz gradient, the convergence rate of the randomized stochastic gradient method is $\mathcal O(T^{-\frac{1}{2}})$ [@ghadimi2013stochastic]. However, the gradient estimate in that problem is unbiased, and the update is one timescale. In our problem, we have a two timescale update rule. Although the fast timescale updates much faster than the slow timescale, there still exists an estimation error, which we call it “tracking error". Specifically, the tracking error is defined as
z\_t=w\_t-w\^\*(\_t).
Moreover, in this paper, we consider the practical scenario where a single sample trajectory with Markovian noise is used. Therefore, for the Greedy-GQ algorithm, there exists bias in the gradient estimate, which justifies the difference in the convergence rate from the one for general non-convex optimization problems [@ghadimi2013stochastic].
Proof Sketch
============
In this section, we provide an outline of the proof, and highlight our major technical contributions. For a complete proof, we refer the readers to the appendix.
The proof can summarized in the following five steps.
1. We first prove that $J(\theta)$ is Lipschitz and smooth.
2. We then decompose the error recursively.
3. We provide a comprehensive characterization of stochastic bias terms and the tracking error in the two timescale updates.
4. We then recursively plug the obtained bound on $\mathbb{E}[\|\nabla J(\theta_M)\|^2]$ back into the analysis, and repeat recursively to obtain the tightest bound.
5. We then optimize the convergence rate over the choice of stepsizes.
In the following, we discuss the proof sketch step by step with more details.
**Step 1.** We first provide a characterization of the geometric property of the objective function $J(\theta)$. Specifically, we show that if $\pi_\theta$ is Lipschitz and smooth (satisfying Assumption \[assump:policy\]), then $J(\theta)$ is also Lipschitz and $K$-smooth for some $K>0$, i.e., for any $\theta_1$ and $\theta_2$,
J(\_1)-J(\_2) K||\_1-\_2||.
Here, larger $k_1$ and $k_2$ imply a larger $K$. As will be seen later in Step 2 and Step 3, a larger $K$ means a looser bound and a higher sample complexity. This theoretical assertion will also be validated in our numerical experiments.
Recall that $J({\theta})$ can be equivalently written as $
\mathbb{E}_{\mu}[\delta_{S,A,S'}({\theta})\phi_{S,A}]^\top \mathbb{E}_{\mu}[\phi_{S,A}\phi_{S,A}^\top ]^{-1}\mathbb{E}_{\mu}[\delta_{S,A,S'}({\theta})\phi_{S,A}],
$ which has a quadratic form in $\mathbb{E}_{\mu}[\delta_{S,A,S'}({\theta})\phi_{S,A}]$. Therefore, it suffices to show that $\mathbb{E}_{\mu}[\delta_{S,A,S'}({\theta})\phi_{S,A}]$ is bounded, Lipschitz and smooth, which is clear from its definition and the fact that $\pi_\theta$ is Lipschitz and smooth.
**Step 2.** Since the object function $J(\theta)$ is Lipschitz and $K$-smooth, then by Taylor expansion, we have that $$\begin{aligned}
\label{eq:1J}
&J(\theta_{t+1})-J(\theta_t)- \langle \theta_{t+1}-\theta_t, \nabla J(\theta_t) \rangle {\nonumber}\\
&\leq \frac{K}{2} \|\theta_{t+1}-\theta_t \|^2.\end{aligned}$$ Denote by $G_{t+1}(\theta,\omega)=(\delta_{t+1}(\theta)\phi_t-\gamma(\omega^\top \phi_t)\hat{\phi}_{t+1}(\theta))$. Then, the difference between $\theta_t$ and $\theta_{t+1}$ is $\alpha_t G_{t+1}(\theta_t,\omega_t)$. The inequality can be further written as $$\begin{aligned}
\label{eq:2}
&J(\theta_{t+1})-J(\theta_t)- \alpha_t\langle G_{t+1}(\theta_t,\omega_t), \nabla J(\theta_t) \rangle{\nonumber}\\
&\leq \frac{K\alpha_t^2}{2} \|G_{t+1}(\theta_t,\omega_t) \|^2.\end{aligned}$$
Note that $G_{t+1}(\theta_t,\omega_t)$ is the stochastic gradient used in the Greedy-GQ algorithm. Due to the two timescale update and the Markovian noise, the stochastic gradient is biased. For a finite-sample analysis, we will then need to characterize the stochastic bias in the gradient estimate $G_{t+1}(\theta_t,\omega_t)$ explicitly.
We first consider the difference between the true gradient $\nabla J(\theta_t)$ and the gradient estimate $G_{t+1}(\theta_t,\omega_t)$ used in the Greedy-GQ algorithm, which is denoted by ${\mathbf\Delta}_t=-2G_{t+1}(\theta_t, \omega_t)-\nabla J(\theta_t)$. Plug this in the inequality , and we obtain that $$\begin{aligned}
\label{eq:3}
&J(\theta_{t+1})-J(\theta_t)+\frac{\alpha_t}{2}\left\langle ({\mathbf\Delta}_t+\nabla J(\theta_t)), \nabla J(\theta_t) \right\rangle{\nonumber}\\
&=J(\theta_{t+1})-J(\theta_t)+\frac{\alpha_t}{2}\|\nabla J(\theta_t) \|^2{\nonumber}\\
&\quad+ \alpha_t\left\langle \frac{1}{2}{\mathbf\Delta}_t, \nabla J(\theta_t) \right\rangle{\nonumber}\\
&\leq \alpha_t^2\frac{K}{2} \|G_{t+1}(\theta_t,\omega_t) \|^2.\end{aligned}$$
Recall the definition of the random variable $M$ in . Applying recursively, we have that
\[eq:4\] &\[J(\_M)\^2\]\
&((J(\_0)-J(\_[T+1]{}))\
&+ \^T\_[t=0]{}\_t\^2 \[ G\_[t+1]{}(\_t,\_t)\^2\]\
&-\^T\_[t=0]{} \_t, J(\_t) ).
From , it can be seen that to understand the convergence rate of $\mathbb{E}[\|\nabla J(\theta_M)\|^2]$, we need to bound the three terms on the right hand side of . The first and second terms are straightforward to bound since $J(\theta)$ is non-negative for any $\theta$, and $\|G_{t+1}\|$ is uniformly bounded by some constant.
For the third term $\left\langle {\mathbf\Delta}_t, \nabla J(\theta_t) \right\rangle$, it can be further decomposed into the following two parts
\[eq:5a\] &J(\_t),-2G\_[t+1]{}(\_t, \_t)+2G\_[t+1]{}(\_t, \^\*(\_t))\
&- J(\_t), J(\_t)+2G\_[t+1]{}(\_t, \^\*(\_t)) ,
where the first part is corresponding to the tracking error, and the second part is corresponding to the stochastic bias caused by the Markovian noise.
**Step 3.** We then provide bounds for each term in and . For the first and second terms in , it is straightforward to develop their upper bounds. For the first term in , it can be upper bounded by exploiting the Lipschitz property of $G_{t+1}(\theta,\omega)$ in $\omega$. Specifically,
\[eq:31a\] &J(\_t),-2G\_[t+1]{}(\_t, \_t)+2G\_[t+1]{}(\_t, \^\*(\_t))\
&\_1 J(\_t) \_t-\^\*(\_t),
for some $\xi_1>0$. Thus, it suffices to bound the tracking error $\|\omega_t-\omega^*(\theta_t)\|$. The bound on the tracking error is difficult due to the complicated coupling between the parameter $\omega_t$, $\theta_t$ and the sample trajectory. We decouple such the dependence between $\omega_t$, $\theta_t$ and the samples by looking $\tau$ steps back, where $\tau$ is the mixing time of the MDP. By the geometric uniform ergodicity, conditioning on $\omega_{t-\tau}$ and $\theta_{t-\tau}$, the distribution of $(s_t,a_t)$ is close to the stationary distribution $\mu$. Thus, the expectation of the tracking error can be bounded. We then bound the second term in . We know that for any fixed $\theta$, ${\mathbb{E}}_\mu[\nabla J(\theta)+2G_{t+1}(\theta, \omega^*(\theta))]=0.$ However, $\theta_t$ and $S_t,A_t,S_{t+1}$ are not independent. Similarly, we exploit the geometric uniform ergodicity of the MDP. For simplicity, we denote by
(\_t,O\_t)=J(\_t), J(\_t)+2G\_[t+1]{}(\_t, \^\*(\_t)) ,
where $O_t=\{S_t,A_t,S_{t+1},r_t\}$. We can show that $\zeta(\theta,O_t)$ is Lipschitz in $\theta$. Thus, if we look $\tau$ step back, then
|(\_t,O\_t)-(\_[t-]{},O\_t)|c\_\_t-\_[t-]{},
for some $c_\zeta>0$. Therefore,
(\_t,O\_t)(\_[t-]{},O\_t) +c\_\_t-\_[t-]{}.
Since we are using small stepsizes, then $\|\theta_t-\theta_{t-\tau}\|$ should be small. In other words, the difference between $\zeta(\theta_t,O_t)$ and $\zeta(\theta_t,O_t)$ is small. By the geometric uniform ergodicity, for any $\theta_{t-\tau}$, the distribution of $O_t$ is close to the stationary distribution $\mu$. Thus, even $\theta_{t-\tau}$ and $O_t$ are not independent, we can still upper bound ${\mathbb{E}}[\zeta(\theta_{t-\tau},O_t)]$. In this way, we decouple the dependence between $\theta_t$ and $O_t$, and we can obtain the bound on the gradient bias.
**Step 4.** After Step 3, we can obtain the following bound on $\mathbb{E}[\|\nabla J(\theta_M)\|^2]$:
\[eq:34a\] \[J(\_M)\^2\]=O(+).
This bound is obtained by upper bounding $\|\nabla J(\theta_t)\|$ on the right hand side of using a constant. Obviously, $\mathbb{E}[\|\nabla J(\theta_M)\|^2]\to 0$ as $T\to \infty$, and thus using a constant to upper bound $\nabla J(\theta_t)$ is not tight.
In this step, we recursively use the obtained bound to further tighten the bound on $\mathbb{E}[\|\nabla J(\theta_M)\|^2]$. Specifically, we plug back into in Step 3. If $1-a> \min\{b,a-b\}$, then the second term on the right hand side of dominates. Plugging back into will further tighten the bound to the following one:
\[J(\_M)\^2\]=O(+).
Repeat this procedure, we can then obtain the following bound:
\[eq:36a\] \[J(\_M)\^2\]=O(+).
If $1-a\leq \frac{1}{2}\min\{b,a-b\}$, then the first term in dominates. Therefore, the above recursive refinement will not improve the convergence rate. If $\frac{1}{2}\min\{b,a-b\}\leq 1-a\leq \min\{b,a-b\}$, we can apply our recursive bounding trick finite times until the first term $\mathcal O\left(\frac{1}{T^{1-a}} \right)$ in dominates. Combining the analyses for the three cases, the overall convergence rate bound can be obtained, which is as in .
**Step 5.** Given the convergence rate bound in , in this step, we optimize over the choice of the stepsizes to obtain the fastest convergence rate. Recall that $\frac{1}{2}< a\leq 1$ and $0<b\leq a$. Then, it can be derived that when $a=\frac{2}{3}$ and $b=\frac{1}{3}$, the best convergence rate that is achievable in is $\mathcal O\left(\frac{\log T}{T^{\frac{1}{3}}}\right)$.
Numerical Experiments
=====================
In this section, we present our numerical experiments. Specifically, we investigate how the Lipschitz and smoothness constants affect the convergence of the Greedy-GQ algorithm. We use the the softmax operator as an example. Recall that in Lemma \[lemma:softmax\_smooth\], the Lipschitz and smoothness constants of the softmax operator is an increasing function of $\sigma$ in .
As has been observed in our finite-sample analysis, the upper bound on the gradient norm increases with $K$, and thus increases with $\sigma$. This suggests a higher sample complexity as the target policy becomes more “greedy". We will numerically validate this observation by simulating the Greedy-GQ algorithm for different values of $\sigma$ in . We consider a simple example: $\mathcal{S}=\{1,2,3,4\}$ and $\mathcal{A}=\{1,2\}$. For the first MDP we consider, taking any action at any state will have the same probability to transit to any state, i.e. $\mathbb{P}(s'|s,a)=\frac{1}{4}$ for any $(s,a,s')$. Five different values of $\sigma$ are considered: $\sigma=1,2,3,15,20$.
We randomly generate two base functions. We initialize $s_0=2$, $\theta_0=(1,2)^\top$ and $\omega_0=(0.1,0.1)^\top$. At each iteration, we choose $A_t \sim \pi_b$, update $\theta_{t+1}$ and $\omega_{t+1}$ according to Algorithm \[al:1\], and compute $\|\nabla J(\theta_t)\|^2$. As for $T$, we consider $T=1000$. For the same state and action spaces, we vary the behavior policy and Markov transition kernel, and repeat our experiment for three more times. We plot the gradient norm as a function of the number of iterations in Fig. \[fig: 1\].
It can be seen from Fig. \[fig: 1\], as $\sigma$ increases, the convergence of the Greedy-GQ algorithm is getting slower. This observation matches with our theoretical bound that the Greedy-GQ algorithm has a higher sample complexity if the targeted policy is less smoother.
Conclusion
==========
In this paper, we developed the first finite-sample analysis for the Greedy-GQ algorithm with linear function approximation under Markovian noise. Our analysis is from a novel optimization perspective to solve RL problems. We comprehensively characterized the stochastic bias in the gradient estimate and designed a novel technique which recursively applies the obtained bound back into the bias analysis to tighten the convergence rate bound. We characterized the convergence rate of the Greedy-GQ algorithm, and provided a general guide for choosing stepsizes in practice. The convergence rate obtained by our analysis is $\mathcal O\left(\frac{\log T}{T^{\frac{1}{3}}}\right)$, and is close to the convergence rate $\mathcal O\left(\frac{1}{T^{\frac{1}{2}}}\right)$ for general non-convex optimization problems with unbiased gradient estimate. Such a different is mainly due to the Markovian noise and the tracking error in the two timescale updates. The techniques developed in this paper may be of independent interest for a wide range of reinforcement learning problems with non-convex objective function and Markovian noise.
In this paper, we provided the finite-sample analysis and the convergence rate for the case with constant stepsizes. The convergence rate for the case with diminishing stepsizes can be derived similarly. One interesting future direction is to investigate the Greedy-GQ algorithm with the greedy policy. Specifically, $$\pi_\theta(a|s)=1 \text{ if } a=\arg\max_{a'\in{\mathcal{A}}} \phi_{s,a}^\top\theta.$$ Due to this max operator, the objective function $J(\theta)$ becomes non-differentiable and non-smooth. To the best of the author’s knowledge, there does not exist a general methodology to analyze non-convex non-differentiable optimization problems. One possible solution is to explore the special geometry of the objective function, i.e., $J(\theta)$ is a piece-wise quadratic function of $\theta$. It is also of further interest to investigate the Greedy-GQ algorithm with general function approximation, e.g., neural network.
[**Supplementary Materials**]{}
Useful Lemmas for Proving Theorem \[thm:main\] {#app:lemmas}
==============================================
In this subsection, we prove some useful Lemmas for our finite-sample analysis.
Before we start, we first introduce some nations. In the following proof, $\|a\|$ denotes the $\ell_2$ norm if $a$ is a vector; and $\|A\|$ denotes the operator norm if $A$ is a matrix. Let $\lambda $ be the smallest eigenvalue of the matrix $C$. Then the operator norm of $C^{-1}$ is $\frac{1}{\lambda}$. We note that the Greedy-GQ algorithm in Algorithm \[al:1\] was shown to converge asymptotically, and $\theta_t$ and $\omega_t$ were shown to be bounded a.s. (see Proposition 4 in [@maei2010toward]). We then define $R$ as the upper bound on both $\theta_t$ and $\omega_t$. Specifically, for any $t$, $\|\theta_t\|\leq R$ and $\|\omega_t\|\leq R$ a.s..
We first prove that if the policy $\pi_\theta$ is smooth in $\theta$, then the object function $J(\theta)$ is also smooth.
\[Lemma:1\] The objective function $J(\theta)$ is $K$-smooth for $\theta \in \{\theta: \|\theta\|\leq R\}$, i.e., for any $\|\theta_1\|,\|\theta_2 \| \leq R$,
J(\_1)-J(\_2) K||\_1-\_2||,
where $
K=2\gamma\frac{1}{\lambda}\left((k_1|{\mathcal{A}}|R+1)(1+\gamma+\gamma Rk_1|{\mathcal{A}}|)+|{\mathcal{A}}|(r_{\max}+R+\gamma R)( 2k_1+ k_2R) \right).$
Recall the expression of $J\left (\theta\right )$: $$\begin{aligned}
J\left (\theta\right )=\mathbb{E}_\mu\left [\delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ]^\top C^{-1}\mathbb{E}_\mu\left [\delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ],\end{aligned}$$ where $\delta_{S,A,S'}=r_{S,A,S'}+\gamma \sum_{a\in \mathcal{A}} \pi_{\theta}\left (a|S'\right )\theta^\top \phi_{S',a}-\theta^\top \phi_{S,A}$. Then, $$\begin{aligned}
\nabla J\left (\theta\right )=2\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ]\right ) C^{-1}\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ],\end{aligned}$$ where $$\begin{aligned}
\label{eq:28}
\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ]\right )&=\mathbb{E}_{\mu}\left [\left (\nabla \gamma\sum_{a\in \mathcal{A}} \pi_{\theta}\left (a|S'\right )\theta^\top \phi_{S',a}\right )\phi_{S,A}^\top \right ]{\nonumber}\\
&=\gamma\mathbb{E}_{\mu}\left [\left (\sum_{a\in\mathcal{A}} \nabla\left (\pi_{\theta}\left (a|S'\right )\right )\theta^\top \phi_{S',a}+\pi_{\theta}\left (a|S'\right )\phi_{S',a}\right )\phi_{S,A}^\top\right ].\end{aligned}$$
It then follows that $$\begin{aligned}
&\nabla J\left (\theta_1\right )-\nabla J\left (\theta_2\right ){\nonumber}\\
&=2\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]\right ) C^{-1}\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]-2\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ]\right ) C^{-1}\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ]{\nonumber}\\
&=2\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]\right ) C^{-1}\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]-2\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]\right ) C^{-1}\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ]{\nonumber}\\
&+2\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]\right ) C^{-1}\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ]-2\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ]\right ) C^{-1}\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ].\end{aligned}$$ Since $C^{-1}$ is positive definite, thus to show $\nabla J(\theta)$ is Lipschitz, it suffices to show both $\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ]\right )$ and $\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ]$ are Lipschitz in $\theta$ and bounded.
We first show that $$\begin{aligned}
\label{eq:30}
\|\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ]\|\leq r_{\max}+(1+\gamma) R,\end{aligned}$$ and $$\begin{aligned}
\|\nabla \mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ] \|=\| \mathbb{E}_{\mu}\left [\nabla \delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ]\|\leq \gamma(k_1|{\mathcal{A}}|R+1).\end{aligned}$$
Following from , we then have that $$\begin{aligned}
\label{eq:32}
&\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]\right )-\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ]\right ){\nonumber}\\
&= \gamma\mathbb{E}_{\mu}\left [\left( \sum_{a\in\mathcal{A}} \nabla\left (\pi_{\theta_1}\left (a|S'\right )\right)\theta_1^\top \phi_{S',a}-\nabla\left (\pi_{\theta_2}\left (a|S'\right )\right)\theta_2^\top \phi_{S',a}+\pi_{\theta_1}\left (a|S'\right )\phi_{S',a}-\pi_{\theta_2}\left (a|S'\right )\phi_{S',a}\right)\phi_{S,A}^\top\right ]{\nonumber}\\
&= \gamma\mathbb{E}_{\mu}\Bigg [\Bigg( \sum_{a\in\mathcal{A}} \nabla\left (\pi_{\theta_1}\left (a|S'\right )\right)\theta_1^\top \phi_{S',a}-\nabla\left (\pi_{\theta_2}\left (a|S'\right )\right)\theta_1^\top \phi_{S',a}+\nabla\left (\pi_{\theta_2}\left (a|S'\right )\right)\theta_1^\top \phi_{S',a}{\nonumber}\\
&\quad-\nabla\left (\pi_{\theta_2}\left (a|S'\right )\right)\theta_2^\top \phi_{S',a}\Bigg)\phi_{S,A}^\top\Bigg ]+\gamma\mathbb{E}_{\mu}\left [\left (\sum_{a\in {\mathcal{A}}} \left ( \pi_{\theta_1}\left (a|S'\right )\phi_{S',a}-\pi_{\theta_2}\left (a|S'\right )\phi_{S',a}\right)\right )\phi_{S,A}^\top\right ].\end{aligned}$$ This implies that $$\begin{aligned}
\label{eq:33}
& \|\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]\right )-\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ]\right ) \|{\nonumber}\\
&\leq \gamma|{\mathcal{A}}|\left(2k_1+ k_2 R \right)\|\theta_1-\theta_2\|,\end{aligned}$$ and thus $\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ]\right )$ is Lipschitz in $\theta$.
Following similar steps, we can also show that $\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta\right )\phi_{S,A}\right ]$ is Lipschitz: $$\begin{aligned}
\label{eq:34}
\|\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]-\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ]\|
\leq \left (\gamma(|{\mathcal{A}}|k_1R+1)+1\right ) \|\theta_1-\theta_2\|.\end{aligned}$$ Now by combining both parts in and , we can show that $$\begin{aligned}
&\|\nabla J\left (\theta_1\right )-\nabla J\left (\theta_2\right )\|{\nonumber}\\
&\leq \|2\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]\right ) C^{-1}\left(\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]-\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ]\right)\|{\nonumber}\\
&\quad+\|2\left(\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_1\right )\phi_{S,A}\right ]\right )-\nabla \left (\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ]\right )\right) C^{-1}\mathbb{E}_{\mu}\left [\delta_{S,A,S'}\left (\theta_2\right )\phi_{S,A}\right ]{\nonumber}\\
&\leq 2\gamma(k_1|{\mathcal{A}}|R+1)\frac{1}{\lambda}(1+\gamma(1+Rk_1|{\mathcal{A}}|)\|\theta_1-\theta_2\|{\nonumber}\\
&\quad+2\frac{1}{\lambda}(r_{\max}+(1+\gamma) R)\gamma|{\mathcal{A}}|(2k_1+k_2R)\|\theta_1-\theta_2\|{\nonumber}\\
&= 2\gamma\frac{1}{\lambda}\left((k_1|{\mathcal{A}}|R+1)(1+\gamma+\gamma Rk_1|{\mathcal{A}}|)+|{\mathcal{A}}|(r_{\max}+R+\gamma R)( 2k_1+ k_2R) \right) \|\theta_1-\theta_2\|,\end{aligned}$$ which implies that $\nabla J\left (\theta\right )$ is Lipschitz. This completes the proof.
Recall that $G_{t+1}(\theta, \omega)=\delta_{t+1}(\theta)\phi_t-\gamma(\omega^T\phi_t)\hat{\phi}_{t+1}(\theta)$, where $\delta_{t+1}(\theta)=r_{t+1}+\gamma\Bar{V}_{t+1}(\theta)-\theta^\top \phi_t$, $\Bar{V}_{t+1}(\theta)=\Bar{V}_{\theta}(S_{t+1})=\sum_{a\in \mathcal{A}}\pi_{\theta}(a|S_{t+1})\theta^\top \phi_{S_{t+1},a}$, and $\hat{\phi}_{t+1}(\theta)=\sum_{a\in\mathcal{A}}\theta^\top\phi_{S_{t+1},a}\nabla \pi_{\theta}(a|S_{t+1})+\pi_{\theta}(a|S_{t+1})\phi_{S_{t+1},a}$. The following Lemma shows that $G_{t+1}(\theta, \omega)$ is Lipschitz in $\omega$, and $G_{t+1}(\theta, \omega^*(\theta))$ is Lipschitz in $\theta$.
\[Lemma:2\] For any $\theta\in \{\theta:\|\theta\|\leq R\}$, $G_{t+1}(\theta, \omega)$ is Lipschitz in $\omega$, and $G_{t+1}(\theta, \omega^*(\theta))$ is Lipschitz in $\theta$. Specifically, for any $w_1,w_2$, $$\begin{aligned}
\|G_{t+1}(\theta,\omega_1)-G_{t+1}(\theta,\omega_2)\|\leq \gamma(|{\mathcal{A}}|Rk_1+1)\|\omega_1-\omega_2\|,\end{aligned}$$ and for any $\theta_1,\theta_2\in \{\theta:\|\theta\|\leq R\}$, $$\begin{aligned}
&\|G_{t+1}(\theta_1,\omega^*(\theta_1))-G_{t+1}(\theta_2,\omega^*(\theta_2))\| \leq k_3\|\theta_1-\theta_2 \|,\end{aligned}$$ where $k_3=(1+\gamma+\gamma R|{\mathcal{A}}|k_1+\gamma\frac{1}{\lambda}|{\mathcal{A}}|(2k_1+k_2R)(r_{\max}+\gamma R+R)+\gamma \frac{1}{\lambda}(1+|{\mathcal{A}}|Rk_1)(1+\gamma+\gamma R|{\mathcal{A}}|k_1)).$
Following similar steps as those in and , we can show that $\hat{\phi}_{t+1}(\theta)$ is Lipschitz in $\theta$, i.e., for any $\theta_1, \theta_2$ $\in \{\theta:\|\theta\|\leq R\}$, $$\begin{aligned}
&\|\hat{\phi}_{t+1}(\theta_1)-\hat{\phi}_{t+1}(\theta_2)\|
\leq | \mathcal{A}| (2k_1 +k_2R)\|\theta_1-\theta_2 \|.\end{aligned}$$ Under Assumption \[assump:policy\], it can be easily shown that
\[eq:39\] \_[t+1]{}() ||Rk\_1+1.
It then follows that for any $\omega_1$ and $\omega_2$, $$\begin{aligned}
&\|G_{t+1}(\theta,\omega_1))-G_{t+1}(\theta,\omega_2))\|{\nonumber}\\
&=\|\gamma(\omega_1-\omega_2)^\top \phi_t) \hat{\phi}_{t+1}(\theta) \|{\nonumber}\\
&\leq \gamma (|{\mathcal{A}}|Rk_1+1)\|\omega_1-\omega_2\|.\end{aligned}$$
To show that $G_{t+1}(\theta,\omega^*(\theta))$ is Lipschitz in $\theta$, we have that $$\begin{aligned}
&\|G_{t+1}(\theta_1,\omega^*(\theta_1))-G_{t+1}(\theta_2,\omega^*(\theta_2)) \|{\nonumber}\\
&\leq |\delta_{t+1}(\theta_1)-\delta_{t+1}(\theta_2) |
+\gamma\|(\omega^*(\theta_2))^\top\phi_t\hat{\phi}_{t+1}(\theta_2)-(\omega^*(\theta_1))^\top\phi_t\hat{\phi}_{t+1}(\theta_1) \|{\nonumber}\\
&\overset{(a)}{\leq}\gamma\|(\omega^*(\theta_2))^\top \phi_t\hat{\phi}_{t+1}(\theta_2)-(\omega^*(\theta_1))^\top\phi_t\hat{\phi}_{t+1}(\theta_1)-(\omega^*(\theta_1))^\top\phi_t\hat{\phi}_{t+1}(\theta_2)+(\omega^*(\theta_1))^\top\phi_t\hat{\phi}_{t+1}(\theta_2) \|{\nonumber}\\
&\quad+(1+\gamma+\gamma R|\mathcal{A}|k_1)\|\theta_1-\theta_2 \|{\nonumber}\\
&\leq \gamma(1+|{\mathcal{A}}|Rk_1)\|\omega^*(\theta_2)-\omega^*(\theta_1) \|+\gamma\|\omega^*(\theta_1) \|\|\hat{\phi}_{t+1}(\theta_1)-\hat{\phi}_{t+1}(\theta_2)\ \|{\nonumber}\\
&\quad+\gamma(1+R|\mathcal{A}|k_1)\|\theta_1-\theta_2 \|+\|\theta_1-\theta_2 \|{\nonumber}\\
&\overset{(b)}{\leq } \left(1+\gamma+\gamma R|{\mathcal{A}}|k_1+\gamma\frac{1}{\lambda}|{\mathcal{A}}|(2k_1+k_2R)(r_{\max}+\gamma R+R)+\gamma \frac{1}{\lambda}(1+|{\mathcal{A}}|Rk_1)(1+\gamma+\gamma R|{\mathcal{A}}|k_1)\right){\nonumber}\\
&\quad\times\|\theta_1-\theta_2 \|{\nonumber}\\
&\triangleq k_3\|\theta_1-\theta_2\|,\end{aligned}$$ where $(a)$ can be shown following steps similar to those in , while $(b)$ can be shown by combining $$\begin{aligned}
\|\omega^*(\theta)\|=\|C^{-1}\mathbb{E}[\delta_{t+1}(\theta)\phi_t]\|\leq \frac{1}{\lambda}(r_{\max}+\gamma R+R),\end{aligned}$$ and $$\begin{aligned}
\label{eq:w*lip}
\|\omega^*(\theta_2)-\omega^*(\theta_1)\|\leq \frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)\|\theta_1-\theta_2\|.\end{aligned}$$
In the following lemma, we provide a decomposition of the stochastic bias, which is essential to our finite-sample analysis.
Consider the Greedy-GQ algorithm (see Algorithm \[al:1\]), when the stepsize $\alpha_t$ is constant, i.e., $\alpha_t=\alpha,\forall t\geq 0$, then $$\begin{aligned}
\label{eq:main}
\sum^T_{t=0} \frac{\alpha_t}{2} \mathbb{E}[\|\nabla J(\theta_t) \|^2] &\leq J(\theta_0)-J(\theta_{T+1})+ \gamma\alpha_t(1+|{\mathcal{A}}|Rk_1)\sqrt{\sum^T_{t=0} \mathbb{E}[\|\nabla J(\theta_t)\|^2]}\sqrt{\sum^T_{t=0}\mathbb{E}[\|\omega^*(\theta_t)-\omega_t\|^2]} {\nonumber}\\
&\quad+\sum^T_{t=0}\alpha_t \mathbb{E}[\langle \nabla J(\theta_t),\frac{\nabla J(\theta_t)}{2}+G_{t+1}(\theta_t, \omega^*(\theta_t)) \rangle]+\frac{K}{2}\sum^T_{t=0}\alpha_t^2 \mathbb{E}[\| G_{t+1}(\theta_t,\omega_t)\|^2].\end{aligned}$$
From Lemma \[Lemma:1\], it follows that $J(\theta)$ is $K$-smooth. Then, by Taylor expansion, for any $\theta_1$ and $\theta_2$, $$\begin{aligned}
|J(\theta_1)-J(\theta_2)-\langle \nabla J(\theta_2), \theta_1-\theta_2 \rangle | \leq \frac{K}{2}||\theta_1-\theta_2||^2.\end{aligned}$$ Then, it can be shown that $$\begin{aligned}
J(\theta_{t+1}) &\leq J(\theta_t) +\langle \nabla J(\theta_t), \theta_{t+1}-\theta_t\rangle + \frac{K}{2} \alpha_t^2||G_{t+1}(\theta_t,\omega_t)||^2{\nonumber}\\
&=J(\theta_t) +\alpha_t \langle \nabla J(\theta_t),G_{t+1}(\theta_t,\omega_t) \rangle + \frac{K}{2} \alpha_t^2||G_{t+1}(\theta_t,\omega_t)||^2{\nonumber}\\
&=J(\theta_t)-\alpha_t\langle \nabla J(\theta_t),-G_{t+1}(\theta_t, \omega_t)-\frac{\nabla J(\theta_t)}{2}+G_{t+1}(\theta_t, \omega^*(\theta_t))-G_{t+1}(\theta_t, \omega^*(\theta_t)) \rangle {\nonumber}\\
&\quad-\frac{\alpha_t}{2}||\nabla J(\theta_t)||^2+\frac{K}{2} \alpha_t^2||G_{t+1}(\theta_t,\omega_t)||^2{\nonumber}\\
&=J(\theta_t)-\alpha_t\langle \nabla J(\theta_t),-G_{t+1}(\theta_t, \omega_t)+G_{t+1}(\theta_t, \omega^*(\theta_t)) \rangle{\nonumber}\\
&\quad+\alpha_t \langle \nabla J(\theta_t), \frac{\nabla J(\theta_t)}{2}+G_{t+1}(\theta_t, \omega^*(\theta_t)) \rangle-\frac{\alpha_t}{2}||\nabla J(\theta_t)||^2+\frac{K}{2} \alpha_t^2||G_{t+1}(\theta_t,\omega_t)||^2{\nonumber}\\
&\overset{(a)}{\leq} J(\theta_t) +\alpha_t \gamma\|\nabla J(\theta_t) \|(1+|{\mathcal{A}}|Rt_1)\|\omega^*(\theta_t)-\omega_t \|+\alpha_t \langle \nabla J(\theta_t), \frac{\nabla J(\theta_t)}{2}+G_{t+1}(\theta_t, \omega^*(\theta_t)) \rangle {\nonumber}\\
&\quad-\frac{\alpha_t}{2}||\nabla J(\theta_t)||^2+\frac{K}{2} \alpha_t^2||G_{t+1}(\theta_t,\omega_t)||^2,\end{aligned}$$ where $(a)$ follows from the fact that $G_{t+1}(\theta,\omega)$ is Lipschitz in $\omega$ (see Lemma \[Lemma:2\]).
By taking expectation of both sides, summing up the inequality from $0$ to $T$, and rearranging the terms, we have that $$\begin{aligned}
\label{eq:47}
&\sum^T_{t=0} \frac{\alpha_t}{2} \mathbb{E}[\|\nabla J(\theta_t) \|^2] {\nonumber}\\
&\leq J(\theta_0)-J(\theta_{T+1})+\sum^T_{t=0} \gamma\alpha_t(1+|{\mathcal{A}}|Rk_1) \mathbb{E}[\|\nabla J(\theta_t)\|\|\omega^*(\theta_t)-\omega_t\|]{\nonumber}\\
&\quad+\sum^T_{t=0}\alpha_t \mathbb{E}[\langle \nabla J(\theta_t),\frac{\nabla J(\theta_t)}{2}+G_{t+1}(\theta_t, \omega^*(\theta_t)) \rangle]+\frac{K}{2}\sum^T_{t=0}\alpha_t^2 \mathbb{E}[\| G_{t+1}(\theta_t,\omega_t)\|^2].\end{aligned}$$ We then apply Cauchy-Schwarz’s inequality, and we have that $$\begin{aligned}
&\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|\|\omega^*(\theta_t)-\theta_t\|]{\nonumber}\\
&\leq \sum^T_{t=0}\sqrt{\mathbb{E}[\|\nabla J(\theta_t)\|^2]\mathbb{E}[\|\omega^*(\theta_t)-\theta_t\|^2]}.\end{aligned}$$ We further define two vectors $a_E$ and $a_z$, where
a\_E&(,,...,)\^,\
a\_z&(,,...,)\^.
Then, it follows that $$\begin{aligned}
\label{eq:64a}
&\sum^T_{t=0}\sqrt{\mathbb{E}[\|\nabla J(\theta_t)\|^2]\mathbb{E}[\|\omega^*(\theta_t)-\theta_t\|^2]}{\nonumber}\\
&=\langle a_E,a_z\rangle {\nonumber}\\
&\leq\|a_E\|\|a_z\|{\nonumber}\\
&=\sqrt{\sum^T_{t=0} \mathbb{E}[\|\nabla J(\theta_t)\|^2]}\sqrt{\sum^T_{t=0}\mathbb{E}[\|\omega^*(\theta_t)-\omega_t\|^2]}.\end{aligned}$$
Thus plugging in , and since $\alpha_t=\alpha, \forall t\geq 0$ is constant, we have that $$\begin{aligned}
\label{eq:recursion0}
&\sum^T_{t=0} \frac{\alpha_t}{2} \mathbb{E}[\|\nabla J(\theta_t) \|^2] {\nonumber}\\
&\leq J(\theta_0)-J(\theta_{T+1})+ \gamma\alpha_t(1+|{\mathcal{A}}|Rk_1) \sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|\|\omega^*(\theta_t)-\omega_t\|]{\nonumber}\\
&\quad+\sum^T_{t=0}\alpha_t \mathbb{E}[\langle \nabla J(\theta_t),\frac{\nabla J(\theta_t)}{2}+G_{t+1}(\theta_t, \omega^*(\theta_t)) \rangle]+\frac{K}{2}\sum^T_{t=0}\alpha_t^2 \mathbb{E}[\| G_{t+1}(\theta_t,\omega_t)\|^2]{\nonumber}\\
&\leq J(\theta_0)-J(\theta_{T+1})+ \gamma\alpha_t(1+|{\mathcal{A}}|Rk_1)\sqrt{\sum^T_{t=0} \mathbb{E}[\|\nabla J(\theta_t)\|^2]}\sqrt{\sum^T_{t=0}\mathbb{E}[\|\omega^*(\theta_t)-\omega_t\|^2]} {\nonumber}\\
&\quad+\sum^T_{t=0}\alpha_t \mathbb{E}[\langle \nabla J(\theta_t),\frac{\nabla J(\theta_t)}{2}+G_{t+1}(\theta_t, \omega^*(\theta_t)) \rangle]+\frac{K}{2}\sum^T_{t=0}\alpha_t^2 \mathbb{E}[\| G_{t+1}(\theta_t,\omega_t)\|^2].\end{aligned}$$
We next derive the bounds on $ \mathbb{E}[\langle \nabla J(\theta_t),\frac{\nabla J(\theta_t)}{2}+G_{t+1}(\theta_t, \omega^*(\theta_t)) \rangle]$ and $\mathbb{E}[\|\omega^*(\theta_t)-\omega_t\|]$, where we refer to the second term as the “tracking error”.
We first define $z_t=\omega_t-\omega^*(\theta_t)$, then the algorithm can be written as: $$\begin{aligned}
\theta_{t+1}&=\theta_t+\alpha_t(f_1(\theta_t,O_t)+g_1(\theta_t,z_t,O_t)), \\
z_{t+1}&=z_t+\beta_t(f_2(\theta_t,O_t)+g_2(\theta_t,O_t))+\omega^*(\theta_t)-\omega^*(\theta_{t+1}), \end{aligned}$$ where $$\begin{aligned}
\left\{
\begin{aligned}
&f_1(\theta_t, O_t) \triangleq \delta_{t+1}(\theta_t)\phi_t-\gamma\phi_t^\top \omega^*(\theta_t)\hat{\phi}_{t+1}(\theta_t), \\
&g_1(\theta_t, z_t, O_t) \triangleq -\gamma\phi_t^\top z_t\hat{\phi}_{t+1}(\theta_t), \\
&f_2(\theta_t,O_t) \triangleq (\delta_{t+1}(\theta_t)-\phi_t^\top \omega^*(\theta_t))\phi_t,\\
&g_2(z_t,O_t) \triangleq -\phi_t^\top z_t\phi_t,\\
&O_t\triangleq(s_t, a_t, r_t,s_{t+1}).
\end{aligned}
\right.\end{aligned}$$ We then develop some upper bounds of functions $f_1,g_1,f_2,g_2$ in the algorithm in the following lemma.
\[Lemma:3\] For $\|\theta\|\leq R$, $\|z\|\leq 2R$, there exist constants $c_{f_1}$, $c_{g_1}$, $c_{g_2}$ and $c_{f_2}$ such that $\|f_1(\theta,O_t)\|\leq c_{f_1},$ $\|g_1(\theta,z,O_t)\|\leq c_{g_1},$ $|f_2(\theta,O_t)|\leq c_{f_2}$ and $|g_2(\theta,O_t)|\leq c_{g_2}$, where $c_{f_1}=r_{\max}+(1+\gamma)R+\gamma\frac{1}{\lambda}(r_{\max}+(1 + \gamma)R)(1+R|\mathcal{A}|k_1)$ , $c_{g_1}=2\gamma R(1+R|\mathcal{A}|k_1)$, $c_{f_2}=r_{\max}+(1+\gamma)R+\frac{1}{\lambda}(r_{\max}+(1 + \gamma)R)$, and $c_{g_2}=2R$.
This Lemma can be shown easily using , and .
We further define $\zeta(\theta, O_t)\triangleq\langle \nabla J(\theta), \frac{\nabla J(\theta)}{2}+G_{t+1}(\theta, \omega^*(\theta)) \rangle$, then we have that $\mathbb{E}_{\mu}[\zeta(\theta, O_t)]=0$ for any fixed $\theta$, where $(S_t,A_t)$ in $O_t$ follow the stationary distribution $\mu$. In the following lemma, we provide upper bound on ${\mathbb{E}}[\zeta(\theta, O_t)]$.
\[thm:zeta\] Let $\tau_{\alpha_T}\triangleq \min \left\{k : m\rho^k \leq \alpha_T \right\}$. If $t \leq \tau_{\alpha_T}$, then $$\begin{aligned}
\mathbb{E}[\zeta(\theta_t,O_t)] \leq c_{\zeta}(c_{f_1}+c_{g_1})\alpha_0\tau_{\alpha_T},\end{aligned}$$ and if $t > \tau_{\alpha_T}$, then $$\begin{aligned}
\mathbb{E}[\zeta(\theta_t, O_t)]\leq k_{\zeta}\alpha_T+c_{\zeta}(c_{f_1}+c_{g_1})\tau_{\alpha_T}\alpha_{t-\tau_{\alpha_T}}.\end{aligned}$$ Where $c_{\zeta}=2\gamma(1+k_1|{\mathcal{A}}|R)\frac{1}{\lambda}(r_{\max}+R+\gamma R)(\frac{K}{2}+k_3)+K(r_{\max}+R+\gamma R)(\gamma \frac{1}{\lambda}(1+k_1|{\mathcal{A}}|R)+1+\gamma \frac{1}{\lambda}(1+Rk_1|{\mathcal{A}}|))$ and $k_{\zeta}=4\gamma(1+k_1R|{\mathcal{A}}|)\frac{1}{\lambda}(r_{\max}+R+\gamma R)^2(2\gamma(1+k_1|{\mathcal{A}}|R)\frac{1}{\lambda}+1)$.
We note that when $\theta$ is fixed, $\mathbb{E}[G_{t+1}(\theta, \omega^*(\theta))]=-\frac{1}{2} \nabla J(\theta)$. We will use this fact and the Markov mixing property to show this Lemma. Note that for any $\theta_1$ and $\theta_2$, it follows that $$\begin{aligned}
\label{eq:52}
&|\zeta(\theta_1,O_t)-\zeta(\theta_2,O_t)|{\nonumber}\\
&=|\langle \nabla J(\theta_1), \frac{\nabla J(\theta_1)}{2}+G_{t+1}(\theta_1, \omega^*(\theta_1)) \rangle-\langle \nabla J(\theta_1), \frac{\nabla J(\theta_2)}{2}+G_{t+1}(\theta_2, \omega^*(\theta_2)) \rangle{\nonumber}\\
&\quad+\langle \nabla J(\theta_1), \frac{\nabla J(\theta_2)}{2}+G_{t+1}(\theta_2, \omega^*(\theta_2)) \rangle-\langle \nabla J(\theta_2), \frac{\nabla J(\theta_2)}{2}+G_{t+1}(\theta_2, \omega^*(\theta_2)) \rangle |.\end{aligned}$$ Since $J(\theta)$ and $\|\nabla J(\theta)\|$ are Lipschitz in $\theta$ by Lemma \[Lemma:1\], thus $\zeta(\theta,O_t)$ is also Lipschitz in $\theta$. We then denote its Lipschitz constant by $c_{\zeta}$, i.e., $$\begin{aligned}
|\zeta(\theta_1,O_t)-\zeta(\theta_2,O_t)| \leq c_{\zeta} \|\theta_1-\theta_2 \|,\end{aligned}$$ where $$\begin{aligned}
c_{\zeta}&=2\gamma(1+k_1|{\mathcal{A}}|R)\frac{1}{\lambda}(r_{\max}+R+\gamma R)(\frac{K}{2}+k_3){\nonumber}\\
&\quad+K(r_{\max}+R+\gamma R)(\gamma \frac{1}{\lambda}(1+k_1|{\mathcal{A}}|R)+1+\gamma \frac{1}{\lambda}(1+Rk_1|{\mathcal{A}}|)).\end{aligned}$$
Thus from , it follows that for any $\tau \geq 0$, $$\begin{aligned}
\label{eq:55}
|\zeta(\theta_t,O_t)-\zeta(\theta_{t-\tau},O_t)| \leq c_{\zeta} \|\theta_t-\theta_{t-\tau} \|\leq c_{\zeta}(c_{f_1}+c_{g_1})\sum^{t-1}_{k=t-\tau}\alpha_k.\end{aligned}$$
We define an independent random variable $\hat O=(\hat S,\hat A,\hat R,\hat S')$, where $(\hat S,\hat A)\sim\mu$, $\hat S'$ is the subsequent state and $\hat R$ is the reward. Then $\mathbb{E}[\zeta(\theta_{t-\tau},\hat O)]=0$ by the fact that $\mathbb{E}_{\mu}[{G_{t+1}(\theta,\omega^*(\theta))}]=-\frac{1}{2}\nabla J(\theta)$. Thus, $$\begin{aligned}
\mathbb{E}[\zeta(\theta_{t-\tau},O_t)] \leq |\mathbb{E}[\zeta(\theta_{t-\tau},O_t)]-\mathbb{E}[\zeta(\theta_{t-\tau},O')]|\leq k_{\zeta}m\rho^{\tau},\end{aligned}$$ which follows from the Markov Mixing property in Assumption \[ass:1\], where $k_{\zeta}=4\gamma(1+k_1R|{\mathcal{A}}|)\frac{1}{\lambda}(r_{\max}+R+\gamma R)^2(2\gamma(1+k_1|{\mathcal{A}}|R)\frac{1}{\lambda}+1)$.
If $t \leq \tau_{\alpha_T}$, then we choose $\tau=t$ in . Then we have that $$\begin{aligned}
\mathbb{E}[\zeta(\theta_t,O_t)] \leq \mathbb{E}[\zeta(\theta_0,O_t)]+c_{\zeta}(c_{f_1}+c_{g_1})\sum^{t-1}_{k=0}\alpha_k\leq c_{\zeta}(c_{f_1}+c_{g_1})t\alpha_0\overset{(a)}{\leq} c_{\zeta}(c_{f_1}+c_{g_1})\alpha_0\tau_{\alpha_T},\end{aligned}$$ where $(a)$ is due to the fact that $\alpha_t$ is non-increasing. If $t > \tau_{\alpha_T}$, we choose $\tau=\tau_{\alpha_T}$, and then $$\begin{aligned}
&\mathbb{E}[\zeta(\theta_t, O_t)]\leq \mathbb{E}[\zeta(\theta_{t-\tau_{\alpha_T}},O_t)]+c_{\zeta}(c_{f_1}+c_{g_1})\sum^{t-1}_{k=t-\tau_{\alpha_T}}\alpha_k{\nonumber}\\
&\leq k_{\zeta}m\rho^{\tau_{\alpha_T}}+c_{\zeta}(c_{f_1}+c_{g_1})\tau_{\alpha_T}\alpha_{t-\tau_{\alpha_T}}\leq k_{\zeta}\alpha_T+c_{\zeta}(c_{f_1}+c_{g_1})\tau_{\alpha_T}\alpha_{t-\tau_{\alpha_T}}.\end{aligned}$$
We next bound the tracking error $\mathbb{E}[\|z_t\|]$. Define $\zeta_{f_2}(\theta,z,O_t)\triangleq\langle z, f_2(\theta,O_t) \rangle $, and $\zeta_{g_2}(z,O_t)\triangleq\langle z,g_2(z,O_t)-\Bar{g_2}(z)\rangle$, where $\Bar{g_2}(z)\triangleq\mathbb{E}[g_2(z,O_t)]=\mathbb{E}[-\phi_t^\top z \phi_t]$.
\[Lemma:5\] Consider any $\theta,\theta_1,\theta_2 \in\{\theta:\|\theta\|\leq R\}$ and any $z,z_1,z_2\in\{z:\|z\|\leq 2R\}$. Then 1) $|\zeta_{f_2}(\theta,z,O_t) | \leq 2Rc_{f_2}$; 2) $|\zeta_{f_2}(\theta_1,z_1,O_t)-\zeta_{f_2}(\theta_2,z_2,O_t) | \leq k_{f_2}\|\theta_1-\theta_2 \|+k'_{f_2}\|z_1-z_2\|$, where $k_{f_2}=2R(1+\gamma+\gamma Rk_1|{\mathcal{A}}|)(1+\frac{1}{\lambda})$ and $k'_{f_2}=c_{f_2}$; 3) $|\zeta_{g_2}(z,O_t) | \leq 8R^2 $; and 4) $|\zeta_{g_2}(z_1,O_t)-\zeta_{g_2}(z_2,O_t) | \leq 8R\|z_1-z_2\|$.
To prove 1), it can be shown that $|\zeta_{f_2}(\theta,z,O_t)|=|\langle z, f_2(\theta,O_t) \rangle|\leq 2Rc_{f_2}$.
For 2), it can be shown that $$\begin{aligned}
&|\zeta_{f_2}(\theta_1,z_1,O_t)-\zeta_{f_2}(\theta_2,z_2,O_t)|{\nonumber}\\
&=|\langle z_1, f_2(\theta_1,O_t) \rangle-\langle z_2, f_2(\theta_2,O_t) \rangle|{\nonumber}\\
&\leq |\langle z_1, f_2(\theta_1,O_t) \rangle-\langle z_1, f_2(\theta_2,O_t)|+|\langle z_1, f_2(\theta_2,O_t)-\langle z_2, f_2(\theta_2,O_t) \rangle|{\nonumber}\\
&\leq 2R \| f_2(\theta_1,O_t)-f_2(\theta_2,O_t)\|+\|f_2(\theta_2,O_t) \| \|z_1-z_2 \|{\nonumber}\\
&\leq 2R(|\delta_{t+1}(\theta_1)-\delta_{t+1}(\theta_2)|+\|\omega^*(\theta_1)-\omega^*(\theta_2) \|)+c_{f_2}\|z_1-z_2 \|{\nonumber}\\
&\overset{(a)}{\leq} k_{f_2}\|\theta_1-\theta_2\|+k'_{f_2}\|z_1-z_2\|,\end{aligned}$$ where $(a)$ is from both $\delta(\theta)$ and $\omega^*(\theta_t)(\theta)$ are Lipschitz, $k_{f_2}=2R(1+\gamma+\gamma Rk_1|{\mathcal{A}}|)(1+\frac{1}{\lambda})$, and $k'_{f_2}=c_{f_2}$.
For 3), we have that $\zeta_{g_2}(z,O_t)=\langle z, -\phi_t^\top z\phi_t+\mathbb{E}[\phi_t^\top z\phi_t]\rangle \leq 8R^2$.
To prove 4), we have that $$\begin{aligned}
&|\zeta_{g_2}(z_1,O_t)-\zeta_{g_2}(z_2,O_t)|{\nonumber}\\
&=|\langle z_1, -\phi_t^\top z_1\phi_t+\mathbb{E}[\phi_t^\top z_1\phi_t]\rangle-\langle z_1, -\phi_t^\top z_2\phi_t+\mathbb{E}[\phi_t^\top z_2\phi_t]\rangle+\langle z_1, -\phi_t^\top z_2\phi_t{\nonumber}\\
&\quad+\mathbb{E}[\phi_t^\top z_2\phi_t]\rangle-\langle z_2, -\phi_t^\top z_2\phi_t+\mathbb{E}[\phi_t^\top z_2\phi_t]\rangle|{\nonumber}\\
&\leq 8R\|z_1-z_2\|.\end{aligned}$$
In the following lemma, we derive bounds on ${\mathbb{E}}[\zeta_{f_2}(\theta_1,z_t,O_t)]$ and ${\mathbb{E}}[\zeta_{g_2}(z_t,O_t)]$.
\[lemma:6\] Define $\tau_{\beta_T}=\min \left\{ k: m\rho^k \leq \beta_T \right\}$. If $t\leq \tau_{\beta_T}$, then $$\begin{aligned}
\mathbb{E}[\zeta_{f_2}(\theta_t,z_t,O_t)]\leq 4Rc_{f_2}\beta_T +a_{f_2}\tau_{\beta_T},\end{aligned}$$ where $a_{f_2}=(k'_{f_2}(c_{f_2}+c_{g_2})\beta_0+ (k_{f_2}(c_{f_1}+c_{g_1})+k'_{f_2}\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1}))\alpha_0)$; and if $t> \tau_{\beta_T}$, then $$\begin{aligned}
\mathbb{E}[\zeta_{f_2}(\theta_t,z_t,O_t)]\leq 4Rc_{f_2}\beta_T+b_{f_2}\tau_{\beta_T}\beta_{t-\tau_{\beta_T}},\end{aligned}$$ where $b_{f_2}=( k'_{f_2}(c_{f_2}+c_{g_2})+ (k_{f_2}(c_{f_1}+c_{g_1})+k'_{f_2}\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1})))$.
We first note that $$\begin{aligned}
&\|z_{t+1}-z_t\|{\nonumber}\\
&=\|\beta_t(f_2(\theta_t,O_t)+g_2(z_t,O_t))+\omega^*(\theta_t)-\omega^*(\theta_{t+1}) \|{\nonumber}\\
&\leq (c_{f_2}+c_{g_2})\beta_t+\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1})\alpha_t,\end{aligned}$$ where the last step is due to . Furthermore, due to part 2) in Lemma \[Lemma:5\], $\zeta_{f_2}$ is Lipschitz in both $\theta$ and $z$, then we have that for any $\tau\geq 0$ $$\begin{aligned}
\label{eq:64}
&|\zeta_{f_2}(\theta_t,z_t,O_t)-\zeta_{f_2}(\theta_{t-\tau},z_{t-\tau},O_t)|{\nonumber}\\
&\overset{(a)}{\leq} k_{f_2}(c_{f_1}+c_{g_1})\sum^{t-1}_{i=t-\tau}\alpha_i+k'_{f_2}(c_{f_2}+c_{g_2})\sum^{t-1}_{i=t-\tau}\beta_i+\sum^{t-1}_{i=t-\tau}k'_{f_2}\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1})\alpha_i{\nonumber}\\
&=k'_{f_2}(c_{f_2}+c_{g_2})\sum^{t-1}_{i=t-\tau}\beta_i+(k_{f_2}(c_{f_1}+c_{g_1})+k'_{f_2}\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1}))\sum^{t-1}_{i=t-\tau}\alpha_i,\end{aligned}$$ where in $(a)$, we apply and Lemma \[Lemma:3\] to obtain the third term.
Define an independent random variable $\hat O=(\hat S,\hat A,\hat R,\hat S')$, where $(\hat S,\hat A)\sim \mu $, $\hat S'\sim\mathsf P(\cdot|\hat S,\hat A)$ is the subsequent state, and $\hat R$ is the reward. Then it can be shown that $$\begin{aligned}
&\mathbb{E}[\zeta_{f_2}(\theta_{t-\tau},z_{t-\tau},O_t)] {\nonumber}\\
&\overset{(a)}{\leq} |\mathbb{E}[\zeta_{f_2}(\theta_{t-\tau},z_{t-\tau},O_t)]-\mathbb{E}[\zeta_{f_2}(\theta_{t-\tau},z_{t-\tau},\hat O)]|{\nonumber}\\
&\leq 4Rc_{f_2}m\rho^{\tau},\end{aligned}$$ where (a) is due to the fact that $\mathbb{E}[\zeta_{f_2}(\theta_{t-\tau},z_{t-\tau},\hat O)]=0$, and the last inequality follows from Assumption \[ass:1\].
If $t\leq \tau_{\beta_T}$, we choose $\tau=t$ in . Then it can be shown that $$\begin{aligned}
&\mathbb{E}[\zeta_{f_2}(\theta_t,z_t,O_t)]{\nonumber}\\
&\leq \mathbb{E}[\zeta_{f_2}(\theta_0,z_0,O_t)]+k'_{f_2}(c_{f_2}+c_{g_2})\sum^{t-1}_{i=0}\beta_i+(k_{f_2}(c_{f_1}+c_{g_1}){\nonumber}\\
&\quad+k'_{f_2}\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1}))\sum^{t-1}_{i=0}\alpha_i{\nonumber}\\
&\leq 4Rc_{f_2}m\rho^t+ k'_{f_2}(c_{f_2}+c_{g_2})t\beta_0+(k_{f_2}(c_{f_1}+c_{g_1})+k'_{f_2}\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1}))t\alpha_0{\nonumber}\\
&\leq 4Rc_{f_2}\beta_T + (k'_{f_2}(c_{f_2}+c_{g_2})\beta_0+ (k_{f_2}(c_{f_1}+c_{g_1})+k'_{f_2}\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1}))\alpha_0)\tau_{\beta_T}.\end{aligned}$$ If $t> \tau_{\beta_T}$, we choose $\tau=\tau_{\beta_T}$ in . Then, it can be shown that $$\begin{aligned}
&\mathbb{E}[\zeta_{f_2}(\theta_t,z_t,O_t)]{\nonumber}\\
&\leq \mathbb{E}[\zeta_{f_2}(\theta_{t-\tau_{\beta_T}},z_{t-\tau_{\beta_T}},O_t)]{\nonumber}\\
&\quad+k'_{f_2}(c_{f_2}+c_{g_2})\sum^{t-1}_{i=t-\tau_{\beta_T}}\beta_i+(k_{f_2}(c_{f_1}+c_{g_1})+k'_{f_2}\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1}))\sum^{t-1}_{i=t-\tau_{\beta_T}}\alpha_i{\nonumber}\\
&\leq 4Rc_{f_2}m\rho^{\tau_{\beta_T}}+k'_{f_2}(c_{f_2}+c_{g_2})\tau_{\beta_T}\beta_{t-\tau_{\beta_T}}+(k_{f_2}(c_{f_1}+c_{g_1})+k'_{f_2}\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1}))\tau_{\beta_T}\alpha_{t-\tau_{\beta_T}}{\nonumber}\\
&\leq 4Rc_{f_2}\beta_T+( k'_{f_2}(c_{f_2}+c_{g_2})+ (k_{f_2}(c_{f_1}+c_{g_1})+k'_{f_2}\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1})))\tau_{\beta_T}\beta_{t-\tau_{\beta_T}},\end{aligned}$$ where in the last step we upper bound $\alpha_t$ using $\beta_t$. Note that this will not change the order of the bound.
Similarly, in the following lemma, we derive a bound on $\mathbb{E}[\zeta_{g_2}(z_t,O_t)]$.
\[lemma:8\] If $t\leq \tau_{\beta_T}$, then
\[\_[g\_2]{}(z\_t,O\_t)\] a\_[g\_2]{}\_[\_T]{};
and if $t> \tau_{\beta_T}$, then
\[\_[g\_2]{}(z\_t,O\_t)\] b\_[g\_2]{}\_T+b’\_[g\_2]{}\_[\_T]{}\_[t-\_[\_T]{}]{},
where $a_{g_2}=8R(c_{f_2}+c_{g_2})\beta_0+\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1})\alpha_0)$, $b_{g_2}=16R^2$, and $b_{g_2}'=8R(c_{f_2}+c_{g_2})\beta_0+\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)(c_{f_1}+c_{g_1})\alpha_0$.
The proof is similar to the one for Lemma \[lemma:6\].
We then bound the tracking error as follows: $$\begin{aligned}
\label{z_1}
&||z_{t+1}||^2\nonumber\\
&=||z_t+\beta_t(f_2(\theta_t,O_t)+g_2(z_t,O_t))+\omega^*(\theta_t)-\omega^*(\theta_{t+1})||^2\nonumber\\
&=||z_t||^2+2\beta_t \langle z_t, f_2(\theta_t,O_t)\rangle +2\beta_t\langle z_t,g_2(z_t,O_t)\rangle +2\langle z_t, \omega^*(\theta_t)-\omega^*(\theta_{t+1})\rangle{\nonumber}\\ &\quad+||\beta_tf_2(\theta_t,O_t)+\beta_tg_2(z_t,O_t)+\omega^*(\theta_t)-\omega^*(\theta_{t+1})||^2\nonumber\\
&\leq||z_t||^2+2\beta_t \langle z_t, f_2(\theta_t,O_t)\rangle+2\beta_t\langle z_t,g_2(z_t,O_t)\rangle +2\langle z_t, \omega^*(\theta_t)-\omega^*(\theta_{t+1})\rangle{\nonumber}\\ &\quad+3\beta_t^2||f_2(\theta_t,O_t)||^2+3\beta_t^2||g_2(z_t,O_t)||^2+3||\omega^*(\theta_t)-\omega^*(\theta_{t+1})||^2\nonumber\\
&\overset{(a)}{\leq}||z_t||^2+2\beta_t\langle z_t, f_2(\theta_t,O_t)\rangle +2\beta_t\langle z_t,\Bar{g_2}(z_t)\rangle +2\langle z_t, \omega^*(\theta_t)-\omega^*(\theta_{t+1})\rangle +2\beta_t\langle z_t,g_2(z_t,O_t)-\Bar{g_2}(z_t)\rangle {\nonumber}\\
&\quad+3\beta_t^2c_{f_2}^2+3\beta_t^2c_{g_2}^2+6\frac{1}{\lambda^2}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)^2\alpha_t^2 (c_{f_1}^2+c_{g_1}^2),\end{aligned}$$ where $(a)$ follows from Lemma \[Lemma:3\] and .
Note that $\langle z_t,\Bar{g_2}(z_t)\rangle=-z_t^\top C z_t$, and $C$ is a positive definite matrix. Recall the minimal eigenvalue of $C$ is denoted by $\lambda$, then can be further bounded as follows: $$\begin{aligned}
\label{eq:69}
||z_{t+1}||^2
&\leq (1-2\beta_t\lambda)\|z_t\|^2+2\beta_t\zeta_{f_2}+2\beta_t\zeta_{g_2}+2\langle z_t,\omega^*(\theta_t)-\omega^*(\theta_{t+1})\rangle+3\beta_t^2c_{f_2}^2\nonumber\\
&\quad+3\beta_t^2c_{g_2}^2+6\frac{1}{\lambda^2}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)^2\alpha_t^2 (c_{f_1}^2+c_{g_1}^2).\end{aligned}$$ Taking expectation on both sides of the , and applying it recursively, we obtain that $$\begin{aligned}
\mathbb{E}[||z_{t+1}||^2]
\leq &\prod_{i=0}^{t}(1-2\beta_i\lambda) ||z_0||^2 \nonumber\\
&+2\sum_{i=0}^{t} \prod_{k=i+1}^{t}(1-2\beta_k\lambda) \beta_i \mathbb{E}[\zeta_{f_2}(z_i,\theta_i,O_i)] \nonumber\\
&+2\sum_{i=0}^{t} \prod_{k=i+1}^{t}(1-2\beta_k\lambda) \beta_i \mathbb{E}[\zeta_{g_2}(z_i,O_i)]\nonumber\\
&+2\sum_{i=0}^{t} \prod_{k=i+1}^{t}(1-2\beta_k\lambda) \mathbb{E}\langle z_i, \omega^*(\theta_i)-\omega^*(\theta_{i+1})\rangle
+3(c_{f_2}^2+c_{g_2}^2)\sum_{i=0}^{t} \prod_{k=i+1}^{t}(1-2\beta_k\lambda) \beta_i^2\nonumber\\
&+6\frac{1}{\lambda^2}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)^2(c_{f_1}^2+c_{g_1}^2)\sum_{i=0}^{t} \prod_{k=i+1}^{t}(1-2\beta_k\lambda) \alpha_i^2.\end{aligned}$$
Also note that $1-2\beta_i\lambda \leq e^{-2\beta_i\lambda}$, which further implies that $$\begin{aligned}
\label{eq:tracking}
\mathbb{E}[||z_{t+1}||^2
&\leq A_t ||z_0||^2+2\sum_{i=0}^t B_{it} +2\sum_{i=0}^t C_{it}+2\sum_{i=0}^t D_{it}{\nonumber}\\
&\quad+3(c_{f_2}^2+c_{g_2}^2+2\frac{1}{\lambda^2}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)^2(c_{f_1}^2+c_{g_1}^2)) \sum_{i=0}^t E_{it},\end{aligned}$$ where $$\begin{aligned}
\label{ABCDE}
A_t&=e^{-2\lambda \sum_{i=0}^t \beta_i}, {\nonumber}\\
B_{it}&=e^{-2\lambda\sum_{k=i+1}^t \beta_k} \beta_i\mathbb{E}[\zeta_{f_2}(z_i,\theta_i,O_i)], {\nonumber}\\
C_{it}&=e^{-2\lambda\sum_{k=i+1}^t \beta_k} \beta_i\mathbb{E}[\zeta_{g_2}(z_i,O_i)],{\nonumber}\\
D_{it}&=e^{-2\lambda\sum_{k=i+1}^t \beta_k} \mathbb{E}[\langle z_t,\omega^*(\theta_i)-\omega^*(\theta_{i+1})\rangle],{\nonumber}\\
E_{it}&=e^{-2\lambda\sum_{k=i+1}^t \beta_k} \beta_i^2.\end{aligned}$$
Consider the second term in . Using Lemma \[lemma:6\], it can be further bounded as follows: $$\begin{aligned}
\label{eq:B}
\sum^t_{i=0} B_{it}
&=\sum^t_{i=0}e^{-2\lambda\sum_{k=i+1}^t \beta_k} \beta_i\mathbb{E}[\zeta_{f_2}(z_i,\theta_i,O_i)]\nonumber\\
&\leq \sum_{i=0}^{\tau_{\beta_T}}(a_{f_2}\tau_{\beta_T}+4Rc_{f_2}\beta_T)e^{-2\lambda\sum_{k=i+1}^t\beta_k}\beta_i+4Rc_{f_2}\beta_T\sum_{i={\tau_{\beta_T}+1}}^te^{-2\lambda\sum_{k=i+1}^t\beta_k}\beta_i{\nonumber}\\
&\quad+b_{f_2}\tau_{\beta_T}\sum_{i={\tau_{\beta_T}+1}}^te^{-2\lambda\sum_{k=i+1}^t\beta_k}\beta_{i-\tau_{\beta_T}}\beta_i.\end{aligned}$$ Further analysis of the bound will be made when we specify the stepsizes $\alpha_t,\beta_t$, which will be provided later.
Similarly, using Lemma \[lemma:8\], we can bound the third term in as follows: $$\begin{aligned}
\label{eq:C}
\sum^t_{i=0}C_{it}&=\sum^t_{i=0} e^{-2\lambda\sum_{k=i+1}^t \beta_k} \beta_i\mathbb{E}[\zeta_{g_2}(z_i,O_i)]{\nonumber}\\
&\leq\tau_{\beta_T}a_{g_2}\sum_{i=0}^{\tau_{\beta_T}}e^{-2\lambda\sum_{k=i+1}^t\beta_k}\beta_i+b_{g_2}\beta_T\sum_{i={\tau_{\beta_T}+1}}^te^{-2\lambda\sum_{k=i+1}^t\beta_k}\beta_i{\nonumber}\\
&\quad+b'_{g_2}\tau_{\beta_T}\sum_{i={\tau_{\beta_T}+1}}^te^{-2\lambda\sum_{k=i+1}^t\beta_k}\beta_{i-\tau_{\beta_T}}\beta_i.\end{aligned}$$
The last step in bounding the tracking error is to bound $\mathbb{E}[\langle z_i,\omega^*(\theta_i)-\omega^*(\theta_{i+1})\rangle]$, which is shown in the following lemma.
\[thm:D\] $$\begin{aligned}
&\sum_{i=0}^{t}e^{-2\lambda\sum_{k=i+1}^t \beta_k} \mathbb{E}[\langle z_i,\omega^*(\theta_i)-\omega^*(\theta_{i+1})\rangle]{\nonumber}\\
&\leq 2\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)R(c_{f_1}+c_{g_1}) \sum_{i=0}^{t}e^{-2\lambda\sum_{k=i+1}^t \beta_k} \alpha_i.\end{aligned}$$
From , we first have that
||\^\*(\_i)-\^\*(\_[i+1]{})|| (1++R||k\_1)||\_i-\_[i+1]{}||.
Then it follows that $$\begin{aligned}
\label{eq:81a}
&\sum_{i=0}^{t}e^{-2\lambda\sum_{k=i+1}^t \beta_k} \mathbb{E}[\langle z_i,\omega^*(\theta_i)-\omega^*(\theta_{i+1})\rangle]{\nonumber}\\
&\leq \sum_{i=0}^{t}e^{-2\lambda\sum_{k=i+1}^t \beta_k} \mathbb{E}[\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1) \|z_i\| \|\theta_i-\theta_{i+1}\|]{\nonumber}\\
&\leq 2\frac{1}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)R(c_{f_1}+c_{g_1}) \sum_{i=0}^{t}e^{-2\lambda\sum_{k=i+1}^t \beta_k} \alpha_i.\end{aligned}$$
Proof of Theorem \[thm:main\]
=============================
In this section, we will use the lemmas in Appendix \[app:lemmas\] to prove Theorem \[thm:main\].
In Appendix \[app:lemmas\], we have developed bounds on both the tracking error and $\mathbb{E}[\zeta(\theta_t,O_t)]$. We then plug them both into , $$\begin{aligned}
\label{eq:80}
&\frac{\sum^T_{t=0}\alpha_t\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{2\sum^T_{t=0} \alpha_t}{\nonumber}\\
&\leq \frac{1}{\sum^T_{t=0} \alpha_t} \bigg( J(\theta_0)-J^*+\gamma\alpha_t(1+|{\mathcal{A}}|Rk_1)\sqrt{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t) \|^2]}\sqrt{\sum^T_{t=0}\mathbb{E}[\|z_t\|^2]}{\nonumber}\\
&\quad+\sum^T_{t=0}\alpha_t \mathbb{E}[\zeta(\theta_t,O_t)]+\sum^T_{t=0}\alpha_t^2 (c_{f_1}+c_{g_1}) \bigg),\end{aligned}$$ where $J^*$ denotes $\min_\theta J(\theta)$, and is positive and finite.
By Lemma \[thm:zeta\], for large $T$, we have that
\[eq:81\] &\_[t=0]{}\^T \_t\[(\_t,O\_t)\]\
&\^[\_[\_T]{}]{}\_[t=0]{}c\_(c\_[f\_1]{}+c\_[g\_1]{})\_0\_t\_[\_T]{}+\^T\_[t=\_[\_T]{}+1]{} k\_\_T\_t+c\_(c\_[f\_1]{}+c\_[g\_1]{})\_[\_T]{}\_[t-\_[\_T]{}]{}\_t.
Here, $\tau_{\alpha_T}=\mathcal O(|\log \alpha_T|)$ by its definition. Therefore, for non-increasing sequence $\{\alpha_t\}_{t=0}^\infty$, can be further upper bounded as follows:
\[eq:79\] &\_[t=0]{}\^T \_t\[(\_t,O\_t)\]=O(|\_T|\^2\_0\^2 + \_[t=0]{}\^T( \_t\_T+||\_t\^2)).
We note that we can also specify the constants for , which, however, will be cumbersome. How those constants affect the finite-sample bound can be easily inferred from , and thus is not explicitly analyzed in the following steps. Also, at the beginning we bound $\sqrt{\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}}$ by some constant that does not scale with $T$: $\gamma \|C^{-1}\|(k_1+|{\mathcal{A}}|R+1)(r_{\max}+R+\gamma R)$.
Hence, we have that $$\begin{aligned}
\label{eq:main0}
&\frac{\sum^T_{t=0}\alpha_t\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{\sum^T_{t=0} \alpha_t}{\nonumber}\\
&=\mathcal O\Bigg( \frac{1}{\sum^T_{t=0} \alpha_t} \Bigg(J(\theta_0)-J^*+\sum^T_{t=0}\alpha_t^2 +\alpha_t\sqrt{T}\sqrt{\sum^T_{t=0}\mathbb{E}[\|z_t\|^2]}+\alpha_0^2|\log (\alpha_T)|^2 +\sum^{T}_{t=0}\alpha_t\alpha_T{\nonumber}\\
&\quad+\sum^T_{t=0}|\log(\alpha_T)|\alpha_t^2 \Bigg)\Bigg).
\end{aligned}$$
In the following, we focus on the case with constant stepsizes. For other possible choices of stepsizes, the convergence rate can also be derived using . Let $\alpha_t=\frac{1}{T^a}=\alpha$ and $\beta_t=\frac{1}{T^b}=\beta$. In this case, can be written as follows: $$\begin{aligned}
\label{eq:main1}
\frac{\sum^T_{t=0}\alpha\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{\sum^T_{t=0} \alpha}&=\mathcal O \left( \frac{1}{T}\left(\sqrt{T}\sqrt{\sum^T_{t=0}\mathbb{E}[\|z_t\|^2]}+\alpha\log(\alpha)^2+T\alpha+T\alpha|\log(\alpha)| \right)+\frac{J(\theta_0)-J^*}{T\alpha} \right){\nonumber}\\
&=\mathcal{O}\left(\sqrt{\frac{\sum^T_{t=0}\mathbb{E}[\|z_t\|^2]}{T}}\right)+\mathcal{O}\left(\frac{\log T^2}{T^{1+a}}+\frac{1}{T^a}+\frac{\log T}{T^a}+\frac{1}{T^{1-a}}\right).\end{aligned}$$
We then consider the tracking error $\mathbb{E}[\|z_{t} \|^2]$. Applying , , and , we obtain that for $t>\tau_{\beta_T}$, $$\begin{aligned}
\label{eq:order}
&\mathbb{E}[\|z_{t} \|^2]{\nonumber}\\
&\leq \|z_0\|^2 e^{-2\lambda t\beta}{\nonumber}\\
&\quad+2(4Rc_{f_2}\beta+(a_{f_2}+a_{g_2})\tau_{\beta_T})\beta\sum_{i=0}^{\tau_{\beta_T}}e^{-2\lambda(t-i)\beta}+(8Rc_{f_2}+2b_{g_2})\beta^2\sum_{i={\tau_{\beta_T}+1}}^t e^{-2\lambda(t-i)\beta}{\nonumber}\\
&\quad+(2b_{f_2}+2b'_{g_2})\tau_{\beta_T}\beta^2\sum_{i={\tau_{\beta_T}+1}}^t e^{-2\lambda(t-i)\beta}+\frac{4}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)R(c_{f_1}+c_{g_1})\alpha\sum_{i=0}^{t}e^{-2\lambda(t-i)\beta}{\nonumber}\\
&\quad+3(c_{f_2}^2+c_{g_2}^2+2\frac{1}{\lambda^2}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)^2(c_{f_1}^2+c_{g_1}^2))\sum^t_{i=0}e^{-2\lambda(t-i)\beta}\beta^2{\nonumber}\\
&=\mathcal{O}\Bigg( e^{-2\lambda t\beta}+\tau\beta\sum^{\tau}_{i=0}e^{-2\lambda(t-i)\beta}+\tau\beta^2\sum^t_{i=1+\tau}e^{-2\lambda(t-i)\beta}+(\alpha+\beta^2)\sum^t_{i=0}e^{-2\lambda(t-i)\beta}\Bigg){\nonumber}\\
&=\mathcal{O}\Bigg( e^{-2\lambda t\beta}+\tau\beta e^{-2\lambda t\beta}\frac{1-e^{2\lambda \beta(\tau+1)}}{1-e^{2\lambda\beta}}+\tau\beta^2(e^{-2\lambda t\beta}-e^{-2\lambda \beta\tau})\frac{e^{2\lambda\beta(\tau+1)}}{1-e^{2\lambda\beta}}+(\alpha+\beta^2)\frac{e^{-2\lambda t\beta}-e^{2\lambda\beta}}{1-e^{2\lambda\beta}}\Bigg).\end{aligned}$$ Similarly, for $t\leq \tau_{\beta_T}$, we obtain that $$\begin{aligned}
\label{eq:order2}
\mathbb{E}[\|z_{t} \|^2]
&\leq \|z_0\|^2 e^{-2\lambda t\beta}+2(4Rc_{f_2}\beta+(a_{f_2}+a_{g_2})\tau_{\beta_T})\beta\sum_{i=0}^{t}e^{-2\lambda(t-i)\beta}{\nonumber}\\
&\quad+\frac{4}{\lambda}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)R(c_{f_1}+c_{g_1})\alpha\sum_{i=0}^{t}e^{-2\lambda(t-i)\beta}{\nonumber}\\
&\quad+3(c_{f_2}^2+c_{g_2}^2+2\frac{1}{\lambda^2}(1+\gamma+\gamma R|{\mathcal{A}}|k_1)^2(c_{f_1}^2+c_{g_1}^2))\sum^t_{i=0}e^{-2\lambda(t-i)\beta}\beta^2{\nonumber}\\
&=\mathcal{O}\bigg( e^{-2\lambda\beta t}+\tau\beta\sum^t_{i=0}e^{-2\lambda(t-i)\beta}\bigg)=\mathcal{O}\Bigg(e^{-2\lambda\beta t}+\tau\beta \frac{e^{-2\lambda\beta t}-e^{2\lambda\beta}}{1-e^{2\lambda\beta}}\Bigg).\end{aligned}$$
We then bound $\sum^T_{t=0}\mathbb{E}[\|z_t\|^2]$. The sum is divided into two parts: $\sum^{\tau}_{t=0}\mathbb{E}[\|z_t\|^2]$ and $\sum^T_{t=\tau+1}\mathbb{E}[\|z_t\|^2]$, thus
$$\begin{aligned}
&\sum^{T}_{t=0}\mathbb{E}[\|z_t\|^2]{\nonumber}\\
&=\sum^{\tau}_{t=0}\mathbb{E}[\|z_t\|^2]+\sum^T_{t=\tau+1}\mathbb{E}[\|z_t\|^2]{\nonumber}\\
&=\sum^{\tau}_{t=0}(e^{-2\lambda\beta t}+\tau\beta \frac{e^{-2\lambda\beta t}-e^{2\lambda\beta}}{1-e^{2\lambda\beta}})+\sum^T_{t=\tau+1}\Bigg(e^{-2\lambda t\beta}+\tau\beta e^{-2\lambda t\beta}\frac{1-e^{2\lambda \beta(\tau+1)}}{1-e^{2\lambda\beta}}{\nonumber}\\
&\quad+\tau\beta^2(e^{-2\lambda t\beta}-e^{-2\lambda \beta\tau})\frac{e^{2\lambda\beta(\tau+1)}}{1-e^{2\lambda\beta}}+(\alpha+\beta^2)\frac{e^{-2\lambda t\beta}-e^{2\lambda\beta}}{1-e^{2\lambda\beta}}\Bigg){\nonumber}\\
&=\frac{1-e^{-2\lambda\beta(T+1)}}{1-e^{-2\lambda\beta}}+\tau\beta\left((\tau+1)\frac{-e^{2\lambda\beta}}{1-e^{2\lambda\beta}}+\frac{1-e^{-2\lambda\beta (\tau+1)}}{(1-e^{2\lambda\beta})(1-e^{-2\lambda\beta})}\right){\nonumber}\\
&\quad+\tau\beta\frac{1-e^{2\lambda\beta(\tau+1)}}{1-e^{2\lambda\beta}}e^{-2\lambda\beta(\tau+1)}\frac{1-e^{-2\lambda\beta(T-\tau)}}{1-e^{-2\lambda\beta}}+\tau\beta^2\frac{e^{2\lambda\beta(\tau+1)}}{1-e^{2\lambda\beta}}\bigg(e^{-2\lambda\beta(\tau+1)}\frac{1-e^{-2\lambda\beta(T-\tau)}}{1-e^{-2\lambda\beta}}{\nonumber}\\
&\quad-(T-\tau)e^{-2\lambda\beta\tau}\bigg) +(\alpha+\beta^2)\frac{1}{1-e^{2\lambda\beta}}\Bigg(e^{-2\lambda\beta(\tau+1)}\frac{1-e^{-2\lambda\beta(T-\tau)}}{1-e^{-2\lambda\beta}}-(T-\tau)e^{2\lambda\beta}\Bigg){\nonumber}\\
&=\mathcal{O}\Bigg( \frac{1}{\beta}+\tau^2+\tau+\tau\beta T+\frac{\alpha+\beta^2}{\beta}T\Bigg).\end{aligned}$$
Thus, we have that $$\begin{aligned}
\label{eq:trackingorder}
\frac{\sum^T_{t=0}\mathbb{E}[\|z_t\|^2]}{T}=\mathcal{O}\Bigg(\frac{1}{T^{1-b}}+\frac{(\log T)^2}{T}+\frac{\log T}{T^b}+\frac{1}{T^{a-b}}+\frac{1}{T^b}\Bigg)=\mathcal{O}\Bigg(\frac{\log T}{T^{\min \left\{a-b,b \right\}}}\Bigg).\end{aligned}$$
We then plug the tracking error in , and we have that $$\begin{aligned}
\frac{\sum^T_{t=0}\alpha\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{\sum^T_{t=0}\alpha}=\mathcal{O}\Bigg(\frac{1}{T^{1-a}}\Bigg)+\mathcal{O}\Bigg(\frac{\log T}{T^{\min \left\{a-b,b \right\}}}\Bigg).\end{aligned}$$
In the following we will recursively refine our bounds on the tracking error using the bound in .
Recall , and denote $D=J(\theta_0-J^*)$, then $$\begin{aligned}
\label{eq:recursion1}
\frac{\sum^T_{t=0} \mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}&= \frac{D}{T\alpha}+\mathcal{O}\Bigg(\frac{\sum^T_{t=0} \sqrt{\mathbb{E}[\|\nabla J(\theta_t)\|^2]\mathbb{E}[\|z_t\|^2]}}{T}\Bigg)
{\nonumber}\\
&=\mathcal{O}\Bigg(\frac{1}{T\alpha}+\sqrt{\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}}\sqrt{\frac{\sum^T_{t=0}\mathbb{E}[\|z_t\|^2]}{T}}\Bigg).\end{aligned}$$ In the first round, we upper bound $\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}$ by a constant. It then follows that $$\begin{aligned}
\label{eq:103}
\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}=\mathcal{O}\Bigg(\frac{1}{T^{1-a}}\Bigg)+\sqrt{\mathcal{O}\Bigg(\frac{\log T}{T^b}+\frac{1}{T^{a-b}}\Bigg)}=\mathcal{O}\Bigg(\frac{1}{T^{1-a}}\Bigg)+\mathcal{O}\Bigg(\frac{\sqrt{\log T}}{T^{\min\left\{ b/2, a/2-b/2\right\}}}\Bigg),\end{aligned}$$ where we denote $\min\left\{ b/2, a/2-b/2 \right\}$ by $c/2$. We then plug into , and we obtain that $$\begin{aligned}
\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}= \mathcal{O}\Bigg(\frac{1}{T^{1-a}}\Bigg)+\mathcal{O}\Bigg(\frac{\sqrt{\log T}}{T^{c/2}}\sqrt{\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}}\Bigg).\end{aligned}$$
**Case 1.** If $1-a<c/2$, then bound in is $\mathcal{O}\Bigg(\frac{1}{T^{1-a}} \Bigg)$: $
\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}=\mathcal{O}\Bigg(\frac{1}{T^{1-a}}\Bigg).
$ Then $$\begin{aligned}
\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T} = \mathcal{O}\Bigg(\frac{1}{T^{1-a}}+\frac{\sqrt{\log T}}{T^{c/2}}\frac{1}{T^{1/2-a/2}}\Bigg).\end{aligned}$$ Note that $c/2>1-a$, then $c/2+1/2-a/2>1-a$, thus the order would be $$\begin{aligned}
\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}=\mathcal{O}\Bigg(\frac{1}{T^{1-a}}\Bigg).\end{aligned}$$
Therefore, such a recursive refinement will not improve the convergence rate if $1-a < \frac{c}{2}$.
**Case 2.** If $c>1-a\geq c/2$, then $$\begin{aligned}
\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}=\mathcal{O}\Bigg(\frac{\sqrt{\log T}}{T^{c/2}}\Bigg).\end{aligned}$$ Also plug this order in , and we obtain that $$\begin{aligned}
&\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}=\mathcal{O}\Bigg(\frac{1}{T^{1-a}}\Bigg)+\mathcal{O}\Bigg(\frac{\sqrt{\log T}}{T^{c/2}}\frac{(\log T)^{1/4}}{T^{c/4}}\Bigg)=\mathcal{O}\Bigg(\frac{1}{T^{1-a}}+\frac{(\log T)^{\frac{3}{4}}}{T^{3c/4}}\Bigg).\end{aligned}$$ Here, we start the second iteration. If $1-a \geq \frac{3c}{4}$, we know that the order is improved as follows $$\begin{aligned}
\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}=\mathcal{O}\Bigg(\frac{(\log T)^{\frac{3}{4}}}{T^{3c/4}}\Bigg).\end{aligned}$$ And if $1-a<\frac{3c}{4}$, then order of will still be $\mathcal{O}\Bigg(\frac{1}{T^{1-a}}\Bigg)$. Thus we will stop the recursion, and we have that $$\begin{aligned}
\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}=\mathcal{O}\Bigg(\frac{1}{T^{1-a}}\Bigg).\end{aligned}$$ This implies that if the recursion stops after some step until there is no further rate improvement, then the convergence rate will be $\mathcal{O}\Bigg(\frac{1}{T^{1-a}} \Bigg)$. Note in this case, since $1-a<c$, then there exists some integral $n$, such that $1-a<\frac{2^n-1}{2^n}c$, and after round $n$, the recursion will stop. Thus the final rate is $\mathcal{O}\Bigg(\frac{1}{T^{1-a}} \Bigg)$.
**Case 3.** If $1-a\geq c$, then after a number of recursions, the order of the bound will be sufficiently close to $\mathcal O \left(\frac{\log T}{T^c}\right)$.
To conclude the three cases, when $1-a<c$, the recursion will stop after finite number of iterations, and the rate would be $\mathcal{O}\Bigg( \frac{1}{T^{1-a}}\Bigg)$; While when $1-a\geq c$, the recursion will always continue, and the fastest rate we can obtain is $\mathcal O \left(\frac{\log T}{T^c}\right)$. Thus the overall rate we can obtain can be written as
( +).
Proof of Corollary \[col:1\]
----------------------------
We next look for suitable $a$ and $b$, such that the rate obtained is the fastest. It can be seen that the best rate is achieved when $1-a=c$, and at the same time $0.5<a\leq 1$ and $0<b<a$. Thus, the best choices are $a=\frac{2}{3}$ and $b=\frac{1}{3}$, and the best rate we can obtain is $$\begin{aligned}
\mathbb{E}[\|\nabla J(\theta_M)\|^2]=\frac{\sum^T_{t=0}\mathbb{E}[\|\nabla J(\theta_t)\|^2]}{T}=\mathcal{O}\Bigg(\frac{\log T}{T^{1-a}}\Bigg)=\mathcal{O}\Bigg(\frac{\log T}{T^{\frac{1}{3}}}\Bigg).\end{aligned}$$
Softmax Is Lipschitz and Smooth
===============================
We first restate Lemma \[lemma:softmax\_smooth\] as follows, and then derive its proof.
The softmax policy is $2\sigma$-Lipschitz and $8\sigma^2$-smooth, i.e., for any $(s,a)\in{\mathcal{S}}\times{\mathcal{A}}$, and for any $\theta_1,\theta_2\in\mathbb R^N$, $|\pi_{\theta_1}(a|s)-\pi_{\theta_2}(a|s)| \leq 2\sigma \|\theta_1-\theta_2 \|$ and $\|\nabla\pi_{\theta_1}(a|s)-\nabla \pi_{\theta_2}(a|s) \|\leq 8\sigma^2 \|\theta_1-\theta_2 \|$.
By the definition of the softmax policy, for any $a\in{\mathcal{A}}$, $s\in {\mathcal{S}}$ and $\theta\in \mathbb R^N$, $$\begin{aligned}
\pi_{\theta}(a|s)=\frac{e^{\sigma {\theta}^\top \phi_{s,a}}}{\sum_{a' \in \mathcal{A}}e^{\sigma {\theta}^\top \phi_{s,a'}}},\end{aligned}$$ where $\sigma>0$ is a constant. Then, it can be shown that $$\begin{aligned}
\nabla \pi_{\theta}(a|s)
&=\frac{1}{\left(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}}\right)^2}
\left(\sigma e^{\sigma{\theta}^\top \phi_{s,a}}\phi_{s,a}\left(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}}\right)-\left(\sum_{a' \in \mathcal{A}}\sigma e^{\sigma{\theta}^\top \phi_{s,a'}}\phi_{s,a'}\right)e^{\sigma{\theta}^\top \phi_{s,a}}\right){\nonumber}\\
&=\frac{\sigma}{(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}})^2} \left(\sum_{a' \in \mathcal{A}} \phi_{s,a} e^{\sigma{\theta}^\top (\phi_{s,a}+\phi_{s,a'})}-\phi_{s,a'}e^{\sigma{\theta}^\top (\phi_{s,a}+\phi_{s,a'})}\right){\nonumber}\\
&=\frac{\sigma\sum_{a' \in \mathcal{A}}(\phi_{s,a}-\phi_{s,a'})e^{\sigma{\theta}^\top (\phi_{s,a}+\phi_{s,a'})}}{(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}})^2}.\end{aligned}$$ Thus, $$\begin{aligned}
||\nabla \pi_{\theta}(a|s)|| \leq 2\sigma \frac{\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top (\phi_{s,a}+\phi_{s,a'})}}{\left(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}}\right)^2}=2\sigma \frac{e^{\sigma{\theta}^\top \phi_{s,a}}}{\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}}}
\leq 2\sigma,\end{aligned}$$ where the last step is due to the fact that $\frac{e^{\sigma{\theta}^\top \phi_{s,a}}}{\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}}} \leq1$.
Note that for any $\theta_1$ and $\theta_2$, there exists some $\alpha \in (0,1)$ and $\bar \theta=\alpha\theta_1+(1-\alpha)\theta_2$, such that $$\begin{aligned}
\|\nabla \pi_{\theta_1}(a|s)-\nabla \pi_{\theta_2}(a|s) \| \leq \|\nabla^2 \pi_{\bar \theta}(a|s)\|\times\|\theta_1-\theta_2 \|.\end{aligned}$$ Here, $\nabla^2 \pi_{\theta}(a|s)$ denotes the Hessian matrix of $\pi_{\theta}(a|s)$ at $\theta$. Thus it suffices to find an universal bound of $\|\nabla^2 \pi_{\theta}(a|s) \|$ for any $\theta$ and $(a,s) \in \mathcal{A}\times\mathcal{S}$.
Note that $\nabla \pi_{\theta}(a|s)=\frac{\sigma\sum_{a' \in \mathcal{A}}(\phi_{s,a}-\phi_{s,a'})e^{\sigma{\theta}^\top (\phi_{s,a}+\phi_{s,a'})}}{\left(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}}\right)^2}$ is a sum of vectors $(\phi_{s,a}-\phi_{s,a'})$ with each entry multiplied by $\frac{\sigma e^{\sigma \theta^\top (\phi_{s,a}+\phi_{s,a'})}}{\left(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}}\right)^2}$. Then it follows that
\^2 \_(a|s) = \_[a’]{} (\_[s,a]{}-\_[s,a’]{}) ()\^.
Thus, to bound $\|\nabla^2 \pi_{\theta}(a|s) \|$, we compute the following: $$\begin{aligned}
\label{eq:37}
&\nabla \frac{e^{\sigma{\theta}^\top (\phi_{s,a}+\phi_{s,a'})}}{(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}})^2}{\nonumber}\\
&=\sigma \frac{e^{\sigma{\theta}^\top (\phi_{s,a}+\phi_{s,a'})}\left((\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}})(\phi_{s,a}+\phi_{s,a'})-2(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}}\phi_{s,a'})\right)}
{(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}})^3}.\end{aligned}$$ Then the norm of can be bounded as follows: $$\begin{aligned}
&\left\|\nabla \left(\frac{e^{\sigma{\theta}^\top (\phi_{s,a}+\phi_{s,a'})}}{(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}})^2}\right)\right\|{\nonumber}\\
&\leq \sigma\frac{2e^{\sigma{\theta}^\top (\phi_{s,a}+\phi_{s,a'})}\left( \sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}} +(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}})\right)}{(\sum_{a' \in \mathcal{A}}e^{{\theta}^\top \phi_{s,a'}})^3}{\nonumber}\\
&=4\sigma\frac{e^{\sigma{\theta}^\top (\phi_{s,a}+\phi_{s,a'})}}{\left(\sum_{a' \in \mathcal{A}}e^{\sigma{\theta}^\top \phi_{s,a'}}\right)^2}{\nonumber}\\
&\leq 4\sigma.\end{aligned}$$
Plug this in the expression of $\nabla^2 \pi_{\theta}(a|s)$, we obtain that $$\begin{aligned}
||\nabla^2\pi_{\theta}(a|s)|| \leq 8\sigma^2. $$
Thus the softmax policy is $2\sigma$-Lipschitz and $8\sigma^2$-smooth. This completes the proof.
|
---
abstract: |
Let ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$ be the hyperspace of nonempty bounded closed subsets of Euclidean space ${\ensuremath{\mathbb R^m}}$ endowed with the Hausdorff metric. It is well known that ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$ is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$ of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert space $\ell_2$. For each $0 {\leqslant}1 < m$, let $$\nu^m_k
= \{x = (x_i)_{i=1}^m \in {\ensuremath{\mathbb R^m}}: x_i \in {\ensuremath{\mathbb R}}\setminus{\mathbb{Q}}\text{ except for at most $k$ many $i$}\},$$ where $\nu^{2k+1}_k$ is the $k$-dimensional N[ö]{}beling space and $\nu^m_0 = ({\ensuremath{\mathbb R}}\setminus{\mathbb{Q}})^m$. It is also proved that the spaces ${\operatorname{Bd}}_H(\nu^1_0)$ and ${\operatorname{Bd}}_H(\nu^m_k)$, $0{\leqslant}k<m-1$, are homeomorphic to $\ell_2$. Moreover, we investigate the hyperspace ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ of all nonempty closed subsets of the real line ${\ensuremath{\mathbb R}}$ with the Hausdorff (infinite-valued) metric. It is shown that a nonseparable component ${\mathcal H}$ of ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ is homeomorphic to the Hilbert space $\ell_2(2^{\aleph_0})$ of weight $2^{\aleph_0}$ in case where ${\mathcal H}\not\ni {\ensuremath{\mathbb R}}, [0,\infty), (-\infty,0]$.
address:
- 'Instytut Matematyki, Akademia Świȩtokrzyska, 25-406 Kielce, Poland'
- 'Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan'
author:
- 'Wies[ł]{}aw Kubiś'
- Katsuro Sakai
title: 'Hausdorff hyperspaces of $\Rm$ and their dense subspaces'
---
Introduction {#introduction .unnumbered}
============
In this paper, we consider metric spaces and their hyperspaces endowed with the Hausdorff metric. Specifically, given a metric space $X = {\langle X, d \rangle}$, we shall denote by ${\operatorname{Cld}}(X)$ and ${\operatorname{Bd}}(X)$ the hyperspaces consisting of all nonempty closed sets and of all nonempty bounded closed sets in $X$ respectively and by $d_H$ the Hausdorff metric, which is infinite-valued on ${\operatorname{Cld}}(X)$ if $X$ is unbounded. We shall sometimes write ${\operatorname{Cld}}_H(X)$ or ${\operatorname{Bd}}_H(X)$ to emphasize the fact that we consider this space with the Hausdorff metric topology.
A theorem of Antosiewicz and Cellina [@AnCe] states that, given a convex set $X$ in a normed linear space, every continuous multivalued map ${{\varphi}\colon Y \to {\operatorname{Bd}}_H(X)}$ from a closed subset $Y$ of a metric space $Z$, can be extended to a continuous map ${{\overline}f\colon Z \to {\operatorname{Bd}}_H(X)}$. Using the language of topology, this theorem says that, under the above assumptions, ${\operatorname{Bd}}_H(X)$ is an absolute extensor or an absolute retract (in the class of metric spaces). In [@CK], it is proved that the above result is still valid when $X$ is replaced by a dense subset of a convex set in a normed linear space. More generally, ${\operatorname{Bd}}_H(X)$ is an absolute retract, whenever the metric on $X$ is [*almost convex*]{} (see §\[whfijfpiapfi\] for the definition). This condition was further weakened in [@KuSaY], which has turned out to be actually a necessary and sufficient one by Banakh and Voytsitskyy [@BaVo]. In the last paper, several equivalent conditions are given, which are too technical to mention them here. We refer to [@BaVo] for the details.
It is a natural question whether ${\operatorname{Bd}}_H(X)$ and some of its natural subspaces are homeomorphic to some standard spaces, like the Hilbert cube/space, etc. Since the Hausdorff metric topology coincides with the Vietoris topology on the hyperspace $\exp(X)$ of nonempty compact sets, the above question was already answered, applying known results, in case where bounded closed sets in $X$ are compact. Among the known results, let us mention the theorem of Curtis and Schori [@CuScho] (cf. [@vMill Chapter 8]), saying that $\exp(X)$ is homeomorphic to (${\approx}$) the Hilbert cube ${\ensuremath{\operatorname{Q}}}= [-1,1]^\omega$ if and only if $X$ is a Peano continuum, that is, it is compact, connected and locally connected. Later, Curtis [@Curtis] characterized non-compact metric spaces $X$ for which $\exp(X)$ is homeomorphic to the Hilbert cube minus a point ${\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}$ ($= {\ensuremath{\operatorname{Q}}}\setminus\{0\}$) or the pseudo-interior ${\ensuremath{\operatorname{s}}}= (-1,1)^\omega$ of ${\ensuremath{\operatorname{Q}}}$.[^1] In particular, ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}}) = \exp({\ensuremath{\mathbb R^m}}) {\approx}{\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}$. For more information concerning Vietoris hyperspaces, we refer to the book [@IlNa].
The aim of this work is to study topological types of some of the natural subspces of the Hausdorff hyperspace. We consider the following subspaces of ${\operatorname{Bd}}_H(X)$:
- ${\operatorname{Nwd}}(X)$ — all nowhere dense closed sets;
- ${\operatorname{Perf}}(X)$ — all perfect sets;
- ${\operatorname{Cantor}}(X)$ — all compact sets homeomorphic to the Cantor set.
In case $X = {\ensuremath{\mathbb R^m}}$ with the standard metric, we can also consider the following subspace:
- ${\mathfrak N}({\ensuremath{\mathbb R^m}})$ — all closed sets of the Lebesgue measure zero.
We show that, in case $X={\ensuremath{\mathbb R^m}}$, the above spaces are homeomorphic to the separable Hilbert space $\ell_2$. Actually, we prove that if ${\mathcal{F}}$ is one of the above spaces then the pair ${\langle {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}}), {\mathcal{F}}\rangle}$ is homeomorphic to ${\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{{\ensuremath{\operatorname{s}}}\setminus0}}\rangle}$.
The completion of a metric space $X = {\langle X, d \rangle}$ is denoted by ${\langle {\tilde}X, d \rangle}$. Then ${\operatorname{Bd}}_H(X,d)$ can be identified with the subspace of ${\operatorname{Bd}}_H({\tilde}X,d)$, via the isometric embedding $A\mapsto {\operatorname{cl}}_{{\tilde}X}A$. Thus we shall often write ${\operatorname{Bd}}(X,d){\subseteq}{\operatorname{Bd}}({\tilde}X,d)$, having in mind this identification. In this case, ${\operatorname{Bd}}({\tilde}X,d)$ is the completion of ${\operatorname{Bd}}(X,d)$. By such a reason, we also consider a dense subspace $D$ of a metric space $X = {\langle X, d \rangle}$. For each $0 {\leqslant}k < m$, let $$\nu^m_k
= \{x = (x_i)_{i=1}^m \in {\ensuremath{\mathbb R^m}}: x_i \in {\ensuremath{\mathbb R}}\setminus{\mathbb{Q}}\text{ except for at most $k$ many $i$}\},$$ which is the universal space for completely metrizable subspaces in ${\ensuremath{\mathbb R^m}}$ of $\dim {\leqslant}k$. In case $2k + 1 < m$, $\nu^m_k$ is homeomorphic to the $k$-dimensional N[ö]{}beling space $\nu^{2k+1}_k$, which is the universal space for all separable completely metrizable spaces. Note that $\nu^m_0 = ({\ensuremath{\mathbb R}}\setminus{\mathbb{Q}})^m{\approx}{\ensuremath{\mathbb R}}\setminus{\mathbb{Q}}$. We show that the pairs ${\langle {\operatorname{Bd}}({\ensuremath{\mathbb R}}), {\operatorname{Bd}}({\ensuremath{\mathbb R}}\setminus{\mathbb{Q}}) \rangle}$ and ${\langle {\operatorname{Bd}}({\ensuremath{\mathbb R^m}}), {\operatorname{Bd}}(\nu^m_k) \rangle}$, $0 {\leqslant}k < m-1$, are homeomorphic to ${\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{{\ensuremath{\operatorname{s}}}\setminus0}}\rangle}$, so we have ${\operatorname{Bd}}_H(\nu^m_k) {\approx}\ell_2$ if ${\langle m, k \rangle} = {\langle 1, 0 \rangle}$ or $0 {\leqslant}k < m-1$.
We also study the space ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$. It is very different from the hyperspace $\exp({\ensuremath{\mathbb R}})$. It is not hard to see that ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ has $2^{\aleph_0}$ many components, ${\operatorname{Bd}}({\ensuremath{\mathbb R}})$ is the only separable one and any other component has weight $2^{\aleph_0}$. We show that a nonseparable component ${\mathcal H}$ of ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ is homeomorphic to the Hilbert space $\ell_2(2^{\aleph_0})$ of weight $2^{\aleph_0}$ in case where ${\mathcal H}\not\ni {\ensuremath{\mathbb R}}, [0,\infty), (-\infty,0]$. This is a partial answer (in case $n = 1$) of Problem 4 in [@KuSaY].
Preliminaries
=============
We use standard notation concerning sets and topology. For example, we denote by $\omega$ the set of all natural numbers. Given a set $X$, we denote by $[X]^{<\omega}$ the family of all finite subsets of $X$.
Given a metric space $X = {\langle X, d \rangle}$ and a set $A{\subseteq}X$, we denote by ${\operatorname{B}}(A,r)$ and ${\overline{{\operatorname{B}}}}(A,r)$ the open and the closed $r$-balls centered at $A$, that is, $$\begin{gathered}
{\operatorname{B}}(A,r)={\{x\in X\colon {\operatorname{dist}}(x,A)<r\}} \quad\text{and}\\
{\overline{{\operatorname{B}}}}(A,r)={\{x\in X\colon {\operatorname{dist}}(x,A){\leqslant}r\}}.\end{gathered}$$ The Hausdorff metric $d_H$ on ${\operatorname{Cld}}(X)$ is defined as follows: $$d_H(A,C) = \inf {\{r>0\colon A{\subseteq}{\operatorname{B}}(C,r)\text{ and }C{\subseteq}{\operatorname{B}}(A,r)\}},$$ where $d_H$ is actually a metric on ${\operatorname{Bd}}(X)$ but $d_H$ is infinite-valued for ${\operatorname{Cld}}(X)$ if ${\langle X, d \rangle}$ is unbounded. The spaces ${\operatorname{Cld}}_H(X)$ and ${\operatorname{Bd}}_H(X)$ are sometimes denoted by ${\operatorname{Cld}}_H(X,d)$ and ${\operatorname{Bd}}_H(X,d)$, to emphasize the fact that they are determined by the metric on $X$. In fact, the metric ${\varrho}(x,y) = d(x,y)/(1 + d(x,y))$ induces the same topology on $X$ as $d$ but the Hausdorff metric ${\varrho}_H$ induces a different one on ${\operatorname{Cld}}(X)$. On the other hand, the Hausdorff metric induced by the metric $\bar d(x,y) = \min\{d(x,y), 1\}$ is finite-valued and induces the same topology on ${\operatorname{Cld}}_H(X)$ as $d_H$; moreover ${\operatorname{Cld}}(X)$ is equal to ${\operatorname{Bd}}(X)$ as sets. Note that the subspace ${\operatorname{Fin}}(X)={[X]^{<\omega}}\setminus{\{\emptyset\}}$ of ${\operatorname{Bd}}_H(X)$ of all nonempty finite subsets of $X$ is dense in ${\operatorname{Bd}}_H(X)$ if and only if every bounded set in $X = {\langle X, d \rangle}$ is totally bounded.
\[complete-separable\] For a metric space $X = {\langle X, d \rangle}$, the following hold:
If $d$ is complete then ${\langle {\operatorname{Bd}}(X,d), d_H \rangle}$ is a complete metric space and the space ${\operatorname{Cld}}_H(X)$ is completely metrizable.
The space ${\operatorname{Bd}}_H(X,d)$ is separable if and only if every bounded set in $X$ is totally bounded.
We use the standard notation $\exp(X)$ for the Vietoris hyperspace of nonempty compact sets in $X$. Note that $\exp(X){\subseteq}{\operatorname{Bd}}(X)$ for every metric space $X = {\langle X, d \rangle}$ and it is well known that the Hausdorff metric induces the Vietoris topology on $\exp(X)$. However, if closed bounded sets of $X$ are not compact, then the space ${\operatorname{Bd}}_H(X)$ is very different from ${\operatorname{Bd}}_V(X)$ endowed with the Vietoris topology. We use the following notation: $$A^-={\{C\in {\operatorname{Cld}}(X)\colon C\cap A{\ne\emptyset}\}} \quad\text{and}\quad A^+={\{C\in {\operatorname{Cld}}(X)\colon C{\subseteq}A\}},$$ where $A{\subseteq}X$. When dealing with ${\operatorname{Bd}}(X)$ (or other subspace of ${\operatorname{Cld}}(X)$), we still write $A^-$ and $A^+$ instead of $A^-\cap {\operatorname{Bd}}(X)$ and $A^+\cap {\operatorname{Bd}}(X)$ respectively.
In the rest of this section, we shall give preliminary results of infinite-dimensional topology. For the details, we refer to the book [@BRZ]. We abbreviate “absolute neighborhood retract” to “ANR”.
Let $X = {\langle X, d \rangle}$ be a metric space. It is said that a map ${f\colon Y \to X}$ can be [*approximated*]{} by maps in a class ${\mathcal{F}}$ of maps if for every map ${\alpha\colon X \to (0,1)}$ there exists a map ${g\colon Y \to X}$ which belongs to ${\mathcal{F}}$ and such that $d(f(y),g(y))<\alpha(f(y))$ for every $y\in Y$. A closed subset $A {\subseteq}X$ is a [*[$\operatorname{Z}$]{}-set*]{} in $X$ if the identity map ${\operatorname{id}}_X$ of $X$ can be approximated by maps ${f\colon X \to X}$ such that ${f[X]}\cap A=\emptyset$. Strengthening the last condition to ${\operatorname{cl}}_X({f[X]})\cap A=\emptyset$, we define the notion of a [*strong [$\operatorname{Z}$]{}-set*]{}. In case $X$ is locally compact, every [$\operatorname{Z}$]{}-set in $X$ is a strong [$\operatorname{Z}$]{}-set. Moreover, it is well known that every [$\operatorname{Z}$]{}-set in an $\ell_2$-manifold is a strong [$\operatorname{Z}$]{}-set. A countable union of (strong) [$\operatorname{Z}$]{}-sets is called a ([*strong*]{}) [*[${\ensuremath{\operatorname{Z}}}_\sigma$]{}-set*]{}. We call $X$ a ([*strong*]{}) [*[${\ensuremath{\operatorname{Z}}}_\sigma$]{}-space*]{} if it is a (strong) [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-set in itself. An embedding ${f\colon X \to Y}$ is called a [*[$\operatorname{Z}$]{}-embedding*]{} if ${f[X]}$ is a [$\operatorname{Z}$]{}-set in $Y$.
It is said that $D{\subseteq}X$ is [*homotopy dense*]{} in $X$ if there exists a homotopy ${h\colon X\times[0,1] \to X}$ such that $h_0 = {\operatorname{id}}$ and ${h_t[X]}{\subseteq}D$ for every $t > 0$, where $h_t(x)=h(x,t)$. The complement of a homotopy dense subset of $X$ is said to be [*homotopy negligible*]{}. If $A{\subseteq}X$ is a homotopy negligible closed set then $A$ is a [$\operatorname{Z}$]{}-set in $X$.
\[Z-set\] For a closed set $A$ in an ANR $X$, the following are equivalent:
$A$ is a [$\operatorname{Z}$]{}-set in $X$;
each map ${f\colon [0,1]^n \to X}$, $n\in{\omega}$, can be approximated by maps into $X\setminus A$;
$A$ is homotopy negligible in $X$.
\[sedgfasf\] Let $D$ be a homotopy dense subset of an ANR $X$. Then the following hold:
$D$ is also an ANR.
A closed set $A{\subseteq}X$ is a [$\operatorname{Z}$]{}-set in $X$ if and only if $A\cap D$ is a [$\operatorname{Z}$]{}-set in $D$.
If $A{\subseteq}X$ is a strong [$\operatorname{Z}$]{}-set in $X$ then $A\cap D$ is a strong [$\operatorname{Z}$]{}-set in $D$.
\[wetafqwtrqf\] Assume that $X$ is a homotopy dense subset of a [$\operatorname{Q}$]{}-manifold $M$. Then $X$ is an ANR and every [$\operatorname{Z}$]{}-set in $X$ is a strong [$\operatorname{Z}$]{}-set. Furthermore, $X$ is a strong [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-space if and only if $X$ is contained in a [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-set in $M$.
We verify only the “furthermore" statement. Assume $X{\subseteq}\bigcup_{{n\in\omega}}Z_n$, where each $Z_n$ is a [$\operatorname{Z}$]{}-set in $M$. Then each $Z_n$ is a strong [$\operatorname{Z}$]{}-set in $M$, because $M$ is locally compact, and therefore by Fact \[sedgfasf\] (iii), each $Z_n\cap X$ is a strong [$\operatorname{Z}$]{}-set in $X$. Conversely, if $X=\bigcup_{{n\in\omega}}X_n$, where each $X_n$ is a (strong) [$\operatorname{Z}$]{}-set in $X$, then by Fact \[sedgfasf\] (ii), $Z_n={\operatorname{cl}}_{M}X_n$ is a [$\operatorname{Z}$]{}-set in $M$. Clearly, $X{\subseteq}\bigcup_{{n\in\omega}}Z_n$.
Let ${\mathcal{C}}$ be a topological class of spaces, that is, if $X$ is homeomorphic to some $Y \in {\mathcal{C}}$ then $X$ also belongs to ${\mathcal{C}}$. It is said that ${\mathcal{C}}$ is [*open*]{} (resp. [*closed*]{}) [hereditary]{} if $X\in{\mathcal{C}}$ whenever $X$ is an open (resp. closed) subspace of some $Y\in{\mathcal{C}}$. A space $X$ is called [*strongly ${\mathcal{C}}$-universal*]{} if for every $Y\in{\mathcal{C}}$ and every closed subset $A {\subseteq}Y$, every map ${f\colon Y \to X}$ such that $f{\restriction}A$ is a [$\operatorname{Z}$]{}-embedding can be approximated by [$\operatorname{Z}$]{}-embeddings ${g\colon X \to Y}$ such that $g{\restriction}A=f{\restriction}A$. Similarly, one defines [*${\mathcal{C}}$-universality*]{}, relaxing the above condition to the case $A=\emptyset$, that is, $X$ is [*${\mathcal{C}}$-universal*]{} if every map ${f\colon Y \to X}$ of $Y\in{\mathcal{C}}$ can be approximated by [$\operatorname{Z}$]{}-embeddings.
\[univ-strong\] Let $X$ be an ANR such that every [$\operatorname{Z}$]{}-set in $X$ is strong and let ${\mathcal{C}}$ be an open and closed hereditary topological class of spaces. If every open subspace $U{\subseteq}X$ is ${\mathcal{C}}$-universal then $X$ is strongly ${\mathcal{C}}$-universal.
Given a topological class ${\mathcal{C}}$ of spaces, we denote by $\sigma{\mathcal{C}}$ the class of all spaces of the form $X=\bigcup_{{n\in\omega}}X_n$, where each $X_n$ is closed in $X$ and $X_n \in{\mathcal{C}}$. Recall that $X$ is a [*${\mathcal{C}}$-absorbing space*]{} if $X \in \sigma{\mathcal{C}}$ is a strongly ${\mathcal{C}}$-universal ANR which is a strong [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-space. In case ${\mathcal{C}}$ is closed hereditary, we can write $X=\bigcup_{{n\in\omega}}X_n$, where each $X_n$ is a strong [$\operatorname{Z}$]{}-set in $X$ and $X_n \in {\mathcal{C}}$.
We shall denote by ${\mathfrak{M}}_0$ and ${\mathfrak{M}}_1$ the classes of all compact metrizable spaces and all Polish spaces[^2] respectively. Let ${\ensuremath{\Sigma}}={\ensuremath{\operatorname{Q}}}\setminus{\ensuremath{\operatorname{s}}}$ denote the pseudo-boundary[^3] of ${\ensuremath{\operatorname{Q}}}$.
\[3e4gwdfs\] If $X$ is an ${\mathfrak{M}}_0$-absorbing homotopy dense subspace of [$\operatorname{Q}$]{}, then ${\langle {\ensuremath{\operatorname{Q}}}, X \rangle}{\approx}{\langle {\ensuremath{\operatorname{Q}}}, {\ensuremath{\Sigma}}\rangle}$. In case $X{\subseteq}{\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}$, ${\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, X \rangle}{\approx}{\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{\Sigma}}\rangle}$.
\[weosaijfpajf\] Assume that $X$ is a both homotopy dense and homotopy negligible subset of a Hilbert cube manifold $M$. If $X$ is $\sigma$-compact then it is a strong [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-space.
Assume $X=\bigcup_{{n\in\omega}}K_n$, where each $K_n$ is compact. Then each $K_n$ is closed in $M$ and therefore it is a strong [$\operatorname{Z}$]{}-set by Fact \[sedgfasf\] (iii).
Borel classes of several Hausdorff hyperspaces {#borelclasses}
==============================================
Let ${\langle {\tilde}X, d \rangle}$ denote the completion of ${\langle X, d \rangle}$. We identify ${\operatorname{Bd}}(X,d)$ with the subspace of ${\operatorname{Bd}}({\tilde}X,d)$, via the isometric embedding $A\mapsto {\operatorname{cl}}_{{\tilde}X}A$. Then, ${\langle {\operatorname{Bd}}({\tilde}X), d_H \rangle}$ is a completion of ${\langle {\operatorname{Bd}}(X), d_H \rangle}$. Moreover, it should be noticed that $A\in {\operatorname{Bd}}({\tilde}X)\setminus {\operatorname{Bd}}(X)$ if and only if $A\ne {\operatorname{cl}}_{{\tilde}X}(A\cap X)$. Saint Raymond proved in [@S Théorème 1] that if $X$ is the union of a Polish subset and a $\sigma$-compact subset then ${\operatorname{Bd}}_H(X)$ is $F_{\sigma\delta}$ (hence Borel) in ${\operatorname{Bd}}_H({\tilde}X)$.[^4] In particular, we have the following:
\[rthsdpio\] If $X = {\langle X, d \rangle}$ is $\sigma$-compact then the space ${\langle {\operatorname{Bd}}(X), d_H \rangle}$ is $F_{\sigma\delta}$ in its completion ${\langle {\operatorname{Bd}}({\tilde}X), d_H \rangle}$.
Moreover, the following can be easily obtained by adjusting the proof of [@S Théorème 1]:
\[owteepgsdgfa\] If $X = {\langle X, d \rangle}$ is Polish ($d$ is not necessarily complete) then the space ${\langle {\operatorname{Bd}}(X), d_H \rangle}$ is $G_\delta$ in its completion ${\langle {\operatorname{Bd}}({\tilde}X), d_H \rangle}$.
For the readers’ convenience, direct short proofs of the above two propositions are given in the Appendix. Combining Fact \[complete-separable\] and Proposition \[owteepgsdgfa\], we have the following:
\[ppojnkjiu\] If $X = {\langle X, d \rangle}$ is Polish in which every bounded set is totally bounded, then the space ${\operatorname{Bd}}_H(X)$ is also Polish.
Concerning the spaces ${\operatorname{Nwd}}(X)$ and ${\operatorname{Perf}}(X)$, we prove here the following:
\[wetgivwet\] For every separable metric space $X$, the space ${\operatorname{Nwd}}(X)$ is $G_\delta$ in ${\operatorname{Bd}}_H(X)$.
Let ${\ensuremath{{\{{U}_n\colon {n\in\omega}\}}}}$ be a countable open base for $X$. For each ${n\in\omega}$, let $${\mathcal{F}}_n={\{A\in{\operatorname{Bd}}(X)\colon U_n{\subseteq}A\}}.$$ Then each ${\mathcal{F}}_n$ is closed in ${\operatorname{Bd}}_H(X)$ and $\bigcup_{{n\in\omega}}{\mathcal{F}}_n={\operatorname{Bd}}(X)\setminus{\operatorname{Nwd}}(X)$.
\[oeihgwef\] If $X$ is locally compact then ${\operatorname{Perf}}(X)$ is $G_\delta$ in ${\operatorname{Bd}}_H(X)$.
Let ${\ensuremath{{\{{U}_n\colon {n\in\omega}\}}}}$ enumerate an open base of $X$ such that ${\operatorname{cl}}U_n$ is compact for every ${n\in\omega}$. Note that, by compactness, $({\operatorname{cl}}U_n)^-$ is closed in ${\operatorname{Bd}}_H(X,d)$. For each $n,m\in{\omega}$ define $$\Phi(n,m)={\{{\langle k, l \rangle}\in{\omega}^2\colon U_k\cap U_l=\emptyset,\ U_k\cup U_l{\subseteq}{\operatorname{B}}(U_n,1/m)\}}.$$ We claim that $${\operatorname{Bd}}(X,d)\setminus{\operatorname{Perf}}(X)=\bigcup_{n,m\in{\omega}}\bigcap_{{\langle k, l \rangle}\in\Phi(n,m)} \Bigl( ({\operatorname{cl}}U_n)^-\setminus(U_k^-\cap U_l^-)\Bigr).$$ The set on the right-hand side is $F_{\sigma}$, so this will finish the proof.
Note that a closed set in a Polish space is perfect if and only if it has no isolated points. If $A\in {\operatorname{Bd}}(X,d)\setminus{\operatorname{Perf}}(X)$ then there is $y\in A$ which is isolated in $A$. We can find $n,m\in{\omega}$ such that $y\in U_n$ and ${\operatorname{B}}(U_n,1/m)\cap A={\{y\}}$. Then $A\in ({\operatorname{cl}}U_n)^-$ and $A\notin U_k^-\cap U_l^-$ whenever ${\langle k, l \rangle}\in\Phi(n,m)$.
Conversely, assume that there are $n,m\in{\omega}$ such that $A\in({\operatorname{cl}}U_n)^-$ and $A\notin U_k^-\cap U_l^-$ for every ${\langle k, l \rangle}\in\Phi(n,m)$. Then $A\cap {\operatorname{B}}(U_n,1/m){\ne\emptyset}$ and the second condition says that $A\cap {\operatorname{B}}(U_n,1/m)$ does not contain two points, so it is a singleton. Thus $A\notin{\operatorname{Perf}}(X)$.
Replacing $({\operatorname{cl}}U_n)^-$ by ${\operatorname{B}}(U_n,1/m)^-$ in the formula from the proof above, we obtain the following:
The space ${\operatorname{Perf}}(X)$ is $F_{\sigma\delta}$ in ${\operatorname{Bd}}_H(X)$ if $X$ is Polish.
Since ${\operatorname{Cantor}}({\ensuremath{\mathbb R}}^m) = {\operatorname{Perf}}({\ensuremath{\mathbb R}}^m) \cap {\operatorname{Nwd}}({\ensuremath{\mathbb R}}^m)$, the following is a combination of Propositions \[wetgivwet\] and \[oeihgwef\]:
The space ${\operatorname{Cantor}}({\ensuremath{\mathbb R}}^m)$ is $G_\delta$ in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R}}^m)$.
Now, we shall prove the following:
The space ${\mathfrak N}({\ensuremath{\mathbb R}}^m)$ is Polish.
Let ${\ensuremath{{\{{I}_n\colon {n\in\omega}\}}}}$ enumerate all open rational cubes (i.e. products of rational intervals) in ${\ensuremath{\mathbb R^m}}$. Given $k\in{\omega}$, we define $$S_k= \Bigl\{s\in{[{\omega}]^{<\omega}} : \sum_{n\in s}|I_n|<1/k\Bigr\},$$ where $|I|$ denotes the volume of the cube $I{\subseteq}{\ensuremath{\mathbb R^m}}$. We claim that $${\mathfrak N}({\ensuremath{\mathbb R}}^m)=\bigcap_{k\in{\omega}}\bigcup_{s\in S_k}\Bigl(\bigcup_{n\in s}I_n\Bigr)^+.$$ Clearly, if $A$ belongs to the right-hand side then for each $k\in{\omega}$ there is $s{\subseteq}{\omega}$ such that $A{\subseteq}\bigcup_{n\in s}I_n$ and $\sum_{n\in s}|I_n|<1/k$; therefore $A$ has Lebesgue measure zero.
Assume now $A$ has Lebesgue measure zero and fix $k<{\omega}$. Then $A{\subseteq}\bigcup_{{n\in\omega}}J_n$, where each $J_n$ is an open rational cube and $\sum_{{n\in\omega}}|J_n|<1/k$. By compactness, $A{\subseteq}J_0\cup\dots\cup J_{l-1}$ for some $m$ and $\{J_0,\dots,J_{l-1}\}={\{I_n\colon n\in s\}}$ for some $s\in S_k$. Thus $A\in\bigcup_{s\in S_k}(\bigcup_{n\in s}I_n)^+$.
Almost convex metric spaces {#whfijfpiapfi}
===========================
Recall that a metric $d$ on $X$ is [*almost convex*]{} if for every $\alpha>0$, $\beta>0$ and for every $x,y\in X$ such that $d(x,y)<\alpha+\beta$, there exists $z\in X$ with $d(x,z)<\alpha$ and $d(z,y)<\beta$.
Fix a dense set $X$ in a separable Banach space $E$. Let $d$ denote the metric on $X$ induced by the norm of $E$. Then ${\langle X, d \rangle}$ is an almost convex metric space and therefore by a result of [@CK] the space ${\operatorname{Bd}}(X,d)$ is an absolute retract. In case where $X$ is $G_\delta$, the space ${\operatorname{Bd}}(X,d)$ is completely metrizable by Proposition \[owteepgsdgfa\]. If additionally $E$ is finite-dimensional then ${\operatorname{Bd}}(X,d)$ is Polish by Corollary \[ppojnkjiu\]. In case where $X$ is $\sigma$-compact, by Proposition \[rthsdpio\], ${\operatorname{Bd}}(X,d)$ is absolutely $F_{\sigma\delta}$. It is natural to ask whether these spaces or their subspaces, discussed in §\[borelclasses\], are homeomorphic to some standard spaces. Such standard spaces appear as homotopy dense subspaces of the Hilbert cube [$\operatorname{Q}$]{}.
Let ${\operatorname{UNb}}(X,d)$ denote the family of all sets of the form ${\overline{{\operatorname{B}}}}(C,t)$, the closed $t$-neighborhood of $C\in{\operatorname{Bd}}(X,d)$, where $t>0$.
\[sdgeriphgpwo\] If ${\langle X, d \rangle}$ is an almost convex metric space then the subspace ${\operatorname{UNb}}(X,d)$ is homotopy dense in ${\operatorname{Bd}}(X,d)$.
Define a homotopy ${h\colon {\operatorname{Bd}}(X,d)\times[0,1] \to {\operatorname{Bd}}(X,d)}$ by the formula: $$h(A,t)={\overline{{\operatorname{B}}}}(A,t).$$ It suffices to verify the continuity of $h$ with respect to Hausdorff metric topology. It has been checked in [@CK] that $d_H({\overline{{\operatorname{B}}}}(A,t),{\overline{{\operatorname{B}}}}(A,s)){\leqslant}|t-s|$. Thus we have $$\begin{aligned}
d_H(h(A,t),h(B,s))&{\leqslant}d_H(h(A,t),h(A,s))+d_H(h(A,s),h(B,s))\\
&{\leqslant}|t-s|+d_H(h(A,s),h(B,s)).\end{aligned}$$ It remains to check that $d_H({\overline{{\operatorname{B}}}}(A,s),{\overline{{\operatorname{B}}}}(B,s)){\leqslant}d_H(A,B)$.
To complete the proof, we show the following: $$r>d_H(A,B),\ {\varepsilon}>0 \Longrightarrow
r+{\varepsilon}{\geqslant}d_H({\overline{{\operatorname{B}}}}(A,s),{\overline{{\operatorname{B}}}}(B,s)),$$ For this aim, it suffices to check that ${\overline{{\operatorname{B}}}}(A,s){\subseteq}{\operatorname{B}}({\overline{{\operatorname{B}}}}(B,s),r+{\varepsilon})$; then by symmetry we shall also get ${\overline{{\operatorname{B}}}}(B,s){\subseteq}{\operatorname{B}}({\overline{{\operatorname{B}}}}(A,s),r+{\varepsilon})$.
For each $x\in {\overline{{\operatorname{B}}}}(A,s)$, choose $a\in A$ such that $d(x,a)<s+{\varepsilon}$. There is $b\in B$ such that $d(a,b)<r$. Then we have $d(x,b)<s+r+{\varepsilon}$. Using the almost convexity of $d$, we can find $y$ such that $d(b,y)<s$ and $d(y,x)<r+{\varepsilon}$. Then $y\in{\operatorname{B}}(B,s)$ and hence $x\in{\operatorname{B}}(y,r+{\varepsilon}){\subseteq}{\operatorname{B}}({\overline{{\operatorname{B}}}}(B,s),r+{\varepsilon})$.
Denote by ${\operatorname{Reg}}(X,d)$ the hyperspace of all nonempty bounded regularly closed subsets of a metric space ${\langle X, d \rangle}$. Clearly, ${\operatorname{UNb}}(X,d){\subseteq}{\operatorname{Reg}}(X,d)$.
\[owehfafafs\] Let ${\langle X, d \rangle}$ be an almost convex metric space and $D{\subseteq}X$ a dense set. Then the spaces ${\operatorname{Reg}}(X,d)$ and ${\operatorname{Bd}}(D,d)$ are homotopy dense in ${\operatorname{Bd}}(X,d)$.
Regarding ${\operatorname{Bd}}(D,d){\subseteq}{\operatorname{Bd}}(X,d)$ via the embedding $A\mapsto {\operatorname{cl}}_X A$, we have ${\operatorname{Reg}}(X,d){\subseteq}{\operatorname{Bd}}(D,d)$. This follows from the fact that ${\operatorname{cl}}(D\cap U)={\operatorname{cl}}U$ for every open set $U{\subseteq}X$. Since ${\operatorname{UNb}}(X,d)$ is homotopy dense in ${\operatorname{Bd}}(X,d)$ by Proposition \[sdgeriphgpwo\] and ${\operatorname{UNb}}(X,d){\subseteq}{\operatorname{Reg}}(X,d)$, we have the result.
Strict deformations
===================
Assume we are looking at certain homotopy dense subspaces of the Hilbert cube [$\operatorname{Q}$]{}. Let $X {\supseteq}X_0$ be such spaces. If $X_0 {\approx}{\ensuremath{\Sigma}}$ then, in order to conclude that ${\langle {\ensuremath{\operatorname{Q}}}, X \rangle}{\approx}{\langle {\ensuremath{\operatorname{Q}}}, {\ensuremath{\Sigma}}\rangle}$, it suffices to check that $X$ is a [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-set in [$\operatorname{Q}$]{}, by applying [@Chapman Theorem 6.6]. However, to see that $X_0 {\approx}{\ensuremath{\Sigma}}$, we have to check that $X_0$ is strongly ${\mathfrak{M}}_0$-universal. Below is a tool which simplifies this step. To formulate it, we need some extra notions concerning homotopies.
A homotopy ${{\varphi}\colon X\times [0,1] \to X}$ is called a [*strict deformation*]{} if ${\varphi}_0={\operatorname{id}}$ and $${\varphi}(x,t)={\varphi}(x',t')\land t>0\land t'>0\implies x=x'.$$ It is said that ${\varphi}$ [*omits*]{} $A{\subseteq}X$ if ${{\varphi}[X\times(0,1]]}\cap A=\emptyset$. Finally, we say that a space $X$ is [*strictly homotopy dense*]{} in $Y$ if $X{\subseteq}Y$ and there exists a strict deformation which omits $Y\setminus X$ (so in particular $X$ is homotopy dense in $Y$).
\[mbysld\] For every ${\ensuremath{\operatorname{Z}}}$-set $A$ in a ${\ensuremath{\operatorname{Q}}}$-manifold $M$, there exists a strict deformation of $M$ which omits $A$.
Find a [$\operatorname{Z}$]{}-embedding ${f_0\colon M \to M}$ which is properly $2^{-2}$-homotopic to the identity and so that ${f_0[M]}\cap A=\emptyset$. Further, find a [$\operatorname{Z}$]{}-embedding ${f_1\colon M \to M}$ which is properly $2^{-3}$-homotopic to the identity and ${f_1[M]}\cap({f_0[M]}\cup A)=\emptyset$. Continuing this way, we find [$\operatorname{Z}$]{}-embeddings ${f_n\colon M \to M}$, ${n\in\omega}$, such that $f_n$ is properly $2^{-n-2}$-homotopic to the identity and $${f_n[M]}\cap ({f_{n-1}[M]}\cup \dots\cup {f_0[M]}\cup A)=\emptyset.$$ Then, we have proper $2^{-(n+1)}$-homotopies ${g^n\colon M\times[0,1] \to M}$, ${n\in\omega}$, such that $g^n_0 = f_n$ and $g^n_1 = f_{n+1}$. We can define a homotopy ${g\colon M\times[0,1] \to M}$ by $g(x,0) = x$ and $$g(x,t) = g^n(x,2 - 2^{n+1}t)
\;\text{ for $2^{-(n+1)} \leqslant t \leqslant 2^{-n}$, ${n\in\omega}$.}$$ Note that $g_{2^{-n}} = f_n$ for each ${n\in\omega}$, each $g{\restriction}M\times[2^{-n-1},2^{-n}]$ is proper and $2^{-n-1}$-close to the projection ${{\operatorname{pr}}_M\colon M\times(0,2^{-n}] \to M}$. The continuity of $g$ at $(x,0)$ is guaranteed by the last fact. Using the strong ${\mathfrak{M}}_0$-universality of $M$ (see [@BRZ Theorem 1.1.26]), we can inductively obtain ${h_n\colon M\times[0,1] \to M}$, ${n\in\omega}$, such that
1. $h_n{\restriction}M\times[2^{-n-1},1]$ is a $Z$-embedding,
2. $h_n{\restriction}M\times[2^{-n},1] = h_{n-1}{\restriction}M\times[2^{-n},1]$,
3. $h_n{\restriction}M\times[0,2^{-n-1}] = g{\restriction}M\times[0,2^{-n-1}]$,
4. $h_n{\restriction}M\times[2^{-n-1},2^{-n}]$ is $2^{-n-1}$-close to $g{\restriction}M\times[2^{-n-1},2^{-n}]$, hence it is $2^{-n}$-close to ${{\operatorname{pr}}_M\colon M\times[2^{-n-1},2^{-n}] \to M}$,
5. ${h_n[M\times[2^{-n-1},1]]}$ is disjoint from $A$.
Finally, the limit $h = \lim_{n\to\infty} h_n$ is the desired one.
\[ejgrpio\] Assume that $X$ is a [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-subset of a ${\ensuremath{\operatorname{Q}}}$-manifold $M$ which is strictly homotopy dense in $M$. Then $X$ is an ${\mathfrak{M}}_0$-absorbing space. In particular, if $M{\approx}{\ensuremath{\operatorname{Q}}}$ then ${\langle M, X \rangle}{\approx}{\langle {\ensuremath{\operatorname{Q}}}, {\ensuremath{\Sigma}}\rangle}$ and if $M{\approx}{\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}$ then ${\langle M, X \rangle}{\approx}{\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{\Sigma}}\rangle}$.
The assumption says in particular that $X$ is homotopy dense in $M$, so it follows from Proposition \[wetafqwtrqf\] that $X$ is an ANR being a strong [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-space. It remains to check that $X$ is strongly ${\mathfrak{M}}_0$-universal. For the additional statement, we can just apply Fact \[3e4gwdfs\].
Fix a map ${f\colon A \to X}$ of a compact metric space such that $f{\restriction}B$ is a [$\operatorname{Z}$]{}-embedding, where $B{\subseteq}A$ is closed. Note that every compact subset of $X$ is a [$\operatorname{Z}$]{}-set in $M$, hence it is a [$\operatorname{Z}$]{}-set in $X$ by Fact \[sedgfasf\] (ii), so we just have to preserve $f{\restriction}B$, not worrying about [$\operatorname{Z}$]{}-sets. We assume that $A$ is endowed with the metric such that ${\operatorname{diam}}(A){\leqslant}1$. Fix ${\varepsilon}>0$. Using the strong ${\mathfrak{M}}_0$-universality of $M$ (see [@BRZ Theorem 1.1.26]), we can find a [$\operatorname{Z}$]{}-embedding ${g\colon A \to M}$ which is ${\varepsilon}/2$-close to $f$ and such that ${g[A\setminus B]}\cap X=\emptyset$ (here we use the fact that $X$ is a ${\ensuremath{\operatorname{Z}}}_{\sigma}$-set in $M$ and also that ${f[B]}$ is a [$\operatorname{Z}$]{}-set in $M$).
By Lemma \[mbysld\], we have a strict deformation ${{\varphi}\colon M\times[0,1] \to M}$ which omits ${f[B]}$. Fix a metric $d$ for $M$ and choose a map ${\gamma\colon A \to [0,1]}$ so that $\gamma^{-1}(0)=B$ and $$d(g(a),{\varphi}(g(a),\gamma(a)))<{\varepsilon}/4 \;\text{ for every $a\in A$.}$$ On the other hand, by the assumption, there is a strict deformation ${\psi\colon M\times[0,1] \to M}$ which omits $M\setminus X$. Define ${h\colon A \to X}$ by setting $$h(a)=\psi({\varphi}(g(a),\gamma(a)),\delta(a)),$$ where ${\delta\colon A \to [0,1]}$ is a map chosen so that $B=\delta^{-1}(0)$ and $$d(h(a),{\varphi}(g(a),\gamma(a)))<\min\{{\varepsilon}/4,\ {\operatorname{dist}}({\varphi}(g(a),\gamma(a)),{f[B]})\}.$$ This ensures us that $h$ is ${\varepsilon}/2$-close to $g$ and that $h(a)\notin {f[B]}$ whenever $a\in A\setminus B$. Then $h$ is a map which is ${\varepsilon}$-close to $f$ and ${h[A]}{\subseteq}X$. Furthermore, $h{\restriction}B=g{\restriction}B=f{\restriction}B$. It remains to check that $h$ is one-to-one (then it is a [$\operatorname{Z}$]{}-embedding, since every compact set in $X$ is a [$\operatorname{Z}$]{}-set).
Suppose $h(a)=h(a')$. If $a,a'\in B$ then $g(a)=g(a')$ and consequently $a=a'$. When $a,a'\in A\setminus B$, since $\psi$ and ${\varphi}$ are strict deformations, $g(a)=g(a')$ and hence $a=a'$. In case $a\in B$ and $a'\notin B$, we have $h(a)=g(a)=f(a)\in{f[B]}$ but $h(a')\notin {f[B]}$ because ${\varphi}$ omits ${f[B]}$. Thus, this case does not occur.
Pseudo-interiors of ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$
=======================================================================
Throughout this section, $m>0$ is a fixed natural number. A particular case of a well known theorem of Curtis [@Curtis] says that ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})=\exp({\ensuremath{\mathbb R^m}})$ is homeomorphic to ${\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}$. We shall consider the standard (convex) Euclidean metric $d$ on ${\ensuremath{\mathbb R^m}}$. In this section, we investigate various $G_\delta$ subspaces of ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$. The main result of this section is the following:
\[pint-hyperspace\] Let ${\mathcal{F}}{\subseteq}{\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$ be one of the subspaces below: $${\operatorname{Nwd}}({\ensuremath{\mathbb R^m}}),\ {\operatorname{Perf}}({\ensuremath{\mathbb R^m}}),\ {\operatorname{Cantor}}({\ensuremath{\mathbb R^m}}),\ {\mathfrak N}({\ensuremath{\mathbb R^m}}),\ {\operatorname{Bd}}(D),$$ where $D$ is a dense $G_\delta$ set in ${\ensuremath{\mathbb R^m}}$ such that ${\ensuremath{\mathbb R^m}}\setminus D$ is also dense in ${\ensuremath{\mathbb R^m}}$ and in case $m > 1$ it is assumed that $D = {p[D]} \times {\ensuremath{\mathbb R}}$, where $p : {\ensuremath{\mathbb R^m}}\to {\ensuremath{\mathbb R}}^{m-1}$ is the projection onto the first $m-1$ coordinates. Then the pair ${\langle {\operatorname{Bd}}({\ensuremath{\mathbb R^m}}), {\mathcal{F}}\rangle}$ is homeomorphic to ${\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{{\ensuremath{\operatorname{s}}}\setminus0}}\rangle}$.
Applying Theorem \[pint-hyperspace\] above, we have
Suppose ${\langle m, k \rangle} = {\langle 1, 0 \rangle}$ or $0 {\leqslant}k < m - 1$. Then, $${\langle {\operatorname{Bd}}({\ensuremath{\mathbb R^m}}), {\operatorname{Bd}}(\nu^m_k) \rangle} {\approx}{\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{{\ensuremath{\operatorname{s}}}\setminus0}}\rangle}.$$ Consequently, ${\operatorname{Bd}}_H(\nu^m_k) {\approx}\ell_2$.
As a direct consequence of Theorem \[pint-hyperspace\], we have $${\langle {\operatorname{Bd}}({\ensuremath{\mathbb R}}), {\operatorname{Bd}}(\nu^1_0) \rangle} = {\langle {\operatorname{Bd}}({\ensuremath{\mathbb R}}), {\operatorname{Bd}}({\ensuremath{\mathbb R}}\setminus{\mathbb{Q}}) \rangle}
{\approx}{\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{{\ensuremath{\operatorname{s}}}\setminus0}}\rangle}.$$ For each $0 {\leqslant}k < m - 1$, observe that ${\ensuremath{\mathbb R^m}}\setminus (\nu^{m-1}_k\times{\ensuremath{\mathbb R}})
= ({\ensuremath{\mathbb R}}^{m-1} \setminus \nu^{m-1}_k)\times{\ensuremath{\mathbb R}}{\subseteq}{\ensuremath{\mathbb R^m}}\setminus \nu^m_k$. Thus, it follows that $${\operatorname{Bd}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Bd}}(\nu^{m-1}_k\times{\ensuremath{\mathbb R}})
{\subseteq}{\operatorname{Bd}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Bd}}(\nu^m_k).$$ By Proposition \[owteepgsdgfa\] and Corollary \[owehfafafs\], ${\operatorname{Bd}}(\nu^m_k)$ is a homotopy dense $G_\delta$ set in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$, which implies that ${\operatorname{Bd}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Bd}}(\nu^m_k)$ is a [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-set in ${\operatorname{Bd}}({\ensuremath{\mathbb R^m}})$. On the other hand, we can apply Theorem \[pint-hyperspace\] to obtain $${\langle {\operatorname{Bd}}({\ensuremath{\mathbb R^m}}), {\operatorname{Bd}}({\ensuremath{\mathbb R^m}})\setminus{\operatorname{Bd}}(\nu^{m-1}_k\times{\ensuremath{\mathbb R}}) \rangle}
{\approx}{\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{\Sigma}}\rangle}.$$ Then, it follows from Theorem 6.6 in [@Chapman] that $${\langle {\operatorname{Bd}}({\ensuremath{\mathbb R^m}}), {\operatorname{Bd}}({\ensuremath{\mathbb R^m}})\setminus{\operatorname{Bd}}(\nu^m_k) \rangle}
{\approx}{\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{\Sigma}}\rangle}.$$ Thus, we have the result.
The conclusion of Theorem \[pint-hyperspace\] is equivalent to $${\langle {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}}), {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})\setminus{\mathcal{F}}\rangle} {\approx}{\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{\Sigma}}\rangle}.$$ We saw in §\[borelclasses\] that the subspace ${\mathcal{F}}{\subseteq}{\operatorname{Bd}}({\ensuremath{\mathbb R^m}})$ in Theorem \[pint-hyperspace\] is $G_\delta$, that is, ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})\setminus{\mathcal{F}}$ is $F_\sigma$ in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$. If ${\mathcal{F}}$ contains a homotopy dense subset of ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$ then the complement ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})\setminus{\mathcal{F}}$ is a [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-set. Thus, in order to apply Theorem \[ejgrpio\] to obtain the result, it suffices to show that ${\mathcal{F}}$ contains a homotopy dense subset of ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$ and the complement ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})\setminus{\mathcal{F}}$ contains a strictly homotopy dense subset of ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$. Observe that $${\operatorname{Fin}}({\ensuremath{\mathbb R^m}}) {\subseteq}{\mathfrak N}({\ensuremath{\mathbb R^m}}) {\subseteq}{\operatorname{Nwd}}({\ensuremath{\mathbb R^m}}) \quad\text{and}\quad
{\operatorname{Cantor}}({\ensuremath{\mathbb R^m}}){\subseteq}{\operatorname{Perf}}({\ensuremath{\mathbb R^m}}).$$
As a special case of a well known result due to Curtis and Nguyen To Nhu [@CuNhu], we have $${\langle {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}}), {\operatorname{Fin}}({\ensuremath{\mathbb R^m}}) \rangle} = {\langle \exp({\ensuremath{\mathbb R^m}}), {\operatorname{Fin}}({\ensuremath{\mathbb R^m}}) \rangle}
{\approx}{\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{\operatorname{Q}}}_f\setminus0 \rangle},$$ where ${\ensuremath{\operatorname{Q}}}_f$ denotes the subspace of ${\ensuremath{\operatorname{Q}}}$ consisting of all eventually zero sequences, which is homotopy dense in ${\ensuremath{\operatorname{Q}}}$. This fact implies the following:
\[fin-h-dense\] The subspace ${\operatorname{Fin}}({\ensuremath{\mathbb R^m}})$ is homotopy dense in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$.
Using Lemma \[fin-h-dense\] above, we can easily show the following:
The space ${\operatorname{Cantor}}({\ensuremath{\mathbb R^m}})$ is homotopy dense in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$.
Let $h$ be a homotopy of ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$ which witnesses that ${\operatorname{Fin}}({\ensuremath{\mathbb R^m}})$ is homotopy dense, i.e., $h(A,t)$ is a finite set for every $t > 0$. Choose a Cantor set $C {\subseteq}[0,1]^m$ with $0 \in C$ and define a homotopy ${{\varphi}\colon {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})\times[0,1] \to {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})}$ by $${\varphi}(A,t) = h(A,t) + tC.$$ Then ${\varphi}_0 = {\operatorname{id}}$ and ${\varphi}(A,t)\in{\operatorname{Cantor}}({\ensuremath{\mathbb R^m}})$ for every $t>0$ because a finite union of Cantor sets is a Cantor set.
Concerning the space ${\operatorname{Bd}}(D)$ in Theorem \[pint-hyperspace\], we have shown in Corollary \[owehfafafs\] that it is homotopy dense in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$. Thus, to complete the proof of Theorem \[pint-hyperspace\], it remains to show the following:
\[strict-homotopy\] Under the same assumption as Theorem \[pint-hyperspace\], each of the following spaces are strictly homotopy dense in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$: $${\operatorname{Bd}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Nwd}}({\ensuremath{\mathbb R^m}}),\ {\operatorname{Bd}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Perf}}({\ensuremath{\mathbb R^m}}),\
{\operatorname{Bd}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Bd}}(D).$$
First, we show the following lemma, which also gives a direct proof of Lemma \[fin-h-dense\]:
\[sdegfaqqfas\] For $D {\subseteq}{\ensuremath{\mathbb R^m}}$, if ${\ensuremath{\mathbb R^m}}\setminus D$ is dense in ${\ensuremath{\mathbb R^m}}$ then ${\operatorname{Fin}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Bd}}(D)$ is homotopy dense in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$.
Let ${\mathcal H}= {\operatorname{Fin}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Bd}}(D)$, that is, ${\mathcal H}$ consists of all nonempty finite sets $A {\subseteq}{\ensuremath{\mathbb R^m}}$ such that $A \setminus D {\ne\emptyset}$. Then ${\mathcal H}$ is dense in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$. Moreover, ${\mathcal H}$ is closed under finite unions, i.e., $A \cup B \in {\mathcal H}$ whenever $A, B \in{\mathcal H}$. Recall that ${\langle {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}}), \cup \rangle}$ is a Lawson semilattice (see [@Lawson]), that is, the union operator ${\langle A, B \rangle} \mapsto A \cup B$ is continuous and ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$ has an open base consisting of subsemilattices; namely, every open ball with respect to the Hausdorff metric is a subsemilattice of ${\langle {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}}), \cup \rangle}$. By virtue of [@KSY Theorem 5.1], it suffices to show that ${\mathcal H}$ is relatively $LC^0$ in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$. Recall that a subspace $Y$ of a space $X$ is [*relatively $LC^0$ in*]{} $X$ if every neighborhood $U$ of each $x \in X$ contains a neighborhood $V$ of $x$ in $X$ such that every $a,b \in V \cap Y$ can be joined by a path in $U \cap Y$.
Fix $A \in {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$ and ${\varepsilon}> 0$. For each $A_0,A_1 \in {\operatorname{B}}_{d_H}(A,{\varepsilon}/2) \cap {\mathcal H}$, we describe how to construct a path in ${\operatorname{B}}_{d_H}(A,{\varepsilon})\cap {\mathcal H}$ which joins $A_0$ to $A_0 \cup A_1$. Let $A_1 = \{p_0,\dots,p_{n-1}\}$. For each $i < n$, find $q_i\in A_0$ such that ${\|p_i-q_i\|} < {\varepsilon}/2$, and define $$h(t) = A_0 \cup {\{(1-t)q_i + tp_i\colon i<n\}}
\quad\text{for each $t \in [0,1]$.}$$ Then $h(t) \in {\mathcal H}$ because $A_0 {\subseteq}h(t) \in {\operatorname{Fin}}({\ensuremath{\mathbb R^m}})$. Further, $d_H(A_0,h(t)) < {\varepsilon}/2$, that is, $h(t)\in {\operatorname{B}}_{d_H}(A,{\varepsilon})$. Finally, $h(0) = A_0$ and $h(1) = A_0\cup A_1$. By the same argument, we can construct a path in ${\operatorname{B}}_{d_H}(A,{\varepsilon}) \cap {\mathcal H}$ which joins $A_0 \cup A_1$ to $A_1$.
First, we show the case $m=1$. It suffices to construct a strict deformation ${{\varphi}\colon {\operatorname{Bd}}_H({\ensuremath{\mathbb R}})\times[0,1] \to {\operatorname{Bd}}_H({\ensuremath{\mathbb R}})}$ which omits ${\operatorname{Nwd}}({\ensuremath{\mathbb R}}) \cup {\operatorname{Perf}}({\ensuremath{\mathbb R}}) \cup {\operatorname{Bd}}(D)$. Let $h$ be a homotopy of ${\operatorname{Bd}}({\ensuremath{\mathbb R}})$ which witnesses that ${\operatorname{Fin}}({\ensuremath{\mathbb R}})\setminus{\operatorname{Bd}}(D)$ is homotopy dense (Lemma \[sdegfaqqfas\]). Since ${\operatorname{Bd}}_H([1,2]) {\approx}{\ensuremath{\operatorname{Q}}}$, we have an embedding $g : {\operatorname{Bd}}_H({\ensuremath{\mathbb R}}) \to {\operatorname{Bd}}_H([1,2])$. The desired ${\varphi}$ can be defined as follows: $${\varphi}(A,t) = h(A,t) \cup \{\max h(A,t) + [t,2t],\
\min h(A,t) - tg(A)\}.$$
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(19.0000,-8.0000)[(0,0)[$*$]{}]{}
(28.8000,-4.5000)[(0,0)\[lb\][$\max h(A,t)$]{}]{}
(18.0000,-4.5000)[(0,0)\[lb\][$\min h(A,t)$]{}]{}(34.4000,-11.4000)[(0,0)\[lt\][$\max h(A,t)+[t,2t]$]{}]{}
(6.2000,-11.4000)[(0,0)\[lt\][$\min h(A,t)-tg(A)$]{}]{}
(4.0000,-4.5000)[(0,0)\[lb\][$\min h(A,t)-[t,2t]$]{}]{}
For each $t > 0$, it is clear that ${\varphi}(A,t) \notin {\operatorname{Nwd}}({\ensuremath{\mathbb R}}) \cup {\operatorname{Perf}}({\ensuremath{\mathbb R}})$. Since $h(A,t)$ contains an isolated point from ${\ensuremath{\mathbb R}}\setminus D$ which remains to be isolated in ${\varphi}(A,t)$, we see that ${\varphi}(A,t) \notin {\operatorname{Bd}}(D)$. Given ${\varphi}(A,t)$ for $t>0$, we can reconstruct $t$ as the length of the interval $J {\subseteq}{\varphi}(A,t)$ with $\max J = \max{\varphi}(A,t)$. Consequently, $g(A)$ can be reconstructed from ${\varphi}(A,t)$. Thus, ${\varphi}$ is a strict deformation.
Next, we show the case $m > 1$. To see that ${\operatorname{Bd}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Perf}}({\ensuremath{\mathbb R^m}})$ and ${\operatorname{Bd}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Bd}}(D)$ are strictly homotopy dense in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$, we shall construct a strict deformation ${{\varphi}\colon {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})\times[0,1] \to {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})}$ which omits ${\operatorname{Perf}}({\ensuremath{\mathbb R^m}}) \cup {\operatorname{Bd}}(D)$. Recall $p : {\ensuremath{\mathbb R^m}}\to {\ensuremath{\mathbb R}}^{m-1}$ is the projection onto the first $m - 1$ coordinates. Note that ${p[D]}$ is a dense $G_\delta$ set in ${\ensuremath{\mathbb R}}^{m-1}$ and ${\ensuremath{\mathbb R}}^{m-1} \setminus {p[D]}$ is also dense in ${\ensuremath{\mathbb R}}^{m-1}$. Let $e_m={\langle 0,0,\dots,0,1 \rangle}\in{\ensuremath{\mathbb R^m}}$.
Since ${\ensuremath{\mathbb R^m}}\setminus ({p[D]} \times {\ensuremath{\mathbb R}})$ is dense in ${\ensuremath{\mathbb R^m}}$, it follows from Lemma \[sdegfaqqfas\] that ${\operatorname{Fin}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Bd}}({p[D]} \times {\ensuremath{\mathbb R}})$ is homotopy dense in ${\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})$. Let $h$ be a homotopy of ${\operatorname{Bd}}({\ensuremath{\mathbb R^m}})$ which witnesses this, i.e., for $t > 0$, $h(A,t)$ is finite and $p[h(A,t)] \not{\subseteq}{p[D]}$. Since ${\operatorname{Bd}}_H([3/5,2/3]) {\approx}{\ensuremath{\operatorname{Q}}}$, we have an embedding $g : {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}}) \to {\operatorname{Bd}}_H([3/5,2/3])$. The desired ${\varphi}$ can be defined as follows: $${\varphi}(A,t) = h(A,t) + t\left(\bigcup_{i\in\omega}
2^{-i}(g(A) \cup [3/4,1])e_m \cup \{2e_m\}\right).$$
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(22.0000,-4.3000)[(0,0)\[lb\][$a+tg(A)$]{}]{}
(30.0000,-4.3000)[(0,0)\[lb\][$a+[3/4,1]te_m$]{}]{}
(44.5000,-9.0000)[(0,0)\[lt\][$a+2te_m$]{}]{}
(22.8000,-12.0000)[(0,0)\[lt\][$a+t(g(A)\cup[3/4,1]e_m)$]{}]{}
(15.4000,-9.0000)[(0,0)[$\cdots$]{}]{}(14.0000,-8.0000)[(0,0)[$*$]{}]{}(14.0500,-10.0200)[(0,0)[$*$]{}]{}(12.0500,-9.0200)[(0,0)[$*$]{}]{}(10.0500,-6.0200)[(0,0)[$*$]{}]{}(14.9500,-5.4200)[(0,0)[$*$]{}]{}
(5.2000,-8.7000)[(0,0)\[lb\][$h(A,t)$]{}]{}(16.8000,-14.8000)[(0,0)\[lt\][$a+2^{-1}t(g(A)\cup[3/4,1]e_m)$]{}]{}
(19.9000,-7.3000)[(0,0)[$*$]{}]{}(17.2000,-6.7000)[(0,0)[$*$]{}]{}
For each $t > 0$, ${\varphi}(A,t)$ has an isolated point because $\max {\operatorname{pr}}_m[{\varphi}(A,t)]$ is attained by an isolated point of ${\varphi}(A,t)$, where ${\operatorname{pr}}_m$ denotes the projection onto the $m$-th coordinate. Hence, ${\varphi}(A,t) \not\in {\operatorname{Perf}}({\ensuremath{\mathbb R^m}})$. Since ${p[\varphi(A,t)]} = {p[h(A,t)]}$ is finite and contains a point of ${\ensuremath{\mathbb R}}^{m-1} \setminus {p[D]}$, it follows that ${\operatorname{cl}}({\varphi}(A,t) \cap ({p[D]} \times {\ensuremath{\mathbb R}})) \not= {\varphi}(A,t)$, which means ${\varphi}(A,t) \not\in {\operatorname{Bd}}({p[D]} \times {\ensuremath{\mathbb R}})$.
Given ${\varphi}(A,t)$ for $t > 0$, we can find $t$ as the distance from $\max{\operatorname{pr}}_m[{\varphi}(A,t)]$ to the interior of ${\operatorname{pr}}_m[{\varphi}(A,t)]$. Let $a_0 \in {\varphi}(A,t)$ be such that $${\operatorname{pr}}_m(a_0) = \min{\operatorname{pr}}_m[{\varphi}(A,t)] = \min{\operatorname{pr}}_m[h(A,t)].$$ Then, for sufficiently large $i$, $$(a_0 + 2^{-i}t(g(A) \cup [3/4,1])e_m) \cap h(A,t) = \emptyset.$$ Thus, we can reconstruct $2^{-i}tg(A)$ and consequently also $g(A)$ from ${\varphi}(A,t)$. This shows that ${\varphi}$ is a strict deformation.
For ${\operatorname{Bd}}({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Nwd}}({\ensuremath{\mathbb R^m}})$, we define a homotopy ${\psi\colon {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})\times[0,1] \to {\operatorname{Bd}}_H({\ensuremath{\mathbb R^m}})}$ as follows: $$\psi(A,t) = h(A,t) + t\left(\bigcup_{i\in\omega}
2^{-i}(g(A) \cup [3/4,1])e_m \cup {\overline{{\operatorname{B}}}}(2e_m,1/2)\right).$$ In other wards, replacing the points $a + 2te_m \in {\varphi}(A,t)$, $a \in h(A,t)$, by the closed balls $$a + t{\overline{{\operatorname{B}}}}(2e_m,1/2) = {\overline{{\operatorname{B}}}}(a + 2te_m,t/2),\ a \in h(A,t),$$ we can obtain $\psi(A,t)$ from ${\varphi}(A,t)$. Evidently $\psi$ omits ${\operatorname{Nwd}}({\ensuremath{\mathbb R^m}})$. Given $\psi(A,t)$ for $t > 0$, let $a_0 \in \psi(A,t)$ be such that $${\operatorname{pr}}_m(a_0) = \min{\operatorname{pr}}_m[\psi(A,t)] = \min{\operatorname{pr}}_m[h(A,t)].$$ Then we can get $t$ as the diameter of the ball ${\overline{{\operatorname{B}}}}(a_0 + 2te_m,t/2)$ (which is equal to $2/3$ of the distance from $a_0$ to this ball). Now, by the same arguments as for ${\varphi}$, we can reconstruct $g(A)$ from $\psi(A,t)$. Thus, $\psi$ is a strict deformation.
Let us note that the subspace ${\operatorname{UNb}}({\ensuremath{\mathbb R}})\cup{\operatorname{Fin}}({\ensuremath{\mathbb R}})$ is actually equal to the space ${\operatorname{Pol}}({\ensuremath{\mathbb R}})$ consisting of all compact polyhedra in ${\ensuremath{\mathbb R}}$. It follows from the result of [@Sakai91] that the pair ${\langle \exp({\ensuremath{\mathbb R}}), {\operatorname{Pol}}({\ensuremath{\mathbb R}}) \rangle}$ is homeomorphic to ${\langle {\ensuremath{\operatorname{Q}}}, {\ensuremath{\operatorname{Q}}}_f \rangle}$.
Nonseparable components of ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$
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In this section, we consider the space ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ of all nonempty closed subsets of ${\ensuremath{\mathbb R}}$. We shall also consider its natural subspaces, using the same notation as before, but having in mind the new setting. For example, ${\operatorname{Perf}}({\ensuremath{\mathbb R}})$ and ${\operatorname{Nwd}}({\ensuremath{\mathbb R}})$ will denote the subspace of ${\operatorname{Cld}}({\ensuremath{\mathbb R}})$ consisting of all perfect closed subsets of ${\ensuremath{\mathbb R}}$ and all closed sets with no interior points, respectively. Now ${\operatorname{Perf}}({\ensuremath{\mathbb R}}) \cap {\operatorname{Nwd}}({\ensuremath{\mathbb R}})$ consists of all nonempty closed (possibly unbounded) subsets of ${\ensuremath{\mathbb R}}$ which have neither isolated points nor interior points. In the new setting, we have $${\operatorname{Cantor}}({\ensuremath{\mathbb R}}) = {\operatorname{Perf}}({\ensuremath{\mathbb R}}) \cap {\operatorname{Nwd}}({\ensuremath{\mathbb R}}) \cap {\operatorname{Bd}}({\ensuremath{\mathbb R}}).$$
As shown in [@KuSaY Proposition 7.2], ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ has $2^{\aleph_0}$ many components, ${\operatorname{Bd}}({\ensuremath{\mathbb R}})$ is the only separable one and any other component has weight $2^{\aleph_0}$. The following is the main theorem in this section:
\[non-sep-compon\] Let ${\mathcal H}$ be a nonseparable component of ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ which does not contain ${\ensuremath{\mathbb R}}$, $[0,+\infty)$, $(-\infty,0]$. Then ${\mathcal H}{\approx}\ell_2(2^{\aleph_0})$.
We shall say that a set $A{\subseteq}{\ensuremath{\mathbb R}}$ [*has infinite uniform gaps*]{} if there are $\delta>0$ and pairwise disjoint open intervals $I_0,I_1,\dots$ such that ${\operatorname{diam}}I_n {\geqslant}\delta$, $A \cap I_n = \emptyset$ and ${\operatorname{bd}}I_n {\subseteq}A$ for every ${n\in\omega}$. Define $${\mathcal{V}}= {\{A\in{\operatorname{Cld}}({\ensuremath{\mathbb R}})\colon A\text{ has infinite uniform gaps }\}}.$$ Clearly, ${\mathcal{V}}$ is open in ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ and ${\mathcal{V}}\cap {\operatorname{Bd}}({\ensuremath{\mathbb R}}) = \emptyset$. For each $A \in {\operatorname{Cld}}({\ensuremath{\mathbb R}}) \setminus {\operatorname{Bd}}({\ensuremath{\mathbb R}})$ and $\varepsilon > 0$, let $D {\subseteq}A$ be a maximal $\varepsilon$-discrete subset. Then $D \in {\mathcal{V}}$ and $d_H(A,D) {\leqslant}\varepsilon$ because $D {\subseteq}A {\subseteq}{\operatorname{B}}(D,\varepsilon)$. Thus, ${\mathcal{V}}$ is dense in ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})\setminus{\operatorname{Bd}}({\ensuremath{\mathbb R}})$.
If ${\mathcal H}$ is a nonseparable component of ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ and ${\ensuremath{\mathbb R}},[0,+\infty),(-\infty,0]\notin {\mathcal H}$ then ${\mathcal H}{\subseteq}{\mathcal{V}}$. Indeed, each $A \in {\mathcal H}$ is unbounded and every component of ${\ensuremath{\mathbb R}}\setminus A$ is an open interval. Let $\mathcal J$ be the set of all bounded component of ${\ensuremath{\mathbb R}}\setminus A$. Assume that ${\{{\operatorname{diam}}I\colon I \in \mathcal J\}}$ is bounded. When $A$ is bounded below (or bounded above), $d_H(A, [0,\infty)) < \infty$ (or $d_H(A, (-\infty,0]) < \infty$), which implies $[0,+\infty) \in {\mathcal H}$ (or $(-\infty,0] \in {\mathcal H}$). When $A$ is not bounded below nor above, $d_H(A,{\ensuremath{\mathbb R}}) < \infty$, which implies ${\ensuremath{\mathbb R}}\in {\mathcal H}$. Therefore, ${\{{\operatorname{diam}}I\colon I \in \mathcal J\}}$ is unbounded. In particular, $A$ has infinite uniform gaps.
Due to Theorem A in [@KuSaY], every component of ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ is an AR, hence it is contractible. Since a contractible $\ell_2(2^{\aleph_0})$-manifold is homeomorphic to $\ell_2(2^{\aleph_0})$, Theorem \[non-sep-compon\] above follows from the following theorem:
The open dense subset ${\mathcal{V}}$ of ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ is an $\ell_2(2^{\aleph_0})$-manifold.
It suffices to show that each $A_0 \in {\mathcal{V}}$ has an open neighborhood ${\mathcal{U}}{\subseteq}{\mathcal{V}}$ which is an $\ell_2(2^{\aleph_0})$-manifold. In this case, ${\mathcal{U}}$ is a completely metrizable ANR because it is an open set in a completely metrizable ANR ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$. Due to Toru[ń]{}czyk characterization of $\ell_2(2^{\aleph_0})$-manifold [@To81] (cf. [@To85]), we have to show that ${\mathcal{U}}$ has the following two properties:
For each maps $f : [0,1]^n \times 2^\omega \to {\mathcal{U}}$ and $\alpha : {\mathcal{U}}\to (0,1)$, there exists a map $g : [0,1]^n \times 2^\omega \to {\mathcal{U}}$ such that $d_H(g(z),f(z)) < \alpha(f(z))$ for each $z \in [0,1]^n \times 2^\omega$ and $\{g[[0,1]^n \times \{x\}] : x \in 2^\omega\}$ is discrete in ${\mathcal{U}}$;
For any finite-dimensional simplicial complexes $K_n$, $n \in \omega$, with ${\operatorname{card}}K_n {\leqslant}2^{\aleph_0}$, for every maps $f : \bigoplus_{n\in\omega} |K_n| \to {\mathcal{U}}$ and $\alpha : {\mathcal{U}}\to (0,1)$, there exists a map $g : \bigoplus_{n\in\omega} |K_n| \to {\mathcal{U}}$ such that $d_H(g(z),f(z))$ $ < \alpha(f(z))$ for each $z \in \bigoplus_{n\in\omega} |K_n|$ and $\{g[|K_n|] : n \in \omega\}$ is discrete in ${\mathcal{U}}$.
In the above, $2^\omega$ is the discrete space of all functions of $\omega$ to $2 = \{0,1\}$. To this end, it suffices to prove the following:
- For each map $\alpha : {\mathcal{U}}\to (0,1)$, there exist maps $f_x : {\mathcal{U}}\to {\mathcal{U}}$, $x \in 2^\omega$, such that $d_H(f_x(A),A) < \alpha(A)$ for every $A \in {\mathcal{U}}$ and $\{f_x[{\mathcal{U}}] : x \in 2^\omega\}$ is discrete.
Fix $A_0 \in {\mathcal{V}}$ and choose open intervals $I_0,I_1,\dots$ such that ${\operatorname{diam}}I_n {\geqslant}\delta$, $A_0 \cap I_n = \emptyset$ and ${\operatorname{bd}}I_n {\subseteq}A_0$ (i.e., $\inf I_n,\ \sup I_n \in A_0$) for every ${n\in\omega}$. Taking a subsequence if necessary, we may assume that either $\sup I_n < \inf I_{n+1}$ for every ${n\in\omega}$ or $\inf I_n > \sup I_{n+1}$ for every ${n\in\omega}$. Because of similarity, we may assume that the first possibility occurs.
Choose intervals $[a_n,b_n] {\subseteq}I_n$, ${n\in\omega}$, so that $b_n - a_n > \delta/4$, $$\begin{gathered}
\inf_{n\in\omega}{\operatorname{dist}}(a_n,{\ensuremath{\mathbb R}}\setminus I_n)
= \inf_{n\in\omega}(a_n - \inf I_n) > \delta/4 \text{ and}\\
\inf_{n\in\omega}{\operatorname{dist}}(b_n,{\ensuremath{\mathbb R}}\setminus I_n)
= \inf_{n\in\omega}(\sup I_n - b_n) > \delta/4.\end{gathered}$$
0.1in
(42.15,5.10)(3.85,-9.70)
(9.0000,-6.3000)[(0,0)\[lb\][$I_{n-1}$]{}]{}(29.0000,-6.3000)[(0,0)\[lb\][$I_n$]{}]{}
(20.0000,-9.7000)[(0,0)\[lt\][$A_0$]{}]{}
(42.0000,-9.7000)[(0,0)\[lt\][$A_0$]{}]{}(25.0000,-9.3000)[(0,0)\[lt\][$a_n$]{}]{}(32.5000,-9.0000)[(0,0)\[lt\][$b_n$]{}]{}(10.5000,-9.0000)[(0,0)\[lt\][$b_{n-1}$]{}]{}(5.5000,-9.3000)[(0,0)\[lt\][$a_{n-1}$]{}]{}
(25.0000,-8.0000)[(0,0)[$\circ$]{}]{}(33.0000,-8.0000)[(0,0)[$\circ$]{}]{}(11.0000,-8.0000)[(0,0)[$\circ$]{}]{}(7.0000,-8.0000)[(0,0)[$\circ$]{}]{}
Observe that if $A \in {\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ and $d_H(A,A_0) < \delta/4$ then $A \cap (b_{n-1},a_n) \not= \emptyset$ for every $n \in \omega$, where $b_{-1} = -\infty$. For each $A \in {\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ with $d_H(A,A_0) < \delta/4$, we can define $$r_n(A) = \max(A \cap (b_{n-1},a_n)),\ {n\in\omega}.$$ For each $A, A' \in {\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ with $d_H(A,A_0), d_H(A',A_0) < \delta/4$, we have $$|r_n(A) - r_n(A')| \leqslant d_H(A,A').$$ Indeed, without loss of generality, we may assume that $r_n(A) < r_n(A')$. Then, the open interval $(r_n(A),b_n)$ contains no points of $A$ and $r_n(A') \in (r_n(A),b_n)$. Since $b_n - r_n(A') > \delta/2$ and $$r_n(A') - r_n(A)
\leqslant |r_n(A') - r_n(A_0)| + |r_n(A) - r_n(A_0)| < \delta/2,$$ we have $|r_n(A') - r_n(A)| \leqslant d_H(A,A')$. Then, it follows that $$\begin{aligned}
\inf_{n\in\omega}(a_n - r_n(A)) - d_H(A,A')
&\leqslant \inf_{n\in\omega}(a_n - r_n(A')) \\
&\leqslant \inf_{n\in\omega}(a_n - r_n(A)) + d_H(A,A').\end{aligned}$$ This means that $A \mapsto \inf_{n\in\omega}(a_n - r_n(A))$ is continuous. Since $r_n(A_0) = \inf I_n$, we have $\inf_{n\in\omega}(a_n - r_n(A_0)) > \delta/4$. Thus, $A_0$ has the following open neighborhood: $${\mathcal{U}}= {\{A\in{\operatorname{Cld}}_H({\ensuremath{\mathbb R}})\colon d_H(A,A_0) < \delta/4,\
\inf_{n\in\omega}(a_n - r_n(A)) > \delta/4\}} {\subseteq}{\mathcal{V}}.$$
Now, for each map ${\alpha\colon {\mathcal{U}}\to (0,1)}$, we define a map $\beta : {\mathcal{U}}\to (0,1)$ as follows: $$\beta(A) = \min\big\{\tfrac12\alpha(A),\ \tfrac14\delta - d_H(A,A_0),\
\inf_{n\in\omega}(a_n - r_n(A)) - \tfrac14\delta\big\}.$$ Given a sequence $x = (x(n))_{{n\in\omega}} \in 2^\omega$, let $$f_x(A) = A \cup \bigcup_{{n\in\omega}}\big(r_n(A) +
\big([0,\tfrac12\beta(A)] \cup \{\beta(A)\cdot x(n)\}\big)\big).$$
0.1in
(29.15,7.30)(4.85,-13.00)
(7.0000,-10.9000)[(0,0)\[lt\][$b_{n-1}$]{}]{}
(30.5000,-11.0000)[(0,0)\[lt\][$a_n$]{}]{}(14.5000,-12.0000)[(0,0)\[lt\][$A\cap(b_{n-1},a_n)$]{}]{}
(20.5000,-7.4000)[(0,0)\[rb\][$r_n(A)$]{}]{}
(24.0000,-13.0000)[(0,0)\[lt\][$r_n(A) + \beta(A)$]{}]{}
(24.2000,-7.4000)[(0,0)\[lb\][$r_n(A) + [0,\frac12\beta(A)]$]{}]{}
(26.4000,-10.0000)[(0,0)[$*$]{}]{}
(8.0000,-10.0000)[(0,0)[$\circ$]{}]{}(30.8000,-10.0000)[(0,0)[$\circ$]{}]{}
This defines a map ${f_x\colon {\mathcal{U}}\to {\mathcal{U}}}$ which is $\alpha$-close to ${\operatorname{id}}$. We claim that if $x \not= y \in 2^\omega$ then $$d_H(f_x(A),f_y(A')) {\geqslant}\min\big\{\tfrac14\beta(A),\tfrac14\beta(A')\big\}
\text{ for every $A, A' \in {\mathcal{U}}$.}$$ Indeed, assume that $x(n) = 1$, $y(n) = 0$ and let $s = \min\{\tfrac14\beta(A),\tfrac14\beta(A')\}$. Then
1. $\max(f_x(A) \cap (b_{n-1},a_n)) = r_n(A)+\beta(A)$;
2. $f_x(A)$ has no points in the open interval $(r_n(A)+\tfrac12\beta(A), r_n(A)+\beta(A))$;
3. $\max(f_y(A') \cap (b_{n-1},a_n)) = r_n(A')+\tfrac12\beta(A')$;
4. $[r_n(A'),r_n(A')+\beta(A')/2] {\subseteq}f_y(A')$.
In case $r_n(A')+\tfrac12\beta(A') {\geqslant}r_n(A)+\beta(A)+s$ or $r_n(A')+\tfrac12\beta(A') {\leqslant}r_n(A)+\beta(A)-s$, we have $$d_H(f_x(A) \cap (b_{n-1},a_n),f_y(A') \cap (b_{n-1},a_n)) {\geqslant}s.$$ In case $r_n(A)+\beta(A)-s < r_n(A')+\tfrac12\beta(A') {\leqslant}r_n(A)+\beta(A)+s$, since $2s {\leqslant}\tfrac12\beta(A')$, we have $r_n(A') < r_n(A)+\beta(A)-s$, hence $r_n(A)+\beta(A)-s \in f_y(A')$. Thus, it follows that $$d_H(f_x(A) \cap (b_{n-1},a_n),f_y(A') \cap (b_{n-1},a_n)) {\geqslant}s.$$
Finally, we show that $\{f_x[{\mathcal{U}}] : x \in 2^\omega\}$ is a discrete collection of ${\mathcal{U}}$. If not, we have $A$, $A_i \in {\mathcal{U}}$ and $x_i \in 2^\omega$, $i \in \omega$, such that $x_i \not= x_j$ if $i \not= j$, and $f_{x_i}(A_i) \to A$ ($i \to \infty$). Then $c = \inf_{i\in\omega}\beta(A_i) = 0$. Indeed, otherwise we could find $i < j$ such that $$d_H(f_{x_i}(A_i),A),\ d_H(f_{x_j}(A_j),A) < c/10$$ and $\beta(A_i),\ \beta(A_j) > 4c/5$. It follows that $d_H(f_{x_i}(A_i),f_{x_j}(A_j)) < c/5$, but $$d_H(f_{x_i}(A_i),f_{x_j}(A_j)) {\geqslant}\min\{\beta(A)/4,\beta(A')/4\}
> c/5,$$ which is a contradiction. Thus, $\inf_{i\in\omega}\beta(A_i) = 0$. Taking a subsequence, we may assume that $\lim_{i\to\infty}\beta(A_i) = 0$. Then $A_i \to A$ ($i \to \infty$) because $d_H(f_{x_i}(A_i),A_i) {\leqslant}\beta(A_i)$. It follows that $\beta(A) = 0$, which is a contradiction. This completes the proof.
Let ${\mathcal{D}}(X)$ be the subspace of ${\operatorname{Cld}}_H(X)$ consisting of all discrete sets in $X$. It follows from the result of [@BaVo] that ${\mathcal{D}}(X)$ is homotopy dense in ${\operatorname{Cld}}_H(X)$ for every almost convex metric space $X$. By the same proof, Lemma \[sdegfaqqfas\] can be extended to ${\operatorname{Cld}}_H({\ensuremath{\mathbb R^m}})$.
\[h-dense-D\] Assume $D{\subseteq}{\ensuremath{\mathbb R^m}}$ is such that ${\ensuremath{\mathbb R^m}}\setminus D$ is dense. Then ${\mathcal{D}}({\ensuremath{\mathbb R^m}})\setminus{\operatorname{Cld}}(D)$ is homotopy dense in ${\operatorname{Cld}}_H({\ensuremath{\mathbb R^m}})$.
Now, we consider the subspaces ${\mathfrak N}({\ensuremath{\mathbb R}})$, ${\operatorname{Nwd}}({\ensuremath{\mathbb R}})$, ${\operatorname{Perf}}({\ensuremath{\mathbb R}})$ and ${\operatorname{Cld}}({\ensuremath{\mathbb R}}\setminus{\mathbb{Q}})$ of ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$. Similarly to ${\operatorname{Bd}}_H({\ensuremath{\mathbb R}})$, the following can be shown:
\[non-sep-Z\_sig\] The sets ${\operatorname{Cld}}({\ensuremath{\mathbb R}}) \setminus {\mathfrak N}({\ensuremath{\mathbb R}})$, ${\operatorname{Cld}}({\ensuremath{\mathbb R}}) \setminus {\operatorname{Nwd}}({\ensuremath{\mathbb R}})$, ${\operatorname{Cld}}({\ensuremath{\mathbb R}}) \setminus {\operatorname{Perf}}({\ensuremath{\mathbb R}})$ and ${\operatorname{Cld}}({\ensuremath{\mathbb R}}) \setminus {\operatorname{Cld}}({\ensuremath{\mathbb R}}\setminus{\mathbb{Q}})$ are [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-sets in the space ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$.
Due to Negligibility Theorem ([@AHW], [@Cut]) if $M$ is an $\ell_2(2^{\aleph_0})$-manifold and $A$ is a [${\ensuremath{\operatorname{Z}}}_\sigma$]{}-set in $M$ then $M \setminus A {\approx}M$. Thus, combining Proposition \[non-sep-Z\_sig\] and Theorem \[non-sep-compon\], we have the following:
Let ${\mathcal H}$ be a nonseparable component of ${\operatorname{Cld}}_H({\ensuremath{\mathbb R}})$ which does not contain ${\ensuremath{\mathbb R}}$, $[0,+\infty)$, $(-\infty,0]$. Then ${\mathcal H}\cap {\mathfrak N}({\ensuremath{\mathbb R}})$, ${\mathcal H}\cap {\operatorname{Nwd}}({\ensuremath{\mathbb R}})$, ${\mathcal H}\cap {\operatorname{Perf}}({\ensuremath{\mathbb R}})$ and ${\mathcal H}\cap {\operatorname{Cld}}({\ensuremath{\mathbb R}}\setminus{\mathbb{Q}})$ are homeomorphic to $\ell_2(2^{\aleph_0})$.
Open problems
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The following questions are left open.
In case $m > 1$, under the only assumption that $D {\subseteq}{\ensuremath{\mathbb R^m}}$ is a dense $G_\delta$ set and ${\ensuremath{\mathbb R^m}}\setminus D$ is also dense in ${\ensuremath{\mathbb R^m}}$, is the pair ${\langle {\operatorname{Bd}}({\ensuremath{\mathbb R}}^m), {\operatorname{Bd}}(D) \rangle}$ homeomorphic to ${\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{{\ensuremath{\operatorname{s}}}\setminus0}}\rangle}$? In particular, is the pair ${\langle {\operatorname{Bd}}({\ensuremath{\mathbb R}}^m), {\operatorname{Bd}}(\nu^m_{m-1}) \rangle}$ homeomorphic to ${\langle {\ensuremath{{\ensuremath{\operatorname{Q}}}\setminus 0}}, {\ensuremath{{\ensuremath{\operatorname{s}}}\setminus0}}\rangle}$?
Does Theorem \[non-sep-compon\] hold even if ${\mathcal H}$ contains ${\ensuremath{\mathbb R}}$, $[0,\infty)$ or $(-\infty,0]$?
For $m > 1$, is ${\operatorname{Cld}}_H({\ensuremath{\mathbb R^m}}) \setminus {\operatorname{Bd}}({\ensuremath{\mathbb R^m}})$ an $\ell_2(2^{\aleph_0})$-manifold?
Appendix
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For the convenience of readers, we give short and straightforward proofs of Propositions \[rthsdpio\] and \[owteepgsdgfa\].
If ${\langle X, d \rangle}$ is $\sigma$-compact then the space ${\langle {\operatorname{Bd}}(X), d_H \rangle}$ is $F_{\sigma\delta}$ in its completion ${\langle {\operatorname{Bd}}({\tilde}X), d_H \rangle}$.
Fix a countable open base ${\{U_n\colon {n\in\omega}\}}$ for ${\tilde}X$. Since $U_n\cap X$ is $F_\sigma$, we have $U_n\cap X=\bigcup_{k\in{\omega}}K_k^n$, where each $K_k^n$ is compact. Observe that, by compactness, the sets $({\tilde}X\setminus K_k^n)^+$ are open in the Hausdorff metric topology. We claim that $${\operatorname{Bd}}({\tilde}X)\setminus {\operatorname{Bd}}(X)=\bigcup_{{n\in\omega}}\Bigl(U_n^-\cap\bigcap_{k\in{\omega}}({\tilde}X\setminus K_k^n)^+\Bigr),$$ which shows that ${\operatorname{Bd}}({\tilde}X)\setminus {\operatorname{Bd}}(X)$ is a countable union of $G_\delta$ sets. This is what we want to prove.
Assume $A\in {\operatorname{Bd}}({\tilde}X)\setminus {\operatorname{Bd}}(X)$, that is, $A\ne {\operatorname{cl}}_{{\tilde}X}(A\cap X)$. Then there is ${n\in\omega}$ such that $U_n\cap A{\ne\emptyset}$ and $U_n\cap A\cap X=\emptyset$, which means that $A\in U_n^-$ and $A\in ({\tilde}X\setminus K_k^n)^+$ for every $k\in {\omega}$. Conversely, if $A\in U_n^-\cap \bigcap_{k\in{\omega}}({\tilde}X\setminus K_k^n)^+$ then $U_n\cap A{\ne\emptyset}$ and $U_n\cap A\cap X=\emptyset$, so $A\ne{\operatorname{cl}}_{{\tilde}X}(A\cap X)$.
If ${\langle X, d \rangle}$ is Polish then the space ${\langle {\operatorname{Bd}}(X), d_H \rangle}$ is $G_\delta$ in its completion ${\langle {\operatorname{Bd}}({\tilde}X), d_H \rangle}$.
Let ${\{W_n\colon {n\in\omega}\}}$ be a family of open subsets of ${\tilde}X$ such that $X=\bigcap_{{n\in\omega}}W_n$. Fix a countable open base ${\{V_n\colon {n\in\omega}\}}$ for ${\tilde}X$. We claim that $${\operatorname{Bd}}({\tilde}X)\setminus {\operatorname{Bd}}(X)=\bigcup_{{n\in\omega}}\bigcup_{k\in{\omega}}\Bigl(V_n^-\setminus(V_n\cap W_k)^-\Bigr).\tag{$*$}$$ As $V^-$ is open in the metric space ${\langle {\operatorname{Bd}}({\tilde}X,d), d_H \rangle}$ whenever $V{\subseteq}{\tilde}X$ is open, it follows that $V_n^-$ is $F_\sigma$ and therefore the set on the right-hand side of ($*$) is $F_\sigma$ in ${\operatorname{Bd}}_H({\tilde}X)$. It remains to prove ($*$).
If $A\in V_n^-\setminus(V_n\cap W_k)^-$ then we have $x\in V_n\cap A$. Since $V_n\cap (A\cap X)=\emptyset$, it follows that $x\notin {\operatorname{cl}}_{{\tilde}X}(A\cap X)$. Thus $A\notin {\operatorname{Bd}}(X)$. Now assume $A\in {\operatorname{Bd}}({\tilde}X)\setminus {\operatorname{Bd}}(X)$, that is, $A\ne{\operatorname{cl}}_{{\tilde}X}(A\cap X)$. Then there exists an open set $U{\subseteq}{\tilde}X$ such that $U\cap A{\ne\emptyset}$ and $U\cap A\cap X=\emptyset$. Hence $\bigcap_{k\in{\omega}}A\cap U\cap W_k=\emptyset$. Note that $A\cap U$ is a Baire space because of the completeness of ${\langle {\tilde}X, d \rangle}$. Thus, by the Baire Category Theorem, there exists $k\in{\omega}$ such that $A\cap U\cap W_k$ is not dense in $A\cap U$. Find a basic open set $V_n{\subseteq}U$ such that $V_n\cap A{\ne\emptyset}$ and $V_n\cap A\cap W_k=\emptyset$. Then $A\in V_n^-\setminus(V_n\cap W_k)^-$.
Let ${\mathfrak B}(X)$ denote the Borel field on a topological space $X$. Given $\mathfrak H {\subseteq}{\operatorname{Cld}}(X)$, the [*Effros $\sigma$-algebra*]{} ${\mathfrak E}(\mathfrak H)$ is the $\sigma$-algebra generated by $${\{U^-\cap\mathfrak H\colon \text{$U$ is open in $X$}\}}.$$ It is well known that ${\mathfrak E}(\exp(X)) = {\mathfrak B}(\exp(X))$ for every separable metric space $X$ (see [@Beer Theorem 6.5.15]).[^5] Whenever $X$ is a separable metric space in which every bounded set is totally bounded, we can regard ${\operatorname{Bd}}_H(X) {\subseteq}\exp(\tilde{X})$ by the identification as in §\[borelclasses\], where $\tilde{X}$ is the completion of $X$. Then, we have not only ${\mathfrak E}({\operatorname{Bd}}(X)) = {\mathfrak B}({\operatorname{Bd}}_H(X))$ but also ${\mathfrak E}(\mathfrak H) = {\mathfrak B}(\mathfrak H)$ for $\mathfrak H {\subseteq}{\operatorname{Bd}}_H(X)$. This implies that ${\mathfrak E}(\mathfrak H)$ is standard if $\mathfrak H$ is absolutely Borel (cf. [@Ke 12.B]). The results in §\[borelclasses\] provide such hyperspaces $\mathfrak H$.
In relation to the results above, we can prove the following:
\[weptjwpf\] Let $X={\langle X, d \rangle}$ be an analytic metric space in which bounded sets are totally bounded. Then, the space ${\operatorname{Bd}}_H(X)$ is analytic.
The completion ${\langle {\tilde}X, d \rangle}$ of ${\langle X, d \rangle}$ is a Polish space in which closed bounded sets are compact. Then ${\operatorname{Bd}}_H({\tilde}X,d)=\exp({\tilde}X)$ is Polish. Fix a countable open base ${\ensuremath{{\{{U}_n\colon {n\in\omega}\}}}}$ for ${\tilde}X$. Since $X$ is analytic, there exists a tree ${\{X_s\colon s\in{\omega}^{<{\omega}}\}}$ of closed subsets of ${\tilde}X$ such that $X=\bigcup_{f\in{\omega}^{\omega}}\bigcap_{{n\in\omega}}X_{f{\restriction}n}$, which is the result of the Suslin operation on the family ${\{X_s\colon s\in{\omega}^{<{\omega}}\}}$ (e.g. see [@Jech Lemma 11.7]). We may assume that $X_s{\supseteq}X_t$ whenever $s{\subseteq}t$. Let $W_s={\operatorname{B}}(X_s,2^{-|s|})$, where $|s|$ denotes the length of the sequence $s$. Then ${\operatorname{cl}}W_s{\supseteq}{\operatorname{cl}}W_t$ whenever $s{\subseteq}t$. Moreover, $\bigcap_{{n\in\omega}}X_{f{\restriction}n}=\bigcap_{{n\in\omega}}{\operatorname{cl}}W_{f{\restriction}n}$ for each $f\in{\omega}^{\omega}$. We claim that $${\operatorname{Bd}}(X,d)=\bigcap_{k\in{\omega}}\bigcup_{f\in{\omega}^{\omega}}\bigcap_{{n\in\omega}}\Bigl(({\operatorname{Bd}}({\tilde}X,d)\setminus U_k^-)\cup(U_k\cap W_{f{\restriction}n})^-\Bigr),\tag{$\sharp$}$$ where, as usual, we regard ${\operatorname{Bd}}(X,d){\subseteq}{\operatorname{Bd}}({\tilde}X,d)$, via the embedding $A\mapsto {\operatorname{cl}}_{{\tilde}X}A$. The above formula ($\sharp$) shows that ${\operatorname{Bd}}(X,d)$ can be obtained from ${\operatorname{Bd}}({\tilde}X,d)$ by using the Suslin operation and countable intersection, which shows that it is analytic. It remains to prove ($\sharp$).
Fix $A\in{\operatorname{Bd}}({\tilde}X,d)\setminus {\operatorname{Bd}}(X,d)$. Then $A\ne{\operatorname{cl}}(A\cap X)$ and hence there exists $k\in{\omega}$ such that $A\in U_k^-$ and ${\operatorname{cl}}U_k\cap A\cap X=\emptyset$. Then $A \notin {\operatorname{Bd}}({\tilde}X,d)\setminus U_k^-$. For each $f\in{\omega}^{\omega}$, we have $$A\cap{\operatorname{cl}}U_k\cap\bigcap_{{n\in\omega}}{\operatorname{cl}}W_{f{\restriction}n}
=A\cap{\operatorname{cl}}U_k\cap\bigcap_{{n\in\omega}}X_{f{\restriction}n}=\emptyset.$$ By compactness, there is ${n\in\omega}$ such that $A\cap{\operatorname{cl}}U_k\cap {\operatorname{cl}}W_{f{\restriction}n}=\emptyset$, hence $A\notin(U_k\cap W_{f{\restriction}n})^-$.
Now assume that $A\in{\operatorname{Bd}}({\tilde}X,d)$ does not belong to the right-hand side of ($\sharp$), that is, there exists $k\in{\omega}$ such that $A\in U_k^-$ and for every $f\in{\omega}^{\omega}$ there is ${n\in\omega}$ with $A\notin(U_k\cap W_{f{\restriction}n})^-$. In particular, $A\cap U_k\cap\bigcap_{{n\in\omega}}X_{f{\restriction}n}=\emptyset$ for every $f\in{\omega}^{\omega}$ and consequently $U_k\cap A\cap X=\emptyset$. On the other hand, $A\cap U_k{\ne\emptyset}$. Thus it follows that $A\ne {\operatorname{cl}}_{{\tilde}X}(A\cap X)$, which means $A\notin{\operatorname{Bd}}(X,d)$.
[99]{}
, [*Negligible subsets of infinite-dimensional manifolds*]{}, Compositio Math. [**21**]{} (1969), 143–150.
, [*Continuous extensions of multifunctions*]{}, Ann. Polon. Math. [**34**]{} (1977), no.1, 107–111.
, [*Absorbing Sets in Infinite-Dimensional Manifolds*]{}, Math. Studies Monog. Ser. [**1**]{}, VNTL Publishers, Lviv, 1996. 232 pp. ISBN: 5-7773-0061-8.
, [*Characterizing metric spaces whose hyperpsaces are absolute neighborhood retracts*]{}, Topology Appl. (to appear).
, [*Topologies on Closed and Closed Convex Sets*]{}, Math. and its Appl. [**268**]{}, Kluwer Academic Publ., Dordrecht, 1993. xii+340 pp. ISBN: 0-7923-2531-1.
, [*Dense sigma-compact subsets of infinite-dimensional manifolds*]{}, Trans. Amer. Math. Soc. [**154**]{} (1971), 399–426.
, [*Every Wijsman topology relative to a Polish space is Polish*]{}, Proc. Amer. Math. Soc. [**123**]{} (1995), no.8, 2569–2574.
, [*Paths in hyperspaces*]{}, Appl. Gen. Topology [**4**]{} (2003), no.2, 377–390.
, [*Hyperspaces of noncompact metric spaces*]{}, Compositio Math. [**40**]{} (1980), 139–152.
, [*Hyperspaces of Peano continua are Hilbert cubes*]{}, Fund. Math. [**101**]{} (1978), 19–38.
, [*Hyperspaces of finite subsets which are homeomorphic to $\aleph_0$-dimensional linear metric spaces*]{}, Topology Appl. [**19**]{} (1985), 251–260.
, [*Negligible subsets of infinite-dimensional Fréchet manifolds*]{}, Proc. Amer. Math. Soc. [**23**]{} (1969), 668–675.
, [*Hyperspaces, Fundamentals and Recent Advances*]{}, Pure and Applied Math. [**216**]{}, Marcel Dekker, Inc., Yew York, 1999. xx+512 pp. ISBN: 0-8247-1282-4.
, [*Set Theory, The third millennium edition, revised and expanded*]{}, Springer Monog. in Math., Springer-Verlag, Berlin, 2003. xiv+769 pp. ISBN: 3-540-44085-2.
, [*Classical Descriptive Set Theory*]{}, Graduate Texts in Math. [**156**]{}, Springer-Verlag, New York, 1995.
, [*Hyperspaces of separable Banach spaces with the Wijsman topology*]{}, Topology Appl. [**148**]{} (2005), 7–32.
, [*Hyperspaces with the Hausdorff metric and uniform ANRs*]{}, J. Math. Soc. Japan [**57**]{}, (2005), no.2, 523–535.
, [*Topological semilattices with small subsemilattices*]{}, J. London Math. Soc. (2) [**1**]{} (1969), 719–724.
, [*Infinite-Dimensional Topology, Prerequisites and Introduction*]{}, North-Holland Math. Library [**43**]{}, Elsevier Science Publisher B.V., Amsterdam, 1989. xii+401 pp. ISBN: 0-444-87133-0.
, [*La structure borélienne d’Effros est-elle standard?*]{}, Fund. Math. [**100**]{} (1978), 201–210.
, [*On hyperspaces of polyhedra*]{}, Proc. Amer. Math. Soc. [**110**]{} (1990), no.4, 1089–1097.
, [*Characterizing Hilbert space topology*]{}, Fund. Math. [**111**]{} (1981), 247–262.
, [*A correction of two papers concerning Hilbert manifolds*]{}, Fund. Math. [**125**]{} (1985), 89–93.
[^1]: It is well known that ${\ensuremath{\operatorname{s}}}$ is homeomorphic to the separable Hilbert space $\ell_2$.
[^2]: I.e., separable completely metrizable spaces.
[^3]: In some articles (e.g. [@BRZ]), $\Sigma$ denotes the [*radial interior*]{} of [$\operatorname{Q}$]{}, i.e., $\Sigma={\{x\in {\ensuremath{\operatorname{Q}}}\colon \sup_{{n\in\omega}}|x(n)|<1\}}$. However, there is an auto-homeomorphism of ${\ensuremath{\operatorname{Q}}}$ which maps the pseudoboundary onto the radial interior.
[^4]: In [@S], $X$ is assumed to be a subspace of a compact metric space, but the proof is valid without this assumption. Moreover, it is also proved in [@S Théorème 6] that if ${\operatorname{Bd}}_H(X)$ is absolutely Borel (i.e., Borel in its completion) then $X$ is the union of a Polish subset and a $\sigma$-compact subset.
[^5]: ${\mathfrak E}({\operatorname{Cld}}(X)) = {\mathfrak B}({\operatorname{Cld}}_H(X))$ for every totally bounded separable metric space $X$ (cf. [@Beer Hess’ Theorem 6.5.14 with Theorem 3.2.3]).
|
---
abstract: 'An LRS Bianchi-I space-time model is studied with constant Hubble parameter in $f(R,T)=R+2\lambda T$ gravity. Although a single (primary) matter source is considered, an additional matter appears due to the coupling between matter and $f(R,T)$ gravity. The constraints are obtained for a realistic cosmological scenario, i.e., one obeying the null and weak energy conditions. The solutions are also extended to the case of a scalar field (normal or phantom) model, and it is found that the model is consistent with a phantom scalar field only. The coupled matter also acts as phantom matter. The study shows that if one expects an accelerating universe from an anisotropic model, then the solutions become physically relevant only at late times when the universe enters into an accelerated phase. Placing some observational bounds on the present equation of state of dark energy, $\omega_0$, the behavior of $\omega(z)$ is depicted, which shows that the phantom field has started dominating very recently, somewhere between $0.2\lesssim z\lesssim0.5$.'
author:
- Vijay Singh
- Aroonkumar Beesham
title: 'LRS Bianchi I model with constant expansion rate in $f(R,T)$ gravity'
---
Introduction {#s:intro}
============
[@Harkoetal2011] proposed a general non-minimal coupling between matter and geometry in the framework of an effective gravitational Lagrangian consisting of an arbitrary function of the Ricci scalar $R$, and the trace $T$ of the energy-momentum tensor, and introduced $f(R,T)$ gravitational theory. An extra acceleration in $f(R,T)$ gravity results not only from a geometrical contribution, but also from the matter content. This extraordinary phenomena of $f(R,T)$ gravity may provide some significance signatures and effects which could distinguish and discriminate between various gravitational models. Therefore, this theory has attracted many researchers to explore different aspects of cosmology and astrophysics in isotropic and as well as in anisotropic space-times (see for example [@Jamiletal2012; @Reddyetal2013; @Azizi2013; @Alvarengaetal2013jmp4; @Alvarengaetal2013prd87; @Sharifetal2013epjp128; @Chakraborty2013; @Houndjoetal2013cjp91; @Pasquaetal2013; @RamPriyankaASS2013; @SinghSinghASS2015; @Baffouetal2015; @BaffouetalPRD2018; @SantosFerstMPLA2015; @NoureenetalEPJC2015; @ShamirEPJC2015; @SinghSingh2016; @AlhamzawiAlhamzawiIJMPD2016; @Yousafetal2016; @Alvesetal2016; @ZubairetalEPJC2016; @SofuogluASS2016; @MomenietalASS2016; @DasetalEPJC2016; @SalehiAftabiJHEP2016; @SahooetalEPJC2018; @MoraesetalEPJC2018; @SinghBeeshamEPJC2018; @SrivastavaSinghASS2018; @SharifAnwarASS2018; @TiwariBeeshamASS2018; @ShabaniZiaieEPJC2018; @RajabiNozariPRD2017; @MoraesetalIJMPD2019; @DebetalMNRAS2019; @LobatoetalEPJP2019; @BaffouetalPRD2018; @DebetalMNRAS2019; @TretyakovEPJC2018; @ElizaldeKhurshudyanPRD2018; @OrdinesCarlsonPRD2019; @MauryaTello-OrtizbJCAP2019; @DebetalMNRAS2019; @EsmaeiliJHEP2018] and references therein).
The first work on any anisotropic model in $f(R,T)$ gravity was done by [@AdhavASS2012] in LRS Bianchi I space-time. The author considered a particular form of $f(R,T)=R+2\lambda T$, where $\lambda$ is an arbitrary constant, and obtained the solutions by assuming a constant expansion rate. This assumption corresponds to the accelerating expansion of the universe. A serious shortcoming in his work is, due to an incorrect field equation, the solutions are mathematically and physically invalid. Our purpose in this paper is to address the correct field equations and explore the geometrical and physical properties of this model.
If one considers any matter in this theory, then due to the coupling between matter and $f(R,T)$ gravity, some extra terms appear on the right hand side of the field equations. These terms must be treated as matter as well and may be called coupled matter. It may act either as a perfect fluid or DE. Therefore, the effective matter in these models is a sum of primary matter and coupled matter. One may ensure a physically viable scenario by demanding the weak energy condition (WEC)[^1] for the primary matter and coupled matter. We have followed this criteria in our recent study [@SinghBeeshamEPJP2020]. We shall follow this criteria in present work too.
In addition to that we also study the physical viability of the model through the energy conditions, and find the constraints for a realistic cosmological scenario. Further, we extend our solutions to the case of a normal/phantom scalar field model to determine the nature of the matter. We also study the behavior of the matter by using the present values of the equation of state parameter consistent with observational constraints. We also examine the role of $f(R,T)$ gravity in this study.
The work is organised as follows. In Sec. 2 we show that the geometrical behavior of the model reported by [@AdhavASS2012] is independent of $f(R,T)$ gravity. In Sec. 3 we present the correct field equations for an LRS Bianchi I spacetime model in $f(R,T)=R+2f(T)$ gravity, and find the constraints for a realistic physical scenario ensuring positivity of the energy density. The scalar field model is considered in Sec. 3.1 followed by a study of the behavior of the effective matter through the equation of state parameter. The findings are summarised in Sec. 4. Note that the equation numbers in round brackets throughout our discussion refer to the equations of our work, whereas the equation numbers in round brackets including the section numbers and some points number mentioned within inverted commas refer to the Ref. [@AdhavASS2012].
Although we consider a single matter source in our model, an extra matter source appears in $f(R,T)=R+2\lambda T$ gravity due to the coupling terms of $T$ with the matter on the right hand side of the field equations. First, we obtain the constraints for the primary matter to obey the weak energy condition. This not only ensures a realistic cosmological scenario, but also helps to identify the various evolutionary phases of the universe, specifically, it distinguishes between early inflation and late time acceleration. Anticipating the dual nature of primary matter, we replace it with a scalar field to know the actual nature (perfect fluid, quintessence or phantom) of primary matter. Further, we analyze the behavior of coupled matter in the various evolutionary phases. In this way, we determine which matter source causes inflation, deceleration and late time acceleration, and what is the role of $f(R,T)$ gravity in the course of evolution of the universe.
The solutions in general relativity
===================================
In this section we show that the geometrical behaviour in points “(i)–(iv)" addressed by [@AdhavASS2012] in section “4", is independent from $f(R,T)$ gravity, and it remains similar to that in general relativity (GR).
A spatially homogenous and anisotropic locally-rotationally-symmetric (LRS) Bianchi I space-time metric is given by $$ds^{2} =dt^{2}-A^2(t)dx^2-B^2(t)(dy^2+dz^2),$$ where $A$ and $B$ are the scale factors, and are functions of cosmic time $t$.
The average scale factor for the metric (1) is defined as $$a=(AB^2)^{\frac{1}{3}}.$$ The average Hubble parameter (average expansion rate) $H$, which is the generalization of the Hubble parameter in the isotropic case, is given by $$H=\frac{1}{3}\left(\frac{\dot A}{A}+2\frac{\dot B}{B}\right).$$ Consider the energy-momentum tensor $$T_{\mu\nu}=(\rho+p)u_\mu u_\nu-pg_{\mu\nu},$$ where $\rho$ is the energy density and $p$ is the thermodynamical pressure of the matter. In comoving coordinates $u^\mu=\delta_0^\mu$, where $u_\mu$ is the four-velocity of the fluid which satisfies the condition $u_\mu u^\nu=1$.
The Einstein field equations read as $$R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}=T_{\mu\nu},$$ where the system of units $8\pi G=1=c$ are used.
The above field equations for the metric (1) and energy-momentum tensor (4), yield $$\begin{aligned}
\left(\frac{\dot B}{B}\right)^2+2\frac{\dot A\dot B}{A B}&=&\rho,\\
\left(\frac{\dot B}{B}\right)^2+2\frac{\ddot B}{ B}&=&-p,\\
\frac{\ddot A}{A}+\frac{\ddot B}{B}+\frac{\dot A \dot B}{AB}&=&-p.\end{aligned}$$ where a dot denotes the derivative with respect to cosmic time $t$. These are three independent equations with four unknowns, namely $A$, $B$, $\rho$ and $p$. Therefore, one requires a supplementary constraint to find the exact solutions of the field equations. [@AdhavASS2012] considered the case of a constant expansion rate $$H=k,$$ where $k>0$ is a constant. Since $H=\dot a/a$, the average scale factor evolves as $$a(t)=a_0e^{k t},$$ where $a_0$ is an integration constant.
The deceleration parameter, $q=-a\ddot a/\dot a^2=-1-\dot H/ H^2$ takes a constant value $$q=-1,$$ which corresponds to an accelerating expansion of the universe.
From (7) and (8), one has $$\frac{\dot A}{A} -\frac{\dot B}{B}=\frac{\beta}{AB^2},$$ where $\beta$ is a constant of integration.
From (3) and (12), by the use of (9), one obtains $$\begin{aligned}
A&=&c_1e^{kt-\frac{2\beta e^{-3kt}}{9k}},\\
B&=&c_1e^{kt+\frac{\beta e^{-3kt}}{9k}},\end{aligned}$$ where $c_1$ is a constant of integration and another integration constant is taken as unity without any loss of generality. In section “3", namely, “physical properties", [@AdhavASS2012] worked out some kinematical parameters and also obtained the expressions for the energy density and pressure. The author in his conclusion mentioned that the scale factors are the solutions of the LRS Bianchi I model in $f(R,T)$ gravity. But here one can see that the scale factors (13) and (14) are obtained in GR. Hence, the behaviour of the kinematical parameters, namely, the expansion scalar, shear scalar, and the anisotropy parameter discussed by Adhav remain is independent of $f(R,T)$ gravity and remain the same as in GR. In our recent work ([@SinghBeeshamGRG2019]), we have explored the features of these parameters.
The main issue in Adhav’s paper is, the expressions for the energy density and pressure are incorrect due to a wrong field equation, namely, equation number “(2.5)". Therefore, the solutions obtained by the author are invalid mathematically. In the next section, we shall reformulate this model. We find the constraints for a physically realistic cosmological scenario and explore the physical behavior of the model. We shall also extend the solutions to a scalar field model.
The solutions in $f(R,T)$ gravity
=================================
In Sect. 2, $\rho$ and $p$, respectively, are the effective energy density and pressure in the model of GR. When one we considers the energy-momentum tensor (4) in $f(R,T)$ gravity then $\rho$ and $p$ no longer correspond to the effective energy density and pressure. As aforementioned in the introduction that due to the coupling between matter and trace some extra terms appear in the right hand side of the field equation in $f(R,T)$ gravity. These terms can also be treated as matter. We may call it coupled matter. The matter given by the energy-momentum tensor (4) should be treated as the primary matter. Let us replace $\rho$ and $p$ with $\rho_m$ and $p_m$, respectively, which represent the energy density and pressure of primary matter. The notations for coupled matter are defined in Sect. 3.2. In this way, $\rho=\rho_m+\rho_f$ and $p=p_m+p_f$ again become the effective energy density and pressure in our model.
The field equations in $f(R, T)=R+2f(T)$ gravity with the system of units $8\pi G=1=c$, are obtained as $$R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}=T_{\mu\nu}+2 (T_{\mu\nu}+p g_{\mu\nu})f'(T)+f(T) g_{\mu\nu}.$$ Adhav (2012) considered the simplest case $f(T)=\lambda T$, i.e., $f(R,T)=R+2\lambda T$, where $T=g^{\mu\nu}T_{\mu\nu}=\rho_m-3p_m$ for which the field equations (15) reduce to $$R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}=(1+2\lambda) T_{\mu\nu}+\lambda(\rho_m-p_m) g_{\mu\nu}.$$ The above field equations for the metric (1), yield $$\begin{aligned}
\left(\frac{\dot B}{B}\right)^2+2\frac{\dot A\dot B}{A B}&=&(1+3\lambda)\rho_m-\lambda p_m,\\
2\frac{\ddot B}{B}+\left(\frac{\dot B}{B}\right)^2&=&-(1+3\lambda)p_m+\lambda \rho_m,\\
\frac{\ddot A}{A}+\frac{\ddot B}{B}+\frac{\dot A \dot B}{AB}&=&-(1+3\lambda)p_m+\lambda \rho_m.\end{aligned}$$ It is to be noted that the last term of equation “(2.5)" in Ahdav’s work is “$+2p\lambda$" which should be “$-\lambda p$".
Using (13), (14) in (17) and (18), for $\lambda=-1/2$ and $\lambda=-1/4$, we obtain $$\begin{aligned}
\rho_m&=&\frac{3 k^2}{1+4 \lambda }-\frac{\beta ^2 e^{-6 k t}}{3(1+2 \lambda) },\\
p_m&=&-\frac{3 k^2}{1+4 \lambda }-\frac{\beta ^2 e^{-6 k t}}{3(1+2 \lambda)}.\end{aligned}$$ These are the correct expressions for the energy density and pressure which are different from those obtained by [@AdhavASS2012]. In both of these expressions the variable term decreases with time, consequently, the energy density and pressure increase with the cosmic evolution and attain a constant value $\rho_m=3 k^2/(4 \lambda +1)=-p_m$ as $t\to\infty$, while both physical quantities are infinite in the infinite past.
The energy density for any physically viable cosmological model must be positive. It is clear from (20) that $\rho_m$ can be positive always if $1+4\lambda>0$ and $1+2\lambda<0$, but it is not possible. Similarly, the models with $-1/2<\lambda<-1/4$ also become physically unrealistic as $\rho_m$ remains negative always in this case. However, the energy density can be positive for some restricted times under the constraints $$t\leq\frac{1}{k}\ln\left[\frac{(1+4\lambda) \beta^2}{9(1+2\lambda) k^2}\right]^\frac{1}{6}\;\; \text{if}\;\; \lambda<-\frac{1}{2},$$ and $$t\geq\frac{1}{k}\ln\left[\frac{(1+4\lambda) \beta^2}{9(1+2\lambda) k^2}\right]^\frac{1}{6} \;\; \text{if}\;\; \lambda>-\frac{1}{4}.$$ The assumption (9)–(11) corresponds to an accelerated universe ($q=-1$). However, the acceleration may be an early inflation or a late time acceleration. Since the model with $\lambda<-1/2$ is physically viable only during early evolution, the acceleration must be an early inflation, whereas the model with $\lambda>-1/4$ is physically viable only at late times, the acceleration must be the present accelerating phase.
The main objective now remains to identify the nature of matter. The EoS parameter which is defined as $\omega_m=p_m/\rho_\mathfrak{m}$, gives $$\omega_m=-1+\frac{2 }{1+\gamma e^{6 \beta t}},$$ where $\gamma=-9(1+2\lambda)k^2/(1+4\lambda)\beta^2$. The expression (24) looks similar to the EoS parameter of the effective matter in GR [@SinghBeeshamGRG2019]. The only difference is that here $\gamma$ contains a term $\lambda$ of $f(R,T)=R+2\lambda T$ gravity. The constraints obtained in (22) and (23) ensure the positivity of $\gamma$. Since $\omega_m\to1$ as $t\to-\infty$, and $\omega_m\to-1$ as $t\to\infty$, the matter behaves as stiff matter in the infinite past while it plays the role of a cosmological constant at late times. We see that $\omega_m$ diverges at a time $t=\frac{1}{k}\ln\left[\frac{(1+4\lambda) \beta^2}{9(1+2\lambda) k^2}\right]^\frac{1}{6}$, so it cannot be used to depict the behavior of the matter during intermediate evolution.
The early and late behavior of primary matter in our model matches with the characteristics of a scalar field. Due to the domination of kinetic energy over the scalar potential at early times, the scalar field acts like stiff matter. A scalar field with a self-interacting potential, due to the domination of the potential term over the kinetic term, gives rise to a negative pressure for driving super fast expansion during inflation. When the field enters into a regime in which the potential energy once again takes over from the kinetic energy, it exerts the same stress as a cosmological constant at late times, which happens however with a different energy density (in comparison to inflation). Therefore, let us substitute the primary matter with a scalar field (quintessence or phantom) for the further investigation.
Scalar field model {#sec:5}
------------------
The energy density and pressure of a minimally coupled normal ($\epsilon=1$) or phantom ($\epsilon=-1$) scalar field, $\phi$ with self-interacting potential, $V(\phi)$ are, respectively, given by $$\begin{aligned}
\rho_\phi&=&\frac{1}{2}\epsilon\dot\phi^2+V(\phi),\\
p_\phi&=&\frac{1}{2}\epsilon\dot\phi^2-V(\phi).\end{aligned}$$ Replacing $\rho_m$ with $\rho_\phi$ and $p_m$ with $p_\phi$, and using (25) and (26) in (20) and (21), the kinetic energy and scalar potential, respectively, are obtained as $$\begin{aligned}
\frac{1}{2}\epsilon\dot\phi^2&=&-\frac{\beta^2 e^{-6 k t}}{3(1+2\lambda)},\\
V(t)&=&\frac{3k^2}{1+4\lambda}.\end{aligned}$$ From (27), for reality of the solutions we must have $\lambda<-1/2$ if $\epsilon=1$ and $\lambda>-1/2$ if $\epsilon=-1$. It is to be note that the requirement of positive kinetic energy is equivalent to obeying the null energy condition (NEC)[^2]. Since we have already ensured the positivity of energy density under the constraints (22) and (23), the WEC is satisfied for $\lambda<-1/2$ and $\lambda>-1/2$. In addition, the scalar potential also must be positive for a physically viable model, which is possible only for $\lambda>-1/4$. Hence, the model is consistent with a phantom scalar field only. It is to be noted that having a negative scalar potential is equivalent to violating the dominant energy condition (DEC)[^3]. Furthermore, the energy density for $\lambda>-1/4$, is positive only after a time (23), therefore, the model accommodates a late time acceleration only and it excludes the possibility of early inflation.
On integration (27), we get $$\phi=\phi_0\pm\left[\frac{2}{3(1+2\lambda)}\right]^\frac{1}{2}\frac{\beta e^{-3 k t}}{3k},$$ where $\phi_0$ is a constant of integration. Only the positive sign is compatible for physical consistency, so we proceed further with positive sign only.
The energy density and pressure of the phantom scalar field are given by (20) and (21), which can be expressed in terms of red shift, $z$ via the relation $a=a_0/(1+z)$ as $$\begin{aligned}
\rho_\phi&=&=\frac{3 k^2}{4 \lambda +1}-\frac{\beta ^2 (1+z)^{6}}{6 \lambda +3},\\
p_\phi&=&-\frac{3 k^2}{4 \lambda +1}-\frac{\beta ^2 (1+z)^{6}}{6 \lambda +3},\end{aligned}$$ respectively, where we have assumed the present scale factor to be unity, i.e., $a_0=1$.
Similarly, the constraints (22) and (23), respectively, can be expressed as $$\begin{aligned}
z&\geq&\gamma^\frac{1}{6}-1\;\; \text{for}\;\; \lambda<-\frac{1}{2},\\
z&\leq&\gamma^\frac{1}{6}-1\;\; \text{for}\;\; \lambda>-\frac{1}{4}.\end{aligned}$$ The EoS parameter (24), takes the form $$\omega_\phi=\left[-1+\frac{2 }{1+\gamma (1+z)^{-6}}\right]^{-1}.$$ As aforesaid, though $\gamma$ contains the parameter $\lambda$ of $f(R,T)=R+2\lambda$ gravity, but being a single parameter expression the above EoS is identical to the EoS of the effective matter in GR [@SinghBeeshamGRG2019]. Hence, the primary matter in this model behaves similar to the effective matter in GR as shown in Fig. 1. An interesting fact here is that though Fig. 1 is identical to one in Ref. [@SinghBeeshamGRG2019] but it represents a different matter content. In present context it describes a part of matter while in the model of GR it describes the effective matter. This difference can also be seen in the expression of scalar field (29) which involve the parameter $\lambda$ of $f(R,T)=R+2\lambda T$ gravity. So we can examine how $f(R,T)$ gravity affect the evolution of scalar field by analyzing its variation against different values of $\lambda$. We pursue further with the approach that we have adopted in our recent study ([@SinghBeeshamGRG2019]).
The present value of the EoS parameter is $$\omega_\phi(z=0)=\frac{1+\gamma}{1-\gamma}.$$ Combined results from cosmic microwave background (CMB) experiments with large scale structure (LSS) data, the $H(z)$ measurement from the Hubble Space Telescope (HST) and luminosity measurements of Type Ia Supernovae (SNe Ia), put the following constraints on the EoS: $-2.68 <\omega_0< -0.78$ [@MelchiorrietalPRD2003]. These bounds become more tight, i.e., $-1.45 <\omega_0< -0.74$ [@HannestadMorstellPRD2002], when the Wilkinson Microwave Anisotropy Probe (WMAP) data is included (also see [@Alametal0311364]; [@AlcanizPRD0401231]). For these observational limits from (35), we get $\gamma>2.19$ for the former bounds, and $\gamma>5.44$ for the latter. Now we can depict the profile of the EoS parameter for some values of $\gamma$ consistent with these observational outcomes.
![$\omega_\phi$ [***versus***]{} $z$ for different values of $\gamma$.[]{data-label="fig:1"}](fig1.eps){width="8cm"}
Figure 1 plots the behavior of the EoS parameter against redshift, under the constraint (33), for some values of $\gamma>2.19$. We see that $\omega_\phi<-1$, which confirms the theoretical outcome that the matter in the present model is of phantom type and becomes the cosmological constant in future.
The phantom scalar field (29) can be given as $$\phi=\phi_0\pm\left[\frac{2}{3\gamma(1+4\lambda)}\right]^\frac{1}{2}(1+z)^{3}.$$
![$\phi(z)$ [***versus***]{} $z$ for different values of $\lambda$ with $\gamma=5$ and $\phi_0=1$.[]{data-label="fig:2"}](fig2.eps){width="8cm"}
The evolution of the scalar field against $z$ for some values of $\lambda$ with $\gamma=5$ (this value is consistent with the observational bounds mentioned above) and $\phi_0=0$ is shown in Fig. 2. The scalar field decreases from an infinite value with the evolution of the universe, and attains a finite minimum value at late times. If $\phi_0=0$, the scalar field vanishes at late times. The flat potential (28) can be identified as a cosmological constant. Moreover, if $\beta=0$ then $\phi=\phi_0$ and $\rho_\phi=3k^2=-p_\phi$, which essentially corresponds to a cosmological constant. If $\lambda=0$, the solutions reduces to the model one in GR [@SinghBeeshamGRG2019]. One may also readily verify that the standard de Sitter solutions for a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) model of GR are recovered when $\beta=0$ and $\lambda=0$.
The behavior of coupled matter
------------------------------
Separating the energy densities and pressures of primary matter and coupled matter the field equations (17)–(19) can be written as $$\begin{aligned}
\left(\frac{\dot B}{B}\right)^2+2\frac{\dot A\dot B}{A B}&=&\rho_\mathfrak{m}+\rho_f,\\
2\frac{\ddot B}{ B}+\left(\frac{\dot B}{B}\right)^2&=&-(p_\mathfrak{m}+p_f),\\
\frac{\ddot A}{A}+\frac{\ddot B}{B}+\frac{\dot A \dot B}{AB}&=&-(p_\mathfrak{m}+p_f),\end{aligned}$$ where $\rho_f=\lambda(3\rho_\mathfrak{m}-p_\mathfrak{m})$ and $p_f=\lambda(3p_\mathfrak{m}-\rho_\mathfrak{m})$, are the energy density and pressure of coupled matter, which are obtained as $$\begin{aligned}
\rho_f&=&\frac{12 \lambda k^2 }{4 \lambda +1}-\frac{2 \lambda \beta ^2 e^{-6 k t}}{6 \lambda +3},\\
p_f&=&-\frac{12 \lambda k^2}{4 \lambda +1}-\frac{2 \lambda \beta ^2 e^{-6 k t}}{6 \lambda +3}.\end{aligned}$$ Although $\rho_f$ remains always positive for $-1/2<\lambda <-1/4$, but the energy density of primary matter becomes negative for these values, we exclude this case. For $\lambda<-1/2$, $\rho_f$ becomes negative at early times, we exclude this case too. Similarly, when $-1/4<\lambda<0$, it becomes negative at late times. Notwithstanding, $\rho_f$ for $\lambda>0$ is positive at late times. Hence, the model in this case provides a realistic scenario. The EoS parameter, $\omega_f=p_f/\rho_f$ diverges at $t=t_\star$, so it is not worthwhile using it to depict the behavior of coupled matter. Therefore, we shall study the nature of coupled matter through the energy conditions. We require $$\begin{aligned}
\rho_f+p_f&=&-\frac{4 \lambda\beta ^2 e^{-6 k t}}{6 \lambda +3},\\
\rho_f-p_f&=&\frac{24 \lambda k^2 }{4 \lambda +1}.\end{aligned}$$ The NEC and DEC can be satisfied, respectively, for $$\begin{aligned}
-\frac{1}{2}&<&\lambda <0, \\
\lambda &<&-\frac{1}{4} ,\;\; \text{or}\;\; \lambda>0.\end{aligned}$$ Since $\rho_f$ is positive only for $\lambda>0$, therefore, the coupled matter violates the NEC but holds the DEC. Hence, the behavior of coupled matter is similar to the primary matter, i.e., it also behaves as phantom DE. Thus, the behavior of coupled matter also shows that the model is viable to describe a late time cosmic acceleration in presence of phantom matter.
Conclusion
==========
[@AdhavASS2012] studied an LRS Bianchi I model with constant expansion rate in $f(R,T)=R+2\lambda T$ gravity. The solutions obtained by the author are mathematically and physically invalid due to an incorrect field equation. We have reconsidered this model in the present paper. We have also extended the solutions to a scalar field (quintessence or phantom) model. While [@AdhavASS2012] discussed only the kinematical behavior of the model, we have also explored the physical properties in detail keeping the physical viability of the solutions at the center. Notwithstanding, a wrong field equation, the behavior of the geometrical parameters is not affected. Moreover, the kinematical parameters in such formulation do not depend on $f(R,T)$ gravity. Firstly, we have shown that the geometrical behavior remains the same as in GR. We have pointed out that [@AdhavASS2012] misunderstood the time of origin of the universe. The author discussed the evolution from $t=0$ to $t\to\infty$. The author probably understood the origin of the universe at $t=0$, while the model has an infinite past. For details, the readers may refer our recent works [@SinghBeeshamGRG2019; @SinghBeeshamEPJP2020]. The solutions in $f(R,T)$ gravity model are valid for all values of $\lambda$ except $\lambda\neq-1/2$ and $\lambda\neq-1/4$. The assumption of constant expansion rate gives a constant value of the deceleration parameter, $q=-1$. Consequently, the model can describe only an accelerating universe. However, the acceleration may be an early inflation or present acceleration. To ensure this we have obtained the constraints for a physically realistic scenario and found that the model is viable to describe only the present accelerating phase.
The evolution of the universe is governed by the effective matter. The behavior of the effective matter has already been studied by us elsewhere recently [@SinghBeeshamGRG2019]. The $f(R,T)$ gravity does not affect the behavior of the effective matter and it remains the same as one in GR. It is important to mention here that $f(R,T)$ gravity does not alter the behavior of effective matter in the formulations where the kinematical behaviour is fixed by some geometrical parameters. The effective matter in the present study thus also remains the same as in GR [@SinghBeeshamGRG2019]. In present study, an extra matter (different from the primary matter) appears due to the coupling between matter and $f(R,T)$ gravity. We have explored the characteristics of primary matter as well as of coupling matter.
The primary matter in this model acts similar to the effective matter in GR. When primary matter is replaced with a scalar field (normal or phantom) model, the model has been found consistent only with a phantom scalar field. The phantom field decreases with the cosmic evolution, while the scalar potential remains flat throughout the comic evolution. The scalar potential can be thought as cosmological constant. The model can also be consistent with normal scalar field, but the scalar potential becomes negative in that case which would be unrealistic. The coupled matter behaves similar to primary matter. The viable models are possible only for $\lambda>0$.
We have also examined the consistency of the behavior of primary matter with the the observational data by borrowing some current values of the EoS parameter from some observational outcomes. The dynamics of the EoS parameter supports the observational results and suggests that the phantom field has started dominating over the other energy contents somewhere between $0.2\lesssim z\lesssim0.5$. The scalar field model also evidences that if one demands an accelerating cosmic expansion from an anisotropic model, then the model represents a viable cosmological scenario (obeying NEC and WEC) only after a time when the universe enters into an accelerating phase.
It is to be noted that [@ShamirJETP2014] obtained solutions of the general Binachi I model with constant expansion rate in $f(R,T)=R+2\lambda$ gravity. Those solutions are also valid for late times only as the energy density is negative at early times. Hence, our results can also be interpreted within the general Binachi I spacetime model. We believe that only the kinematical behavior would be different, but the physical behavior will remain the same.
As the model is capable to explain the present cosmic acceleration without the use any hypothetical exotic matter, $f(R,T)=R+2\lambda T$ gravity can be a good alternative to GR.
@thebibliography@page=
Adhav, K.S.: **339**, 365–369 (2012)
Alam, U., Sahni, V., Saini, T.D., Starobinsky, A.A.: **354**, 275 (2004) arXiv:astro-ph/0311364
Alcaniz, J.S.: **69**, 083521 (2004) arXiv:astro-ph/0312424
Jamil, M., Momeni, D., Raza, M., Myrzakulov, R.: EPJC **72**, 1999 (2012) arXiv:gen-ph/1107.5807
Alhamzawi, A., Alhamzawi, R. Int. J. Mod. Phys. D **35**, 1650020 (2016)
Alvarenga, F.G., Cruz-Dombriz, A. de la, Houndjo, M.J.S., Rodrigues, M.E., Sáez-Gómez, D.: **87**, 103526 (2013) arXiv:gr-qc/1302.1866
Alvarenga, F.G., Houndjo, M.J.S., Monwanou, A.V., Oron, J.B.C.: JMP **4**, 130 (2013) arXiv:gr-qc/1205.4678
Alves, M.E.S., Moraes, P.H.R.S., Araujo, J.C.N. de, Malheiro, M.: **94**, 024032 (2016) arXiv:gr-qc/1604.03874
Azizi, T.: IJTP **52**, 3486 (2013) arXiv:gr-qc/1205.6957
Baffou, E.H., Kpadonou, A.V., Rodrigues, M.E., Houndjo, M.J.S., Tossa,J.: **356**, 173–180 (2015) arXiv:gr-qc/1312.7311
Baffou, E.H., Houndjo, M.J.S., Kanfon, D.A., Salako, I.G.: **98**, 124037 (2018) arXiv:gr-qc/1808.01917
Chakraborty, S.: Gen. Rel. Grav. **45**, 2039 (2013) arXiv:gen-ph/1212.3050
Deb, D., et al.: **485**, 5652 (2019) arXiv:gr-qc/1810.07678 (check)
Das, A., Rahaman, F., Guha, B.K., Ray, S. Eur. Phys. J. C **76**, 654 (2016) arXiv:gr-qc/1608.00566
Deb, D., Guha, B.K., Rahaman, F., Ray, S.: **97**, 084026 (2018) arXiv:gr-qc/1810.01409
Debnath, P.S.: Int. J. Geom. Meth. Mod. Phys. **16**, 1950005 (2019) arXiv:gr-qc/1907.02238
Elizalde, E., Khurshudyan, M.: **98**, 123525 (2018) arXiv:gr-qc/1811.11499
Esmaeili, F.M.: J. of High Energ. Phys., Gravit. and Cosmo. **4**, 716 (2018)
Harko, T., Lobo, F.S.N.,Nojiri, S.,Odintsov, S.D.: **84**, 024020 (2011) arXiv:gr-qc/1104.2669
Hannestad, S., Morstell, E.: **66**, 063508 (2002) arXiv:astro-ph/0205096
Houndjo, M.J.S., Batista, C.E.M., Campos, J.P., Piattella, O.F.: Can. J. Phys. **91**, 548 (2013) arXiv:gr-qc/1203.6084
Kumar, P., Singh, C.P.: **357**, 120 (2015)
Lobato, R.V., Carvalho, G.A., Martins, A.G. Moraes, P.H.R.S.: Eur. Phys. J. Plus **134**, 132 (2019) arXiv:gr-qc/1803.08630
Maurya, S.K., Tello-Ortizb, F.: **28**, 1950056, (2019) arXiv:gr-qc/1905.13519
Melchiorri, A., Mersini, L., Odman, C.J., Trodden, M.: **68** 043509 (2003) arXiv:astro-ph/021152
Momeni, D., Moraes, P.H.R.S., Myrzakulov, R.: **361**, 228 (2016) arXiv:gr-qc/1512.04755
Moraes, P.H.R.S., Correa, R.A.C., Ribeiro, G.: Eur. Phys. J. C **78**, 192 (2018) arXiv:gr-qc/1606.07045
Moraes, P.H.R.S., Paula, W.de, Correa, R.A.C.: Int. J. Mod. Phys. D **28**, 1950098 (2019) arXiv:gr-qc/1710.07680
Noureen, I., Zubair, M., Bhatti, A.A., Abbas, G.: Eur. Phys. J. C **75**, 323 (2015) arXiv:gr-qc/1504.01251
Ordines, T.M., Carlson, E.D.: **99**, 104052 (2019) arXiv:gr-qc/1902.05858
Pasqua, A., Chattopadhyay, S., Khomenkoc,I.: Can. J. Phys. **91**, 632 (2013) arXiv:gen-ph1305.1873
Rajabi, F. Nozari, K.: **96**, 084061 (2017) arXiv:gr-qc/1710.01910
Ram, S., Priyanka: **347**, 389 (2013)
Reddy, D.R.K., Naidu, R.L., Satyanarayana, B.: IJTP **51**, 3222 (2013)
Tiwari, R.K., Beesham, A.: **363**, 234 (2018)
Tretyakov, P.V.: Eur. Phys. J. C **78**, 896 (2018) arXiv:gr-qc/1810.11313 (check)
Salehi, A., Aftabi, S.: J. High Energ. Phys. **09**, 140 (2016) arXiv:gr-qc/1502.04507
Sahoo, P.K., Moraes, P.H.R.S., Sahoo, P.: Eur. Phys. J. C **78**, 46 (2018) arXiv:gr-qc/1709.07774
Santos, A.F., Ferst, C.J.: Mod. Phys. Lett. A **30**, 1550214 (2015)
Shabani, H., Farhoudi, M.: **88**, 044048 (2013) arXiv:gr-qc/1306.3164
Sharif, M., Rani, S., Myrzakulov, R.: Eur. Phys. J. Plus **128**, 123 (2013) arXiv:gr-qc/1210.2714
Singh, C.P., Singh, V.: **356**, 153 (2015)
Singh, V., Singh,C.P.: Int. J. Theor. Phys. **55**, 1257 (2016)
Singh, C.P., Singh, V.: Gen. Relativ. Grav. **46**, 1696 (2014)
Shamir, M.F.: Eur. Phys. J. C **75**, 354 (2015) arXiv:gen-ph/1507.08175
Sofuoglu, D.: **361**, 12 (2016)
Singh, V., Beesham, A.: Eur. Phys. J. C **78**, 564 (2018)
Sharif, M., Anwar, A.: **363**, 123 (2018)
Shabani, H., Ziaie, A.H.: Eur. Phys. J. C **78**, 397 (2018) arXiv:gr-qc/1708.07874
Singh, V., Beesham, A.: Int. J. Mod. Phys. D **28**, 1950056 (2019)
Singh, V., Beesham, A.: Gen. Relativ. Grav. **51**, 166 (2019)
Singh, V., Beesham, A.: Eur. Phys. J. Plus****, DOI: 10.1140/epjp/s13360-020-00314-x (2020)
Shamir, M.F.: J. Exp. Theor. Phys. **119**, 242 (2014)
Srivastava, M., Singh, C.P.: **363**, 117 (2018)
Yousaf, Z., Bamba, K., Bhatti, M.Z.: **93**, 124048 (2016) arXiv:gr-qc/1606.00147
Zubair, M., Waheed, S., Ahmad, Y.: Eur. Phys. J. C **76**, 444 (2016) arXiv:gr-qc/1607.05998
Melchiorri, A. Mersini, L., Odman, C.J., Trodden, M. **68**, 043509 (2003) arXiv:astro-ph/0211522
Hannestad, S., Morstell, E. **66**, 063508 (2002)
Hannestad, S., Morstell, E. **69**, 083521 (2004) arXiv:astro-ph/0312424
[^1]: $\rho\geq0$, $\rho+p\geq0$
[^2]: $\rho+p\geq0$
[^3]: $|\rho|\geq p$ or $\rho+p\geq0$ and $\rho-p\geq0$
|
---
abstract: 'Speaker separation refers to isolating speech of interest in a multi-talker environment. Most methods apply real-valued Time-Frequency (T-F) masks to the mixture Short-Time Fourier Transform (STFT) to reconstruct the clean speech. Hence there is an unavoidable mismatch between the phase of the reconstruction and the original phase of the clean speech. In this paper, we propose a simple yet effective phase estimation network that predicts the phase of the clean speech based on a T-F mask predicted by a chimera++ network. To overcome the label-permutation problem for both the T-F mask and the phase, we propose a mask-dependent permutation invariant training (PIT) criterion to select the phase signal based on the loss from the T-F mask prediction. We also propose an Inverse Mask Weighted Loss Function for phase prediction to focus the model on the T-F regions in which the phase is more difficult to predict. Results on the WSJ0-2mix dataset show that the phase estimation network achieves comparable performance to models that use iterative phase reconstruction or end-to-end time-domain loss functions, but in a more straightforward manner.'
address: |
City University of New York\
365 Fifth Avenue, New York, NY 10016, USA\
bibliography:
- 'refs19.bib'
title: 'Mask-dependent Phase Estimation for Monaural Speaker Separation'
---
Speech separation, phase estimation, permutation invariant training, chimera++, deep learning
Introduction {#sec:intro}
============
Recently, many deep-learning approaches have been proposed for the monaural multi-talker speech separation problem. [@hershey2016deep] proposed a deep clustering method that projects each time-frequency bin to a high-dimensional vector space using a bi-direction long short-term memory (BLSTM) network. T-F masks can be generated by clustering these vectors using the k-means algorithm. While deep clustering can separate mixtures when the number of speakers is unknown without changing the model architecture, it produces binary time-frequency masks, which may hurt the intelligibility of the speech. [@erdogan2015phase] showed that phase-sensitive masks perform better than ideal binary masks on the speech enhancement task. To overcome the label-permutation problem in multi-talker speech separation, [@yu2017permutation] proposed the permutation invariant training (PIT) criterion which lets the deep neural network select the assignment of output sources to ground truth sources using the lowest loss value among all permutations. [@luo2017deep] combined deep clustering with a mask-estimation model in a single hybrid network, called the chimera network. Their results show that adding deep clustering leads to better mask predictions, thus achieving better separation performance. [@wang2018alternative] made slight modifications to the original chimera network and developed several alternative loss functions for both deep clustering and mask-inference outputs. The new network (dubbed “chimera++”) boosted separation performance using a much simpler architecture.
While there have also been many recent developments in mask-based separation approaches, there are two main issues remaining to be solved:
- Since most masks are restricted to be between 0 and 1, there will be an unavoidable error if the magnitude of the clean speech is greater than that of the mixture due to phase cancellation.
- Most of these methods utilize the phase of the mixture as that of the separated speech, which causes an unavoidable phase difference error.
The phase-sensitive mask (PSM) was introduced [@erdogan2015phase] to decrease the impact of such phase differences. The PSM as a mask prediction target is defined as: $$M_{\text{PSM}} = \frac{|S|}{|X|} \odot \cos{(\angle{S}-\angle{X})},$$ where $S$ and $X$ are the STFT of the clean speech and the speech mixture respectively at any particular time-frequency point. By applying the PSM to the mixture, the real component of the estimated STFT is close to that of the ground truth.
Besides the PSM, [@williamson2016complex] introduced the complex ratio mask (cRM) to enhance both the magnitude and the phase spectra by operating in the complex domain. The real component of the cRM is equivalent to the PSM. However, [@williamson2017speech] showed that the imaginary component is difficult to predict directly from the noisy magnitude due to the randomness of the phase pattern.
To estimate the phase of each separated source, [@wang2018alternative] started with the estimated magnitude and the noisy phase and jointly reconstructed the phase of all sources using the multiple-input spectrogram inversion (MISI) algorithm [@gunawan2010iterative]. Later, [@wang2018end] unfolded the iterations in the MISI algorithm into separate STFT and iSTFT layers on top of a chimera++ network and trained the network to minimize the error in the time domain. Their results showed that it is possible to predict the phase of separated sources from a magnitude estimate and the noisy phase.
Instead of using an iterative STFT and iSTFT to reconstruct the phase, the current paper proposes an approach to reconstruct the phase directly from the magnitude estimate and the noisy phase. The problem can thus be formulated as estimating the magnitude and phase of each source from the noisy STFT, $X$ $$\label{eq:formulation}
S_c = |X| \odot M_c \odot \cos{\theta_c} + j |X| \odot M_c \odot \sin{\theta_c},$$ where $|X|$ is the noisy magnitude, $M_c$ and $\theta_c$ are the mask and phase estimates for source $c$, respectively.
Some studies have tried to predict the mask and phase directly without iterative algorithms. In [@ephrat2018looking] and [@afouras2018conversation], the authors utilized visual feature to drive the model to predict the corresponding mask and phase. However, some datasets may not contain visual data, or the video may not capture the speaker of interest. Though the PIT criterion solves the label-permutation problem for mask estimates, integrating PIT with both mask and phase estimates is still an open problem. Thus, it is still essential to figure out how to apply PIT criterion in phase estimation with only the audio data.
Another problem in phase estimation is that its difficulty varies across regions. Since the noisy STFT $X = \sum_c{S_c}$, where $S_c$ represents the clean STFT for source $c$, if a T-F bin is only dominated by one speaker (i.e., the mask value is close to one), the phase difference between noisy and clean is close to zero. In regions where the mask is not close to one, the phase difference is influenced by the magnitude and phase of all significant sources, and is thus more difficult to estimate.
Hence this study proposes a joint-training algorithm that estimates the ideal ratio mask (IRM) by using a chimera++ network and then estimates the clean phase based on the mask estimate and the noisy phase for each source. The label permutation problem is solved by using a PIT criterion based only on the estimated masks, with the phase estimated after this matching. Three different weighed loss functions are proposed to compare the influence of different T-F regions over the training of the phase estimation model.
Chimera++ Network {#sec:chimera++}
=================
This section describes the architecture of the chimera++ network and its loss function, which is also incorporated into our algorithm. The left half of Figure \[fig:phase-net\] shows the architecture of the chimera++ network. Each time-frequency bin is projected to a D-dimensional vector $v_i \in \mathbb{R}^{1\times D}$ via a deep clustering layer[@hershey2016deep], where $i$ corresponds to a particular pair of time and frequency indices. Each T-F bin has a one-hot label vector $y_i \in \mathbb{R}^{1 \times C}$ indicating the speaker among $C$ speakers who dominates this bin. Stacking all embedding vectors and labels produces the embedding matrix $V \in R^{TF\times D}$ and the label matrix $Y \in R^{TF\times C}$. The objective of deep clustering is to group together the embedding vectors of T-F bins from the same speaker and make those from different speakers orthogonal. The loss function is defined as: $$\begin{aligned}
L_{\text{DC, classic}} &= &{\left\lVertVV^T - YY^T \right\rVert}_F^2 $$ [@wang2018alternative] showed reducing the influence of silence T-F bins benefits training the deep clustering network. They introduced binary voice activity weights $W_{\text{VA}}$ to the loss function: $$\begin{aligned}
L_{\text{DC, classic, W}} &= & {\left\lVertW^{\frac{1}{2}}(VV^T - YY^T)W^{\frac{1}{2}}\right\rVert}_F^2 \end{aligned}$$ In terms of the loss function for the mask inference layer, [@wang2018alternative] recommends using a loss function based on the truncated phase-sensitive spectrum approximation (tPSA), defined as: $$\begin{gathered}
L_{\text{MI}, \text{tPSA}} = \min_{\pi \in P}\sum_{c} \left\| \hat{M_{c}}\odot \left| X\right| \right. \\ \left. -
T_0^{\left| X\right|}(\left| S_{\pi(c)}\right| \odot \cos (\theta_{X} - \theta_{\pi(c)})) \right\|_F^{1},\end{gathered}$$ where $P$ is the set of permutations on $\{1,\dots,C\}$, $X$ and $S$ are the STFT of the noisy and clean speech respectively, $T$ is the truncation function defined as $T_a^b(x) = \min(\max(x,a),b)$, $\theta_X$ and $\theta_c$ are the noisy phase and the true phase of source $c$, respectively.
However, estimating tPSA contradicts with estimating the phase vectors in our problem formulation since both consider the cosine of the clean phase. To adapt the estimate into our problem formulation, we change the tPSA loss function to the magnitude spectrum approximation (MSA) for magnitude reconstruction. $$L_{\text{MI}, \text{MSA}} = \min_{\pi \in P}\sum_{c}{{\left\lVert\hat{M_{c}}\odot \left| X\right| -
\left| S_{\pi(c)}\right|\right\rVert}_F^{1}}.$$
We compare our chimera++ implementation with that in [@wang2018alternative] in Table \[tab:chimeraComparison\], which shows that we have successfully reproduced the reported result. We find that our implementation achieves slightly better performance by using the classic loss function based on tPSA, compared that using a whitened k-means loss function (W) [@wang2018alternative] based on tPSA. This indicates we can fairly compare our phase estimation network with the other methods which are also based on the chimera++ network.
Model Loss Function Mask Type SDR
---------------------------------- -------------------------------------------------- ----------- ------
Chimera++ [@wang2018alternative] $\textrm{DC}(W, W_{\textrm{VA}})$ tPSA 10.9
Our implementation $\textrm{DC}(\textrm{classic}, W_{\textrm{VA}})$ tPSA 11.0
: Comparison between the chimera++ in [@wang2018alternative] and our implementation[]{data-label="tab:chimeraComparison"}
Phase Estimation Network {#sec:phase_network}
========================
![Model architecture of the proposed phase estimation network. Note that the left tower is the original chimera++ network.[]{data-label="fig:phase-net"}](phase_network.pdf){width="\columnwidth"}
Figure \[fig:phase-net\] shows the architecture of the proposed phase network. First the magnitudes of each of the sources are estimated by applying the masks estimated by the chimera++ network to the noisy magnitude. Each magnitude estimate is concatenated with the real and imaginary components of the noisy STFT as the input feature for the phase inference (PI) sub-network. The output is a $T\times F\times 2$ matrix representing the cosine and sine of the phase estimate for each T-F bin. Each phase estimate is generated based on one of the masks. The output of the final linear layer is added to the original real and imaginary components of the noisy phase as a residual connection before unit-normalization. The loss function for this prediction is $$L_{\text{PI}} = -\sum_{c,t,f}{ \langle \hat{p}_{t,f}^c, p_{t,f}^c \rangle}$$ where $p_{t,f}^c$ is the vector $\langle \cos{\theta_c}, \sin{\theta_c} \rangle$ representing the true phase of the $t,f$ bin in source $c$ and $\hat{p}_{t,f}^c$ is the corresponding phase estimate.
Mask-dependent PIT Criterion {#subsec:mask-dependent-loss}
----------------------------
The PIT criterion [@yu2017permutation] has been successful in solving the label-permutation problem. To overcome the permutation problem for both the mask and the phase, several combinations of mask and phases losses could be used for selecting the best permutation. Since the phase estimate is dependent on the mask, the simplest combined PIT is based on a combination of both the losses of the MI and PI layers together. We use $L_{\text{PIT}}$ to represent this combined loss $$\begin{gathered}
L_{\text{PIT, MP}} = \min_{\pi \in P}\sum_{c} \Big( {\left\lVert\hat{M_{c}}\odot \left| X\right| - \left| S_{\pi(c)}\right|\right\rVert}_F^{1} \\
- \sum_{t,f} \langle \hat{p}_{t,f}^c, p_{t,f}^{\pi(c)} \rangle \Big)\end{gathered}$$ But since the phase is more difficult to learn to predict than the mask, perhaps because of a larger number of local minima, the phase tends to be misleading to this permutation matching in early epochs. Instead we only utilize the mask for matching, and continue this throughout training. Hence we propose a Mask-dependent PIT criterion defined as $$\begin{aligned}
L_{\text{PIT, MD}} &= &\sum_{c}{{\left\lVert\hat{M_{c}}\odot \left| X\right| -
\left| S_{\pi(c)}\right|\right\rVert}_F^{1}} - \sum_{c,t,f}{ \langle \hat{p}_{t,f}^c, p_{t,f}^{\pi(c)} \rangle}, \nonumber \\
&&\pi =\operatorname*{arg\,min}_{\pi}{\sum_{c}{{\left\lVert\hat{M_{c}}\odot \left| X\right| -
\left| S_{\pi(c)}\right|\right\rVert}_F^{1}}}.\end{aligned}$$
Weighted Loss Functions for Phase Estimation
--------------------------------------------
Phase is harder to predict in certain T-F regions. Figure \[fig:phase-visual\] shows for one source in a mixture the log magnitude of the clean speech, IRM, $\cos$, and $\sin$ of the phase difference $\angle\theta_{\text{S}} - \angle\theta_{\text{X}}$. If the mask value is close to one, meaning the phase of the mixture mostly comes from this source, the $\cos$ of the phase difference is close to 1 and the $\sin$ of the phase difference is close to 0. Hence the phase difference between the clean and the noisy STFT is very small. When the mask value is small, the phase differences are larger, making it more challenging to predict the phase.
/011a010g_0.16366_40pc0204_-0.16366.wav))[]{data-label="fig:phase-visual"}](phase_visualization.png){width="\columnwidth"}
In the MI loss of the chimera++ network, an MSA or tPSA loss function is introduced to increase the weight of the losses where the energy of the noisy magnitude is higher. However, the same weight may not be suitable for estimating the phase. We propose three different weighted loss function for the PI loss to differently weight the contributions to the phase estimates.
We define the Magnitude Weighted Loss Function (MWL) as $$L_{\text{PI, MWL}} = -\sum_{c,t,f}{(\gamma + M_{c, t,f}) \langle \hat{p}_{t,f}^c, p_{t,f}^{\pi(c)} \rangle},$$ where $M_{c, t,f}$ represents the magnitude of source $c$ on the $t,f$ bin, $\pi$ is identified by the PIT loss, and $\gamma$ is a tunable parameter that avoids applying 0 loss to points with 0 magnitude. We set it to 0.2 for our experiments. The motivation for this is that higher magnitude values should be emphasized because they are more reliable in training the phase network.
We define the Inverse Magnitude Weighted Loss Function (I-MWL) as $$L_{\text{PI, I-MWL}} = -\sum_{c,t,f}{(\gamma + M_{\neg c,t,f})\langle \hat{p}_{t,f}^c, p_{t,f}^{\pi(c)} \rangle},$$ where $M_{\neg c,t,f} = \sum_{i \neq c}^C M_{i, t,f}$ and $\pi$ is identified by the PIT loss. The motivation for this is that lower mask values are more difficult to predict, so should be emphasized.
We define the Joint Weighted Loss Function (Joint) as $$L_{\text{PI, Joint}} = -\sum_{c,t,f} \langle \hat{p}_{t,f}^c, p_{t,f}^{\pi(c)} \rangle \sum_i M_{i,t,f}.$$ The motivation for this weighting is to emphasize all active regions in the spectrogram. Only silent regions are ignored, similarly to the loss function of the deep clustering layer. Note that if the noisy magnitude is small while there are other sources whose magnitudes are large, the bin is emphasized more.
The loss function for the whole network thus becomes $$L_{\text{Phase Network}} = \alpha L_{\text{DC}} + (1-\alpha) L_{\text{PIT}},$$ where $\alpha$ is a weight parameter that balances the loss of the DC layers and the MI and PI layers. We use 0.975 for $\alpha$, following [@wang2018alternative].
Experiments
===========
To evaluate our proposed method, the models are trained and validated on the WSJ0-2mix dataset [@hershey2016deep], which is a publicly available dataset for multi-talker speech separation. There are 20,000 mixtures ($\sim$30 hours) in the training data, 5,000 mixtures ($\sim$10 hours) in the validation data, and 3,000 mixtures ($\sim$5 hours) in the test data. The training data and the validation data are generated by randomly mixing two utterances of two random speakers in the WSJ0 training data (si\_tr\_s). Some speakers in the validation data are included in the training data with different utterances, thus the validation data is considered to be the closed speaker condition (CSC). The test data is generated by mixing two random utterances of two random speakers from the WSJ0 test data (si\_et\_05). None of the speakers in the test data are included in either the training or the validation data, thus it is considered to be the open speaker condition (OSC). The sampling rate is 8 kHz and the Signal to Noise Ratio (SNR) of utterances is in a range between 0 and 10 dB. The chimera++ network and the proposed phase network are implemented in PyTorch 1.0[^1] [@paszke2017automatic]. Both the chimera++ network and the phase network have 4 BLSTM layers, each with 600 neural units in each direction. A 0.3 dropout rate is applied to the outputs of the BLSTM layers. Random chunks of 400 consecutive frames from each utterance are extracted to train the model with a batch size of 16 such chunks. The STFT is calculated by the librosa library [@mcfee2015librosa] utilizing a Hann window. The FFT window size is 256 samples (32 ms) and the hop length is 64 samples (8 ms). The Adam optimizer with a $10^{-3}$ initial learning rate is used. Training is stopped when the validation loss does not decrease for 10 epochs. The model is trained for a total of 100 epochs if no early stopping mechanism is activated.
To evaluate the separation performance, the Signal to Distortion Ratio (SDR) computed with the mir\_eval library [@raffel2014mir_eval] as the major evaluation metric. The baseline method is the noisy phase with the MSA magnitude estimated by the chimera++ network.
Method SDR SI-SDR
-------------------------------------------------- ------ --------
Chimera++, MSA 10.5 -
+ tPSA [@wang2018alternative] 11.5 11.2
+ MISI-5 [@wang2018alternative] 11.8 11.5
+ WA-MISI-5 [@wang2018end] 12.9 12.6
Phasebook, MISI-0 [@le2019phasebook] - 12.6
+ MISI-5 [@le2019phasebook] - 12.8
Chimera++(Encoder-BLSTM-Decoder) [@wang2018deep] - 11.9
Sign prediction network [@wang2018deep] 15.6 15.3
: Published SDR/SI-SDR improvements of different phase estimation methods on the open speaker condition (OSC) of the WSJ0-2mix dataset.[]{data-label="tab:sdr-baseline"}
PIT Criterion Mask Activation Phase Loss SDR
--------------- ----------------- ---------------------- ------
MP ReLU MWL 11.0
MP Sigmoid MWL 11.5
MD Sigmoid I-MWL 12.0
MD Sigmoid MWL 12.6
MD Sigmoid Joint 13.0
MD Sigmoid Joint,$\alpha = 0.5$ 13.6
: SDR improvements of the proposed method with different settings on the OSC of the WSJ0-2mix dataset. PIT criteria Mask+Phase (MP) and Mask-dependent (MD). Phase losses magnitude-weighted loss (MWL), inverse magnitude-weighted loss (I-MWL), and joint weighted loss (Joint).[]{data-label="tab:sdr-propose"}
Results {#sec:results}
=======
Tables \[tab:sdr-baseline\] and \[tab:sdr-propose\] show the SDR and scale-invariant SDR (SI-SDR) [@le2018sdr] improvements of recently published methods and those of the proposed phase estimation networks, respectively. The chimera++ network with MSA loss achieves 10.5 dB SDR. Adding our phase estimation network with $\text{PIT}_{\text{MP}}$ criterion improves the performances by 1.0 dB, which is the same as that of the chimera++ network with the tPSA loss. By comparing the activation functions of the MI layer, ReLU does not improve the performance over Sigmoid.
Comparing rows 1 and 3 of Table \[tab:sdr-propose\], using the mask-dependent $\text{PIT}_{\text{MD}}$ criterion achieves 12.6 dB SDR, which is 1.1 dB higher than using the $\text{PIT}_{\text{MP}}$ criterion. This shows that the loss for the mask estimate is more reliable and stable than combining it with the phase loss for the PIT criterion. In terms of different weighted loss functions, the weight of all magnitudes (Joint) achieves the best result of 13.0 dB SDR, followed by T-F regions of the clean speech (MWL) and then noisy speech (I-MWL). Inspired by [@isik2016single], we also apply a curriculum training strategy [@bengio2009curriculum] that trains the network using $\alpha=0.975$ first, then retrains the model using $\alpha=0.5$, increasing the performance to 13.6 dB.
Comparing our results with published phase-based methods in Table \[tab:sdr-baseline\], our result is better than the chimera++ with 5 iterations of the MISI algorithm (MISI-5). We get competitive result to the chimera++ with 5 MISI iterations and an end-to-end waveform approximate loss function (WA-MISI-5) without applying the curriculum training strategy. This indicates the proposed network can predict phase directly from the mask generated by chimera++. We do not currently use the waveform approximate loss function in our model, but our mask-dependent PIT criterion is applicable to the such losses. Instead of choosing the minimum waveform difference, we choose the permutation that gives the minimum mask loss. The model still backpropagates to minimize the waveform difference, while the mask-dependent PIT criterion helps train the model in a more reliable way. Our future research will analyze how the mask-dependent PIT criterion can contribute to such end-to-end approaches.
Though we do not achieve better performance than [@wang2018deep], it is good to mention the chimera++ network applied in [@wang2018deep] is different. It adds several convolutional encoder and decoder layers before and after the chimera++ network to achieve 11.9 dB SI-SDR ($\sim$12.2 to 12.4 dB SDR). Also, since the model applies the waveform approximate loss function to reconstruct the time-domain signal, similarly to Phasebook, we can adapt our proposed method to the model to improve the phase estimation accuracy further. [@liu2019divide] show that frame-level PIT criterion can find a better local minimum hence significantly improves the performance comparing than the utterance-level PIT. Our future plan is to integrate the frame-level PIT into our mask-dependent criterion and conduct the evaluation.
Conclusion {#sec:conc}
==========
This paper proposed a phase estimation method using the mask estimate from the chimera++ network. A mask-dependent PIT criterion is applied to solve the label-permutation problem for both the mask and the phase estimates. The mask-dependent PIT criterion significantly improved the separation performance compared with the PIT over the whole loss. Future study will focus on applying the proposed PIT criterion to end-to-end phase estimation methods (e.g., phasebook [@le2019phasebook] and the sign prediction network [@wang2018deep]).
[^1]: https://github.com/speechLabBcCuny/onssen
|
---
abstract: 'Let $(\mL;C)$ be the (up to isomorphism unique) countable homogeneous structure carrying a binary branching C-relation. We study the reducts of $(\mL;C)$, i.e., the structures with domain $\mL$ that are first-order definable in $(\mL;C)$. We show that up to existential interdefinability, there are finitely many such reducts. This implies that there are finitely many reducts up to first-order interdefinability, thus confirming a conjecture of Simon Thomas for the special case of $(\mL;C)$. We also study the endomorphism monoids of such reducts and show that they fall into four categories.'
author:
- |
Manuel Bodirsky\
Institut für Algebra, TU Dresden, Germany\
[manuel.bodirsky@tu-dresden.de]{}\
- |
Peter Jonsson\
Department of Computer and System Science\
Linköpings Universitet, Linköping, Sweden\
[peter.jonsson@liu.se]{}\
- |
Trung Van Pham\
Institut für Computersprachen, Theory and Logic Group\
TU Wien, Austria\
[pvtrung@logic.at]{}
bibliography:
- 'local.bib'
title: |
The Reducts of the Homogeneous\
Binary Branching C-relation[^1]
---
intro.tex phylogeny.tex autos.tex ramsey.tex canonical.tex two-constants.tex classification.tex ending.tex
[^1]: The first and third author have received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 257039). The third author has received funding from the project P27600 of the Austrian Science Fund (FWF) and the project “Models on graphs: enumerative combinatorics and algebraic structures” of the Vietnam National Foundation for Science and Technology Development (NAFOSTED). The second author is partially supported by the [*Swedish Research Council*]{} (VR) under Grant 621-2012-3239.
|
---
abstract: |
We provide a uniform construction of $L^2$-models for all small unitary representations in degenerate principal series of semisimple Lie groups which are induced from maximal parabolic subgroups with abelian nilradical. This generalizes previous constructions to the case of a maximal parabolic subgroup which is not necessarily conjugate to its opposite, and hence the previously used Jordan algebra methods have to be generalized to Jordan pairs.\
The crucial ingredients for the construction of the $L^2$-models are the Lie algebra action and the spherical vector. Working in the so-called Fourier transformed picture of the degenerate principal series, the Lie algebra action is given in terms of Bessel operators on Jordan pairs. We prove that precisely for those parameters of the principal series where small quotients occur, the Bessel operators are tangential to certain submanifolds. Further, we show that the small quotients are unitarizable if and only if these submanifolds carry equivariant measures. In this case we can express the spherical vectors in terms of multivariable K-Bessel functions, and some delicate estimates for these Bessel functions imply the existence of the $L^2$-models.
address:
- 'Department Mathematik, FAU Erlangen–Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany'
- 'Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany'
author:
- Jan Möllers
- Benjamin Schwarz
title: Bessel operators on Jordan pairs and small representations of semisimple Lie groups
---
Introduction {#introduction .unnumbered}
============
In the representation theory of semisimple Lie groups the principal series plays a central role. It is constructed by parabolic induction from minimal parabolic subgroups, inducing from finite-dimensional representations of the Levi factor, and forms a standard family of representations of the group. The celebrated Harish-Chandra Subquotient Theorem asserts that every irreducible representation on a Banach space, in particular every irreducible unitary representation, is equivalent to a subquotient of a representation in the principal series. In this sense, every irreducible unitary representation can be constructed through the principal series.
However, it turns out to be a highly non-trivial problem to find the irreducible unitarizable constituents of the principal series, in particular when the rank of the group is large. Moreover, the unitary realizations of representations obtained in this way are of an algebraic nature, and often rather complicated and difficult to work with.
A more accessible setting arises if one replaces the minimal parabolic subgroup by a larger parabolic subgroup, for instance a maximal one. The corresponding parabolically induced representations are called *degenerate principal series representations* and they essentially depend on a single complex parameter. This simplifies the problem of finding the irreducible constituents and determining the unitarizable ones among them. In various cases this problem has been solved, and in this way particularly small unitarizable irreducible representations are obtained, among them so-called *minimal representations*.
If additionally the nilradical of the parabolic subgroup is abelian, it turns out that the Euclidean Fourier transform on the nilradical can be used to construct very explicit models of the small unitarizable quotients of the degenerate principal series, realized on Hilbert spaces of $L^2$-functions on certain orbits of the Levi factor of the parabolic. These so-called *$L^2$-models* have been obtained by Rossi–Vergne [@RV76], Sahi [@Sah92], Dvorsky–Sahi [@DS99; @DS03], Kobayashi–[Ø]{}rsted [@KO03c] and Barchini–Sepanski–Zierau [@BSZ06] in all cases where the maximal parabolic subgroup is conjugate to its opposite parabolic subgroup. Their constructions are different in nature, and in particular do not easily generalize to the case where the parabolic subgroup is not conjugate to its opposite.
In this paper we provide a uniform approach to $L^2$-models of small representations that works for all maximal parabolic subgroups with abelian nilradical. Our key ingredients are twofold:
1. \[Ingredients:1\] We extend the Bessel operators, previously defined for Jordan algebras, to the context of Jordan pairs. These operators form the crucial part of the Lie algebra action.
2. \[Ingredients:2\] We express the spherical vectors in the degenerate principal series and its Fourier transformed picture in terms of multivariable K-Bessel functions.
That the Bessel operators on Jordan algebras appear in the Lie algebra action of small representations was first observed by Mano [@Man08] and later used by Hilgert–Kobayashi–Möllers [@HKM14] to uniformly construct $L^2$-models for minimal representations. The occurrence of classical one-variable K-Bessel functions as spherical vectors appears first in the work by Dvorsky–Sahi [@DS99] (see also [@HKM14; @KO03c]). In this sense, our ingredients and are generalizations of previously used techniques.
In this setting, we show that small constituents arise precisely for those parameters where the Bessel operators are tangential to certain submanifolds. Further, these small constituents turn out to be unitarizable if and only if there exist equivariant measures on these submanifolds. In this case the Bessel operators are shown to be symmetric with respect to these measures, and the Lie algebra acts by formally skew-adjoint operators with respect to the $L^2$-inner product. To integrate this Lie algebra representation to the group level, we use the explicit description of the spherical vector in terms of a multivariable K-Bessel function and prove some delicate estimates for this Bessel function that ensure that the Lie algebra representation indeed integrates to an irreducible unitary representation on the $L^2$-space of the submanifold, providing an $L^2$-model of the small representations.\
We now explain our results in more detail. Let $G$ be a connected simple real non-compact Lie group with maximal parabolic subgroup $P=MAN\subseteq G$ whose nilradical $N$ is abelian. Then $G/P$ is a compact Riemannian symmetric space. Write $\overline{P}=MA\overline N$ for the opposite parabolic and ${\mathfrak{a}}$, ${\mathfrak{n}}$ and $\overline{\mathfrak{n}}$ for the Lie algebras of $A$, $N$ and $\overline N$, then $({\mathfrak{n}},\overline{\mathfrak{n}})$ naturally carries the structure of a *Jordan pair*. This Jordan pair is associated to a Jordan algebra if and only if $P$ and $\overline P$ are conjugate. The rank $r$ of $({\mathfrak{n}},\overline{\mathfrak{n}})$ agrees with the rank of the compact symmetric space $G/P$.
For $\nu\in{\mathfrak{a}}_{\mathbb{C}}^*$ we form the degenerate principal series (smooth normalized parabolic induction) $$\pi_\nu = {\textup{Ind}}_P^G({\mathbf{1}}\otimes e^\nu\otimes{\mathbf{1}})$$ and realize it on a space $I(\nu)\subseteq C^\infty(\overline{\mathfrak{n}})$ of smooth functions on $\overline{\mathfrak{n}}$. The structure of these representations has been completely determined by Sahi [@Sah92] for $G$ Hermitian (see also Johnson [@Joh90; @Joh92] and [Ø]{}rsted–Zhang [@OZ95]), by Sahi [@Sah95] and Zhang [@Zha95] for $G$ non-Hermitian and $P$ and $\overline P$ conjugate, and by the authors [@MS12] for the remaining cases. The precise results differ from case to case, but they all have a common feature: There exist $$0<\nu_{r-1}<\ldots<\nu_1<\nu_0=\rho=\tfrac{1}{2}\operatorname{Tr}{\textup{ad}}_{\mathfrak{n}}\in{\mathfrak{a}}^*$$ such that the representation $\pi_\nu$ is reducible for $\nu=\nu_k$, $0\leq k\leq r-1$, and has a unique irreducible quotient $J(\nu_k)$ which is a small representation of rank $k$ (see Theorem \[thm:StructureDegPrincipalSeries\]). Here a representation is said to have *rank $k$* if its $K$-types, identified with their highest weights, are contained in a single $k$-dimensional affine subspace. In particular, $J(\nu_0)$ is the trivial representation.
Strictly speaking, there are two cases in which this statement only holds if slightly modified. More precisely, if $G$ is Hermitian or if $G=SO_0(p,q)$, $p\neq q$, one has to allow parabolic induction from a possibly non-trivial unitary character of $M$ as well. Since these two special cases were treated before in great detail (see [@HKM14; @KM08; @RV76]), we exclude them in the introduction, but we remark that similar statements hold and therefore our construction is uniform. In the rest of the paper it is clearly stated which results hold in general and which need slight modifications, and we also provide the corresponding references for the respective statements.
To construct $L^2$-models of the representations $J(\nu_k)$ we use the Euclidean Fourier transform ${\mathcal{F}}:{\mathcal{S}}'(\overline{\mathfrak{n}})\to{\mathcal{S}}'({\mathfrak{n}})$ corresponding to the non-degenerate pairing ${\mathfrak{n}}\times\overline{\mathfrak{n}}\to{\mathbb{R}}$ given by the Killing form. Twisting with the Fourier transform we obtain a new realization $\tilde\pi_\nu={\mathcal{F}}\circ\pi_\nu\circ{\mathcal{F}}^{-1}$ of $\pi_\nu$ on $\tilde I(\nu)={\mathcal{F}}(I(\nu))\subseteq{\mathcal{S}}'({\mathfrak{n}})$. This realization is called the *Fourier transformed picture*.
In the realization $\tilde\pi_\nu$ the Levi factor $L=MA$ essentially acts by the left-regular representation of the adjoint action of $L$ on ${\mathfrak{n}}$. Under the $L$-action ${\mathfrak{n}}$ decomposes into $r+1$ orbits ${\mathcal{O}}_0,\ldots,{\mathcal{O}}_r$, ordered such that ${\mathcal{O}}_j$ is contained in the closure of ${\mathcal{O}}_k$ if $j\leq k$.
In Proposition \[prop:FTpicturealgebraaction\] we also compute the Lie algebra action of $\tilde\pi_\nu$, showing that the non-trivial part is given in terms of so-called *Bessel operators* ${\mathcal{B}}_\lambda$ on the Jordan pair $({\mathfrak{n}},\overline{\mathfrak{n}})$ for $\lambda=2(\rho-\nu)$. These are $\overline{\mathfrak{n}}$-valued second order differential operators on ${\mathfrak{n}}$ and can be defined by the simple formula $${\mathcal{B}}_\lambda=Q\left(\frac{\partial}{\partial x}\right)x+\lambda\frac{\partial}{\partial x}, \qquad \lambda\in{\mathbb{C}},$$ where $\frac{\partial}{\partial x}:C^\infty({\mathfrak{n}})\to C^\infty({\mathfrak{n}})\otimes\overline{\mathfrak{n}}$ denotes the gradient with respect to the (renormalized) Killing form ${\mathfrak{n}}\times\overline{\mathfrak{n}}\to{\mathbb{R}}$ and $Q:\overline{\mathfrak{n}}\to{\textup{Hom}}({\mathfrak{n}},\overline{\mathfrak{n}})$ the quadratic operator of the Jordan pair $({\mathfrak{n}},\overline{\mathfrak{n}})$.
We first relate the existence of small quotients of $\tilde I(\nu)$ to the Bessel operators.
\[thm:IntroTangential\] Let $0\leq k\leq r-1$, then $\tilde I(\nu)$ has an irreducible quotient of rank $k$ if and only if the Bessel operator ${\mathcal{B}}_\lambda$ for $\lambda=2(\rho-\nu)$ is tangential to the orbit ${\mathcal{O}}_k$. This is precisely the case for $\nu=\nu_k$.
The previous theorem implies that the Lie algebra representation ${\mathrm{d}}\tilde\pi_{\nu_k}$ has $\tilde I_0(\nu_k)={\{f\in\tilde I(\nu)\,|\,f|_{{\mathcal{O}}_k}=0\}}$ as a subrepresentation and hence induces a representation on the quotient $\tilde J(\nu_k)=\tilde I(\nu_k)/\tilde I_0(\nu_k)$ which is identified with ${\{f|_{{\mathcal{O}}_k}\,|\,f\in\tilde I(\nu)\}}$. This is the unique irreducible quotient of $\tilde I(\nu_k)$ which is a small representation of rank $k$. We now turn to the question whether this quotient is unitarizable.
\[thm:IntroMeasures\] Let $0\leq k\leq r-1$, then the irreducible quotient $\tilde J(\nu_k)$ is unitarizable if and only if the orbit ${\mathcal{O}}_k$ carries an $L$-equivariant measure ${\mathrm{d}}\mu_k$. This is precisely the case for $({\mathfrak{g}},{\mathfrak{l}})\not\simeq({\mathfrak{sl}}(p+q,{\mathbb{F}}),{\mathfrak{s}}({\mathfrak{gl}}(p,{\mathbb{F}})\oplus{\mathfrak{gl}}(q,{\mathbb{F}})))$, ${\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}}$ with $p\neq q$, where ${\mathfrak{l}}$ denotes the Lie algebra of $L$.
The space $L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$ is an obvious candidate for the Hilbert space completion of the unitarizable quotient $\tilde J(\nu_k)$. In this case the Lie algebra has to act by skew-adjoint operators on $L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$. The cruciual part of this action is given in terms of the Bessel operators which have to be self-adjoint on $L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$.
\[thm:IntroSymmetric\] Let $0\leq k\leq r-1$ and assume that the orbit ${\mathcal{O}}_k$ carries an $L$-equivariant measure ${\mathrm{d}}\mu_k$. Then the Bessel operator ${\mathcal{B}}_\lambda$, $\lambda=2(\rho-\nu_k)$, is symmetric on $L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$.
We finally obtain the $L^2$-model for $\tilde J(\nu_k)$:
\[thm:IntroL2Models\] Let $0\leq k\leq r-1$, and assume that the orbit ${\mathcal{O}}_k$ carries an $L$-equivariant measure ${\mathrm{d}}\mu_k$. Then the operator $$T_k:\tilde I(\nu_k)\to\tilde I(-\nu_k), \quad f\mapsto f|_{{\mathcal{O}}_k}{\mathrm{d}}\mu_k,$$ is defined and intertwining for $\tilde\pi_{\nu_k}$ and $\tilde\pi_{-\nu_k}$. Its kernel is the subrepresentation $\tilde I_0(\nu_k)$ and its image is a subrepresentation $\tilde I_0(-\nu_k)\subseteq\tilde I(-\nu_k)$ isomorphic to the irreducible quotient $\tilde J(\nu_k)=\tilde I(\nu_k)/\tilde I_0(\nu_k)$. The subrepresentation $\tilde I_0(-\nu_k)$ is contained in $L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$ and the $L^2$-inner product is a $\tilde\pi_{-\nu_k}$-invariant Hermitian form, so that the representation $(\tilde\pi_{-\nu_k},\tilde I_0(-\nu_k))$ extends to an irreducible unitary representation of $G$ on $L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$.
For $k=1$ the representation $J(\nu_1)$ is in many cases the minimal representation of $G$ (see Corollary \[cor:MinRep\] for the precise statement).
We remark that in the case where $P$ and $\overline P$ are conjugate, the operator $T_k$ is a regularization of the Knapp–Stein family of standard intertwining operators $\tilde\pi_\nu\to\tilde\pi_{-\nu}$. However, in the case where $P$ and $\overline P$ are not conjugate, such a family does not exist and the operators $T_k$ are a geometric version of the non-standard intertwining operators constructed in [@MS12].
In addition to the previously stated results on the Bessel operators, the proof of Theorem \[thm:IntroL2Models\] consists of a detailed analysis of the $K$-spherical vector in $\tilde I(\nu)$, where $K$ is a maximal compact subgroup of $G$. In Theorem \[thm:PsiInL1\] we express the spherical vector $\psi_\nu\in\tilde I(\nu)$ for $\operatorname{Re}\nu>-\nu_{r-1}$ in terms of a multivariable K-Bessel function. It is supported on the whole vector space ${\mathfrak{n}}$ and depends holomorphically on $\nu\in{\mathbb{C}}$. For its values at $\nu=-\nu_k$, $0\leq k\leq r-1$, we have the following result:
\[thm:IntroSphericalVector\] Let $0\leq k\leq r-1$, and assume that the orbit ${\mathcal{O}}_k$ carries an $L$-equivariant measure ${\mathrm{d}}\mu_k$. Then the spherical vector $\psi_{-\nu_k}\in\tilde I(-\nu_k)$ is given by $\psi_{-\nu_k}={\textup{const}}\times\Psi_k\,{\mathrm{d}}\mu_k$, where $\Psi_k\in C^\infty({\mathcal{O}}_k)$ is in polar coordinates given by $$\Psi_k(mb_t) = K_{\mu_k}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right), \qquad m\in M\cap K,\,t\in C_k^+.$$ We further have $\Psi_k\in L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$ and hence the representation $(\tilde\pi_{-\nu_k},L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k))$ is spherical.
Here we use polar coordinates $(M\cap K)\times C_k^+\to{\mathcal{O}}_k,\,(m,t)\mapsto mb_t$ where $C_k^+={\{t\in{\mathbb{R}}^k\,|\,t_1>\ldots>t_k>0\}}$, and $K_\mu(s_1,\ldots,s_k)$ is a multivariable K-Bessel function of parameter $\mu=\mu_k\in{\mathbb{R}}$. The K-Bessel function can be obtained from the theory of Bessel functions on symmetric cones (see Appendix \[app:KBesselFunctions\]), and we prove some involved estimates for these functions that guarantee for instance that $\Psi_k\in L^1({\mathcal{O}}_k,{\mathrm{d}}\mu_k)\cap L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$. These estimates are essential in the proof of Theorem \[thm:IntroL2Models\].
We remark that for $k=1$ the function $K_\mu(s_1)$ is essentially the classical K-Bessel function in one variable $s_1\in{\mathbb{R}}_+$. Its occurrence as the spherical vector in $L^2$-models for small representations was first observed by Dvorsky–Sahi [@DS99] (see also Kobayashi–[Ø]{}rsted [@KO03c]).\
Let us comment on the validity of the stated results in the cases of $G$ Hermitian and $G=SO_0(p,q)$. Theorems \[thm:IntroTangential\], \[thm:IntroMeasures\] and \[thm:IntroSymmetric\] are in fact proven in general, i.e. also for $G$ Hermitian and $G=SO_0(p,q)$. The proofs use a description of the orbits ${\mathcal{O}}_k$ by local coordinates and are carried out in the framework of Jordan pairs. Theorem \[thm:IntroL2Models\] also holds for $G$ Hermitian as stated here, but for $G=SO_0(p,q)$ one has to assume that $p+q$ is even. The latter case is treated in detail in [@HKM14] which is why we omit the details here. Theorem \[thm:IntroSphericalVector\] is not true in the two special cases, in fact the corresponding small representations for $G$ Hermitian and $G=SO_0(p,q)$, $p\neq q$, are not spherical. For $G$ Hermitian they have a non-trivial scalar minimal $K$-type given by an exponential function in the $L^2$-model, and for $G=SO_0(p,q)$, $p\neq q$, the minimal $K$-type is of dimension $>1$ (see Remark \[rem:MinKtypeHermitianAndOpq\] for more explanations). The comparable statements for $G$ Hermitian are well-known and can e.g. be found in [@Moe13 Section 2.1], and for $G=SO_0(p,q)$ we refer to [@KM08 Chapter 3] or [@HKM14 Theorem 2.19 (c) and Proposition 2.24 (b)] for a detailed treatment of the $L^2$-model.\
Theorem \[thm:IntroTangential\], \[thm:IntroMeasures\] and \[thm:IntroSymmetric\] were previously shown in [@HKM14] for the case where $P$ and $\overline P$ are conjugate, or equivalently ${\mathfrak{n}}$ is a Jordan algebra. For Theorems \[thm:IntroTangential\] and \[thm:IntroSymmetric\] the proofs use zeta functions on Jordan algebras which are not available for Jordan pairs. The proofs we present here work uniformly for all Jordan pairs since they merely use a local description of the orbits ${\mathcal{O}}_k$. Theorem \[thm:IntroL2Models\] was previously obtained by Rossi–Vergne [@RV76] for $G$ Hermitian (see also Sahi [@Sah92]), and by Dvorsky–Sahi [@DS99; @DS03] for $G$ non-Hermitian and $P$ and $\overline P$ conjugate (see also Barchini–Sepanski–Zierau [@BSZ06]), and is new for $P$ and $\overline P$ not conjugate. Theorem \[thm:IntroSphericalVector\] seems to be new in this general form, but for $k=1$ the K-Bessel function is (up to renormalization) the classical one-variable K-Bessel function and in this case the corresponding result was first obtained by Dvorsky–Sahi [@DS99]. We remark that multivariable K-Bessel functions also appear in other contexts in representations theory, for instance in the construction of Fock models for scalar type unitary highest weight representations, see [@Moe13].\
The structure of this paper is as follows. In Section \[sec:degenerateprincipalseries\] we recall the construction of the degenerate principal series, the main results about its structure, and the relation to Jordan pairs. In particular, we obtain an explicit formula for the spherical vector in the non-compact picture, see Proposition \[prop:SphericalVectorNonCptPicture\]. Section \[sec:OrbitStructure\] provides details on the local and global structure of the orbits ${\mathcal{O}}_k$, in particular we find explicit local coordinates in Proposition \[prop:LOrbits\] and determine the cases where the orbits admit equivariant measures in Theorem \[thm:equivariantmeasures\]. Bessel operators on Jordan pairs are defined in Section \[sec:Besseloperator\] where we further show that for certain parameters these operators are tangential to the orbits ${\mathcal{O}}_k$ (see Theorem \[thm:pullbackBessel\]) and symmetric with respect to the equivariant measures (see Theorem \[thm:BesselSymmetricOnOrbits\]). Finally, in Section \[sec:L2models\] we construct the $L^2$-models by a detailed analysis of the spherical vectors in terms of multivariable K-Bessel functions. Appendix \[app:KBesselFunctions\] provides some details about these Bessel functions in the framework of symmetric cones.
Degenerate principal series and Jordan pairs {#sec:degenerateprincipalseries}
============================================
In this section we fix the necessary notation, recall the relevant reducibility and unitarizability results for degenerate principal series representations and give a description of the representations in terms of Jordan pairs.
Degenerate principal series {#subsec:degenerateprincipalseries}
---------------------------
Let $G$ be a connected simple real non-compact Lie group with finite center. Assume that $G$ has a maximal parabolic subgroup $P=MAN\subseteq G$ whose nilradical $N\subseteq P$ is abelian. Let $\theta$ be a Cartan involution of $G$ which leaves the Levi subgroup $L=MA$ invariant and denote by $K=G^\theta$ the corresponding maximal compact subgroup of $G$. Then $M\cap K=M^\theta$ is maximal compact in $M$ and $L$, and the generalized flag variety $X=G/P$ is a compact Riemannian symmetric space $X=K/(M\cap K)$, sometimes referred to as a *symmetric $R$-space*. Let ${\mathfrak{g}}$, ${\mathfrak{k}}$, ${\mathfrak{l}}$, ${\mathfrak{m}}$, ${\mathfrak{a}}$ and ${\mathfrak{n}}$ be the Lie algebras of $G$, $K$, $L$, $M$, $A$ and $N$, respectively, and let $\theta$ also denote the Cartan involution on ${\mathfrak{g}}$ with respect to ${\mathfrak{k}}$. There exists a (unique) grading element $Z_0$ in ${\mathfrak{a}}$ such that ${\mathfrak{g}}$ decomposes under the adjoint action of $Z_0$ into $${\mathfrak{g}}= \overline{\mathfrak{n}}\oplus{\mathfrak{l}}\oplus{\mathfrak{n}}$$ with eigenvalues $-1$, $0$, $1$. Then $\overline{\mathfrak{n}}= \theta{\mathfrak{n}}$, and ${\mathfrak{l}}\oplus{\mathfrak{n}}$ is the Lie algebra of $P$. The dimension of $X$ is denoted by $n=\dim\overline{\mathfrak{n}}=\dim{\mathfrak{n}}$. Concerning root data, we fix a maximal abelian subalgebra ${\mathfrak{t}}$ in ${\mathfrak{k}}\cap(\overline{\mathfrak{n}}\oplus{\mathfrak{n}})$. The dimension $r=\dim{\mathfrak{t}}$ is called the *rank* of the symmetric space $X$. The adjoint action on ${\mathfrak{g}}_{\mathbb{C}}$ yields a (restricted) root system $\Phi({\mathfrak{g}}_{\mathbb{C}},{\mathfrak{t}}_{\mathbb{C}})$ which is either of type $C$ or of type $BC$. We write $$\begin{aligned}
\Phi({\mathfrak{g}}_{\mathbb{C}},{\mathfrak{t}}_{\mathbb{C}}) =
\begin{cases}
{\left\{\tfrac{1}{2}(\pm\gamma_j\pm\gamma_k)\,\middle|\,1\leq j,k\leq r\right\}}\setminus\{0\}
&\text{for type $C_r$}\\
{\left\{\tfrac{1}{2}(\pm\gamma_j\pm\gamma_k),\ \pm\tfrac{1}{2}\,\gamma_j\,\middle|\,1\leq j,k\leq r\right\}}\setminus\{0\} &\text{for type $BC_r$},
\end{cases}\end{aligned}$$ where $\gamma_1,\ldots,\gamma_r$ are strongly orthogonal roots. Depending on the type of this root system, $X$ is called *unital* if $\Phi({\mathfrak{g}}_{\mathbb{C}},{\mathfrak{t}}_{\mathbb{C}})$ is of type $C_r$, and *non-unital* otherwise. These notions are due to the Jordan theoretic description of symmetric $R$-spaces, see Section \[sec:JordanTheoryOfSymmRSpaces\]. More precisely, in the unital case ${\mathfrak{n}}$ is a Jordan algebra whereas in the non-unital case it is only a Jordan triple system.
The adjoint action of ${\mathfrak{t}}_{\mathbb{C}}$ on ${\mathfrak{k}}_{\mathbb{C}}$ also yields a root system $\Phi({\mathfrak{k}}_{\mathbb{C}},{\mathfrak{t}}_{\mathbb{C}})$, and the *structure constants* of the symmetric $R$-space $X$ are defined by the multiplicities of the roots, $$\begin{aligned}
\label{eq:structureconstants}
\begin{aligned}
e&=\dim({\mathfrak{k}}_{\mathbb{C}})_{\pm\gamma_j}, &
d_+&=\dim({\mathfrak{k}}_{\mathbb{C}})_{\tfrac{\gamma_j-\gamma_k}{2}},\\
b&=\dim({\mathfrak{k}}_{\mathbb{C}})_{\pm\tfrac{\gamma_j}{2}}, &
d_-&=\dim({\mathfrak{k}}_{\mathbb{C}})_{\pm\tfrac{\gamma_j+\gamma_k}{2}}.
\end{aligned}\end{aligned}$$ These dimensions are independent of the choice of sign $\pm$ and $1\leq j,k\leq r$, $j\neq k$, and the possible combinations of root systems and structure constants are given in Table \[tab:RootSystems\]. Depending on $\Phi({\mathfrak{k}}_{\mathbb{C}},{\mathfrak{t}}_{\mathbb{C}})$, we say $X$ is of type $A$, $B$, $BC$, $C$ or $D$, and if necessary also refer to the rank of the root system.
[llll]{} $\Phi({\mathfrak{g}}_{\mathbb{C}},{\mathfrak{t}}_{\mathbb{C}})$ & $\Phi({\mathfrak{k}}_{\mathbb{C}},{\mathfrak{t}}_{\mathbb{C}})$ & structure constants & naming\
$C_r$ & $A_{r-1}$ & $b=0,\ e=0,\ d_-=0$ & Euclidean, in particular unital\
$C_r$ & $C_r$ & $b=0,\ e\neq 0$ & unital non-reduced\
$C_r$ & $D_r$ & $b=0,\ e=0,\ d_-\neq0$ & unital reduced\
$BC_r$ & $B_r$ & $b\neq0,\ e=0$ & non-unital reduced\
$BC_r$ & $BC_r$ & $b\neq0,\ e\neq 0$ & non-unital non-reduced\
For convenience, we set $$\begin{aligned}
\label{eq:dDef}
d=\frac{d_++d_-}{2}.\end{aligned}$$ If $d_+,d_->0$ then $d=d_+=d_-$ unless the root system is reducible which happens in case $D_2$. Here ${\mathfrak{g}}={\mathfrak{o}}(p,q)$ with $d_+=p-1$, $d_-=q-1$. Further, it can happen that $d_-=0$, then the root system is of type $A$ and $d_+=2d$. In this case ${\mathfrak{g}}$ is a Hermitian Lie algebra. This yields:
Precisely one of the following holds:
1. (Type $A$) ${\mathfrak{g}}$ is Hermitian of tube type (in this case $d_-=b=e=0$),
2. (Type $D_2$) ${\mathfrak{g}}={\mathfrak{o}}(p,q)$ with $p\neq q$, $p,q\geq3$ (in this case $d_+=p-1$, $d_-=q-1$ and $b=e=0$),
3. \[d+=d-\] $d_+=d_-$.
In those statements that only hold in case we will state $d_+=d_-$ as an assumption. In fact, this is the algebraic reason why modifications of the statements are necessary in type $A$ and $D_2$.
For each strongly orthogonal root $\gamma_k$, we fix an ${\mathfrak{sl}}_2$-triple $(E_k,H_k,F_k)$, $E_k\in{\mathfrak{n}}$, $F_k\in\overline{\mathfrak{n}}$, with $\theta(E_k) = -F_k$ and $(E_k-F_k)\in{\mathfrak{t}}$, $(E_k+F_k\pm iH_k)\in({\mathfrak{g}}_{\mathbb{C}})_{\pm\gamma_k}$. Let $\kappa$ denote the Killing form of ${\mathfrak{g}}$. Then, $$\begin{aligned}
\label{eq:pDefinition}
p=\tfrac{1}{8}\kappa(H_k,H_k) = (e+1) + (r-1)\,d + \tfrac{b}{2},\end{aligned}$$ is independent of $1\leq k\leq r$, and is called the *genus* of the symmetric $R$-space $X$.
For $\nu\in{\mathfrak{a}}_{\mathbb{C}}^*$ we consider the degenerate principal series $\pi_\nu={\textup{Ind}}_P^G({\bf 1}\otimes e^\nu\otimes{\bf 1})$, realized on the space $$I(\nu)={\left\{f\in C^\infty(G)\,\middle|\,f(gman) = a^{-\nu-\rho}f(g)\,\forall\,g\in G,\,man\in MAN\right\}},$$ where $\rho=\frac{1}{2}\operatorname{Tr}{\textup{ad}}_{\mathfrak{n}}\in{\mathfrak{a}}^*$ denotes half the sum of all positive roots. Identifying ${\mathfrak{a}}_{\mathbb{C}}^*\simeq{\mathbb{C}}$ by $\nu\mapsto\frac{p}{n}\,\nu(Z_0)$ we have $\rho=\frac{p}{2}$. The restriction of $\pi_\nu$ to $K$ decomposes into a multiplicity-free direct sum of $K$-modules of the form $V_{{\bf m}}$ with highest weight $\sum_{k=1}^r m_k\gamma_k$, where ${{\bf m}}=(m_1,\ldots,m_r)$ is a non-increasing $r$-tuple of integers. More precisely, $$I(\nu)_{{K-\textup{finite}}}= \bigoplus_{{{\bf m}}\in\Lambda} V_{{\bf m}},$$ where $$\begin{aligned}
\Lambda=\begin{cases}
{\left\{{{\bf m}}\in{\mathbb{Z}}^r\,\middle|\,m_1\geq\cdots\geq m_r\right\}} & \text{for type $A$,}\\
{\left\{{{\bf m}}\in{\mathbb{Z}}^r\,\middle|\,m_1\geq\cdots\geq m_{r-1}\geq|m_r|\right\}} &
\text{for type $D$,}\\
{\left\{{{\bf m}}\in{\mathbb{Z}}^r\,\middle|\,m_1\geq\cdots\geq m_r\geq 0\right\}} & \text{else.}\\
\end{cases}\end{aligned}$$
The questions of reducibility, composition series and unitarizability of the composition factors of $I(\nu)$ were completely answered by Johnson [@Joh90; @Joh92], [Ø]{}rsted–Zhang [@OZ95] and Sahi [@Sah92] for case $A$, by Sahi [@Sah95] and Zhang [@Zha95] for the cases $C$ and $D$, and by the authors [@MS12] for the cases $B$ and $BC$. The precise results differ from case to case, but they all have a common feature. Let $$\nu_k=\frac{p}{2}-k\frac{d}{2}, \qquad \mbox{for }k=0,\ldots,r-1.$$ We say an irreducible constituent of $I(\nu)$ has *rank $k$*, $0\leq k\leq r$, if its $K$-types, identified with their heighest weights in $\Lambda\subseteq{\mathbb{Z}}^r$, are contained in a single $k$-dimensional subspace of ${\mathbb{R}}^r$.
\[thm:StructureDegPrincipalSeries\]
1. \[StructureDegPrincipalSeries1\] For $X$ not of type $A$ or $D_2$, and for any $k=0,\ldots,r-1$ the representation $I(\nu)$ has an irreducible quotient of rank $k$ if and only if $\nu=\nu_k$. This quotient $J(\nu_k)$ has the $K$-type decomposition $$J(\nu_k)_{K-\textup{finite}} = \bigoplus_{{{\bf m}}\in\Lambda,\,m_{k+1}=0}\hspace{-.5cm}V_{{\bf m}}.$$ Moreover, $J(\nu_k)$ is unitarizable if and only if either $k=0$ or $k>0$ and the pair $({\mathfrak{g}},{\mathfrak{l}})$ is not of the form $({\mathfrak{sl}}(p+q,{\mathbb{F}}),{\mathfrak{s}}({\mathfrak{gl}}(p,{\mathbb{F}})\times{\mathfrak{gl}}(q,{\mathbb{F}})))$, ${\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}}$ with $p\neq q$.
2. For $X$ of type $A$ or $D_2$ a similar statement as in (\[StructureDegPrincipalSeries1\]) holds if one replaces $I(\nu)$ by the induced representation ${\textup{Ind}}_P^G(\sigma\otimes e^\nu\otimes{\mathbf{1}})$ for a certain unitary character $\sigma$ of $M$. In type $A$ the corresponding quotient is always unitarizable, whereas in type $D_2$ we have ${\mathfrak{g}}={\mathfrak{o}}(p,q)$ and the quotient is unitarizable if and only if $p+q$ is even.
Because of the special role of type $A$ and $D_2$ we exclude these from some of our statements by assuming $d_+=d_-$. We remark that modifications of the main statements also hold in type $A$ or $D_2$. In fact, for $X$ of type $A$ the Lie algebra ${\mathfrak{g}}$ is Hermitian of tube type and the quotients $J(\nu_k)$ are contained in the analytic continuation of the holomorphic discrete series (the discrete part of the so-called Wallach or Berezin–Wallach set). $L^2$-models for these representations were constructed by Rossi–Vergne [@RV76] (see also Sahi [@Sah92]). For $X$ of type $D_2$ the Lie algebra ${\mathfrak{g}}$ is the indefinite orthogonal algebra ${\mathfrak{o}}(p,q)$ and the irreducible quotient $J(\nu_1)$ is the minimal representation of $O(p,q)$. An $L^2$-model was constructed by Kobayashi–[Ø]{}rsted [@KO03c] (see also Kobayashi–Mano [@KM08] for a detailed analysis of this model). A construction of these models by the same methods as used in this paper can be found in [@Moe13 Section 2.1] for the Hermitian cases and in [@HKM14] for the case $k=1$ (including ${\mathfrak{g}}={\mathfrak{o}}(p,q)$).
Jordan pairs {#sec:JordanTheoryOfSymmRSpaces}
------------
Recall that the pair $V=(V^+,V^-)=({\mathfrak{n}},\overline{\mathfrak{n}})$ turns into a real Jordan pair when equipped with the trilinear products $${\left\{\,,\,,\,\right\}}\colon V^\pm\times V^\mp\times V^\pm\to V^\pm, \quad (x,y,z)\mapsto{\left\{x,y,z\right\}}=-[[x,y],z].$$ We refer to [@Loo75] for a detailed introduction to Jordan pairs and to [@Ber00] for a classification. Simplicity of $G$ implies that $V$ is in fact a simple real Jordan pair. As usual, we define additional operators $Q_x$, $Q_{x,z}$, and $D_{x,y}$ by $$Q_xy=\tfrac{1}{2}{\left\{x,y,x\right\}},\qquad Q_{x,z}y=D_{x,y}z={\left\{x,y,z\right\}}.$$ The following three identities are needed later and are contained in the Appendix of [@Loo77] as JP$7$, JP$8$ and JP$16$ (note that we use different normalizations): $$\begin{aligned}
& D_{{\left\{x,y,z\right\}},y} = D_{z,Q_yx} + D_{x,Q_yz},\label{JP7}\\
& D_{x,{\left\{y,x,z\right\}}} = D_{Q_xy,z} + D_{Q_xz,y},\label{JP8}\\
& {\left\{{\left\{x,y,u\right\}},v,z\right\}} - {\left\{u,{\left\{y,x,v\right\}},z\right\}} = {\left\{x,{\left\{v,u,y\right\}},z\right\}} - {\left\{{\left\{u,v,x\right\}},y,z\right\}}.\label{JP16}\end{aligned}$$
The trace of the operator $D_{x,y}$ on $V^+$ defines a non-degenerate pairing, $$\begin{aligned}
\label{eq:traceform}
\tau\colon V^+\times V^-\to{\mathbb{R}}, \quad (x,y)\mapsto \tfrac{1}{2p}\operatorname{Tr}_{V^+}(D_{x,y}),\end{aligned}$$ the *trace form* on $(V^+,V^-)$, where $p$ is the structure constant defined in . The trace form is related to the Killing form $\kappa$ of ${\mathfrak{g}}$ by $$\tau(x,y) = -\tfrac{1}{4p}\,\kappa(x,y),$$ see [@Sat80 I.§7], and the normalization is chosen such that $$\begin{aligned}
\label{eq:traceformonprimitive}
\tau(E_k,-F_k) = \tfrac{1}{8p}\kappa(H_k,H_k)=1,\end{aligned}$$ where $E_k$ and $F_k$ are the root vectors corresponding to strongly orthogonal roots.
In the Jordan setting, we abbreviate the adjoint action of $T\in{\mathfrak{l}}$ on $x\in V^\pm$ by $T x=[T,x]$. Likewise, $h x={\textup{Ad}}(h)x$ denotes the adjoint action of $h\in L$ on $x\in V^\pm$. We note that $$h {\left\{x,y,z\right\}} = {\left\{hx,hy,hz\right\}}$$ for all $x,z\in V^\pm$ and $y\in V^\mp$, i.e., the pair $(h|_{V^+},h|_{V^-})\in{\textup{GL}}(V^+)\times{\textup{GL}}(V^-)$ is an automorphism of the Jordan pair $V$. There are automorphisms of particular importance: For $(x,y)\in V^\pm\times V^\mp$ the *Bergman operator* $B_{x,y}\in{\textup{End}}(V^\pm)$ is defined by $${B_{x,\,y}}={\textup{id}}_{V^\pm}-D_{x,y}+Q_xQ_y.$$ It is well-know that ${B_{x,\,y}}$ is invertible if and only if ${B_{y,\,x}}$ is invertible, and in this case, $({B_{x,\,y}},{B_{y,\,x}}^{-1})$ is an automorphism of $V$. Moreover, there is an element $h\in L$ such that $h|_{V^+}={B_{x,\,y}}$ and $h|_{V^-}={B_{y,\,x}}^{-1}$. By abuse of notation, we simply write $({B_{x,\,y}},{B_{y,\,x}}^{-1})\in L$. The benefit of the Bergman operator becomes apparent in the following identity, which describes the decomposition of particular group elements $g\in G$ according to $\overline NLN\subseteq G$, see [@Loo77 Theorem 8.11]:
Let $x\in V^+$, $y\in V^-$ be such that ${B_{x,\,y}}$ is invertible. Then, $$\begin{aligned}
\label{eq:BergmanIdentity}
\exp(x)\exp(y) = \exp(y^x)({B_{x,\,y}},{B_{y,\,x}}^{-1})\exp(x^y)
\end{aligned}$$ where $x^y={B_{x,\,y}}^{-1}(x-Q_xy)$, $y^x={B_{y,\,x}}^{-1}(y-Q_yx)$.
The determinant $\operatorname{Det}_{V^+}{B_{x,\,y}}$ of the Bergman operator turns out to be the $2p$’th power of an irreducible polynomial $\Delta\colon V^+\times V^-\to{\mathbb{R}}$, $$\begin{aligned}
\label{eq:JPDet}
\operatorname{Det}_{V^+}{B_{x,\,y}} = \Delta(x,y)^{2p},\end{aligned}$$ where $p$ is the structure constant of $X$ defined in . The polynomial $\Delta$ is called the *Jordan pair determinant* (or the *generic norm*, cf. [@Loo75]).
The Cartan involution $\theta$ on ${\mathfrak{g}}$ induces an involution $V^\pm\to V^\mp$ on the Jordan pair $(V^+,V^-)$, which for simplicity is denoted by $$\overline x=\theta(x)$$ for $x\in V^\pm$. For later calculations, we note the following useful Lemma.
\[lem:expDecomp\] For $y\in V^-$, the Bergman operator ${B_{\overline y,\,-y}}$ is positive definite with respect to the trace form $\tau$. Moreover, the element $\exp(y)$ admits a decomposition $\exp(y)=k_y\ell_yn_y$ according to $G=KLN$ with $$\ell_y =({B_{-\overline{y},\,y}}^{{\nicefrac{1}{2}}}, {B_{y,\,-\overline y}}^{-{{\nicefrac{1}{2}}}}).$$
Write $\exp(y)=k_y\ell_yn_y$ with $k_y\in K$, $\ell_y\in L$ and $n_y\in N$. We may choose $\ell_y$ such that $\theta(\ell_y)=\ell_y^{-1}$ by putting the $(M\cap K)$-part of $\ell_y$ into $k_y\in K$. Then, on the one hand $$\begin{aligned}
\theta(\exp(y)^{-1})\exp(y) &= \theta(n_y)^{-1}\ell_yk_y^{-1}k_y\ell_yn_y
= \theta(n_y)^{-1}\ell_y^2n_y.\end{aligned}$$ On the other hand, $\theta(\exp(y)^{-1}) = \exp(-\overline y)$ with $\overline y\in V^+$, so yields $$\theta(\exp(y)^{-1})\exp(y)
= \exp(y^{-\overline y})({B_{-\overline{y},\,y}}, {B_{y,\,-\overline y}}^{-1})\exp((-\overline y)^y).$$ Since the decomposition of an element in $G$ according to $\overline{N}LN\subseteq G$ is unique, this implies $\ell_y^2 = ({B_{-\overline{y},\,y}}, {B_{y,\,-\overline y}}^{-1})$.
The non-compact picture
-----------------------
The character $\chi_\nu={\mathbf{1}}\otimes e^\nu$ of $L=MA$ which is used to define the principal series $I(\nu)$ in Section \[sec:degenerateprincipalseries\] is recovered in the Jordan setting as $$\begin{aligned}
\label{eq:characterformula}
\chi_\nu(h) = |\operatorname{Det}_{V^+}(h)|^{\frac{\nu}{p}} = |\operatorname{Det}_{V^-}(h)|^{-\frac{\nu}{p}} \qquad \text{for $h\in L$.}\end{aligned}$$ This follows from the observation that for $t\in{\mathbb{R}}$ the central element $\exp(tZ_0)\in L$ acts on $V^\pm$ by $e^{\pm t}\,{\textup{id}}_{V^\pm}$. Up to powers, there exists only one positive character of $L=MA$, since $\dim A=1$ and $M$ is reductive with compact center.
Since $\overline NP\subseteq G$ is dense, the restriction of functions in $I(\nu)$ to $\overline N$ is an injective map. Moreover, since the exponential map $\exp\colon V^-\to\overline N$ is a diffeomorphism, it follows that $$I(\nu)\hookrightarrow C^\infty(V^-), \quad f\mapsto f_{V^-}(y)=f(\exp(y))$$ is an injective map. In the following, we identify $I(\nu)$ with the corresponding subspace of $C^\infty(V^-)$, and by abuse of notation we write $f(y)=f_{V^-}(y)$ for $f\in I(\nu)$ and $y\in V^-$. This is called the *non-compact picture* of $\pi_\nu$. We note that $${\mathcal{S}}(V^-)\subseteq I(\nu)\subseteq C^\infty(V^-)\cap{\mathcal{S}}'(V^-),$$ where ${\mathcal{S}}(V^-)$ denotes the Schwartz space of rapidly decreasing functions and ${\mathcal{S}}'(V^-)$ its dual, the space of tempered distributions.
\[prop:noncompactgroupaction\] The action of $\exp(a)\in\overline N$, $h\in L$ and $\exp(b)\in N$ on $f\in I(\nu)$ is given by ($y\in V^-$) $$\begin{aligned}
\pi_\nu(\exp(a))f(y) &= f(y-a),\\
\pi_\nu(h)f(y) &= \chi_{\nu+\tfrac{p}{2}}(h)f(h^{-1}y),\\
\pi_\nu(\exp(b))f(y) &= |\Delta(-b,y)|^{-2\nu-p}f(y^{-b}).
\end{aligned}$$
This is an easy consequence of the identities and .
The infinitesimal action of ${\mathfrak{g}}$ on $I(\nu)$ is determined in a straightforward way from this by using and the derivatives $$\left.{\mathrm{d}}_b(x\mapsto y^x)\right|_{x=0} = Q_yb, \qquad \left.{\mathrm{d}}_b(x\mapsto\Delta(x,y))\right|_{x=0} = -\tau(b,y).$$ Here ${\mathrm{d}}_v$ denotes the differential of a map in the direction of $v$. We thus obtain the following result.
\[prop:liealgaction\] The infinitesimal action ${\mathrm{d}}\pi_\nu$ of $\pi_\nu$ is given by $$\begin{aligned}
{\mathrm{d}}\pi_\nu(a)f(y) &= -{\mathrm{d}}_af(y), & &a\in V^-,\\
{\mathrm{d}}\pi_\nu(T)f(y) &= -{\mathrm{d}}_{Ty}f(y)+(\tfrac{\nu}{p}+\tfrac{1}{2})\,\operatorname{Tr}_{V^+}(T)\,f(y),
& &T\in{\mathfrak{l}},\\
{\mathrm{d}}\pi_\nu(b)f(y) &= -{\mathrm{d}}_{Q_yb}f(y)-(2\nu+p)\,\tau(b,y)\,f(y), & &b\in V^+.\end{aligned}$$
The representations $(I(\nu),\pi_\nu)$ are spherical and the trivial representation of $K$ occurs with multiplicity one. We determine an explicit expression for the $K$-invariant vector in $I(\nu)$:
\[prop:SphericalVectorNonCptPicture\] The unique $K$-invariant vector $\phi_{\nu}$ in $I(\nu)$ with $\phi_{\nu}(0)=1$ is given by $$\phi_{\nu}(y) = \Delta(-\overline{y},y)^{-\nu-\frac{p}{2}}, \qquad y\in V^-.$$
Let $f\colon G\to{\mathbb{C}}$ denote the $K$-invariant vector in the induced picture of $I(\nu)$, normalized by $f({\mathbf{1}})={\mathbf{1}}$. Then $$\begin{aligned}
\phi_{\nu}(y) &= f(\exp(y)).\end{aligned}$$ We write $\exp(y)=k_y\ell_yn_y$ with $k_y\in K$, $\ell_y\in L$ and $n_y\in N$ as in Lemma \[lem:expDecomp\]. Due to $K$-invariance of $f$ this yields $$\begin{aligned}
\phi_{\nu}(y) = \chi_{\nu+\tfrac{p}{2}}(\ell_y)^{-1}.\end{aligned}$$ Finally, applying to $\ell_y=({B_{-\overline{y},\,y}}^{{\nicefrac{1}{2}}}, {B_{y,\,-\overline y}}^{-{{\nicefrac{1}{2}}}})$ and using proves the assertion.
Structure and geometry of the orbits {#sec:OrbitStructure}
====================================
It is well-known that the action of $L$ decomposes $V^+$ into finitely many orbits, which can be described explicitly. For this recall the ${\mathfrak{sl}}_2$-triples $(E_k,H_k,F_k)$, $1\leq k\leq r$, with $E_k\in{\mathfrak{n}}$, $H_k\in{\mathfrak{l}}$, $F_k\in\overline{\mathfrak{n}}$ defined in Section \[subsec:degenerateprincipalseries\]. For $0\leq k,\ell\leq r$ with $k+\ell\leq r$ put $$o_{k,\ell} = \sum_{i=1}^k E_i - \sum_{j=k+1}^{k+\ell} E_j\in{\mathfrak{n}},$$ and let ${\mathcal{O}}_{k,\ell}=L\cdot o_{k,\ell}$ denote the $L$-orbit of $o_{k,\ell}$. The following result is due to Kaneyuki [@Kan98], see also [@FaEtAl00 Part II]:
Every $L$-orbit in $V^+$ is of the form ${\mathcal{O}}_{k,\ell}$ for some $0\leq k,\ell\leq r$ with $k+\ell\leq r$. The orbit ${\mathcal{O}}_{k,\ell}$ is open if and only if $k+\ell=r$. Moreover, the non-open orbits in $V^+$ are given by $$\begin{aligned}
&{\left\{{\mathcal{O}}_{k,\ell}\,\middle|\,k,\ell\geq 0,\ k+\ell\leq r-1\right\}} & &\text{for type $A$,}\\
&{\left\{{\mathcal{O}}_{k,0}\,\middle|\,0\leq k\leq r-1\right\}} & &\text{for type $B$, $BC$, $C$ or $D$.}\end{aligned}$$
In this section, we find local charts for the orbits ${\mathcal{O}}_{k,\ell}$, which are used in Section \[sec:tangency\] to show that for certain parameters the Bessel operators act tangentially along these orbits. We further determine those orbits which carry an $L$-equivariant measure, and provide integral formulas for these measures. This will be carried out in the framework of Jordan pairs and idempotents.
Idempotents, Peirce decomposition, and rank
-------------------------------------------
For convenience to the reader, we recall some basic notions from Jordan theory used in the sequel. Most statements are valid for arbitrary simple real Jordan pairs $V=(V^+,V^-)$.
An *idempotent* is a pair ${{\bf e}}=(e,e')\in V^+\times V^-$ satisfying $$Q_ee'=e \qquad \text{and} \qquad Q_{e'}e=e'.$$ Then, ${{\bf e}}$ induces a *Peirce decomposition* of $V^\pm$ into eigenspaces of $D_{e,e'}$ and $D_{e',e}$, respectively, $$\begin{aligned}
V^\pm=V_2^\pm({{\bf e}})\oplus V_1^\pm({{\bf e}})\oplus V_0^\pm({{\bf e}})
\qquad\text{with}\quad
\left\{\begin{aligned}
&V^+_k({{\bf e}}) = {\{x\in V^+\,|\,D_{e,e'}(x) = kx\}},\\
&V^-_k({{\bf e}}) = {\{y\in V^-\,|\,D_{e',e}(y) = ky\}}.
\end{aligned}\right.\end{aligned}$$ For a given ${{\bf e}}$, we often simply write $V^\pm_k=V^\pm_k({{\bf e}})$. The *Peirce rules* $$\begin{aligned}
\label{eq:peircerules}
{\left\{V^\pm_k,V^\mp_\ell,V^\pm_m\right\}}\subseteq V^\pm_{k-\ell+m}\qquad\text{and}\qquad
{\left\{V^\pm,V^\mp_2,V^\pm_0\right\}}=0\end{aligned}$$ describe the algebraic relations between Peirce spaces, see [@Loo75 §5] for details. In particular, each $V_k=(V^+_k,V^-_k)$ is a Jordan subpair of $V=(V^+,V^-)$. Two idempotents ${{\bf e}},{{\bf c}}$ are *orthogonal*, if ${{\bf e}}\in V^+_0({{\bf c}})\times V^-_0({{\bf c}})$. In this case, the sum ${{\bf e}}+{{\bf c}}=(e+c,e'+c')$ is also idempotent. An idempotent ${{\bf e}}$ is called *primitive*, if it cannot be decomposed into the sum of non-zero orthogonal idempotents. A maximal system of orthogonal primitive idempotents is called a *frame*. Any frame has the same number of elements, called the *rank* of the Jordan pair $(V^+,V^-)$. The rank of an idempotent ${{\bf e}}$ is by definition the rank of $(V_2^+({{\bf e}}),V_2^+({{\bf e}}))$. Any element $e\in V^+$ admits a completion to an idempotent ${{\bf e}}=(e,e')$ with $e'\in V^-$. The rank of ${{\bf e}}$ is independent of the choice of $e'$, so ${\textup{rank}}(e)={\textup{rank}}({{\bf e}})$ is well-defined.
\[prop:ClassificationV1\] Let $V=(V^+,V^-)$ be a simple real Jordan pair, ${{{\bf e}}}=(e,e')\in V^+\times V^-$ an idempotent of rank $1\leq k\leq r-1$ and $V^\pm=V_2^\pm\oplus V_1^\pm\oplus V_0^\pm$ the corresponding Peirce decomposition.
1. \[ClassificationV1-1\] The subpair $V_1=(V_1^+,V_1^-)$ is either a simple real Jordan pair or the direct sum of two simple real Jordan pairs. In the latter case, the two simple summands are isomorphic if and only if $V\not\simeq(M(p\times q,{\mathbb{F}}),M(q\times p,{\mathbb{F}}))$, ${\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}}$ with $p\neq q$.
2. \[ClassificationV1-2\] We have $\operatorname{Tr}_{V^+}(T)=0$ for every $T\in{\mathfrak{l}}$ with $T|_{V_2^+\oplus V_0^+}=0$ and $TV_1^+\subseteq V_1^+$ if and only if $V\not\simeq(M(p\times q,{\mathbb{F}}),M(q\times p,{\mathbb{F}}))$, ${\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}}$ with $p\neq q$.
We first prove the statements for simple complex Jordan pairs. These are classified, and we check \[ClassificationV1-1\] and \[ClassificationV1-2\] for all pairs separately. Regarding statement \[ClassificationV1-2\], we note that $[T,D_{u,v}]=D_{Tu,v}+D_{u,Tv}=0$ if $(u,v)\in V_2^+\times V_2^-$ or $(u,v)\in V_0^+\times V_0^-$. Hence $T$ commutes with the Lie subalgebra ${\mathfrak{l}}_{0,2}$ of ${\mathfrak{l}}$ generated by $D_{u,v}$ with $(u,v)\in V_2^+\times V_2^-$ and $(u,v)\in V_0^+\times V_0^-$. If ${\mathfrak{l}}_{0,2}$ acts irreducibly on each simple factor of $V_1^+$, then $T$ has to be scalar on each factor. For $V_1$ simple this scalar $\lambda$ has to be zero, because the Peirce rule $0\neq{\left\{V_2^+,V_1^-,V_0^+\right\}}\subseteq V_1^+$ implies $-\lambda=\lambda$. (Note that if $T$ acts on $V_1^+$ by $\lambda$ then it acts on $V_1^-$ by $-\lambda$.) For $V_1$ the sum of two simple pairs $U$ and $W$ the scalars have to add up to zero since $0\neq{\left\{U^+,W^-,V_2^+\right\}}\subseteq V_2^+$. Hence the trace of $T$ is zero if and only if the dimensions of $U$ and $W$ agree. We will see that this is the case if and only if $U\simeq W$.\
By this previous observation, for statement \[ClassificationV1-2\] it is sufficient to show that ${\mathfrak{l}}_{0,2}$ acts irreducibly on each simple factor of $V_1^+$. This is true for all but one simple complex Jordan pair (see case for the exception). We now proceed to show \[ClassificationV1-1\] and \[ClassificationV1-2\] for all simple complex Jordan pairs case by case.
1. $V^+=M(p\times q,{\mathbb{C}})$. Let ${{\bf e}}$ be an idempotent of rank $0\leq k\leq r=\min(p,q)$, then $V_1^+\simeq M(k\times(q-k),{\mathbb{C}})\times M((p-k)\times k,{\mathbb{C}})$ which is the direct sum of two simple ideals. Note that these are isomorphic if and only if they have the same dimension. Further, the structure algebra of $V_2^+\simeq M(k\times k,{\mathbb{C}})$ is isomorphic to ${\mathfrak{sl}}(k,{\mathbb{C}})\times{\mathfrak{sl}}(k,{\mathbb{C}})\times{\mathbb{C}}$ and the structure algebra of $V_0^+\simeq M((p-k)\times(q-k),{\mathbb{C}})$ is isomorphic to ${\mathfrak{sl}}(p-k,{\mathbb{C}})\times{\mathfrak{sl}}(q-k,{\mathbb{C}})\times{\mathbb{C}}$. Clearly ${\mathfrak{sl}}(k,{\mathbb{C}})\times{\mathfrak{sl}}(q-k,{\mathbb{C}})$ acts irreducibly on $M(k\times(q-k),{\mathbb{C}})$ and ${\mathfrak{sl}}(p-k,{\mathbb{C}})\times{\mathfrak{sl}}(k,{\mathbb{C}})$ acts irreducibly on $M((p-k)\times k,{\mathbb{C}})$.
2. $V^+={\textup{Sym}}(r,{\mathbb{C}})$. Let ${{\bf e}}$ be an idempotent of rank $0\leq k\leq r$, then $V_1^+\simeq M(k\times(r-k),{\mathbb{C}})$ which is simple. Further, the structure algebra of $V_2^+\simeq{\textup{Sym}}(k,{\mathbb{C}})$ is isomorphic to ${\mathfrak{gl}}(k,{\mathbb{C}})$ and the structure algebra of $V_0^+\simeq{\textup{Sym}}(r-k,{\mathbb{C}})$ is isomorphic to ${\mathfrak{gl}}(r-k,{\mathbb{C}})$. Clearly ${\mathfrak{gl}}(k,{\mathbb{C}})\times{\mathfrak{gl}}(r-k,{\mathbb{C}})$ acts irreducibly on $M(k\times(r-k),{\mathbb{C}})$.
3. $V^+={\textup{Skew}}(m,{\mathbb{C}})$, $m=2r$ or $m=2r+1$. Let ${{\bf e}}$ be an idempotent of rank $0\leq k\leq r$, then $V_1^+\simeq M(2k\times(m-2k),{\mathbb{C}})$ which is simple. Further, the structure algebra of $V_2^+\simeq{\textup{Skew}}(2k,{\mathbb{C}})$ is isomorphic to ${\mathfrak{gl}}(2k,{\mathbb{C}})$ and the structure algebra of $V_0^+\simeq{\textup{Skew}}(m-2k,{\mathbb{C}})$ is isomorphic to ${\mathfrak{gl}}(m-2k,{\mathbb{C}})$. Clearly ${\mathfrak{gl}}(2k,{\mathbb{C}})\times{\mathfrak{gl}}(m-2k,{\mathbb{C}})$ acts irreducibly on $M(2k\times(m-2k),{\mathbb{C}})$,
4. \[exceptionalcase\] $V^+={\mathbb{C}}^n$. Let ${{\bf e}}$ be an idempotent of rank $0\leq k\leq 2$. If $k=0$ or $k=2$ then $V_1^+=\{0\}$. If $k=1$ then $V_1^+\simeq{\mathbb{C}}^{n-2}$ which is simple. Further, the subalgebra of $T\in{\mathfrak{l}}={\mathfrak{so}}(n,{\mathbb{C}})\oplus{\mathbb{C}}$ with $T|_{V_2^+}=0$ and $T|_{V_0^+}=0$ is isomorphic to ${\mathfrak{so}}(n-2,{\mathbb{C}})$ and hence $\operatorname{Tr}_{V_1^+}(T)=0$.
5. $V^+={\textup{Herm}}(3,{\mathbb{O}}_{\mathbb{C}})$. Let ${{\bf e}}$ be an idempotent of rank $0\leq k\leq3$. If $k=0$ or $k=3$ then $V_1^+=\{0\}$. In the cases $k=1$ and $k=2$ the subpair $V_1$ and the Lie algebra ${\mathfrak{str}}(V_2)\times{\mathfrak{str}}(V_0)$ is the same and therefore it suffices to treat $k=1$. In this case $V_1^+\simeq M(1\times 2,{\mathbb{O}}_{\mathbb{C}})$ which is simple. Further, the structure group of $V_0^+\simeq{\textup{Herm}}(2,{\mathbb{O}}_{\mathbb{C}})\simeq{\mathbb{C}}^{10}$ is isomorphic to ${\mathfrak{so}}(10,{\mathbb{C}})\oplus{\mathbb{C}}$ which acts irreducibly on $M(1\times 2,{\mathbb{O}}_{\mathbb{C}})$ (which is isomorphic to ${\mathbb{C}}^{16}$ as a vector space) by the spin representation.
6. $V^+=M(1\times2,{\mathbb{O}}_{\mathbb{C}})$. Let ${{\bf e}}$ be an idempotent of rank $0\leq k\leq 2$. If $k=0$ or $k=2$ then $V_1^+\simeq{\mathbb{O}}_{\mathbb{C}}\simeq{\mathbb{C}}^8$ on which the structure group ${\mathfrak{so}}(8,{\mathbb{C}})\oplus{\mathbb{C}}$ of $V_2^+\oplus V_0^+\simeq{\mathbb{O}}_{\mathbb{C}}\simeq{\mathbb{C}}^8$ acts irreducibly by the standard representation. If $k=1$ then $V_1^+\simeq{\textup{Skew}}(5,{\mathbb{C}})$ which is simple (see e.g. [@Roo08]). Further, the structure algebra of $V_0^+\simeq M(1\times5,{\mathbb{C}})$ is isomorphic to ${\mathfrak{gl}}(5,{\mathbb{C}})$ which acts irreducibly on ${\textup{Skew}}(5,{\mathbb{C}})=\bigwedge^2{\mathbb{C}}^5$.
Next assume $V$ is not complex, then its complexification $V_{\mathbb{C}}$ is a simple complex Jordan pair and the idempotent ${{\bf e}}$ is idempotent in $V_{\mathbb{C}}$. Hence $(V_1)_{\mathbb{C}}$ is either a simple complex Jordan pair, whence $V_1$ is also simple, or the direct sum of two simple complex Jordan pairs, whence $V_1$ is either simple or the direct sum of two simple real Jordan pairs. These are non-isomorphic if and only if $(V_1)_{\mathbb{C}}$ is the sum of two non-isomorphic simple pairs which happens only in the case $V_{\mathbb{C}}\simeq M(p\times q,{\mathbb{C}})$, $p\neq q$. Further, any $T\in{\mathfrak{l}}$ extends ${\mathbb{C}}$-linearly to $T\in{\mathfrak{l}}_{\mathbb{C}}$ and the second statement follows.\
It only remains to show that $V^+=M(p\times q,{\mathbb{F}})$, ${\mathbb{F}}={\mathbb{R}},{\mathbb{H}}$, are the only real forms of $V_{\mathbb{C}}^+=M(p\times q,{\mathbb{C}})$ for which $V_1^+$ is the sum of two simples. But by classification (see e.g. [@Ber00]) there is only one more real form, namely $V^+={\textup{Herm}}(m,{\mathbb{C}})$, $m=p=q$, and for this real form $V_1^+\simeq M(k\times(m-k),{\mathbb{C}})$ which is simple. This finishes the proof.
The rank of $(V^+,V^-)$ coincides with the rank of the symmetric space $X=K/(M\cap K)$. Indeed, recall that the root vectors $E_k$ and $F_k$ of strongly orthogonal roots define a particular frame $({{\bf e}}_1,\ldots,{{\bf e}}_r)$, given by ${{\bf e}}_k=(E_k,-F_k)$, $k=1,\ldots, r$. This frame has the additional property of being compatible with the Cartan involution, i.e., $\overline E_k=-F_k$. More generally, an element $e\in V^+$ is called *tripotent*, if the pair $(e,\overline e)$ is an idempotent. Accordingly, we call $e$ *primitive*, if $(e,\overline e)$ is primitive, and two tripotents $e,c$ are *orthogonal*, if $e\in V_0^+(c,\overline c)$. A maximal system of othogonal primitive tripotents is called a *frame of tripotents*.
Orthogonal idempotents yield compatible Peirce decompositions. Therefore, a frame $({{\bf e}}_1,\ldots,{{\bf e}}_r)$ induces a *joint Peirce decomposition* $$V^{\pm} = \bigoplus_{0\leq i\leq j\leq r} V_{ij}^\pm,$$ where $$\begin{aligned}
V_{ij}^+ &= {\{x\in V^\pm\,|\,{\left\{e_k,e_k',x\right\}} = (\delta_{k i}+\delta_{kj})x\}},\\
V_{ij}^- &= {\{y\in V^\pm\,|\,{\left\{e_k',e_k,y\right\}} = (\delta_{k i}+\delta_{kj})y\}}.\end{aligned}$$ This corresponds to the root space decomposition of ${\mathfrak{n}}$ and $\overline{\mathfrak{n}}$ with respect to ${\mathfrak{t}}_{\mathbb{C}}$ of Section \[sec:degenerateprincipalseries\]. For a frame $(e_1,\ldots,e_r)$ of tripotents, the Cartan involution relates the positive and negative joint Peirce spaces: $$\overline{V_{ij}^+}=\theta V_{ij}^+=V_{ij}^-.$$ For $e=e_1+\cdots+e_r$, the maps $x\mapsto Q_e\overline x$ and $y\mapsto Q_{\overline e}\overline y$ define involutions on $V_{ij}^+$ and $V_{ij}^-$ for $1\leq i,j\leq r$ with $\pm 1$-eigenspace decomposition $$V_{ij}^\pm = A_{ij}^\pm\oplus B_{ij}^\pm.$$ The structure constants of the symmetric $R$-space $X$ are related to these refined Peirce spaces via $$\begin{aligned}
\label{eq:Jordanstructureconstants}
\dim B_{ii}^\pm=e,\qquad \dim A_{ij}^\pm=d_+,\qquad \dim B_{ij}^\pm=d_-,\qquad \dim V_{0i}^\pm=b.\end{aligned}$$ Moreover, $A_{ii}^+={\mathbb{R}}e_i$ and $A_{ii}^-={\mathbb{R}}\overline e_i$. The constant $p$ defined in is also called the *genus* of the Jordan pair $V$.
We prove some summation formulas which are needed later on.
\[lem:basessums\] Set $I=\{1,\ldots, n\}$, and let $\{c_\alpha\}_{\alpha\in I}$ be a basis of $V^+$, and $\{\widehat{c}_\alpha\}_{\alpha\in I}$ be the basis of $V^-$ dual to $\{c_\alpha\}_{\alpha\in I}$ with respect to the trace form $\tau$.
1. \[basessums1\] We have $$\sum_{\alpha\in I} D_{c_\alpha,\widehat{c}_\alpha} = 2p\cdot{\textup{id}}_{V^+}.$$
2. \[basessums2\] If ${{\bf e}}=(e,e')$ is an idempotent of rank $k$, and the basis $\{c_\alpha\}_{\alpha\in I}$ is compatible with the Peirce decomposition with respect to ${{\bf e}}$, i.e., $I=I_2\sqcup I_1\sqcup I_0$ with $c_\alpha\in V_\ell^+$ if and only if $\alpha\in I_\ell$, then $$\begin{aligned}
\sum_{\alpha\in I_2} D_{c_\alpha,\widehat{c}_\alpha} &= p_2\cdot D_{e,e'},\\
\sum_{\alpha\in I_1}\left. D_{c_\alpha,\widehat{c}_\alpha}\right|_{V_2^+} &= 2(p-p_2)\cdot{\textup{id}}_{V_2^+} = (2(r-k)d+b)\cdot{\textup{id}}_{V_2^+},\\
\sum_{\alpha\in I_1}\left. D_{c_\alpha,\widehat{c}_\alpha}\right|_{V_0^+} &= 2(p-p_0)\cdot{\textup{id}}_{V_0^+} = 2kd\cdot{\textup{id}}_{V_0^+}.
\end{aligned}$$ Here, $p_2=(e+1)+(k-1)d$ resp. $p_0=(e+1)+(r-k-1)d+\frac{b}{2}$ is the structure constant defined as in but with respect to the simple Jordan pair $(V_2^+,V_2^-)$ resp. $(V_0^+,V_0^-)$.
3. If further $V\not\simeq(M(p\times q,{\mathbb{F}}),M(q\times p,{\mathbb{F}}))$, ${\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}}$ with $p\neq q$,, then $$\begin{aligned}
\sum_{\alpha\in I_1}\left. D_{c_\alpha,\widehat{c}_\alpha}\right|_{V_1^+} &= (2p-p_2-p_0)\cdot{\textup{id}}_{V_1^+} = \left(rd+\tfrac{b}{2}\right)\cdot{\textup{id}}_{V_1^+}.
\end{aligned}$$
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1. This identity is a consequence of the following calculation. For arbitrary $v\in V^+$, $w\in V^-$, associativity of the trace form yields $$\begin{aligned}
\tau(\sum_{\alpha\in I} D_{c_\alpha,\widehat{c}_\alpha}v, w) &= \sum_{\alpha\in I}\tau({\left\{c_\alpha,\widehat{c}_\alpha,v\right\}},w) = \sum_{\alpha\in I}\tau({\left\{v,w,c_\alpha\right\}},\widehat{c}_\alpha)\\
&=\operatorname{Tr}_{V^+}(D_{v,w}) = 2p\,\tau(v,w).\end{aligned}$$
2. For the first identity, we apply to the Jordan pair $(V_2^+,V_2^-)$ and obtain $$\begin{aligned}
D_{e,e'}=\frac{1}{2p_2}\sum_{\alpha\in I_2}D_{{\left\{c_\alpha,\widehat c_\alpha,e\right\}},e'}\end{aligned}$$ Since $e=Q_ee'$, ${\left\{e',e,\widehat c_\alpha\right\}}=2\widehat c_\alpha$, and $$\begin{aligned}
{\left\{c_\alpha,\widehat c_\alpha,Q_ee'\right\}}
&\stackrel{\eqref{JP8}}{=}
{\left\{c_\alpha,{\left\{e',e,\widehat c_\alpha\right\}},e\right\}}-{\left\{c_\alpha,e',Q_e\widehat c_\alpha\right\}}\\
&\stackrel{\hphantom{\eqref{JP8}}}{=} 2{\left\{c_\alpha,\widehat c_\alpha,e\right\}} - {\left\{c_\alpha,e',Q_e\widehat c_\alpha\right\}},\end{aligned}$$ hence ${\left\{c_\alpha,\widehat c_\alpha,e\right\}}={\left\{c_\alpha,e',Q_e\widehat c_\alpha\right\}}$, we obtain $$\begin{aligned}
D_{e,e'}
=\frac{1}{2p_2}\sum_{\alpha\in I_2} D_{{\left\{c_\alpha,e',Q_e\widehat c_\alpha\right\}},e'}
\stackrel{\eqref{JP7}}{=}
\frac{1}{2p_2}\sum_{\alpha\in I_2} \left(D_{Q_e\widehat c_\alpha,Q_{e'}c_\alpha}
+ D_{c_\alpha,Q_{e'}Q_e\widehat c_\alpha}\right).\end{aligned}$$ Since $\{Q_e\widehat c_\alpha\}_{\alpha\in I_2}$, $\{Q_{e'}c_\alpha\}_{\alpha\in I_2}$ is another pair of dual bases for the Jordan pair $(V_2^+,V_2^-)$, and the operator $\sum_{\alpha\in I_2}D_{c_\alpha,\widehat c_\alpha}$ is easily seen to be independent of the choice of such bases, we conclude that $$D_{e,e'}=\frac{1}{p_2}\sum_{\alpha\in I_2} D_{c_\alpha,\widehat c_\alpha}.$$ Concerning the second identity, we note that $$\sum_{\alpha\in I_1} D_{c_\alpha,\widehat{c}_\alpha}
=\sum_{\alpha\in I}D_{c_\alpha,\widehat{c}_\alpha}
-\sum_{\alpha\in I_2}D_{c_\alpha,\widehat{c}_\alpha}
-\sum_{\alpha\in I_0}D_{c_\alpha,\widehat{c}_\alpha}.$$ Since for $\alpha\in I_0$, $D_{c_\alpha,\widehat{c}_\alpha}$ vanishes on $V_2^+$ due to the Peirce rules, the second identity follows from the previous ones.\
Finally, for the last identity note that $\sum_{\alpha\in I_1}D_{c_\alpha,\widehat{c}_\alpha}$ commutes with the group $L_{[e]}={\{g\in L\,|\,gV_k^\pm\subseteq V_k^\pm\,\forall\,k=0,1,2\}}$. Since $L_{[e]}\to{\textup{Str}}(V_0),\,g\mapsto g|_{V_0}$ is surjective and ${\textup{Str}}(V_0)$ acts irreducibly on the simple Jordan pair $V_0$, the operator $\sum_{\alpha\in I_1}D_{c_\alpha,\widehat{c}_\alpha}$ acts on $V_0^+$ by a scalar. To determine this scalar we note that for $c_\alpha\in V_{0i}^+$ ($1\leq i\leq k$) and $x\in V_{jk}$ ($k+1\leq j,k\leq r$) we have $D_{c_\alpha,\widehat{c}_\alpha}x=0$. Hence, it suffices to compute the scalar for the Jordan algebra $\widetilde{V}^+=\bigoplus_{1\leq i,j\leq r}V_{ij}^+$. Using the previous identity this shows that the scalar is given by $2(\widetilde{p}-\widetilde{p}_0)$, where $\widetilde{p}$ and $\widetilde{p}_0$ are the structure constants for $\widetilde{V}^+$ and $\widetilde{V}_0^+=\bigoplus_{k+1\leq i,j\leq r}V_{ij}^+$. Then $$\widetilde{p} = e+1+(r-1)d, \qquad \widetilde{p}_0 = e+1+(r-k-1)d$$ and hence $$2(\widetilde{p}-\widetilde{p}_0) = 2(k+1)d = 2(p-p_0).$$
3. Note that by Proposition \[prop:ClassificationV1\] the Jordan pair $V_1^+$ is the direct sum of isomorphic simple Jordan pairs. Hence, thanks to , $T=\sum_{\alpha\in I_1}D_{c_\alpha,\widehat{c}_\alpha}$ acts on $V_1^+$ by a fixed scalar $\lambda$. We further know by that $T$ acts on $V_2^+$ by $2(p-p_2)$ and on $V_0^+$ by $2(p-p_0)$. Since ${\left\{V_2^+,V_1^-,V_0^+\right\}}=V_1^+$ and $T$ acts on $V_1^\pm$ by $\pm\lambda$ this shows the claim.
We state another summation formula for which we did not find a direct proof, but which can be verified case by case using the classification.
\[lem:basessums2\] Let ${{\bf e}}=(e,e')$ be an idempotent with Peirce decomposition $V=V_2\oplus V_1\oplus V_0$, and let $\{c_\alpha\}_{\alpha\in I_1}$ be a basis of $V_1^+$ and $\{\widehat{c}_\alpha\}_{\alpha\in I_1}$ the dual basis of $V_1^-$ with respect to the trace form $\tau$. Then $$\sum_{\alpha,\beta\in I_1} \left.Q_{c_\alpha,c_\beta}Q_{\widehat{c}_\alpha,\widehat{c}_\beta}\right|_{V_2^+} = 2p_0(p-p_2)\cdot{\textup{id}}_{V_2^+}.$$
Orbit decomposition
-------------------
The notion of rank for elements in $V^+$ introduced in the last section yields the decomposition $$V^+ = \bigsqcup_{k=0}^r {\mathcal{V}}_k,$$ where ${\mathcal{V}}_k\subseteq V^+$ denotes the subset of elements of rank $k$. We construct local charts for ${\mathcal{V}}_k$ to show that ${\mathcal{V}}_k$ is an embedded submanifold. For any $e\in{\mathcal{V}}_k$ let ${{\bf e}}=(e,e')$ be a completion to an idempotent ${{\bf e}}$, and denote by $V^\pm = V^\pm_2\oplus V^\pm_1\oplus V^\pm_0$ the Peirce decomposition with respect to ${{\bf e}}$. For $x\in V^+$ we write $x=x_2+x_1+x_0$ according to this Peirce decomposition. Recall that the Jordan pair $(V^+_2,V^-_2)$ admits invertible elements, i.e., elements $x\in V^+_2$ such that $Q_x\colon V^-_2\to V^+_2$ is invertible. In this case, $x^{-1}=Q_x^{-1}(x)$ denotes the inverse of $x$, which is an element in $V^-_2$. Then, $$N_e={\left\{x\in V^+\,\middle|\,x_2\text{ invertible in $V^+_2$}\right\}}$$ is open and dense in $V^+$. We consider the map $$\begin{aligned}
\label{eq:diffeo}
\varphi_e\colon N_e\to N_e, \quad x\mapsto x+Q_{x_1}x_2^{-1}.\end{aligned}$$
\[prop:LOrbits\] For $0\leq k\leq r$ and $e\in{\mathcal{V}}_k$ the map $\varphi_e:N_e\to N_e$ is a diffeomorphism which maps $N_e\cap(V_2^+\oplus V_1^+)$ onto an open subset of ${\mathcal{V}}_k$. In particular, ${\mathcal{V}}_k$ is an $L$-invariant (embedded) submanifold of $V^+$, and each $L$-orbit in $V^+$ is a union of connected components of ${\mathcal{V}}_k$ for some fixed $k$.
Standard arguments show that $\varphi_e$ is smooth, and since $Q_{x_1}x_2^{-1}$ is an element of $V_0^+$ according to the Peirce rules, a straightforward computation shows that $(x\mapsto x-Q_{x_1}x_2^{-1})$ is a smooth inverse of $\varphi_e$. Next we note that $$\varphi_e(x) = {B_{x_1,\,-x_2^{-1}}}(x_2+x_0)\qquad\text{and}\qquad
({B_{x_1,\,-x_2^{-1}}},{B_{-x_2^{-1},\,x_1}}^{-1})\in L.$$ Since $L$ acts by automorphisms on $(V^+,V^-)$, it follows that $\varphi_e(x)$ has the same rank as $x_2+x_0$. Since the rank of orthogonal idempotents is additive, and since $x_2$ is invertible in $V_2^+$, it follows that $${\textup{rank}}(\varphi_e(x)) = {\textup{rank}}(x_2+x_0) = {\textup{rank}}(e)+{\textup{rank}}(x_0).$$ Therefore, $\varphi_e(x)$ is in ${\mathcal{V}}_k$ if and only if $x_0=0$. It remains to consider the action of $L$ on ${\mathcal{V}}_k$. Since $L$ acts by automorphisms, it is clear that ${\mathcal{V}}_k$ is $L$-invariant. In order to prove that the $L$-orbits consist of connected components of ${\mathcal{V}}_k$, it suffices to show that the derived map ${\mathfrak{l}}\to T_e{\mathcal{V}}_k$ is surjective, where $T_e{\mathcal{V}}_k\simeq V_2^+\oplus V_1^+$. This immediately follows from the fact that ${\mathfrak{l}}$ acts by Jordan pair derivations, and applying the derivation $(D_{x,e'},-D_{e',x})$ to $e$ yields $D_{x,e'}(e) = \ell\cdot x$ for $x\in V_\ell^+$, $\ell\in\{0,1,2\}$.
Fibration and polar decomposition
---------------------------------
For $e\in V^+$, the subspace $[e]=Q_eV^-$ is called the *principal inner ideal* associated to $e$. If ${{\bf e}}=(e,e')$ is a completion to an idempotent, the Peirce rules imply that $[e] = V_2^+({{\bf e}})$. In particular, this Peirce space is independent of the choice of $e'$. Moreover, the product $x\circ y=\tfrac{1}{2}{\left\{x,e',y\right\}}$ turns $[e]$ into a Jordan algebra with unit element $e$. This Jordan algebra structure is in fact also independent of the choice of $e'$, since for $x=Q_eu\in[e]$, we have $$x^2 = Q_xe' = Q_{Q_eu}{e'}=Q_eQ_uQ_ee' = Q_eQ_ue$$ by the fundamental formula, and polarization of this identity also shows independence of the product $x\circ y$ with respect to the choice of $e'$. For a detailed introduction to Jordan algebras, we refer to [@BK66; @FK94].
Let ${{\bf e}}=(e,e')$ be a completion of $e\in{\mathcal{V}}_k$ to an idempotent, and let ${{\bf e}}={{\bf e}}_1+\cdots+{{\bf e}}_k$ be a decomposition into primitive idempotents, ${{\bf e}}_j=(e_j,e_j')$. Then, the restriction of the operator $D_{e_j,e_j'}$ to $[e]$ coincides with (left) multiplication by $e_j$ with respect to the Jordan algebra structure on $[e]$. Therefore, the Peirce decomposition $$[e]=\bigoplus_{i\leq j} V_{ij}^+$$ coincides with the Jordan algebraic Peirce decomposition given by the left multiplication operators $L_{e_j}$ on $[e]$. Moreover, if $e$ and $e_1,\ldots,e_k$ are tripotents, the map $x\mapsto Q_e\overline x$ is a Cartan involution of the Jordan algebra $[e]$, and the refined Peirce decomposition $V_{ij}^+=A_{ij}^+\oplus B_{ij}^+$ coincides with the respective refined Jordan algebraic Peirce decomposition. By these considerations, it follows that the structure constants of the Jordan algebra (namely the dimensions of the various refined Peirce spaces) coincide with the structure constants of the Jordan pair $(V^+,V^-)$.
In what follows, we fix $0\leq k\leq r$. Let $${\mathcal{P}}_k={\{[e]\,|\,e\in{\mathcal{V}}_k\}}$$ denote the space of principal inner ideals in $V^+$ generated by elements of rank $k$. We call ${\mathcal{P}}_k$ the *$k$’th Peirce manifold* associated to $(V^+,V^-)$.
\[prop:Peircemanifold\]
1. \[Peircemanifold1\] The $k$’th Peirce manifold ${\mathcal{P}}_k$ is a smooth compact manifold. Moreover, ${\mathcal{P}}_k$ is an $L$-homogeneous space, and the stabilizer subgroup $Q_{[e]}$ of $[e]\in{\mathcal{P}}_k$ in $L$ is parabolic in $L$. A Levi decomposition of $Q_{[e]}$ is given by $Q_{[e]}=L_{[e]}U_{[e]}$ with $$L_{[e]}=Z_L(D_{e,e'}),\qquad U_{[e]}={\{B_{e,v}\,|\,v\in V_1^-({{\bf e}})\}},$$ where $Z_L(D_{e,e'})$ denotes the centralizer of $D_{e,e'}$ in $L$, i.e., $L_{[e]}$ consists of elements preserving the Peirce decomposition with respect to ${{\bf e}}$.
2. \[Peircemanifold2\] The action of $L_{[e]}$ on $[e]$ is given by elements of the structure group ${\textup{Str}}([e])$ of the Jordan algebra $[e]$, and $U_{[e]}$ acts trivially on $[e]$. Moreover, the induced Lie algebra homomorphism ${\textup{Lie}}(L_{[e]})\to{\textup{Lie}}({\textup{Str}}([e]))$ is onto.
Clearly, $L$ acts on ${\mathcal{P}}_k$, since $h[e]=[he]$ for $e\in{\mathcal{V}}_k$ and $h\in L$. Recall from [@Loo77 §11.8] that $M\cap K\subseteq L$ acts transitively on the set of frames of tripotents (modulo signs) in $V^+$. Since each principal inner ideal is also generated by a maximal tripotent element, it follows that ${\mathcal{P}}_k$ is $(M\cap K)$-homogeneous. In particular, ${\mathcal{P}}_k$ is a compact, $L$-homogeneous manifold. For the following, we fix $[e]\in{\mathcal{P}}_k$ with representative $e\in{\mathcal{V}}_k$, and let ${{\bf e}}=(e,e')$ be a completion to an idempotent with $e'\in V^-$. We first show that the adjoint action of $D_{e,e'}\in{\mathfrak{l}}$ induces the eigenspace decomposition $$\begin{aligned}
\label{eq:ldecomposition}
{\mathfrak{l}}={\mathfrak{l}}_-\oplus{\mathfrak{l}}_0\oplus{\mathfrak{l}}_+\qquad\text{with}\quad
\left\{
\begin{aligned}
{\mathfrak{l}}_-&={\{D_{u,e'}\,|\,u\in V_1^+({{\bf e}})\}},\\
{\mathfrak{l}}_0&={\{T\in{\mathfrak{l}}\,|\,[D_{e,e'},T]=0\}},\\
{\mathfrak{l}}_+&={\{D_{e,v}\,|\,v\in V_1^-({{\bf e}})\}}.
\end{aligned}\right.\end{aligned}$$ Indeed, by the relation $[D_{e,e'},D_{u,v}] = D_{{\left\{e,e',u\right\}},v}-D_{u,{\left\{e',e,v\right\}}}$, it immediately follows that ${\mathfrak{l}}_\pm$ is the $(\pm1)$-eigenspace of ${\textup{ad}}(D_{e,e'})$. Now consider $T\in{\mathfrak{l}}$, and let $Te=u_2+u_1+u_0$ and $Te'=v_2+v_1+v_0$ be the decompositions according to the Peirce decomposition of $V$ with respect to ${{\bf e}}$. Since $$Te = TQ_ee' = {\left\{Te,e',e\right\}} + Q_eTe',$$ it follows that $u_0=0$ and $u_2=-Q_ev_2$. The same argument applied to $Te'$ yields $v_0=0$. Setting $T'=T-D_{u_1,e'}+D_{e,v_1}$, we thus obtain $$\begin{aligned}
[D_{e,e'},T'] &= -D_{Te,e'}-D_{e,Te'} +D_{u_1,e'} + D_{e,v_1} \\
&= -D_{u_2,e'} - D_{e,v_2}
= D_{Q_ev_2,e'}-D_{e,v_2}=0.\end{aligned}$$ Here, the last step follows from the relation $D_{Q_xy,z}=D_{x,{\left\{y,x,z\right\}}} -D_{Q_xz,y}$. This proves . This implies that $Q_{[e]}=N_L({\mathfrak{l}}_0\oplus{\mathfrak{l}}_+)$ is a parabolic subgroup of $L$ with Levi decomposition $Q_{[e]}=L_{[e]}U_{[e]}$ given by $$L_{[e]}=Z_L(D_{e,e'}), \qquad U_{[e]}=\exp({\mathfrak{l}}_+).$$ Since $\exp(D_{e,v})={B_{e,\,-v}}$ due to the Peirce rules, this completes the proof of .\
We next consider the action of $h\in Q_{[e]}$ on $[e]$. Due to the Peirce rules, it is clear that $h\in U_{[e]}$ acts as the identity on $[e]$. Now let $h\in L_{[e]}$. Recall that the quadratic representation $P_x$ of $x\in[e]$ is given by $P_x=Q_xQ_{e'}$, and the Jordan algebra trace form of $[e]$ is a constant multiple of $\tau_{[e]}(x,y)=\tau(x,Q_{e'}y)$. Moreover, $h$ acts on $[e]$ by structure automorphisms if and only if $P_{hx}=hP_xh^\#$, where $h^\#$ denotes the adjoint of $h$ with respect to $\tau_{[e]}$. One easily checks that $h^\#=Q_eh^{-1}Q_{e'}$, and it follows that $P_{hx}=hP_xh^\#$.\
We finally note that the Lie algebra of ${\textup{Str}}([e])$ is generated by all $D_{x,y}$ with $x\in V_2^+$, $y\in V_2^-$, which also belong to ${\mathfrak{l}}_0\subseteq{\mathfrak{q}}_{[e]}$. Since ${\textup{Lie}}(L_{[e]})={\mathfrak{l}}_0$, it follows that ${\textup{Lie}}(L_{[e]})\to{\textup{Lie}}({\textup{Str}}([e]))$ is onto.
We next consider the relation between the Peirce manifold ${\mathcal{P}}_k$ and the manifold ${\mathcal{V}}_k$.
\[prop:fiberbundle\] The canonical projection $$\pi_k\colon{\mathcal{V}}_k\to{\mathcal{P}}_k, \quad e\mapsto [e]$$ is an $L$-equivariant fiber bundle with fiber over $[e]\in{\mathcal{P}}_k$ consisting of the set $[e]^\times$ of invertible elements in the Jordan algebra $[e]$. Moreover, each connected component of $[e]^\times$ is a reductive symmetric space.
Fix $e\in{\mathcal{V}}_k$, and let $L^0$ denote the identity component of $L$. By Proposition \[prop:LOrbits\], the orbit $L^0\cdot e$ is the connected component of ${\mathcal{V}}_k$ containing $e$. Likewise, $L^0\cdot[e]$ is the connected component of ${\mathcal{P}}_k$ containing $[e]$. Since $\pi_k$ clearly is $L$-equivariant, it follows that $\pi_k$ is locally given as a projection of $L^0$-homogeneous spaces. Hence, ${\mathcal{V}}_k$ is a fiber bundle over ${\mathcal{P}}_k$. Concerning the fiber over $[e]$, recall that $x\in[e]$ is invertible in the Jordan algebra $[e]$ if and only if the quadratic operator $P_x=Q_xQ_{e'}$ is invertible. Equivalently, $[x]=[e]$, which amounts to the condition that $x$ has the same rank as $e$, so $x\in{\mathcal{V}}_k$.\
Let $Y\subseteq[e]^\times$ be a connected component of $[e]^\times$. We may assume that $e\in Y$, otherwise consider the mutation of the Jordan algebra $[e]$ by an element $y\in Y$, see [@BK66] for reference. Recall that $L_{[e]}$ denotes the Levi subgroup of the stabilizer $Q_{[e]}$ of $[e]\in{\mathcal{P}}_k$ in $L$. Due to Proposition \[prop:Peircemanifold\], the restriction map $$\begin{aligned}
\label{eq:restictionmap}
\rho_{[e]}\colon L_{[e]}\to{\textup{Str}}([e]), \quad h\mapsto h|_{[e]}\end{aligned}$$ identifies the identity component of $L_{[e]}$ with the identity component of the Jordan algebraic structure group of $[e]$, denoted by $L'_{[e]}={\textup{Str}}([e])^0$. It follows, that $Y$ is the orbit of $L'_{[e]}$ through $e$, i.e. $$Y=L'_{[e]}\cdot e\cong L'/H'_e,$$ where $H'_e$ denotes the stabilizer subgroup of $e$ in $L'_{[e]}$. This orbit is symmetric, since $$\begin{aligned}
\label{eq:symmetrycondition}
L_{[e]}'^{\sigma,0}\subseteq H_e'\subseteq L_{[e]}'^\sigma,\end{aligned}$$ where $\sigma$ is the involution on $L_{[e]}'$ given by $h\mapsto h^{-\#}$, where $h^\#=Q_eh^{-1}Q_{e'}$ is the adjoint of $h$ with respect to the Jordan trace form $\tau_{[e]}$ on $[e]$.
Let ${\mathcal{E}}_k$ be the tautological vector bundle of ${\mathcal{P}}_k$, $${\mathcal{E}}_k={\{([e],x)\,|\,e\in{\mathcal{V}}_k,\, x\in[e]\}}\subseteq{\mathcal{P}}_k\times V^+,$$ and let ${\mathcal{E}}_k^\times\subseteq{\mathcal{E}}_k$ denote the fiber bundle over ${\mathcal{P}}_k$ with fiber $[e]^\times$ over $[e]$. By means of Proposition \[prop:fiberbundle\], ${\mathcal{V}}_k$ is naturally identified with the fiber bundle ${\mathcal{E}}_k^\times$. Moreover, the topological closure ${\mathcal{V}}_k^{\textup{cl}}$ of ${\mathcal{V}}_k$ in $V^+$ is a real algebraic variety, since it is the zero-set of the ideal generated by Jordan minors associated to idempotents of rank $k+1$. It follows that $${\mathcal{V}}_k^{\textup{cl}}=\bigsqcup_{j=0}^k{\mathcal{V}}_j,$$ and the projection map $${\mathcal{E}}_k\to{\mathcal{V}}_k^{\textup{cl}}, \quad ([e],x)\mapsto x$$ is an $L$-equivariant resolution of singularities.
The preceding propositions yield the following description of the $L$-orbits on $V^+$. Fix a frame $(e_1,\ldots, e_r)$ of tripotents in $V^+$, and let $e=e_1+\cdots+e_k\in{\mathcal{V}}_k$ be the base element of the orbit $${\mathcal{O}}_e=L\cdot e\subseteq{\mathcal{V}}_k,\qquad {\mathcal{O}}_e\cong L/H_e,$$ where $H_e$ denotes the stabilizer of $e$ in $L$. The fibration of ${\mathcal{V}}_k$ over ${\mathcal{P}}_k$ now corresponds to the fibration $$\begin{aligned}
\label{eq:fibration}
L/H_e\cong L\times_{Q_{[e]}} Q_{[e]}/H_e \qquad \text{with fiber} \qquad
Q_{[e]}/H_e\cong L_{[e]}/(L_{[e]}\cap H_e),\end{aligned}$$ where $Q_{[e]}$ and $L_{[e]}$ are given as in Proposition \[prop:Peircemanifold\]. Geometrically, the base of this fibration coincides with the Peirce manifold ${\mathcal{P}}_k$, and the canonical fiber is realized as the $L_{[e]}$-orbit of $e$ in the Jordan algebra $[e]$. Let $L_{[e]}'$ denote the image of $L_{[e]}$ under the restriction map $\rho_{[e]}$ given in . Then, by Proposition \[prop:Peircemanifold\] $L_{[e]}'$ is an open subgroup of the Jordan algebraic structure group ${\textup{Str}}([e])$. In general, $L$ has finitely many connected components, so we note that $L_{[e]}'$ might differ from the corresponding group in the proof of Proposition \[prop:fiberbundle\]. In any case, the canonical fiber $$\begin{aligned}
\label{eq:fiber}
L_{[e]}/(L_{[e]}\cap H_e)\cong L_{[e]}'/H_e',\end{aligned}$$ is a reductive symmetric space, where $H_e'$ denotes the stabilizer of $e\in[e]$ in $L_{[e]}'$, which still satisfies with respect to the involution $\sigma$ on $L_{[e]}'$.
The structure theory of reductive symmetric spaces yields a polar decomposition of the canonical fiber. Recall that $(M\cap K)$ denotes the maximal compact subgroup of $L$. Since $e$ is tripotent, $L_{[e]}$ is $\theta$-stable, so $(M\cap K)_{[e]}=(M\cap K)\cap L_{[e]}$ is maximal compact in $L_{[e]}$, and the image $(M\cap K)_{[e]}'$ of $(M\cap K)_{[e]}$ under the restriction map $\rho_{[e]}$ is maximal compact in $L_{[e]}'$. Define $$\begin{aligned}
\label{eq:fiberCartan}
{\mathfrak{a}}_k=\bigoplus_{j=1}^k {\mathbb{R}}D_{e_j,\overline e_j},\qquad A_k=\exp({\mathfrak{a}}_k).\end{aligned}$$ Since $D_{e_j,\overline e_j}$ acts on $[e]$ by the Jordan algebraic multiplication with $e_j$, we may consider ${\mathfrak{a}}_k$ by abuse of notation also as a subalgebra of ${\textup{Lie}}(L_{[e]}')={\mathfrak{str}}([e])$, and $A_k$ can be considered as a subgroup of $L_{[e]}'$. Then, ${\mathfrak{a}}_k$ is maximal abelian in the subspace of elements $X\in{\mathfrak{str}}([e])$ satisfying $\sigma(X)=-X$, $\theta(X)=-X$. This yields the following polar decompositions of $L_{[e]}$ and $L$, which induce polar decompositions of the corresponding orbits.
\[prop:polardecomposition\] With the above notation, $$L_{[e]}'=(M\cap K)_{[e]}'A_kH_e',\qquad L =(M\cap K) A_k H_e.$$
The first identity is a standard result for reductive symmetric spaces. For the decomposition of $L$, note that Proposition \[prop:Peircemanifold\] yields $L=(M\cap K)Q_{[e]}$, since $Q_{[e]}$ is parabolic. Recall that $Q_{[e]}=L_{[e]}U_{[e]}$. Since $L_{[e]}'=L_{[e]}/\ker\rho_{[e]}$ with $U_{[e]}\subseteq\ker\rho_{[e]}\subseteq H_e$, this yields the second identity.
For later use, we also note the following local description of the $L$-orbit ${\mathcal{O}}_e$. Let $\overline Q_{[e]}=\theta(Q_{[e]})$ be the parabolic subgroup opposite to $Q_{[e]}$. Then, $$\begin{aligned}
\label{eq:oppositeParabolic}
\overline Q_{[e]}=L_{[e]}\overline U_{[e]}\qquad\text{with}\qquad
\overline U_{[e]}=\theta(U_{[e]})={\{{B_{v,\,\overline e}}\,|\,v\in V_1^+\}}\end{aligned}$$ is the corresponding Levi decomposition, and since $\overline U_{[e]}L_{[e]}U_{[e]}\subseteq L$ is open and dense, the $\overline Q_{[e]}$-orbit of $e\in{\mathcal{O}}_e$ is open and dense in ${\mathcal{O}}_e$. In the following, we aim for a description of this part of ${\mathcal{O}}_e$ with respect to the parametrization of ${\mathcal{O}}_e$ induced by the diffeomorphism $\varphi_e$ in .
Let $V_\ell^+$, $\ell=0,1,2$, denote the Peirce spaces with respect to $(e,\overline e)$, and recall that $N_e\subseteq V^+$ denotes the open set of elements $x=x_2+x_1+x_0$ with invertible $x_2\in [e]$. Then $\varphi_e\colon N_e\to N_e$ is a diffeomorphism, and its restriction to $N_e\cap(V_2^+\oplus V_1^+)$ is a diffeomorphism onto an open and dense subset of ${\mathcal{O}}_e$. We note that $V_2^+=[e]$. By means of it is straightforward to check that $N_e\subseteq V^+$ is preserved under the standard action of $\overline Q_{[e]}$ on $V^+$, and hence the $\overline Q_{[e]}$-orbit of $e\in{\mathcal{O}}_e$ is contained in $N_e\cap{\mathcal{O}}_e$.
\[lem:pullbackaction\] The open subset $N_e\subseteq V^+$ is invariant under the action of $\overline Q_{[e]}\subseteq L$, and the pullback of this action along $\varphi_e$ is given and denoted by $$\xi(h)(x)=hx,\qquad
\xi({B_{v,\,\overline e}})(x)=x-{\left\{v,\overline e,x_2\right\}}$$ for $h\in L_{[e]}$, $v\in V_1^+$ and $x\in N_e$. In particular, the pullback action of $\overline Q_{[e]}$ is linear again.
Since $\overline Q_{[e]}$ is generated by $L_{[e]}$ and elements of the form ${B_{v,\,\overline e}}$, the invariance of $N_e$ under the action of $\overline Q_{[e]}$ follows immediately from the Peirce rules. By definition, the pullback of the $\overline Q_{[e]}$-action is given by $\xi(q)(x)=\varphi_e^{-1}(q\cdot\varphi_e(x))$. Since $h\in L_{[e]}$ acts on each Jordan pair $(V_\ell^+, V_\ell^-)$, $\ell=0,1,2$, by automorphisms, it follows that $h(x_2^{-1})=(hx_2)^{-1}$, which implies the formula for $\xi(h)$. For $q={B_{v,\,\overline e}}$, we obtain $$\begin{aligned}
\xi({B_{v,\,\overline e}})(x)
&= x_2+(x_1-{\left\{v,\overline e,x_2\right\}}) \\
&\quad+
(x_0 + Q_vQ_{\overline e}x_2-{\left\{v,\overline e,x_1\right\}}+Q_{x_1}x_2^{-1}-Q_{x_1-{\left\{v,\overline e,x_2\right\}}}x_2^{-1}),\end{aligned}$$ where the terms are sorted with respect to the Peirce space decomposition. Recall from Proposition \[prop:LOrbits\] that the restriction of $\varphi_e$ to $N_e\cap(V_2^+\oplus V_1^+)$ is a diffeomorphism onto an open subset of $N_e\cap{\mathcal{V}}_k$. Since ${\mathcal{V}}_k$ is $\overline Q_{[e]}$-invariant, it follows that $N_e\cap(V_2^+\oplus V_1^+)$ is invariant for the pullback action of $\overline Q_{[e]}$. Therefore, if $x_0=0$, the $V_0^+$-part of $\xi(q)$ must vanish for all $q$. Applied to $\xi({B_{v,\,\overline e}})$, it follows that $\xi({B_{v,\,\overline e}})(x_2+x_1)=x_2+(x_1-{\left\{v,\overline e,x_2\right\}})$, and comparing this with the formula above completes the proof. (We note that using it can also be seen directly that the $V_0^+$-part of $\xi({B_{v,\,\overline e}})(x_2+x_1)$ vanishes.)
This immediately implies:
\[prop:orbitcoordinates\] Let $\Omega_e$ denote the $L_{[e]}$-orbit of $e\in[e]$. Then the restriction of $\varphi_e$ to the open subset $\Omega_e+V_1^+\subseteq N_e$ is a diffeomorphism onto the $\overline Q_{[e]}$-orbit of $e$ in ${\mathcal{O}}_e$. The pullback of the $\overline Q_{[e]}$-action along this diffeomorphism is given and denoted by $$\xi(h)(x_2+x_1)=hx_2+hx_1,\qquad
\xi({B_{v,\,\overline e}})(x_2+x_1)=x_2 + (x_1-{\left\{v,\overline e,x_2\right\}})$$ for $h\in L_{[e]}$, $v\in V_1^+$ and $x_2+x_1\in\Omega_e+V_1^+\subseteq V_2^+\oplus V_1^+$.
Equivariant measures
--------------------
In this section we determine which of the $L$-orbits in $V^+$ carry an $L$-equivariant measure. Fix $0\leq k\leq r$, and let ${\mathcal{O}}_e=L\cdot e$ be the $L$-orbit through $e\in{\mathcal{V}}_k$. We may assume that $e$ is tripotent, and $e=e_1+\cdots+e_k$, where $e_1,\ldots,e_r$ is a frame of tripotents. We adopt all notations from the last section, so in particular, $H_e$ denotes the stabilizer of $e$ in $L$, so ${\mathcal{O}}_e\cong L/H_e$. Recall that a measure ${\mathrm{d}}\mu$ on ${\mathcal{O}}_e$ is called $\chi$-equivariant, $\chi$ a positive character of $L$, if $${\mathrm{d}}\mu(hx)=\chi(h){\mathrm{d}}\mu(x) \qquad \text{for }h\in L.$$ A simple criterion for the existence and uniqueness of equivariant measures is given in terms of the modular functions $\Delta_L$ and $\Delta_{H_e}$ of $L$ and $H_e$: The orbit ${\mathcal{O}}_e$ admits a $\chi$-equivariant measure if and only if $$\chi(h)=\frac{\Delta_L(h)}{\Delta_{H_e}(h)} \qquad \text{for all $h\in H_e$.}$$ Since $L$ is reductive and hence unimodular we have $\Delta_L=1$, and therefore ${\mathcal{O}}_e$ carries an $L$-equivariant measure if and only if the modular function $\Delta_{H_e}$ extends to a positive character of $L$. Moreover, uniqueness of the equivariant measure corresponds to uniqueness of this extension. Recall that all positive characters of $L$ are of the form $$\chi_\lambda(h)=|\operatorname{Det}_{V^+}(h)|^{\frac{\lambda}{p}}$$ for some $\lambda\in{\mathbb{R}}$, see .
The group $H_e\subseteq L$ decomposes as the semidirect product $H_e=H_{e,\overline e}\ltimes U_{[e]}$, where $H_{e,\overline e}=H_e\cap L_{[e]}$ and $L_{[e]}$, $U_{[e]}$ are given in Proposition \[prop:Peircemanifold\]. Moreover, the modular function $\Delta_{H_e}$ of $H_e$ is given by $$\Delta_{H_e}(h) = |\operatorname{Det}_{V_1^+}(h')|^{-1} = \frac{|\operatorname{Det}_{V_0^+}(h')|}{|\operatorname{Det}_{V^+}(h')|}\qquad\text{for }h=h'u\in H_{e,\overline e}U_{[e]},$$ where $V^\pm=V_2^\pm\oplus V_1^\pm\oplus V_0^\pm$ denotes the Peirce decomposition with respect to $(e,\overline e)$.
Note that $H_{e,\overline e}\subseteq H_e$ is the subgroup that fixes $e\in V^+$ as well as $\overline e\in V^-$. Then each Peirce space $V_k^\pm$ is invariant under the action of $H_{e,\overline e}$. Recall from Proposition \[prop:Peircemanifold\] the subgroups $L_{[e]},U_{[e]}\subseteq L$. Then due to , $H_e$ decomposes into the semidirect product $H_e=H_{e,\overline e}\ltimes U_{[e]}$. We may identify the Lie algebra of $U_{[e]}$ with $V_1^-$. Since ${\textup{Ad}}(h)D_{e,v}=D_{e,hv}$ for $h\in H_{e,\overline e}$ and $v\in V_1^-$, this identification is $H_{e,\overline e}$-equivariant. Then, the modular function of the semidirect product $H_e=H_{e,\overline e}U_{[e]}$ is given by $$\Delta_{H_e}(h)=|\operatorname{Det}_{V_1^-}(h')|\Delta_{H_{e,\overline e}}(h')\Delta_{U_{[e]}}(u),$$ where $h=h'u\in H_{e,\overline e}U_{[e]}$. Since $H_{e,\overline e}$ is $\theta$-stable, it is reductive, and hence unimodular. Moreover, $U_{[e]}$ is abelian. Therefore, the modular function of $H_e$ simplifies to $$\begin{aligned}
\label{eq:modularfunction}
\Delta_{H_e}(h)=|\operatorname{Det}_{V_1^-}(h')|\qquad\text{for }h=h'u\in H_{e,\overline e}U_{[e]}.\end{aligned}$$ Since the trace form $\tau$ gives an $H_{e,\overline e}$-invariant, non-degenerate pairing of $V_1^+$ and $V_1^-$, it follows that $$\operatorname{Det}_{V_1^-}(h)=\operatorname{Det}_{V_1^+}(h)^{-1}\qquad\text{for }h\in L\label{eq:determinantVpm}$$ and hence the first formula for $\Delta_{H_e}$ follows. Moreover, since $h'\in H_{e,\overline e}$ preserves the Peirce spaces $V_\ell^+$, $\ell=0,1,2$, we obtain $$\begin{aligned}
\label{eq:determinantproduct}
\operatorname{Det}_{V^+}(h') = \operatorname{Det}_{V^+_2}(h')\cdot\operatorname{Det}_{V^+_1}(h')\cdot\operatorname{Det}_{V^+_0}(h').\end{aligned}$$ Furthermore, due to Proposition \[prop:Peircemanifold\], $h'$ acts on the simple Jordan algebra $V_2^+=[e]$ as a structure automorphism preserving the identity element $e$. Recall that the determinant of such an automorphism on a simple Jordan algebra is of absolute value $1$ (see e.g. [@FK94]), hence $|\operatorname{Det}_{V^+_2}(h')|=1$ for all $h'\in H_{e,\overline e}$. In combination with , and the second formula for $\Delta_{H_e}$ follows.
\[thm:equivariantmeasures\] For $0\leq k\leq r$ the $L$-orbit ${\mathcal{O}}_e=L.e$ with $e\in{\mathcal{V}}_k$ carries an $L$-equivariant measure with positive character $\chi_\lambda$ if and only if one of the following is satisfied
1. \[equivariantmeasures1\] $k=0$ and $\lambda=0$,
2. \[equivariantmeasures2\] $1\leq k\leq r-1$, $V\not\simeq(M(p\times q,{\mathbb{F}}),M(q\times p,{\mathbb{F}}))$, ${\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}}$ with $p\neq q$, and $\lambda=kd$,
3. \[equivariantmeasures3\] $k=r$, $V$ is unital and $\lambda\in{\mathbb{R}}$,
4. \[equivariantmeasures4\] $k=r$, $V$ is non-unital and $\lambda=p$.
Moreover, the equivariant measure (if it exists) is unique up to scalars.
By the previous lemma ${\mathcal{O}}_e$ admits an $L$-equivariant measure if and only if $|\operatorname{Det}_{V_0^+}|$ (or equivalently $|\operatorname{Det}_{V_1^+}|$) is a power of $|\operatorname{Det}_{V^+}|$ on $H_{e,\overline e}$.\
For $k=0$ we have ${\mathcal{O}}_e=\{0\}$ and follows. Next assume that $k=r$, i.e., ${\mathcal{O}}_e$ is an open orbit. Then $V_0^+=\{0\}$, and $\Delta_{H_e}$ extends to the character $\chi_{-p}$. If, in addition, $V$ is unital, then $V_1^+=\{0\}$, and $\Delta_{H_e}(h)=1$ for all $h\in H_e$. In this case, any character $\chi_\lambda$ is an extension of $\Delta_{H_e}$. If $V$ is non-unital, $\Delta_{H_e}(h)$ is non-trivial and the extension to $\chi_{-p}$ is unique. This implies and , so that it remains to show .\
Let $1\leq k\leq r-1$. Then, $V_0^+$ is non-trivial. If $|\operatorname{Det}_{V_0^+}(h)|$ is a power of $|\operatorname{Det}_{V^+}(h)|$, evaluation at $h=\exp(tD_{e_{k+1},\overline e_{k+1}})$, $t\in{\mathbb{R}}$, shows that $$\Delta_{H_e}(h)=|\operatorname{Det}_{V^+}(h)|^{-\frac{kd}{p}}=\chi_{-kd}(h),$$ which is non-trivial on $H_e$. This shows that an $L$-equivariant measure on ${\mathcal{O}}_e$ (if it exists) is unique up to normalization, and the corresponding positive character is $\chi_{kd}$. It remains to determine those cases, in which $|\operatorname{Det}_{V_0^+}(h)|$ is a power of $|\operatorname{Det}_{V^+}(h)|$. Since $|\operatorname{Det}_{V^+}|$ and $|\operatorname{Det}_{V^+_0}|$ are positive characters it is sufficient to show that $\operatorname{Tr}_{V^+_0}$ is a scalar multiple of $\operatorname{Tr}_{V^+}$ on ${\mathfrak{h}}_{e,\overline e}$ if and only if $V\not\simeq M(p\times q,{\mathbb{F}})$, ${\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}}$ with $p\neq q$. Let $T\in{\mathfrak{h}}_{e,\overline e}$. Since $T|_{V_2}$ is a structure endomorphism of the simple Jordan pair $V_2$ and the structure endomorphisms are generated by $D_{u,v}$ with $(u,v)\in V_2^+\times V_2^-$, there exists $T_2\in{\mathfrak{l}}$ which is a linear combination of $D_{u,v}$, $(u,v)\in V_2^+\times V_2^-$, such that $T|_{V_2}=T_2|_{V_2}$. The same is true for $T|_{V_0}$, so there exists $T_0\in{\mathfrak{l}}$ which is a linear combination of $D_{u,v}$, $(u,v)\in V_0^+\times V_0^-$, such that $T|_{V_0}=T_0|_{V_0}$. Since $D_{u,v}|_{V_0^+}=0$ for $(u,v)\in V_2^+\times V_2^-$ and $D_{u,v}|_{V_2^+}=0$ for $(u,v)\in V_0^+\times V_0^-$ the structure endomorphism $T_1=T-T_2-T_0$ vanishes on both $V_2$ and $V_0$. Note that $T_2e=0$ since $Te=0$. Now, we have $$\operatorname{Tr}_{V^+}(T) = \operatorname{Tr}_{V^+}(T_2)+\operatorname{Tr}_{V^+}(T_1)+\operatorname{Tr}_{V^+}(V_0).$$ We claim that $\operatorname{Tr}_{V^+}(T_2)=\frac{p}{p_2}\operatorname{Tr}_{V^+_2}(T_2)$. In fact, the bilinear forms $$B(u,v)=\operatorname{Tr}_{V^+}(D_{u,v}) \qquad \mbox{and} \qquad B_2(u,v)=\operatorname{Tr}_{V_2^+}(D_{u,v})$$ on $V_2^+\times V_2^-$ are both $L_{[e]}$-invariant. Since $$L_{[e]}\to{\textup{Str}}(V_2), \quad g\mapsto g|_{V_2}$$ is surjective, these forms are also ${\textup{Str}}(V_2)$-invariant. The form $B_2$ is non-degenerate since $V_2$ is simple, hence $B(u,v)=B_2(Au,v)$ for some $A\in{\textup{End}}(V_2^+)$. Both forms being ${\textup{Str}}(V_2)$-invariant, the endomorphism $A$ commutes with ${\textup{Str}}(V_2)$. Since ${\textup{Str}}(V_2)$ acts irreducibly on $V_2$, the map $A$ is a scalar multiple of the identity (allowing the scalar to be complex if $V_2$ is a complex Jordan pair). Evaluating both forms at $(u,v)=(e,\overline e)$ yields $B(u,v)=\frac{p}{p_2}B_2(u,v)$. Now, $T_2$ is a linear combination of operators of the form $D_{u,v}$, $(u,v)\in V_2^+\times V_2^-$ and hence $\operatorname{Tr}_{V^+}(T_2)=\frac{p}{p_2}\operatorname{Tr}_{V_2^+}(T_2)$. The same argument can be applied to $T_0$ acting on $V_0^+$ to obtain $\operatorname{Tr}_{V^+}(T_0)=\frac{p}{p_0}\operatorname{Tr}_{V_0^+}(T_0)$. Now, $\operatorname{Tr}_{V_2^+}(T_2)=0$ since $T_2$ is a structure automorphism of the simple Jordan algebra $V_2^+$ which annihilates the identity element $e$ (see e.g. [@FK94]). Hence $$\operatorname{Tr}_{V^+}(T) = \operatorname{Tr}_{V^+}(T_1)+\frac{p}{p_0}\operatorname{Tr}_{V_0^+}(T_0) = \operatorname{Tr}_{V_1^+}(T_1)+\frac{p}{p_0}\operatorname{Tr}_{V_0^+}(T).$$ Therefore, $\operatorname{Tr}_{V^+_0}$ is a scalar multiple of $\operatorname{Tr}_{V^+}$ if and only if $\operatorname{Tr}_{V_1^+}(T_1)=0$ for every $T_1\in{\mathfrak{l}}_{[e]}$ with $T|_{V_2^+\oplus V_0^+}=0$. By Proposition \[prop:ClassificationV1\] this is the case if and only if $V\not\simeq(M(p\times q,{\mathbb{F}}),M(q\times p,{\mathbb{F}}))$, ${\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}}$ with $p\neq q$.
The next goal is to determine an explicit formula for integration over ${\mathcal{O}}_e$ with respect to an $L$-equivariant measure by means of the polar decomposition given in Proposition \[prop:polardecomposition\]. Recall from the fibration of ${\mathcal{O}}_e$ over the Peirce manifold ${\mathcal{P}}_k$ with canonical fiber $L_{[e]}'/H_e'$, which is a symmetric space. We first determine a polar decomposition of the integral over the fiber $L_{[e]}'/H_e'$. With ${\mathfrak{a}}_k$ as in , we define $${\mathfrak{a}}_k^+={\left\{\sum_{j=1}^k \frac{\tau_j}{2}\,D_{e_j,\overline e_j}\,\middle|\,\tau_1>\cdots>\tau_k\right\}}\subseteq{\mathfrak{a}}_k,$$ and for $(\tau_1,\ldots,\tau_r)\in{\mathbb{R}}^r$, we set $$a_\tau=\exp(\tfrac{\tau_1}{2}D_{e_1,\overline e_1}+\cdots+\tfrac{\tau_k}{2}D_{e_k,\overline e_k}).$$ We endow ${\mathfrak{a}}_k^+$ with a Lebesgue measure ${\mathrm{d}}\tau$.
\[prop:fiberintegration\] With the notation as above, and after suitable normalization of the measures, $$\begin{aligned}
\label{eq:fiberintegralformula}
\int_{L_{[e]}'} f(h)\,{\mathrm{d}}h=\int_{(M\cap K)_{[e]}'}\int_{{\mathfrak{a}}_k^+}\int_{H_e'}
f(m'a_\tau\ell')\cdot J_e(\tau)\,{\mathrm{d}}\ell'\,{\mathrm{d}}\tau\,{\mathrm{d}}m'
\end{aligned}$$ for all $f\in L^1(L_{[e]}')$, where the Jacobian $J_e(\tau)$ is given by $$J_e(\tau_1,\ldots,\tau_k)
=\prod_{1\leq i<j\leq k}
\sinh^{d_+}\left(\frac{\tau_i-\tau_j}{2}\right)
\cosh^{d_-}\left(\frac{\tau_i-\tau_j}{2}\right).$$
This integral formula is a standard result from the theory of semisimple symmetric spaces, see e.g. [@HS94], based on the following data: The root system $\Phi(\mathfrak{str}([e]),{\mathfrak{a}}_k)$ is of type $A_{k-1}$. More precisely, let $\gamma_i\in{\mathfrak{a}}_k^*$ be defined by $\gamma_i(D_{e_j,\overline e_j})=\delta_{ij}$, then $$\Phi(\mathfrak{str}([e]),{\mathfrak{a}}_k) = {\left\{\frac{\gamma_i-\gamma_j}{2}\,\middle|\,1\leq i\neq j\leq k\right\}}$$ with root spaces $$\mathfrak{str}([e])_{\frac{\gamma_i-\gamma_j}{2}}={\{D_{x,\overline e_j}\,|\,x\in V_{ij}^+\}}.$$ Therefore, the multiplicity of $\frac{\gamma_i-\gamma_j}{2}$ is $2d=d_++d_-$. Moreover, $V_{ij}^+=A_{ij}^+\oplus B_{ij}^+$ corresponds to the decomposition of $\mathfrak{str}([e])_{\frac{\gamma_i-\gamma_j}{2}}$ into $\sigma\theta$-stable subspaces with according multiplicities $\dim A_{ij}^+=d_+$ and $\dim B_{ij}^+=d_-$. Fix the ordering $\gamma_1>\gamma_2>\cdots>\gamma_k$. Then, ${\mathfrak{a}}_k^+$ is the positive Weyl chamber. Moreover, this also coincides with the Weyl chamber of $\Sigma(\mathfrak{str}([e])^{\sigma\theta},{\mathfrak{a}}_k)$.
For $0\leq k\leq r$ let $$C_k^+ = {\left\{t\in{\mathbb{R}}^k\,\middle|\,t_1>\ldots>t_k>0\right\}}\label{eq:Defbt}$$ and put $$b_t = t_1e_1+\cdots+t_ke_k, \qquad t\in C_k^+.\label{eq:DefCk+}$$ Then by Proposition \[prop:polardecomposition\] the map $$(M\cap K)\times C_k^+\to{\mathcal{O}}_k, \quad (m,t)\mapsto mb_t$$ is onto an open dense subset of ${\mathcal{O}}_k$.
\[prop:equivariantmeasure\] Assume that ${\mathcal{O}}_e=L\cdot e\subseteq{\mathcal{V}}_k$, $0\leq k\leq r$, admits an $L$-equivariant measure ${\mathrm{d}}\mu$ with positive character $\chi_\lambda$. Then, for all $f\in L^1({\mathcal{O}}_e,{\mathrm{d}}\mu)$, $$\begin{aligned}
\int_{{\mathcal{O}}_e} f(x)\,{\mathrm{d}}\mu(x)
&= \int_{M\cap K}\int_{C_k^+} f(mb_t)\cdot J(t)\,{\mathrm{d}}t\,{\mathrm{d}}m,
\end{aligned}$$ where $$J(t)=\prod_{j=1}^k t_j^{\lambda+(r-2k+1)d+\frac{b}{2}-1}
\prod_{1\leq i< j\leq k}(t_i-t_j)^{d_+}(t_i+t_j)^{d_-}.$$
Recall that the equivariant measure ${\mathrm{d}}\mu$ on ${\mathcal{O}}_e=L/H_e$ is uniquely determined by the relation $$\begin{aligned}
\label{eq:equivariantintegral}
\int_{L/H_e}\int_{H_e} f(gh)\,{\mathrm{d}}h\,{\mathrm{d}}\mu(g H_e) = \int_L\chi_\lambda(g)f(g)\,{\mathrm{d}}g\end{aligned}$$ for all $f\in L^1(L)$. We determine a suitable decomposition of the right hand side. Then, the comparision with the left hand side proves the statement. In the following, all measures on Lie groups are meant to be left-invariant.\
We first use the fibration of ${\mathcal{O}}_e$. Recall that $Q_{[e]}\subseteq L$ is a parabolic subgroup of $L$, hence $L=(M\cap K)Q_{[e]}$, and $Q_{[e]}=L_{[e]}U_{[e]}$ is the Levi decomposition of $Q_{[e]}$. Therefore, $$\begin{aligned}
\int_L\chi_\lambda(g)f(g)\,{\mathrm{d}}g
&= \int_{M\cap K}\int_{Q_{[e]}}\Delta_{Q_{[e]}}(q)\chi_\lambda(mq)f(mq)\,{\mathrm{d}}q\,{\mathrm{d}}m\\
&= \int_{M\cap K}\int_{L_{[e]}}\int_{U_{[e]}}
\frac{\Delta_{U_{[e]}}(u)}{\Delta_{Q_{[e]}}(u)}\Delta_{Q_{[e]}}(hu)
\chi_\lambda(h)f(mhu)\,{\mathrm{d}}u\,{\mathrm{d}}h\,{\mathrm{d}}m\\
&= \int_{M\cap K}\int_{L_{[e]}}\int_{U_{[e]}}
\Delta_{Q_{[e]}}(h)\chi_\lambda(h)f(mhu)\,{\mathrm{d}}u\,{\mathrm{d}}h\,{\mathrm{d}}m.\end{aligned}$$ Here, we used that $\chi_\lambda(mhu)=\chi_\lambda(h)$, since $\chi_\lambda$ is a positive character on $L$, and hence $\chi_\lambda$ is trivial on the compact subgroup $M\cap K\subseteq L$ and the unipotent subgroup $U_{[e]}\subseteq L$. The modular function of $Q_{[e]}=L_{[e]}U_{[e]}$ is given by $$\Delta_{Q_{[e]}}(hu)=|\operatorname{Det}_{{\textup{Lie}}(U_{[e]})}{\textup{Ad}}(h)|,\qquad hu\in L_{[e]}U_{[e]}.$$ As in , we may identify the Lie algebra of $U_{[e]}$ with $V_1^-$. Then, the adjoint action of $h\in L_{[e]}$ on $v\in V_1^-$ is given by ${\textup{Ad}}(h)v={\left\{\overline e,he,hv\right\}}$. We thus obtain $$\Delta_{Q_{[e]}}(hu)=|\operatorname{Det}_{V_1^-}(D_{\overline e,he})|\cdot|\operatorname{Det}_{V_1^-}(h)|,
\qquad hu\in L_{[e]}U_{[e]}.$$ Now recall from that the fiber $$L_{[e]}/H_{e,\overline e}\cong L_{[e]}'/H_e'$$ is a symmetric space, where $L_{[e]}'$ and $H_e'$ is the image of $L_{[e]}$ and $H_e'$ under the restriction map $\rho_{[e]}$ defined in . We may identify $L_{[e]}'$ and $H_e'$ with the quotients $L_{[e]}/R$ and $H_{e,\overline e}/R$, where $R=\ker\rho_{[e]}$. Since normal subgroups of unimodular Lie groups are unimodular, we thus obtain $$\int_{L_{[e]}} F(h)\,{\mathrm{d}}h = \int_{L_{[e]}/R}\int_R F(hr)\,{\mathrm{d}}r\,{\mathrm{d}}(hR)
= \int_{L_{[e]}'}\int_R F(hr)\,{\mathrm{d}}r\,{\mathrm{d}}h.$$ Now, applying Proposition \[prop:fiberintegration\] yields $$\begin{aligned}
\int_L&\chi_\lambda(g)f(g)\,{\mathrm{d}}g\\
&= \int_{M\cap K}\int_{{\mathfrak{a}}_k^+}\int_{H_e'}\int_R\int_{U_{[e]}}
\Delta_{Q_{[e]}}(a_\tau\ell'r)\chi_\lambda(a_\tau\ell'r)J_e(\tau)
f(ma_\tau\ell'ru)\,{\mathrm{d}}u\,{\mathrm{d}}r\,{\mathrm{d}}\ell'\,{\mathrm{d}}\tau\,{\mathrm{d}}m.\end{aligned}$$ Since $H_e'=H_{e,\overline e}/R$ and $H_e=H_{e,\overline e}U_{[e]}$, we thus obtain $$\begin{aligned}
\int_L\chi_\lambda(g)f(g)\,{\mathrm{d}}g
&=\int_{M\cap K}\int_{{\mathfrak{a}}_k^+}\int_{H_{e,\overline e}}\int_{U_{[e]}}
\Delta_{Q_{[e]}}(a_\tau\ell)\chi_\lambda(a_\tau\ell)J_e(\tau)f(ma_\tau\ell u)\,{\mathrm{d}}u\,{\mathrm{d}}\ell\,{\mathrm{d}}\tau\,{\mathrm{d}}m\\
&=\int_{M\cap K}\int_{{\mathfrak{a}}_k^+}\int_{H_e} \Delta_{Q_{[e]}}(a_\tau h)
\chi_\lambda(a_\tau h)J_e(\tau)f(ma_\tau h)\,{\mathrm{d}}h\,{\mathrm{d}}\tau\,{\mathrm{d}}m.\end{aligned}$$ We note that $\Delta_{Q_{[e]}}(h)=\Delta_{H_e}(h)$ for all $h\in H_e$. Moreover, $\Delta_{H_e}(h)=\chi_\lambda(h)^{-1}$ by equivariance of ${\mathrm{d}}\mu$. This eventually implies $$\begin{aligned}
\int_L\chi_\lambda(g)f(g)\,{\mathrm{d}}g
=\int_{M\cap K}\int_{{\mathfrak{a}}_k^+}\left(\int_{H_e} f(ma_\tau h)dh\right)\tilde J(\tau)\,{\mathrm{d}}\tau\,{\mathrm{d}}m\end{aligned}$$ with $$\tilde J(\tau)=\Delta_{Q_{[e]}}(a_\tau)\chi_\lambda(a_\tau)J_e(\tau).$$ Comparing this integral formula with shows that $$\int_{{\mathcal{O}}_e} f(x)\,{\mathrm{d}}\mu(x)
= \int_{M\cap K}\int_{{\mathfrak{a}}_k^+}f(ma_\tau\cdot e)\tilde J(\tau)\,{\mathrm{d}}\tau\,{\mathrm{d}}m$$ for $f\in L^1({\mathcal{O}}_e,{\mathrm{d}}\mu)$. For $a_\tau=\exp(\frac{\tau_1}{2}D_{e_1,\overline e_1}+\cdots+\frac{\tau_k}{2}D_{e_k,\overline e_k})$, the joint Peirce decomposition with respect to the frame $(e_1,\ldots, e_r)$ shows that $$\Delta_{Q_{[e]}}(a_\tau) = \prod_{j=1}^k e^{((r-k)d+\frac{b}{2})\tau_j} \qquad \mbox{and} \qquad \chi_\lambda(a_\tau) = \prod_{j=1}^k e^{\lambda \tau_j}.$$ Therefore, $$\tilde J(\tau) = \prod_{j=1}^k e^{(\lambda+(r-k)d+\frac{b}{2})\tau_j}
\prod_{1\leq i<j\leq k}\sinh^{d_+}\left(\frac{\tau_i-\tau_j}{2}\right)
\cosh^{d_-}\left(\frac{\tau_i-\tau_j}{2}\right).$$ Finally, the change of coordinates $t_j=e^{\tau_j}$ yields the proposed integration formula.
Finally, we determine a formula for the $L$-equivariant measure ${\mathrm{d}}\mu$ in the coordinates of ${\mathcal{O}}_e$ given by the diffeomorphism $\varphi_e$ in . Recall from Proposition \[prop:orbitcoordinates\] that the restriction of $\varphi_e$ to $\Omega_e+V_1^+$ is a diffeomorphism onto the $\overline Q_{[e]}$-orbit of $e\in{\mathcal{O}}_e$, which is an open and dense subset of ${\mathcal{O}}_e$. Here, $\Omega_e$ is the $L_{[e]}$-orbit of $e\in[e]$, which is open in $[e]=V_2^+$.
\[prop:PullbackMeasure\] Let $e\in{\mathcal{V}}_k$, $1\leq k\leq r-1$, and let ${\mathrm{d}}\mu$ be a $\chi_{kd}$-equivariant measure on ${\mathcal{O}}_e$. Then, the pullback of ${\mathrm{d}}\mu$ along $\varphi_e$ is given by $$\int_{{\mathcal{O}}_e} f(x) {\mathrm{d}}\mu(x)
= \int_{\Omega_e+V_1^+}f(\varphi_e(x_2+x_1))|\Delta(x_2)|^{kd-p}\,{\mathrm{d}}\lambda(x_2+x_1),$$ where ${\mathrm{d}}\lambda$ denotes a suitably normalized Lebesgue measure on $V_2^+\oplus V_1^+$.
First we note that the pullback of ${\mathrm{d}}\mu$ is absolutely continuous with respect to the Lebesgue measure on $V_2^+\oplus V_1^+$. Therefore, $\varphi^*{\mathrm{d}}\mu=g\,{\mathrm{d}}\lambda$ for a nowhere vanishing continuous map $g$. Now, $\overline Q_{[e]}$-equivariance of $\varphi_e$ with respect to the actions of $\overline Q_{[e]}$ given as in Proposition \[prop:orbitcoordinates\] implies that $$g(qe)=\frac{\chi_{kd}(q)}{|\operatorname{Det}_{V_2^+\oplus V_1^+}(d_e\xi(q))|}\,g(e)$$ for all $q\in\overline Q_{[e]}$. With $q=h{B_{v,\,\overline e}}$ according to , we obtain $$\operatorname{Det}_{V_2^+\oplus V_1^+}(d_e\xi(q))=\operatorname{Det}_{V_2^+\oplus V_1^+}(h), \qquad \chi_{kd}(q)=\operatorname{Det}_{V^+}(h)^{\frac{kd}{p}}.$$ Hence $g$ is independent of $x_1$, and as a function of $x_2$ it satisfies $$g(hx_2) = \frac{|\operatorname{Det}_{V^+}(h)|^{\frac{kd}{p}}}{|\operatorname{Det}_{V_2^+\oplus V_1^+}(h)|}g(x_2).$$ Similar to the proof of Theorem \[thm:equivariantmeasures\] (but now on the group level) we decompose $h\in L_{[e]}$ as $h=h_2h_1h_0$ with $$h|_{V_2^+}=h_2|_{V_2^+}, \quad h_2|_{V_0^+}={\textup{id}}_{V_0^+}, \qquad \mbox{and} \qquad h|_{V_0^+}=h_0|_{V_0^+}, \quad h_0|_{V_2^+}={\textup{id}}_{V_2^+}.$$ Then $h_1$ acts trivially on $V_2^+\oplus V_0^+$ and by Proposition \[prop:ClassificationV1\] \[ClassificationV1-2\] we have $\operatorname{Det}_{V_2^+\oplus V_1^+}(h_1)=\operatorname{Det}_{V^+}(h_1)=1$. Further, by the proof of Theorem \[thm:equivariantmeasures\] we have $$\begin{aligned}
\operatorname{Det}_{V_0^+}(h_0) &= \operatorname{Det}_{V^+}(h_0)^{\frac{p_0}{p}} = \operatorname{Det}_{V^+}(h_0)^{1-\frac{kd}{p}},\\
\operatorname{Det}_{V_2^+}(h_2) &= \operatorname{Det}_{V^+}(h_2)^{\frac{p_2}{p}}.\end{aligned}$$ Together this gives $$\begin{aligned}
\frac{\operatorname{Det}_{V^+}(h)^{\frac{kd}{p}}}{\operatorname{Det}_{V_2^+\oplus V_1^+}(h)} &= \operatorname{Det}_{V^+}(h_2)^{\frac{kd}{p}-1}\operatorname{Det}_{V^+}(h_1)^{\frac{kd}{p}-1}\operatorname{Det}_{V^+}(h_0)^{\frac{kd}{p}-1}\operatorname{Det}_{V_0^+}(h_0)\\
&= \operatorname{Det}_{V_2^+}(h_2)^{\frac{1}{p_2}(kd-p)} = \operatorname{Det}_{V_2^+}(h)^{\frac{1}{p_2}(kd-p)}\end{aligned}$$ and hence $$g(hx_2) = |\operatorname{Det}_{V_2^+}(h)|^{\frac{1}{p_2}(kd-p)}g(x_2).$$ Since $L_{[e]}\to{\textup{Str}}(V_2^+),\,h\mapsto h|_{V_2^+}$ is surjective and the (absolute value of the) Jordan determinant $|\Delta(x_2)|$ is the (up to scalar multiples) unique function on $V_2^+$ such that $|\Delta(gx_2)|=|\operatorname{Det}(g)|^{\frac{1}{p_2}}|\Delta(x_2)|$ we find that $$g(x_2) = |\Delta(x_2)|^{kd-p}. \qedhere$$
Bessel operators on Jordan pairs {#sec:Besseloperator}
================================
Bessel operators were first defined for Euclidean Jordan algebras by Dib [@Dib90] (see also Faraut–Koranyi [@FK94]) and later generalized to arbitrary semisimple Jordan algebras by Mano [@Man08] and Hilgert–Kobayashi–Möllers [@HKM14]. We extend this definition to Jordan pairs and show that for certain parameters the operators are tangential to the rank submanifolds ${\mathcal{V}}_k$ in $V^+$. In the case where ${\mathcal{V}}_k$ carries an $L$-equivariant measure we further prove that the Bessel operators are symmetric with respect to the corresponding $L^2$-inner product. For Jordan algebras these results were obtained by Hilgert–Kobayashi–Möllers [@HKM14] using zeta functions which are not available for Jordan pairs. The proofs we give here work uniformly for all Jordan pairs, since they merely use local parametrizations of the submanifolds ${\mathcal{V}}_k$ and the explicit form of the measures ${\mathrm{d}}\mu_k$ in these parametrizations.
Definition, equivariance, and symmetry {#sec:BesselDefinition}
--------------------------------------
We fix a basis $\{c_\alpha\}_{\alpha=1,\ldots, n}$ of $V^+$, and let $\{\widehat c_\alpha\}_{\alpha=1,\ldots, n}$ be the dual basis of $V^-$ with respect to the trace form $\tau$ defined in . Then, for any $\lambda\in{\mathbb{C}}$ the *Bessel operator* $${\mathcal{B}}_\lambda\colon C^\infty(V^+)\to C^\infty(V^+)\otimes V^-$$ is defined by the formula $${\mathcal{B}}_\lambda f(x)
=\frac{1}{2}\sum_{\alpha,\beta} \frac{\partial^2 f}{\partial x_\alpha\partial x_\beta}(x)
{\left\{\widehat c_\alpha,x,\widehat c_\beta\right\}} +
\lambda\,\sum_\alpha\frac{\partial f}{\partial x_\alpha}(x)\,\widehat c_\alpha,$$ which is easily seen to be independent of the choice of $\{c_\alpha\}_{\alpha=1,\ldots, n}$. On a formal level, this is also sometimes denoted as $${\mathcal{B}}_\lambda = Q\left(\frac{\partial}{\partial x}\right)x + \lambda\frac{\partial}{\partial x},$$ where $\tfrac{\partial}{\partial x}\colon C^\infty(V^+)\to C^\infty(V^+)\otimes V^-$ denotes the gradient with respect to $\tau$, which is defined by the condition $${\mathrm{d}}_vf(x) = \tau\left(v,\frac{\partial f}{\partial x}(x)\right)
\qquad\text{for all }v\in V^+.$$ One of the basic properties of the Bessel operator is its equivariance under the action of $L\subseteq G$. For $h\in L$, let $\rho(h)$ denote the action on functions on $V^+$, $$(\rho(h)f)(x) = f(h^{-1}x).$$ Recall that $hx={\textup{Ad}}(h)x$ denotes the adjoint action of $h\in L$ on $x\in V^+={\mathfrak{n}}$ resp. $V^-=\overline{\mathfrak{n}}$.
\[lem:BesselEquivariance\] For any $\lambda\in{\mathbb{C}}$ and $h\in L$, $${\mathcal{B}}_\lambda\circ\rho(h) = (\rho(h)\otimes h)\circ{\mathcal{B}}_\lambda.$$
It suffices to note that the action of $h\in L$ on $V^\pm$ is by Jordan pair automorphisms, i.e., satisfies $h{\left\{x,y,z\right\}} = {\left\{hx,hy,hz\right\}}$ for all $x,z\in V^+$, $y\in V^-$. In particular, the pairing $\tau$ is $L$-invariant, hence $\{hc_\alpha\}_{\alpha=1,\ldots,n}$ and $\{h\widehat c_\alpha\}_{\alpha=1,\ldots,n}$ is another pair of dual bases for $V^+$ and $V^-$. The equivariance of ${\mathcal{B}}_\lambda$ now follows from standard transformation rules.
For later use we note the following symmetry property of the Bessel operator.
\[prop:BesselPartialIntegration\] Let $\lambda\in{\mathbb{C}}$, then $$\int_{V^+}{\mathcal{B}}_\lambda f(x)\cdot g(x)\,{\mathrm{d}}x = \int_{V^+}f(x)\cdot{\mathcal{B}}_{2p-\lambda}g(x)\,{\mathrm{d}}x.$$
Integrating by parts and using Lemma \[lem:basessums\] we obtain $$\begin{aligned}
& \int_{V^+}{{\mathcal{B}}_\lambda f(x)\cdot g(x)\,{\mathrm{d}}x}\\
={}& \int_{V^+}{f(x)\cdot\left(
\frac{1}{2}\sum_{\alpha,\beta}\frac{\partial^2}{\partial x_\alpha\partial x_\beta}
\Big(g(x){\left\{\widehat{c}_\alpha,x,\widehat{c}_\beta\right\}}\Big)
-\lambda\sum_\alpha\frac{\partial g}{\partial x_\alpha}(x)\,\widehat c_\alpha\right){\mathrm{d}}x}\\
={}& \int_{V^+}{f(x)\cdot\left(
{\mathcal{B}}_0g(x) + \sum_{\alpha,\beta}\frac{\partial g}{\partial x_\alpha}(x)
{\left\{\widehat{c}_\alpha,c_\beta,\widehat{c}_\beta\right\}}
-\lambda\sum_\alpha\frac{\partial g}{\partial x_\alpha}(x)\,\widehat c_\alpha\right){\mathrm{d}}x}\\
={}& \int_{V^+}{f(x)\cdot\left(
{\mathcal{B}}_0g(x) + 2p\sum_\alpha\frac{\partial g}{\partial x_\alpha}(x)\,\widehat{c}_\alpha
-\lambda\sum_\alpha\frac{\partial g}{\partial x_\alpha}(x)\,\widehat c_\alpha\right){\mathrm{d}}x}\\
={}& \int_{V^+}{f(x)\cdot{\mathcal{B}}_{2p-\lambda}g(x)\,{\mathrm{d}}x}.\qedhere\end{aligned}$$
Restriction to submanifolds {#sec:tangency}
---------------------------
We now turn to tangential differential operators. Recall that a differential operator $D$ on $V^+$ is *tangential* to a submanifold $S\subseteq V^+$, if for any smooth function $f\in C^\infty(V^+)$ with $f|_S=0$ we have $(Df)|_S=0$. For a linear subspace $S\subseteq V^+$ this means that $D$ only contains derivatives in the direction of $S$.
For an idempotent $e\in{\mathcal{V}}_k$ recall the map $\varphi_e$ from . The pullback of the Bessel operator is defined by $$(\varphi_e^*{\mathcal{B}}_\lambda f)(x) = {\mathcal{B}}_\lambda(f\circ\varphi_e^{-1})(\varphi_e(x)).$$ We fix a pair of dual bases $\{c_\alpha\}_{\alpha\in I}$ of $V^+$ and $\{\widehat c_\alpha\}_{\alpha\in I}$ of $V^-$ with respect to the trace form $\tau$. Choose these bases compatible with the Peirce decomposition $V^\pm=V_2^\pm\oplus V_1^\pm\oplus V_0^\pm$ with respect to $e$, i.e. $I=I_2\sqcup I_1\sqcup I_0$ with $c_\alpha\in V_\ell^+$ if and only if $\alpha\in I_\ell$, $\ell=0,1,2$. We further write $\nabla$ for the gradient with respect to the trace form, and $\nabla_\ell$ for its projection to $V_\ell^-$, $\ell=0,1,2$.
\[thm:pullbackBessel\] The pullback of ${\mathcal{B}}_\lambda$ along $\varphi_e$ at $x=x_2+x_1\in(V_2^+)^\times\times V_1^+$ is given by $$\begin{aligned}
(\varphi_e^*{\mathcal{B}}_\lambda f)(x)
={}&\frac{1}{2}\sum_{\alpha,\beta\in I_2\sqcup I_1}
\frac{\partial^2f}
{\partial x_\alpha\partial x_\beta}(x)
{\left\{\widehat{c}_\alpha,x,\widehat{c}_\beta\right\}} +\frac{1}{2}\sum_{\alpha,\beta\in I_1}
\frac{\partial^2f}
{\partial x_\alpha\partial x_\beta}(x)
{\left\{\widehat{c}_\alpha,Q_{x_1}x_2^{-1},\widehat{c}_\beta\right\}}\\
&+\lambda\nabla_2f(x)+\lambda\nabla_1f(x)+(\lambda-kd){B_{x_2^{-1},\,x_1}}\nabla_0f(x).
\end{aligned}$$ In particular, ${\mathcal{B}}_\lambda$ is tangential to ${\mathcal{V}}_k$ if and only if $\lambda=kd$.
For simplicity, we write ${\mathcal{B}}_\lambda'=\varphi_e^*{\mathcal{B}}_\lambda$. We first prove the claimed formula for $x\in V_2^+$ and then use the equivariance of the Bessel operator to determine the general formula. Set $\varphi=\varphi_e$, and recall that for general $x\in V^+$, $\varphi(x)=x+Q_{x_1}x_2^{-1}$ and $\varphi^{-1}(x)=x-Q_{x_1}x_2^{-1}$. Therefore, $${\mathrm{d}}\varphi^{\pm1}(x)(u)=u\pm{\left\{x_1,x_2^{-1},u_1\right\}} + Q_{x_1}Q_{x_2^{-1}}u_2.$$ For the following, we assume that $x\in(V_2^+)^\times$. Then, it follows that $$\begin{aligned}
\label{eq:derivatives}
{\mathrm{d}}\varphi^{\pm1}(x)(u)=u,\qquad
{\mathrm{d}}^2\varphi^{\pm1}(x)(u,v)=\pm{\left\{u_1,x^{-1},v_1\right\}}.\end{aligned}$$ Write ${\mathcal{B}}_\lambda={\mathcal{B}}_0+\lambda\nabla$. The pullback of the gradient $\nabla$ is easily shown to be $$\begin{aligned}
\label{eq:gradienttrafo}
(\varphi^*\nabla)(x)=\nabla(f\circ\varphi^{-1})(\varphi(x)) = {\mathrm{d}}\varphi(x)^{-*}(\nabla f)(x)
=(\nabla f)(x).\end{aligned}$$ It remains to determine ${\mathcal{B}}'_0$ which is $${\mathcal{B}}'_0f(x) = \frac{1}{2}\sum_{\alpha,\beta\in I}
\frac{\partial^2(f\circ\varphi^{-1})}
{\partial x_\alpha\partial x_\beta}(\varphi(x))
{\left\{\widehat{c}_\alpha,\varphi(x),\widehat{c}_\beta\right\}}.$$ Due to the chain rule, $$\begin{aligned}
({\mathcal{B}}'_0 f)(x)
&= \frac{1}{2}\sum_{\gamma,\delta\in I}\left(\sum_{\alpha,\beta\in I}
\frac{\partial\varphi_\gamma^{-1}}{\partial x_\alpha}(\varphi(x))
\frac{\partial\varphi_\delta^{-1}}{\partial x_\beta}(\varphi(x))
{\left\{\widehat{c}_\alpha,\varphi(x),\widehat{c}_\beta\right\}}\right)
\frac{\partial^2 f}{\partial x_\gamma\partial x_\delta}(x)\\
&\quad + \frac{1}{2}\sum_{\gamma\in I}\left(\sum_{\alpha,\beta\in I}
\frac{\partial^2\varphi_\gamma^{-1}}
{\partial x_\alpha\partial x_\beta}(\varphi(x))
{\left\{\widehat{c}_\alpha,\varphi(x),\widehat{c}_\beta\right\}} \right)\,
\frac{\partial f}{\partial x_\gamma}(x),\end{aligned}$$ where $\varphi^{-1}_\gamma(x) = \tau(\varphi^{-1}(x),\widehat{c}_\gamma)$. Applying yields $$\begin{aligned}
({\mathcal{B}}'_0 f)(x)
&= \frac{1}{2}\sum_{\gamma,\delta\in I}{\left\{\widehat{c}_\gamma,x,\widehat{c}_\delta\right\}}
\frac{\partial^2 f}{\partial x_\gamma\partial x_\delta}(x) - \frac{1}{2}\sum_{\gamma\in I_0}\left(\sum_{\beta\in I_1}
{\left\{{\left\{x^{-1},c_\beta,\widehat{c}_\gamma\right\}},x,\widehat{c}_\beta\right\}}\right)\,
\frac{\partial f}{\partial x_\gamma}(x).\end{aligned}$$ We determine the $\beta$-sum. According to and the Peirce rules (more precisely, with $(u,v,x,y,z)\widehat=(x^{-1},x,\widehat{c}_\gamma,c_\beta,\widehat{c}_\beta)$): $${\left\{{\left\{x^{-1},c_\beta,\widehat{c}_\gamma\right\}},x,\widehat{c}_\beta\right\}}
={\left\{\widehat{c}_\gamma,{\left\{x,x^{-1},c_\beta\right\}},\widehat{c}_\beta\right\}}.$$ Using $D_{x,x^{-1}}=D_{e,e'}$ and Lemma \[lem:basessums\] , it follows that the $\beta$-sum evaluates to $$\sum_{\beta\in I_1}{\left\{\widehat{c}_\gamma,c_\beta,\widehat{c}_\beta\right\}}
=2p\,\widehat{c}_\gamma - \sum_{\beta\in I_0} D_{\widehat{c}_\beta,c_\beta}\widehat{c}_\gamma.$$ The last sum only involves terms with $c_\beta\in V_0^+$ since ${\left\{\widehat c_\beta,c_\beta,\widehat c_\gamma\right\}}=0$ for $c_\beta\in V_2^+$. This sum evaluates to $2p_0\,\widehat c_\gamma$, where $p_0$ is the structure constant defined as in but with respect to the subpair $V_0=(V_0^+,V_0^-)$. Since $p-p_0=kd$, we conclude that $$\begin{aligned}
({\mathcal{B}}'_0 f)(x)
&= \frac{1}{2}\sum_{\gamma,\delta\in I}{\left\{\widehat{c}_\gamma,x,\widehat{c}_\delta\right\}}
\frac{\partial^2 f}{\partial x_\gamma\partial x_\delta}(x)
-kd\sum_{\gamma\in I_0}\widehat{c}_\gamma
\frac{\partial f}{\partial x_\gamma}(x).\end{aligned}$$ In combination with the gradient term, we thus obtain $$({\mathcal{B}}'_\lambda f)(x) = {\mathcal{B}}_0 f(x) +\lambda\nabla f(x)- kd\nabla_0f(x)\qquad\text{for }x\in(V_2^+)^\times.$$ We next extend this formula to elements in $(V_2^+)^\times\times V_1^+$. Recall from Lemma \[lem:pullbackaction\] that the action of $\overline Q_{[e]}=V_1^+\rtimes L_{[e]}$ on $V^+$ admits a pullback along $\varphi$ which is given by $$\begin{aligned}
\xi(h)(x)&=hx,\qquad
\xi({B_{v,\,b}})(x)=x - {\left\{v,b,x_2\right\}}\end{aligned}$$ for any $h\in L_{[e]}$ and $v\in V_1^+$, $b\in V_2^-$, and $x\in N_e$. The equivariance of the Bessel operator ${\mathcal{B}}_\lambda$ given by Lemma \[lem:BesselEquivariance\] translates to the following equivariance of the pullback Bessel operator ${\mathcal{B}}'_\lambda$: $${\mathcal{B}}'_\lambda(f\circ\xi(q)^{-1})(x)=q\cdot({\mathcal{B}}'_\lambda f)(\xi(q)^{-1}x).$$ In order to avoid confusion, in the following we write $a\in(V_2^+)^\times$ instead of $x$ for the fixed element. For $q={B_{v,\,a^{-1}}}$ with $v\in V_1^+$, we note that $q^{-1}={B_{-v,\,a^{-1}}}$, and the action of $q^{-1}$ on element of $V^-$ is given by ${B_{a^{-1},\,-v}}^{-1}={B_{a^{-1},\,v}}$. Since $$\xi(q)^{-1}a=\xi({B_{-v,\,a^{-1}}})a=a+{\left\{v,a^{-1},a\right\}}=a+v,$$ equivariance yields $${\mathcal{B}}'_\lambda f(a+v)={B_{a^{-1},\,v}}\,{\mathcal{B}}'_\lambda(f\circ\xi({B_{-v,\,a^{-1}}}))(a).$$ For the gradient part of ${\mathcal{B}}'_\lambda$, we obtain $$\nabla(f\circ\xi({B_{-v,\,a^{-1}}}))(a) = \xi({B_{-v,\,a^{-1}}})^*\nabla f(x+v),$$ where adjoint $\xi({B_{-v,\,a^{-1}}})^*$ of $\xi({B_{-v,\,a^{-1}}})$ with respect to the trace form $\tau$ is given by $$\begin{aligned}
\label{eq:adjointaction}
\xi({B_{-v,\,a^{-1}}})^*(w) = \begin{cases}
{B_{a^{-1},\,-v}}(w) &,\ w\in V_2^-\oplus V_1^-,\\
w &,\ w\in V_0^-.
\end{cases}\end{aligned}$$ The $V_0^+$-grandient is invariant for the action of $\xi({B_{-v,\,a^{-1}}})$ since $V_0^+$ is fixed under this action. We thus conclude that $$\left(\varphi^*(\lambda\nabla-kd\nabla_0)\right)f(x)=\lambda\nabla_2f(x)+\lambda\nabla_1f(x)
+(\lambda-kd){B_{a^{-1},\,v}}\nabla_0f(x).$$ It remains to determine ${\mathcal{B}}'_0f$ at $a+v$. Since $$\sum_{\alpha,\beta\in I}
\frac{\partial^2(f\circ T)}
{\partial x_\alpha\partial x_\beta}(a){\left\{\widehat{c}_\alpha,a,\widehat{c}_\beta\right\}}
= \sum_{\alpha,\beta\in I}
\frac{\partial^2f}
{\partial x_\alpha\partial x_\beta}(T(a)){\left\{T^*\widehat{c}_\alpha,a,T^*\widehat{c}_\beta\right\}}$$ holds for any linear isomorphism $T\in{\textup{GL}}(V^+)$, it follows that $$\label{eq:besseltrafo}
{\mathcal{B}}'_0f(a+v) = \frac{1}{2}\sum_{\alpha,\beta\in I}
\frac{\partial^2f}
{\partial x_\alpha\partial x_\beta}(a+v)\cdot{B_{a^{-1},\,v}}
{\left\{\xi({B_{-v,\,a^{-1}}})^*\widehat{c}_\alpha,a,\xi({B_{-v,\,a^{-1}}})^*\widehat{c}_\beta\right\}}.$$ If either $\widehat{c}_\alpha$ or $\widehat{c}_\beta$ is in $V_0^-$, the Jordan product in the sum of vanishes due to the Peirce rules since $a\in V_2^+$. Therefore, we may assume that $\alpha,\beta\in I_2\sqcup I_1$, and due to , $$\begin{aligned}
{\left\{\xi({B_{-v,\,a^{-1}}})^*\widehat{c}_\alpha,a,\xi({B_{-v,\,a^{-1}}})^*\widehat{c}_\beta\right\}}
&= {B_{a^{-1},\,-v}}{\left\{\widehat{c}_\alpha,{B_{-v,\,a^{-1}}}(a),\widehat{c}_\beta\right\}}\\
&= {B_{a^{-1},\,-v}}{\left\{\widehat{c}_\alpha,a+v+Q_va^{-1},\widehat{c}_\beta\right\}}.\end{aligned}$$ According to the Peirce rules we thus obtain $$\begin{aligned}
{\mathcal{B}}'_0f(a+v) &= \frac{1}{2}\sum_{\alpha,\beta\in I_2\sqcup I_1}
\frac{\partial^2f}
{\partial x_\alpha\partial x_\beta}(a+v)
{\left\{\widehat{c}_\alpha,a+v,\widehat{c}_\beta\right\}}\\
&\quad+\frac{1}{2}\sum_{\alpha,\beta\in I_1}
\frac{\partial^2f}
{\partial x_\alpha\partial x_\beta}(a+v)
{\left\{\widehat{c}_\alpha,Q_va^{-1},\widehat{c}_\beta\right\}}.\end{aligned}$$ Collecting all terms, this completes the proof.
\[thm:BesselSymmetricOnOrbits\] Let $\lambda=kd$, $0\leq k\leq r-1$, so that ${\mathcal{B}}_\lambda$ is tangential to ${\mathcal{V}}_k$, and assume that ${\mathcal{V}}_k$ carries an $L$-equivariant measure ${\mathrm{d}}\mu_k$ with corresponding character $\chi_{kd}$. Then ${\mathcal{B}}_\lambda$ is symmetric on $L^2({\mathcal{V}}_k,{\mathrm{d}}\mu_k)$.
We prove this result using the parametrization of ${\mathcal{V}}_k$ by $\varphi=\varphi_e$, see . By Proposition \[prop:PullbackMeasure\] the $L^2$-inner product associated to the pullback measure $\varphi^*{\mathrm{d}}\mu_k$ is given by $$(f,g)\mapsto\int_{(V_2^+)^\times\times V_1^+} f(x)\overline{g(x)}\Delta(x)^{kd-p}\,{\mathrm{d}}\lambda(x),$$ and by Theorem \[thm:pullbackBessel\] the pullback of the Bessel operator ${\mathcal{B}}_\lambda$ along $\varphi$ is given by $$\begin{gathered}
(\varphi^*{\mathcal{B}}_\lambda)f(x) = \frac{1}{2}\sum_{\alpha,\beta\in I_2\sqcup I_1}\frac{\partial^2f}{\partial x_\alpha\partial x_\beta}(x){\left\{\widehat{c}_\alpha,x,\widehat{c}_\beta\right\}}\\
+\frac{1}{2}\sum_{\alpha,\beta\in I_1}\frac{\partial^2f}{\partial x_\alpha\partial x_\beta}(x){\left\{\widehat{c}_\alpha,Q_{x_1}x_2^{-1},\widehat{c}_\beta\right\}}+\lambda\nabla_2f(x)+\lambda\nabla_1f(x).\end{gathered}$$ Recall that the formal adjoint $D^{\textup{ad}}$ of an operator $$D = \sum_{\alpha,\beta}a_{\alpha\beta}(x)\frac{\partial^2}{\partial x_\alpha\partial x_\beta} + \sum_\alpha a_\alpha(x)\frac{\partial}{\partial x_\alpha}$$ with coefficient functions $a_\alpha(x)$ and $a_{\alpha\beta}(x)$ is given by $$D^{\textup{ad}}f(x)
=\frac{1}{\Delta^{kd-p}}\sum_{\alpha,\beta}\frac{\partial^2}{\partial x_\alpha\partial x_\beta}\left(a_{\alpha\beta}\cdot\Delta^{kd-p}\cdot f\right)-\frac{1}{\Delta^{kd-p}}
\sum_{\alpha}\frac{\partial}{\partial x_\alpha}\left(a_\alpha\cdot\Delta^{kd-p}\cdot f\right).$$ In case of the Bessel operator we obtain $$\begin{aligned}
(\varphi^*{\mathcal{B}}_\lambda)^{\textup{ad}}f(x) =
&\underbrace{\frac{1}{2\Delta^{kd-p}}\sum_{\alpha,\beta\in I_2}
\frac{\partial^2}{\partial x_\alpha\partial x_\beta}
\left({\left\{\widehat c_\alpha,x_2,\widehat c_\beta\right\}}\cdot\Delta^{kd-p}\cdot f\right)}_{(1)}\\
&\underbrace{+\frac{1}{\Delta^{kd-p}}\sum_{\alpha\in I_1,\beta\in I_2}
\frac{\partial^2}{\partial x_\alpha\partial x_\beta}
\left({\left\{\widehat c_\alpha,x,\widehat c_\beta\right\}}\cdot\Delta^{kd-p}\cdot f\right)}_{(2)}\\
&\underbrace{+\frac{1}{2\Delta^{kd-p}}\sum_{\alpha,\beta\in I_1}
\frac{\partial^2}{\partial x_\alpha\partial x_\beta}
\left({\left\{\widehat c_\alpha,x,\widehat c_\beta\right\}}\cdot\Delta^{kd-p}\cdot f\right)}_{(3)}\\
&\underbrace{+\frac{1}{2\Delta^{kd-p}}\sum_{\alpha,\beta\in I_1}
\frac{\partial^2}{\partial x_\alpha\partial x_\beta}
\left({\left\{\widehat c_\alpha,Q_{x_1}x_2^{-1},\widehat c_\beta\right\}}
\cdot\Delta^{kd-p}\cdot f\right)}_{(4)}\\
&\underbrace{-\frac{\lambda}{\Delta^{kd-p}}\sum_{\alpha\in I_1\sqcup I_2}
\frac{\partial}{\partial x_\alpha}
\left(\widehat c_\alpha\cdot\Delta^{kd-p}\cdot f\right).}_{(5)}\end{aligned}$$ The Jordan algebra determinant $\Delta$ is independent of $x_1$, and satisfies $$\frac{1}{\Delta}\frac{\partial}{\partial x_\alpha}(\Delta) = \tau(c_\alpha,x_2^{-1})\qquad
\text{for }\alpha\in I_2,$$ hence $$\frac{1}{\Delta^\sigma}\frac{\partial}{\partial x_\alpha}(\Delta^\sigma)
=\sigma\cdot\tau(c_\alpha,x_2^{-1})$$ and $$\begin{aligned}
\frac{1}{\Delta^\sigma}\frac{\partial^2}{\partial x_\alpha\partial x_\beta}(\Delta^\sigma)
&=\sigma^2\cdot\tau(c_\alpha,x_2^{-1})\cdot\tau(c_\beta,x_2^{-1})
+ \sigma\cdot\tau(c_\alpha,\frac{\partial x_2^{-1}}{\partial x_\beta})\\
&=\sigma^2\cdot\tau(c_\alpha,x_2^{-1})\cdot\tau(c_\beta,x_2^{-1})
- \sigma\cdot\tau(c_\alpha,Q_{x_2^{-1}}c_\beta),\end{aligned}$$ which follows from differentiating the relation $Q_{x_2} x_2^{-1} = x_2$. Using Lemma \[lem:basessums\], it follows that $$\begin{aligned}
(1) ={}& (kd-p)\,f(x)\sum_{\alpha\in I_2}{\left\{\widehat c_\alpha,c_\alpha,x_2^{-1}\right\}}+\sum_{\alpha\in I_2}{\left\{\widehat c_\alpha,c_\alpha,\nabla_2f\right\}}+(kd-p){\left\{x_2^{-1},x_2,\nabla_2 f\right\}}\\
&+\frac{(kd-p)^2}{2}\,f(x)\cdot{\left\{x_2^{-1},x_2,x_2^{-1}\right\}}-\frac{kd-p}{2}\,f(x)\sum_{\alpha\in I_2}{\left\{Q_{x_2^{-1}}c_\alpha,x_2,\widehat c_\alpha\right\}}\\
&+\frac{1}{2}\sum_{\alpha,\beta\in I_2}\frac{\partial^2 f}{\partial x_\alpha\partial x_\beta}(x){\left\{\widehat c_\alpha,x_2,\widehat c_\beta\right\}}\\
={}& (kd-p)(kd+p_2-p)\,f(x)\cdot x_2^{-1}+2\,(kd+p_2-p)\nabla_2 f(x)\\
& +\frac{1}{2}\sum_{\alpha,\beta\in I_2}\frac{\partial^2 f}{\partial x_\alpha\partial x_\beta}(x){\left\{\widehat c_\alpha,x_2,\widehat c_\beta\right\}},\\
\intertext{where the relation ${\left\{Q_{x_2^{-1}}c_\alpha,x_2,\widehat c_\alpha\right\}}={\left\{x_2^{-1},c_\alpha,\widehat c_\alpha\right\}}$ follows from \eqref{JP8},}
(2) ={}& \sum_{\alpha\in I_1}
(kd-p)\cdot{\left\{\widehat c_\alpha,c_\alpha,x_2^{-1}\right\}}\cdot f(x)
+\sum_{\alpha\in I_1}{\left\{\widehat c_\alpha,c_\alpha,\nabla_2 f\right\}}\\
&+\sum_{\beta\in I_2}{\left\{\nabla_1 f,c_\beta,\widehat c_\beta\right\}}
+(kd-p){\left\{\nabla_1 f,x,x_2^{-1}\right\}}\\
&+\sum_{\alpha\in I_1,\beta\in I_2}
\frac{\partial^2 f}{\partial x_\alpha\partial x_\beta}(x)\,
{\left\{\widehat c_\alpha,x,\widehat c_\beta\right\}}\\
={}&2(p-p_2)\,\big((kd-p)\cdot f(x)\cdot x_2^{-1} + \nabla_2 f(x)\big)
+(kd+p_2-p)\,\nabla_1f(x)\\
&+(kd-p){\left\{\nabla_1 f,x_1,x_2^{-1}\right\}}
+ \sum_{\alpha\in I_1,\beta\in I_2}
\frac{\partial^2 f}{\partial x_\alpha\partial x_\beta}(x)\,
{\left\{\widehat c_\alpha,x,\widehat c_\beta\right\}},\\[2mm]
(3) ={}& \sum_{\alpha\in I_1}{\left\{\widehat c_\alpha,c_\alpha,\nabla_1 f\right\}}+\frac{1}{2}\sum_{\alpha,\beta\in I_1}\frac{\partial^2 f}{\partial x_\alpha\partial x_\beta}(x)
{\left\{\widehat c_\alpha,x_2,\widehat c_\beta\right\}}\\
={}& (2p-p_2-p_0)\nabla_1 f+\frac{1}{2}\sum_{\alpha,\beta\in I_1}\frac{\partial^2 f}{\partial x_\alpha\partial x_\beta}(x)
{\left\{\widehat c_\alpha,x_2,\widehat c_\beta\right\}},\\[2mm]
(4) ={}& \frac{1}{2}\sum_{\alpha,\beta\in I_1}f(x)\cdot
{\left\{\widehat c_\alpha,{\left\{c_\alpha,x_2^{-1},c_\beta\right\}},\widehat c_\beta\right\}}
+\sum_{\alpha\in I_1}
{\left\{\widehat c_\alpha,{\left\{c_\alpha,x_2^{-1},x_1\right\}},\nabla_1 f\right\}}\\
&+\frac{1}{2}\sum_{\alpha,\beta\in I_1}
\frac{\partial^2 f}{\partial x_\alpha\partial x_\beta}(x)
{\left\{\widehat c_\alpha,Q_{x_1}x_2^{-1},\widehat c_\beta\right\}},\end{aligned}$$ $$\begin{aligned}
(5) ={}& -\lambda\nabla_2 f(x)-\lambda\nabla_1 f(x)-\lambda(kd-p)\,f(x)\cdot x_2^{-1}.\end{aligned}$$ Since $\varphi^*{\mathcal{B}}_\lambda$ and the measure ${\mathrm{d}}\mu_k$ are $\overline Q_{[e]}$-equivariant, the adjoint operator $(\varphi^*{\mathcal{B}}_\lambda)^{\textup{ad}}$ has the same equivariance property. Now, $\overline Q_{[e]}$ acts transitively on $\Omega_e\times V_1^+$, and therefore it suffices to show that $(\varphi^*{\mathcal{B}}_\lambda)^{\textup{ad}}f(x)=\varphi^*{\mathcal{B}}_\lambda f(x)$ for $x\in(V_2^+)^\times$. This follows by putting $x_1=0$ in the above formulas, regrouping the various terms, and using Lemma \[lem:basessums2\].
Action on radial functions
--------------------------
We calculate the action of ${\mathcal{B}}_\lambda$ on $(M\cap K)$-invariant functions on the orbits ${\mathcal{O}}_k=L\cdot(e_1+\cdots+e_k)$, where $e_1,\ldots,e_r$ is a fixed frame of tripotents in $V^+$. Recall that the map $$(M\cap K)\times C_k^+\to{\mathcal{O}}_k, \quad (m,t)\mapsto mb_t$$ is a diffeomorphism onto an open dense subset of ${\mathcal{O}}_k$ where $b_t$ and $C_k^+$ are defined in and .
We first discuss the action of ${\mathcal{B}}_\lambda$ on $(M\cap K)$-invariant functions on the open orbit ${\mathcal{O}}_r\subseteq V^+$. In this case, there is no restriction on $\lambda$ concerning the tangentiality of ${\mathcal{B}}_\lambda$ to ${\mathcal{O}}_r$.
\[prop:radialBessel\] Suppose $f\in C^\infty({\mathcal{O}}_r)$ is $(M\cap K)$-invariant, and put $F(t_1,\ldots,t_r)=f(b_t)$, where $b_t=t_1e_1+\cdots+t_re_r\in{\mathcal{O}}_r$. Then $$\begin{aligned}
{\mathcal{B}}_\lambda f(b_t) &= \sum_{i=1}^r{{\mathcal{B}}_\lambda^iF(t_1,\ldots,t_r)\,\overline e_i}
\end{aligned}$$ with $$\label{eq:DefinitionBlambdaI}
{\mathcal{B}}_\lambda^i = t_i\frac{\partial^2}{\partial t_i^2} + \left(\lambda - e - (r-1)d\right)\,\frac{\partial}{\partial t_i} + \frac{1}{2}\sum_{j\neq i}\left(\frac{d_+}{t_i-t_j} + \frac{d_-}{t_i+t_j}\right)\left(t_i\frac{\partial}{\partial t_i}-t_j\frac{\partial}{\partial t_j}\right),$$ where $d_+,d_-,d,e$ are the structure constants defined in and , see also .
Let $\{c_\alpha\}_\alpha$ be an orthonormal basis of $V^+$ with respect to the inner product $(x|y)=\tau(x,\overline y)$, which is compatible with the Peirce decomposition associated to the frame $e_1,\ldots,e_r$, $$V^+ = \bigoplus_{1<i\leq j\leq r}(A_{ij}^+\oplus B_{ij}^+)\oplus\bigoplus_{i = 1}^r V^+_{i0}.$$ Then, $\{\overline c_\alpha\}_\alpha$ is an orthonormal basis of $V^-$ which is compatible with the corresponding decomposition of $V^-$ and dual to $\{c_\alpha\}_\alpha$ with respect to the trace form. Since $\dim A_{ii}^+=1$, we note that $c_\alpha=e_i$ for $c_\alpha\in A_{ii}^+$ due to the appropriate normalization of the trace form $\tau$.\
We note that an $(M\cap K)$-invariant function satisfies for any $X,Y\in{\mathfrak{m}}\cap{\mathfrak{k}}$ and $a\in V^+$ the relations $${\mathrm{d}}f(a)(Xa) = 0 \qquad \mbox{and} \qquad {\mathrm{d}}^2f(a)(Xa,Ya) + {\mathrm{d}}f(a)(YXa) = 0.\label{eq:derivativeRelation2}$$ We use these idenities to determine the first and second derivatives of $f$ at $a$. For any $x,y\in V^+$ the operator $D_{x,\overline y}-D_{y,\overline x}$ is an element of ${\mathfrak{m}}\cap{\mathfrak{k}}$. Consider $b_t = t_1e_1+\ldots+t_r e_r$ with $t\in C_r^+$, and $X = D_{e_i,\overline\eta} - D_{\eta,\overline e_i}$ with $0\neq\eta\in V_{ij}^+$, more precisely if $j\neq 0$ we assume $\eta\in A_{ij}^+$ or $\eta\in B_{ij}^+$. For convenience, we introduce the following symbols: Let $\epsilon_{ij} = 1+\delta_{ij}$, where $\delta_{ij}$ is Kronecker’s delta, and let $\sigma\in\{+1,-1,0\}$ be defined by $Q_e\overline\eta = \sigma\cdot\eta$, where $e=e_1+\cdots+e_r$. Then, $Xb_t$ evaluates to $$\begin{aligned}
Xb_t &= {\left\{e_i,\overline\eta,b_t\right\}} - {\left\{\eta,\overline e_i,b_t\right\}} = t_j {\left\{e_i,\overline\eta,e_j\right\}} - t_i{\left\{\eta,\overline e_i,e_i\right\}} = \epsilon_{ij}t_j Q_e\overline\eta - \epsilon_{ij}t_i\eta\\
&= -\epsilon_{ij}(t_i-\sigma\,t_j)\eta.\end{aligned}$$ Therefore, $Xb_t = 0$ if and only if $\eta\in A_{ii}^+ = {\mathbb{R}}\,e_i$, and we thus obtain $$\begin{aligned}
\label{eq:FirstDerivatives}
\frac{\partial f}{\partial x_\alpha}(b_t) = \begin{cases}
\frac{\partial F}{\partial t_i}(t_1,\ldots,t_r)
& \text{if $c_\alpha\in A_{ii}^+$ for some $i$,} \\
0 &\text{else.}
\end{cases}\end{aligned}$$ Now we turn to the discussion of second derivatives. Let $Y = D_{e_k,\overline\zeta} - D_{\zeta,\overline e_k}$ be another element of ${\mathfrak{m}}\cap{\mathfrak{k}}$ with $\zeta\in V_{k\ell}^+$ satisfying $Q_e\zeta = \sigma'\cdot\zeta$ where $\sigma'\in\{+1,-1,0\}$, then $$\begin{aligned}
Yb_t = -\epsilon_{k\ell}(t_k-\sigma'\,t_\ell)\,\zeta.\end{aligned}$$ Due to and , ${\mathrm{d}}^2f(b_t)(Xb_t,Yb_t)$ vanishes if the orthogonal projection of $YXb_t$ onto $\bigoplus_{i = 1}^r A_{ii}^+$ vanishes. Since $$YXb_t = \gamma_{ij}\big({\left\{e_k,\overline\zeta,\eta\right\}} - {\left\{\zeta,\overline e_k,\eta\right\}}\big) \qquad\text{with}\qquad \gamma_{ij} = -\epsilon_{ij}(t_i-\sigma\,t_j),$$ the Peirce rules imply that $YXb_t$ is an element of $V_{ik}^++ V_{jk}^++ V_{i\ell}^++V_{j\ell}^+$. Therefore, ${\mathrm{d}}^2f(b_t)(Xb_t,Yb_t)$ vanishes if the orthogonal projection $\pi_{k\ell}(YXb_t)$ of $YXb_t$ onto $A_{kk}^++ A_{\ell\ell}^+ = {\mathbb{R}}\,e_k+{\mathbb{R}}\,e_\ell$ vanishes. We therefore compute $(YXb_t|e_k)$ and $(YXb_t|e_\ell)$. Since $$\begin{aligned}
&\big({\left\{e_k,\overline\zeta,\eta\right\}}|e_k\big)
= \big(\eta|{\left\{\zeta,\overline e_k,e_k\right\}}\big)
= \epsilon_{kl}(\eta|\zeta), \\
&\big({\left\{\zeta,\overline e_k,\eta\right\}}|e_k\big)
= \big(\eta|{\left\{e_k,\overline\zeta,e_k\right\}}\big )
= 2\delta_{kl}\sigma'(\eta|\zeta),\end{aligned}$$ we obtain $(YXb_t|e_k) = \gamma_{ij}(\epsilon_{kl} -2\,\delta_{kl}\sigma')\,(\eta|\zeta)$, and a similar calculation yields $(YXb_t|e_\ell) = \gamma_{ij}\epsilon_{k\ell}(\delta_{k\ell}-\sigma')\,(\eta|\zeta)$. In any case, we conclude that if $\eta\perp\zeta$ with respect to the inner product $(-|-)$, then ${\mathrm{d}}^2f(b_t)(\eta,\zeta) = 0$. For $\zeta = \eta$, i.e. in particular $i = k$, $j = \ell$ and $\sigma' = \sigma$, the orthogonal projection $\pi_{ij}(YXb_t)$ of $YXb_t$ onto $A_{ii}^++ A_{jj}^+$ is given by $$\begin{aligned}
\pi_{ij}(YXb_t) = \epsilon_{ij}\gamma_{ij}(\eta|\eta)(e_i-\sigma e_j).\end{aligned}$$ Now assume $\eta\notin A_{ii}^+$. Then, $\gamma_{ij}\neq0$, and the second derivative ${\mathrm{d}}^2f(b_t)(\eta,\eta) = -\tfrac{1}{\gamma_{ij}^2}\,{\mathrm{d}}f(b_t)(YXb_t)$ is given by $$\begin{aligned}
{\mathrm{d}}^2f(b_t)(\eta,\eta)
= \frac{(\eta|\eta)}{t_i-\sigma t_j}\,\left(\frac{\partial F}{\partial t_i}(t)-
\sigma\frac{\partial F}{\partial t_j}(t)\right).\end{aligned}$$ We also evaluate ${\left\{\eta,\overline b_t,\eta\right\}}$. For $\eta\in V_{i0}^+$, this term vanishes due to the Peirce rules. For $\eta\in A_{ij}^+$ or $\eta\in B_{ij}^+$, we have $$Q_e\overline{{\left\{\eta,\overline b_t,\eta\right\}}} = 2\,Q_eQ_{\overline\eta}Q_e\overline b_t
= 2\,Q_{Q_e\overline\eta}\overline b_t = 2\,Q_{\pm\eta}\overline b_t
= {\left\{\eta,\overline b_t,\eta\right\}}.$$ Therefore, ${\left\{\eta,\overline b_t,\eta\right\}}\in A_{ii}^++A_{jj}^+$, and hence ${\left\{\eta,\overline b_t,\eta\right\}} =\alpha_i\,e_i+\alpha_j e_j$, where the coefficients $\alpha_i,\alpha_j$ are obtained by evalutation of the inner products of ${\left\{\eta,\overline b_t,\eta\right\}}$ with $e_i$ and $e_j$. This yields $${\left\{\eta,\overline b_t,\eta\right\}}
= \sigma(\eta|\eta)(t_je_i + t_i e_j).$$ Collecting everything, the sum over all second derivatives is given by $$\begin{aligned}
\sum_{\alpha,\beta} &\frac{\partial^2 f}{\partial x_\alpha\partial x_\beta}(b_t)
{\left\{\overline c_\alpha,b_t,\overline c_\beta\right\}}
= \sum_{\alpha}\frac{\partial^2f}{\partial x_\alpha^2}(b_t)
{\left\{\overline c_\alpha,b_t,\overline c_\alpha\right\}}\\
&= \underbrace{\sum_{i=1}^r \sum_{c_\alpha\in A_{ii}^+}(..)}_{(1)}
+ \underbrace{\sum_{i=1}^r \sum_{c_\alpha\in B_{ii}^+}(..)}_{(2)}
+ \underbrace{\sum_{1\leq i<j\leq r} \sum_{c_\alpha\in A_{ij}^+}(..)}_{(3)}
+ \underbrace{\sum_{1\leq i<j\leq r} \sum_{c_\alpha\in B_{ij}^+}(..)}_{(4)},\end{aligned}$$ and the single terms evaluate to $$\begin{aligned}
(1) &= \sum_{i=1}^r 2t_i\,\frac{\partial^2 F}{\partial t_i^2}
\,\overline e_i, \\
(2) &= \sum_{i=1}^r \sum_{c_\alpha\in B_{ii}^+}
\frac{1}{t_i}\frac{\partial F}{\partial t_i}\,(-2t_i\cdot\overline e_i)
= -2\dim B_{ii}^+\cdot\sum_{i=1}^r \frac{\partial F}{\partial t_i}
\,\overline e_i,\\
(3) &= \sum_{i<j}^r \sum_{c_\alpha\in A_{ij}^+}\frac{1}{t_i-t_j}
\left(\frac{\partial F}{\partial t_i}-\frac{\partial F}{\partial t_j}\right)
(t_i\overline e_j + t_j\overline e_i) \\
&= \sum_{i=1}^r \dim A_{ij}^+\cdot\left((1-r)\frac{\partial F}{\partial t_i} +
\sum_{j\neq i}\frac{1}{t_i-t_j}
\left(t_i\,\frac{\partial F}{\partial t_i}
-t_j\frac{\partial F}{\partial t_j}\right)\right)\overline e_i,\\
(4) &= \sum_{i<j}^r \sum_{e_\alpha\in B_{ij}^+}\frac{1}{t_i+t_j}
\left(\frac{\partial F}{\partial t_i}+\frac{\partial F}{\partial t_j}\right)
(-t_i\overline c_j - t_j\overline c_i) \\
&= \sum_{i=1}^r\dim B_{ij}^+\cdot\left((1-r)\frac{\partial F}{\partial t_i} +
\sum_{j\neq i}\frac{1}{t_i+t_j}
\left(t_i\,\frac{\partial F}{\partial t_i}
-t_j\frac{\partial F}{\partial t_j}\right)\right)\overline e_i.\end{aligned}$$ In combination with , this completes the proof.
As a consequence of Proposition \[prop:radialBessel\], we also obtain the action of ${\mathcal{B}}_\lambda$ on $(M\cap K)$-invariant functions on the lower dimensional orbits ${\mathcal{O}}_k$. Here, $\lambda$ needs to be fixed such that ${\mathcal{B}}_\lambda$ is tangential to ${\mathcal{O}}_k$.
\[cor:radialBessel\] Let $0\leq k\leq r-1$ and $\lambda=kd$. Suppose $f\in C^\infty({\mathcal{O}}_k)$ is $(M\cap K)$-invariant, and put $F(t_1,\ldots,t_k)=f(b_t)$, where $b_t=t_1e_1+\cdots+t_ke_k\in{\mathcal{O}}_k$. Then $$\begin{aligned}
{\mathcal{B}}_\lambda f(b_t) &= \sum_{i=1}^r{{\mathcal{B}}_k^iF(t_1,\ldots,t_k)\overline e_i}
\end{aligned}$$ with $$\begin{aligned}
{\mathcal{B}}_k^i &= t_i\frac{\partial^2}{\partial t_i^2}
+(d-e)\,\frac{\partial}{\partial t_i} +\frac{1}{2}\sum_{\substack{j=1\\j\neq i}}^k\left(\frac{d_+}{t_i-t_j} + \frac{d_-}{t_i+t_j}\right)\left(t_i\frac{\partial}{\partial t_i}-t_j\frac{\partial}{\partial t_j}\right),\label{eq:DefBLambdaK}
\intertext{for $1\leq i\leq k$, and}
{\mathcal{B}}_k^i &= \frac{d_+-d_-}{2}\sum_{j=1}^k\frac{\partial}{\partial t_j},\label{eq:DefBLambdaK2}
\end{aligned}$$ for $k<i\leq r$.
Let $\tilde f$ be the extension of $f$ to an $(M\cap K)$-invariant function on ${\mathcal{O}}_r$ defined by $\tilde f(mb_t)=F(t_1,\ldots, t_k)$ for $b_t=t_1e_1+\cdots+t_re_r$ with $t\in C_r^+$. Then, Proposition \[prop:radialBessel\] yields a formula for ${\mathcal{B}}_\lambda\tilde f$, which simplifies since $\frac{\partial F}{\partial t_j}=0$ for $j>k$. Taking limits $t_j\to0$ for $j>k$ proves our statement.
$L^2$-models for small representations {#sec:L2models}
======================================
In this section we give a different proof of the statement in Theorem \[thm:StructureDegPrincipalSeries\] that the representation $I(\nu_k)$ is reducible and has an irreducible unitarizable quotient $J(\nu_k)$. We show unitarity by constructing an intrinsic invariant inner product. This inner product is most explicit in the so-called Fourier transformed picture where it simply is the $L^2$-inner product of the $L$-equivariant measure ${\mathrm{d}}\mu_k$ on the $L$-orbit ${\mathcal{O}}_k$.
The Fourier-transformed picture {#sec:FourierTranform}
-------------------------------
The Fourier transform under consideration is the map $$\begin{aligned}
{\mathcal{F}}\colon{\mathcal{S}}'(V^-)\to{\mathcal{S}}'(V^+),\quad{\mathcal{F}}f(x) &= \int_{V^-}{e^{i\tau(x,y)}f(y)\,{\mathrm{d}}y}.\end{aligned}$$ Here, ${\mathrm{d}}y$ is any fixed Lebesgue measure on $V^-$, and $\tau$ is the trace form of the Jordan pair $(V^+,V^-)$. For simplicity we normalize Lebesgue measure ${\mathrm{d}}x$ on $V^+$ such that $${\mathcal{F}}^{-1}f(y) = \int_{V^+} e^{-i\tau(x,y)}f(x)\,{\mathrm{d}}x, \qquad y\in V^-.$$ Since $I(\nu)\subseteq{\mathcal{S}}'(V^-)$ we can apply the Fourier transform to the principal series $I(\nu)$, and hence call $$\tilde I(\nu)={\mathcal{F}}(I(\nu))\subseteq{\mathcal{S}}'(V^+)$$ the *Fourier-transformed picture* of $I(\nu)$. We define a representation $\tilde\pi_\nu$ on $\tilde I(\nu)$ by twisting $\pi_\nu$ with the Fourier transform: $$\begin{aligned}
\tilde\pi_\nu(g) &= {\mathcal{F}}\circ\pi_\nu(g)\circ{\mathcal{F}}^{-1}\qquad\text{for }g\in G.\end{aligned}$$ From Proposition \[prop:noncompactgroupaction\] it is easy to deduce the following explicit formulas for the action $\tilde\pi_\nu|_{\overline P}$:
\[prop:FTpicturegroupaction\] The action of $\exp(a)\in\overline N$ and $h\in L$ on $f\in\tilde I(\nu)$ is given by $$\begin{aligned}
\tilde\pi_\nu(\exp(a))f(x) &= e^{i\tau(x,a)}f(x),\\
\tilde\pi_\nu(h)f(x) &= \chi_{\nu-\frac{p}{2}}(h)f(h^{-1}y).
\end{aligned}$$
Since $N$ does not act by affine linear transformations in $\pi_\nu$, the action of $N$ in the Fourier-transformed picture $\tilde\pi_\nu$ is hard to determine. However, the infinitesimal action ${\mathrm{d}}\tilde\pi_\nu$ of $\tilde\pi_\nu$ can be expressed in terms of the Bessel operators:
\[prop:FTpicturealgebraaction\] The infinitesimal action ${\mathrm{d}}\tilde\pi_\nu$ of $\tilde\pi_\nu$ extends to ${\mathcal{S}}'(V^+)$ and is given by $$\begin{aligned}
{\mathrm{d}}\tilde\pi_\nu(a)f(x) &= i\tau(x,a)\,f(x), & &a\in\overline{\mathfrak{n}}= V^-,\\
{\mathrm{d}}\tilde\pi_\nu(T)f(x) &= -{\mathrm{d}}_{Tx}f(x)+(\tfrac{\nu}{p}-\tfrac{1}{2})\,\operatorname{Tr}_{V^+}(T)\,f(x),
& &T\in{\mathfrak{l}},\\
{\mathrm{d}}\tilde\pi_\nu(b)f(x) &= \tfrac{1}{i}\,\tau(b,{\mathcal{B}}_\lambda f(x)), & &b\in{\mathfrak{n}}=V^+,\end{aligned}$$ where $\lambda=p-2\nu$.
The first two formulas are easily deduced from Proposition \[prop:FTpicturegroupaction\]. For the last formula, a short calculation shows that for fixed $y\in V^-$, $${\mathcal{B}}_\lambda(e^{-i\tau(x,y)})(x) = -e^{-i\tau(x,y)}(Q_yx+i\lambda y).$$ Using this identity, the formula $\tau(x,Q_yb) = \tau(b,Q_yx)$, and Propositions \[prop:liealgaction\] and \[prop:BesselPartialIntegration\], we obtain $$\begin{aligned}
{\mathrm{d}}\pi_\nu(b){\mathcal{F}}^{-1}f(y)
&= \left(-{\mathrm{d}}_{Q_yb}-(2\nu+p)\tau(b,y)\right)
\int_{V^+}{e^{-i\tau(x,y)}f(x)\,{\mathrm{d}}x}\\
&= \int_{V^+}\left(i\tau(x,Q_yb)-(2\nu+p)\tau(b,y)\right)e^{-i\tau(x,y)}f(x)\,{\mathrm{d}}x\\
&= \tfrac{1}{i}\int_{V^+}
\tau\left(b,-\left(Q_yx+i(2\nu+p)y\right)e^{-i\tau(x,y)}\right)f(x)\,{\mathrm{d}}x\\
&= \tfrac{1}{i}\,\tau\left(b,\int_{V^+}\left({\mathcal{B}}_{2\nu+p}e^{-i\tau(x,y)}\right)(x)
f(x)\,{\mathrm{d}}x\right)\\
&= \tfrac{1}{i}\,
\tau\left(b,\int_{V^+}e^{-i\tau(x,y)}{\mathcal{B}}_{p-2\nu}f(x)\,{\mathrm{d}}x\right)\\
&= \int_{V^+}e^{-i\tau(x,y)}
\cdot\tfrac{1}{i}\,\tau\left(b,{\mathcal{B}}_{p-2\nu}f(x)\right){\mathrm{d}}x,\end{aligned}$$ and the proof is complete.
The spherical vector
--------------------
The $K$-spherical vector $\phi_\nu\in I(\nu)$ was explicitly computed in Proposition \[prop:SphericalVectorNonCptPicture\] in the non-compact picture. We now find its Fourier transform $$\psi_\nu={\mathcal{F}}\phi_\nu\in\tilde I(\nu)\subseteq{\mathcal{S}}'(V^+).$$ Note that since the family of distributions $\phi_\nu\in{\mathcal{S}}'(V^-)$ is holomorphic in the parameter $\nu\in{\mathbb{C}}$ the same is true for the Fourier transforms $\psi_\nu\in{\mathcal{S}}'(V^+)$. We assume $d_+=d_-$ throughout the whole section.
We make use of the following lemma which follows by the same arguments as [@DS99 Lemma 2.3]:
\[lem:KInvariantDistributions\] For any $\nu\in{\mathbb{C}}$, $\psi_\nu$ is the unique (up to scalar multiples) ${\mathrm{d}}\tilde\pi_\nu({\mathfrak{k}})$-invariant tempered distribution on $V^+$.
We now express $\psi_\nu$ in terms of a K-Bessel function on a symmetric cone. For details about K-Bessel functions on symmetric cones we refer the reader to Appendix \[app:KBesselFunctions\].
Fix $0\leq k\leq r$ and let $e=e_1+\cdots+e_k\in V^+$ be the standard rank $k$ idempotent. Then $[e]\subseteq V^+$ is a Jordan algebra and we consider its fixed points $A^{(k)}\subseteq[e]$ under the involution $x\mapsto Q_e\overline x$. The subalgebra $A^{(k)}$ is Euclidean of rank $k$ and $e_1,\ldots,e_k$ is a Jordan frame of $A^{(k)}$, whence the Peirce decomposition of $A^{(k)}$ is given by $$A^{(k)}=\bigoplus_{1\leq i\leq j\leq k}A_{ij}^+.$$ Denote by $\Omega^{(k)}\subseteq A^{(k)}$ the symmetric cone of $A^{(k)}$. Then by [@FK94 Theorem VII.1.1] the *Gindikin Gamma function of $\Omega^{(k)}$* is given by $$\Gamma_{k,d}(\lambda) = (2\pi)^{k(k-1)\frac{d}{4}}\prod_{i=1}^k\Gamma(\lambda-(i-1)\tfrac{d}{2}),$$ where $\Gamma(\lambda)$ denotes the classical gamma function. Hence, $\Gamma_{k,d}(\lambda)$ is holomorphic for $\operatorname{Re}\lambda>(k-1)\frac{d}{2}$. In the case $k=r$ we abbreviate $A=A^{(r)}$ and $\Omega=\Omega^{(r)}$.
Now let first $k=r$ and consider for $\mu\in{\mathbb{C}}$ the radial part $K_\mu(t_1,\ldots,t_r)$ of the K-Bessel function ${\mathcal{K}}_\mu(x)$ on the symmetric cone $\Omega$ (see Appendix \[app:KBesselFunctions\] for its definition). Define an $(M\cap K)$-invariant function $\Psi_\nu$ on the open dense orbit ${\mathcal{O}}_r\subseteq V^+$ by $$\Psi_\nu(mb_t) = K_{\frac{p-e+1}{2}-\nu}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_r}{2})^2\right), \qquad m\in M\cap K,\,t\in C_r^+.$$ Then $\Psi_\nu$ defines a measurable function on $V^+$.
Assume that $d_+=d_-$. Then for any $\nu\in{\mathbb{C}}$ with $\operatorname{Re}\nu>-\nu_{r-1}$ the function $\Psi_\nu$ belongs to $L^1(V^+)$ and hence defines a tempered distribution $\Psi_\nu\in{\mathcal{S}}'(V^+)$. We have $$\psi_\nu = {\textup{const}}\times\frac{\Psi_\nu}{\Gamma_{r,d}(\nu+\frac{p}{2})} \qquad \mbox{for $\operatorname{Re}\nu>-\nu_{r-1}$,}$$ where the constant only depends on the structure constants and the normalization of the measures. In particular, the family $\Psi_\nu\in{\mathcal{S}}'(V^+)$ extends meromorphically in the parameter $\nu\in{\mathbb{C}}$.
We first show that $\Psi_\nu\in L^1(V^+)$, this implies $\Psi_\nu\in{\mathcal{S}}'(V^+)$. Using the integral formula of Proposition \[prop:equivariantmeasure\] we find $$\begin{aligned}
\int_{V^+}|\Psi_\nu(x)|\,{\mathrm{d}}x &= \int_{M\cap K}\int_{C_r^+}\Psi_\nu(mb_t)\prod_{i=1}^rt_i^{e+b}\prod_{1\leq i<j\leq r}(t_i^2-t_j^2)^d\,{\mathrm{d}}t\,{\mathrm{d}}m\\
&= \int_{C_r^+}K_{\frac{p-e+1}{2}-\nu}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\prod_{i=1}^rt_i^{e+b}\prod_{1\leq i<j\leq r}(t_i^2-t_j^2)^d\,{\mathrm{d}}t\\
&= {\textup{const}}\times\int_{C_r^+}K_{\frac{p-e+1}{2}-\nu}(s_1,\ldots,s_k)\prod_{i=1}^rs_i^{\frac{e+b-1}{2}}\prod_{1\leq i<j\leq k}(s_i-s_j)^d\,{\mathrm{d}}s.\end{aligned}$$ By [@FK94 Theorem VI.2.3] the last integral is equal to a constant multiple of $$\int_\Omega{\mathcal{K}}_{\frac{p-e+1}{2}-\nu}(x)\Delta(x)^{\frac{e+b-1}{2}}\,{\mathrm{d}}x.$$ This integral is by Lemma \[lem:L1L2KBessel\] finite if and only if $$\frac{e+b-1}{2}>-1 \qquad \mbox{and} \qquad \operatorname{Re}\nu+e-1+\frac{b-p}{2}>-2-(r-1)\frac{d}{2},$$ which is satisfied since $b,e\geq0$ and $\operatorname{Re}\nu>-\nu_{r-1}=-\frac{1}{2}(e+\frac{b}{2}+1)$. Next we show that $\Psi_\nu$ is ${\mathrm{d}}\tilde\pi_\nu({\mathfrak{k}})$-invariant. Since $\Psi_\nu$ is $(M\cap K)$-invariant by definition, it suffices to show that ${\mathrm{d}}\tilde\pi_\nu({\mathfrak{k}}\cap(\overline{\mathfrak{n}}\oplus{\mathfrak{n}}))\Psi_\nu=0$. By Proposition \[prop:FTpicturealgebraaction\] this is equivalent to $${\mathcal{B}}_\lambda\Psi_\nu(x)=\Psi_\nu(x)\cdot\overline x$$ for $\lambda=p-2\nu$. In view of the equivariance property of the Bessel operator (see Lemma \[lem:BesselEquivariance\]) we may assume $x=b_t$, $t\in C_r^+$. According to Proposition \[prop:radialBessel\] we have $${\mathcal{B}}_\lambda\Psi_\nu(b_t) = \sum_{i=1}^r {\mathcal{B}}_\lambda^i\left[K_{\frac{p-e+1}{2}-\nu}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\right]\overline{e}_i$$ with ${\mathcal{B}}_\lambda^i$ as in . Then it remains to show $$\label{eq:DiffEqRadialKBesselRankR}
{\mathcal{B}}_\lambda^i\left[K_{\frac{p-e+1}{2}-\nu}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\right]=t_i\,K_{\frac{p-e+1}{2}-\nu}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right) \qquad \forall\,1\leq i\leq r.$$ Since $d_+=d_-$ the operator ${\mathcal{B}}_\lambda^i$ becomes $$\begin{aligned}
{\mathcal{B}}_\lambda^i = t_i\frac{\partial^2}{\partial t_i^2}
+\left(\lambda-e-(r-1)d\right)\,\frac{\partial }{\partial t_i}
+d\hspace{-4mm}\sum_{1\leq j\leq r,\ j\neq i}\frac{t_i}{t_i^2-t_j^2}
\left(t_i\frac{\partial}{\partial t_i}-t_j\frac{\partial}{\partial t_j}\right).\end{aligned}$$ Substituting $s_i=(\frac{t_i}{2})^2$ we find that $$\frac{1}{t_i}{\mathcal{B}}_\lambda^i = s_i\frac{\partial^2}{\partial s_i^2}
+\frac{1}{2}\left(\lambda-e-(r-1)d+1\right)\,\frac{\partial }{\partial s_i}
+\frac{d}{2}\sum_{1\leq j\leq r,\ j\neq i}\frac{1}{s_i-s_j}
\left(s_i\frac{\partial}{\partial s_i}-s_j\frac{\partial}{\partial s_j}\right),$$ so that is equivalent to . Thus $\Psi_\nu\in{\mathcal{S}}'(V^+)$ is ${\mathrm{d}}\tilde\pi_\nu({\mathfrak{k}})$-invariant, and by Lemma \[lem:KInvariantDistributions\] $\Psi_\nu=c(\nu)\psi_\nu$ for some constant $c(\nu)$. We determine $c(\nu)$ by evaluating the identity $c(\nu)\phi_\nu(y)={\mathcal{F}}^{-1}\Psi_\nu(y)$ at $y=0$. As above, using the integral formula of Proposition \[prop:equivariantmeasure\] and Lemma \[lem:L1L2KBessel\] we find $$c(\nu) = {\mathcal{F}}^{-1}\Psi_\nu(0) = \int_{V^+}\Psi_\nu(x)\,{\mathrm{d}}x = \int_\Omega{\mathcal{K}}_{\frac{p-e+1}{2}-\nu}(x)\Delta(x)^{\frac{e+b-1}{2}}\,{\mathrm{d}}x = {\textup{const}}\times\Gamma_{r,d}(\tfrac{2\nu+p}{2}).\qedhere$$
Now let $0\leq k\leq r-1$. For $\mu\in{\mathbb{C}}$ we consider the radial part $K_\mu(t_1,\ldots,t_k)$ of the K-Bessel function ${\mathcal{K}}_\mu(x)$ on the symmetric cone $\Omega^{(k)}\subseteq A^{(k)}$. Define an $(M\cap K)$-invariant function $\Psi_k$ on ${\mathcal{O}}_k$ by $$\Psi_k(mb_t) = K_{\frac{1}{2}(kd-e+1)}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right), \qquad m\in M\cap K,\,t\in C_k^+.$$
\[thm:PsiInL1\] Assume that $d_+=d_-$. Then for any $0\leq k\leq r-1$, we have $\Psi_k\in L^1({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$ and hence $\Psi_k\,{\mathrm{d}}\mu_k\in{\mathcal{S}}'(V^+)$. Moreover, $$\psi_{-\nu_k} = {\textup{const}}\times\Psi_k\,{\mathrm{d}}\mu_k.$$
We first show that $\Psi_k\in L^1({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$, this implies $\Psi_k\,{\mathrm{d}}\mu_k\in{\mathcal{S}}'(V^+)$. Using the integral formula of Proposition \[prop:equivariantmeasure\] we find $$\begin{aligned}
& \int_{{\mathcal{O}}_k}|\Psi_k(x)|\,{\mathrm{d}}\mu_k(x)\\
& \hspace{1cm} =\int_{C_k^+}K_{\frac{1}{2}(kd-e+1)}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\prod_{i=1}^kt_i^{(r-k+1)d+\frac{b}{2}-1}\prod_{1\leq i<j\leq k}(t_i^2-t_j^2)^d\,{\mathrm{d}}t\\
& \hspace{1cm} ={\textup{const}}\times\int_{C_k^+}K_{\frac{1}{2}(kd-e+1)}(s_1,\ldots,s_k)\prod_{i=1}^ks_i^{(r-k+1)\frac{d}{2}+\frac{b}{4}-1}\prod_{1\leq i<j\leq k}(s_i-s_j)^d\,{\mathrm{d}}s.\end{aligned}$$ By [@FK94 Theorem VI.2.3] the last integral is equal to a constant multiple of $$\int_{\Omega^{(k)}}{\mathcal{K}}_{\frac{kd-e+1}{2}}(x)\Delta(x)^{(r-k+1)\frac{d}{2}+\frac{b}{4}-1}\,{\mathrm{d}}x.$$ This integral is by Lemma \[lem:L1L2KBessel\] finite if and only if $$(r-k+1)\frac{d}{2}+\frac{b}{4}-1>-1 \qquad \mbox{and} \qquad (r-2k+1)\frac{d}{2}+\frac{e}{2}+\frac{b}{4}-\frac{3}{2}>-2-(k-1)\frac{d}{2},$$ which is satisfied since $0\leq k\leq r-1$, $d>0$ and $b,e\geq0$. Next we show that $\Psi_k\,{\mathrm{d}}\mu_k$ is ${\mathrm{d}}\tilde\pi_\nu({\mathfrak{k}})$-invariant for $\nu=-\nu_k$. Since $\Psi_k\,{\mathrm{d}}\mu_k$ is $(M\cap K)$-invariant by definition, it suffices to show that ${\mathrm{d}}\tilde\pi_\nu({\mathfrak{k}}\cap(\overline{\mathfrak{n}}\oplus{\mathfrak{n}}))(\Psi_k\,{\mathrm{d}}\mu_k)=0$. By Proposition \[prop:FTpicturealgebraaction\] this is equivalent to $${\mathcal{B}}_\lambda\left[\Psi_k(x)\,{\mathrm{d}}\mu_k(x)\right]=\left[\Psi_k(x)\,{\mathrm{d}}\mu_k(x)\right]\cdot\overline x,\qquad x\in{\mathcal{O}}_k,$$ for $\lambda=p+2\nu_k=2p-kd$. Using Proposition \[prop:BesselPartialIntegration\] we have by duality for any $\varphi\in{\mathcal{S}}(V^+)$: $$\begin{aligned}
\langle({\mathcal{B}}_\lambda-\overline x)\left[\Psi_k\,{\mathrm{d}}\mu_k\right],\varphi\rangle &= \langle\Psi_k\,{\mathrm{d}}\mu_k,({\mathcal{B}}_{kd}-\overline x)\varphi\rangle = \int_{{\mathcal{O}}_k}\Psi_k(x)({\mathcal{B}}_{kd}-\overline x)\varphi(x)\,{\mathrm{d}}\mu_k(x).\end{aligned}$$ Now ${\mathcal{B}}_{kd}$ is by Theorem \[thm:BesselSymmetricOnOrbits\] symmetric on $L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$ and hence $$\langle({\mathcal{B}}_\lambda-\overline x)\left[\Psi_k\,{\mathrm{d}}\mu_k\right],\varphi\rangle = \int_{{\mathcal{O}}_k}({\mathcal{B}}_{kd}-\overline x)\Psi_k(x)\varphi(x)\,{\mathrm{d}}\mu_k(x).$$ However, since $\Psi_k$ is not compactly supported on ${\mathcal{O}}_k$ we have to assure that during the integration by parts (see the proof of Theorem \[thm:BesselSymmetricOnOrbits\]) no boundary terms appear. The Bessel operator is of order $2$ and Euler degree $-1$ and therefore it suffices to show that all first partial derivatives of $\Psi_k$ are still contained in $L^1({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$. This follows if for every $\ell=1,\ldots,k$ we have $$\int_{C_k^+} \left|\frac{\partial}{\partial t_\ell}\left[K_{\frac{1}{2}(kd-e+1)}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\right]\right|\prod_{i=1}^kt_i^{(r-k+1)d+\frac{b}{2}-1}\prod_{1\leq i<j\leq k}(t_i^2-t_j^2)^d\,{\mathrm{d}}t < \infty.\label{eq:L1IntegralFirstDerivativePsik}$$ We compute the derivative using : $$\begin{aligned}
& \frac{\partial}{\partial t_\ell}\left[K_{\frac{1}{2}(kd-e+1)}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\right] = \frac{\partial}{\partial t_\ell}\left[\prod_{i=1}^k(\tfrac{t_i}{2})^{\frac{n}{r}-\lambda}\int_{\Omega^{(k)}} e^{-\frac{1}{2}(b_t|v+v^{-1})}\Delta(v)^{-\lambda}\,{\mathrm{d}}v\right]\\
& \hspace{1cm} = \prod_{i=1}^k(\tfrac{t_i}{2})^{\frac{n}{r}-\lambda}\int_{\Omega^{(k)}}\left(\frac{(\frac{n}{r}-\lambda)}{t_\ell}-\frac{1}{2}(e_\ell|v+v^{-1})\right)e^{-\frac{1}{2}(b_t|v+v^{-1})}\Delta(v)^{-\lambda}\,{\mathrm{d}}v.\end{aligned}$$ Using $(e_i|v+v^{-1})\geq2$ for all $i=1,\ldots,k$ and $v\in\Omega^{(k)}$, we can estimate $$\frac{1}{t_\ell} \leq 2^{1-k}(t_1\cdots t_k)^{-1}\cdot\prod_{\substack{i=1\\i\neq\ell}}^k(t_ie_i|v+v^{-1}) \leq 2^{1-k}(t_1\cdots t_k)^{-1}(b_t|v+v^{-1})^{k-1}$$ and $$(e_\ell|v+v^{-1}) \leq 2^{1-k}(t_1\cdots t_k)^{-1}\prod_{i=1}^k(t_ie_i|v+v^{-1}) \leq 2^{1-k}(t_1\cdots t_k)^{-1}(b_t|v+v^{-1})^k.$$ Now $(x^{k-1}+x^k)e^{-\frac{1}{2}x}\leq{\textup{const}}\times e^{-\frac{1}{4}x}$ for $x\geq0$, and hence $$\left|\frac{\partial}{\partial t_\ell}\left[K_{\frac{1}{2}(kd-e+1)}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\right]\right| \leq {\textup{const}}\times(t_1\cdots t_k)^{-1}K_{\frac{1}{2}(kd-e+1)}\left((\tfrac{t_1}{4})^2,\ldots,(\tfrac{t_k}{4})^2\right).$$ As above this shows that the integral is bounded by a constant times $$\int_{\Omega^{(k)}}{\mathcal{K}}_{\frac{kd-e+1}{2}}(x)\Delta(x)^{(r-k+1)\frac{d}{2}+\frac{b}{4}-\frac{3}{2}}\,{\mathrm{d}}x,$$ which is by Lemma \[lem:L1L2KBessel\] finite if and only if $$(r-k+1)\frac{d}{2}+\frac{b}{4}-\frac{3}{2}>-1 \qquad \mbox{and} \qquad (r-2k+1)\frac{d}{2}+\frac{e}{2}+\frac{b}{4}-2>-2-(k-1)\frac{d}{2}.$$ These inequalities hold since $0\leq k\leq r-1$, $d\geq1$ and $b,e\geq0$. This justifies the use of Theorem \[thm:BesselSymmetricOnOrbits\] and it remains to show that ${\mathcal{B}}_{kd}\Psi_k(x)=\Psi_k(x)\cdot\overline x$ for any $x\in{\mathcal{O}}_k$. In view of the equivariance property of the Bessel operator (see Lemma \[lem:BesselEquivariance\]) we may assume $x=b_t$, $t\in C_k^+$. According to Corollary \[cor:radialBessel\] we have $${\mathcal{B}}_{kd}\Psi_k(b_t)=\sum_{i=1}^k {\mathcal{B}}_k^i\left[K_{\frac{1}{2}(kd-e+1)}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\right]\overline{e}_i$$ with ${\mathcal{B}}_k^i$ as in and . Then it remains to show $$\begin{aligned}
\label{eq:radialPDE}
{\mathcal{B}}_k^i\left[K_{\frac{kd-e+1}{2}}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\right] = \begin{cases} t_iK_{\frac{kd-e+1}{2}}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right) & \mbox{for }1\leq i\leq k,\\0 & \mbox{for }k<i\leq r.\end{cases}\end{aligned}$$ Due to the assumption $d_+=d_-$, the second condition is automatic, and for $1\leq i\leq k$ the operator ${\mathcal{B}}_k^i$ becomes $$\begin{aligned}
{\mathcal{B}}_k^i=t_i\frac{\partial^2}{\partial t_i^2}
+(d-e)\,\frac{\partial }{\partial t_i}
+d\hspace{-4mm}\sum_{1\leq j\leq k,\ j\neq i}\frac{t_i}{t_i^2-t_j^2}
\left(t_i\frac{\partial}{\partial t_i}-t_j\frac{\partial}{\partial t_j}\right).\end{aligned}$$ Substituting $s_i=(\frac{t_i}{2})^2$ it is easy to see that this is equivalent to the differential equation for the K-Bessel function. Hence $\Psi_k\,{\mathrm{d}}\mu_k$ is a ${\mathrm{d}}\tilde\pi_\nu({\mathfrak{k}})$-invariant tempered distribution on $V^+$ for $\nu=-\nu_k$ and therefore a constant multiple of $\psi_\nu$.
\[thm:PsiInL2\] Assume $d_+=d_-$. Then $\Psi_k\in L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$ for any $0\leq k\leq r-1$.
A similar computation as in the proof of Theorem \[thm:PsiInL1\] shows that $$\int_{{\mathcal{O}}_k} |\Psi_k(x)|^2\,{\mathrm{d}}\mu_k(x) = {\textup{const}}\times\int_{\Omega^{(k)}} |{\mathcal{K}}_{\frac{kd-e+1}{2}}(x)|^2 \Delta(x)^{(r-k+1)\frac{d}{2}+\frac{b}{4}-1}\,{\mathrm{d}}x$$ which is by Lemma \[lem:L1L2KBessel\] finite if and only if $$(r-k+1)\frac{d}{2}+\frac{b}{4}-1>-1 \qquad \mbox{and} \qquad (r-3k+1)\frac{d}{2}+e+\frac{b}{4}-2>-3-(k-1)d.$$ This is satisfied since $0\leq k\leq r-1$, $d>0$ and $b,e\geq0$.
Construction of the $L^2$-model
-------------------------------
By Theorem \[thm:StructureDegPrincipalSeries\] we know that for $0\leq k\leq r-1$ the $({\mathfrak{g}},K)$-module $${\mathrm{d}}\tilde\pi_{-\nu_k}({\mathcal{U}}({\mathfrak{g}}))\psi_{-\nu_k} \subseteq \tilde I(-\nu_k)_{{K-\textup{finite}}}$$ generated by the spherical vector $\psi_{-\nu_k}$ is irreducible and unitarizable. We give a new proof of this fact which also provides an explicit invariant Hermitian form. Let us write $\tilde I_0(-\nu_k)$ for the corresponding subrepresentation of $\tilde I(-\nu_k)$ with underlying $({\mathfrak{g}},K)$-module ${\mathrm{d}}\tilde\pi_{-\nu_k}({\mathcal{U}}({\mathfrak{g}}))\psi_{-\nu_k}$.
By Theorem \[thm:PsiInL1\] we have $\psi_{-\nu_k}={\textup{const}}\times\Psi_k\,{\mathrm{d}}\mu_k$. For $0\leq k\leq r-1$ we let $$(M\cap K)_k={\left\{m\in M\cap K\,\middle|\,me_i=e_i\,\forall\,i=1,\ldots,k\right\}},$$ then the map $$(M\cap K)/(M\cap K)_k\times C_k^+\to{\mathcal{O}}_k, \quad (m,t)\mapsto mb_t,$$ is a diffeomorphism onto an open dense subset of ${\mathcal{O}}_k$. In these coordinates the function $\Psi_k$ takes the form $$\Psi_k(mb_t) = F(t_1,\ldots,t_k), \qquad m\in M\cap K\,t\in C_k^+,$$ where $F(t_1,\ldots,t_k)=K_{\frac{kd-e+1}{2}}((\frac{t_1}{2})^2,\ldots,(\frac{t_k}{2})^2)$.
Let $0\leq k\leq r-1$, then any $f\in\tilde I_0(-\nu_k)_{{K-\textup{finite}}}$ can be written as $$f(mb_t) = \sum_\alpha \varphi_\alpha(m)f_\alpha(t), \qquad m\in M\cap K,\,t\in C_k^+,\label{eq:FormWk}$$ where $\varphi_\alpha\in C^\infty((M\cap K)/(M\cap K)_k)$ and $f_\alpha$ is of the form $$t_{i_1}\cdots t_{i_p}t_{j_1}\cdots t_{j_q}\frac{\partial^p F}{\partial t_{i_1}\cdots\partial t_{i_p}}(t_1,\ldots,t_k)\label{eq:Formfalpha}$$ with $i_1,\ldots,i_p,j_1,\ldots,j_q\in\{1,\ldots,k\}$ and $p,q\geq0$.
By the Poincaré–Birkhoff–Witt Theorem we have $${\mathcal{U}}({\mathfrak{g}}) = {\mathcal{U}}(\overline {\mathfrak{n}})\,{\mathcal{U}}({\mathfrak{l}})\,{\mathcal{U}}({\mathfrak{k}})$$ and hence $$\tilde I_0(-\nu_k)_{{K-\textup{finite}}}= {\mathrm{d}}\tilde\pi_\nu\big({\mathcal{U}}(\overline{\mathfrak{n}})\,{\mathcal{U}}({\mathfrak{l}})\,{\mathcal{U}}({\mathfrak{k}})\big)(\Psi_k\,{\mathrm{d}}\mu_k) = {\mathrm{d}}\tilde\pi_\nu\big({\mathcal{U}}(\overline{\mathfrak{n}})\,{\mathcal{U}}({\mathfrak{l}})\big)(\Psi_k\,{\mathrm{d}}\mu_k).$$ Since $\overline{\mathfrak{n}}$ acts by multiplication with polynomials $\tau(mb_t,u)=\sum_{i=1}^kt_i\tau(me_i,u)$, $u\in V^-$, the action of ${\mathcal{U}}(\overline{\mathfrak{n}})$ leaves the space of functions of the form invariant. Hence, it suffices to show that every $f\in{\mathrm{d}}\tilde\pi_\nu({\mathcal{U}}({\mathfrak{l}}))(\Psi_k\,{\mathrm{d}}\mu_k)$ is of the form . We show this for ${\mathrm{d}}\tilde\pi_\nu({\mathcal{U}}_n({\mathfrak{l}}))(\Psi_k\,{\mathrm{d}}\mu_k)$ by induction on $n$, where $\{{\mathcal{U}}_n({\mathfrak{l}})\}_{n\geq0}$ is the natural filtration of ${\mathcal{U}}({\mathfrak{l}})$. For $n=0$ this is clear since ${\mathcal{U}}_0({\mathfrak{l}})={\mathbb{C}}$. We also carry out the proof for $n=1$. Let $T\in{\mathfrak{l}}$, then ${\mathrm{d}}\tilde\pi_\nu(T)=-{\mathrm{d}}_{Tx}+(\frac{\nu}{p}-\frac{1}{2})\operatorname{Tr}_{V^+}(T)$. We now compute ${\mathrm{d}}_{Tx}\Psi_k(x)$. Note that since $\Psi_k$ is $(M\cap K)$-invariant, implies that for $x=b_t$ we have $${\mathrm{d}}_{Tx}\Psi_k(x) = \sum_{j=1}^k\tau(Tb_t,\overline{e}_j)\cdot\frac{\partial F}{\partial t_j}(t_1,\ldots,t_k).$$ By the standard transformation rules this implies for $x=mb_t$: $${\mathrm{d}}_{Tx}\Psi_k(x) = \sum_{i,j=1}^k \tau(Tme_i,m\overline{e}_j)\cdot t_i\frac{\partial F}{\partial t_j}(t_1,\ldots,t_k).\label{eq:LactionOnMinvariants}$$ Now, $\tau(Se_i,\overline{e}_j)=0$ for $i\neq j$ and any $S\in{\mathfrak{l}}$. In fact, ${\mathfrak{l}}$ is generated by the operators $D_{u,v}$, $u\in V^+$, $v\in V^-$, and by : $$\tau(D_{u,v}e_i,\overline{e}_j) = \tau(u,{\left\{v,e_i,\overline{e}_j\right\}}) = 0.$$ Therefore $\tau(Tme_i,m\overline{e}_j)=\tau(m^{-1}Tme_i,\overline{e}_j)=0$ whenever $i\neq j$ so that $${\mathrm{d}}_{Tx}\Psi_k(x) = \sum_{i=1}^k \tau(Tme_i,m\overline{e}_i)\cdot t_i\frac{\partial F}{\partial t_i}(t_1,\ldots,t_k)$$ which clearly is of the form with $$\varphi_i(m)=\tau(Tme_i,m\overline e_j) \qquad \mbox{and} \qquad f_i(t_1,\ldots,t_k)=t_i\frac{\partial F}{\partial t_i}(t_1,\ldots,t_k).$$ Now let us complete the induction step. Note that $${\mathrm{d}}_{Tx}f(x)=\left.\frac{{\mathrm{d}}}{{\mathrm{d}}s}\right|_{s=0}f(e^{sT}x).$$ For $x=mb_t$ we write $e^{sT}mb_t=m_sb_{t,s}$, where $m_s\in M\cap K$ and $b_{t,s}$ depend differentiably on $s\in(-\varepsilon,\varepsilon)$ and $m_0=m$, $b_{t,0}=b_t$. For $f$ of the form we obtain $${\mathrm{d}}_{Tx}f(mb_t) = \sum_\alpha\left.\frac{{\mathrm{d}}}{{\mathrm{d}}s}\right|_{s=0}\varphi_\alpha(m_s)f_\alpha(b_t)+\sum_\alpha\varphi_\alpha(m)\left.\frac{{\mathrm{d}}}{{\mathrm{d}}s}\right|_{s=0}f_\alpha(b_{t,s}).$$ Clearly the first summand is again of the form . To treat the second summand we note that for $f=\Psi_k$ this expression has to agree with and hence it is of the form $$\sum_\alpha\sum_{i=1}^k\varphi_\alpha(m)\varphi_i(m)\cdot t_i\frac{\partial f_\alpha}{\partial t_i}(t_1,\ldots,t_k).$$ Clearly $t_i\frac{\partial}{\partial t_i}$ leaves the space of functions of the form invariant so that this expression is again of the form . This completes the induction step and the proof.
\[prop:gKmoduleL2\] For any $0\leq k\leq r-1$ we have $\tilde I_0(-\nu_k)_{{K-\textup{finite}}}\subseteq L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$.
It remains to show that every function of the form is contained in $L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$. In view of the integral formula of Proposition \[prop:equivariantmeasure\] this amounts to showing that $$\int_{C_k^+} \left|t_{i_1}\cdots t_{i_p}t_{j_1}\cdots t_{j_q}\frac{\partial^p F}{\partial t_{i_1}\cdots t_{i_p}}(t_1,\ldots,t_k)\right|^2\prod_{i=1}^kt_i^{(r-k+1)d+\frac{b}{2}-1}\prod_{1\leq i<j\leq k}(t_i^2-t_j^2)^d\,{\mathrm{d}}t < \infty$$ for all $i_1,\ldots,i_p,j_1,\ldots,j_q\in\{1,\ldots,k\}$ with $p,q\geq0$. By the chain rule $$t_{i_1}\cdots t_{i_p}t_{j_1}\cdots t_{j_q}\frac{\partial^p F}{\partial t_{i_1}\cdots t_{i_p}}(t_1,\ldots,t_k) = 2^{-p}t_{i_1}^2\cdots t_{i_p}^2t_{j_1}\cdots t_{j_q}\frac{\partial^p K_\lambda}{\partial s_{i_1}\cdots s_{i_p}}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)$$ with $\lambda=\frac{kd-e+1}{2}$. For $p=q=0$ we showed in Theorem \[thm:PsiInL2\] that $$\int_{C_k^+} \left|K_\lambda\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\right|^2\prod_{i=1}^kt_i^{(r-k+1)d+\frac{b}{2}-1}\prod_{1\leq i<j\leq k}(t_i^2-t_j^2)^d\,{\mathrm{d}}t < \infty.$$ Now let $s_i=(\frac{t_i}{2})^2$ and consider $s_i\frac{\partial K_\lambda}{\partial s_i}(s_1,\ldots,s_k)$. By we have $$s_i\frac{\partial K_\lambda}{\partial s_i}(s_1,\ldots,s_k) = -\int_{\Omega^{(k)}} (s_ie_i|v^{-1})e^{-\operatorname{tr}(v)-(b_s|v^{-1})}\Delta(v)^{-\lambda}\,{\mathrm{d}}v$$ so that $$\left|s_{i_1}\cdots s_{i_p}\frac{\partial^p K_\lambda}{\partial s_{i_1}\cdots\partial s_{i_p}}(s_1,\ldots,s_k)\right| = \int_{\Omega^{(k)}}(s_{i_1}e_{i_1}|v^{-1})\cdots(s_{i_p}e_{i_p}|v^{-1})e^{-\operatorname{tr}(v)-(b_s|v^{-1})}\Delta(v)^{-\lambda}\,{\mathrm{d}}v.$$ Now for any $u\in\Omega^{(k)}$ we have $(u|e_i)\geq0$ for all $i=1,\ldots,k$ and hence $$(s_{i_1}e_{i_1}|v^{-1})\cdots(s_{i_p}e_{i_p}|v^{-1}) \leq (b_s|v^{-1})^p \qquad \forall\,v\in\Omega^{(k)}.$$ Let $C>0$ such that $x^pe^{-x}\leq Ce^{-\frac{1}{4}x}$ for $x\geq0$, then $$\left|s_{i_1}\cdots s_{i_p}\frac{\partial^p K_\lambda}{\partial s_{i_1}\cdots\partial s_{i_p}}(s_1,\ldots,s_k)\right| \leq C\int_{\Omega^{(k)}} e^{-\operatorname{tr}(v)-\frac{1}{4}(b_s|v^{-1})}\Delta(v)^{-\lambda}\,{\mathrm{d}}v = C\cdot K_\lambda(\tfrac{1}{4}s_1,\ldots,\tfrac{1}{4}s_k).$$ Finally consider $t_{j_1}\cdots t_{j_q}K_\lambda\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)$. Using we have $$t_{j_1}\cdots t_{j_q}K_\lambda\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right) = t_{j_1}\cdots t_{j_q}\int_{\Omega^{(k)}} e^{-\frac{1}{2}(b_t|v+v^{-1})}\Delta(v)^{-\lambda}\,{\mathrm{d}}v.$$ Since $(e_i|v+v^{-1})\geq2$ for all $v\in\Omega^{(k)}$ we find $$t_{j_1}\cdots t_{j_q} \leq (t_{j_1}e_{j_1}|v+v^{-1})\cdots(t_{j_q}e_{j_q}|v+v^{-1}) \leq (b_t|v+v^{-1})^q.$$ Let $C'>0$ such that $x^qe^{-\frac{1}{2}x}\leq C'e^{-\frac{1}{4}x}$ for $x\geq0$, then $$\left|t_{j_1}\cdots t_{j_q}K_\lambda\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\right| \leq C'\int_{\Omega^{(k)}} e^{-\frac{1}{4}(b_t|v+v^{-1})}\Delta(v)^{-\lambda}\,{\mathrm{d}}v = C'\cdot K_\lambda\left((\tfrac{t_1}{4})^2,\ldots,(\tfrac{t_k}{4})^2\right).$$ Combining both arguments we obtain $$\left|t_{i_1}^2\cdots t_{i_p}^2t_{j_1}\cdots t_{j_q}\frac{\partial^p K_\lambda}{\partial s_{i_1}\cdots s_{i_p}}\left((\tfrac{t_1}{2})^2,\ldots,(\tfrac{t_k}{2})^2\right)\right| \leq {\textup{const}}\times K_\lambda\left((\tfrac{t_1}{8})^2,\ldots,(\tfrac{t_k}{8})^2\right).$$ The same computation as in the proof of Theorem \[thm:PsiInL2\] (substituting $s_i=(\frac{t_i}{8})^2$ instead of $s_i=(\frac{t_i}{2})^2$) shows that $$\int_{C_k^+} \left|K_\lambda\left((\tfrac{t_1}{8})^2,\ldots,(\tfrac{t_k}{8})^2\right)\right|^2\prod_{i=1}^kt_i^{(r-k+1)d+\frac{b}{2}-1}\prod_{1\leq i<j\leq k}(t_i^2-t_j^2)^d\,{\mathrm{d}}t < \infty$$ and the proof is complete.
\[thm:UniRepOnL2\] Assume $d_+=d_-$. Then $\tilde I_0(-\nu_k)\subseteq L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$ and the $L^2$-inner product is a $\tilde\pi_{-\nu_k}$-invariant Hermitian form. The representation $(\tilde\pi_{-\nu_k},\tilde I_0(-\nu_k))$ integrates to an irreducible unitary representation of $G$ on $L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$.
This is the same argument as in [@HKM14 Theorem 2.30]. By Proposition \[prop:gKmoduleL2\] we have $\tilde I_0(-\nu_k)_{{K-\textup{finite}}}\subseteq L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$, and by Theorem \[thm:BesselSymmetricOnOrbits\] and Proposition \[prop:FTpicturealgebraaction\] the $L^2$-inner product is an invariant Hermitian form on $\tilde I_0(-\nu_k)_{{K-\textup{finite}}}$. Hence, the $({\mathfrak{g}},K)$-module $\tilde I_0(-\nu_k)_{{K-\textup{finite}}}$ integrates to a unitary representation $(\tau,{\mathcal{H}})$ on a Hilbert space ${\mathcal{H}}\subseteq L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$. Since the Lie algebra actions of $\tau$ and $\tilde\pi_{-\nu_k}$ agree on the Lie algebra of $\overline P$, the group actions $\tau$ and $\tilde\pi_{-\nu_k}$ agree on ${\mathcal{H}}$. But in view of Proposition \[prop:FTpicturegroupaction\] the subgroup $\overline P$ acts by Mackey theory irreducibly on $L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$ and hence ${\mathcal{H}}=L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$ and $\tau$ is irreducible as $G$-representation.
\[rem:MinKtypeHermitianAndOpq\] Let us comment on the two cases excluded in Theorem \[thm:UniRepOnL2\].
1. In the case where $G$ is Hermitian, the K-Bessel function has to be replaced by the exponential function $$\Psi_k(mb_t) = e^{-(t_1+\cdots+t_k)}, \qquad m\in M\cap K,t\in C_k^+,$$ then one can show that $\Psi_k\,{\mathrm{d}}\mu_k$ transforms under ${\mathrm{d}}\tilde\pi_\nu({\mathfrak{k}})$ by a unitary character of ${\mathfrak{k}}$ for $\nu=-\nu_k$. Note that here the Lie algebra ${\mathfrak{k}}$ has a one-dimensional center. In the same way as above one then shows that $\tilde I_0(-\nu_k)$ extends to an irreducible unitary representation of $G$ on $L^2({\mathcal{O}}_k,{\mathrm{d}}\mu_k)$ (see e.g. [@Moe13 Section 2.1]). We remark that in this case the degenerate principal series has to be formed by inducing from a possibly non-trivial unitary character of $M$.
2. If $G=SO_0(p,q)$, $p\leq q$, then $r=2$ and the K-Bessel function $\Psi_1$ is essentially a classical K-Bessel function. In [@HKM14] it is shown that, after modifying the parameter of the Bessel function, $\Psi_1\,{\mathrm{d}}\mu_1$ generates a finite-dimensional ${\mathfrak{k}}$-representation in ${\mathrm{d}}\tilde\pi_\nu$, $\nu=-\nu_k$, if and only if $p+q$ is even. This ${\mathfrak{k}}$-representation is isomorphic to the representation ${\mathcal{H}}^{\frac{q-p}{2}}({\mathbb{R}}^p)$ on spherical haromonics of degree $\frac{q-p}{2}$ in $p$ variables, and hence non-trivial for $p\neq q$. Also in this case the same methods as above show that the corresponding $({\mathfrak{g}},K)$-module integrates to an irreducible unitary representation of $G$ on $L^2({\mathcal{O}}_1,{\mathrm{d}}\mu_1)$ (see [@HKM14] for details). We note that here one has to form the degenerate principal series by inducing from a non-trivial unitary character of $M$ if and only if $p$ and $q$ are both odd and $p-q\equiv2\mod4$.
\[cor:MinRep\] Assume $d_+=d_-$. If $e=0$ or $V$ is complex, and ${\mathfrak{g}}_{\mathbb{C}}$ is not a Lie algebra of type $A$, then $J(\nu_1)$ is the unique minimal representation of $G$ (in the sense of [@HKM14 Definition 2.16]).
For $P$ and $\overline P$ conjugate this is shown in [@HKM14 Corollary 2.32]. For $P$ and $\overline P$ not conjugate we showed in [@MS12 Theorem 5.3 (2)] that the associated variety of $J(\nu_1)$ is the minimal nilpotent $K_{\mathbb{C}}$-orbit. Since the Joseph ideal is the unique completely prime ideal in ${\mathcal{U}}({\mathfrak{g}})$ with associated variety the minimal nilpotent $K_{\mathbb{C}}$-orbit, it remains to show that the annihilator ideal of $J(\nu_1)$ is completely prime. We employ the same argument as in [@HKM14 Theorem 2.18] using the explicit $L^2$-model. The Lie algebra acts by regular differential operators on the irreducible variety ${\mathcal{O}}_1={\mathcal{V}}_1$ and this induces an algebra homomorphism from ${\mathcal{U}}({\mathfrak{g}})$ to the algebra of regular differential operators on ${\mathcal{V}}_1$. The latter algebra has no zero-divisors which implies that the kernel of this homomorphism (which is the annihilator of $J(\nu_1)$) is completely prime. This finishes the proof.
If we include the cases of type $A$ or $D_2$ (for which the same results hold), then we obtain $L^2$-models for the minimal representations of the groups $$\begin{aligned}
& Sp(n,{\mathbb{R}}),\,Sp(n,{\mathbb{C}}),\,SO^*(4n),\,SO(p,q)\,(p+q\mbox{ even}),\,SO(n,{\mathbb{C}})\\
& E_{6(6)},\,E_6({\mathbb{C}}),\,E_{7(7)},\,E_{7(-25)},\,E_7({\mathbb{C}}).\end{aligned}$$ This is a complete list of all groups having a maximal parabolic subalgebra with abelian nilradical, which admit a minimal representation (see [@HKM14]).
The non-standard intertwining operator
--------------------------------------
For any unitarizable quotient $J(\nu)=I(\nu)/I_0(\nu)$ in a (degenerate) principal series representation $I(\nu)$ there exists an intertwining operator $T:I(\nu)\to I(-\nu)$ with kernel $I_0(\nu)$ and the property that the invariant inner product on $J(\nu)$ is given by $$J(\nu)\times J(\nu)\to{\mathbb{C}}, \quad (f,g)\mapsto\int_{V^-} Tf(x)\overline{g(x)}\,{\mathrm{d}}x.$$ In many cases these operators can be obtained from standard families of intertwining operators such as the Knapp–Stein intertwiners. However, as observed in [@MS12] in the case where $P$ and $\overline P$ are not conjugate such families do not exist. Still, unitarizable quotients can occur in this setting, and hence the corresponding intertwiners cannot be obtained from standard families by regularization. In [@MS12] we constructed non-standard intertwining operators between $I(\nu_k)$ and $I(-\nu_k)$ for $0\leq k\leq r-1$ in the case where $P$ and $\overline P$ are not conjugate, using algebraic methods. Here we provide a geometric version of these non-standard intertwiners in the Fourier transformed realization $\tilde I(\nu)$.
\[thm:Intertwiner\] For any $0\leq k\leq r-1$ the map $$T_k:\tilde I(\nu_k) \to \tilde I(-\nu_k), \quad f\mapsto f|_{{\mathcal{O}}_k}\,{\mathrm{d}}\mu_k$$ is an intertwining operator $\tilde\pi_{\nu_k}\to\tilde\pi_{-\nu_k}$. Its kernel is the subrepresentation $$\tilde I_0(\nu_k) = {\left\{f\in\tilde I(\nu_k)\,\middle|\,f|_{{\mathcal{O}}_k}=0\right\}} \subseteq \tilde I(\nu_k)$$ and its image is the subrepresentation $\tilde I_0(-\nu_k)\subseteq\tilde I(-\nu_k)$.
We first show that the spherical vector $\psi_{\nu_k}\in\tilde I(\nu_k)$ is mapped to the spherical vector $\psi_{-\nu_k}\in\tilde I(-\nu_k)$. Recall that $\psi_{\nu_k}=\Psi_{\nu_k}$ is given by $$\Psi_{\nu_k}(mb_t) = K_{\frac{p-e+1}{2}-\nu}(t_1,\ldots,t_r), \qquad m\in M\cap K,\,t\in C_r^+.$$ Due to a theorem of Clerc [@Cle88 Theorem 4.1] (see also [@Moe13 Proposition 3.10]) the restriction of $K_\mu(t_1,\ldots,t_r)$ to $t_{k+1}=\ldots=t_r$ is defined and gives the K-Bessel function of $k$ variables with the same parameter: $$K_\mu(t_1,\ldots,t_k,0,\ldots,0) = {\textup{const}}\times K_\mu(t_1,\ldots,t_k)$$ whenever $\operatorname{Re}\mu<1+k\frac{d}{2}$. For $\nu=\nu_k$ we have $\mu=\frac{p-e+1}{2}-\nu_k=\frac{kd-e+1}{2}$ which is less than $1+k\frac{d}{2}$ since $e\geq0$. Hence $$T_k\psi_{\nu_k} = {\textup{const}}\times\psi_{-\nu_k}.$$ Now, $\tilde I(\nu_k)_{{K-\textup{finite}}}$ is generated by $\psi_{\nu_k}$. Further, the map $T_k$ is ${\mathfrak{g}}$-intertwining by Proposition \[prop:FTpicturealgebraaction\] and the fact that the Bessel operator is symmetric with respect to the measure ${\mathrm{d}}\mu_k$ (see Theorem \[thm:BesselSymmetricOnOrbits\]). Hence, $T_k$ is defined on the whole $\tilde I(\nu_k)_{{K-\textup{finite}}}$ and its image is $\tilde I_0(-\nu_k)_{{K-\textup{finite}}}={\mathrm{d}}\tilde\pi_{-\nu_k}({\mathcal{U}}({\mathfrak{g}}))\psi_{-\nu_k}$. That its kernel is equal to $\tilde I_0(\nu_k)_{{K-\textup{finite}}}$ is clear by the definition of $T_k$. Finally, $T_k$ extends to $\tilde I(\nu_k)$ by the Casselman–Wallach Globalization Theorem.
In the case where $P$ and $\overline P$ are conjugate, the intertwiner $T_k$ simply is a regularization of the standard family of Knapp–Stein intertwining operators as observed in [@BSZ06].
K-Bessel functions on symmetric cones {#app:KBesselFunctions}
=====================================
We recall the definition and the differential equation of the K-Bessel function on a symmetric cone and prove some integrability results. For more details we refer to [@Dib90], [@FK94 XVI.§3] or [@Moe13].
Let $A$ be a Euclidean Jordan algebra of dimension $n$ and rank $r$ (see [@FK94] for details). Denote by $\operatorname{tr}$ and $\Delta$ its Jordan algebra trace and determinant, and let $(x|y)=\operatorname{tr}(xy)$ be the trace form. Let $\Omega$ by the symmetric cone in $A$ which is the interior of the cone of squares, or equivalently the connected component of the invertible elements containing the identity element $e$.
For $\lambda\in{\mathbb{C}}$ the K-Bessel function ${\mathcal{K}}_\lambda$ on $\Omega$ is defined by $${\mathcal{K}}_\lambda(x) = \int_\Omega e^{-\operatorname{tr}(u^{-1})-(x|u)}\Delta(u)^{\lambda-\frac{2n}{r}}\,{\mathrm{d}}u = \int_\Omega e^{-\operatorname{tr}(v)-(x|v^{-1})}\Delta(v)^{-\lambda}\,{\mathrm{d}}v, \qquad x\in\Omega.\label{eq:IntFormulaKBessel1}$$ Evaluated at a square the K-Bessel function can be expressed as $${\mathcal{K}}_\lambda(x^2) = \Delta(x)^{\frac{n}{r}-\lambda}\int_\Omega e^{-(x|u+u^{-1})}\Delta(u)^{\lambda-\frac{2n}{r}}\,{\mathrm{d}}u = \Delta(x)^{\frac{n}{r}-\lambda}\int_\Omega e^{-(x|v+v^{-1})}\Delta(v)^{-\lambda}\,{\mathrm{d}}v.\label{eq:IntFormulaKBessel2}$$ All integrals converge for any $\lambda\in{\mathbb{C}}$ and $x\in\Omega$ and we have ${\mathcal{K}}_\lambda(x)>0$. If ${\mathcal{B}}_\lambda$ is the Bessel operator on $A$ then ${\mathcal{K}}_\lambda$ solves the differential equation (see e.g [@Dib90 Proposition 7.2]) $${\mathcal{B}}_\lambda{\mathcal{K}}_\lambda(x) = {\mathcal{K}}_\lambda(x)\cdot e,$$ where $e$ is the unit element of $A$. Choose a Jordan frame $e_1,\ldots,e_r\in A$ such that $e=e_1+\cdots+e_r$ and let $$A = \bigoplus_{1\leq i\leq j\leq r} A_{ij}$$ be the corresponding Peirce decomposition. Write $K_\lambda$ for the radial part of ${\mathcal{K}}_\lambda$, i.e., $$K_\lambda(t_1,\ldots,t_r)={\mathcal{K}}_\lambda(t_1e_1+\cdots+t_re_r), \qquad t_1,\ldots,t_r>0.$$ According to Proposition \[prop:radialBessel\] (cf. [@FK94 XV.2.8]), $K_\lambda$ satisfies the differential equation $$\label{eq:DiffEqKBesselRadial}
{\mathcal{B}}_\lambda^iK_\lambda(t) = K_\lambda(t) \qquad \forall\,1\leq i\leq r,$$ where $${\mathcal{B}}_\lambda^i=t_i\frac{\partial^2}{\partial t_i^2}
+\left(\lambda-(r-1)\frac{d}{2}\right)\,\frac{\partial}{\partial t_i}\\
+\frac{d}{2}\sum_{j\neq i}\frac{1}{t_i-t_j}
\left(t_i\frac{\partial}{\partial t_i}-t_j\frac{\partial}{\partial t_j}\right)$$ and $d=\dim A_{ij}$ for any $i<j$.
\[lem:L1L2KBessel\] We have $$\int_\Omega |{\mathcal{K}}_\lambda(x)|^2 \Delta(x)^\mu\,{\mathrm{d}}x < \infty \qquad \Leftrightarrow \qquad \begin{cases}\mu>-1\qquad\mbox{and}\\\mu-2\lambda>-3-(r-1)d,\end{cases}$$ and $$\int_\Omega {\mathcal{K}}_\lambda(x) \Delta(x)^\mu\,{\mathrm{d}}x < \infty \qquad \Leftrightarrow \qquad \begin{cases}\mu>-1\qquad\mbox{and}\\\mu-\lambda>-2-(r-1)\frac{d}{2},\end{cases}$$ where $dx$ is a Lebesgue measure on the open set $\Omega\subseteq A$. The latter integral evaluates to $$\int_\Omega{\mathcal{K}}_\lambda(x)\Delta(x)^\mu\,{\mathrm{d}}x = \Gamma_{r,d}(\mu+\tfrac{n}{r})\Gamma_{r,d}(\mu-\lambda+\tfrac{2n}{r}),$$ where $\Gamma_{r,d}$ denotes the Gamma function of the symmetric cone $\Omega$.
We make use of the following integral formula (see [@FK94 Proposition VII.1.2]): $$\int_\Omega e^{-(x|y)}\Delta(x)^{\lambda-\frac{n}{r}}\,{\mathrm{d}}x = \Gamma_{r,d}(\lambda)\Delta(y)^{-\lambda},$$ where the integral converges (absolutely) if and only if $\lambda>(r-1)\frac{d}{2}$. We further need the identites (see [@FK94 Proposition III.4.2 and Lemma X.4.4]) $$\Delta(u^{-1}+v^{-1}) = \Delta(u+v)\Delta(u)^{-1}\Delta(v)^{-1}, \qquad \Delta(P(x)y)=\Delta(x)^2\Delta(y).$$ Then we have $$\begin{aligned}
& \int_\Omega |{\mathcal{K}}_\lambda(x)|^2 \Delta(x)^\mu\,{\mathrm{d}}x\\
={}& \int_\Omega\int_\Omega\int_\Omega e^{-\operatorname{tr}(u)-\operatorname{tr}(v)-(x|u^{-1})-(x|v^{-1})}\Delta(u)^{-\lambda}\Delta(v)^{-\lambda}\Delta(x)^\mu\,{\mathrm{d}}u\,{\mathrm{d}}v\,{\mathrm{d}}x\\
={}& \Gamma_{r,d}(\mu+\tfrac{n}{r})\int_\Omega\int_\Omega e^{-\operatorname{tr}(u)-\operatorname{tr}(v)}\Delta(u^{-1}+v^{-1})^{-\mu-\frac{n}{r}}\Delta(u)^{-\lambda}\Delta(v)^{-\lambda}\,{\mathrm{d}}u\,{\mathrm{d}}v\\
={}& \Gamma_{r,d}(\mu+\tfrac{n}{r})\int_\Omega\int_\Omega e^{-\operatorname{tr}(u)-\operatorname{tr}(v)}\Delta(u+v)^{-\mu-\frac{n}{r}}\Delta(u)^{\mu-\lambda+\frac{n}{r}}\Delta(v)^{\mu-\lambda+\frac{n}{r}}\,{\mathrm{d}}u\,{\mathrm{d}}v\\
={}& \Gamma_{r,d}(\mu+\tfrac{n}{r})\int_\Omega\int_\Omega e^{-\operatorname{tr}(u)-\operatorname{tr}(v)}\Delta(e+P(u^{\frac{1}{2}})^{-1}v)^{-\mu-\frac{n}{r}}\Delta(u)^{-\lambda}\Delta(v)^{\mu-\lambda+\frac{n}{r}}\,{\mathrm{d}}u\,{\mathrm{d}}v\\
={}& \Gamma_{r,d}(\mu+\tfrac{n}{r})\int_\Omega\int_\Omega e^{-\operatorname{tr}(u)-(u|v')}\Delta(e+v')^{-\mu-\frac{n}{r}}\Delta(u)^{\mu-2\lambda+\frac{2n}{r}}\Delta(v')^{\mu-\lambda+\frac{n}{r}}\,{\mathrm{d}}u\,{\mathrm{d}}v'\\
={}& \Gamma_{r,d}(\mu+\tfrac{n}{r})\Gamma_{r,d}(\mu-2\lambda+\tfrac{3n}{r})\int_\Omega \Delta(e+v')^{2\lambda-2\mu-\frac{4n}{r}}\Delta(v')^{\mu-\lambda+\frac{n}{r}}\,{\mathrm{d}}v',\end{aligned}$$ where we have substituted $v=P(u^{\frac{1}{2}})v'$ with ${\mathrm{d}}v=\Delta(u)^{\frac{n}{r}}\,{\mathrm{d}}v'$. Note that we have used $$\mu+\tfrac{n}{r} > (r-1)\tfrac{d}{2} \qquad \mbox{and} \qquad \mu-2\lambda+\tfrac{3n}{r} > (r-1)\tfrac{d}{2}.\label{eq:L2Condition1}$$ The final integral is finite if and only if $$\mu-\lambda+\tfrac{2n}{r}>(r-1)\tfrac{d}{2} \qquad \mbox{and} \qquad \lambda-\mu-\tfrac{3n}{r}<-1-(r-1)d.\label{eq:L2Condition2}$$ Note that is equivalent to $\mu>-1$ and $\mu-2\lambda>-3-(r-1)d$. Further, is implied by which proves the first claim. For the second claim we compute similarly $$\begin{aligned}
\int_\Omega {\mathcal{K}}_\lambda(x) \Delta(x)^\mu\,{\mathrm{d}}x &= \int_\Omega\int_\Omega e^{-\operatorname{tr}(u)-(x|u^{-1})}\Delta(u)^{-\lambda}\Delta(x)^\mu\,{\mathrm{d}}u\,{\mathrm{d}}x\\
&= \Gamma_{r,d}(\mu+\tfrac{n}{r})\int_\Omega e^{-\operatorname{tr}(u)}\Delta(u)^{\mu-\lambda+\frac{n}{r}}\,{\mathrm{d}}u\end{aligned}$$ which is finite if and only if $$\mu+\tfrac{n}{r}>(r-1)\tfrac{d}{2} \qquad \mbox{and} \qquad \mu-\lambda+\tfrac{2n}{r}>(r-1)\tfrac{d}{2}.$$ These conditions are equivalent to $\mu>-1$ and $\mu-\lambda>-2-(r-1)\frac{d}{2}$ which shows the second claim. The last claim follows by using the integral formula [@FK94 Proposition VII.1.2] once more.
[10]{}
L. Barchini, M. Sepanski, and R. Zierau, *Positivity of zeta distributions and small unitary representations*, The ubiquitous heat kernel, Contemp. Math., vol. 398, Amer. Math. Soc., Providence, RI, 2006, pp. 1–46.
W. Bertram, *The geometry of [J]{}ordan and [L]{}ie structures*, Lecture Notes in Mathematics, vol. 1754, Springer-Verlag, Berlin, 2000.
H. Braun and M. Koecher, *Jordan-[A]{}lgebren*, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 128, Springer-Verlag, Berlin, 1966.
J.-L. Clerc, *Fonctions [$K$]{} de [B]{}essel pour les algèbres de [J]{}ordan*, Harmonic analysis ([L]{}uxembourg, 1987), Lecture Notes in Math., vol. 1359, Springer, Berlin, 1988, pp. 122–134.
H. Dib, *Fonctions de [B]{}essel sur une algèbre de [J]{}ordan*, J. Math. Pures Appl. (9) **69** (1990), no. 4, 403–448.
A. Dvorsky and S. Sahi, *Explicit [H]{}ilbert spaces for certain unipotent representations. [II]{}*, Invent. Math. **138** (1999), no. 1, 203–224.
[to3em]{}, *Explicit [H]{}ilbert spaces for certain unipotent representations. [III]{}*, J. Funct. Anal. **201** (2003), no. 2, 430–456.
J. Faraut, S. Kaneyuki, A. Kor[á]{}nyi, Q. Lu, and G. Roos, *Analysis and geometry on complex homogeneous domains*, Progress in Mathematics, vol. 185, Birkhäuser Boston, Inc., Boston, MA, 2000.
J. Faraut and A. Kor[á]{}nyi, *Analysis on symmetric cones*, The Clarendon Press, Oxford University Press, New York, 1994.
G. Heckman and H. Schlichtkrull, *Harmonic analysis and special functions on symmetric spaces*, Perspectives in Mathematics, vol. 16, Academic Press Inc., San Diego, CA, 1994.
J. Hilgert, T. Kobayashi, and J. M[ö]{}llers, *Minimal representations via [B]{}essel operators*, J. Math. Soc. Japan **66** (2014), no. 2, 349–414.
K. D. Johnson, *Degenerate principal series and compact groups*, Math. Ann. **287** (1990), no. 4, 703–718.
[to3em]{}, *Degenerate principal series on tube type domains*, Hypergeometric functions on domains of positivity, [J]{}ack polynomials, and applications ([T]{}ampa, [FL]{}, 1991), Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 175–187.
S. Kaneyuki, *The [S]{}ylvester’s law of inertia in simple graded [L]{}ie algebras*, J. Math. Soc. Japan **50** (1998), no. 3, 593–614.
T. Kobayashi and G. Mano, *The [S]{}chrödinger model for the minimal representation of the indefinite orthogonal group [${\rm O}(p,q)$]{}*, Mem. Amer. Math. Soc. **213** (2011), no. 1000.
T. Kobayashi and B. [Ø]{}rsted, *Analysis on the minimal representation of [${\rm O}(p,q)$]{}. [III]{}. [U]{}ltrahyperbolic equations on [$\mathbb{R}^{p-1,q-1}$]{}*, Adv. Math. **180** (2003), no. 2, 551–595.
O. Loos, *[Jordan Pairs]{}*, Lecture notes in Mathematics, vol. 460, Springer-Verlag, Berlin-New York, 1975.
[to3em]{}, *Bounded symmetric domains and [J]{}ordan pairs*, Lecture notes, University of California, Irvine, 1977.
G. Mano, *[S]{}emisimple [J]{}ordan algebras*, unpublished notes, 2008.
J. M[ö]{}llers, *A geometric quantization of the [K]{}ostant–[S]{}ekiguchi correspondence for scalar type unitary highest weight representations*, Doc. Math. **18** (2013), 785–855.
J. M[ö]{}llers and B. Schwarz, *Structure of the degenerate principal series on symmetric [$R$]{}-spaces and small representations*, J. Funct. Anal. **266** (2014), no. 6, 3508–3542.
B. [Ø]{}rsted and G. Zhang, *Generalized principal series representations and tube domains*, Duke Math. J. **78** (1995), no. 2, 335–357.
G. Roos, *Exceptional symmetric domains*, Symmetries in complex analysis, Contemp. Math., vol. 468, Amer. Math. Soc., Providence, RI, 2008, pp. 157–189.
S. Sahi, *Explicit [H]{}ilbert spaces for certain unipotent representations*, Invent. Math. **110** (1992), no. 2, 409–418.
[to3em]{}, *Jordan algebras and degenerate principal series*, J. Reine Angew. Math. **462** (1995), 1–18.
I. Satake, *Algebraic structures of symmetric domains*, Kanô Memorial Lectures, vol. 4, Iwanami Shoten, Tokyo, 1980.
M. Vergne and H. Rossi, *Analytic continuation of the holomorphic discrete series of a semi-simple [L]{}ie group*, Acta Math. **136** (1976), no. 1-2, 1–59.
G. Zhang, *[Jordan algebras and generalized principal series representations]{}*, Math. Ann. **786** (1995), 773–786.
|
---
abstract: 'A $k$-ary charm bracelet is an equivalence class of length $n$ strings with the action on the indices by the additive group of the ring of integers modulo $n$ extended by the group of units. By applying an $O(n^3)$ amortized time algorithm to generate charm bracelet representatives with a specified content, we construct 29 new periodic Golay pairs of length $68$.'
author:
- 'Dragomir [Ž ]{}okovi[ć]{}[^1] Ilias Kotsireas[^2] Daniel Recoskie[^3] Joe Sawada[^4]'
bibliography:
- 'refs.bib'
title: Charm bracelets and their application to the construction of periodic Golay pairs
---
Introduction
============
One of the most natural groups acting on $k$-ary strings $a_0 a_1 \cdots a_{n-1}$ of length $n$ is the group of rotations. A generator of this group acts on the indices by sending $i\to i+1 \pmod{n}$, and so sends the string $a_0 a_1 \cdots a_{n-1} \to a_1 \cdots a_{n-1} a_0$. Applying this action partitions the set of $k$-ary strings into equivalence classes that are called *necklaces*. When the action of reversal is composed with rotations, the resulting dihedral groups partition $k$-ary strings into equivalence classes called *bracelets*. Generally, we will refer only to the lexicographically smallest element in each respective equivalence class as a necklace or a bracelet. For example, consider the bracelet equivalence class for the string $12003$:
-- ------- ------- -----------------------------------------------
12003 30021
20031 $\leftarrow$ *[bracelet (necklace)]{}\
*[necklace]{} $\rightarrow$ & & 02130\
& 03120 & 21300\
& 31200 & 13002\
**
-- ------- ------- -----------------------------------------------
Observe that this class contains two necklaces 00312 and 00213, the lexicographically smallest being the bracelet representative.
In this paper we generalize the notion of bracelets by considering the action of the group of affine transformations $j\to a+dj \pmod{n}$ on the indices. Here we consider the indices as elements of the ring of integers modulo $n$ denoted by $\bZ_n:=\bZ/n\bZ$. The coefficients $a$ and $d$ also belong to $\bZ_n$ and $d$ is relatively prime to $n$. We call the resulting equivalence classes *charm bracelets*. Note that if $d \in \{1\}$ we get necklaces, and if $d \in \{1,n-1\}$ we get bracelets. As an example, consider the charm bracelet equivalence class for the string $\alpha = a_0a_1a_2a_3a_4 = 12003$:
------------------ ------- ------- ------- ------- -----------------------------------
12003 10320 10230 13002
20031 03201 02301 30021
32010 23010 $\leftarrow$ *[charm bracelet]{}\
& 03120 &20103 & 30102 & 02130\
& 31200 & & & 21300\
*
------------------ ------- ------- ------- ------- -----------------------------------
Observe that the first strings in each column are the result of the application of the multiplicative group mapping corresponding to $d=1,2,3$ and $4$ respectively. The subsequent strings in each column correspond to a rotation of the previous string. Thus, each column will have one necklace representative: 00312, 01032, 01023, and 00213 respectively. The lexicographically smallest necklace 00213 is a charm bracelet. Note that if we take $a=d=n-1$ then the above affine transformation is just the reversal. In general, the maximum number of necklaces in each charm bracelet equivalence class is given by Euler’s totient function $\phi(n)$, which denotes the number of positive integers less than $n$ that are relatively prime to $n$. Also, observe that each charm bracelet class will have at most $\phi(n)/2$ bracelets. In particular, observe that the first and last columns of our charm bracelet example correspond to the strings in our previous bracelet example for the string 12003.
Both necklaces and bracelets have been well studied. Enumeration formulae are well known and efficient algorithms to list necklaces have been given by Fredricksen, Kessler and Maiorana [@fred1; @fred2] and Cattell *et al*. [@cattell]. An efficient algorithm to list bracelets is given in [@brac_cat]. Very little is known about charm bracelets except for an enumeration formula presented by Titsworth [@titsworth]. Its binary enumeration sequence was one of the original 2372 sequences presented in 1973 by Sloane in *A Handbook of Integer Sequences* [@handbook]. In Section \[sec:charm\], we discuss charm bracelets in more detail, presenting a known enumeration formula along with an algorithm to generate them.
An application
--------------
This study of charm bracelets was motivated by the difficult task of deciding the existence of periodic Golay pairs of length 68. Using our charm bracelet algorithm as step in a searching process we discover 29 new (pairwise nonequivalent) periodic Golay pairs of length 68. This process is outlined in detail in Section \[sec:app\].
Since our discovery, two separate techniques were discovered to multiply a Golay pair of length $g$ and a periodic Golay pair of length $v$, and obtain as a result a periodic Golay pair of length $gv$. We refer loosely to this operation as “multiplication by $g$”. For more details on these multiplications see the recent preprint [@PerGol72]. A special case to multiply by $g=2$ was discovered long ago [@ba90 Theorems 2 and 3][^5]. Applying the two multiplications by two, the periodic Golay pairs of length 34 presented in [@FSQ99 Theorem 3.1]) allows us to construct two nonequivalent periodic Golay pair of length 68; however, we have verified that these pairs are not equivalent to any of the $29$ new pairs discovered in this paper (listed in the appendix).
Finally, we mention that eight non-equivalent periodic Golay pairs of length 72 have been constructed recently [@PerGol72]. Consequently, the smallest length for which the existence of periodic Golay pairs is undecided is now 90.
Charm Bracelets {#sec:charm}
===============
Enumeration
-----------
An enumeration formula for the number of $k$-ary charm bracelets of length $n$, denoted $CB(n,k)$, was derived in [@titsworth]: $$CB(n,k) = \frac{1}{n\cdot\phi(n)} \sum_{t=0}^{n-1} \mathop{\sum_{j=1}^{n-1}} \ [\![ \ gcd(n,j)=1 \ ]\!] \ k^{c(j,t)} \ \mbox{ where }$$
$$c(j, t) = \sum_{u=0}^{n-1} \frac{1}{M\left(j, \frac{n}{gcd(n, u(j-1)+t)}\right)}$$ and where $M(j, L)$ is the smallest positive integer $m$ such that $1+j+\cdots+j^{(m-1)} = 0$ (mod $L$). The Iverson bracket \[\] evaluates to 1 if $condition$ is true, and 0 otherwise. The enumeration sequence of $CB(n,2)$ corresponds to sequence A002729 in Sloane’s [*The On-Line Encyclopedia of Integer Sequences*]{} [@sloane]. Additionally, the sequences for $CB(n,k)$ for $k=3,4,5$, and 6 correspond to sequences A056411, A056412, A056413, A056414.
Generation algorithm
--------------------
Before outlining an algorithm to generate charm bracelets, we first introduce some notation. Let $\Phi(n)$ denote the set of positive integers less than $n$ that are relatively prime to $n$. Let ${\tau}(d,\alpha)$ denote the mapping of $j$ to $dj \bmod n$ acting on the indices of the string $\alpha = a_0a_1\cdots a_{n-1}$. Let $neck(\alpha)$ denote the necklace representative of the string $\alpha$. Let $\mathbf{N}_k(n)$ denote the set of all $k$-ary necklaces of length $n$ and let $\mathbf{CB}_k(n)$ denote the set of all $k$-ary charm bracelets of length $n$.
When developing algorithms to exhaustively list combinatorial objects, one of the primary goals is to achieve a CAT algorithm: one that generates each object in constant amortized time. For charm bracelets this does not appear to be a trivial task. In this section we outline an algorithm that runs in $O(n^3)$ time per charm bracelet generated.
Perhaps the most straightforward way to exhaustively list $\mathbf{CB}_k(n)$ is by the following approach:
1. Generate all the $k$-ary necklaces $\mathbf{N}_k(n)$.
2. For each necklace $\alpha \in \mathbf{N}_k(n)$ compute $S(\alpha) = \{ {\tau}(\alpha, d) \ | \ d \in \Phi(n) \}$.
3. Compute the necklace of each string in $S(\alpha)$ to get $T(\alpha) = \{ neck(s) \ | \ s \in S(\alpha) \}$.
4. Test if $\alpha$ is lexicographically less than or equal to every string in $T(\alpha)$. If it is, a charm bracelet is found and process $\alpha$.
As mentioned earlier, necklaces can be generated in constant amortized time. Step 2 requires $O(n^2)$ time to compute the set of $\phi(n)$ strings. Since the necklace of each string can be computed in $O(n)$ time (see p.222 from [@combgen]), the set $T$ can also be computed in $O(n^2)$ time. The third step trivially takes $O(n^2)$ time. Thus the resulting algorithm runs in $O(n^2)$ time *per necklace*. Since there are $\phi(n) = O(n)$ necklaces in each charm bracelet class, each charm bracelet gets generated in $O(n^3)$ time.
More detailed pseudocode is given in Algoirthm \[alg:min\]. The function [GenCharm]{} generates the necklaces using the algorithm from [@cattell; @combgen]. For each necklace $\alpha$ generated, the function [IsCharm]{}$(\alpha)$ returns whether or not $\alpha$ is a charm bracelet. It in turn, applies the function [Necklace]{}$(\beta)$ that returns the necklace of the string $\beta$ by applying a simple modification of the technique given in [@combgen]. The initial call is [GenCharm]{}(1,1) initializing $a_0 = 0$. A complete C implementation is given in the Appendix.
$b_1b_2\cdots b_{2n} \gets \beta \beta$ $t \gets j \gets p \gets 1$
$t \gets t+ p \lfloor \frac{j-t}{p} \rfloor$ $j \gets t+1$ $p \gets 1$ $p \gets j-t+1$ $j \gets j + 1$ $b_{t} b_{t+1} \cdots b_{t+n-1}$
$a_t \gets i$
The algorithm [GenCharm]{} generates all length $n$ charm bracelets in $O(n^3)$-amortized time.
As mentioned earlier, the ultimate goal is an algorithm that runs in $O(1)$-amortized time. However, this appears a very difficult task for charm bracelets. Any improvement on the $O(n^3)$ algorithm presented here would be a very nice result. The algorithm can be slightly improved by generating bracelets [@brac_cat] instead of necklaces. For the application discussed in the next section, only charm bracelets with a specified content are required. They can also be generated in $O(n^3)$-amortized time by replacing the function [GenCharm]{} with the CAT algorithm for fixed content necklaces [@neck_fixed_c] or fixed content bracelets [@brac_fixed].
Application: Periodic Golay pairs {#sec:app}
=================================
Periodic Golay pairs (also known as “periodic complementary sequences”) will be defined formally in Section \[sec:pg\]. Early research by Yang [@ya76] used an exhaustive computer search to show that there are no periodic Golay pairs of length 18. Subsequently, this case was ruled out by the non-existence result of Arasu and Xiang [@ax92]. For an up-to-date listing of lengths of known periodic Golay pairs which are not Golay pairs see [@PerGol; @PerGol72]. As mentioned earlier, the smallest length for which the existence of periodic Golay pairs is undecided is now 90. The periodic Golay pairs can be used to construct Hadamard matrices (see [@SY:1992 p. 468]).
By applying the (fixed-content) charm bracelet algorithm described in the previous section along with a compression of complementary sequences, we construct 29 periodic Golay pairs of length 68. One of them will be discussed in more detail in Section \[PerGolExample\]. The full listing of the 29 solutions is given in Appendix A.
For the remainder of this section, we use $v$ for the string/sequence lengths rather than the $n$ we used in the previous section, as $v$ is the standard in design theory.
Periodic Golay pairs vs. Golay pairs {#sec:pg}
------------------------------------
The symbols $\bZ,\bR,\bC$ will denote the set of integers, real numbers and complex numbers, respectively. Binary sequences will have terms $\pm1$. A pair of binary sequences of length $v$, say, $$\label{seqAB}
A=[a_0,a_1,\ldots,a_{v-1}],\quad
B=[b_0,b_1,\ldots,b_{v-1}]$$ is a [*Golay pair*]{} if for each $k=1,2,\ldots,v-1$: $$\sum_{i=0}^{v-k-1} (a_ia_{i+k}+b_ib_{i+k})=0.
$$ It is well known that Golay pairs exist for all lengths $v=2^a 10^b 26^c$ where $a,b,c$ are nonnegative integers. For convenience, we shall refer to integers $v$ having this form as [*Golay numbers*]{}. No Golay pairs of other lengths are presently known [@Borwein:Ferguson:2003].
We are interested in an analogue of Golay pairs to which we refer as periodic Golay pairs. They can be defined over any finite abelian group, but we will consider only the finite cyclic groups. To be specific, we shall use only the cyclic group $\bZ_v=\{0,1,\ldots,v-1\}$ of integers modulo $v$. The group operation is addition modulo $v$. From now on we shall consider the indices of sequences as members of $\bZ_v$. A [*periodic Golay pair*]{} is a pair of binary sequences (\[seqAB\]) such that for each $k=1,2,\ldots,v-1$: $$\label{PAF=0}
\sum_{i=0}^{v-1} (a_ia_{i+k}+b_ib_{i+k})=0.
$$
Since for any sequence $x_0,x_1,\ldots,x_{v-1}$ we have $$\sum_{i=0}^{v-1} x_ix_{i+k} =
\sum_{i=0}^{v-k-1} x_ix_{i+k}+\sum_{i=0}^{k-1} x_ix_{i+v-k},$$ any Golay pair is also a periodic Golay pair. Therefore periodic Golay pairs of length $v$ exist whenever $v$ is a Golay number. However, it is known that they also exist for some other lengths as well. The first such example was of length 34 (see [@Dj-1998]). At the present time, only finitely many periodic Golay pairs are known whose length $v$ is not a Golay number. The smallest length $v$ for which the existence of periodic Golay pairs of length $v$ is undecided is $v=68$. In this note we show that such pairs exist.
The role of charm bracelets in the search for periodic Golay pairs
------------------------------------------------------------------
Our objective in this subsection is to explain the role of bracelets in the search for Golay pairs. In order to do that, we first briefly review some background material.
For an integer sequence $A=[a_0,a_1,\ldots,a_{v-1}]$ of length $v$, the function $\bZ_v\to\bZ$ which sends $s\to\sum_{i=0}^{v-1} a_ia_{i+s}$ is known as the [*periodic autocorrelation function*]{} (PAF) of $A$. If $(A,B)$ is a periodic Golay pair of length $v$, then the equation (\[PAF=0\]) can be written as $$\label{PAF=nula}
(\PAF_A+\PAF_B)(s)=0, \quad s=1,2,\ldots,v-1.$$
The [*discrete Fourier transform*]{} (DFT) of the above sequence $A$ is the function $\bZ_v\to\bC$ which sends $s\to\sum_{k=0}^{v-1} a_k \omega^{ks}$, where $\omega=e^{2\pi i/v}$. The [*power spectral density*]{} (PSD) of the sequence $A$ is the function $\bZ_v\to\bR$ defined by $\PSD_A(s)=|\DFT_A(s)|^2$. By using [@Compress Theorem 2], we deduce that (\[PAF=nula\]) implies $$\label{PSD=nula}
(\PSD_A+\PSD_B)(s)=2v, \quad s=0,1,2,\ldots,v-1.$$ Occasionally we shall write $\PSD(A,s)$ instead of $\PSD_A(s)$, and similarly for the $\PAF$ function.
Our search for a periodic Golay pair $(A,B)$ is based on the compression method which is described in detail in the very recent paper of two of the authors [@Compress]. We refer the reader to this paper also for some additional facts concerning $A$ and $B$ that we shall use below. In this computation we used the compression factor $m=2$, and so the compressed sequences have length $d=v/m=34$. If $a$ and $b$ are the sums of the terms of the sequence $A$ and $B$, respectively, it is known that $a^2+b^2=4v=136$, and so we may assume that $a=6$ and $b=10$.
In the first stage of the computation we search for suitable compressed sequences $(A^{(34)},B^{(34)})$. This is a pair of ternary sequences of length 34, $$A^{(34)}=[a_0+a_{34},a_1+a_{35},\ldots,a_{33}+a_{67}], \quad
B^{(34)}=[b_0+b_{34},b_1+b_{35},\ldots,b_{33}+b_{67}],$$ whose terms $a^{(34)}_i=a_i+a_{i+34}$ and $b^{(34)}_i=b_i+b_{i+34}$ belong to the set $\{0,2,-2\}$. Another known fact that we need is that the total number of 0 terms in these two compressed sequences is equal to 34. For instance, we can choose the case where each of $A$ and $B$ has seventeen 0 terms. As $a=6$ the sequence $A^{(34)}$ must have the content $(17,10,7)$, i.e., it has seventeen terms equal to 0, ten terms equal to 2, and seven terms equal to $-2$. Similarly, $B^{(34)}$ must have the content $(17,11,6)$.
We can perform on $(A,B)$, as well as on the compressed sequences, the following operations which preserve the set of periodic Golay pairs. First, we can permute cyclically $A$ or $B$ (independently of one another). Second, we can reverse independently the sequence $A$ or $B$. Third, we can apply the transformation $x_i\to x_{ki \pmod{v}}$ to both $A$ and $B$ simultaneously, where $k$ is a fixed integer relatively prime to $v$. By using these transformations on the compressed sequences, we deduce that we can restrict our search for the pairs $(A^{(34)},B^{(34)})$ to the case where $A^{(34)}$ is a charm bracelet and $B^{(34)}$ is an ordinary bracelet. (The alphabet used for these bracelets is $\{0,2,-2\}$.) Since the number of bracelets is much smaller than the number of all sequences with the same content, our search will be much faster. There is an additional speed-up when we restrict (as we may) $A^{(34)}$ to be a charm bracelet. The searches for the bracelets $A^{(34)}$ and $B^{(34)}$ are performed separately and the bracelets are written in two files. The search is aborted if the output file becomes too large. Some of the bracelets do not need to be recorded. This happens when they fail the so called PSD test. In our case this test is based on the fact that we must have $\PSD(A^{(34)},s)+\PSD(B^{(34)},s)=136$. Hence, the bracelets for which one of its PSD values is larger than 136 can be safely discarded. By implementing this test into the search for (charm) bracelets, the size of the output file is considerably reduced.
Periodic Golay pairs of length $68$ {#PerGolExample}
-----------------------------------
In this section we present one of the periodic Golay pairs that we found for length $v = 68$.
Consider the following two sequences of length $34$ each, with $\{-2,0,+2\}$ elements: $$\begin{aligned}
A^{(34)} & = & [0,0,0,2,0,0,-2,0,0,0,2,-2,0,0,-2,0,0,2,0,0,0,2,2,-2,0,0,-2,0,0,2,0,2,0,2] \\
B^{(34)} & = & [0,0,-2,2,0,2,0,-2,-2,0,2,2,0,2,-2,0,2,0,-2,2,0,2,2,0,2,0,2,2,0,-2,2,0,-2,-2]\end{aligned}$$ These two sequences satisfy the following properties:
1. $\PAF(A^{(34)},s) + \PAF(B^{(34)},s)=0, s=0,1,\ldots,33$;
2. $\PSD(A^{(34)},s) + \PSD(B^{(34)},s)=2\cdot68=136,~
s = 0,1, \ldots, 33$;
3. $\PSD(A^{(34)},17) = 100$ and $\PSD(B^{(34)},17) = 36$;
4. $\displaystyle\sum_{i=1}^{34} A^{(34)}_i = 6$ and $\displaystyle\sum_{i=1}^{34} B^{(34)}_i = 10$;
5. The total number of $0$ elements in $A^{(34)}$ and $B^{(34)}$ is equal to $34$;
6. The total number of $\pm 2$ elements in $A^{(34)}$ and $B^{(34)}$ is equal to $34$;
7. $A^{(34)}$ contains $21$ zeros and $B^{(34)}$ contains $13$ zeros.
We claim that the sequences $A^{(34)}$ and $B^{(34)}$ are in fact the 2-compressed sequences of two $\{-1,+1\}$ sequences of length $68$ each, that form a particular periodic Golay pair. Here is this particular periodic Golay pair of length $68$: $$\begin{aligned}
A &=&
\begin{array}{l}
--++-+-+-++--+--++---++------+-+++ \\
++-++---+-+-+--+-++++++-++-+++++-+ \\
\end{array} \\
B &=&
\begin{array}{l}
---+++---+++++--++-+-+++++++--+--- \\
++-+-++---++-+-++--++++-+-+++-++-- \\
\end{array}\end{aligned}$$ In the above periodic Golay pair we use the customary notation of representing $-1$ by $-$ and $+1$ by $+$, so as to achieve a constant length encoding of the sequences.
In order to find the periodic Golay pair given above, starting from the two sequences $A^{(34)}$ and $B^{(34)}$, we needed to write a program that looks at every individual element of $A^{(34)}$ and $B^{(34)}$ and generates all corresponding potential $\{-1,+1\}$ sequences of length $68$. If we encounter an element equal to $-2$ then this implies that we can set two elements of the length $68$ sequences equal to $-1$. If we encounter an element equal to $+2$ then this implies that we can set two elements of the length $68$ sequences equal to $+1$. If we encounter an element equal to $0$, then this implies that we have two possibilities for the two elements of the length $68$ sequences, either $(-1,+1)$ or $(+1,-1)$. Therefore $A^{(34)}$ generates $2^{21}$ sequences of length $68$ and $A^{(34)}$ generates $2^{13}$ sequences of length $68$. Subsequently we filter these two sets of sequences using the PSD test with PSD constant equal to $136$, since we know from compression theory [@Compress] that the PSD constants of the compressed sequences and the original sequences are equal. The PSD test typically eliminates anywhere between $95$% to $99$% of the sequences, so we are left with a very small number of sequences and then it is easy to locate a solution.
Note that there are several thousands (possibly several millions) of pairs of sequences that satisfy properties 1 to 6 (and a variant of property 7) of the pair $A^{(34)}, B^{(34)}$, but which do not correspond (via 2-compression) to periodic Golay pairs of order $68$. Both bracelets and charm bracelets are an essential tool for locating such pairs in a systematic manner. On the other hand, all periodic Golay pairs of order $68$ must necessarily be obtained from a pair of sequences of length $34$ that satisfies properties 1 up to 6 and an appropriate version of property 7. Note that property 7 reflects the distribution of the $34$ zeros in $A^{(34)}, B^{(34)}$ and is directly related with the corresponding bracelets content.
Connection with supplementary difference sets {#SDS}
---------------------------------------------
The periodic Golay pairs of fixed length $v$ are in one-to-one correspondence with a special class of combinatorial objects known as supplementary difference sets (SDS). For the definition of SDSs in general we refer the reader to [@Compress]. Here we shall just explain, in the context of this paper, the meaning of SDSs with parameters $(v;r,s;\lambda)=(68;31,29;26)$. Each of our SDSs consists of two base blocks, say $X$ and $Y$. They are subsets of the additive group $\bZ_v=\bZ_{68}=\{0,1,\ldots,67\}$ of sizes $|X|=r=31$ and $|Y|=s=29$. Each nonzero integer in $\bZ_v$ can be represented as a difference $x_1-x_2$ with $x_1,x_2\in X$ or as a difference $y_1-y_2$ with $y_1,y_2\in Y$ in total in exactly $\lambda=26$ ways. These particular SDSs are in one-to-one correspondence with periodic Golay pairs of length $v=68$. Let us make this correspondence explicit. Given an SDS $(X,Y)$ with the above parameters, we associate to it a periodic Golay pair $(A,B)$. The first binary sequence $A=[a_0,a_1,\ldots,a_{v-1}]$ is constructed from the set $X$ by setting $a_j=-1$ if $j\in X$ and $a_j=1$ otherwise. The sequence $B$ is constructed from the set $Y$ in the same way.
We point out that there exist SDSs with two base blocks which do not correspond to periodic Golay pairs. The SDS’s which do correspond to periodic Golay pairs are exactly those whose parameters satisfy the condition $v=2(r+s-\lambda)$.
Acknowledgements
================
The first two authors wish to acknowledge generous support by NSERC. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET) and Compute/Calcul Canada. We thank a referee for his suggestions.
For convenience we list only the 29 SDSs which correspond to the 29 periodic Golay sequences that we found. All 29 SDSs are given in the normal form defined in [@Dj-2011]. The solution discussed in Section \[PerGolExample\] is equivalent to the solution no. 15 in the list below.
$$\begin{aligned}
1) && [[0,1,2,3,4,5,6,7,9,10,12,14,15,20,21,25,28,31,33,34,40,
41,42,45,46,50,52,54,56,57,60], \\
&& [0,2,3,4,6,7,10,11,13,16,18,20,21,23,25,27,28,29,35,36,38,
40,44,45,50,51,58,59,62]],\\
2) && [[0,1,2,3,5,6,7,9,10,11,12,13,17,19,20,21,25,28,31,33,
34,35,40,45,48,49,50,55,58,61], \\
&& [0,1,2,4,7,8,9,12,13,16,18,19,20,22,27,30,35,37,39,41,42,
43,48,50,52,53,56,59,62]],\\
3) && [[0,1,2,3,4,5,6,9,11,15,16,20,22,23,27,29,30,32,36,38,
39,42,43,44,47,48,52,54,55,60,62], \\
&& [0,1,2,3,4,7,8,10,11,12,13,15,16,18,21,22,25,31,33,35,38,
42,44,50,52,55,56,57,60]],\\
4) && [[0,1,2,3,4,5,7,10,11,12,13,15,19,20,21,24,25,27,30,31,
32,37,39,42,46,48,52,55,56,57,59], \\
&& [0,1,2,3,5,6,8,9,10,14,17,20,23,24,27,29,31,33,34,35,39,40,
42,43,47,52,55,57,63]],\\
5) && [[0,1,2,3,4,5,7,11,13,16,19,21,22,27,28,29,30,31,33,35, 38,39,42,43,46,48,49,51,56,58,64], \\
&& [0,1,2,3,7,8,11,12,13,14,15,17,19,21,24,26,27,31,32,35,36,
39,45,46,48,49,53,55,64]],\\
6) && [[0,1,2,3,4,5,8,10,12,13,17,18,19,21,22,24,28,31,32,33, 35,37,38,41,43,45,49,56,57,58,63], \\
&& [0,1,2,3,5,6,7,8,11,12,14,16,20,23,24,26,27,33,34,36,41,42,
45,48,50,52,53,58,65]],\\
7) && [[0,1,2,3,4,5,8,10,14,15,17,23,24,25,26,27,28,29,32,33, 35,36,40,42,43,47,52,54,56,60,62], \\
&& [0,1,2,3,5,7,8,10,13,14,16,18,19,22,25,26,30,31,34,35,36,
39,41,46,49,50,52,56,63]],\\
8) && [[0,1,2,3,4,5,9,11,13,14,15,17,21,22,23,26,28,31,33,35,
38,39,41,42,46,47,50,53,54,56,63],\\
&& [0,1,2,3,4,6,7,9,10,11,12,14,15,20,24,26,27,30, 31,36,41,43,46,47,49,54,56,60,61]],\\
9) && [[0,1,2,3,4,6,7,12,14,15,16,20,22,23,25,26,27,30,32, 34,38,39,40,41,43,44,47,49,52,55,62],\\
&& [0,1,2,4,6,8,9,11,13,14,15,18,19,21,23,29,30,33, 35,36,39,40,44,45,52,53,55,56,63]],\\
10) && [[0,1,2,3,4,6,8,9,10,14,16,17,19,20,23,24,26,28,30,31,35,
36,37,41,45,46,48,49,54,57,58],\\
&& [0,1,2,6,9,10,12,13,15,16,17,19,20,21,25,30,32,35, 37,38,40,42,44,45,51,53,54,56,62]],\\
11) && [[0,1,2,3,4,7,8,10,11,12,13,14,17,21,22,26,27,29,30,33,
38,39,41,46,47,52,54,56,57,58,62],\\
&& [0,1,3,4,5,6,8,9,11,12,15,17,19,21,24,26,27,32,34, 37,38,39,41,46,48,49,53,55,59]],\\
12) && [[0,1,2,3,5,6,7,8,9,14,16,17,20,22,24,26,27,30,31, 33,38,40,43,46,47,48,50,55,58,59,63],\\
&& [0,1,2,4,5,6,7,10,11,15,16,18,19,21,22,27,28,30,34, 36,37,38,40,41,46,48,50,55,61]],\\
13) && [[0,1,2,3,5,6,8,10,12,14,15,17,18,19,23,25,26,29,32, 33,37,40,41,42,43,45,49,52,55,61,62],\\
&& [0,1,2,3,5,6,7,8,10,13,14,18,20,22,23,24,27,28,36, 38,39,41,47,48,49,52,54,58,63]],\\
14) && [[0,1,2,3,5,6,9,10,12,14,15,18,20,21,23,24,25,26,31, 32,33,38,42,43,47,48,50,54,57,58,61],\\
&& [0,1,3,5,6,7,8,9,11,13,15,16,22,25,27,29,30,31,33, 35,40,42,43,44,47,50,56,60,61]],\\
15) && [[0,1,2,3,5,7,8,10,11,17,18,20,21,25,26,27,30,33,34, 38,40,43,44,45,46,47,49,55,57,61,65], \\
&& [0,1,2,3,7,8,10,12,13,14,18,19,21,22,23,25,28,30, 34,36,37,39,40,42,44,47,51,55,56]],\\\end{aligned}$$
$$\begin{aligned}
16) && [[0,1,2,3,5,8,10,11,12,15,19,20,21,24,25,27,28,33,
35,36,39,40,41,43,45,46,47,50,51,57,60],\\
&& [0,1,2,4,5,6,7,10,12,15,18,19,21,22,26,27,28,30, 32,34,36,39,41,43,46,49,55,56,57]],\\
17) && [0,2,4,6,7,8,9,11,12,13,17,18,19,22,23,25,27,33,34, 35,36,37,39,42,43,46,49,51,56,57,59]],\\
&& [[0,1,2,3,7,8,9,11,12,14,15,18,21,25,27,28,29,30, 33,35,39,40,44,47,48,52,55,57,60],\\
18) && [[0,1,2,4,5,8,9,10,11,15,16,19,20,21,22,23,29,31,33, 35,37,40,41,46,48,51,53,55,56,60,63],\\
&& [0,1,2,3,5,6,7,10,11,13,14,16,22,24,26,27,29,30, 32,33,36,39,40,41,45,48,50,56,57]],\\
19) && [[0,1,2,4,5,8,9,11,12,15,17,20,21,22,23,27,28,29,30, 33,37,39,44,45,46,49,50,53,55,58,59],\\
&& [0,1,3,5,6,7,8,9,11,13,14,17,19,20,22,26,27,29,31, 37,40,41,42,44,45,52,54,56,61]],\\
20) && [[0,1,2,5,6,7,8,10,12,13,16,17,18,20,24,28,29,30, 31,33,34,38,40,42,47,48,49,53,59,61,62],\\
&& [0,1,2,4,6,7,9,10,11,13,14,16,19,20,23,26,28,34, 36,37,39,45,49,50,52,53,54,57,60]],\\
21) && [[0,1,2,5,6,7,9,10,11,15,17,19,20,24,26,27,31,32,33, 35,38,39,42,44,45,47,52,55,56,58,59],\\
&& [0,1,2,3,4,5,6,9,10,11,12,13,16,19,22,24,25,27,32, 35,39,40,44,46,50,52,54,56,61]],\\
22) && [0,1,2,3,4,7,8,10,12,13,14,16,19,20,22,23,25,27,32, 33,34,36,38,41,48,49,50,53,54,58,61]],\\
&& [[0,1,3,4,5,6,7,8,11,12,14,19,20,21,26,29,31,35,36, 39,44,45,47,48,50,52,58,62,64],\\
23) && [0,1,2,3,4,6,8,9,10,13,17,21,22,25,26,28,32,33,34, 35,38,40,42,45,48,49,50,51,56,59,62]],\\
&& [[0,1,3,4,5,6,10,13,14,15,18,19,20,22,24,26,29,30,33, 36,38,39,40,41,44,46,51,57,59],\\
24) && [[0,1,3,4,5,7,8,9,10,12,14,15,16,17,24,25,26,27,30, 33,35,36,39,40,43,50,51,55,56,57,63],\\
&& [0,1,2,5,7,8,9,11,13,17,18,20,21,23,25,31,32,35, 36,38,40,42,45,46,49,51,54,57,59]],\\
25) && [[0,1,3,4,5,7,8,11,13,14,15,19,20,21,24,27,29,30, 31,33,34,36,38,39,44,46,47,48,51,57,60],\\
&& [0,2,3,4,6,8,9,10,14,15,17,19,22,24,25,26,28, 35,36,40,43,45,48,49,53,54,55,56,60]],\\
26) && [[0,1,3,4,5,7,9,10,11,12,15,16,18,19,20,26,27,29,31, 33,34,36,38,39,42,43,51,52,56,57,59],\\
&& [0,2,3,4,5,6,10,11,12,13,16,17,22,23,25,27,31, 34,38,40,41,44,46,48,51,54,56,59,60]],\\
27) && [[0,1,3,4,6,7,9,10,11,12,14,18,19,20,22,26,30,32,33, 34,35,37,39,42,47,49,50,51,55,56,60],\\
&& [0,1,2,3,6,9,10,12,14,15,16,17,18,20,23,25,27,30, 34,37,38,43,44,49,50,54,56,59,60]],\\
28) && [[0,2,3,4,5,6,7,9,11,14,15,18,19,21,24,31,32,33,35, 39,40,41,45,47,50,51,52,57,59,60,64],\\
&& [0,1,2,3,4,5,6,9,10,11,15,16,19,22,24,25,26,27,30, 33,36,38,40,44,46,49,53,56,61]],\\
29) && [[0,2,3,4,6,8,9,10,11,12,15,17,18,21,25,28,29,30, 34,38,39,41,44,46,48,49,53,54,56,60,61],\\
&& [0,1,2,3,6,7,8,9,11,13,17,18,21,22,24,27,29,30, 31,33,35,38,41,42,43,52,55,56,58]].\end{aligned}$$
\#include <stdio.h> int a\[100\],b\[100\]; int N,K,total=0; //————————————————————- int Gcd(int x, int y)
int t;
while( y != 0 ) [ t = y; y = x ]{} return x;
//————————————————————- void Print()
int i;
total++; for (i=1; i<=N; i++) printf(“ printf(”");
//————————————————————- // Find the necklace of the string b\[1..n\] by concatenating two // copies of b\[1..n\] together. The necklace will be start at // index t. O(n) time. //————————————————————- int Necklace()
int j,t,p;
for (j=1; j<=N; j++) b\[N+j\] = b\[j\];
j=t=p=1; do [ t = t + p\*((j-t)/p); j = t + 1; p = 1; while (j <= 2\*N && b\[j-p\] <= b\[j\]) [ if (b\[j-p\] < b\[j\]) p = j-t+1; j++; ]{} ]{} while (p \* ((j-t)/p) < N);
return t;
//————————————————————- // For each i relatively prime to N, map index j to (ij mod N) // Then find the necklace of the resulting string, if that // necklace is less than the necklace a\[1..n\] - reject //————————————————————- int IsCharm()
int i,j,offset;
for(i=2; i<=N-1; i++)
if ( Gcd(i,N) == 1)
// Perform the mapping then determine the necklace for(j=0; j<N; j++) b\[(j\*i) offset = Necklace();
for (j=1; j<=N; j++)[ if (a\[j\] < b\[offset + j-1\]) break; else if (a\[j\] > b\[offset + j-1\]) return 0; ]{}
return 1;
//————————————————————– // Generate necklaces and then check if they are charm bracelets //————————————————————– int GenCharm(int t, int p)
int i;
if (t > N) [ if (N ]{} else [ for (i=a\[t-p\]; i<K; i++) [ a\[t\] = i; if (i == a\[t-p\]) GenCharm(t+1,p); else GenCharm(t+1,t); ]{} ]{}
//————————————————————– int main()
printf(“Enter N K: ”); scanf(“ a\[0\] = 0; GenCharm(1,1); printf(”Total =
[^1]: University of Waterloo, Department of Pure Mathematics and Institute for Quantum Computing, Waterloo, Ontario, N2L 3G1, Canada e-mail: `djokovic@uwaterloo.ca`
[^2]: Wilfrid Laurier University, Department of Physics & Computer Science, Waterloo, Ontario, N2L 3C5, Canada e-mail: `ikotsire@wlu.ca`
[^3]: School of Computer Science, University of Guelph, Canada. email: `drecoski@uoguelph.ca`
[^4]: School of Computer Science, University of Guelph, Canada. Research supported by NSERC. email: `jsawada@uoguelph.ca`
[^5]: We are grateful to an anonymous referee for pointing this out.
|
---
abstract: 'Can multicellular life be distinguished from single cellular life on an exoplanet? We hypothesize that abundant upright photosynthetic multicellular life (trees) will cast shadows at high sun angles that will distinguish them from single cellular life and test this using Earth as an exoplanet. We first test the concept using Unmanned Arial Vehicles (UAVs) at a replica moon landing site near Flagstaff, Arizona and show trees have both a distinctive reflectance signature (red edge) and geometric signature (shadows at high sun angles) that can distinguish them from replica moon craters. Next, we calculate reflectance signatures for Earth at several phase angles with POLDER (Polarization and Directionality of Earth’s reflectance) satellite directional reflectance measurements and then reduce Earth to a single pixel. We compare Earth to other planetary bodies (Mars, the Moon, Venus, and Uranus) and hypothesize that Earth’s directional reflectance will be between strongly backscattering rocky bodies with no weathering (like Mars and the Moon) and cloudy bodies with more isotropic scattering (like Venus and Uranus). Our modelling results put Earth in line with strongly backscattering Mars, while our empirical results put Earth in line with more isotropic scattering Venus. We identify potential weaknesses in both the modelled and empirical results and suggest additional steps to determine whether this technique could distinguish upright multicellular life on exoplanets.'
author:
- 'Christopher E. Doughty, Andrew Abraham, James Windsor, Michael Mommert, Michael Gowanlock, Tyler Robinson, and David Trilling [^1] [^2] [^3] [^4]'
bibliography:
- 'astro.bib'
title: Distinguishing multicellular life on exoplanets by testing Earth as an exoplanet
---
Introduction
============
Recently, a 1.3 Earth mass planet only $\sim$4 light years from Earth was found within the habitable zone of the red dwarf Proxima Centauri [@Anglada-Escude2016ACentauri]. According to the NASA Exoplanet Archive, by Dec 5 of 2019, 4104 exoplanets, have been confirmed (https://exoplanetarchive.ipac.caltech.edu/), including one in the habitable zone with water vapor in its atmosphere [@Tsiaras2019WaterB]. Do these exoplanets have life and if so, what type of life might it be? A number of techniques have been proposed to test whether life exists on exoplanets and many of these are summarized in recent reviews by [@Schwieterman2018ExoplanetLife][@Catling2018ExoplanetAssessment]. The goal of all this is, of course, to be able to use next generation astronomical facilities (recently reviewed by Fujii et al. [@Fujii2018ExoplanetProspects]) to detect life on the recently discovered exoplanets [@Fujii2018ExoplanetProspects]. However, such reviews have missed a critical stage – distinguishing an exoplanet with single cellular life from that of multicellular life. Some have hypothesized that single cellular life may be abundant in the universe, but multicellular life may be rare [@Brownlee20000RareEarth]. We clearly need a technique to distinguish between the two types of life.
Since photosynthesis could be abundant in the universe, what techniques, for example, could we use to distinguish the change between land covered with abundant terrestrial single-celled photosynthetic organisms like those in the Precambrian [@Kenny2001StablePrecambrian] and the rise of multicellular life, like the land plants that occupied Earth from the Mid-Ordovician (490–430 million years ago) to today [@Graham2000TheRadiation]? Previous work has proposed that the most abundant multicellular life on an exoplanet would likely be vertical photosynthetic organisms – trees [@Doughty2010DetectingPlanets]. The need to transport water and nutrients and competition for light in multicellular photosynthetic organisms has led to the tree-like structure on Earth characterized by hierarchical branching networks [@Brown2000ScalingBiology; @West1997ABiology]. In fact, the “tree shape” evolved independently many times throughout Earth’s history likely as a consequence of the previously mentioned biomechanical and evolutionary constraints [@Donoghue2005KeyPhylogeny]. Such biomechanical constraints combined with Darwinian evolution will also make tree-like photosynthetic structures the most abundant evidence of multicellular life on exoplanets.
Earth has more than 3 trillion trees [@Crowther2015MappingScale], each with a vertical structure that casts shadows differently than objects on a lifeless planet with weather and climate. Almost all trees are at a 90$^\circ$ angle to the ground while less than 1 percent of the surface of the Earth has with a slope greater than 45$^\circ$ [@Hall2005CharacterizationData]. This is simply because weather and climate, which are thought to be necessary on any planet capable of sustaining multicellular life [@Kasting2003EvolutionPlanet] will erode much abiotic topography over time. For instance, one study suggested a lifeless planet with weather will be very similar to Earth topologically [@Dietrich2006TheLife]. Therefore, shadows at certain sun angles may be indicative of multicellular life, but could we detect them on an exoplanet?
Earth Scientists know a great deal about tree shadows because to accurately estimate terrestrial reflectance (with, for example, Landsat or Modis satellite data) shadows at different sun angles must be removed. Therefore, a great deal of effort has been put into developing a quantitative framework to predict shadows at different sun angles. This framework, called the bidirectional reflectance distribution function (BRDF), is the change in observed reflectance with changing view angle or illumination direction [@Schaepman-Strub2006ReflectanceStudies]. Forests seen from different sensor sun angles have predictable differences in reflectance [@Breon2002AnalysisSpace; @Breon2006SpaceborneDistributions; @Li1992Geometric-OpticalShadowing; @Wolf2010AllometricMeasurements]. Previous work used a semi-empirical BRDF model [@Bacour2005VariabilityPOLDER; @Maignan2004BidirectionalSpot] at the global scale to explore whether, in theory, Earth with vegetation would have different albedo at different sensor sun angles versus an Earth without vegetation [@Doughty2010DetectingPlanets]. They found that even if the entire planetary albedo were rendered to a single pixel, the rate of increase of albedo as a planet approaches full illumination would be comparatively greater on a vegetated planet than on a non-vegetated planet. It was hypothesized that the technique would work at 4 light years (and greater depending on knowledge on cloud abundance and a coronagraph design) meaning it could be tested on the recently discovered planet in the habitable zone of Proxima Centauri.
The method was then tested empirically [@Doughty2016DetectingEarth] using the Galileo space probe data and first principles, in a similar methodology to Sagan et al. [@Sagan1993ASpacecraft]. Sagan et al. [@Sagan1993ASpacecraft] detected multiple stages of life on Earth, but they did not have a technique to distinguish between single and multicellular life on Earth. Doughty and Wolf [@Doughty2016DetectingEarth] used the Galileo space probe data but because the Galileo dataset had only a small change ($<2^\circ$) in phase angle (sun-satellite position), the observed anisotropy signal was small, and they could not detect multicellular life on Earth. In contrast, in this paper, we propose to use to the POLDER satellite (Polarization and Directionality of Earth’s reflectance) data to test this question. This dataset gives global reflectance, directionality (BRDF), and polarization measurements at 20km resolution and phase angles of $>60^\circ$ [@Bicheron2000BidirectionalSpace]. Therefore, we can create a view of Earth at different phase angles and determine empirically if, even scaling to a single pixel, we could distinguish between single and multicellular life on Earth.
However, could the BRDF technique distinguish between abundant vertical structures like moon craters and abundant vegetation on an exoplanet? Most such craters would in theory be eroded on a lifeless planet with weather and climate. However, we test the BRDF of craters on Earth to understand how they cast shadows at different sun angles. We took advantage of moon-like craters near our university that were created by the USGS in 1967 to help Apollo astronauts train by simulating different-sized lunar impact craters. A total of 497 craters were made within two sites comprising 2,000 square feet. We fly a UAV above a cratered landscape at different sun angles meant to replicate the moon landing site.
We can also use detection of the red edge as corroborating evidence for the existence of vegetation. Our goal is to compare the reflectance properties at the red edge of plants with the BRDF or geometric optics, for example, the shape and arrangement of objects within a pixel that transmit or block light [@Torrance1967TheorySurfaces], using Earth as an Exoplanet at various scales (Figure 1). We propose to test this at the following scales: at the replica moon landing crater field, at the Amazon basin and the Sahara Desert, on all of Earth’s cloud free continental terrestrial surface and for the Earth as a whole. We will then compare the phase function of the Earth as a single pixel to phase functions of other planets in the solar system. We will compare Earth empirically (with POLDER data) and for Earth modelled with and without vegetation with a BRDF model [@Maignan2004BidirectionalSpot; @Bacour2005VariabilityPOLDER] (Figure \[fig:fig1\]).
![Our conceptual design of a distant observer monitoring Earth and the change in backscattering as it revolves around the sun. $\Theta$ is the azimuth angle, $\Omega$ i is the solar zenith angle, $\Omega$ v is the view angle, and $\Psi$ is the phase angle.[]{data-label="fig:fig1"}](fig1.PNG){width="45.00000%"}
Methods {#sec:examples}
=======
Site information
-----------------
To test NDVI and BRDF as biosignatures, we took advantage of an “extraterrestrial landscape” near our university that we call the replica moon landing crater site (35.30594920 lon, -111.50617530 lat). Moon-like craters were created by the USGS in 1967 by digging holes and filling them with various amounts of explosives, which were detonated to simulate different-sized lunar impact craters. The human-made craters range in size from 1.5–12 meters in diameter. This area was chosen for the craters because of the basaltic cinders from an eruption of the Sunset Crater Volcano 950 years ago. After the explosions, the excavated lighter clay material spread out from the blast craters and across the fields, like ejecta from actual meteorite impacts. A total of 497 craters were made within two sites comprising 2,000 square feet (Figure \[fig:fig2\]).
UAV data acquisition
---------------------
We flew the Parrot Bluegrass (Parrot) UAV with 4 wavelengths (green 550 nm (40nm bandwidth (bw)), red – 660 nm(40nm bw), red edge 735nm (10 nm bw), and NIR 790nm (40 nm bw)) above the replica moon landing crater site described above. We flew at various times to get different sun instrument angles (5:30, 7:30, 9:00 and 11:30 am) comparing three landscape types (bare ground, craters, and ponderosa pine trees). The Parrot takes $\sim$200 photos at a height of 50m in each of the wavebands which are combined to form a map of $\sim$300 m^2^ (88 by 338m or 6ha) with a resolution of 4.7cm/pixel. We use the program Pix4DCapture to plan the flight paths and Pix4Dmapper to orthomosaic the raw images into reflectance values (WGS 84 coordinate system). This program created geotiffs for each band which we uploaded into the Google Earth Engine. We used matlab (Mathworks) to further analyze this data.
Empirical Earth at different phase angles with POLDER data
----------------------------------------------------------
POLDER (Polarization and directionality of Earth’s reflectance) gives global reflectance, directionality (BRDF), and polarization measurements[@Bacour2005VariabilityPOLDER; @Bicheron2000BidirectionalSpace]. The ground size or resolution of a POLDER-measured pixel is 6x7 km$^2$ at nadir. 12 directional radiance measurements at each spectral band are taken for each point on Earth. We downloaded data that capture the period from October 30, 1996 to February 28, 1997. During that period, we chose 21 days interspersed within this broader period and aggregated data from those days (Specifically – Oct 30,31, Nov 1–6 and Dec 30–31 1996 and Jan 8, 9, 10, 11, 12, 14, 16, 17, 22, 23, 27 1997). We also collected solar zenith angle (which is relative to the local zenith and may vary between 0$^\circ$ (sun at zenith) and approximately 80$^\circ$) and view zenith angle, (which is relative to the local zenith and may vary between 0$^\circ$ (POLDER at zenith) and approximately 75$^\circ$) (see Figure \[fig:fig1\] for an example of the geometries). For each day, we subtracted the view zenith angle from the solar zenith angle (but we did not control for azimuth angle) to estimate phase angle for the wavelengths 565 nm (20 nm bandwidth) and 763 nm (10 nm bandwidth). These two wavelengths were then used to create NDVI (Normalized difference vegetation index) according to the following equation:
$$NDVI = (763nm-565nm)/(763nm+565nm)$$
We then created separate data maps for $<1^\circ$ phase angle ranges, then 1–3$^\circ$, then 3–6$^\circ$, 6–20$^\circ$, and 20–30$^\circ$. We aggregated all available data for these five different phase angles and created cloud free land images of the Amazon basin, the Sahara Desert region, and all regions combined together. We averaged these maps as a single pixel at the different phase angles to replicate what Earth might look like to a distant observer as it circles the sun at different phase angles.
Modelled Earth at different phase angles with a BRDF model
----------------------------------------------------------
We used simulations of Earth with and without vegetation from Doughty et al 2010 at different phase angles. In that paper, they used a semi-empirical BRDF model [@Bacour2005VariabilityPOLDER; @Bicheron2000BidirectionalSpace]. It combines a geometric kernel (F1), which models a flat Lambertian surface covered with randomly distributed spheroids with the same optical properties as soil [@Lucht2000AnModels], with a volumetric kernel (F2), which models a theoretical turbid vegetation canopy with high leaf density [@Maignan2004BidirectionalSpot]. They simulated global cloud cover with CAM 3.0; http://www.ccsm.ucar.edu/models/atm-cam) [@Collins2006TheCAM3], and combined simulated cloud height (low, medium, and high) and total percent cover with albedo values for low, medium, and high clouds (strato-cumulous, alto-stratus, and cirrus) at several planetary phase angles [@Tinetti2006DetectabilityModel].
Other planets
--------------
To compare the how Earth would look circling the sun at a distance to other planetary bodies, we digitized data from Sudarsky et al. [@Sudarsky2005PhasePlanets] where they aggregated data for optical phase functions for Mars, Venus, the moon, and Uranus along with a Lambert model where radiation is scattered isotropically off a surface regardless of its angle of incidence [@Sudarsky2005PhasePlanets]. A classical phase function normalizes planetary albedo to 1 at a phase angle of 0$^\circ$. Data for Mars is originally from [@Thorpe1977Viking1976], for the Moon from [@Lane1973MonochromaticDisk], for Uranus from [@Pollack1986EstimatesNeptune] and Sudarsky et al. [@Sudarsky2005PhasePlanets] does not state where the Venus data is originally from.
We normalized all the datasets (Earth-POLDER, Earth no vegetation, Earth with vegetation, Mars, Venus, Uranus, and the moon) so that the albedo at phase angle of 0$^\circ$ was one. We then subtracted these from a Lambert curve to highlight the impact of directional scattering from each of these bodies.
Results
=======
The Apollo astronaut training ground offers a unique opportunity to compare NDVI and BRDF in an “extraterrestrial landscape” with trees. In 1967, a flyover of the area early in the morning shows large shadows for both the craters and the local ponderosa pine trees (Figure \[fig:fig2\](a)). It is therefore conceivable that craters could replicate the shadows and BRDF is not a good multicellular life biosignature. However, our UAV demonstrates why at later times of the day (at lower phase angles) the story changes. Figure \[fig:fig2\](b) shows our UAV NDVI image for the region at 5am. The trees clearly have a higher NDVI and the craters still have shadows. However, Figure \[fig:fig2\](c) shows strong shadows with the craters at 5:30am but not at 9am and 11am. In contrast, the trees show clear shadows at all times even towards noon (at lower phase angles). This effect will change slightly with latitude [@Doughty2010DetectingPlanets].
We can quantify these qualitative observations with our UAV collected reflectance data. Figure \[fig:fig3\] shows the reflectance histograms for trees and craters in the NIR (790 nm) at different times of the day. Because the UAV flew overhead, the daytimes correspond with high (5am), medium (9am) and low phase angles (11am). In Figure \[fig:fig3\](a), at 5:30 am the histogram of the crater shows a strong shadow peak at $\sim$0.01 reflectance and another reflectance peak at $\sim$0.05 reflectance. However, by 9am the shadow peak disappears and there is only the ground reflectance peak at $\sim$0.05 reflectance. In Figure \[fig:fig3\](b), at 9am there are reflectance peaks for shadows at $\sim$0.01 reflectance, at the ground at $\sim$0.05 reflectance, and for the tree canopy which was scattered but for clarity we reduced to 0.15 reflectance. At 11am, there are similar peaks, but with a small number of shadow pixels at 0.01 reflectance as expected. The difference between the peak brightness at 0$^\circ$ phase angle and reduced brightness at higher phase angles is our hypothesized “multicellular life biosignature”.
NDVI showed different reflectance peaks for trees than for bare ground and craters. The “tree” NDVI signal included shadows and bare ground which reduced the overall NDVI signal. However, even with the mixed signal, NDVI also showed a clear signal that could distinguish between the three areas with NDVI’s median histogram of 0.06 for the trees and $\sim$0 for both the crater and bare ground (Figure \[fig:fig3\](c)). Was the NDVI or BRDF signal greater? For example, a typical region of interest with 50 percent tree cover, 50 percent ground at 9am might have 25 percent of the ground covered in shadow. At 9am, our scene might have an NIR reflectance of 0.09 (0.15\*0.5 (tree)+0.01\*0.25 (shadow)+0.05\*0.25 (ground)) while at noon, as the shadows are masked, it would change to 0.10 (0.15\*0.5 (tree)+0.05\*0.50 (ground)). This is a relatively small change of 0.01. We have shown that moon craters would not show this change and the 0.01 signal is the “multicellular life biosignature”. However, the NDVI signal of $\sim$0.06 is clearly larger.
Next, we scaled up to the regional and global scale with POLDER data. We first created cloud free terrestrial maps of Earth at 5 different phase angles. We found that the $<1^\circ$ phase angle contained many regional blank areas, especially tropical regions with great cloud cover, and we did not include it in our final analysis. We discuss this more in the discussion section. Therefore, we focused on the phase angle ranges of 1–3$^\circ$, 3–6$^\circ$, 6–20$^\circ$, and 20–30$^\circ$. Averaging over 21 days gave sufficient cloud free images to create maps for most of the planet. There were still gaps in our coverage, both at high latitudes, where POLDER did not cover, and in parts of the tropics where clouds were very abundant.
Amazon Sahara All land world
------------- --------- --------- ---------- ---------
$765 nm$ $0.016$ $0.007$ $0.015$ $0.012$
$NDVI$ $0.055$ $0.009$ $0.043$ $0.033$
$per765 nm$ $8.5$ $3.8$ $10.9$ $8.2$
: **Absolute change of reflectance (between 1–3$^\circ$ phase angle and 20–30$^\circ$ phase angle) for band 765nm, NDVI and the percent change for band 765nm for the Amazon, Sahara, all land and the world.**
\[tab:shapefunctions\]
These cloud free images allowed us to compare two multicellular life endmembers – the Amazon basin, with abundant tree cover, and the Sahara Desert, with very few trees. In Figure \[fig:fig7\](a), we show the average reflectance for these two regions at both 565 and 763nm at several different phase angles. The changes were smaller than we had hypothesized with our BRDF model possibly because we missed the large change between 0–1$^\circ$ phase angle. At 763nm between phase angle 1–3$^\circ$ and 20–30$^\circ$ there was a difference of 0.016 reflectance units or $\sim$9 percent for the Amazon versus 0.007 reflectance units or $\sim$4 percent for the Sahara (Table 1). There were only minor changes for the Sahara or for the Amazon at 565nm.
We next created a global view of Earth (including land, clouds and oceans) at the different phase angles (Figure \[fig:fig5\]) and a NDVI of the entire Earth at different phase angles (Figure \[fig:fig6\]). In Figure \[fig:fig7\](b) and (c), we average Figure \[fig:fig4\], \[fig:fig5\], and \[fig:fig6\] as a single pixel at the different phase angles. As a single pixel, at 565nm, there are only minor reflectance changes between phase angle 1–3$^\circ$ and 20–30$^\circ$. However, at 763 nm, the land only had reflectance changes $\sim$0.015 or $\sim$12 percent and the whole world had a slightly smaller change of 0.011 or $\sim$8 percent (Table 1). We also compared averaged NDVI for the Amazon, the Sahara, all land and the averaged planet to combine information on the red edge with BRDF. As expected, the Amazon had the highest NDVI followed by all land, the Sahara and the whole world. The decrease in NDVI across phase angles was similar (0.06) for the Amazon, the land (0.04) and the world (0.03) but stayed flat for the Sahara (0.01) (Table 1).
Finally, we combined information from POLDER for Earth and compared this to measured estimates for other planetary bodies such as Mars, Venus, the Moon, and Uranus. We also added estimates of a Lambert body (a body with perfect isotropic reflectance) and modelled Earth with and without vegetation [@Doughty2010DetectingPlanets]. All planetary bodies have very different albedos, but for comparison purposes, we standardized the average albedo to 1 at a phase angle of 0. We initially hypothesized that Earth would have a phase function between Mars and Venus (with both POLDER and the vegetation model in agreement). In other words, Earth might be a partially cloudy planet with some directional reflectance. However, our modeled estimates of Earth, with and without vegetation showed similar directional reflectance to Mars but our empirical results using POLDER data showed Earth was more similar to Venus (Figure \[fig:fig8\]).
Discussion
==========
Why there was a large divergence between our modelled results of Earth at different phase angles and our empirical ones? To review, modelled Earth’s reflectance at different phase angles is similar to Mars while empirical POLDER data of Earth’s reflectance at different phase angles are similar to Venus (Figure \[fig:fig8\]). We hypothesize that both the model and empirical data have issues that make them not align. For instance, our model uses the best vegetation BRDF model, but it did not have a good BRDF model for other components of the Earth, such as oceans, clouds and atmosphere. Therefore, it likely missed key components of atmospheric scattering and cloud directional reflectance. In contrast, we hypothesize that there were also issues with the empirical data because by excluding our phase angle data of $<1^\circ$ degree in our empirical analysis, we missed the largest change in BRDF. Our BRDF model suggests the largest change in reflectance from vegetation will be between phase angles of 0-1$^\circ$ and 1–3$^\circ$. Therefore, by missing this peak, and showing little change $<10^\circ$, our phase curve is more like an isotropic body like Venus.
Mars and the moon both have greater backscattering than Earth. For solid bodies with thin atmospheres like Mars, previous work has shown that backscattering can be significant [@Thorpe1977Viking1976]. This is because Mars (currently) has no liquid water to erode and smooth its rough edges. Our phase curve (Figure \[fig:fig8\]), shows that the moon has even stronger backscattering than Mars, which is initially surprising [@Lane1973MonochromaticDisk]. However, this is due to a phenomenon called coherent backscatter which occurs on very dry soils where particles have a diameter that is similar to the wavelength of the photon used to view them [@Hapke1993TheBackscatter]. A planet with climate like Earth does not exhibit coherent backscatter, even in dry areas, such as deserts, because the particle sizes are too big (generally between 0.05 to 2mm) at 800 nm or less [@Tarbuck2008EarthGeology]. Therefore, Earth shows less backscattering than Mars or the moon because of the presence of abundant isotropic clouds. The presence of craters on the moon and Mars also affects backscattering. At low phase angles the BRDF of craters is substantially different than that of trees (Figures \[fig:fig2\] and \[fig:fig3\]). Earth has few craters due to abundant erosion caused by climate. It is interesting to note the large amount of erosion of the craters at the replica moon landing site that has already occurred due to weather and climate in the 50 years since the craters were first formed.
In contrast, Venus and Uranus have scattering more similar to Lambert scattering where radiation is scattered isotropically off a surface. Lambert scattering is a good approximation for objects such as Uranus [@Pollack1986EstimatesNeptune], and to a lesser extent Venus [@Sudarsky2005PhasePlanets]. Surprisingly, our empirically derived phase function for Earth was less steep than either Venus or Uranus (Figure \[fig:fig8\]). This is surprising because Earth has many strong backscattering surfaces like trees. We hypothesize that this is due to excluding our phase angle data of $<1^\circ$ in our empirical analysis.
To improve our future empirical analysis, we need to better capture low phase angles. With the POLDER data, averaging for phase angles of 1 degree or less was inherently more patchy because it was averaging over a smaller dataset. Key regions, like Amazonia were missing because of high cloud cover. In fact, the cloudier terrestrial areas, and the regions less represented at $<1^\circ$ phase angle, were those most likely to have abundant tree cover (like Amazonia). For this reason, we were not confident including our maps of $<1^\circ$ phase angle. POLDER was only available for a few months during 1996-1997 and it is currently the only satellite of its kind to capture the Earth at all phase angles. Capturing planets at low phase angles will also be a problem with any viewing of an exoplanet because it could be washed out by the light of its star, even with the most advanced coronagraph design [@Guyon2006TheoreticalCoronagraphs]. However, in theory, we could observe the planet during continuous rotation cycles which could increase the amount of data available to analyze the exoplanet for vegetation structure.
To improve our modelling analysis, we need to better model the BRDF of non-vegetated surfaces. We used a state of the art BRDF model for vegetation [@Bicheron2000BidirectionalSpace; @Bacour2005VariabilityPOLDER], but only averaged BRDF values for clouds, atmosphere and oceans. With this improved model, how do we envision using the model in the future to distinguish a planet with multicellular life versus just single cellular life? We could create a model of an exoplanet based on the exoplanet’s size, density, cloud cover, distance to star, and the star’s irradiance. For instance, let us imagine we had the proper technology and coronagraph to observe the 1.3 Earth mass planet only $\sim$4 light years from Earth within the habitable zone of the red dwarf Proxima Centauri [@Anglada-Escude2016ACentauri]. We would then create three versions of the model, first a relatively smooth, eroded, planetary surface, one covered with single cellular slime exhibiting NDVI and one with 3D vegetation structure. We would look for evidence of which model better fit observations of the exoplanet over years. False positives caused by instrument error or intermittent events such as volcanic activity or changing cloud cover could be determined by observing the planet during continuous rotation cycles. Multicellular life would continuously demonstrate the BRDF signal, while other causes would demonstrate it only intermittently. In practice, this will be difficult with the next generation potential space telescopes for directly imaging exoplanets such as HabEx and LUVOIR. These are predicted to have 10–20 signal to noise ratio (SNR) for exoplanet spectroscopy but if an exciting target were to be discovered, more telescope time could increase this to $\sim$20–100 SNRs.
Conclusions and future directions
=================================
Overall, in theory, BRDF could distinguish between multicellular and single cellular life on exoplanets, but we have recognized issues with both our models and our empirical observations that must be improved before this technique could be used with confidence. The easiest short-term step is to improve the modelling by combining the various BRDF models. Further empirical validation will be more challenging as POLDER is a unique satellite. Here we demonstrate that BRDF is challenging to detect and will be a smaller signal than NDVI, which has already proven to be challenging to detect with Earth as an exoplanet [@MontanesRodriguez2006VegetationPlanets]. Should this line of research therefore be abandoned? Theoretically, it could still work and since we are not aware of other techniques to distinguish an exoplanet with multicellular life, we believe further work should still continue.
Acknowledgments {#acknowledgments .unnumbered}
===============
This project was funded by NASA’s Habitable World’s program with the project name: “Testing methods to detect 3D vegetation structure on exoplanets” (16-HW16-2-0025).
[^1]: Christopher E. Doughty, Andrew Abraham, and Michael Gowanlock are with the School of Informatics, Computing, and Cyber Systems, Northern Arizona University, Flagstaff, AZ. 86011, USA
[^2]: James Windsor, Tyler Robinson, and David Trilling are with the Department of Astronomy and Planetary Science, Northern Arizona University, Flagstaff, AZ. 86011, USA
[^3]: Michael Mommert is with Lowell Observatory, 1400 W Mars Hill Rd, Flagstaff, AZ 86001
[^4]: Corresponding author: chris.doughty@nau.edu
|
---
abstract: 'In the late $1980$’s, it was shown that the Casson invariant appears in the difference between the two filtrations of the Torelli group: the lower central series and the Johnson filtration, and that its core part was identified with the secondary characteristic class $d_1$ associated with the fact that the first $\mathrm{MMM}$ class vanishes on the Torelli group (however it turned out that Johnson proved the former part highly likely prior to the above, see Remark \[rem:j\]). This secondary class $d_1$ is a rational generator of $H^1(\mathcal{K}_g;{\mathbb{Z}})^{\mathcal{M}_g}\cong{\mathbb{Z}}$ where $\mathcal{K}_g$ denotes the Johnson subgroup of the mapping class group $\mathcal{M}_g$. Hain proved, as a particular case of his fundamental result, that this is the only difference in degree $2$. In this paper, we prove that no other invariant than the above gives rise to new rational difference between the two filtrations up to degree $6$. We apply this to determine $H_1(\mathcal{K}_g;{\mathbb{Q}})$ explicitly by computing the description given by Dimca, Hain and Papadima. We also show that any finite type rational invariant of homology $3$-spheres of degrees up to $6$, including the second and the third Ohtsuki invariants, can be expressed by $d_1$ and lifts of Johnson homomorphisms.'
address:
- 'Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan'
- 'Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan'
- 'Department of Frontier Media Science, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan'
author:
- Shigeyuki Morita
- Takuya Sakasai
- Masaaki Suzuki
title: ' Torelli group, Johnson kernel and invariants of homology spheres'
---
Introduction and statements of the main results {#sec:intro}
===============================================
Let $\mathcal{M}_g$ be the mapping class group of a closed oriented surface $\Sigma_g$ of genus $g$ and let $\mathcal{I}_g\subset \mathcal{M}_g$ be the Torelli subgroup. Namely, it is the subgroup of $\mathcal{M}_g$ consisting of all the elements which act on the homology $H:=H_1(\Sigma_g;{\mathbb{Z}})$ trivially.
There exist two filtrations of the Torelli group. One is the lower central series which we denote by $\mathcal{I}_g(k)\ (k=1,2,\ldots)$ where $\mathcal{I}_g(1)=\mathcal{I}_g$, $\mathcal{I}_g(2)=[\mathcal{I}_g, \mathcal{I}_g]$ and $\mathcal{I}_g(k+1)=[\mathcal{I}_g(k), \mathcal{I}_g]$ for $k\geq 1$. The other is called the Johnson filtration $\mathcal{M}_g(k)\ (k=1,2,\ldots)$ of the mapping class group where $\mathcal{M}_g(k)$ is defined to be the kernel of the natural homomorphism $$\rho_k: \mathcal{M}_g\rightarrow \mathrm{Out} (N_k(\pi_1\Sigma_g)).$$ Here $N_k(\pi_1\Sigma_g)$ denotes the $k$-th nilpotent quotient of the fundamental group of $\Sigma_g$ and $\mathrm{Out} (N_k(\pi_1\Sigma_g))$ denotes its outer automorphism group. $\mathcal{M}_g(1)$ is nothing other than the Torelli group $\mathcal{I}_g$ so that $\mathcal{M}_g(k)\ (k=1,2,\ldots)$ is a filtration of $\mathcal{I}_g$. This filtration was originally introduced by Johnson [@johnson] for the case of a genus $g$ surface with one boundary component. The above is the one adapted to the case of a closed surface (see [@morita99] for details). It can be shown that $\mathcal{I}_g(k)\subset \mathcal{M}_g(k)$ for all $k\geq 1$. Johnson showed in [@j3] that $\mathcal{I}_g(2)$ is a finite index subgroup of $\mathcal{M}_g(2)$ and asked whether this will continue to hold for the pair $\mathcal{I}_g(k)\subset \mathcal{M}_g(k)\ (k\geq 3)$. He also showed in [@j2] that $\mathcal{M}_g(2)$ is equal to the subgroup $\mathcal{K}_g$, called the Johnson subgroup or Johnson kernel, consisting of all the Dehn twists along separating simple closed curves on $\Sigma_g$.
The above question was answered negatively for the case $k=3$ in [@morita87][@morita89] (however, it turned out that Johnson proved this fact highly likely prior to the above , see Remark \[rem:j\] below). More precisely, a homomorphism $d_1: \mathcal{K}_g\rightarrow {\mathbb{Z}}$ was constructed which is non-trivial on $\mathcal{M}_g(3)$ while it vanishes on $\mathcal{I}_g(3)$ so that the index of the pair $\mathcal{I}_g(3)\subset \mathcal{M}_g(3)$ was proved to be infinite. Furthermore it was shown in [@morita91] that there exists an isomorphism $$H^1(\mathcal{K}_g;{\mathbb{Z}})^{\mathcal{M}_g}\cong {\mathbb{Z}}\quad (g\geq 2)$$ where the homomorphism $d_1$ serves as a rational generator. It is characterized by the fact that its value on a separating simple closed curve on $\Sigma_g$ of type $(h,g-h)$ is $h(g-h)$ up to non-zero constants. This homomorphism was defined as the secondary characteristic class associated with the fact that the first $\mathrm{MMM}$ class, which is an element of $H^2(\mathcal{M}_g;{\mathbb{Z}})$, vanishes in $H^2(\mathcal{I}_g;{\mathbb{Z}})$. It was also interpreted as a manifestation of the Casson invariant $\lambda$, which is an invariant defined for homology $3$-spheres, in the structure of the Torelli group.
In a note of Johnson opened as [@jnote], he studied the influence of the Casson invariant on the structure of the Johnson kernel. It turned out that he proved a negative answer for the case $k=3$ to his question mentioned above and also obtained a large part of those results of [@morita87][@morita89] concerning the above topic. Although his note is undated, it seems highly likely that his work was done prior to the works of the above papers. This is very surprising. On the other hand, the main result of the above papers is not covered, which expresses the Casson invariant as the secondary invariant associated to the fact that the first MMM class vanishes on the Torelli group. \[rem:j\]
Now let us consider the following two graded Lie algebras $$\begin{aligned}
\mathrm{Gr}\,\mathfrak{t}_g&=\bigoplus_{k=1}^\infty \mathfrak{t}_g(k),\quad \mathfrak{t}_g(k)
=(\mathcal{I}_g(k)/\mathcal{I}_g(k+1))\otimes{\mathbb{Q}}\\
\mathfrak{m}_g&=\bigoplus_{k=1}^\infty \mathfrak{m}_g(k),\quad
\mathfrak{m}_g(k)=(\mathcal{M}_g(k)/\mathcal{M}_g(k+1))\otimes{\mathbb{Q}}\end{aligned}$$ associated to the above two filtrations of the Torelli group. Here $\mathfrak{t}_g$ denotes the Malcev Lie algebra of the Torelli group and $\mathrm{Gr}\,\mathfrak{t}_g$ denotes its associated graded Lie algebra. Hain [@hain] obtained fundamental results about the structure of these Lie algebras. He gave an explicit finite presentation of them (see Theorem \[th:hain\] below) which implies that $\mathrm{Ker}(\mathfrak{t}_g(2)\rightarrow \mathfrak{m}_g(2))\cong{\mathbb{Q}}$. Furthermore, he proved that the natural homomorphism $$\mathfrak{t}_g(k)\rightarrow \mathfrak{m}_g(k)
\label{eq:nh}$$ is surjective for any $k$ which implies that the index of the pair $\mathcal{I}_g(k)\subset \mathcal{M}_g(k)$ remains infinite for any $k\geq 4$ extending the above mentioned result. He also showed that all the higher Massey products of the Torelli group vanish for $g\geq 4$.
On the other hand, Ohtsuki [@ohtsukip] defined a series of invariants $\lambda_k$ for homology $3$-spheres the first one being the Casson invariant $\lambda$. He also initiated a theory of finite type invariants for homology $3$-spheres in [@ohtsuki]. Then Garoufalidis and Levine [@gl] studied the relation between this theory and the structure of the Torelli group extending the case of the Casson invariant mentioned above extensively.
In these situations, it would be natural to ask whether there exists any other difference between the two filtrations of the Torelli group than the Casson invariant, in particular whether any finite type rational invariant of homology $3$-spheres of degree greater than $2$ appears there or not. This is equivalent to asking whether the natural homomorphism is an isomorphism for $k=3,4,\ldots$ or not.
Now it was proved in [@morita99] that $\mathfrak{t}_g(3)\cong \mathfrak{m}_g(3)$. The main theorem of the present paper is the following.
For any $k=4,5,6,$ we have $$\mathfrak{t}_g(k)\cong \mathfrak{m}_g(k).$$ \[th:main\]
As a corollary to the above theorem, we obtain the cases $k=5,6,7$ of the following result. The case $k=3$ follows from Hain’s theorem (Theorem \[th:hain\]) combined with a result of [@morita89] and the case $k=4$ follows from a result of [@morita99] mentioned above.
For any $k=3,4,5,6,7$, the $k$-th group $
\mathcal{I}_g(k)
$ in the lower central series of the Torelli group is a finite index subgroup of the kernel of the non-trivial homomorphism $$d_1: \mathcal{M}_g(k)\rightarrow {\mathbb{Z}}.$$ \[cor:fi\]
Recall here that Johnson [@j3] proved that $\mathcal{I}_g(2)=[\mathcal{I}_g,\mathcal{I}_g]$ is a finite index subgroup of $\mathcal{M}_g(2)=\mathcal{K}_g$.
Next we present two applications of Theorem \[th:main\]. First, we give the explicit form of the rational abelianization $H_1(\mathcal{K}_g;{\mathbb{Q}})$ of the Johnson subgroup. Dimca and Papadima [@dp] proved that $H_1(\mathcal{K}_g;{\mathbb{Q}})$ is finite dimensional for $g\geq 4$. Then Dimca, Hain and Papadima [@dhp] gave a description of it. However they did not give the final explicit form. Here we compute their description by combining the case $k=4$ of Theorem \[th:main\] and former results concerning the Johnson homomorphisms to obtain the following result.
The secondary class $d_1$ together with the refinement $\tilde{\tau}_2$ of the second Johnson homomorphism gives the following isomorphism for $g\geq 6$. $$H_1(\mathcal{K}_g;{\mathbb{Q}})\cong {\mathbb{Q}}\oplus [2^2]\oplus [31^2].$$ \[th:kab\]
Here and henceforth, for a given Young diagram $\lambda=[\lambda_1\cdots \lambda_h]$, we denote the irreducible representation of $\mathrm{Sp}(2g,{\mathbb{Q}})$ corresponding to $\lambda$ simply by $[\lambda_1\cdots \lambda_h]$. As for the refinements $\tilde{\tau}_k$ of Johnson homomorphisms, see Section \[sec:ft\] for details. By making use of recent remarkable results of Ershov-He [@eh] and Church-Ershov-Putman [@cep], we obtain the following.
$\mathrm{(i)}\ $ Two subgroups $[\mathcal{K}_g,\mathcal{K}_g]$ and $\mathcal{I}_g(4)$ of the Torelli group $\mathcal{I}_g$ are commensurable for $g\geq 6$.
$\mathrm{(ii)}\ $ $[\mathcal{K}_g,\mathcal{K}_g]$ is finitely generated for $g\geq 7$. \[cor:ki\]
Johnson [@j3] determined the abelianization $H_1(\mathcal{I}_g;{\mathbb{Z}})$ of the Torelli group completely where the Birman-Craggs homomorphisms introduced in [@bc] played an essential role in describing its torsion part. Although some of the Birman-Craggs homomorphisms restrict non-trivially on $\mathcal{K}_g$, they are mod $2$ reductions of integral ones because the Casson invariant defines an integer valued homomorphism on $\mathcal{K}_g$. Therefore, no $2$-torsion class in $H_1(\mathcal{I}_g;{\mathbb{Z}})$ can be lifted to $H_1(\mathcal{K}_g;{\mathbb{Z}})$ as a torsion class. Thus, at present, there is no known information about the torsion part of $H_1(\mathcal{K}_g;{\mathbb{Z}})$. It should be an important problem to determine it.
Another application of our main theorem is the following.
Any finite type rational invariant of homology $3$-spheres of degrees $4$ and $6$, including the Ohtsuki invariants $\lambda_2$ and $\lambda_3$, can be expressed by $d_1$ and (lifts of) Johnson homomorphisms. \[th:fti\]
In Section \[sec:ft\], we give more detailed statements Theorem \[th:type4\] and Theorem \[th:type6\].
Based on the above result, we would like to propose the following conjecture (see Problem 6.2 of [@morita99]).
For any $k\not=2$, the equality $
\mathfrak{t}_g(k)\cong \mathfrak{m}_g(k)
$ holds so that $$\mathrm{Ker}(\mathrm{Gr}\,\mathfrak{t}_g\twoheadrightarrow \mathfrak{m}_g)\cong {\mathbb{Q}}.$$
\[rem:aftercor16\] This is equivalent to the statement that Corollary \[cor:fi\] continues to hold for all $k \geq 3$.
In the context of characteristic classes of the mapping class group, the above conjecture can be translated as follows.
The Lie algebra $\mathfrak{t}_g$ is isomorphic to the completion of the central extension of $\mathfrak{m}_g$ associated to the infinitesimal first $\mathrm{MMM}$ class defined in $H^2(\mathfrak{m}_g)$.
Hain [@hain] considered the relative completion of the mapping class group with respect to the classical homomorphism $\mathcal{M}_g\rightarrow\mathrm{Sp}(2g,{\mathbb{Z}})$. He proved that the kernel of the natural surjective homomorphism $\mathfrak{t}_g\rightarrow \mathfrak{u}_g$ is isomorphic to ${\mathbb{Q}}$ where $\mathfrak{u}_g$ denotes the graded Lie algebra associated to the Lie algebra of his relative completion. In this terminology, the above conjecture is also equivalent to saying that the natural homomorphism $$\mathrm{Gr}\,\mathfrak{u}_g\rightarrow \mathfrak{m}_g,$$ which exists because of the universality of $\mathfrak{u}_g$, is an isomorphism.
The content of the present paper is roughly as follows. In Section $2$, we relate the difference between the two filtrations of the Torelli group to the second homology group $H_2(\mathfrak{m}_g)$ of the Lie algebra $\mathfrak{m}_g$. In Section $3$, we explain the method of proving the main Theorem \[th:main\]. Then in Sections $4, 5$ and $6$, we prove the vanishing of the weight $4$, $5$ and $6$ parts of $H_2(\mathfrak{m}_g)$, respectively. Finally in Section $7$, we prove the main results.
In this paper, whenever we mention groups like $\mathcal{M}_g, \mathcal{I}_g,\mathcal{M}_g(k),\mathcal{I}_g(k)$, modules like $\mathfrak{t}_g(k), \mathfrak{m}_g(k)$ and also homomorhpsims like $\tau_g(k)$, which depend on the genus $g$, we always assume that it is sufficiently large, more precisely in a stable range with respect to the property we consider, unless we describe the range of $g$ explicitly.
The second homology groups of the Lie algebras $\mathfrak{t}_g, \mathfrak{m}_g$ {#sec:h2}
===============================================================================
In this section, we reduce the problem of determining the kernel of the surjective homomorphism $\mathrm{Gr}\,\mathfrak{t}_g\twoheadrightarrow \mathfrak{m}_g$ to the computation of the second homology group $H_2(\mathfrak{m}_g)$ of the Lie algebra $\mathfrak{m}_g$. Let us denote $\mathrm{Ker}(\mathrm{Gr}\,\mathfrak{t}_g\rightarrow \mathfrak{m}_g)$ by $\mathfrak{i}_g$ so that we have a short exact sequence $$0\rightarrow \mathfrak{i}_g
\rightarrow \mathrm{Gr}\,\mathfrak{t}_g\rightarrow\mathfrak{m}_g\rightarrow 0
\label{eq:itm}$$ of the three graded Lie algebras $$\mathfrak{i}_g=\bigoplus_{k=1}^\infty\mathfrak{i}_g(k),\
\mathrm{Gr}\,\mathfrak{t}_g=\bigoplus_{k=1}^\infty\mathfrak{t}_g(k),\
\mathfrak{m}_g=\bigoplus_{k=1}^\infty\mathfrak{m}_g(k).$$
Now it is a classical result of Johnson [@ja][@j3] that $\mathfrak{t}_g(1)\cong \mathfrak{m}_g(1)\cong\wedge^3 H_{\mathbb{Q}}/H_{\mathbb{Q}}$ and hence $\mathfrak{i}_g(1)=0$. The module $\wedge^3 H_{\mathbb{Q}}/H_{\mathbb{Q}}$ is an irreducible representation of $\mathrm{Sp}(2g,{\mathbb{Q}})$ corresponding to the Young diagram $[1^3]$. Hain proved the following fundamental result.
The Lie algebra $\mathfrak{t}_g\ (g\geq 6)$ is isomorphic to its associated graded $\mathrm{Gr}\,\mathfrak{t}_g$ which has presentation $$\mathrm{Gr}\,\mathfrak{t}_g=\mathcal{L}(
[1^3])/\langle[1^6]+[1^4]+[1^2]+[2^21^2]\rangle$$ where $\mathcal{L}([1^3])$ denotes the free Lie algebra generated by $[1^3]$ and $\langle[1^6]+[1^4]+[1^2]+[2^21^2]\rangle$ is the ideal generated by $$[1^6]+[1^4]+[1^2]+[2^21^2]\subset \wedge^2[1^3].$$ \[th:hain\]
Here we recall a few facts about the homology of graded Lie algebras briefly. Let us consider a graded Lie algebra $$\mathfrak{g}=\bigoplus_{k=1}^\infty \mathfrak{g}(k)$$ satisfying the condition that $
H_1(\mathfrak{g})\cong \mathfrak{g}(1).
$ Namely, we assume that $\mathfrak{g}$ is generated by the degree $1$ part $\mathfrak{g}(1)$ as a Lie algebra. Both the Lie algebras $\mathfrak{t}_g, \mathfrak{m}_g$ satisfy this condition. The homology group of a graded Lie algebra is bigraded, where the first grading is the usual homology degree while the second grading is induced from the grading of the graded Lie algebra. We call the latter grading by weight, denoted simply by the subscript $w$. In particular, the second homology group has the following direct sum decomposition. $$H_2(\mathfrak{g})=\bigoplus_{w=2}^\infty H_2(\mathfrak{g})_w$$ where $$H_2(\mathfrak{g})_w=
\frac{\mathrm{Ker}\left(\partial: (\wedge^{2}\mathfrak{g})_w
\twoheadrightarrow \mathfrak{g}(w)\right)}{\mathrm{Im}\left(\partial: (\wedge^3\mathfrak{g})_w\rightarrow
(\wedge^{2}\mathfrak{g})_w\right)}.
\label{eq:hw}$$
In this terminology, the following is an immediate consequence of Theorem \[th:hain\].
\[cor:22\] $H_2(\mathrm{Gr}\,\mathfrak{t}_g)_2\cong [1^6]+[1^4]+[1^2]+[2^21^2]$ and for any $w\geq 3$, $H_2(\mathrm{Gr}\,\mathfrak{t}_g)_w=0$. \[cor:hain\]
Now we consider the short exact sequence .
The following short exact sequence $$0\rightarrow H_2(\mathrm{Gr}\,\mathfrak{t}_g)_2\rightarrow H_2(\mathfrak{m}_g)_2\rightarrow
(H_1(\mathfrak{i}_g)_{\mathfrak{m}_g})_2\cong{\mathbb{Q}}\rightarrow
0$$ holds and for any $w\geq 3$ $$H_2(\mathfrak{m}_g)_w\cong (H_1(\mathfrak{i}_g)_{\mathfrak{m}_g})_w.$$ \[prop:mi\]
The $5$-term exact sequence of the Lie algebra extension is given by $$H_2(\mathrm{Gr}\,\mathfrak{t}_g)\overset{\text{Cor. \ref{cor:22}}}{=}H_2(\mathrm{Gr}\,\mathfrak{t}_g)_2
\rightarrow H_2(\mathfrak{m}_g)\rightarrow
H_1(\mathfrak{i}_g)_{\mathfrak{m}_g}
\rightarrow
H_1(\mathrm{Gr}\,\mathfrak{t}_g)
\overset{\cong}{\rightarrow} H_1(\mathfrak{m}_g).$$ The result follows from this.
$$H_2(\mathfrak{m}_g)_3=0 ,\quad
\mathfrak{i}_g(4)\cong H_2(\mathfrak{m}_g)_4$$ and for any $w\geq 4$, we have the following exact sequence. $$0\rightarrow \sum_{k=3}^{w-1}\ [\mathfrak{i}_g(k), \mathfrak{t}_g(w-k)]
\rightarrow\mathfrak{i}_g(w)\rightarrow H_2(\mathfrak{m}_g)_w\rightarrow 0.$$ \[cor:h2\]
It was shown in [@morita99] (see also a related result of [@sakasai]) that $\mathfrak{t}_g(3)\cong\mathfrak{m}_g(3)$ and hence $\mathfrak{i}_g(3)=0$. It follows that $H_2(\mathfrak{m}_g)_3\cong (H_1(\mathfrak{i}_g)_{\mathfrak{m}_g})_3=0$. As for the case $w=4$, we have $$H_2(\mathfrak{m}_g)_4\cong (H_1(\mathfrak{i}_g)_{\mathfrak{m}_g})_4
=\mathfrak{i}_g(4)$$ because $\mathfrak{i}_g(2)\cong{\mathbb{Q}}$ is contained in the center of $\mathfrak{t}_g$. Finally, for any $w\geq 4$, we have $$H_2(\mathfrak{m}_g)_w\cong (H_1(\mathfrak{i}_g)_{\mathfrak{m}_g})_w
\cong \mathfrak{i}_g(w)/ \sum_{k=3}^{w-1}\ [\mathfrak{i}_g(k), \mathfrak{t}_g(w-k)].$$ This completes the proof.
In general, we have the following
Assume $\mathfrak{i}_g(k)=0\ $ for $k=3,4,\ldots,w-1\ (w\geq 4)$. Then we have $$\mathfrak{i}_g(w)\cong H_2(\mathfrak{m}_g)_w.$$ \[prop:iw\]
This is an immediate consequence of Corollary \[cor:h2\].
Method of proving Theorem \[th:main\] {#sec:method}
=====================================
To prove Theorem \[th:main\], it is enough to show that $$H_2(\mathfrak{m}_g)_w=0 \ (w=4,5,6)$$ by Proposition \[prop:iw\]. We prove this in Sections $4,5,6$. To do so, for technical reasons regarding computer computations, we consider the following variants of our Lie algebra. Let $\mathcal{M}_{g,1}$ be the mapping class group of a genus $g$ compact oriented surface with one boundary component and let $\mathcal{I}_{g,1}$ be its Torelli subgroup. Then we have the Johnson filtration $\{\mathcal{M}_{g,1}(k)\}_k$ for the former and the lower central series $\{\mathcal{I}_{g,1}(k)\}_k$ for the latter. We set $$\begin{aligned}
\mathrm{Gr}\,\mathfrak{t}_{g,1}&=\bigoplus_{k=1}^\infty \mathfrak{t}_{g,1}(k),\quad \mathfrak{t}_{g,1}(k)
=(\mathcal{I}_{g,1}(k)/\mathcal{I}_{g,1}(k+1))\otimes{\mathbb{Q}}\\
\mathfrak{m}_{g,1}&=\bigoplus_{k=1}^\infty \mathfrak{m}_{g,1}(k),\quad
\mathfrak{m}_{g,1}(k)=(\mathcal{M}_{g,1}(k)/\mathcal{M}_{g,1}(k+1))\otimes{\mathbb{Q}}\end{aligned}$$ where $\mathfrak{t}_{g,1}$ denotes the Malcev Lie algebra of $\mathcal{I}_{g,1}$. Define $\mathfrak{i}_{g,1}=\mathrm{Ker}(\mathrm{Gr}\,\mathfrak{t}_{g,1}\twoheadrightarrow\mathfrak{m}_{g,1})$ so that we have a short exact sequence $$0\rightarrow \mathfrak{i}_{g,1}\rightarrow \mathrm{Gr}\,\mathfrak{t}_{g,1}\rightarrow\mathfrak{m}_{g,1}\rightarrow 0
\label{eq:itm1}$$ of graded Lie algebras, where the surjectivity of the natural homomorphism $\mathrm{Gr}\,\mathfrak{t}_{g,1}\rightarrow\mathfrak{m}_{g,1}$ is again due to Hain.
The natural homomorphism $\mathfrak{i}_{g,1}\rightarrow \mathfrak{i}_{g}$ is an isomorphism so that $\mathfrak{i}_{g,1}(k)\cong \mathfrak{i}_{g}(k)$ for any $k$. \[prop:ii\]
Recall that the kernel of the natural surjection $\mathcal{I}_{g,1}\rightarrow \mathcal{I}_g$ is known to be isomorphic to $\pi_1 T_1\Sigma_g$ where $T_1\Sigma_g$ denotes the unit tangent bundle of $\Sigma_g$. Thus we have a short exact sequence $1\rightarrow \pi_1 T_1\Sigma_g\rightarrow \mathcal{I}_{g,1}\rightarrow \mathcal{I}_g\rightarrow 1$. Let $\mathfrak{p}_{g,1}$ be the Malcev Lie algebra of $\pi_1 T_1\Sigma_g$ which is a central extension of the Malcev Lie algebra $\mathfrak{p}_g$ of $\pi_1\Sigma_g$. Hain [@hain] showed that this induces a short exact sequence $0\rightarrow \mathfrak{p}_{g,1}\rightarrow \mathfrak{t}_{g,1}\rightarrow \mathfrak{t}_{g}\rightarrow 0$ which in turn induces a short exact sequence of their associated graded Lie algebras as depicted in . $$\begin{CD}
@. @. 0 @. 0@.\\
@. @. @VVV @VVV @.\\
@. @. \mathfrak{i}_{g,1} @>{\cong}>> \mathfrak{i}_{g}
@.\\
@. @. @VVV @VVV @.\\
0 @>>> \mathrm{Gr}\,\mathfrak{p}_{g,1} @>>> \mathrm{Gr}\,\mathfrak{t}_{g,1} @>>> \mathrm{Gr}\,\mathfrak{t}_{g}
@>>> 0\\
@. @| @VVV @VVV @.\\
0 @>>> \mathrm{Gr}\,\mathfrak{p}_{g,1} @>>> \mathfrak{m}_{g,1} @>>>
\mathfrak{m}_{g} @>>> 0\\
@. @. @VVV @VVV @.\\
@. @. 0 @. 0
\end{CD}
\label{eq:cd}$$ On the other hand, a result of Asada and Kaneko (and Labute) in [@ak] implies that the kernel of the natural surjection $\mathfrak{m}_{g,1}\rightarrow\mathfrak{m}_{g}$ is isomorphic to $\mathrm{Gr}\,\mathfrak{p}_{g,1}$ (see [@morita99][@mss4] for details). Thus the two rows as well as the two columns of the commutative diagram are all short exact sequences. Then it is easy to see from this fact that the homomorphism $\mathfrak{i}_{g,1}\rightarrow\mathfrak{i}_g$ is an isomorphism. This completes the proof.
Now we consider the short exact sequence .
For any $w\geq 3$, $$H_2(\mathfrak{m}_{g,1})_w\cong (H_1(\mathfrak{i}_{g,1})_{\mathfrak{m}_{g,1}})_w.$$ \[prop:mi1\]
In [@hain Corollary 7.8], Hain showed that $\mathrm{Gr}\,\mathfrak{t}_{g,1}$ is quadratically presented. This implies that $H_2(\mathrm{Gr}\,\mathfrak{t}_{g,1}) \cong H_2(\mathrm{Gr}\,\mathfrak{t}_{g,1})_2 $. Then the $5$-term exact sequence of the Lie algebra extension is given by $$\begin{aligned}
H_2(\mathrm{Gr}\,\mathfrak{t}_{g,1})=H_2(\mathrm{Gr}\,\mathfrak{t}_{g,1})_2
\rightarrow H_2(\mathfrak{m}_{g,1})\rightarrow
H_1(\mathfrak{i}_{g,1})_{\mathfrak{m}_{g,1}}
\rightarrow
H_1(\mathrm{Gr}\,\mathfrak{t}_{g,1})
\overset{\cong}{\rightarrow} H_1(\mathfrak{m}_{g,1}).\end{aligned}$$ The result follows from this.
The natural homomorphism $$H_2(\mathfrak{m}_{g,1})_w\rightarrow H_2(\mathfrak{m}_{g})_w$$ is an isomorphism for all $w\geq 3$. \[prop:h2g1\]
By Proposition \[prop:mi1\] and Proposition \[prop:mi\], the natural homomorphism $
H_2(\mathfrak{m}_{g,1})_w\rightarrow H_2(\mathfrak{m}_{g})_w
$ can be identified with the natural homomorphism $$(H_1(\mathfrak{i}_{g,1})_{\mathfrak{m}_{g,1}})_w\rightarrow
(H_1(\mathfrak{i}_{g})_{\mathfrak{m}_{g}})_w$$ for any $w\geq 3$. Then it is easy to see that this is an isomorphism by Proposition \[prop:ii\].
As mentioned in the beginning of this section, to prove Theorem \[th:main\], it is enough to show that $
H_2(\mathfrak{m}_g)_w=0 \ (w=4,5,6).
$ By the above Proposition \[prop:h2g1\], this is equivalent to showing that $$H_2(\mathfrak{m}_{g,1})_w=0 \ (w=4,5,6).$$ We consider this case of one boundary component which is best suited for our computer computation because of the following reason. By virtue of the Johnson homomorphisms (see [@ja][@johnson][@morita93]), the Lie algebra $\mathfrak{m}_{g,1}$ can be embedded into the Lie algebra $\mathfrak{h}_{g,1}$ consisting of all the symplectic derivations of the free Lie algebra generated by $H_{\mathbb{Q}}=H_1(\Sigma_g,{\mathbb{Q}})$ as a Lie subalgebra. More precisely, we have an embedding $$\tau_{g,1}=\bigoplus_{k=1}^\infty \tau_{g,1}(k):
\mathfrak{m}_{g,1}=\bigoplus_{k=1}^\infty
\mathfrak{m}_{g,1}(k)\rightarrow
\mathfrak{h}_{g,1}=\bigoplus_{k=1}^\infty
\mathfrak{h}_{g,1}(k)
\subset \bigoplus_{k=1}^\infty H_{\mathbb{Q}}^{\otimes (k+2)}.$$ Thus each graded piece $\mathfrak{m}_{g,1}(k)$ can be identified with $\mathrm{Im}\,\tau_{g,1}(k)$, which is a submodule of $H_{\mathbb{Q}}^{\otimes (k+2)}$, so that we can make explicit computer computations on this space of tensors.
Proof of $H_2(\mathfrak{m}_g)_4=0$ {#sec:w4}
==================================
In this section, we prove the following.
$H_2(\mathfrak{m}_g)_4\cong H_2(\mathfrak{m}_{g,1})_4=0.$ \[prop:w4\]
It is sufficient to show that $H_2(\mathfrak{m}_{g,1})_4=0$, by Proposition \[prop:h2g1\]. By equality , we have $$H_2(\mathfrak{m}_{g,1})_4=
\frac{\mathrm{Ker}\left((\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3))\oplus \wedge^2 \mathfrak{m}_{g,1}(2)
\twoheadrightarrow \mathfrak{m}_{g,1}(4)\right)}{\mathrm{Im}\left(\wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(2)\rightarrow
(\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3))\oplus \wedge^2 \mathfrak{m}_{g,1}(2)\right)}.$$ Here the boundary operator $$\partial : \wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(2)\rightarrow
(\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3))\oplus \wedge^2 \mathfrak{m}_{g,1}(2)
\label{eq:b}$$ is given by $$\begin{aligned}
&\wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(2)\ni (u\land v)\otimes w\longmapsto\\
&(u\otimes [v,w]-v\otimes [u,w],-[u,v]\land w)\in (\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3))\oplus \wedge^2 \mathfrak{m}_{g,1}(2).\end{aligned}$$
In our paper [@mss4], we gave the irreducible decompositions of $\mathfrak{m}_{g,1}(k)\cong \mathrm{Im}\,\tau_{g,1}(k)$ for all $k\leq 6$ as shown in Table \[tab:6\].
[|l|l|]{} $k$ & $\mathfrak{m}_{g,1}(k)$\
$1$ & $[1^3]\ [1]$\
$2$ & $[2^2]\ [1^2]\ [0]$\
$3$ & $[31^2]\ [21]$\
$4$ & $[42][31^3][2^3]\ 2[31][21^2]\ 2[2]$\
$5$ & $[51^2][421][3^21][321^2][2^21^3]\ 2[41]2[32]2[31^2]2[2^21]2[21^3]\ [3]3[21]2[1^3]\ [1]$\
$6$ & $[62] [521][51^3][4^2][431]2[42^2][421^2][41^4]2[3^21^2][32^21][321^3][2^4]][2^21^4]$\
& $3[51]3[42]4[41^2]3[3^2]7[321]3[31^3][2^3]5[2^21^2]2[21^4][1^6]$\
& $4[4]6[31]9[2^2]6[21^2]4[1^4]\ 3[2]6[1^2]\ 2[0]$\
\[tab:6\]
By using this and applying our techniques described in [@mss2], we determine the space $$Z_2(4)=\mathrm{Ker}\left((\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3))\oplus
\wedge^2 \mathfrak{m}_{g,1}(2)
\twoheadrightarrow \mathfrak{m}_{g,1}(4)\right)$$ of $2$-cycles for the weight $4$ homology group $H_2(\mathfrak{m}_{g,1})_4$ as shown in Table \[tab:z24\].
[|c|l|]{} $Z_2(4)$ & $[431][42^2][421^2][41^4]2[32^21][321^3][31^5]$\
& $[42]2[41^2][3^2]6[321]4[31^3][2^3]3[2^21^2]2[21^4]$\
& $[4]5[31]5[2^2]7[21^2][1^4]\ 3[2]4[1^2]$\
$\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3)$ & $[42^2][421^2][41^4][32^21][321^3][31^5]$\
& $[42]2[41^2]4[321]4[31^3][2^3]2[2^21^2]2[21^4]$\
& $[4]5[31]3[2^2]5[21^2][1^4]\ 3[2]2[1^2]$\
$\wedge^2 \mathfrak{m}_{g,1}(2)$ & $[431][32^21]\ [42][3^2]2[321][31^3][2^3][2^21^2]\ 2[31]2[2^2]3[21^2]\ 2[2]2[1^2]$\
$\mathfrak{m}_{g,1}(4)$ & $[42][31^3][2^3]\ 2[31][21^2]\ 2[2]$\
\[tab:z24\]
Thus we can write $$H_2(\mathfrak{m}_{g,1})_4\cong
\mathrm{Coker}\, \left(
\wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(2)
\overset{\partial}{\rightarrow}
Z_2(4) \right).$$ Now we show that the cokernel is trivial for each irreducible component. As an example, we pick up $[21^2]$. The multiplicity of $[21^2]$ in $Z_2(4)$ is $7$, which is the biggest (see Table \[tab:z24\]). We have checked that all of these $[21^2]$ components are hit by the boundary operator $$\partial : \wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(2)\rightarrow
Z_2(4)\subset
(\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3))\oplus \wedge^2 \mathfrak{m}_{g,1}(2)$$ as follows. First, $\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3)$ and $\wedge^2 \mathfrak{m}_{g,1}(2)$ are regarded as subspaces of $H_{\mathbb{Q}}^{\otimes 8}$ via the natural embeddings $$\begin{aligned}
\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3)
&\subset \mathfrak{h}_{g,1}(1)\otimes \mathfrak{h}_{g,1}(3)
\subset H_{\mathbb{Q}}^{\otimes 3}\otimes H_{\mathbb{Q}}^{\otimes 5}
\subset H_{\mathbb{Q}}^{\otimes 8}, \\
\wedge^2 \mathfrak{m}_{g,1}(2)
&\subset \wedge^2 \mathfrak{h}_{g,1}(2)
\subset \wedge^2 H_{\mathbb{Q}}^{\otimes 4}
\subset H_{\mathbb{Q}}^{\otimes 8}.\end{aligned}$$ Then we constructed $7$ $\mathrm{Sp}$-projections $D_i: H_{\mathbb{Q}}^{\otimes 8}\rightarrow [21^2]\ (i=1,\ldots,7)$ which detect the $7$ copies of $[21^2]$ in $Z_2(4)$. Here we denote by $\mu_{(i_1,j_1)\cdots(i_k,j_k)}$ and $p_{(i_1,\ldots,i_k) \cdots (j_1,\ldots, j_l)}$ [*multiple contractions*]{} and [*projections*]{} respectively: $$\mu_{(i_1,j_1)\cdots(i_k,j_k)} : H^{\otimes n}_{\mathbb{Q}}\to H^{\otimes (n-2k)}_{\mathbb{Q}}, \qquad
p_{(i_{1},\ldots,i_{k}) \cdots (j_1,\ldots, j_l)} : H^{\otimes n}_{\mathbb{Q}}\to \wedge^k H_{\mathbb{Q}}\otimes \cdots \otimes \wedge^l H_{\mathbb{Q}}.$$ For example, $$\begin{aligned}
\mu_{(13)(25)} (x_1 \otimes x_2 \otimes x_3 \otimes x_4 \otimes x_5 \otimes x_6)
&= \mu(x_1, x_3) \mu(x_2, x_5) x_4 \otimes x_6, \\
p_{(124)(35)(6)} (x_1 \otimes x_2 \otimes x_3 \otimes x_4 \otimes x_5 \otimes x_6)
&= (x_1 \wedge x_2 \wedge x_4) \otimes (x_3 \wedge x_5) \otimes x_6 ,\end{aligned}$$ where $\mu : H_{\mathbb{Q}}\otimes H_{\mathbb{Q}}\to {\mathbb{Q}}$ is a natural non-degenerate antisymmetric bilinear form. It is known that any subrepresentation of $H^{\otimes n}_{\mathbb{Q}}$ can be detected by a combination of these multiple contractions and projections. Such $\mathrm{Sp}$-projections are called [*detectors*]{}. The following are detectors for this case $[21^2]$: $$\begin{aligned}
&p_{(123)(4)} \circ \mu_{(12)(34)} \circ \Phi_1, \quad
p_{(123)(4)} \circ \mu_{(12)(45)} \circ \Phi_1, \quad
p_{(123)(4)} \circ \mu_{(14)(25)} \circ \Phi_1, \\
&p_{(123)(4)} \circ \mu_{(14)(56)} \circ \Phi_1, \quad
p_{(124)(3)} \circ \mu_{(14)(56)} \circ \Phi_1, \\
& p_{(123)(4)} \circ \mu_{(12)(35)} \circ \Phi_2, \quad
p_{(123)(4)} \circ \mu_{(12)(56)} \circ \Phi_2 , \end{aligned}$$ by using projections $$\begin{aligned}
\Phi_1 &: \mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3) \oplus \wedge^2 \mathfrak{m}_{g,1}(2)
\xrightarrow{pr_1} \mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3) \subset H_{\mathbb{Q}}^{\otimes 8}, \\
\Phi_2 & : \mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3) \oplus \wedge^2 \mathfrak{m}_{g,1}(2)
\xrightarrow{pr_2} \wedge^2 \mathfrak{m}_{g,1}(2) \subset H_{\mathbb{Q}}^{\otimes 8} .\end{aligned}$$ where $pr_i : A_1 \oplus A_2 \to A_i$ is the projection to the $i$th component. Next, we take the following $7$ elements $v_j$ of $\wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(2)$. In general, $\mathfrak{m}_{g,1}(k)$ is contained in $ \mathfrak{h}_{g,1}(k)~ (\subset H^{\otimes (k+2)}_{\mathbb{Q}})$. Furthermore, any element of $ \mathfrak{h}_{g,1}(k)$ can be expressed by using a Lie spider with $(k+2)$ legs $$\begin{aligned}
&S(u_1, u_2, u_3, \ldots, u_{k+2}) =\\
&u_1 \otimes [u_2, [u_3,[\cdots [ u_{k+1},u_{k+2}]\cdots]]]
+ u_2 \otimes [[u_3, [u_4,[\cdots [ u_{k+1},u_{k+2}]\cdots]]],u_1] \\
&+u_3 \otimes [[u_4, [u_5,[\cdots [ u_{k+1},u_{k+2}]\cdots]]],[u_1,u_2]]
+ \cdots + u_{k+2} \otimes [[[\cdots [ u_1,u_2],\cdots],u_k],u_{k+1}] ,\end{aligned}$$ where $u_i \in H_{\mathbb{Q}}$. We fix a symplectic basis $\{a_1, b_1, \ldots, a_g,b_g \}$ of $H_{\mathbb{Q}}$ with respect to $\mu$ so that $$\mu(a_i,a_j) = \mu(b_i,b_j) = 0, \quad
\mu(a_i, b_j) = \delta_{i,j} .$$ Then we write $v_j \in \wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(2)$ as Lie spiders $$\begin{aligned}
&v_1 = S(a_1,a_2,a_4) \wedge S(a_1,b_4,a_5) \otimes S(a_3,b_5,a_6,b_6), \\
&v_2 = S(a_1,a_2,a_4) \wedge S(a_1,a_5,b_5) \otimes S(a_3,b_4,a_6,b_6), \\
&v_3 = S(a_1,a_2,a_4) \wedge S(a_1,a_5,a_6) \otimes S(a_3,b_4,b_5,b_6), \\
&v_4 = S(a_1,a_2,a_4) \wedge S(a_3,b_4,a_5) \otimes S(a_1,b_5,a_6,b_6), \\
&v_5 = S(a_1,a_2,a_4) \wedge S(b_4,a_5,b_5) \otimes S(a_1,a_3,a_6,b_6), \\
&v_6 = S(a_1,a_2,a_4) \wedge S(b_4,a_5,a_6) \otimes S(a_1,a_3,b_5,b_6), \\
&v_7 = S(a_1,a_2,a_4) \wedge S(a_5,b_5,a_6) \otimes S(a_1,a_3,b_4,b_6).\end{aligned}$$ Finally, we computed the boundary operator $\partial$ on these vectors $v_j$ and applied the $7$ detectors $D_i$. Then we checked that the rank of the matrix $D_i(\partial (v_j))$ is $7$. This means that $\partial (v_j)$’s generate the space of the highest weight vectors corresponding to $7[21^2]$ in $Z_2(4)$. This is the process of proving that the $[21^2]$ component of $H_2(\mathfrak{m}_{g,1})_4$ vanishes.
In this way, we checked that all the $\mathrm{Sp}$-irreducible components of $Z_2(4)$ are boundaries. This finishes the proof of $H_2(\mathfrak{m}_{g,1})_4=0$ and hence $H_2(\mathfrak{m}_{g})_4=0$.
All the other data are shown in the web [@szkHP].
Proof of $H_2(\mathfrak{m}_g)_5=0$ {#sec:w5}
==================================
In this section, we prove the following.
$H_2(\mathfrak{m}_g)_5\cong H_2(\mathfrak{m}_{g,1})_5=0.$ \[prop:w5\]
By equality , we have $$H_2(\mathfrak{m}_{g,1})_5=\frac{Z_2(5)}{B_2(5)}$$ where $$Z_2(5)=
\mathrm{Ker}\left((\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(4))\oplus
(\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(3))
\twoheadrightarrow \mathfrak{m}_{g,1}(5)\right)$$ and $$\begin{aligned}
B_2(5)=
\mathrm{Im}
&(\wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3))
\oplus (\mathfrak{m}_{g,1}(1)\otimes \wedge^2 \mathfrak{m}_{g,1}(2)) \\
&
\overset{\partial}{\rightarrow}
(\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(4))\oplus
(\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(3))
).\end{aligned}$$ Here the boundary operator $$\begin{split}
\partial :(\wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3))
& \oplus (\mathfrak{m}_{g,1}(1)\otimes \wedge^2 \mathfrak{m}_{g,1}(2)) \\
&\rightarrow
(\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(4))\oplus
(\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(3))
\label{eq:b5}
\end{split}$$ is given by $$\begin{aligned}
&\wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3)\ni (u\land v)\otimes w\longmapsto\\
&(u\otimes [v,w]-v\otimes [u,w],-[u,v]\otimes w)\in (\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(4))\oplus
(\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(3))\\
&\mathfrak{m}_{g,1}(1)\otimes \wedge^2 \mathfrak{m}_{g,1}(2)\ni u\otimes (v\land w)\longmapsto\\
&(u\otimes [v,w],-v\otimes [u,w]+w\otimes [u,v])\in (\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(4))\oplus
(\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(3)).\end{aligned}$$ We can write $$H_2(\mathfrak{m}_{g,1})_5\cong
\mathrm{Coker}\, \left(
(\wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(3))
\oplus (\mathfrak{m}_{g,1}(1)\otimes \wedge^2 \mathfrak{m}_{g,1}(2))
\overset{\partial}{\rightarrow}
Z_2(5) \right).$$ As in the preceding section, by using Table \[tab:6\] and applying our techniques described in [@mss2], we have determined the space $Z_2(5)$ of $2$-cycles for the weight $5$ homology group $H_2(\mathfrak{m}_{g,1})_5$, see Table \[tab:z25\].
[|c|l|]{} $Z_2(5)$ & $2[531]2[521^2][432]2[431^2]2[42^21]3[421^3][41^5]2[3^3]2[3^221][32^3]$\
& $3[32^21^2][321^4][31^6][2^31^3]\ 3[52][51^2]4[43]10[421]8[41^3]4[3^21]8[32^2]$\
& $12[321^2]8[31^4]6[2^31]3[2^21^3]2[21^5]\ 9[41]12[32]23[31^2]12[2^21]13[21^3]$\
& $11[3]17[21]6[1^3]\ 4[1]$\
$\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(4)$ & $[531][521^2][431^2][42^21]2[421^3][41^5][3^3][3^221][32^3]2[32^21^2][321^4][31^6][2^31^3]$\
& $2[52][51^2]2[43]6[421]5[41^3]2[3^21]4[32^2]8[321^2]6[31^4]4[2^31]3[2^21^3]2[21^5]$\
& $7[41]8[32]15[31^2]8[2^21]10[21^3]\ 8[3]12[21]5[1^3]\ 3[1]$\
$\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(3)$ & $[531][521^2][432][431^2][42^21][421^3][3^3][3^221][32^21^2]$\
& $[52][51^2]2[43]6[421]3[41^3]3[3^21]4[32^2]5[321^2]2[31^4]2[2^31][2^21^3]$\
& $4[41]6[32]10[31^2]6[2^21]5[21^3]\ 4[3]8[21]3[1^3]\ 2[1]$\
$\mathfrak{m}_{g,1}(5)$ & $[51^2][421][3^21][321^2][2^21^3]\ 2[41]2[32]2[31^2]2[2^21]2[21^3]\ [3]3[21]2[1^3]\ [1]$\
\[tab:z25\]
Moreover, we have computed the boundary operator explicitly and checked that all the $2$-cycles ($35$-types of Young diagrams with multiplicities) are boundaries. This finishes the proof of $H_2(\mathfrak{m}_{g,1})_5=0$ and hence $H_2(\mathfrak{m}_{g})_5=0$.
Proof of $H_2(\mathfrak{m}_g)_6=0$ {#sec:w6}
==================================
In this section, we prove the following.
$H_2(\mathfrak{m}_g)_6\cong H_2(\mathfrak{m}_{g,1})_6=0.$ \[prop:w6\]
By equality , we have $$H_2(\mathfrak{m}_{g,1})_6=\frac{Z_2(6)}{B_2(6)}$$ where $$Z_2(6)=\mathrm{Ker}\left(
(\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(5))\oplus
(\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(4))\oplus
\wedge^2\mathfrak{m}_{g,1}(3)
\rightarrow \mathfrak{m}_{g,1}(6)\right)$$ and $$\begin{aligned}
B_2(6)=
\mathrm{Im}
&((\wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(4))
\oplus (\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(2)\otimes\mathfrak{m}_{g,1}(3))
\oplus
\wedge^3 \mathfrak{m}_{g,1}(2))\\
&
\overset{\partial}{\rightarrow}
(\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(5))\oplus
(\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(4))\oplus
\wedge^2\mathfrak{m}_{g,1}(3)
).\end{aligned}$$
Here the boundary operator $$\begin{split}
\partial :(\wedge^2\mathfrak{m}_{g,1}(1)&\otimes \mathfrak{m}_{g,1}(4))
\oplus (\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(2)\otimes\mathfrak{m}_{g,1}(3))
\oplus
\wedge^3 \mathfrak{m}_{g,1}(2))\\
&\rightarrow
(\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(5))\oplus
(\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(4))\oplus
\wedge^2\mathfrak{m}_{g,1}(3)
\end{split}
\label{eq:b6}$$ is given by $$\begin{aligned}
&\wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(4)\ni (u\land v)\otimes w\longmapsto\\
&\hspace{3mm}(u\otimes [v,w]-v\otimes [u,w],-[u,v]\otimes w,0)
\in (\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(5))\oplus
(\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(4))\oplus
\wedge^2\mathfrak{m}_{g,1}(3)\\
&(\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(2)\otimes\mathfrak{m}_{g,1}(3))
\ni u\otimes v\otimes w\longmapsto\\
&\hspace{3mm}(u\otimes [v,w],-v\otimes [u,w],- [u,v]\land w)
\in (\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(5))\oplus
(\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(4))\oplus
\wedge^2\mathfrak{m}_{g,1}(3)\\
&\wedge^3 \mathfrak{m}_{g,1}(2)\ni u\land v\land w\longmapsto\\
&\hspace{3mm}(0, u\otimes [v,w]+v\otimes [w,u]+w\otimes [u,v],0)
\in (\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(5))\oplus
(\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(4))\oplus
\wedge^2\mathfrak{m}_{g,1}(3).\end{aligned}$$
Thus we can write $$\begin{aligned}
&H_2(\mathfrak{m}_{g,1})_6\cong \mathrm{Coker}\, \\
& \left(
(\wedge^2\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(4))
\oplus (\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(2)\otimes\mathfrak{m}_{g,1}(3))
\oplus
\wedge^3 \mathfrak{m}_{g,1}(2))
\overset{\partial}{\rightarrow}
Z_2(6)\right).\end{aligned}$$ As in the preceding two sections, by using Table \[tab:6\] and applying our techniques described in [@mss2], we can determine the space $Z_2(6)$ of $2$-cycles for the weight $6$ homology group $H_2(\mathfrak{m}_{g,1})_6$ as in Table \[tab:z26\].
[|c|l|]{} $Z_2(6)$ & $[{64}][{631}] 2[{62^2}] 2[{621^2}][{61^4}][{541}]3[{532}]3[{531^2}]4[{52^21}]4[{521^3}]2[{51^5}]3[{4^22}]$\
& $5[{431^3}]2[{4^21^2}]7[{4321}]3[{42^3}]4[{42^21^2}]3[{421^4}][{3^31}]2[{3^22^2}]5[{3^221^2}]4[{3^21^4}]$\
& $3[{32^31}]3[{32^21^3}] 2[{321^5}]2[{2^5}][{2^41^2}][{2^31^4}][{2^21^6}]\ 2[{62}]3[{61^2}]6[{53}]17[{521}]$\
& $10[{51^3}]2[{4^2}]20[{431}]16[{42^2}]29[{421^2}]12[{41^4}]14[{3^22}]19[{3^21^2}]25[{32^21}]25[{321^3}]$\
& $8[{31^5}]5[{2^4}]12[{2^31^2}]9[{2^21^4}]3[{21^6}]\ 2[{6}]13[{51}]34[{42}]35[{41^2}]17[{3^2}]63[{321}]42[{31^3}]$\
& $25[{2^3}]40[{2^21^2}]25[{21^4}]5[{1^6}]\ 13[{4}]52[{31}]33[{2^2}]56[{21^2}]18[{1^4}]\ 26[{2}]21[{1^2}]\ 3[{0}]$\
$\mathfrak{m}_{g,1}(1)\otimes \mathfrak{m}_{g,1}(5)$ & $[62^2][621^2][61^4][532][531^2]2[52^21]2[521^3][51^5][4^22][4^21^2]3[4321]3[431^3]$\
& $[42^3]2[42^21^2]2[421^4][3^22^2]3[3^221^2]3[3^21^4]2[32^31]2[32^21^3]2[321^5][2^5]$\
& $[2^41^2][2^31^4][2^21^6]\ [62]2[61^2][53]8[521]6[51^3][4^2]9[431]8[42^2]15[421^2]$\
& $7[41^4]6[3^22]12[3^21^2]13[32^21]17[321^3]6[31^5]4[2^4]8[2^31^2]8[2^21^4]3[21^6]$\
& $[6]8[51]16[42]19[41^2]10[3^2]36[321]25[31^3]13[2^3]28[2^21^2]$\
& $19[21^4]5[1^6]\ 8[4]28[31]23[2^2]35[21^2]15[1^4]\ 14[2]16[1^2]\ 3[0]$\
$\mathfrak{m}_{g,1}(2)\otimes \mathfrak{m}_{g,1}(4)$ & $[64][631][62^2][541][532]2[531^2][52^21][521^3]2[4^22]3[4321][431^3]2[42^3]$\
& $[42^21^2][421^4][3^31]2[3^221^2][32^21^3][2^21^4][2^5]\ [62][61^2]5[53]7[521]2[51^3][4^2]$\
& $9[431]6[42^2]11[421^2]3[41^4]7[3^22]5[3^21^2][32^31]10[32^21]6[321^3]2[31^5][2^4]$\
& $4[2^31^2]\ [6]5[51]18[42]13[41^2]5[3^2]25[321]16[31^3]12[2^3]10[2^21^2]6[21^4]$\
& $5[4]25[31]11[2^2]22[21^2]3[1^4]\ 14[2]6[1^2]$\
$\wedge^2\mathfrak{m}_{g,1}(3)$ & $[621^2][532][52^21][521^3][51^5][4^21^2][4321][431^3][42^21^2][3^22^2][3^21^4]$\
& $[62]3[521]3[51^3][4^2]3[431]4[42^2]4[421^2]3[41^4][3^22]4[3^21^2]3[32^21]3[321^3][2^4][2^21^4]$\
& $3[51]3[42]7[41^2]5[3^2]9[321]4[31^3][2^3]7[2^21^2]2[21^4][1^6]$\
& $4[4]5[31]8[2^2]5[21^2]4[1^4]\ [2]5[1^2]\ 2[0]$\
$\mathfrak{m}_{g,1}(6)$ & $[62][521][51^3][431][4^2]2[42^2][421^2][41^4]2[3^21^2][32^21][321^3][2^4]$\
& $3[51]3[42]4[41^2]3[3^2]7[321]3[31^3][2^3][2^21^4]5[2^21^2]2[21^4][1^6]$\
& $4[4]6[31]9[2^2]6[21^2]4[1^4]\ 3[2]6[1^2]\ 2[0]$\
\[tab:z26\]
We have computed the boundary operator explicitly and checked that all the $2$-cycles ($67$-types of Young diagrams with multiplicities) are boundaries. In this way, we checked that all the $\mathrm{Sp}$-irreducible components of $Z_2(6)$ are boundaries. This finishes the proof of $H_2(\mathfrak{m}_{g,1})_6=0$ and hence $H_2(\mathfrak{m}_{g})_6=0$.
The size of our computer computation grows very rapidly with respect to weights and, in particular, the weight $6$ case is approximately $1000$ times as large as the weight $4$ case.
Proofs of the main results {#sec:ft}
===========================
By Corollary \[cor:h2\], we have $\mathfrak{i}_g(4)\cong H_2(\mathfrak{m}_g)_4$. On the other hand, we have $H_2(\mathfrak{m}_g)_4=0$ by Proposition \[prop:w4\]. Hence, we conclude that $\mathfrak{i}_g(4)=0$, namely $\mathfrak{t}_g(4)\cong \mathfrak{m}_g(4)$. Then, if we combine Proposition \[prop:iw\] with Proposition \[prop:w5\] and Proposition \[prop:w6\], we conclude that $\mathfrak{i}_g(5)=\mathfrak{i}_g(6)=0$ so that $\mathfrak{t}_g(5)\cong \mathfrak{m}_g(5)$ and $\mathfrak{t}_g(6)\cong \mathfrak{m}_g(6)$. This finishes the proof.
Observe first that, for any $k,\ell$ with $1\leq k\leq\ell$, the quotient group $\mathcal{M}_g(k)/\mathcal{I}_g(\ell)$ is a finitely generated nilpotent group because it is a subgroup of $\mathcal{I}_g/\mathcal{I}_g(\ell)$ which is finitely generated by Johnson [@jfg] and nilpotent. Hence we can consider the rational form $(\mathcal{M}_g(k)/\mathcal{I}_g(\ell))\otimes{\mathbb{Q}}$ of $\mathcal{M}_g(k)/\mathcal{I}_g(\ell)$.
Now the case $k=3$ is a direct consequence of Theorem \[th:hain\] combined with a result in [@morita89] because of the following reason. Since $\mathcal{I}_g(2)$ is a finite index subgroup of $\mathcal{M}_g(2)=\mathcal{K}_g$ by Johnson, we have a short exact sequence $$0\rightarrow(\mathcal{M}_g(3)/\mathcal{I}_g(3))\otimes{\mathbb{Q}}\rightarrow(\mathcal{I}_g(2)/\mathcal{I}_g(3))\otimes {\mathbb{Q}}\rightarrow
(\mathcal{M}_g(2)/\mathcal{M}_g(3))\otimes{\mathbb{Q}}\rightarrow 0.$$ Here $(\mathcal{I}_g(2)/\mathcal{I}_g(3))\otimes {\mathbb{Q}}=\mathfrak{t}_g(2)$ and $(\mathcal{M}_g(2)/\mathcal{M}_g(3))\otimes {\mathbb{Q}}=\mathfrak{m}_g(2)$ by definition. Hence we conclude $(\mathcal{M}_g(3)/\mathcal{I}_g(3))\otimes{\mathbb{Q}}\cong \mathfrak{i}_g(2)\cong {\mathbb{Q}}$. The result follows from this because we know that the homomorphism $d_1:\mathcal{M}_g(3)\rightarrow {\mathbb{Q}}$ is non-trivial whereas its restriction to $\mathcal{I}_g(3)$ is trivial as shown in [@morita89].
Next, we consider the cases $k\geq 4$. We recall here how the non-triviality of the homomorphism $d_1:\mathcal{M}_g(k)\rightarrow {\mathbb{Q}}$ for all $k\geq 4$ follows immediately from Hain’s result that the homomorphism $\mathfrak{t}_g(k) \to \mathfrak{m}_g(k)$ is surjective for any $k$. Assume that $d_1:\mathcal{M}_g(k)\rightarrow {\mathbb{Q}}$ were trivial for some $k\geq 4$ and let $m$ be the smallest such. Then consider the homomorphism $$\mathfrak{t}_g(m-1)=(\mathcal{I}_g(m-1)/\mathcal{I}_g(m)) \otimes {\mathbb{Q}}\rightarrow
\mathfrak{m}_g(m-1)=(\mathcal{M}_g(m-1)/\mathcal{M}_g(m)) \otimes {\mathbb{Q}}.
\label{eq:m}$$ By the assumption, the non-trivial homomorphism $d_1:\mathcal{M}_g(m-1)\rightarrow{\mathbb{Q}}$ factors through $d_1:\mathcal{M}_g(m-1)/\mathcal{M}_g(m)\rightarrow{\mathbb{Q}}$. On the other hand, we know that the restriction of $d_1$ on $\mathcal{I}_g(k)$ is trivial for all $k\geq 3$. We now conclude that the above homomorphism is [*not*]{} surjective which is a contradiction.
Now the case $k=4$ follows from the fact $\mathfrak{t}_g(3)\cong \mathfrak{m}_g(3)$ proved in [@morita99] as follows. We have the following two exact sequences. $$0\rightarrow\mathfrak{t}_g(3)=(\mathcal{I}_g(3)/\mathcal{I}_g(4))\otimes {\mathbb{Q}}\rightarrow
(\mathcal{M}_g(3)/\mathcal{I}_g(4))\otimes{\mathbb{Q}}\rightarrow(\mathcal{M}_g(3)/\mathcal{I}_g(3))\otimes{\mathbb{Q}}\rightarrow 0,$$ $$0\rightarrow(\mathcal{M}_g(4)/\mathcal{I}_g(4))\otimes{\mathbb{Q}}\rightarrow(\mathcal{M}_g(3)/\mathcal{I}_g(4))\otimes {\mathbb{Q}}\rightarrow
\mathfrak{m}_g(3)=(\mathcal{M}_g(3)/\mathcal{M}_g(4))\otimes{\mathbb{Q}}\rightarrow 0.$$ By the case $k=3$ above, we have $(\mathcal{M}_g(3)/\mathcal{I}_g(3))\otimes{\mathbb{Q}}\cong{\mathbb{Q}}$. If we put this to the first exact sequence, we obtain $$\mathrm{rank}\, (\mathcal{M}_g(3)/\mathcal{I}_g(4))\otimes {\mathbb{Q}}=\dim \mathfrak{t}_g(3)+1.$$ Here $\mathrm{rank}\, (\mathcal{M}_g(3)/\mathcal{I}_g(4))\otimes{\mathbb{Q}}$ means the rank of the nilpotent group $(\mathcal{M}_g(3)/\mathcal{I}_g(4))\otimes{\mathbb{Q}}$ over ${\mathbb{Q}}$. On the other hand, from the second exact sequence, we have $$\mathrm{rank}\, (\mathcal{M}_g(3)/\mathcal{I}_g(4))\otimes {\mathbb{Q}}=\dim \mathfrak{m}_g(3)+
\mathrm{rank}\, (\mathcal{M}_g(4)/\mathcal{I}_g(4))\otimes{\mathbb{Q}}.$$ Since $\mathfrak{t}_g(3)\cong \mathfrak{m}_g(3)$, we conclude that $$\mathrm{rank}\, (\mathcal{M}_g(4)/\mathcal{I}_g(4))\otimes{\mathbb{Q}}=1$$ finishing the proof of the case $k=4$.
The remaining cases $k=5,6,7$ follow from similar arguments as above by using Theorem \[th:main\].
Next we prove Theorem \[th:kab\], Corollary \[cor:ki\], Theorem \[th:fti\] and its refinements. For that, we recall a few facts about the relation between the Torelli group and homology spheres. Let $S^1\times D^2$ denote a framed solid torus and let $H_g=\natural^g (S^1\times D^2)$ (boundary connected sum of $g$-copies of $S^1\times D^2$) denote a handlebody of genus $g$. We identify $\partial H_g$ with $\Sigma_g$ equipped with a system of $g$ meridians and longitudes. Let $\iota_g\in \mathcal{M}_g$ be the mapping class which exchanges each meridian and longitude curves so that the manifold $H_g\cup_{\iota_g} -H_g$ obtained by identifying the boundaries of $H_g$ and $-H_g$ by $\iota_g$ is $S^3$. Now for each element $\varphi\in\mathcal{I}_g$, we consider the manifold $M_\varphi=H_g\cup_{\iota_g\varphi} -H_g$ which is a homology $3$-sphere. It was shown in [@morita89] that any homology sphere can be expressed as $M_\varphi$ for some $\varphi\in\mathcal{K}_g=\mathcal{M}_g(2)$ and Pitsch [@pitsch] further proved that $\varphi$ can be taken in $\mathcal{M}_g(3)$.
Now we recall the relation between the Casson invariant $\lambda$ and the structure of the Torelli group as revealed in [@morita89] briefly (see [@morita91] for further results). We defined $\lambda^*:\mathcal{K}_g\rightarrow{\mathbb{Z}}$ by setting $\lambda^*(\varphi)=\lambda(M_{\varphi})\ (\varphi\in\mathcal{K}_g)$ and then proved that it is a homomorphism. On the other hand, we have the following two abelian quotients of the group $\mathcal{K}_g$ $$\begin{aligned}
\tau_g(2)&: \mathcal{K}_g\rightarrow\mathfrak{h}_g(2)\\
d_1&: \mathcal{K}_g\rightarrow{\mathbb{Z}}\end{aligned}$$ where the first one is the second Johnson homomorphism and the second one is constructed in the above cited paper. Then we have the following.
The homomorphism $\lambda^*:\mathcal{K}_g\rightarrow{\mathbb{Z}}$ is expressed as $$\lambda^*=\frac{1}{24} d_1+\bar{\tau}_g(2)$$ where $\bar{\tau}_g(2)$ denotes a certain quotient of the second Johnson homomorphism. Furthermore, the restriction of $\lambda^*$ to the subgroup $\mathcal{M}_g(3)\subset \mathcal{K}_g$ is given by $$\lambda^*=\frac{1}{24} d_1: \mathcal{M}_g(3)\rightarrow{\mathbb{Z}}.$$ \[th:core\]
Ohtsuki initiated a theory of finite type invariants for homology $3$-spheres in [@ohtsuki] and in [@ohtsukip] he constructed a series of such invariants $\lambda_k\ (k=1,2,\ldots)$ the first one being ($6$ times) the Casson invariant. They are now called the Ohtsuki invariants. Garoufalidis and Levine [@gl] studied the relation between the finite type invariants of homology spheres and the structure of the Torelli group, particularly its lower central series. This work extended the case of the Casson invariant mentioned above extensively.
Now, let $v$ be a rational invariant of homology spheres of finite type $k$. Then we define a mapping $$v^*: \mathcal{I}_g\rightarrow {\mathbb{Q}}$$ by setting $v^*(\varphi)=v(M_\varphi)$. By a result of Garoufalidis and Levine [@gl], it vanishes on $\mathcal{I}_g(k+1)$. On the other hand, the following result is known.
Let $v$ be an invariant of homology spheres of finite type $k$. Then for any $\varphi\in\mathcal{I}_g(k_1), \psi\in\mathcal{I}_g(k_2)$ with $k_1+k_2>k$, the equality $$v(M_{\varphi\psi})=v(M_{\varphi})+v(M_{\psi})$$ holds.
As a direct corollary, we obtain the following.
Let $v$ be a rational invariant of homology spheres of finite type $k$. Then the mapping $$v^*: \mathcal{I}_{g} /\mathcal{I}_{g}(k+1)\rightarrow {\mathbb{Q}}$$ is induced and its restriction to $\mathcal{I}_{g} (m)/\mathcal{I}_{g}(k+1)$ $$v^*: \mathcal{I}_{g}(m)/\mathcal{I}_{g}(k+1)\rightarrow {\mathbb{Q}}$$ is a homomorphism if $2m > k$. \[cor:levine\]
Since the Ohtsuki invariant $\lambda_k$ is of finite type $2k$, it induces a homomorphism $$\lambda^*_{k}: \mathcal{I}_g(k+1)/\mathcal{I}_g(2k+1)\rightarrow{\mathbb{Q}}.$$
If we put $k=1$ here, then we obtain that $$\lambda^*: \mathcal{I}_g(2)/\mathcal{I}_g(3)\rightarrow{\mathbb{Q}}$$ is a homomorphism. However, this follows from a fact already proved in [@morita89] because $\mathcal{I}_g(2)$ is a finite index subgroup of $\mathcal{K}_g$ by Johnson as mentioned above.
In view of Corollary \[cor:levine\], it should be meaningful to consider abelian quotients of the group $\mathcal{I}_g(k)$. Here we recall known abelian quotients of a larger group $\mathcal{M}_g(k) \supset \mathcal{I}_g(k)$. First, we have the $k$-th Johnson homomorphism $$\tau_g(k): \mathcal{M}_g(k)\rightarrow \mathfrak{h}_g(k)$$ and secondly we have its lift $$\tilde{\tau}_g(k):\mathcal{M}_g(k)\rightarrow \bigoplus_{i=k}^{2k-1} \mathfrak{h}_g(i)
\label{eq:ttau}$$ defined as follows. In [@morita93], a homomorphism $$\tilde{\tau}_{g,1}(k): \mathcal{M}_{g,1}(k)\rightarrow H_3(N_k(\pi_1\Sigma_g^0))\quad
(\Sigma_g^0=\Sigma_g\setminus\mathrm{Int}\, D^2)$$ was defined which is a refinement of $\tau_{g,1}(k)$. Heap [@heap] studied this homomorphism by giving a geometric construction of it and, in particular, proved that $\mathrm{Ker}\,\tilde{\tau}_{g,1}(k)=\mathcal{M}_{g,1}(2k)$. Comparing his result with the description of $H_3(N_k (\pi_1 \Sigma_g^0))$ by Igusa-Orr [@igusaorr], we have an embedding $$\tilde{\tau}_{g,1}(k): \mathcal{M}_{g,1}(k)/\mathcal{M}_{g,1}(2k)\hookrightarrow
\bigoplus_{i=k}^{2k-1} \mathfrak{h}_{g,1}(i),$$ though the direct sum decomposition is not canonical except for the lowest part $i=k$. Massuyeau [@massuyeau] (see also Habiro-Massuyeau [@habiromassuyeau]) further studied this homomorphism by an infinitesimal approach. The above homomorphism is obtained from this by passing from $\mathcal{M}_{g,1}, \mathfrak{h}_{g,1}$ to $\mathcal{M}_{g}, \mathfrak{h}_{g}$ along the lines described in [@morita99][@mss4]. Here, if we add the homomorphism $d_1$ and modding out the smaller subgroup $\mathcal{I}_g(2k)\subset \mathcal{M}_{g}(2k)$, then we obtain a homomorphism $$(d_1,\tilde{\tau}_{g}(k)): \mathcal{M}_{g}(k)/\mathcal{I}_{g}(2k)\rightarrow
{\mathbb{Z}}\oplus \bigoplus_{i=k}^{2k-1} \mathfrak{h}_{g}(i)\quad (k\geq 2)
\label{eq:dttau}$$ which is conjecturally an embedding modulo torsion elements (see Remark \[rem:aftercor16\]). As far as the authors understand, this homomorphism $(d_1,\tilde{\tau}_{g}(k))$ gives the known largest free abelian quotient of the group $\mathcal{M}_{g}(k)$ for $k\geq 2$. The case $k=2$ gives an abelian quotient $$(d_1,\tilde{\tau}_g(2)):\mathcal{M}_g(2)=\mathcal{K}_g\rightarrow {\mathbb{Z}}\oplus \mathfrak{h}_{g}(2)\oplus\mathfrak{h}_{g}(3)
={\mathbb{Z}}\oplus [2^2]\oplus [31^2]$$ which is rationally surjective.
Before considering the cases of $k=3,4$, here we prove Theorem \[th:kab\] which is equivalent to the statement that the above homomorphism gives the whole rational abelianization of $\mathcal{K}_g$.
Dimca, Hain and Papadima [@dhp Theorem C] proved that there exists an isomorphim $$H_1(\mathcal{K}_g;{\mathbb{Q}})\cong H_1([\mathrm{Gr}\,\mathfrak{t}_g,\mathrm{Gr}\,\mathfrak{t}_g]).$$ Since the projection to the degree $1$ part $
\mathrm{Gr}\,\mathfrak{t}_g\rightarrow \mathfrak{t}_g(1)=[1^3]
$ gives the abelianization of $\mathrm{Gr}\,\mathfrak{t}_g$, we have $$[\mathrm{Gr}\,\mathfrak{t}_g,\mathrm{Gr}\,\mathfrak{t}_g]=\bigoplus_{k=2}^\infty \mathfrak{t}_g(k).$$ Hence we can write $$H_1([\mathrm{Gr}\,\mathfrak{t}_g,\mathrm{Gr}\,\mathfrak{t}_g])=\bigoplus_{k=2}^\infty
H_1([\mathrm{Gr}\,\mathfrak{t}_g,\mathrm{Gr}\,\mathfrak{t}_g])_k.$$ The results of [@hain] and [@morita99] imply that $$\begin{aligned}
H_1([\mathrm{Gr}\,\mathfrak{t}_g,\mathrm{Gr}\,\mathfrak{t}_g])_2&=\mathfrak{t}_g(2)\cong {\mathbb{Q}}\oplus [2^2]\\
H_1([\mathrm{Gr}\,\mathfrak{t}_g,\mathrm{Gr}\,\mathfrak{t}_g])_3&=\mathfrak{t}_g(3)\cong\mathfrak{m}_g(3)\cong [31^2].\end{aligned}$$ By the definition of the first homology group of Lie algebras, we can write $$H_1([\mathrm{Gr}\,\mathfrak{t}_g,\mathrm{Gr}\,\mathfrak{t}_g])_4
=\mathrm{Coker}\left(\wedge^2 \mathfrak{t}_g(2)\overset{[\ ,\ ]}{\rightarrow} \mathfrak{t}_g(4)\right).$$ On the other hand, it was proved in [@sakasai2] that the homomorphism $$[\ ,\ ]: \wedge^2 \mathfrak{m}_g(2)\rightarrow \mathfrak{m}_g(4)$$ is surjective. Here we use the case $k=4$ of Theorem \[th:main\], namely the fact that $\mathfrak{t}_g(4)\cong \mathfrak{m}_g(4)$. This is the key point of our proof of Theorem \[th:kab\]. Then we conclude that the homomorphism $[\ ,\ ]: \wedge^2 \mathfrak{t}_g(2)\rightarrow \mathfrak{t}_g(4)$ is also surjective. It follows that $$H_1([\mathrm{Gr}\,\mathfrak{t}_g,\mathrm{Gr}\,\mathfrak{t}_g])_4=0.$$ To finish the proof, it remains to prove that $$H_1([\mathrm{Gr}\,\mathfrak{t}_g,\mathrm{Gr}\,\mathfrak{t}_g])_k=0$$ for all $k\geq 5$. In other words, we have to prove that the homomorphism $$\bigoplus_{i+j=k, i, j>1} \mathfrak{t}_g(i)\otimes \mathfrak{t}_g(j)\rightarrow \mathfrak{t}_g(k)
\label{eq:bracket}$$ induced by the bracket operation is surjective. Note that the above homomorphism is surjective for $k=4$ as already mentioned above. We use induction on $k\geq 4$. Assuming that is surjective for $k$, we prove the surjectivity for $k+1$. Since the Torelli Lie algebra is generated by its degree $1$ part $\mathfrak{t}_g(1)$, if we delete the condition $i, j >1$ on the left hand side of , then this map is surjective. Hence it is enough to show that any element of the form $$[\alpha, \xi]\in \mathfrak{t}_g(k+1)\quad (\alpha \in \mathfrak{t}_g(1), \xi\in \mathfrak{t}_g(k))$$ is contained in the image of . By the induction assumption, we can write $$\xi=\sum_{s}\, [\beta_s,\gamma_s]\quad (\beta_s \in \mathfrak{t}_g(k_s),
\gamma_s\in \mathfrak{t}_g(k-k_s), 2\leq k_s\leq k-2).$$ Then, by the Jacobi identity, we have $$\begin{aligned}
[\alpha, \xi]&=\sum_{s}\, [\alpha, [\beta_s,\gamma_s]]\\
&=-\sum_{s}\, \left([\beta_s, [\gamma_s,\alpha]]+[\gamma_s, [\alpha,\beta_s]]\right).\end{aligned}$$ This element is clearly contained in the image of proving that it is surjective for $k+1$. This completes the proof.
Here we mention the relation between our computation and the statement $$\begin{aligned}
&H_1(\mathcal{K}_g;{\mathbb{Q}})\cong {\mathbb{Q}}\oplus \bigoplus_{k=0}^\infty\mathrm{Coker} \,q_k\\
& q_k:\mathrm{Sym}^{k-1} [1^3]\otimes \wedge^3 [1^3] \rightarrow \mathrm{Sym}^{k}[1^3] \otimes [2^2]\end{aligned}$$ given by Dimca, Hain and Papadima [@dhp Theorem B]. The homomorphism $q_k$ is defined by $$q_k(f\otimes (a\wedge b\wedge c))=fa\otimes \pi(b\wedge c)+fb\otimes \pi(c\wedge a)+fc\otimes \pi(a\wedge b)$$ where $f\in \mathrm{Sym}^{k-1} [1^3], a,b,c\in [1^3]$ and $\pi:\wedge^2 [1^3]\cong\wedge^2 \mathfrak{h}_g(1)\rightarrow [2^2]\cong\mathfrak{h}_g(2)$ denotes the bracket operation.
Now the factor ${\mathbb{Q}}$ is detected by $d_1$ and $\mathrm{Coker}\, q_0=[2^2]$ is detected by the second Johnson homomorphism $\tau_g(2)$. These two summands correspond to $\mathfrak{t}_g(2)={\mathbb{Q}}\oplus [2^2]$ determined by Hain [@hain]. The homomorphism $q_1: \wedge^3 [1^3]\rightarrow [1^3]\otimes [2^2]$ appeared already in [@morita99] (Proposition 6.3) and it was proved that $\mathrm{Coker}\, q_1=\mathfrak{t}_g(3)\cong [31^2]$. Our computation $H_1([\mathrm{Gr}\,\mathfrak{t}_g,\mathrm{Gr}\,\mathfrak{t}_g])_4=0$ corresponds to the fact that the homomorphism $q_2$ is surjective, namely $\mathrm{Coker}\,q_2=0$. Then, by the definition of the homomorphisms $q_k$ mentioned above, it is easy to see that they are surjective for all $k\geq 3$ as well.
$\mathrm{(i)}\ $ The result of Heap mentioned above implies that $\mathrm{Ker}\,\tilde{\tau}_g(2)=\mathcal{M}_g(4)$. On the other hand, the case $k=4$ of Corollary \[cor:fi\] shows that $\mathcal{I}_g(4)$ is a finite index subgroup of $$\mathrm{Ker}(d_1: \mathcal{M}_g(4)\rightarrow {\mathbb{Q}})=\mathrm{Ker}\, (d_1,\tilde{\tau}_2).$$ Now Theorem \[th:kab\] implies that $$\mathrm{Ker}\, (d_1,\tilde{\tau}_2)/[\mathcal{K}_g,\mathcal{K}_g]\cong
\mathrm{Torsion} (H_1(\mathcal{K}_g;{\mathbb{Z}})).$$ According to Ershov-He [@eh] and Church-Ershov-Putman [@cep], $\mathcal{K}_g$ is finitely generated. It follows that the torsion subgroup $\mathrm{Torsion} (H_1(\mathcal{K}_g;{\mathbb{Z}}))$ of $H_1(\mathcal{K}_g;{\mathbb{Z}})$ is a finite group and hence $[\mathcal{K}_g,\mathcal{K}_g]$ is a finite index subgroup of $\mathrm{Ker}\, (d_1,\tilde{\tau}_2)$. Thus both the groups $\mathcal{I}_g(4)$ and $[\mathcal{K}_g,\mathcal{K}_g]$ are finite index subgroups of the same group $\mathrm{Ker}\, (d_1,\tilde{\tau}_2)$. Therefore they are commensurable.
$\mathrm{(ii)}\ $ According to [@cep], $\mathcal{I}_g(4)$ is finitely generated for $g\geq 7$. On the other hand, $\mathcal{I}_g(4)$ is a finite index subgroup of $\mathrm{Ker}\, (d_1,\tilde{\tau}_2)$ as shown in $\mathrm{(i)}\ $. It follows that $\mathrm{Ker}\, (d_1,\tilde{\tau}_2)$ is finitely generated. Since $[\mathcal{K}_g,\mathcal{K}_g]$ is a finite index subgroup of $\mathrm{Ker}\, (d_1,\tilde{\tau}_2)$ as above, we conclude that it is also finitely generated.
This finishes the proof.
For a given $k\geq 3$, determine whether the rationally surjective homomorphisms $$\begin{aligned}
(d_1,\tilde{\tau}_{g}(k))&: \mathcal{M}_{g}(k)\rightarrow
{\mathbb{Q}}\oplus\bigoplus_{i=k}^{2k-1} \mathfrak{m}_{g}(i)\\
\tilde{\tau}_{g}(k)&: \mathcal{I}_{g}(k)\rightarrow
\bigoplus_{i=k}^{2k-1} \mathfrak{m}_{g}(i)\end{aligned}$$ give the whole of $H_1(\mathcal{M}_{g}(k);{\mathbb{Q}})$ and $H_1(\mathcal{I}_{g}(k);{\mathbb{Q}})$ or not.
In view of the result of [@cep] that $\mathcal{M}_{g}(k)$ and $\mathcal{I}_{g}(k)$ are all finitely generated for $g\geq 2k-1$, a positive solution to the above problem would imply that the subgroups $[\mathcal{M}_{g}(k),\mathcal{M}_{g}(k)], [\mathcal{I}_{g}(k),\mathcal{I}_{g}(k)],
\mathcal{I}_g(2k)$ of the Torelli group $\mathcal{I}_g$ are commensurable for $g\geq 4k-1$. It would then follow that the groups $[\mathcal{M}_{g}(k),\mathcal{M}_{g}(k)], [\mathcal{I}_{g}(k),\mathcal{I}_{g}(k)]$ are finitely generated in the same range.
Now we go back to the homomorphism for the cases $k=3,4$.
There exist isomorphisms $$(d_1,p\circ \tilde{\tau}_g(3)): (\mathcal{M}_g(3)/\mathcal{I}_g(5))\otimes{\mathbb{Q}}\cong
{\mathbb{Q}}\oplus\mathfrak{m}_g(3)\oplus \mathfrak{m}_g(4),$$ $$(d_1,q\circ \tilde{\tau}_g(4)):(\mathcal{M}_g(4)/\mathcal{I}_g(7))\otimes{\mathbb{Q}}\cong
{\mathbb{Q}}\oplus\mathfrak{m}_g(4)\oplus \mathfrak{m}_g(5)\oplus \mathfrak{m}_g(6)$$ where $p, q$ are the projections $$p:\mathfrak{m}_g(3)\oplus \mathfrak{m}_g(4)\oplus \mathfrak{m}_g(5)\rightarrow
\mathfrak{m}_g(3)\oplus \mathfrak{m}_g(4),$$ $$q:\mathfrak{m}_g(4)\oplus \mathfrak{m}_g(5)\oplus \mathfrak{m}_g(6)\oplus \mathfrak{m}_g(7)\rightarrow
\mathfrak{m}_g(4)\oplus \mathfrak{m}_g(5)\oplus \mathfrak{m}_g(6).$$ \[prop:tildet\]
This follows by combining the facts that $\mathcal{I}_g(5), \mathcal{I}_g(7)$ are finite index subgroups of the kernel of the homomorphism $d_1$ on $\mathcal{M}_g(5), \mathcal{M}_g(7)$, respectively (see Corollary \[cor:fi\]) and the above homomorphism .
By using Corollary \[cor:levine\] and Proposition \[prop:tildet\], we obtain the following.
Let $v$ be a rational invariant of homology spheres of finite type $4$, including the second Ohtsuki invariant $\lambda_2$, and let $v^*:\mathcal{I}_g\rightarrow{\mathbb{Q}}$ be the associated mapping. Then we have the following.
$\mathrm{(i)}\, $ The restriction $v^*:\mathcal{I}_g(3)\rightarrow{\mathbb{Q}}$, which is a homomorphism, is a quotient of the homomorphism $$p \circ\tilde{\tau}_g(3):\mathcal{I}_g(3)\rightarrow \bigoplus_{i=3}^{5} \mathfrak{m}_g(i)
\overset{p}{\rightarrow} \mathfrak{m}_g(3)\oplus \mathfrak{m}_g(4)$$ where $p$ denotes the natural projection.
$\mathrm{(ii)}\, $ The restriction map $v^*:\mathcal{M}_g(3)\rightarrow{\mathbb{Q}}$ factors through the homomorphism $$(d_1,p\circ \tilde{\tau}_g(3)):\mathcal{M}_g(3)\rightarrow {\mathbb{Q}}\oplus \mathfrak{m}_g(3)\oplus \mathfrak{m}_g(4).$$
$\mathrm{(iii)}\, $ The restriction map $v^*:\mathcal{M}_g(5)\rightarrow{\mathbb{Q}}$ factors through the homomorphism $$d_1:\mathcal{M}_g(5)\rightarrow {\mathbb{Q}}.$$ \[th:type4\]
Note that the restriction maps $v^*:\mathcal{M}_g(3)\rightarrow{\mathbb{Q}}$ and $v^*:\mathcal{M}_g(5)\rightarrow{\mathbb{Q}}$ might not be homomorphisms.
Let $v$ be a rational invariant of homology spheres of finite type $6$, including the third Ohtsuki invariant $\lambda_3$, and let $v^*:\mathcal{I}_g\rightarrow{\mathbb{Q}}$ be the associated mapping. Then we have the following.
$\mathrm{(i)}\, $ The restriction $v^*:\mathcal{I}_g(4)\rightarrow{\mathbb{Q}}$, which is a homomorphism, is a quotient of the homomorphism $$q \circ\tilde{\tau}_g(4):\mathcal{I}_g(4)\rightarrow \bigoplus_{i=4}^{7} \mathfrak{m}_g(i)
\overset{q}{\rightarrow} \mathfrak{m}_g(4)\oplus \mathfrak{m}_g(5)\oplus \mathfrak{m}_g(6)$$ where $q$ denotes the natural projection.
$\mathrm{(ii)}\, $ The restriction map $v^*:\mathcal{M}_g(4)\rightarrow{\mathbb{Q}}$ factors through the homomorphism $$(d_1,q\circ \tilde{\tau}_g(4)):\mathcal{M}_g(4)\rightarrow {\mathbb{Q}}\oplus \mathfrak{m}_g(4)\oplus \mathfrak{m}_g(5)
\oplus \mathfrak{m}_g(6).$$
$\mathrm{(iii)}\, $ The restriction map $v^*:\mathcal{M}_g(7)\rightarrow{\mathbb{Q}}$ factors through the homomorphism $$d_1:\mathcal{M}_g(7)\rightarrow {\mathbb{Q}}.$$ \[th:type6\]
Note also that the restriction maps $v^*:\mathcal{M}_g(4)\rightarrow{\mathbb{Q}}$ and $v^*:\mathcal{M}_g(7)\rightarrow{\mathbb{Q}}$ might not be homomorphisms.
Determine the precise formulae for the expressions of $\lambda^*_2, \lambda^*_3$ in terms of $d_1$ on the subgroups $\mathcal{M}_g(5), \mathcal{M}_g(7)$, respectively. Recall here that $\lambda^*=\frac{1}{24} d_1$ on $\mathcal{M}_g(3)$ (Theorem \[th:core\]) and based on this, we call $d_1$ the core of the Casson invariant. It seems reasonable to imagine that $\lambda^*_2, \lambda^*_3$ are constant times $d^2_1, d^3_1$, respectively, including the cases where the constant vanishes.
[**Acknowledgements**]{} The authors would like to express their hearty thanks to\
Richard Hain, Gwénaël Massuyeau, Masatoshi Sato and Shunsuke Tsuji for enlightening discussions. Also, they would like to thank Joan Birman, Robion Kirby and Sheila Newbery for their great assistance and help on the notes [@jnote]. Finally, they appreciate the referee’s careful reading of the previous version and helpful comments.
[30]{}
M. Asada, M. Kaneko, *On the automorphism group of some pro-$l$ fundamental groups*, Adv. Studies in Pure Math. 12 (1987), 137–159.
J. Birman, R. Craggs, *The $\mu$-invariant of $3$-manifolds and certain structural properties of the group of homeomorphisms of a closed oriented $2$-manifold*, Trans. Amer. Math. Soc. 237 (1978), 283–309.
T. Church, M. Ershov, A. Putman, *On finite generation of the Johnson filtrations*, preprint, arXiv:1711.04779 \[math.GR\].
A. Dimca, R. Hain, S. Papadima, *The abelianization of the Johnson kernel*, J. Eur. Math. Soc. 16 (2014), 805–822.
A. Dimca, S. Papadima, *Arithmetic group symmetry and finiteness properties of Torelli groups*, Ann. Math. 177 (2013), 395–423.
M. Ershov, S. He, *On finiteness properties of the Johnson filtrations*, preprint, arXiv:1703.04190 \[math.GR\].
S. Garoufalidis, J. Levine, *Finite type $3$-manifold invariants and the structure of the Torelli group. I*, Invent. Math. 131 (1998), 541–594.
K. Habiro, G. Massuyeau, *From mapping class groups to monoids of homology cobordisms: a survey*, Handbook of Teichmüller theory volume III, edited by A. Papadopoulos, 465–529, IRMA Lect. Math. Theor. Phys., 17, Eur. Math. Soc., Zürich, 2012.
R. Hain, *Infinitesimal presentations of the Torelli groups*, J. Amer. Math. Soc. 10 (1997), 597–651.
A. Heap, *Bordism invariants of the mapping class group*, Topology 45 (2006), 851–866.
K. Igusa, K. Orr, *Links, pictures and the homology of nilpotent groups*, Topology 40 (2001), 1125–1166.
D. Johnson, *An abelian quotient of the mapping class group $\mathcal{I}_g$*, Math. Ann. 249 (1980), 225–242.
D. Johnson, *A survey of the Torelli group*, in: Low-dimensional topology (San Francisco, Calif., 1981) , Contemp. Math. [**20**]{}, 165–179. Amer. Math. Soc., Providence, RI, 1983.
D. Johnson, *The structure of the Torelli group I: A finite set of generators for $\mathcal{I}_g$*, Ann. Math. 118 (1983), 423–442.
D. Johnson, *The structure of the Torelli group II: A characterization of the group generated by twists on bounding simple closed curves*, Topology 24 (1985), 113–128.
D. Johnson, *The structure of the Torelli group III: The abelianization of $\mathcal{I}_g$*, Topology 24 (1985), 127–144.
D. Johnson, *About the space of Casson homomorphisms for a surface $K_g$*, Celebratio Mathematica, the volume of Joan S. Birman, Correspondence with Dennis Johnson \[7\].\
http://celebratio.org/cmmedia/docs/BirmanCorrespondence.Johnson.170726[\_]{}10.pdf
J. Levine, *The Lagrangian filtration of the mapping class group and finite-type invariants of homology spheres*, Math. Proc. Camb. Phil. Soc. 141 (2006), 303–315.
G. Massuyeau, *Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant*, Bull. Soc. Math. France 140 (2012), 101–161.
S. Morita, *Casson’s invariant for homology $3$-spheres and characteristic classes of surface bundles*, Proc. Japan Acad. 63 (1987), 229–232.
S. Morita, *Casson’s invariant for homology $3$-spheres and characteristic classes of surface bundles I*, Topology 28 (1989), 305–323.
S. Morita, *On the structure of the Torelli group and the Casson invariant*, Topology 30 (1991), 603–621.
S. Morita, *Abelian quotients of subgroups of the mapping class group of surfaces*, Duke Math. J. 70 (1993), 699–726.
S. Morita, *Structure of the mapping class groups of surfaces: a survey and a prospect*, Geometry and Topology monographs 2, *Proceedings of the Kirbyfest* (1999), 349–406.
S. Morita, T. Sakasai, M. Suzuki, *Computations in formal symplectic geometry and characteristic classes of moduli spaces*, Quantum Topology 6 (2015), 139–182.
S. Morita, T. Sakasai, M. Suzuki, *Structure of symplectic invariant Lie subalgebras of symplectic derivation Lie algebras*, Advances in Mathematics 282 (2015), 291–334.
S. Morita, T. Sakasai, M. Suzuki, https://www.fms.meiji.ac.jp/macky/torelliinvariant.html
T. Ohtsuki, *A polynomial invariant of rational homology $3$-spheres*, Invent. Math. 123 (1996), 241–257.
T. Ohtsuki, *Finite type invariants of integral homology $3$-spheres*, J. Knot Theory and its Rami. 5 (1996), 101–115.
W. Pitsch, *Integral homology $3$-spheres and the Johnson filtration*, Trans. Amer. Math. Soc. 360 (2008), 2825–2847. T. Sakasai, *The Johnson homomorphism and the third rational cohomology group of the Torelli group*, Topol. Appl. 148 (2005), 83–111.
T. Sakasai, *The second Johnson homomorphism and the second rational cohomology of the Johnson kernel*, Math. Proc. Camb. Phil. Soc. 143 (2007), 627–648.
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abstract: 'We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters [[$\alpha_-$,$\alpha_+$,$\beta_-$,$\beta_+$]{}]{} are nonzero. A generalization of the Baxter $T-Q$ equation that involves more than one independent $Q$ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.'
title: 'Bethe Ansatz and boundary energy of the open spin-$1/2$ XXZ chain'
---
Introduction
============
While the open spin-$1/2$ XXZ quantum spin chain (with diagonal boundary terms) has been solved and well understood [@Ga; @ABBBQ; @Sk], the solution for the corresponding XXZ chain with general integrable boundary terms, has remained unsolved. The Hamiltonian for this model is given by [@dVGR; @GZ] & = & \_[n=1]{}\^[N-1]{}[1 2]{} (\_n\^x\_[n+1]{}\^x+\_n\^y\_[n+1]{}\^y+\_n\^z\_[n+1]{}\^z)\
& & +[12]{} where $\sigma^x$,$\sigma^y$,$\sigma^z$ are Pauli matrices, $\eta$ is the bulk anisotropy parameter, $\alpha_{\pm}$,$\beta_{\pm}$,$\theta_{\pm}$ are the boundary parameters, and $N$ is the number of spins. However, the case of nondiagonal boundary terms with the boundary parameters satisfying certain constraint has been solved recently [@Ne2; @CLSW]. Hence, it would be desirable to find the solution for the general case, with such constraint removed.\
The outline of this paper is as follows. In section 2, we review our Bethe Ansatz solutions for special case at roots of unity [@MN2] which we utilize to compute the boundary (surface) energy of the XXZ chain [@MNS]. Next, in section 3, we present the Bethe Ansatz solution for the general case [@MNS2]. This is followed by a brief conclusion of the paper together with some outline of possible future works on the subject in section 4.
Special case
============
Here, the results are presented for odd values of $p$ [^1], where $p$ is related to $\eta$ through $\eta={i\pi\over p+1}$.
Bethe Ansatz
------------
We consider the case with the following choice of boundary parameters; $\beta_{\pm} = 0$,$\theta_{-}=\theta_{+}$,$\alpha_{\pm}$ arbitrary. One crucial step here is to notice that certain functional relation which the transfer matrix, $t(u)$ and its eigenvalues, $\Lambda(u)$ obey at roots of unity [@NPB], can be written as [@BR] (u) = 0 , \[detzero\] We give an example of the functional relation below, for $p=3$ & &(u) (u+) (u+2) (u+3) - (u) (u+2) (u+3) - (u+) (u) (u+3)\
& &-(u+2) (u) (u+) - (u+3) (u+) (u+2)\
& &+(u) (u+2) + (u+) (u+3) = f(u) . $\delta(u)$ and $f(u)$ are known scalar functions in terms of boundary parameters [@MN1; @MN2]. The matrix ${\mathcal M}(u)$ is given by [@MN2], (u) = (
[cccccccc]{} (u) & -[(u)h\^[(1)]{}(u)]{} & 0 & …& 0 & -[(u-)h\^[(2)]{}(u-)]{}\
-h\^[(1)]{}(u) & (u+) & -h\^[(2)]{}(u+) & …& 0 & 0\
& & & & &\
-h\^[(2)]{}(u-) & 0 & 0 & …& -h\^[(1)]{}(u+(p-1)) & (u+p)
) , where $h^{(1)}(u)$ and $h^{(2)}(u)$ are functions which are $i\pi$-periodic. Comparing (2) to the functional relation for the eigenvalues, one can solve for $h^{(1)}(u)$. Also, $h^{(2)}(u) = h^{(1)}(-u-2\eta)$, as one would conclude from the crossing properties of $\Lambda(u)$ and (8) below. The matrix above has the following symmetry, $T {\cal M}(u) T^{-1} = {\cal M}(u+2\eta)$ and $T \equiv S^{2}$. Other ${\mathcal M}(u)$ matrices we found with stronger symmetry yield inconsistent results. Details on this argument can be found in [@MN2]. Here $S$ is S = (
[cccccccc]{} 0 & 1 & 0 & …& 0 & 0\
0 & 0 & 1 & …& 0 & 0\
& & & & &\
0 & 0 & 0 & …& 0 & 1\
1 & 0 & 0 & …& 0 & 0
) , S\^[p+1]{} = I. \[Smatrix\] Hence, we have h\^[(1)]{}(u) &=& [8\^[2N+1]{}(u+2)\^[2]{}(u+) (u+2) (2u+3)]{} , The above symmetry for the present ${\cal M}(u)$ suggests a null eigenvector, $v(u)= \left( Q_{1}(u)\,, Q_{2}(u+\eta) \,, \ldots \,, Q_{1}(u-2\eta)
\,, Q_{2}(u-\eta)\right)$ with the following ansatz for $Q_{a}(u)$ Q\_[a]{}(u) = \_[j=1]{}\^[M\_[a]{}]{} (u - u\_[j]{}\^[(a)]{}) (u + u\_[j]{}\^[(a)]{} + ) , a = 1, 2, Note that there are two $Q(u)$ functions, a direct consequence of the weaker symmetry mentioned above. Thus, we have the following $T-Q$ relations (u) &=& [(u)h\^[(1)]{}(u)]{} [Q\_[2]{}(u+)Q\_[1]{}(u)]{} + [(u-)h\^[(2)]{}(u-)]{} [Q\_[2]{}(u-) Q\_[1]{}(u)]{} , \[TQ1\]\
&=& h\^[(1)]{}(u-) [Q\_[1]{}(u-)Q\_[2]{}(u)]{} + h\^[(2)]{}(u) [Q\_[1]{}(u+) Q\_[2]{}(u)]{} . with $M_{1} = {1\over 2}(N+p+1)$ and $M_{2} = {1\over 2}(N+p-1) \,.$
We see that (8) is a coupled equation in terms of $Q_{1}(u)$ and $Q_{2}(u)$, hence exhibiting a generalized structure of $T-Q$ relation. Bethe Ansatz follows directly by demanding analyticity for the $\Lambda(u)$. &=&-[Q\_[2]{}(u\_[j]{}\^[(1)]{}-)Q\_[2]{}(u\_[j]{}\^[(1)]{}+)]{} , j = 1, 2, …, M\_[1]{} ,\
[h\^[(1)]{}(u\_[j]{}\^[(2)]{}-)h\^[(2)]{}(u\_[j]{}\^[(2)]{})]{} &=&-[Q\_[1]{}(u\_[j]{}\^[(2)]{}+)Q\_[1]{}(u\_[j]{}\^[(2)]{}-)]{} , j = 1, 2, …, M\_[2]{} .
Boundary energy
---------------
The energy for the chain of finite length is given by E=[12]{} \^[2]{} \_[a=1]{}\^[2]{}\_[j=1]{}\^[M\_[a]{}]{}[1 (u\_[j]{}\^[(a)]{} - [2]{}) (u\_[j]{}\^[(a)]{} + [2]{})]{} + [12]{}(N-1) where $\tilde u_{j}^{(a)} \equiv u_{j}^{(a)} + {\eta\over2}$, We make the string hypothesis that, for suitable values of boundary parameters [^2], the ground state roots,$\{ \tilde u_{j}^{(1)} \}$ and $\{ \tilde
u_{j}^{(2)} \}$ have the following form as $N\rightarrow\infty$. {
[c@[: ]{} l]{} v\_[j]{}\^[(1,1)]{} & j = 1, 2, …, [N2]{}\
v\_[j]{}\^[(1,2)]{} + [i 2]{} , & j = 1, 2, …, [p+12]{}
. , {
[c@[: ]{} l]{} v\_[j]{}\^[(2,1)]{} & j = 1, 2, …, [N2]{}\
v\_[j]{}\^[(2,2)]{} + [i 2]{} , & j = 1, 2, …, [p-12]{}
. ,
$\{ v_{j}^{(a,b)} \}$ are all real and positive. The logarithm of the Bethe equations for both sets of sea roots, $\{v_{j}^{(1,1)} \}$ and $\{
v_{j}^{(2,1)} \}$ gives the ground state root densities, $\rho^{(1)}(\lambda)$ and $\rho^{(2)}(\lambda)$ with $v_{j}^{(a,b)} = \mu \lambda_{j}^{(a,b)}
$. The energy depends only on the sum of root densities computed from the counting functions. Further, using (10) (where $\sum\ldots\rightarrow $N$\int(\rho^{(1)}(\lambda)+\rho^{(2)}(\lambda))d\lambda\ldots$ in the thermodynamic limit, $N\rightarrow\infty$) and keeping term of order 1, we have the following E\_[boundary]{}\^&=& - [2]{} \_[-]{}\^ d [12(/ 2)]{} { [((-2)/4) 2(/4)]{} -[12]{}\
&+& [(/2) ((-2|a\_|)/2) (/2)]{} } -[14]{}. where $\alpha_{\pm}=i\mu a_{\pm}$ and $\mu=-i\eta$. + and - refer to right and left boundary respectively.
General case ($p>1$)
====================
Finally, we present the solution for the case of general nondiagonal boundary terms. We first present the matrix, ${\cal M}(u)$, for this case (u) = (
[ccccccccc]{} (u) & -m\_[1]{}(u) & 0 & …& 0 & 0 & -n\_[p+1]{}(u)\
-n\_[1]{}(u) & (u+) & -m\_[2]{}(u) & …& 0 & 0 & 0\
& & & & & &\
0 & 0 & 0 & …& -n\_[p-1]{}(u) & (u+(p-1) ) & -m\_[p]{}(u)\
-m\_[p+1]{}(u) & 0 & 0 & …& 0 & -n\_[p]{}(u) & (u+p )
) where the matrix elements,$\{ m_{j}(u) \,, n_{j}(u) \}$ are given by [@MNS2] m\_[j]{}(u) &=& h(-u-j ) , n\_[j]{}(u) = h(u+j ) , j=1, 2, …, p ,\
m\_[p+1]{}(u) &=& [z\^[-]{}(u) \_[k=1]{}\^[p]{}h(-u-k)]{} , n\_[p+1]{}(u) = [z\^[+]{}(u) \_[k=1]{}\^[p]{}h(u+k)]{}, where h(u) = -4\^[2N]{}(u+)[(2u+2)(2u+)]{}\
(u+\_[-]{}) (u+\_[-]{})(u+\_[+]{}) (u+\_[+]{}) and z\^(u)=[12]{}(f(u) g(u) Y(u) ) Explicit expressions for $g(u)$ and $Y(u)$ (a non-analytic function) and their properties are given in [@MNS2]. The null eigenvector is $v(u) = (v_{1}(u) \,, v_{2}(u)\,, \ldots \,, v_{p+1}(u))$[^3]. Periodicity of ${\cal M}(u)$ makes it reasonable to assume the same $i\pi$ periodicity for $v(u)$. Utilizing ${\cal M}(u)v(u)=0$ together with the following ansatz for $v_{j}(u)$ [^4], v\_[j]{}(u) = a\_[j]{}(u) + b\_[j]{}(u) Y(u) , j = 1, 2 , … , +1 , where a\_[j]{}(u) &=& A\_[j]{} \_[k=1]{}\^[2M\_[a]{}]{} (u-u\_[k]{}\^[(a\_[j]{})]{}) , b\_[j]{}(u) = B\_[j]{}\_[k=1]{}\^[2M\_[b]{}]{} (u-u\_[k]{}\^[(b\_[j]{})]{}), j +1 ,\
a\_[[p2]{}+1]{}(u) &=& A\_[[p2]{}+1]{} \_[k=1]{}\^[M\_[a]{}]{} (u-u\_[k]{}\^[(a\_[[p2]{}+1]{})]{}) (u+u\_[k]{}\^[(a\_[[p2]{}+1]{})]{}) ,\
b\_[[p2]{}+1]{}(u) &=& B\_[[p2]{}+1]{} \_[k=1]{}\^[M\_[b]{}]{} (u-u\_[k]{}\^[(b\_[[p2]{}+1]{})]{}) (u+u\_[k]{}\^[(b\_[[p2]{}+1]{})]{}) , and equating analytic and non-analytic terms separately, one would derive a set of generalized $T-Q$ equations [@MNS2]. Here $M_{a} = \lfloor {N-1\over 2} \rfloor + 2p+1$ and $M_{b} = \lfloor {N-1\over 2} \rfloor + p$.
Using similar arguments as in section 2.1, and invoking analyticity of $\Lambda(u)$, we get the following Bethe-Ansatz like equations for the zeros $\{ u_{l}^{(a_{j})} \}$ and $\{ u_{l}^{(b_{j})} \}$ of the functions $\{ a_{j}(u)\}$ and $\{ b_{j}(u)\}$ respectively, h(-u\_[l]{}\^[(a\_[1]{})]{}-) &=&-[f(u\_[l]{}\^[(a\_[1]{})]{}) a\_[1]{}(-u\_[l]{}\^[(a\_[1]{})]{}) + g(u\_[l]{}\^[(a\_[1]{})]{}) Y(u\_[l]{}\^[(a\_[1]{})]{})\^[2]{} b\_[1]{}(-u\_[l]{}\^[(a\_[1]{})]{})2a\_[2]{}(u\_[l]{}\^[(a\_[1]{})]{})\_[k=1]{}\^[p]{}h(u\_[l]{}\^[(a\_[1]{})]{}+k)]{} ,\
[h(-u\_[l]{}\^[(a\_[j]{})]{}-j) h(u\_[l]{}\^[(a\_[j]{})]{}+(j-1))]{}&=&-[a\_[j-1]{}(u\_[l]{}\^[(a\_[j]{})]{})a\_[j+1]{}(u\_[l]{}\^[(a\_[j]{})]{})]{} , j = 2 , …, +1 , and h(-u\_[l]{}\^[(b\_[1]{})]{}-) &=&-[f(u\_[l]{}\^[(b\_[1]{})]{}) b\_[1]{}(-u\_[l]{}\^[(b\_[1]{})]{}) + g(u\_[l]{}\^[(b\_[1]{})]{}) a\_[1]{}(-u\_[l]{}\^[(b\_[1]{})]{})2b\_[2]{}(u\_[l]{}\^[(b\_[1]{})]{}) \_[k=1]{}\^[p]{}h(u\_[l]{}\^[(b\_[1]{})]{}+k)]{} ,\
[h(-u\_[l]{}\^[(b\_[j]{})]{}-j) h(u\_[l]{}\^[(b\_[j]{})]{}+(j-1))]{}&=&-[b\_[j-1]{}(u\_[l]{}\^[(b\_[j]{})]{})b\_[j+1]{}(u\_[l]{}\^[(b\_[j]{})]{})]{} , j = 2 , …, +1 . Normalization contants $\{ A_{j}, B_{j} \}$,$j=1 \,, \ldots \,,\lfloor{p\over 2}\rfloor+1 \,,$ can be determined by noting the poles at $u = -{\eta\over 2}$ and $u = -\alpha_{-}-\eta$, and from the analyticity of $\Lambda(u)$. This yields few extra Bethe-Ansatz like equations that can be solved for these normalization constants.
Conclusion
==========
We have presented Bethe Ansatz solutions for both special and general cases. The solutions presented here have been verified for completeness numerically. However, these solutions do not hold for generic values of $\eta$, but only for special values, ${i \pi\over p+1}$. Further, we have demonstrated the use of string hypothesis to compute the ground state boundary energy for a special case. The Bethe Ansatz equations appear in a generalized form due to the appearance of multiple $Q(u)$ (or $a(u)$ and $b(u)$). Few questions needed to be answered here. Firstly, having found the general solutions, can they lead to some other interesting results, e.g. finite size effects? Secondly, do solutions exist for generic values of $\eta$? These are certainly questions of utmost importance that we hope to pursue and address in future.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank R. I. Nepomechie for his invaluable help and advice during the course of completing this work. I also thank the Department of Physics and Graduate School (Arts and Sciences), University of Miami for their financial support, and C. Shi for his useful suggestions and comments. This work was supported in part by the NSF under Grant PHY-0244261.
[99]{}
M. Gaudin, [*Phys. Rev.*]{} [**A4**]{} (1971) 386; [*La fonction d’onde de Bethe*]{} (Masson, 1983).
F.C. Alcaraz, M.N. Barber, M.T. Batchelor, R.J. Baxter and G.R.W. Quispel, [*J. Phys.*]{} [**A20**]{} (1987) 6397.
E.K. Sklyanin, [*J. Phys.*]{} [**A21**]{} (1988) 2375.
H.J. de Vega and A. González-Ruiz, [*J. Phys.*]{} [**A26**]{} (1993) L519. \[[hep-th/9211114]{}\]
S. Ghoshal and A. B. Zamolodchikov, [*Int. J. Mod. Phys.*]{} [**A9**]{} (1994) 3841. \[[hep-th/9306002]{}\]
R.I. Nepomechie, Nucl. Phys. B [**622**]{} (2002) 615; Addendum, Nucl. Phys. B [**631**]{} (2002) 519.
R.I. Nepomechie, [*J. Stat. Phys.*]{} [**111**]{} (2003) 1363. \[[hep-th/0211001]{}\]; R.I. Nepomechie, [*J. Phys.*]{} [**A37**]{} (2004) 433. \[[hep-th/0304092]{}\]
J. Cao, H.-Q. Lin, K.-J. Shi and Y. Wang, \[[cond-mat/0212163]{}\]; [*Nucl. Phys.*]{} [**B663**]{} (2003) 487.
R. Murgan and R.I. Nepomechie, [*J. Stat. Mech.*]{} [**P05007**]{} (2005); Addendum, [*J. Stat. Mech.*]{} [**P11004**]{} (2005) \[[hep-th/0504124]{}\]
R. Murgan and R.I. Nepomechie, [*J. Stat. Mech.*]{} [**P08002**]{} (2005) \[[hep-th/0507139]{}\]
V.V. Bazhanov and N.Yu. Reshetikhin, [*Int. J. Mod. Phys.*]{} [**A4**]{} (1989) 115.
R. Murgan, R.I. Nepomechie and C. Shi, Ann. Henri Poincaré, in press \[[hep-th/0512058]{}\]
R. Murgan, R.I. Nepomechie and C. Shi, \[[hep-th/0605223]{}\]
[^1]: for even $p$ values,we refer readers to [@MN1]
[^2]: Readers are urged to refer to [@MNS] for a detail discussion on this matter
[^3]: $v_{j+p+1} = v_{j}$
[^4]: with following crossing property, $v_{j}(-u) = v_{p+2-j}(u)$ , j = 1, 2, …, p+1 .
|
---
abstract: 'The high energy radiation emitted by black hole X-ray binaries originates in an accretion disk, hence the variability of the lightcurves mirrors the dynamics of the disc. We study the time evolution of the emitted flux in order to find evidences, that low dimensional non-linear equations govern the accretion flow. Here we test the capabilities of our novel method to find chaotic behaviour on the two numerical time series describing the motion of a test particle around a black hole surrounded by a thin massive disc, one being regular and the other one chaotic.'
author:
- 'P. Suková$^1$ and A. Janiuk$^1$'
title: Numerical test of the method for revealing traces of deterministic chaos in the accreting black holes
---
1.0cm
Introduction
============
In this paper we test our method for revealing the traces of non-linear dynamics in the observed X-ray lightcurves on two numerical trajectories, from which one is regular and the other one is chaotic. We developed and described in details the method in the paper [@velkyChaos]. We also refer the reader to the article by A. Janiuk et al in these proceedings to learn more about astrophysical applications of this method. Here we only briefly summarize its key features.
We compute the estimate of Rényi’s entropy $K_2$ [@Grassberger1983227] using the recurrence analysis[^1] for the time series. This value is compared with the values obtained for $N^{\rm surr} = 100$ surrogates made in such a way, that they share the value distribution and power spectra with the original series (IAAFT surrogates). We use the software package TISEAN [@1999chao.dyn.10005H; @Schreiber2000346]. The significance of the non-linearity is given by $$\mathcal{S}(\epsilon) = \frac{N_{\rm sl}(\epsilon)}{N^{\rm surr}} \mathcal{S}_{\rm sl} - {\rm sign}( Q^{\rm obs} (\epsilon) - \bar{Q}^{\rm surr}(\epsilon) ) \frac{N^{\rm surr} - N_{\mathcal{S}_{\rm sl}} (\epsilon)}{N^{\rm surr}} \mathcal{S}_{K_2}(\epsilon) , \label{significance}$$ for chosen recurrence threshold $\epsilon$, where $N_{\rm sl}$ is the number of surrogates, which have only short diagonal lines in their recurrence matrix, $Q^{\rm obs}$ and ${Q}^{\rm surr}_i$ are the natural logarithms of $K_2$ for the observed and surrogate data, respectively, $\bar{Q}^{\rm surr}$ is the averaged value of the set ${Q}^{\rm surr}_i$, $\mathcal{S}_{\rm sl} =3$ and $\mathcal{S}_{K_2}$ is the significance computed from surrogates with enough long lines according to the relation $$\mathcal{S}_{K_2} (\epsilon) = \frac{| Q^{\rm obs} (\epsilon) - \bar{Q}^{\rm surr}(\epsilon) |}{\sigma_{Q^{\rm surr}(\epsilon)}}.$$
We quantize the results with $\bar{\mathcal{S}}_{K_2}$ – the average of ${\mathcal{S}}_{K_2}$ over a range of $\epsilon$.
Testing the method with simulated time series {#sect:poincare}
=============================================
{width="\textwidth"}
In general our method can be applied to different kinds of time series, which are produced by some dynamical system. Here we test the method applying it on time series, whose nature is known. We choose the numerical time series, which describe the motion of geodesic test particle in the field of a static black hole surrounded by a massive thin disc. The background metric is given by an exact solution of Einstein equations and is described in details in [@semerak2010free]. The time series are obtained as the numerical solution to the geodesic equation with this metric using the 6th order Runge-Kutta method As the input data we use the time dependence of the particle’s $z$-coordinate.
We study two numerical trajectories, orbit $o_1$ being regular and orbit $o_2$ chaotic, whose Poincaré surface of section[^2] and time dependence of $z$ coordinate is depicted in Fig. \[fig:Poincare\]. The two selected trajectories belong to the regular island ($o_1$) and chaotic sea ($o_2$) depicted in Fig. 19 of [@witzany2015free]. We sample the trajectory with ${\rm d}\tau=10{\,{\rm M}}$ for $\tau_{\rm max}=50\,000{\,{\rm M}}$ yielding the data set of $N=5000$ points. The first minimum of mutual information is at $\Delta t =90{\,{\rm M}}=k \Delta \tau, k=9$ for $o_1$ and $\Delta t =110{\,{\rm M}}, k=11$ for $o_2$, hence we adopt $\Delta t = 100{\,{\rm M}}, k=10$ for both orbits. We generate the set of surrogates and perform the analysis in the same way as for the observed X-ray lightcurves in [@velkyChaos].
At first we investigate the dependence of the length of the longest diagonal line present in the recurrence matrix $L_{\rm max}$ on $\epsilon$. As expected, the regular trajectory yields very long diagonal lines for small thresholds and $L_{\rm max}$ goes up almost to the maximal value $N$. The surrogates behave in a similar way for a little bit higher threshold. This is due to the way how the surrogate data are constructed, as they have exactly the same value distribution but they reproduce the spectrum only approximately depending also on the available length of the data set. In case of regular motion, very narrow peaks are in the spectrum and the error in reproducing such spectrum causes the very long diagonal lines to be broken. This higher value of $\epsilon$ for surrogates corresponds to the size of the neighbourhood needed for covering the small discrepancies of the surrogates. The chaotic orbit $o_2$ provides shorter lines, so that $L_{\rm max}< 2000$ for the range of thresholds we used. Yet it is significantly larger than the corresponding values for surrogates. Only for very high thresholds, the difference decreases.
{width="0.495\columnwidth"} {width="0.495\columnwidth"}
Because in reality the data always contain some level of noise, we take the normalized times series for $o_1$ and $o_2$ and we add a white noise with zero mean and increasing variance $\sigma_n$ and rescale the resulting data back to zero mean and unit variance. The surrogates created from the regular orbit with added noise reproduce the spectrum better than for the regular orbit alone (normalised rms discrepancy between the exact spectrum and the exact amplitude stage reported by the [surrogates]{} procedure decreases from $\sigma_n=0$ to $\sigma_n=0.25$).
In Fig. \[fig:Preg\_Lmax\] the plots of $L_{\rm max}$ versus $\epsilon$ for the added white noise with $\sigma_n = 0.4$ are shown. The presence of noise shifts up the needed threshold for some lines to occur in RP. For the regular orbit $o_1$ there is no significant difference from the surrogates. For chaotic orbit $o_2$ the threshold is also shifted to higher values, but the difference from the surrogates remains.
Our posed null hypothesis is that the data are the product of linearly autocorrelated process and because the regular trajectory can be treated as such (e.g. in the case of a periodic orbit, the points separated by the period $T$ are the same), the significance is small. On the other hand, the chaotic trajectory cannot be treated as linear dynamics and yields high significance.
In Fig. \[fig:Preg\_Lmax2\] the estimate of $K_2$ and the significance of its comparison with the surrogates is given for the increasing level of noise ($\sigma_n =0, 0.05, \dots, 1.00$). We note, that for low levels of noise the regular orbit yields much lower value of $K_2$, which can serve as the differentiation between chaotic and regular motion (see [@semerak2012free]). However for increasing strength of the noise, the regular orbit $o_1$ seems to be more affected than the chaotic one, providing higher $K_2$ for the noise levels $\sigma_n>0.25$. Therefore, the significance for the regular orbit drops down bellow one quickly, while the significance for chaotic orbit reaches values around 10 for low noise levels, near 6 for intermediate noise levels and stays around 4 for high noise levels, even up to the case, when the variance of the noise is the same as the variance of the data.
{width="0.495\columnwidth"} {width="0.495\columnwidth"}
Conclusions {#discussion}
===========
The test of the method for finding non-linear dynamics in dynamical systems based on the observed time series shows that for a given length of observational data set (5000 points) we can expect that chaotic dynamics would yield values of significance between 2-10 depending on the strength of the noise. Regular motion would not provide significant result, because even low level of noise present in the measured data destroys the differences with respect to the surrogates.
This work was supported in part by the grant DEC-2012/05/E/ST9/03914 from the Polish National Science Center.
[6]{} natexlab\#1[\#1]{}
Grassberger, P. 1983, Physics Letters A, 97, 227
, R., [Kantz]{}, H., & [Schreiber]{}, T. 1999, Chaos, 9, 413
, T. & [Schmitz]{}, A. 2000, Physica D: Nonlinear Phenomena, 142, 346
Semer[á]{}k, O. & Sukov[á]{}, P. 2010, MNRAS, 404, 545
Semer[á]{}k, O. & Sukov[á]{}, P. 2012, MNRAS, 425, 2455
Sukov[á]{}, P., Grzedzielski, M., & Janiuk, A. 2016, A&A, 586, A143
Witzany, V., Semerák, O., & Suková, P. 2015, MNRAS, 451, 1770
[^1]: Using software package provided at `http://tocsy.pik-potsdam.de/commandline-rp.php`.
[^2]: Poincaré surface of section is a method for visualization of the features in the phase space of dynamical system, which is very useful for low-dimensional systems. The surface of section is a chosen surface in the phase space, on which the intersections of the phase trajectory are plotted. The regular orbits draw a closed curve whereas chaotic orbits fill some non-zero area. More details can be found e.g. in [@semerak2010free].
|
---
abstract: 'The interaction between phonons and high-energy excitations of electronic origin in cuprates and their role on the superconducting phenomenon is still controversial. Here, we use coherent vibrational time-domain spectroscopy together with density functional and dynamical mean-field theory calculations to establish a direct link between the c-axis phonon modes and the in-plane electronic charge excitations in optimally doped YBCO. Our findings clarify the nature of the anomalous high-energy response associated to the formation of the superconducting phase in the cuprates.'
author:
- Daniele Fausti
- Fabio Novelli
- Gianluca Giovannetti
- Adolfo Avella
- Federico Cilento
- Luc Patthey
- Milan Radovic
- Massimo Capone
- Fulvio Parmigiani
bibliography:
- 'references.bib'
title: 'Dynamical coupling between off-plane phonons and in-plane electronic excitations in superconducting YBCO'
---
The role played by phonons in cuprates superconductivity is still controversial. While on one hand a purely phononic mechanism can hardly account for the superconducting properties[@Chubkov2004; @DalConte2012], on the other hand strong anomalies are visible in the phonon sub-system upon entering the superconducting phase in most cuprates[@Johnston2010; @Kresin2009]. Furthermore, non-standard fingerprints of electron-phonon interaction have been observed as a result of the interplay with strong electron-electron correlations[@Capone2010]. A common feature of the cuprates, confirming the complexity of the riddle posed by these materials, is that the onset of the superconducting phases is accompanied by large changes in the optical properties up to energies of order of few electronvolts[@Molegraaf2002; @Boris2004; @Lee2004; @Basov2005], ten times larger than the typical superconducting gaps (10-100 meV). This is in striking contrast with a BCS scenario in which the variations of the optical properties at the superconducting phase transition are limited to energies of the order of the gap. Such an unusual behavior reveals the strong interconnection between high-energy processes, mainly controlled by electron-electron correlations, and the low-energy excitations relevant in the pairing mechanism of the cuprates[@Norman2003].
Here, we report on a novel approach, based on time-domain broadband spectroscopy, density functional theory (DFT), and dynamical mean field theory (DMFT)[@Georges1996RMP], to address the interplay between vibrational modes and the high-energy electronic response in the cuprates. Our measurements and calculations rationalize the different coupling of the barium (Ba) and copper (Cu) c-axis A1g modes to the superconducting phase and highlight the importance of the coupling between electrons and specific lattice modes in the low-energy dynamics, which underlies the superconducting transition.
![(a) Transient reflectivity representative of the normal (red) and superconducting phase (black) at a probe wavelenght of 560 nm (2.2 eV). (b) Relative variation of the incoherent reflectivity for a delay of 0.3 ps as a function of temperature and (c) ratio between the amplitudes of the Ba and Cu phonon modes vs. temperature. (d) Coherent contribution to the reflectivity as a function of frequency for $T$=5 K and 120 K. The peaks at 3.5 THz and 4.5 THz consist mainly of c-axis oscillations of the Ba and Cu positions, respectively (sketched in e).[]{data-label="fig1"}](fig1.png)
The peculiar changes in the optical properties of the cuprates at the onset of the superconducting phase[@Molegraaf2002; @Boris2004; @Lee2004; @Basov2005] are particularly evident in pump and probe (p&p) experiments[@Han1990; @Chekalin1991; @Reitze1992; @Stevens1997; @Demsar1999; @Segre2002], where the pump-driven charge excitations modify the electron distribution resulting in a sudden destabilization of the condensate[@Howell2004; @Nicol2003; @kabanov2005]. In such conditions, the *anomalous* connection, also observed at equilibrium, between the formation of the condensate and the high-energy optical transitions leads to the large changes in the time-resolved reflectivity (TRR) $R(\lambda,t)$ observed on the visible range[@Elbert2007; @gedik2005; @Kaindl2000; @Liu2008; @Giannetti2011; @eesley1990; @gedik2004].
Figure \[fig1\] shows our reflectivity data for 100 nm thick optimally doped YBCO films, grown by pulsed laser deposition on STO(001)[@sassa2011], having a critical temperature T$_c$=88 K. The pump pulses are $\approx$80 fs long with central wavelength $\lambda_{pump}$ = 1300 nm. The TRR was measured on a large spectral region with broadband white-light probes generated in a sapphire crystal (400 nm $<\, \lambda_{probe}\,<$ 1000 nm; Energy $<$ 5 $\mu$J/cm$^2$, see supplementary material for details and results). The transient reflectivity measurements, as reported in Fig.\[fig1\]a for $\lambda_{probe}$= 560 nm, manifestly show an oscillating component, which we associate to “coherent” phonon excitations. This oscillation, which is the main target of the present work, is superimposed to a decaying function of time, which we label as “incoherent”.
The coherent contribution is extracted by first fitting $R(\lambda,t)$ for each $\lambda$ with a multi-exponential decay convoluted with a step function, which provides the incoherent part, then a Fourier analysis of the difference between the data and the coherent part is performed. In Fig.\[fig1\]b we show the temperature evolution of the relative variation of the incoherent reflectivity $\Delta$R/R for a 0.3 ps delay, while Fig. \[fig1\]d reports the Fourier analysis of the oscillating part. The coherent response is indeed dramatically sensitive to the onset of superconductivity, as it consists of one single frequency (4.5 THz) above the critical temperature, while a second oscillation frequency (3.5 THz) appears below T$_c$ (Fig.\[fig1\]d). The appearence of a second component below T$_c$ is shown in Fig. \[fig1\]c, where the ratio between the amplitude of the two modes is plotted. These frequencies correspond to two phonon modes observed in static Raman[@Friedl1991; @henn1997] and time-resolved experiments[@albrecht1992; @misochko2002; @lobad2001; @misochko2000], namely A$_{1g}$ modes involving almost pure Ba and Cu off-plane vibrations respectively [@henn1997]. A displacive excitation of coherent phonon mechanism[@zeiger1992] has been invoked in order to describe the TRR in YBCO[@mazin1994].
In order to rationalize the different behavior of the two low-frequency modes, we investigated the effect of coherent phonon distortions on the electronic structure by means of DFT calculations. We employed the generalized gradient approximation (PBE) using Quantum Espresso[@QE]. The experimental lattice parameters and ionic positions are taken from Ref. [@Pickett1989].
To highlight the effect of the coherent phonon excitation, we compare the electronic density of states calculated for YBCO at equilibrium with the ones obtained by artificially moving the Cu or Ba ions along the eigenvectors of the A$_{1g}$ c-axis phonons of 0.116 $\AA$ (1/100 of the unit cell). The calculation reveals that the density of states at the Fermi level is heavily modified by the Ba displacement while it is not considerably perturbed by the c-axis rearrangement of the Cu ions. The 4.52 states/eV value of the undistorted system becomes 3.26 and 4.82 for Ba and Cu displacement respectively. This result confirms that a perturbation of the low-energy electronic properties, such as the photo-induced quench of the superconducting gap is more coupled with a displacement of Ba atom, while the Cu displacement is essentially insensitive to the low-energy electrodynamics.
![(a) Amplitude of the coherent transient reflectivity (colours) at different wavelengths at 5 K and 20 $\mu$J/cm$^2$ pump fluence (the black line is a representative cut at 560 nm). (b) Amplitude in the frequency-domain obtained by Fourier transforming the time traces. (c) Amplitude of the two relevant phonon modes at the different wavelengths, (d) line-shape representative of the different responses (dashed lines in b) with an offset added for clarity, and with the dotted lines obtained by a fit with Fano lineshape.[]{data-label="fig2"}](fig2.png)
We now focus on the dependence on the probe wavelength (energy) of the coherent response in order to reveal the interplay between the displacement and the different electronic excitations. Fig.\[fig2\]a depicts, in a false color plot, the residual energy-dependent coherent contribution for the measurements at 5 K and 20 $\mu$J/cm$^2$ of pump fluence. In Fig.\[fig2\]b, we plot the amplitude of the phonon modes at each photon energy. The Fourier transforms for representative energies are shown in Fig.\[fig2\]d. The wavelength-dependent coherent phonon response (CPR) amplitude for the two phonon modes plotted in Fig.\[fig2\]c are vertical cuts of Fig.\[fig2\]b taken at 4.5 THz for Cu and 3.5 THz for Ba. We have checked that these low-fluence CPR are in good agreement with the wavelength dependence of the Raman tensor[@henn1997].
![ (a) Density of states obtained in DMFT (red), compared with the DFT result (dashed black). (b) DMFT Optical conductivity (OC) (c) Calculated OC changes driven by the vibrational modes in the visible range (highlited in grey in panel in (b)). The Ba response has been multiplied by 2.5, see text. (d) Energy-dependent amplitude of the coherent response measured in p&p experiments.[]{data-label="fig3"}](fig3.png)
In order to theoretically account for such a wavelenght dependence, it is necessary to include the strong electron-electron correlations which are responsible for the insulating state of the parent compounds and of the optical spectral weight distribution[@Toschi2008]. This is implemented combining the DFT bandstructure with a DMFT treatment of the Coulomb interaction. We used a Wannier projection [@wannier90] including dx2-y2 orbital and an on-site Coulomb repulsion on copper of U = 4.8 eV. We used exact diagonalization as the impurity solver with 8 sites in the bath.
In Fig.\[fig3\]a we compare the DMFT density of states with the uncorrelated results from DFT. The prime effect of the electronic correlation is to shift spectral weight to the lower Hubbard band (LHB) and an upper Hubbard band (UHB), which coexists with a low-energy structure corresponding to itinerant carriers with a renormalized bandwidth. In Fig.\[fig3\]b the real part of the optical conductivity $\sigma(\omega)$ calculated with DMFT is shown. As underlined by the gray shaded areas in Figs.\[fig3\]a and \[fig3\]b, $\sigma(\omega)$ in the visible region is dominated by optical transitions between the LHB and the electronic states at the Fermi level. This observation enucleates the origin of the conductivity variations up to energies as high as few eV following the opening of a superconducting gap of the order of tens of meV.
Here we rationalize the CPR($\lambda_{probe}$) amplitude by means of a differential approach. We argue that the CPR($\lambda_{probe}$) can be described qualitatively by the difference between $\sigma(\omega)$ calculated with the ions in the equilibrium position and that obtained with the ions displaced along the phonon eigenvector. Fig.\[fig3\]c displays the optical conductivity changes driven by the displacement of Cu (red) and Ba (black) ions along the c-axis. These differences are compared with the CPR($\lambda_{probe}$) amplitude as a function of the probe energy for both phonon modes. Such a comparison shows that a proper treatment of the on-site Coulomb interaction on copper allows to correctly reproduce the dependence on the wavelenght of the coherent response even in the single-site DMFT approximation without including other bands, longer-range interactions and explicit coupling to phonon modes. The quantitative mismatch in the amplitude could be due to the arbitrary absolute values for the distortion introduced in the calculation or, more interestingly, it could be ascribed to the anomalous Ba response, enhanced by superconductivity, which is not taken into account into the calculation, performed in the paramagnetic phase.
![ (a-d) Amplitudes of the two vibrational modes as a function of wavelength for different temperatures (a,c) and excitation densities (b,d). (e) Amplitude of the incoherent part of $\Delta$R/R and (f) ratio between the amplitudes of the coherent response of the two phonons. The amplitudes of the coherent phonon responses as a function of fluence have been renormalized in b and d to stress the wavelength dependence.[]{data-label="fig4"}](fig4.png)
As we discussed above, the Cu mode is unaffected if we increase the base temperature of the sample. On the contrary the Ba mode disappears suddenly at T$>$T$_C$ (Fig.\[fig1\]c). In Fig.\[fig4\]a and \[fig4\]c), we show that this behavior extends to the full wavelength-dependent CPR($\lambda_{probe}$). The data for low pump fluence confirm that the Cu mode (Fig.\[fig4\]a) is essentially unaffected by raising the temperature above Tc while the fingerprint of the Ba mode (Fig.\[fig4\]b) on the optical properties disappears in the whole energy range.
We can further strenghten the connection between the phonon coherent response and the electronic properties by increasing the pump fluence in our p&p measurements, reaching high-intensity perturbations, where the photo-excitation results in a partial or complete collapse of the superconducting gap[@kusar2008; @giannetti2009]. In Fig. \[fig4\]e we show the effect of the increased fluence on the relative variation of the reflectivity $\Delta$R/R at $\lambda_{probe}$ = 560 nm after 0.3 ps delay (Fig. \[fig4\]b) and 5 K. At low fluence this quantity is negative and increases in absolute value as the fluence grows. At the critical value $\Phi_{C}\approx 50 \mu$ J/cm$^2$ a sharp kink is observed, followed by an increase of $\Delta$R/R which eventually turns positive for fluences larger than 100 $\mu$J/cm$^2$. This trend is a well-known signature of the photo-induced collapse of the superconducting gap observed in various compounds of the cuprate family[@kusar2008; @coslovich2011; @giannetti2009].
Once again, the light-driven collapse of the superconducting phase is mirrored in the amplitude of the CPR($\lambda_{probe}$) for the Ba and Cu modes. Figure \[fig4\]f depicts the ratio between the amplitudes of the two modes as a function of fluence. Even if the behavior is less sharp than the temperature dependence of Fig.\[fig1\]c, the melting of the superconducting state is associated with the disappearance of the Ba peak. The wavelength-dependent CPR($\lambda_{probe}$) are shown in Figs. \[fig4\]b and \[fig4\]d. The insensitivity of the Cu mode is confirmed, while the Ba mode is influenced also by the non-adiabatic melting of the superconducting gap (Fig.\[fig4\]c and Fig.\[fig4\]a, bottom right).
However, the effect of an increased pump fluence on the Ba mode does not mirror the effect of temperature. While the temperature essentially washes out the Ba phonon mode, an increased intensity rather leads to a change in the energy dependence of the CPR, even if the overall weight is reduced. This is suggestive of a non-thermal character of the pump-driven excitations, in agreement with recent time-domain photoemission experiments indicating that the pump pulse gives rise to excitations mainly in the antinodal regions[@cortes2009; @Cilento2014]. On the other hand a thermal excitation results in a large increase of excitations around the nodal points, where low-energy states are available, while no significant population changes are observed at the antinode[@damascelli2003]. Taking into account that the apical oxigens in YBCO are excited in p&p experiment within 150 fs, i.e. faster than the quasiparticles thermalization time[@pashkin2010], we can interpret our findings in terms of an effective momentum-dependent light driven dynamics of the superconducting gap. Further theoretical studies in the superconducting state including out-of-plane degrees of freedom are needed to substantiate this interesting scenario.
In summary, we addressed the interaction mechanism between the low-energy off-plane vibrations and the high-energy electronic in-plane excitations in superconducting cuprates by means of time-resolved ultra-fast reflectivity measurements, density functional theory, and dynamical mean field theory. The broadband optical probe allowed for the full dynamical characterization of the Raman tensor, confirming the existence of a c-axis barium vibration that is strictly related to superconductivity in the copper-oxygen planes. With the aid of DFT and DMFT calculations we could identify the microscopical link between the off-plane modes and the electronic density of states.
D. F., M.C., F. N., F. C. and F.P. acknowledge support by the European Union under FP7 GO FAST, grant agreement no. 280555. M.C. and G. G. are financed by the European Research Council through the ERC Starting Grant n.240524 SUPERBAD.
|
---
abstract: 'A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. A threshold graph can be obtained from a chain graph by making adjacent all pairs of vertices in one color class. Given a graph $G$, let ${\lambda}$ be an eigenvalue (of the adjacency matrix) of $G$ with multiplicity $k \geq 1$. A vertex $v$ of $G$ is a downer, or neutral, or Parter depending whether the multiplicity of ${\lambda}$ in $G-v$ is $k-1$, or $k$, or $k+1$, respectively. We consider vertex types in the above sense in threshold and chain graphs. In particular, we show that chain graphs can have neutral vertices, disproving a conjecture by Alazemi [*et al.*]{}'
address:
- 'Department of Mathematics, Kuwait University, Safat 13060, Kuwait'
- 'Department of Mathematics, K. N. Toosi University of Technology, P. O. Box 16315-1618, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran'
- 'State University of Novi Pazar, Vuka Karadžića bb, 36 300 Novi Pazar, Serbia, and Mathematical Institute SANU, Kneza Mihaila 36, 11 000 Belgrade, Serbia'
author:
- 'M. Anelić'
- 'E. Ghorbani'
- 'S.K. Simić'
title: '**Vertex types in threshold and chain graphs**'
---
Introduction
============
This paper is a successor of [@AFGS] in which vertex types (see the Abstract) in the lexicographic products of an arbitrary graph over cliques and/or co-cliques were investigated. Such class of graphs includes threshold graphs and chain graphs as particular instances. Both of these types (or classes) of graphs were discovered, and also rediscovered by various researchers in different contexts (see, for example, [@bcrs; @bfp; @hpx], and references therein). Needles to say, they were named by different names mostly depending on applications in which they arise. It is also noteworthy that threshold graphs are subclass of cographs, i.e. of $P_4$-free graphs. Recall that threshold graphs are $\{P_4,2K_2,C_4\}$-free graphs, while chain graphs are $\{2K_2,C_3,C_5\}$-free graphs - see [@AlAS; @Pt-srb] for more details. Note, if these graphs are not connected then (since $2K_2$ is forbidden) at most one of its components is non-trivial (others are trivial, i.e. isolated vertices). Moreover, stars are the only connected graphs which belong to both of two classes of graphs under consideration.
Recall, these graphs play a very important role in Spectral Graph Theory, since the maximizers for the largest eigenvalue of the adjacency matrix (for graphs of fixed order and size, either connected or disconnected) belong to these classes (threshold graphs in general case, and chain graphs in bipartite case). Such graphs (in both classes) have a very specific structure (embodied in nesting property), and this fact enables us to tell more on the type of certain vertices. Here, we also disprove Conjecture 3.1 from [@Pt-srb].
Throughout, we will consider simple graphs, i.e. finite undirected graphs without loops or multiple edges. In addition, without loss of generality, we will assume that any such graph is connected. For a graph $G$ we denote its vertex set by $V(G)$, and by $n=|V(G)|$ its [*order*]{}. An $n \times
n$ matrix $A(G) = [a_{ij}]$ is its [*adjacency matrix*]{} if $a_{ij}
= 1$ whenever vertices $i$ and $j$ are adjacent, or $a_{ij} = 0$ otherwise. For a vertex $v$ of $G$, let $N(v)$ denote the [*neighborhood*]{} of $v$, i.e. the set of all vertices of $G$ adjacent to $v$. The eigenvalues of $G$ are the eigenvalues of its adjacency matrix. In non-increasing order they are denoted by $$\lambda_1(G)\geq \lambda_2(G) \geq \cdots \geq \lambda_n(G),$$ or by $$\mu_1(G) > \mu_2(G)> \cdots > \mu_r(G)$$ if only distinct eigenvalues are considered. If understandable from the context we will drop out graph names from the notation of eigenvalues (or other related objects). The eigenvalues comprise (together with multiplicities, say $k_1,k_2,\ldots,k_r$, respectively) the spectrum of $G$, denoted by ${{\rm Spec}}(G)$. The [*characteristic polynomial*]{} of $G$, denoted by $\phi(x;G)$, is the characteristic polynomial of its adjacency matrix. Both, the spectrum and characteristic polynomial of a graph $G$ are its invariants. Further on, all spectral invariants (and other relevant quantities) associated to the adjacency matrix will be prescribed to the corresponding graph. For a given eigenvalue ${\lambda}\in {{\rm Spec}}(G)$, ${{\rm mult}}({\lambda},G)$ denotes its multiplicity, while ${\mathcal E}({\lambda};G)$ its eigenspace (provided $G$ is a labeled graph). The equation $A
{\mathbf x} = {\lambda}{\mathbf x}$, is called the eigenvalue equation for ${\lambda}$. Here $A$ is the adjacency matrix, while ${\mathbf x}$ a ${\lambda}$-eigenvector also of the labeled graph $G$. If $G$ is of order $n$, then ${\mathbf x}$ can be seen as an element of ${\mathbb
R}^n$, or a mapping ${\mathbf x} : V(G) \rightarrow {\mathbb R}^n$ (so its $i$-th entry can be denoted by $x_i$ or ${\mathbf x}(i)$). Eigenspaces (as the eigenvector sets) are not graph invariants, since the eigenvector entries become permuted if the vertices of $G$ are relabeled.
An eigenvalue ${\lambda}\in {{\rm Spec}}(G)$ is [*main*]{} if the corresponding eigenspace ${\mathcal E}({\lambda};G)$ is not orthogonal to all-$1$ vector $\mathbf{j}$; otherwise, it is [*non-main*]{}.
Given a graph $G$, let ${\lambda}$ be its eivgenvalue of multiplicity $k
\geq 1$ and $v\in V(G)$. Then $v$ is a [*downer*]{}, or [*neutral*]{}, or [*Parter*]{} vertex of $G$, depending whether the multiplicity of ${\lambda}$ in $G-v$ is $k-1$, or $k$, or $k+1$, respectively. Recall, neutral and Parter vertices of $G$ are also called [*Fiedler*]{} vertices. For more details, about the above vertex types see, for example, [@SAFZ].
Sum rule: Let ${{\bf x}}$ be a ${\lambda}$-eigenvector of a graph $G$. Then the entries of ${{\bf x}}$ satisfy the following equalities: $$\label{sumrule}
{\lambda}{{\bf x}}(v)=\sum_{u\sim v}{{\bf x}}(u),~~\hbox{for all}~v\in V(G).$$ From it follows that if ${\lambda}\ne0$, then $N(u)=N(v)$ implies that ${{\bf x}}(u)={{\bf x}}(v)$ and if ${\lambda}\ne-1$, $N(u)\cup\{u\}=N(v)\cup\{v\}$ implies that ${{\bf x}}(u)={{\bf x}}(v)$.
In sequel, we will need the following interlacing property for graph eigenvalues (or, eigenvalues of Hermitian matrices, see [@bh Theorem 2.5.1]).
\[inter\] Let $G$ be a graph of order $n$ and $G'$ be an induced subgraph of $G$ of order $n'$. If ${\lambda}_1 \ge {\lambda}_2\ge \cdots \ge {\lambda}_n$ and ${\lambda}'_1 \ge {\lambda}'_{2} \ge \cdots \ge {\lambda}'_{n'}$ are their eigenvalues respectively, then $${\label{jed}}
{\lambda}_i \ge {\lambda}'_i \ge {\lambda}_{n-n'+i}~~~\hbox{for}~ i=1,2, \ldots,n'.$$ In particular, if $n'=n-1$, then $${\lambda}_1\ge{\lambda}'_1\ge{\lambda}_2\ge{\lambda}'_2\ge\cdots\ge{\lambda}_{n-1}\ge{\lambda}'_{n-1}\ge{\lambda}_n.$$
In the case of equality in (\[jed\]) (see [@bh Theorem 2.5.1]) the following holds.
\[eqinter\] If ${\lambda}'_i={\lambda}_i$ or ${\lambda}'_i={\lambda}_{n-n'+i}$ for some $i \in \{1,2,\ldots,n'\}$, then $G'$ has an eigenvector ${{\bf x}}'$ for ${\lambda}'_i$ such that $\begin{pmatrix}\bf0 \\ {{\bf x}}'\end{pmatrix}$ is an eigenvector of $G$ for ${\lambda}'_i$, where $\bf0$ is a zero vector whose entries correspond to the vertices from $V(G)\setminus V(G')$.
\[downer\] A vertex $v$ is a downer for a fixed eigenvalue $\lambda$, if there exists in the corresponding eigenspace an eigenvector whose $v$-th component is non-zero. Otherwise, it is a Fiedler vertex. Let $W$ be the eigenspace corresponding to ${\lambda}$. If for each ${{\bf x}}\in W$, we have ${{\bf x}}(v)=0$, then $v$ cannot be a downer vertex as for any ${{\bf x}}\in W$, the vector ${{\bf x}}'$ obtained by deleting the $v$-th component, is a ${\lambda}$-eigenvector of $G-v$, and therefore we have $${{\rm mult}}({\lambda},G-v)\ge\dim\,\{{{\bf x}}':{{\bf x}}\in W\}=\dim W={{\rm mult}}({\lambda},G).$$ From this and Lemma \[eqinter\] it follows, if ${{\rm mult}}({\lambda}, G)=1$ that there exists a ${\lambda}$-eigenvector ${{\bf x}}$ with ${{\bf x}}(v)=0$ if and only if $v$ is not a downer vertex for ${\lambda}$.
The rest of the paper is organized as follows: in Section \[th\] we give some particular results about vertex types in threshold graphs, while in Section \[cg\] we put focus on chain graphs, and among others we disprove Conjecture 3.1 from [@AlAS], which states that in any chain graph, every vertex is a downer with respect to every non-zero eigenvalue. Besides we point out that some weak versions of the same conjecture are true.
Vertex types in threshold graphs {#th}
================================
Any (connected) threshold graph $G$ is a split graph i.e., it admits a partition of its vertex set into two subsets, say $U$ and $V$, such that the vertices of $ U$ induce a co-clique, while the vertices of $ V$ induce a clique. All other edges join a vertex in $U$ with a vertex in $V$. Moreover, if $G$ is connected, then both $U$ and $V$ are partitioned into $h\geq 1$ non-empty cells such that $U=\bigcup_{i=1}^h U_i$ and $V=\bigcup_{i=1}^h V_i$ and the following holds for (cross) edges: each vertex in $U_i$ is adjacent to all vertices in $V_1\cup\cdots\cup V_i$ (a nesting property). Accordingly, connected threshold graphs are also called [*nested split graphs*]{} (or NSG for short). If $m_i = |U_i|$ and $n_i =|V_i|$, then we write $${\label{nsg}}
G ={{\rm NSG}}(m_1,\ldots , m_h; n_1, \ldots, n_h),$$ (see Fig. \[thrg\]). We denote by $M_h$ ($=\sum_{i=1}^h m_i$) the size of $U$, and by $N_h$ ($=\sum_{i=1}^h n_i$) the size of $V$.
The following Theorem states the essential spectral properties of threshold graphs (see [@AlAS; @trev2; @SciFar]).
\[nsg\_mult\] Let $G={{\rm NSG}}(m_1, \ldots, m_h; n_1, \ldots, n_h)$. Then the spectrum of $G$ contains:
- $h$ positive simple eigenvalues;
- $h-1$ simple eigenvalues less than $-1$ if $m_h=1$, or otherwise if $m_h\geq 2$, $h$ simple eigenvalues less than $-1$;
- eigenvalue $0$ of multiplicity $M_h-h$, and $-1$ of multiplicity $N_h-h+1$ if $m_h=1$, or of multiplicity $N_h-h$ if $m_h > 1$.
In addition, if $\lambda \neq 0,-1$ then $\lambda$ is a main eigenvalue.
If $\lambda \neq 0,-1$, then any vertex of a threshold graph is either downer or neutral. Parter vertices may arise only for $\lambda = 0$ or $-1$. Any vertex deleted subgraph $G-v$ of a threshold graph $G$ is a threshold graph as well. By Theorem \[nsg\_mult\] one can easily determine the multiplicities of $0$ and $-1$ in both $G$ and $G-v$ and, consequently, the vertex type for $v$ of ${\lambda}=0$ or $-1$.
Recall that any vertex of a connected graph is downer for the largest eigenvalue, see [@crs Proposition 1.3.9.]. In addition, if ${\lambda}\neq 0,-1$, then the corresponding eigenvector ${{\bf x}}$ is unique (up to scalar multiple) and constant on each of the sets $U_i$ and $V_i$ ($i=1,\ldots, h$); in particular, if $m_h=1$ then it is constant on the set $U_h\cup V_h$. These facts will be used repeatedly further on without any recall.
\[downers\] Let $G={{\rm NSG}}(m_1, \ldots, m_h; n_1, \ldots, n_h)$ and $\lambda \neq
0,-1$ its eigenvalue other than the largest one. Then all vertices in $U_1 \cup V_1$ are downers for $\lambda$. The same holds for vertices in $U_h$, and also in $V_h$ unless $\lambda = -m_h$ and $m_h\geq 2$.
Let $u_1 \in U_1$ and $v_1 \in V_1$. Then, by the sum rule, $\lambda
{{\bf x}}(u_1) = n_1 {{\bf x}}(v_1)$. Since $\lambda \neq 0,-1$, $u_1$ and $v_1$ are both downer or Fiedler vertices (see Remark \[downer\]). Let $X = \sum_{w \in V(G)} {{\bf x}}(w)$, and by the way of contradiction assume that $u_1$ and $v_1$ are both Fiedler vertices, i.e. ${{\bf x}}(u_1) = {{\bf x}}(v_1) = 0$. Again, by the sum rule, we have $\lambda
{{\bf x}}(v_1) =X - {{\bf x}}(v_1)$, and therefore $X = 0$, a contradiction since $\lambda\neq 0,-1$ is a simple and non-main eigenvalue (see Theorem \[nsg\_mult\]).
Let $u_h\in U_h$, $v_h \in V_h$ and $Y = \sum_{w \in V_1 \cup
\cdots \cup V_h} {{\bf x}}(w)$. Then, $\lambda {{\bf x}}(u_h) = Y$ and $\lambda
{{\bf x}}(v_h) = Y - {{\bf x}}(v_h) + m_h {{\bf x}}(u_h)$. For a contradiction, let ${{\bf x}}(u_h) = 0$. Then it easily follows that ${{\bf x}}(v_h) = 0$. We next claim that for $2\le i \le h$, ${{\bf x}}(u_i) ={{\bf x}}(v_i) = 0$ implies ${{\bf x}}(u_{i-1}) ={{\bf x}}(v_{i-1}) = 0$. To see this, since ${{\bf x}}(v_{i})=0$ by the sum rule we obtain $$\lambda {{\bf x}}(u_{i})= Y-\sum_{j=i+1}^h n_j{{\bf x}}(v_j)= Y-\sum_{j=i}^h n_j{{\bf x}}(v_j)=\lambda{{\bf x}}(u_{i-1}),$$ and therefore $\lambda{{\bf x}}(u_{i-1}) = 0$. Similarly, since ${{\bf x}}(u_i)=0$ and $$\begin{aligned}
\lambda{{\bf x}}(v_{i})&=&Y-{{\bf x}}(v_{i})+\sum_{j=i}^hm_j{{\bf x}}(u_{j})\\
&=&Y-{{\bf x}}(v_{i})+\sum_{j=i-1}^hm_j{{\bf x}}(u_{j})=({\lambda}+1){{\bf x}}(v_{i-1}),\end{aligned}$$ it follows ${{\bf x}}(v_{i-1})=0$. Consequently, we obtain ${{\bf x}}(u_h)=\cdots={{\bf x}}(u_1)=0$ and ${{\bf x}}(v_h)=\cdots={{\bf x}}(v_1)=0$, i.e. ${{\bf x}}=\mathbf{0}$, a contradiction. This proves that all vertices in $U_h$ are downers for $\lambda$.
For the last part of the theorem, let $\lambda \neq - m_h$. Then we have $${\lambda}{{\bf x}}(u_h)=Y,~~{\lambda}{{\bf x}}(v_h)=Y-{{\bf x}}(v_h)+m_h{{\bf x}}(u_h),$$ and so $({\lambda}+1){{\bf x}}(v_h)=({\lambda}+m_h){{\bf x}}(u_h)$. Hence, if ${{\bf x}}(v_h) = 0$, then ${{\bf x}}(u_h) =0$ and we reach a contradiction as above. Consequently, all vertices in $V_h$ are downers.
[\[mh\]]{} The following example shows that in unresolved case when $\lambda=-m_h$ and $m_h\geq 2$ vertices in $V_h$ may be neutral.
Let $G={{\rm NSG}}(2,2,2;2,3,2)$. Then all vertices in $U_3$ are downers, while all vertices in $V_3$ are neutral for $\lambda =-2$. So, an unresolved case from Theorem \[downers\] can be an exceptional one.
So, the following question arises: Can we find an example when ${\lambda}=-m_h$, $m_h\geq 2$ and that each vertex in $V_h$ is a downer?
Let $G={{\rm NSG}}(m_1, \ldots, m_h; n_1, \ldots, n_h)$ and let $\lambda
\neq 0,-1$ be its eigenvalue. Then, for any $i=1,\ldots, h-1$, at least one of $U_i$, $U_{i+1}$ (resp. $V_i$, $V_{i+1}$) contains only downer vertices for $\lambda$.
Recall first that all vertices within $U_k$ or $V_k$ $(k=1,\ldots,
h)$ are of the same type for ${\lambda}$, and that ${\lambda}$ is a simple eigenvalue. Assume on the contrary that all vertices in $U_i$ and $U_{i+1}$ are neutral and let ${{\bf x}}$ be a ${\lambda}$-eigenvector. Then, for $u_i \in U_i$ and $u_{i+1}\in U_{i+1}$, ${{\bf x}}(u_i)={{\bf x}}(u_{i+1})=0$. By the sum rule it easily follows that for any $v_{i+1}\in V_{i+1},$ ${{\bf x}}(v_{i+1})=0$. Next, we have $$\begin{aligned}
\lambda {{\bf x}}(v_{i})=\sum_{j=1}^h
n_j{{\bf x}}(v_j)-{{\bf x}}(v_{i})+\sum_{j=i}^h
m_j{{\bf x}}(u_j)\label{eq_i},\\
\lambda {{\bf x}}(v_{i+1})=\sum_{j=1}^h
n_j{{\bf x}}(v_j)-{{\bf x}}(v_{i+1})+\sum_{j=i+1}^h
m_j{{\bf x}}(u_j)\label{eq_i+1}.\end{aligned}$$ By subtracting (\[eq\_i+1\]) from (\[eq\_i\]) we obtain $\lambda
{{\bf x}}(v_{i})=-{{\bf x}}(v_{i})$. Since $\lambda \neq -1$, ${{\bf x}}(v_{i})=0$ and consequently ${{\bf x}}(u_{i-1})=0$. Proceeding in the similar way, we conclude that ${{\bf x}}(u_1)=0$, which contradicts Theorem \[downers\].
The proof for vertices in $V_i, V_{i+1}$ is similar, and therefore omitted.
Next examples show that in an nested split graph $G$ neutral vertices for the same eigenvalue may be distributed in different $U_i$’s, $V_i$’s and at the same time in both $U$ and $V$.
If $G={{\rm NSG}}(4,1,3,1,1; 1,1,1,2,1)$, then all vertices in $U_2$ and $U_4$ are neutral vertices for $\lambda_3=1$.
If $G={{\rm NSG}}(2,4,4,2; 1,1,1,2)$, then all vertices in $V_2$ and $V_4$ are neutral for $\lambda_{16}=-2$.
In $G={{\rm NSG}}(2,2,5,1;1,1,1,1)$ all vertices in $U_3$ and in $V_2$ are neutral vertices for $\lambda_2=1$.
In what follows we assume that all vertices in $U_{s}$ (resp. $V_{s}$) of a nested split graph $G$ are neutral for some $s$ with respect to some $\lambda_i \neq 0,-1$. If so, we will show that this assumption imply some restrictions on position of $\lambda_i$ in the spectrum of $G$.
\[strict-inter\] Let $G={{\rm NSG}}(m_1, \ldots, m_h; n_1, \ldots, n_h)$ such that all vertices in $U_{s}$ for some $2\leq s\leq h-1$ are neutral for $\lambda_i\neq 0,-1$. If $G'={{\rm NSG}}(m_{s+1}, \ldots, m_h; n_{s+1},
\ldots, n_h)$, $n'=|V(G')|$ and ${{\rm Spec}}(G')=\{{\lambda}'_1, \ldots,
{\lambda}'_{n'}\}$ then $\lambda_i={\lambda}'_j$ for some $j\in \{1,
\ldots,n'\}$. Moreover, $j<i<n-n'+j$, and if $i\leq n'$ then $\lambda_i\neq {\lambda}'_{n'}$.
Let $G'$ be the induced subgraph of $G$ obtained by deleting all vertices in $U_1, \ldots, U_s, V_1, \ldots, V_s$ i.e. $$G'={{\rm NSG}}(m_{s+1},\ldots, m_h; n_{s+1},\ldots, n_h).$$ Let $n'=|V(G')|=\sum_{j=s+1}^h(m_j+n_j)$ and let ${{\bf x}}$ be a ${\lambda}$-eigenvector of $G$. Denote by $\mathbf{x'}$ the vector obtained from ${{\bf x}}$ by deleting all entries corresponding to deleted vertices from $G$. Since $$0=\lambda_i {{\bf x}}(u_s)=\sum_{j=1}^s n_j{{\bf x}}(v_j),$$ for any $k\geq s+1$, we obtain $$\begin{aligned}
\lambda_i\mathbf{x'}(u_k)&=&\lambda_i{{\bf x}}(u_k)=\sum_{j=1}^kn_j{{\bf x}}(v_j)=\sum_{j=s+1}^k n_j{{\bf x}}(v_j)=\sum_{j=s+1}^k n_j\mathbf{x'}(v_j)\\
\lambda_i\mathbf{x'}(v_k)&=&\lambda_i{{\bf x}}(v_k)=\sum_{j=1}^hn_j{{\bf x}}(v_j)-{{\bf x}}(v_k)+\sum_{j=k}^hm_j{{\bf x}}(u_j)\\
&=&\sum_{j=s+1}^h n_j
\mathbf{x'}(v_j)-m_j\mathbf{x'}(v_k)+\sum_{j=k}^h \mathbf{x'}(u_j)\end{aligned}$$ and therefore $\mathbf{x'}$ is an eigenvector of $G'$ for $\lambda_i$, i.e. $\lambda_i\in{{\rm Spec}}(G')$. Suppose $\lambda_i={\lambda}'_j$ for some $j\in \{1,\ldots, n'\}$. From interlacing it follows that $$\label{inter1}
\lambda_{n-n'+i}\leq {\lambda}'_i\leq \lambda_i={\lambda}'_j, \quad
\mbox{if}\quad i\leq n'.$$ as well as $$\label{inter1'}
\lambda_{n-n'+j}\leq \lambda_i={\lambda}'_j\leq \lambda_j.$$
If in , at least one of inequalities holds as an equality then, by Lemma \[eqinter\], $G'$ has an eigenvector $\mathbf{y'}$ for ${\lambda}'_i$ such that $\left(
\begin{matrix}
\mathbf{0}\\
\mathbf{y'}
\end{matrix}
\right)$ is an eigenvector of $G$ for ${\lambda}'_i$. By the sum rule for any vertex in $V_{s}$ we obtain that the sum of all entries of $\mathbf{y'}$ is $0$ and accordingly that ${\lambda}'_i$ is non-main eigenvalue of $G'$. Hence, ${\lambda}'_i=0$ or ${\lambda}'_i=-1$ which implies ${\lambda}'_i<{\lambda}_i$. Similarly, in we conclude that ${\lambda}'_j={\lambda}_i$ is a non-main eigenvalue of $G'$, a contradiction, by Theorem \[nsg\_mult\]. Therefore, the interlacing in these cases reads $$\begin{aligned}
\lambda_{n-n'+i}\leq{\lambda}'_i< \lambda_i, \quad i\leq n',\\
\lambda_{n-n'+j}< {\lambda}'_j={\lambda}_i<\lambda_j\label{inter2}.\end{aligned}$$ Moreover, (\[inter2\]) implies $j<i<n-n'+j$. Also, if $i\leq n'$, $\lambda_i\neq {\lambda}'_{n'}$ holds. Otherwise, $\lambda_{n-n'+i}\leq{\lambda}'_{i}<\lambda_i= {\lambda}'_{n'}$, a contradiction.
If all vertices in $V_{s}$ for some $s$ are neutral for $\lambda_i\neq 0,-1$, then bearing in mind that $$G-V_s={{\rm NSG}}(m_1, \ldots, m_{s-1}+m_{s}, \ldots, m_h; n_1,
\ldots, n_{s-1}, n_{s+1}, \ldots, n_h)$$ we can similarly conclude the following.
\[strict-inter2\] Let $G={{\rm NSG}}(m_1, \ldots, m_h; n_1, \ldots, n_h)$ such that all vertices in $V_{s}$ for some $2\leq s\leq h$ are neutral for $\lambda_i\neq 0,-1$. If $$H_s={{\rm NSG}}(m_1, \ldots, m_{s-1}+m_{s}, \ldots, m_h; n_1,
\ldots, n_{s-1}, n_{s+1}, \ldots, n_h),$$ and ${{\rm Spec}}(H_{s})=\{{\lambda}'_1,
\ldots, {\lambda}'_{n-n_{s}}\}$, then $\lambda_i={\lambda}'_j$ for some $j\in
\{1, \ldots, n-n_{s}\}$. Moreover, $j<i<n_{s}+j$ and if $i\leq
n-n_s$ then $\lambda_i\neq {\lambda}'_{n-n_s}$.
Let $G={{\rm NSG}}(m_1, \ldots, m_h; n_1, \ldots, n_h)$ of order $n$. Then all vertices in $V(G)$ are downer vertices for $\lambda_n$.
If ${\lambda}_n=-1$, then $G$ is a complete graph and all vertices are downers for it. So, we assume that ${\lambda}_n\ne0,-1$. Suppose on the contrary that there exists at least one neutral vertex $u$ for $\lambda_n$. If $u\in U_s$, then $\mathbf{x}(u)=0$, where $\mathbf{x}$ is a ${\lambda}$-eigenvector of $G$. As shown in the proof of Theorem \[strict-inter\], $\lambda_n={\lambda}'_j\in{{\rm Spec}}(G')$, for some $j\in\{1, \ldots, n'\}$ and $\lambda_{n-n'+j}<{\lambda}'_j< \lambda_j$, i.e. $\lambda_{n-n'+j}<\lambda_n< \lambda_j$, a contradiction.
The proof is similar if $v \in V_s$ for some $s$ and hence omitted here.
Let $G={{\rm NSG}}(m_1, \ldots, m_h; n_1, \ldots, n_h)$ such that all vertices in $U_{s}$ are neutral for $\lambda\neq 0,-1$ and $G''={{\rm NSG}}(m_1, \ldots, m_{s}; n_1, \ldots, n_{s})$. Then $$\lambda_{n''}(G'')<\lambda <\lambda_1(G''),$$ where $n''=|V(G'')|=\sum_{i=1}^s (m_i+n_i)$.
The graph $G''$ is an induced subgraph of $G$ with vertex set $V(G'')=\bigcup_{j=1}^{s} (U_j\cup V_j)$. The adjacency matrix $A$ of the whole graph is equal to: $$\left[\begin{matrix}
A'' & B\\
B^T& A'
\end{matrix}\right],$$ where $A', A''$ are adjacency matrices of $$G'={{\rm NSG}}(m_{s+1},\ldots,
m_h; n_{s+1},\ldots, n_h)$$ and $G''$, respectively, and $$B=\left[
\begin{matrix}
O_{M_s,n'}\\
J_{N_s, n'}
\end{matrix}\right],$$ where $M_s=\sum_{j=1}^s m_j$, $N_s=\sum_{j=1}^s n_j$ and $n'=|V(G')|$. The corresponding eigenvector ${{\bf x}}$ can be represented as ${{\bf x}}=\left(
\begin{matrix}
{{\bf x}}_1\\
{{\bf x}}_2 \end{matrix} \right)$ and the eigenvalue system reads: $$\begin{aligned}
A''{{\bf x}}_1+B{{\bf x}}_2&=&\lambda {{\bf x}}_1\label{eq*}\\
B^T{{\bf x}}_1+A'{{\bf x}}_2&=&\lambda {{\bf x}}_2\label{eq**}.\end{aligned}$$ As we have seen in the proof of Theorem \[strict-inter\], $\lambda$ is an eigenvalue of $A'$, the corresponding eigenvector is ${{\bf x}}_2$ and further ${{\bf x}}_1\neq 0$. Therefore, it follows that $B^T{{\bf x}}_1=0$, i.e. the sum of some entries of ${{\bf x}}_1$ is $0$. From (\[eq\*\]), we obtain $$(\lambda I- A''){{\bf x}}_1=B{{\bf x}}_2$$ and then by multiplying by ${{\bf x}}_1^T$ from the left we obtain $${{\bf x}}_1^T(\lambda I- A''){{\bf x}}_1=0$$ and consequently $$\min_{{{\bf y}}\neq 0}\frac{{{\bf y}}^T(\lambda I- A''){{\bf y}}}{{{\bf y}}^T{{\bf y}}}\leq
\frac{{{\bf x}}_1^T(\lambda I-A''){{\bf x}}_1}{{{\bf x}}_1^T{{\bf x}}_1}\leq \max_{{{\bf y}}\neq 0}\frac{{{\bf y}}^T(\lambda I- A''){{\bf y}}}{{{\bf y}}^T{{\bf y}}}.$$ Hence, $$\lambda_{n''} (\lambda I- A'')\leq 0 \leq \lambda_1 (\lambda
I- A''),$$ where $n''=|V(G'')|=M_s+N_s$. Since, $\lambda_{n''}
(\lambda I- A'')=\lambda-\lambda_1(G'')$ and $\lambda_1 (\lambda I-
A'')=\lambda-\lambda_{n''}(G'')$ it follows $$\label{ineq}
\lambda_{n''}(G'')\leq \lambda\leq \lambda_1(G'').$$ Moreover, $\lambda\neq \lambda_1(G'')$. Equality holds if and only if ${{\bf x}}_1$ is an eigenvector of $G''$ for $\lambda_1(G'')$, that is not possible due to the condition (\[eq\*\*\]) and positivity of ${{\bf x}}_1$ as an eigenvector corresponding to the largest eigenvalue of a connected graph. Similarly, if $\lambda=\lambda_{n''}(G'')$, then ${{\bf x}}_1$ is the corresponding eigenvector and from (\[eq\*\]) it follows $B{{\bf x}}_2=0$. This implies that $\lambda$ is a non-main eigenvalue of a nested split graph $G'$, a contradiction by Theorem \[nsg\_mult\].
Let $$\begin{aligned}
G&=&{{\rm NSG}}(m_1, \ldots, m_h; n_1, \ldots, n_h),\\
G_s''&=&{{\rm NSG}}(m_1, \ldots, m_s; n_1, \ldots, n_s),\end{aligned}$$ $I_s=(\lambda_{n_s''}(G_s''),\lambda_1(G_s''))$, where $n_{s''}=|(V(G_s'')|$ and $\lambda\in {{\rm Spec}}(G)$. If $\lambda\notin
\bigcup_{s=2}^{h-1} I_s$, then all vertices in $U$ are downer vertices for $\lambda$.
Let $G={{\rm NSG}}(1,1,5; 1,1,8)$. Then $I_2=(-1.48,2.17)$ and besides $\lambda_1$ and $\lambda_n$ all vertices in $U$ are downer for $\lambda_{n-2}$ and $\lambda_{n-1}$, as well.
Vertex types in chain graphs {#cg}
============================
Chain graph can be defined as follows: a graph is a chain graph if and only if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. For this reason, if connected (as was the case with threshold graphs), it is also called [*double nested graph*]{} [@bcrs].
Non-zero eigenvalues of chain graphs are simple (see Theorem \[dng\_mult\] below). As the subgraphs of any chain graph are also chain graphs, it follows that there is no Parter vertex in any chain graphs with respect to non-zero eigenvalues. A question raises whether they can have neutral vertices. In [@AlAS] it is conjectured that this cannot be the case.
\[conj\] [([@AlAS])]{} In any chain graph, every vertex is downer with respect to every non-zero eigenvalue.
We disprove Conjecture \[conj\] in this section. Indeed, Theorems \[1\] and \[om\] will show that there are infinitely many counterexamples for this conjecture. In spite of that, a couple of weak versions of the conjecture are true.
\[struc\] [*(Structure of chain graphs)*]{} As it was observed in [@bcrs], the color classes of any chain graph $G$ can be partitioned into $h$ non-empty cells $U_1,\ldots, U_h$ and $V_1,\ldots, V_h$ such that $N(u)=V_1\cup\cdots\cup V_{h-i+1}~~\hbox{for any}~ u\in U_i,~1\le
i\le h.$ If $m_i=|U_i|$ and $n_i=|V_i|$, then we write ${{\rm DNG}}(m_1,\ldots, m_h;n_1, \ldots, n_h)$ (see Fig. \[chgr\]).
The spectrum of any chain graph has the following properties (see [@AlAS]):
\[dng\_mult\] Let $G={{\rm DNG}}(m_1, \ldots, m_h; n_1, \ldots, n_h)$. Then the spectrum of $G$ is symmetric about the origin and it contains:
- $h$ positive simple eigenvalues greater then $\frac{1}{2}$;
- $h$ negative simple eigenvalues less than $-\frac{1}{2}$;
- eigenvalue $0$ of multiplicity $M_h+N_h-2h$.
On the contrary from threshold graphs nonzero eigenvalues of chain graphs need not be main. For more information see [@Pt-srb].
\[chain\] Let $G={{\rm DNG}}(m_1, \ldots, m_h; n_1, \ldots, n_h)$ be a chain graph. Then the vertices in $U_1\cup U_h\cup V_1\cup V_h$ are downer for any non-zero eigenvalue.
Let ${{\bf x}}$ be any ${\lambda}$-eigenvector of $G$. Assume that $u_1\in U_1$ and $v_h\in V_h$. By the sum rule $\lambda {{\bf x}}(v_h)=m_1{{\bf x}}(u_1)$. Since, ${\lambda}\neq 0$, $u_1$ and $v_h$ are both downer or neutral. Let $X=\sum_{w\in V}{{\bf x}}(w)$ and assume on the contrary that ${{\bf x}}(u_1)={{\bf x}}(v_h)=0$. Again, by the sum rule ${\lambda}{{\bf x}}(u_2)=X-n_h{{\bf x}}(v_h)=0$ and consequently ${{\bf x}}(u_2)=0,$ for any $u_2\in U_2$ as well as ${{\bf x}}(v_{h-1})=0$ for any $v_{h-1}\in
V_{h-1}$. Next, for any $u_3\in U_3$, $${\lambda}{{\bf x}}(u_3)=X-n_{h-1}{{\bf x}}(v_{h-1})-n_h{{\bf x}}(v_h)=0$$ It follows that ${{\bf x}}$ is zero on $U_3$, too. Continuing this argument, it follows that ${{\bf x}}=\bf0$, a contradiction.
The following proposition states some facts related to vertex types in chain graphs. The proofs are similar to those in Section \[th\] and therefore omitted here.
Let $$\begin{aligned}
G&=&{{\rm DNG}}(m_1, \ldots, m_h;n_1, \ldots, n_h),\\
G_s'&=&{{\rm DNG}}(m_1, \ldots,m_{s-1}; n_{h-s+2}, \ldots, n_h),\\
G_s''&=&{{\rm DNG}}(m_s, \ldots, m_h; n_1, \ldots, n_{h-s+1}),\end{aligned}$$ $1<s<h$, ${\lambda}_i\in {{\rm Spec}}(G)\setminus\{0\}$, $n_s'=\sum_{j=1}^{s-1}
(m_j+n_{h-j+1})$, $n_s''=n-n_s'$, ${{\rm Spec}}(G_{s}')=\{{\lambda}'_1,\ldots,
{\lambda}'_{n_s'}\}$. Then
- For any $j=1,\ldots, h-1$ at least one of $U_j$, $U_{j+1}$ contains only downer vertices for $\lambda_i$.
- If all vertices in $U_s$ for some $2<s<h-1$ are neutral for $\lambda_i$, then
- $\lambda_i$ is an eigenvalue of $G_s'$ and ${\lambda}_i={\lambda}'_j$, for some $j\in \{1,\ldots, n_{s}'\}$. If $\lambda_i$ is main, then $j<i< n-n_s'+j$. If $i\leq n_s'$ then $\lambda_i\neq {\lambda}'_{n_s'}$.
- $\lambda_i\in[\lambda_{n_s''}(G_s''),\lambda_1(G_s''))$.
- If ${\lambda}_i$ is a main eigenvalue then $\lambda_i\in (\lambda_{n_s''}(G_s''),\lambda_1(G_s''))$.
- If $\lambda_i\notin \bigcup_{s=2}^{h-1}
[\lambda_{n_s''}(G_s''),\lambda_1(G_s''))$ then all vertices in $V(G)$ are downer vertices for $\lambda_i$.
A chain graph for which $|U_1|=\cdots=|U_h|=|V_1|=\cdots=|V_h|=1$ is called a [*half graph*]{}. Here we denote it by $H(h)$. As we will see in what follows, specific half graphs provide counterexamples to Conjecture \[conj\]. Let $$(a_1,\ldots,a_6):=(1,0,-1,-1,0,1).$$
In what follows, for convenience, we will instead of column vectors use row vectors, especially for eigenvectors.
Let $${{\bf x}}:=(x_1,\ldots,x_h)$$ where $x_i=a_s~\hbox{if}~i\equiv s\hspace{-0.25cm}\pmod6.$ In the next theorem, we show that the vector $({{\bf x}}, {{\bf x}})$ (each ${{\bf x}}$ corresponds to a color class) is an eigenvector of a non-zero eigenvalue of $H(h)$ for some $h$. In view of Remark \[downer\], this disproves Conjecture \[conj\] .
\[1\] In any half graph $H(h)$, the vector $({{\bf x}}, {{\bf x}})$ is an eigenvector for ${\lambda}=1$ if $h\equiv1\pmod6$ and it is an eigenvector for ${\lambda}=-1$ if $h\equiv4\pmod6$.
From Table \[a\_i\], we observe that for $1\le s\le6$, $$\sum_{i=1}^{5-s}a_i=-a_s~~\hbox{and}~~\sum_{i=1}^{2-s}a_i=a_s,$$ where we consider $5-s$ and $2-s$ modulo 6 as elements of $\{1,\ldots,6\}.$
$$\begin{array}{cccccc}
\hline
s&a_s & 5-s& \sum_{i=1}^{5-s}a_i & 2-s& \sum_{i=1}^{2-s}a_i \\
\hline
1 & 1 & 4 & -1 & 1 & 1 \\
2& 0 & 3 & 0 & 6 & 0 \\
3 &-1 & 2 & 1 & 5 & -1 \\
4 &-1 & 1 & 1 & 4 & -1 \\
5 &0 & 6 & 0 & 3 & 0 \\
6 & 1 & 5 & -1 & 2 & 1 \\
\hline
\end{array}$$
Note that, since $\sum_{i=1}^6a_i=0$, if $1\le\ell\le h$, $1\le s\le6$ and $\ell\equiv s\pmod6$, then $$\sum_{i=1}^\ell x_i=\sum_{i=1}^s a_i.$$ Let $\{u_1,\ldots,u_h\}$ and $\{v_1,\ldots,v_h\}$ be the color classes of $H(h)$. Let $h=6t+4$. We show that $({{\bf x}}, {{\bf x}})$ satisfies the sum rule for ${\lambda}=-1$. By the symmetry, we only need to show this for $u_i$’s. Let $i=6t'+s$ for some $1\le s\le6$. Then $n-i+1=6(t-t')+5-s$. $$\sum_{j:\,v_j\sim u_i}x_j=\sum_{j=1}^{n-i+1}x_j=\sum_{j=1}^{5-s}a_j=-a_s=-x_i.$$
Now, let $h=6t+1$. We show that in this case $({{\bf x}},~{{\bf x}})$ satisfies the sum rule for ${\lambda}=1$. Let $i=6t'+s$ for some $1\le s\le6$. Then $n-i+1=6(t-t')+2-s$. $$\sum_{j:\,v_j\sim u_i}x_j=\sum_{j=1}^{n-i+1}x_j=\sum_{j=1}^{2-s}a_j=a_s=x_i.$$
Now we give another class of counterexamples to Conjecture \[conj\]. For this, let $${\omega}^2+{\omega}-1=0,$$ and $$(b_1,\ldots,b_{10}):=({\omega},-1,0,1,-{\omega},-{\omega},1,0,-1,{\omega}).$$ Let $${{\bf x}}:=(x_1,\ldots,x_h)$$ where $x_i=b_s~\hbox{if}~i\equiv
s\,\hspace{-0.25cm}\pmod{10}.$
\[om\] In any half graph $H(h)$, the vector $({{\bf x}}, {{\bf x}})$ is an eigenvector for ${\lambda}={\omega}$ if $h\equiv7\pmod{10}$ and it is an eigenvector for ${\lambda}=-{\omega}$ if $h\equiv2\pmod{10}$.
From Table \[b\_i\], we observe that for $1\le s\le10$, $$\sum_{i=1}^{8-s}b_i={\omega}b_s~~\hbox{and}~~\sum_{i=1}^{3-s}b_i=-{\omega}b_s,$$ where we consider $8-s$ and $3-s$ modulo 10 as elements of $\{1,\ldots,10\}$.
$${\small\begin{array}{cccccc}
\hline
s &b_s &8-s &\sum_{i=1}^{8-s}b_i & 3-s&\sum_{i=1}^{3-s}b_i \\ \hline
1 & {\omega}&7 &1-{\omega}&2 & {\omega}-1\\
2 & -1& 6&-{\omega}&1 &{\omega}\\
3 &0 & 5& 0 &10 &0\\
4 &1 &4 &{\omega}&9 &-{\omega}\\
5 & -{\omega}& 3&{\omega}-1 & 8& 1-{\omega}\\
6 &-{\omega}&2 & {\omega}-1&7 & 1-{\omega}\\
7 &1 &1 &{\omega}&6 &-{\omega}\\
8 &0 &10 &0 &5 &0\\
9 &-1 & 9 &-{\omega}&4 &{\omega}\\
10 &{\omega}&8 & 1-{\omega}&3 &{\omega}-1 \\ \hline
\end{array}}$$
Note that, since $\sum_{i=1}^{10}b_i=0$, if $1\le\ell\le k$, $1\le s\le10$ and $\ell\equiv s\pmod{10}$, then $$\sum_{i=1}^\ell x_i=\sum_{i=1}^s b_i.$$ Let $k=10t+7$. Then $({{\bf x}}, {{\bf x}})$ satisfies the sum rule for ${\lambda}={\omega}$. Let $i=10t'+s$ for some $1\le s\le10$. Then $n-i+1=10(t-t')+8-s$. $$\sum_{j:\,v_j\sim u_i}x_j=\sum_{j=0}^{n-i+1}x_j=\sum_{j=1}^{8-s}b_j={\omega}b_s={\omega}x_i.$$ Now, let $h=10t+2$. Assume that $i=10t'+s$ for some $1\le s\le10$. Then $n-i+1=6(t-t')+3-s$. $$\sum_{j:\,v_j\sim u_i}x_j=\sum_{j=1}^{n-i+1}x_j=\sum_{j=1}^{3-s}b_j=-{\omega}b_s=-{\omega}x_i.$$ It follows that in this case $({{\bf x}}, {{\bf x}})$ satisfies the sum rule for ${\lambda}=-{\omega}$.
The following two facts deserve to be mentioned:
\(i) Given $({{\bf x}},{{\bf x}})$ as eigenvector of $H(h)$ for ${\lambda}\in\{\pm1,\pm{\omega}\}$, then $({{\bf x}},-{{\bf x}})$ is an eigenvector of $H(h)$ for $-{\lambda}$. This gives more eigenvalues of $H(h)$ with eigenvectors containing zero components.
\(ii) Let ${{\bf x}}$ be an eigenvector for eigenvalue ${\lambda}\neq 0$ of a graph $G$ with ${{\bf x}}_v=0$, for some vertex $v$. If we add a new vertex $u$ with $N(u)=N(v)$ and add a zero component to ${{\bf x}}$ corresponding to $u$, then the new vector is an eigenvector of $H$ for ${\lambda}$. So, we can extend any graph presented in Theorems \[1\] or \[om\] to construct infinitely many more counterexamples for Conjecture \[conj\].
Acknowledgments {#acknowledgments .unnumbered}
===============
The research of the second author was in part supported by a grant from IPM.
[00]{}
A. Alazemi, M. Anelić, S.K. Simić, Eigenvalue location for chain graphs, *Linear Algebra Appl.* [**505**]{} (2016), 194–210. M. Anelić, S.K. Simić, Some notes on the threshold graphs, [*Discrete Math.*]{} [**310**]{} (2010), 2241–2248.
M. Anelić, E. Andrade, D.M. Cardoso, C.M. da Fonseca, S.K. Simić, D.V. Tošić, Some new considerations about double nested graphs, *Linear Algebra Appl.* [**483**]{} (2015), 323–341.
M. Anelić, F. Ashraf, C.M. da Fonseca, S.K. Simić, Vertex types in some lexicographic products of graphs, submitted.
F.K. Bell, D. Cvetković, P. Rowlinson, S.K. Simić, Graphs for which the least eigenvalue is minimal, II, [*Linear Algebra Appl.*]{} [**429**]{} (2008), 2168–2179. A. Bhattacharya, S. Friedland, U.N. Peled, On the first eigenvalue of bipartite graphs, [*Electron. J. Combin.*]{} [**15**]{} (2008), \#R144. A. Brandstädt, V.B. Le, J.P. Spinrad, [*Graph Classes: A Survey*]{}, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. A.E. Brouwer, W.H. Haemers, [*Spectra of Graphs*]{}, Springer, New York, 2012. D.M. Cardoso, M.A.A. Freitas, E.A. Martins, M. Robbiano, Spectra of graphs obtained by a generalization of the join graph operation, *Discrete Math.* [**313**]{} (2013), 733–741. D. Cvetković, P. Rowlinson, S. Simić, [*An Introduction to the Theory of Graph Spectra*]{}, London Mathematical Society Student Texts, 75, Cambridge University Press, Cambridge, 2010. C.D. Godsil, B.D. McKay, A new graph product and its spectrum, *Bull. Austral. Math. Soc.* [**18(1)**]{} (1978), 21–28.
P.L. Hammer, U.N. Peled, X. Sun, Difference graphs, [*Discrete Appl. Math.*]{} [**28**]{} (1990), 35–44. F. Harary, The structure of threshold graphs, [*Riv. Mat. Sci. Econom. Social.*]{} [**2**]{} (1979), 169–172. R.A. Horn, C.R. Johnson, [*Matrix Analysis*]{}, Cambridge University Press, 2013. D. Jacobs, V. Trevisan, F. Tura, Eigenvalues and energy in threshold graphs, [*Linear Algebra Appl.*]{} [**465**]{} (2015), 412–425. P. Manca, On a simple characterisation of threshold graphs, [*Riv. Mat. Sci. Econom. Social.*]{} [**2**]{} (1979), 3–8.
N.V.R. Mahadev, U.N. Peled, [*Threshold Graphs and Related Topics*]{}, Annals of Discrete Mathematics, North�Holland Publishing Co., Amsterdam, 1995. I. Sciriha, S. Farrugia, On the Spectrum of Threshold Graphs, [*ISRN Discrete Mathematics*]{}, [**2011**]{} (2011), Article ID 108509, 21 pages. S.K.Simić, M. Anelić, C.M. da Fonseca, D. Živković, On the multiplicties of eigenvalues of graphs and their vertex deleted subgraphs: old and new results, [*Electron. J. Linear Algebra*]{} [**30**]{} (2015), 85–105.
|
---
author:
- |
C. Zălinescu\
Institute of Mathematics Octav Mayer, Iasi, Romania
title: On constrained optimization problems solved using CDT
---
**Abstract** DY Gao together with some of his collaborators applied his Canonical duality theory (CDT) for solving a class of constrained optimization problems. Unfortunately, in several papers on this subject there are unclear statements, not convincing proofs, or even false results. It is our aim in this work to study rigorously these class of constrained optimization problems in finite dimensional spaces and to discuss several results published in the last ten years.
Introduction
============
In the preface of the book *Canonical Duality Theory. Advances in Mechanics and Mathematics, vol 37, Springer, Cham (2017)*, edited by DY Gao, V Latorre and N Ruan, one says:
Canonical duality theory is a breakthrough methodological theory that can be used not only for modeling complex systems within a unified framework, but also for solving a large class of challenging problems in multidisciplinary fields of engineering, mathematics, and sciences. ...
This theory is composed mainly of
\(1) a canonical dual transformation, which can be used to formulate perfect dual problems without duality gap;
\(2) a complementary-dual principle, which solved the open problem in finite elasticity and provides a unified analytical solution form for general nonconvex/nonsmooth/discrete problems;
\(3) a triality theory, which can be used to identify both global and local optimality conditions and to develop powerful algorithms for solving challenging problems in complex systems."
It is our aim in this work to present rigorously this methodological theory for constrained optimization problems in finite dimensional spaces. It is not the most general framework, but it covers all the situations met in the examples provided in DY Gao and his collaborators’ works on constrained optimization problems in finite dimensions. We also point out some drawbacks and not convincing arguments from some of those papers.
Preliminaries
=============
We consider the following minimization problem with equality and inequality constraints
$(P_{J})$ $\min$ $f(x)$ s.t. $x\in X_{J}$,
where $J\subset\overline{1,m}$, $$X_{J}:=\big\{x\in\mathbb{R}^{n}\mid\left[ \forall j\in J:g_{j}(x)=0\right]
~~\wedge~~\left[ \forall j\in J^{c}:g_{j}(x)\leq0\right] \big\}$$ with $J^{c}:=\overline{1,m}\setminus J$, and $$f(x):=g_{0}(x):=q_{0}(x)+V_{0}\left( \Lambda_{0}(x)\right) ,\quad
g_{j}(x):=q_{j}(x)+V_{j}\left( \Lambda_{j}(x)\right) \quad\left(
x\in\mathbb{R}^{n},~~j\in\overline{1,m}\right) ,$$ $q_{k}$ and $\Lambda_{k}$ being quadratic functions on $\mathbb{R}^{n}$, and $V_{k}\in\Gamma_{sc}:=\Gamma_{sc}(\mathbb{R})$ for $k\in\overline{0,m}$; note that $$X_{J\cup K}=X_{J}\cap X_{K}\quad\forall J,K\subset\overline{1,m}.
\label{r-xjk}$$
Before giving the precise definition of $\Gamma_{sc}$ we recall some notions and results from convex analysis we shall use in the sequel.
Having $h:\mathbb{R}^{p}\rightarrow\overline{\mathbb{R}}:=\mathbb{R}\cup\{-\infty,+\infty\}$, its domain is $\operatorname*{dom}h:=\{y\in
\mathbb{R}^{p}\mid h(y)<\infty\}$; $h$ is proper if $\operatorname*{dom}h\neq\emptyset$ and $h(y)\neq-\infty$ for $y\in\mathbb{R}^{p}$. The Fenchel conjugate $h^{\ast}:\mathbb{R}^{p}\rightarrow\overline{\mathbb{R}}$ of the proper function $h$ is defined by $$h^{\ast}(\sigma):=\sup\{\left\langle y,\sigma\right\rangle -h(y)\mid
y\in\mathbb{R}^{p}\}=\sup\{\left\langle y,\sigma\right\rangle -h(y)\mid
y\in\operatorname*{dom}h\}\quad(\sigma\in\mathbb{R}^{p}),$$ while its subdifferential at $y\in\operatorname*{dom}h$ is $$\partial h(y):=\left\{ \sigma\in\mathbb{R}^{p}\mid\left\langle y^{\prime
}-y,\sigma\right\rangle \leq h(y^{\prime})-h(y)~\forall y^{\prime}\in\mathbb{R}^{p}\right\} ,$$ and $\partial h(y):=\emptyset$ if $y\notin\operatorname*{dom}h$; clearly, $$h(y)+h^{\ast}(\sigma)\geq\left\langle y,\sigma\right\rangle ~~\wedge~~\left[
\sigma\in\partial h(y)\Longleftrightarrow h(y)+h^{\ast}(\sigma)=\left\langle
y,\sigma\right\rangle \quad\forall(y,\sigma)\in\mathbb{R}^{p}\times
\mathbb{R}^{p}\right] . \label{r-fen}$$ The class of proper convex lower semicontinuous (lsc for short) functions $h:\mathbb{R}^{p}\rightarrow\overline{\mathbb{R}}$ is denoted by $\Gamma(\mathbb{R}^{p})$. It is well known that for $h\in\Gamma(\mathbb{R}^{p})$ one has $h^{\ast}\in\Gamma(\mathbb{R}^{p})$, $(h^{\ast})^{\ast}=h$, and $\sigma\in\partial h(y)$ iff $y\in\partial h^{\ast}(\sigma)$; moreover, $\partial h(y)\neq\emptyset$ for every $y\in\operatorname*{ri}(\operatorname*{dom}h)$ and $h(\overline{y})=\inf_{y\in\mathbb{R}^{p}}h(y)$ iff $0\in\partial h(\overline{y})$.
We denote by $\Gamma_{sc}(\mathbb{R}^{p})$ the class of those $h\in
\Gamma(\mathbb{R}^{p})$ which are essentially strictly convex and essentially smooth, that is the class of proper lsc convex functions of Legendre type (see [@Roc:72 Sect. 26]). For $h\in\Gamma_{sc}(\mathbb{R}^{p})$ we have: $h^{\ast}\in\Gamma_{sc}(\mathbb{R}^{p})$, $\operatorname*{dom}\partial
h=\operatorname*{int}(\operatorname*{dom}h)$, and $h$ is differentiable on $\operatorname*{int}(\operatorname*{dom}h)$; moreover, $\nabla
h:\operatorname*{int}(\operatorname*{dom}h)\rightarrow\operatorname*{int}(\operatorname*{dom}h^{\ast})$ is bijective and continuous with $\left(
\nabla h\right) ^{-1}=\nabla h^{\ast}$. Having in view these properties and (\[r-fen\]), for $h\in\Gamma_{sc}(\mathbb{R}^{p})$ and $(y,\sigma
)\in\mathbb{R}^{p}\times\mathbb{R}^{p}$ we have that $$\begin{aligned}
h(y)+h^{\ast}(\sigma)=\left\langle y,\sigma\right\rangle &
\Longleftrightarrow\left[ y\in\operatorname*{int}(\operatorname*{dom}h)~\wedge~\sigma=\nabla h(y)\right] \nonumber\\
& \Longleftrightarrow\left[ \sigma\in\operatorname*{int}(\operatorname*{dom}h^{\ast})~\wedge~y=\nabla h^{\ast}(\sigma)\right] . \label{r-nggs}$$ It follows that $\Gamma_{sc}:=\Gamma_{sc}(\mathbb{R})$ is the class of those $h\in\Gamma(\mathbb{R})$ which are strictly convex and derivable on $\operatorname*{int}(\operatorname*{dom}h)$, assumed to be nonempty; hence $h^{\prime}:\operatorname*{int}(\operatorname*{dom}h)\rightarrow
\operatorname*{int}(\operatorname*{dom}h^{\ast})$ is continuous, bijective and $(h^{\prime})^{-1}=(h^{\ast})^{\prime}$ whenever $h\in\Gamma_{sc}$. The problem $(P_{\overline{1,m}})$ \[resp. $(P_{\emptyset})$\], denoted by $(P_{e})$ \[resp. $(P_{i})$\], is a minimization problem with equality \[resp. inequality\] constraints whose feasible set is $X_{e}:=X_{\overline
{1,m}}$ \[resp. $X_{i}:=X_{\emptyset}$\]. From (\[r-xjk\]) we get $X_{e}\subset X_{J}\subset X_{i}$, each inclusion being generally strict for $J\notin\{\emptyset,\overline{1,m}\}$.
In many examples considered by DY Gao and his collaborators, some functions $g_{k}$ are quadratic, that is $g_{k}=q_{k}$; we set$$Q:=\{k\in\overline{0,m}\mid g_{k}=q_{k}\},\quad Q_{0}:=Q\setminus
\{0\}=\overline{1,m}\cap Q. \label{r-Q}$$ For $k\in Q$ we take $\Lambda_{k}:=0$ and $V_{k}(t):=\tfrac{1}{2}t^{2}$ for $t\in\mathbb{R}$; then clearly $V_{k}^{\ast}=V_{k}\in\Gamma_{sc}$. To be more precise, we take$$q_{k}(x):=\tfrac{1}{2}\left\langle x,A_{k}x\right\rangle -\left\langle
b_{k},x\right\rangle +c_{k}~~\wedge~~\Lambda_{k}(x):=\tfrac{1}{2}\left\langle
x,C_{k}x\right\rangle -\left\langle d_{k},x\right\rangle +e_{k}\quad\left(
x\in\mathbb{R}^{n}\right)$$ with $A_{k},C_{k}\in\mathfrak{S}_{n}$, $b_{k},d_{k}\in\mathbb{R}^{n}$ (seen as column matrices), and $c_{k},e_{k}\in\mathbb{R}$ for $k\in\overline{0,m}$, where $\mathfrak{S}_{n}$ denotes the set of $n\times n$ real symmetric matrices; of course, $c_{0}$ can be taken to be $0$. Clearly, $C_{k}=0\in\mathfrak{S}_{n}$, $b_{k}=0\in\mathbb{R}^{n}$ and $c_{k}=0\in\mathbb{R}$ for $k\in Q$. We use also the notations$$I_{k}:=\operatorname*{dom}V_{k},\quad I_{k}^{\ast}:=\operatorname*{dom}V_{k}^{\ast}\quad(k\in\overline{0,m}),\quad I^{\ast}:={\textstyle\prod
\nolimits_{k=0}^{m}} I_{k}^{\ast}; \label{r-s2}$$ of course, $I_{k}=I_{k}^{\ast}=\mathbb{R}$ for $k\in Q$. In order to simplify the writing, in the sequel $$\lambda_{0}:=\overline{\lambda}_{0}:=1.$$
To the functions $f$ $(=g_{0})$ and $(g_{j})_{j\in\overline{1,m}}$ we associate several sets and functions. The Lagrangian $L$ is defined by $$L:X\times\mathbb{R}^{m}\rightarrow\mathbb{R},\quad L(x,\lambda):=f(x)+\sum
\nolimits_{j=1}^{m}\lambda_{j}g_{j}(x)=\sum\nolimits_{k=0}^{m}\lambda
_{k}\left[ q_{k}(x)+V_{k}\left( \Lambda_{k}(x)\right) \right]
,\label{r-l2}$$ where $\lambda:=(\lambda_{1},...,\lambda_{m})^{T}\in\mathbb{R}^{m}$, and $$\begin{gathered}
X:=\left\{ x\in\mathbb{R}^{n}\mid\forall k\in\overline{0,m}:\Lambda_{k}(x)\in\operatorname*{dom}V_{k}\right\} ={\textstyle\bigcap\nolimits_{k=0}^{m}}\Lambda_{k}^{-1}\left( \operatorname*{dom}V_{k}\right) ,\\
X_{0}:=\left\{ x\in\mathbb{R}^{n}\mid\forall k\in\overline{0,m}:\Lambda
_{k}(x)\in\operatorname*{int}(\operatorname*{dom}V_{k})\right\}
\subset\operatorname*{int}X;\end{gathered}$$ clearly $X_{0}$ is open and $L$ is differentiable on $X_{0}$. Using Gao’s procedure, we consider the extended Lagrangian" $\Xi$ associated to $f$ and $(g_{j})_{j\in\overline{1,m}}$: $$\Xi:\mathbb{R}^{n}\times\mathbb{R}^{1+m}\times I^{\ast}\rightarrow
\mathbb{R},\quad\Xi(x,\lambda,\sigma):=\sum\nolimits_{k=0}^{m}\lambda
_{k}\left[ q_{k}(x)+\sigma_{k}\Lambda_{k}(x)-V_{k}^{\ast}(\sigma_{k})\right]
,\label{r-xi2}$$ where $I^{\ast}$ is defined in (\[r-s2\]) and $\sigma:=(\sigma_{0},\sigma_{1},...,\sigma_{m})\in\mathbb{R}\times\mathbb{R}^{m}=\mathbb{R}^{1+m}$. Clearly, $\Xi(\cdot,\lambda,\sigma)$ is a quadratic function for every fixed $(\lambda,\sigma)\in\mathbb{R}^{m}\times I^{\ast}$.
In the sequel we shall use frequently the following sets associated to $\lambda\in\mathbb{R}^{m}$ and $J\subset\overline{1,m}$:$$\begin{aligned}
M_{\neq}(\lambda) & :=\{j\in\overline{1,m}\mid\lambda_{j}\neq0\},\quad
M_{\neq}^{0}(\lambda):=M_{\neq}(\lambda)\cup\{0\}\\
\Gamma_{J} & :=\big\{\lambda\in\mathbb{R}^{m}\mid\lambda_{j}\geq0~\forall
j\in J^{c}\big\}\supset\mathbb{R}_{+}^{m},\end{aligned}$$ respectively; clearly, $$\Gamma_{\emptyset}=\mathbb{R}_{+}^{m},\quad\Gamma_{\overline{1,m}}=\mathbb{R}^{m},\quad\Gamma_{J\cap K}=\Gamma_{J}\cap\Gamma_{K}\quad\forall
J,K\subset\overline{1,m}.$$
Taking into account the convexity of the functions $V_{k}$ we obtain useful relations between $L$ and $\Xi$ in the next result.
\[lem-xiL\]Let $x\in X$ and $J\subset\overline{1,m}$. Then $$L(x,\lambda)=\sup_{\sigma\in I_{J,Q}}\Xi(x,\lambda,\sigma)\quad\forall
\lambda\in\Gamma_{J\cap Q},\label{r-LXi}$$ where$$I_{J,Q}:={\prod\nolimits_{k=0}^{m}}I_{k}^{\ast\ast}\text{~~with~~}I_{k}^{\ast\ast}:=\left\{
\begin{array}
[c]{ll}\{0\} & \text{if }k\in J\cap Q,\\
I_{k}^{\ast} & \text{if }k\in\overline{0,m}\setminus(J\cap Q),
\end{array}
\right. \label{r-ikss}$$ and $$\sup_{(\lambda,\sigma)\in\Gamma_{J\cap Q}\times I_{J,Q}}\Xi(x,\lambda
,\sigma)=\sup_{\lambda\in\Gamma_{J\cap Q}}L(x,\lambda)=\left\{
\begin{array}
[c]{ll}f(x) & \text{if }x\in X_{J\cap Q},\\
\infty & \text{if }x\in X\setminus X_{J\cap Q}.
\end{array}
\right.$$
Proof. Let us set $K:=J\cap Q=J\cap Q_{0}$. It is convenient to observe that $\Gamma_{K}= {\textstyle\prod\nolimits_{j=1}^{m}} \Gamma_{j}$, where $\Gamma_{j}:=\mathbb{R}$ for $j\in K$ and $\Gamma_{j}:=\mathbb{R}_{+}$ for $j\in K^{c}$; moreover, we set $\Gamma_{0}:=\mathbb{R}_{+}$. Take $x\in X$, $\lambda\in\Gamma_{K}$ and $k\in\overline{0,m}$. Using the fact that $V_{k}^{\ast\ast}=V_{k}$, we have that $$\begin{aligned}
g_{k}(x) & =q_{k}(x)+V_{k}\left( \Lambda_{k}(x)\right) =q_{k}(x)+\sup_{\sigma_{k}\in I_{k}^{\ast}}\left[ \sigma_{k}\Lambda_{k}(x)-V_{k}^{\ast}(\sigma_{k})\right] \\
& =\sup_{\sigma_{k}\in I_{k}^{\ast}}\left[ q_{k}(x)+\sigma_{k}\Lambda
_{k}(x)-V_{k}^{\ast}(\sigma_{k})\right] ,\end{aligned}$$ whence, because $g_{k}(x)\in\mathbb{R}$, $$\mu g_{k}(x)=\sup_{\sigma_{k}\in I_{k}^{\ast}}\mu\left[ q_{k}(x)+\sigma
_{k}\Lambda_{k}(x)-V_{k}^{\ast}(\sigma_{k})\right] \quad\forall\mu
\in\mathbb{R}_{+},~\forall k\in\overline{0,m}. \label{r-lgk}$$ Assume, moreover, that $k\in K$ $(\subset Q_{0}\subset Q)$; then $g_{k}(x)=q_{k}(x)$, and so $$\mu g_{k}(x)=\mu q_{k}(x)=\mu\left[ q_{k}(x)+0\cdot\Lambda_{k}(x)-V_{k}^{\ast}(0)\right] =\sup_{\sigma_{k}\in I_{k}^{\ast\ast}}\mu\left[
q_{k}(x)+\sigma_{k}\Lambda_{k}(x)-V_{k}^{\ast}(\sigma_{k})\right]$$ for every $\mu\in\mathbb{R}$. Therefore, $$\begin{aligned}
L(x,\lambda)= & \sum_{k\in K}\sup_{\sigma_{k}\in\{0\}}\lambda_{k}\left[
q_{k}(x)+\sigma_{k}\Lambda_{k}(x)-V_{k}^{\ast}(\sigma_{k})\right] \\
& +\sum_{k\in\overline{0,m}\setminus K}\sup_{\sigma_{k}\in I_{k}^{\ast}}\lambda_{k}\left[ q_{k}(x)+\sigma_{k}\Lambda_{k}(x)-V_{k}^{\ast}(\sigma
_{k})\right] \\
= & \sum_{k\in\overline{0,m}}\sup_{\sigma_{k}\in I_{k}^{\ast\ast}}\lambda_{k}\left[ q_{k}(x)+\sigma_{k}\Lambda_{k}(x)-V_{k}^{\ast}(\sigma
_{k})\right] \\
= & \sup_{\sigma\in I_{J,Q}}\sum_{k\in\overline{0,m}}\lambda_{k}\left[
q_{k}(x)+\sigma_{k}\Lambda_{k}(x)-V_{k}^{\ast}(\sigma_{k})\right]
=\sup_{\sigma\in I_{J,Q}}\Xi(x,\lambda,\sigma);\end{aligned}$$ hence, (\[r-LXi\]) holds. Using (\[r-LXi\]) we get $$\sup_{(\lambda,\sigma)\in\Gamma_{K}\times I_{J,Q}}\Xi(x,\lambda,\sigma
)=\sup_{\lambda\in\Gamma_{K}}\sup_{\sigma\in I_{J,Q}}\Xi(x,\lambda
,\sigma)=\sup_{\lambda\in\Gamma_{K}}L(x,\lambda). \label{r-LXiJ}$$ Since$$\sup_{\lambda\in\mathbb{R}_{+}}\lambda\alpha=\iota_{\mathbb{R}_{-}}(\alpha),\quad\sup_{\lambda\in\mathbb{R}}\lambda\alpha=\iota_{\{0\}}(\alpha),$$ where the indicator function $\iota_{E}:Z\rightarrow\overline{\mathbb{R}}$ of $E\subset Z$ is defined by $\iota_{E}(z):=0$ for $z\in E$, $\iota
_{E}(z):=+\infty$ for $z\in Z\setminus E$, we get$$\sup_{\lambda\in\Gamma_{K}}L(x,\lambda)=f(x)+\sum_{j\in\overline{1,m}}\sup_{\lambda_{j}\in\Gamma_{j}}\lambda_{j}g_{j}(x)=\left\{
\begin{array}
[c]{ll}f(x) & \text{if }x\in X_{K},\\
\infty & \text{if }x\in X\setminus X_{K}.
\end{array}
\right.$$ Using (\[r-LXiJ\]) and the previous equalities, the conclusion follows. $\square$
Another useful result in this context is the following.
\[lem-gj\]Let $\overline{x}\in\mathbb{R}^{n}$, $\overline{\sigma}\in\mathbb{R}^{m}$ and $k\in\overline{0,m}$. Then $$\begin{aligned}
\Lambda_{k}(\overline{x})=V_{k}^{\ast\prime}(\overline{\sigma}_{k}) &
\Longleftrightarrow\overline{\sigma}_{k}=V_{k}^{\prime}(\Lambda_{k}(\overline{x}))\Longleftrightarrow V_{k}(\Lambda_{k}(\overline{x}))+V_{k}^{\ast}(\overline{\sigma}_{k})=\overline{\sigma}_{k}\Lambda
_{k}(\overline{x})\\
& \Longleftrightarrow g_{k}(\overline{x})=q_{k}(\overline{x})+\overline
{\sigma}_{k}\Lambda_{k}(\overline{x})-V_{k}^{\ast}(\overline{\sigma}_{k})\label{r-gkqk}\\
& \Longrightarrow\left[ \overline{\sigma}_{k}\in\operatorname*{int}(\operatorname*{dom} V_{k}^{\ast})~~\wedge~~\Lambda_{k}(\overline{x})\in\operatorname*{int} (\operatorname*{dom}V_{k})\right] .\end{aligned}$$
In particular, for $k\in Q$, $\Lambda_{k}(\overline{x})=V_{k}^{\ast\prime
}(\overline{\sigma}_{k})$ if and only if $\overline{\sigma}_{k}=0$.
Proof. Because $V_{k}\in\Gamma_{sc}$, (\[r-nggs\]) holds. Since $g_{k}(\overline{x})=q_{k}(\overline{x})+V_{k}(\Lambda_{k}(\overline{x}))$, we obtain that $g_{k}(\overline{x})=q_{k}(\overline{x})+\overline{\sigma}_{k}\Lambda_{k}(\overline{x})-V_{k}^{\ast}(\overline{\sigma}_{k})$ if and only $V_{k}(\Lambda_{k}(\overline{x}))=\overline{\sigma}_{k}\Lambda_{k}(\overline{x})-V_{k}^{\ast}(\overline{\sigma}_{k})$, and so the conclusion follows. The case $k\in Q$ follows immediately. $\square$
\[cor-LXi\]Let $(\overline{x},\overline{\lambda},\overline{\sigma})\in
X\times\mathbb{R}^{m}\times I^{\ast}$ with $\overline{\sigma}_{k}=0$ for $k\in
Q$. If $$\forall k\in M_{\neq}^{0}(\overline{\lambda})\setminus Q:\left[ \Lambda
_{k}(\overline{x})\in\operatorname*{int}(\operatorname*{dom}V_{k})~\wedge~\overline{\sigma}_{k}\in\operatorname*{int}(\operatorname*{dom}V_{k}^{\ast})~\wedge~\overline{\sigma}_{k}=V_{k}^{\prime}\left( \Lambda
_{k}(\overline{x})\right) \right] , \label{r-ml0}$$ then $L(\overline{x},\overline{\lambda})=\Xi(\overline{x},\overline{\lambda
},\overline{\sigma})$. Conversely, if $L(\overline{x},\overline{\lambda})=\Xi(\overline{x},\overline{\lambda},\overline{\sigma})$ and $\overline
{\lambda}\in\Gamma_{Q_{0}}$, then (\[r-ml0\]) holds.
Proof. Assume first that (\[r-ml0\]) holds. Using Lemma \[lem-gj\] we obtain that $V_{k}\left( \Lambda_{k}(\overline{x})\right) =\overline{\sigma
}_{k}\Lambda_{k}(\overline{x})-V_{k}^{\ast}(\overline{\sigma}_{k})$ for $k\in
M_{\neq}^{0}(\overline{\lambda})$, and so $$\overline{\lambda}_{k}\left[ q_{k}(\overline{x})+V_{k}\left( \Lambda
_{k}(\overline{x})\right) \right] =\overline{\lambda}_{k}\left[
q_{k}(\overline{x})+\overline{\sigma}_{k}\Lambda_{k}(\overline{x})-V_{k}^{\ast}(\overline{\sigma}_{k})\right] \quad\forall k\in\overline
{0,m}\label{r-kLXi}$$ because $\overline{\lambda}_{k}=0$ for $k\not \in \overline{0,m}\setminus
M_{\neq}^{0}(\overline{\lambda})$. Then the equality $L(\overline{x},\overline{\lambda})=\Xi(\overline{x},\overline{\lambda},\overline{\sigma})$ follows from de definitions of $L$ and $\Xi$.
Conversely, assume that $L(\overline{x},\overline{\lambda})=\Xi(\overline
{x},\overline{\lambda},\overline{\sigma})$ and $\overline{\lambda}\in
\Gamma_{Q_{0}}$; hence $\overline{\lambda}_{k}\geq0$ for all $k\in Q_{0}^{c}$. Clearly, $g_{k}(\overline{x})=q_{k}(\overline{x})=q_{k}(\overline
{x})+\overline{\sigma}_{k}\Lambda_{k}(\overline{x})-V_{k}^{\ast}(\overline{\sigma}_{k})$ for $k\in Q$. Because $g_{k}(\overline{x})=q_{k}(\overline{x})+V_{k}\left( \Lambda_{k}(\overline{x})\right) \geq
q_{k}(\overline{x})+\overline{\sigma}_{k}\Lambda_{k}(\overline{x})-V_{k}^{\ast}(\overline{\sigma}_{k})$ and $\overline{\lambda}_{k}\geq0$ for all $k\in\{0\}\cup Q_{0}^{c}\supset\overline{0,m}\setminus Q$, from $L(\overline
{x},\overline{\lambda})=\Xi(\overline{x},\overline{\lambda},\overline{\sigma
})$ we obtain that $$\overline{\lambda}_{k}\left[ q_{k}(\overline{x})+V_{k}\left( \Lambda
_{k}(\overline{x})\right) \right] =\overline{\lambda}_{k}\left[
q_{k}(\overline{x})+\overline{\sigma}_{k}\Lambda_{k}(\overline{x})-V_{k}^{\ast}(\overline{\sigma}_{k})\right] \quad\forall k\in\overline
{0,m}\setminus Q;$$ hence $g_{k}(\overline{x})=q_{k}(\overline{x})+\overline{\sigma}_{k}\Lambda_{k}(\overline{x})-V_{k}^{\ast}(\overline{\sigma}_{k})$, that is $V_{k}\left( \Lambda_{k}(\overline{x})\right) =\overline{\sigma}_{k}\Lambda_{k}(\overline{x})-V_{k}^{\ast}(\overline{\sigma}_{k})$, for $k\in
M_{\neq}^{0}(\overline{\lambda})\setminus Q$. Using (\[r-nggs\]) we obtain that (\[r-ml0\]) is verified. $\square$
Let us consider $G:\mathbb{R}^{m}\times\mathbb{R}^{1+m}\rightarrow
\mathfrak{S}_{n}$, $F:\mathbb{R}^{m}\times\mathbb{R}^{1+m}\rightarrow
\mathbb{R}^{n}$, $E:\mathbb{R}^{m}\times\mathbb{R}^{1+m}\rightarrow\mathbb{R}$ defined by $$G(\lambda,\sigma):=\sum_{k=0}^{m}\lambda_{k}(A_{k}+\sigma_{k}C_{k}),~~F(\lambda,\sigma):=\sum_{k=0}^{m}\lambda_{k}(b_{k}+\sigma_{k}d_{k}),~~E(\lambda,\sigma):=\sum_{k=0}^{m}\lambda_{k}(c_{k}+\sigma_{k}e_{k}).
\label{r-cls}$$ Hence, for $(\lambda,\sigma)\in\mathbb{R}^{m}\times I^{\ast}$ we have that $$\Xi(x,\lambda,\sigma)=\tfrac{1}{2}\left\langle x,G(\lambda,\sigma
)x\right\rangle -\left\langle F(\lambda,\sigma),x\right\rangle +E(\lambda
,\sigma)-\sum\nolimits_{k=0}^{m}\lambda_{k}V_{k}^{\ast}(\sigma_{k}).
\label{r-x2}$$
\[rem-afin\]Note that $G$, $F$ and $E$ do not depend on $\sigma_{k}$ for $k\in Q$. Moreover, $G$, $F$ and $E$ are affine functions when $\overline
{1,m}\subset Q$, that is $Q_{0}=\overline{1,m}$.
For $(\lambda,\sigma)\in\mathbb{R}^{m}\times I^{\ast}$ we have that $$\begin{gathered}
\nabla_{x}\Xi(x,\lambda,\sigma)=G(\lambda,\sigma)x-F(\lambda,\sigma
),\quad\nabla_{xx}^{2}\Xi(x,\lambda,\sigma)=G(\lambda,\sigma), \label{r-d1xx2}\\
\nabla_{\lambda}\Xi(x,\lambda,\sigma)=\left( q_{1}(x)+\sigma_{1}\Lambda
_{1}(x)-V_{1}^{\ast}(\sigma_{1}),...,q_{m}(x)+\sigma_{m}\Lambda_{m}(x)-V_{m}^{\ast}(\sigma_{m})\right) ^{T}, \label{r-d1mx2}$$ while for $(x,\lambda,\sigma)\in\mathbb{R}^{n}\times\mathbb{R}^{m}\times\operatorname*{int}I^{\ast}$ we have that $$\nabla_{\sigma}\Xi(x,\lambda,\sigma)=\left( \lambda_{0}\left[ \Lambda
_{0}(x)-V_{0}^{\ast\prime}(\sigma_{0})\right] ,\lambda_{1}\left[ \Lambda
_{1}(x)-V_{1}^{\ast\prime}(\sigma_{1})\right] ,...,\lambda_{m}\left[
\Lambda_{m}(x)-V_{m}^{\ast\prime}(\sigma_{m})\right] \right) ^{T}.
\label{r-d1sx2}$$
Other relations between $L$ and $\Xi$ are provided in the next result.
\[lem-nXiL\]Let $(\overline{x},\overline{\lambda},\overline{\sigma})\in
X_{0}\times\mathbb{R}^{m}\times\operatorname*{int}I^{\ast}$ be such that $\nabla_{\sigma}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})=0$ and $\overline{\sigma}_{k}=0$ for $k\in Q$. Then $L(\overline{x},\overline
{\lambda})=\Xi(\overline{x},\overline{\lambda},\overline{\sigma})$ and $\nabla_{x}L(\overline{x},\overline{\lambda})=\nabla_{x}\Xi(\overline
{x},\overline{\lambda},\overline{\sigma})$. Moreover, for $j\in\overline{1,m}$, $\frac{\partial L}{\partial\lambda_{j}}(\overline{x},\overline{\lambda
})\geq\frac{\partial\Xi}{\partial\lambda_{j}}(\overline{x},\overline{\lambda
},\overline{\sigma})$, with equality if $j\in M_{\neq}(\overline{\lambda})\cup
Q_{0}$; in particular $\nabla_{\lambda}L(\overline{x},\overline{\lambda
})=\nabla_{\lambda}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})$ if $M_{\neq}(\overline{\lambda})\supset Q_{0}^{c}$ $(=\overline{1,m}\setminus Q)$.
Proof. For $k\in M_{\neq}^{0}(\overline{\lambda})$ we have that $\overline
{\lambda}_{k}\neq0$; using (\[r-d1sx2\]) and Lemma \[lem-gj\] we get $\Lambda_{k}(\overline{x})-V_{k}^{\ast\prime}(\overline{\sigma}_{k})=0$, and so $\overline{\sigma}_{k}\in\operatorname*{int}(\operatorname*{dom}V_{k}^{\ast})$, $\Lambda_{k}(\overline{x})\in\operatorname*{int}(\operatorname*{dom}V_{k})$, $\overline{\sigma}_{k}=V_{k}^{\prime}(\Lambda
_{k}(\overline{\overline{x}}))$ for $k\in M_{\neq}^{0}(\overline{\lambda})$. Hence (\[r-ml0\]) is verified, and so $L(\overline{x},\overline{\lambda
})=\Xi(\overline{x},\overline{\lambda},\overline{\sigma})$ by Corollary \[cor-LXi\]. Moreover,$$\begin{aligned}
\nabla_{x}L(\overline{x},\overline{\lambda}) & =\sum\nolimits_{k\in
\overline{0,m}}\overline{\lambda}_{k}\left[ A_{k}\overline{x}-b_{k}+V_{k}^{\prime}(\Lambda_{k}(\overline{x}))(C_{k}\overline{x}-d_{k})\right] \\
& =\sum\nolimits_{k\in M_{\neq}^{0}(\overline{\lambda})}\overline{\lambda
}_{k}\left[ A_{k}\overline{x}-b_{k}+V_{k}^{\prime}(\Lambda_{k}(\overline
{x}))(C_{k}\overline{x}-d_{k})\right] ,\\
\nabla_{x}\Xi(\overline{x},\overline{\lambda},\overline{\sigma}) &
=\sum\nolimits_{k\in\overline{0,m}}\overline{\lambda}_{k}\left[
A_{k}\overline{x}-b_{k}+\overline{\sigma}_{k}(C_{k}\overline{x}-d_{k})\right]
\\
& =\sum\nolimits_{k\in M_{\neq}^{0}(\overline{\lambda})}\overline{\lambda
}_{k}\left[ A_{k}\overline{x}-b_{k}+\overline{\sigma}_{k}(C_{k}\overline
{x}-d_{k})\right] ,\end{aligned}$$ and so $\nabla_{x}L(\overline{x},\overline{\lambda})=\nabla_{x}\Xi
(\overline{x},\overline{\lambda},\overline{\sigma})$. Clearly, from the definitions of $L$, $\Xi$ and the inequality in (\[r-fen\]), we have that $$\frac{\partial L}{\partial\lambda_{j}}(\overline{x},\overline{\lambda})=g_{j}(\overline{x})=q_{j}(\overline{x})+\overline{\sigma}_{j}V_{j}(\Lambda_{j}(\overline{x}))\geq q_{j}(\overline{x})+\overline{\sigma}_{j}\Lambda_{j}(\overline{x})-V_{j}^{\ast}(\overline{\sigma}_{j})=\frac{\partial\Xi}{\partial\lambda_{j}}(\overline{x},\overline{\lambda
},\overline{\sigma}).$$ Using Lemma \[lem-gj\] we obtain that $g_{k}(\overline{x})=q_{k}(\overline{x})+\overline{\sigma}_{k}\Lambda_{k}(\overline{x})-V_{k}^{\ast
}(\overline{\sigma}_{k})$ for $k\in M_{\neq}(\overline{\lambda})\cup Q$ and so $\frac{\partial L}{\partial\lambda_{j}}(\overline{x},\overline{\lambda})=\frac{\partial\Xi}{\partial\lambda_{j}}(\overline{x},\overline{\lambda
},\overline{\sigma})$ for $j\in M_{\neq}(\overline{\lambda})\cup Q_{0}$. $\square$
We consider also the sets $$\begin{gathered}
T_{Q}:=\left\{ (\lambda,\sigma)\in\mathbb{R}^{m}\times I^{\ast}\mid\det
G(\lambda,\sigma)\neq\emptyset\wedge\lbrack\forall k\in Q:\sigma
_{k}=0]\right\} ,\\
T_{Q,\operatorname{col}}:=\left\{ (\lambda,\sigma)\in\mathbb{R}^{m}\times
I^{\ast}\mid F(\lambda,\sigma)\in\operatorname{Im}G(\lambda,\sigma
)\wedge\lbrack\forall k\in Q:\sigma_{k}=0]\right\} \supseteq T_{Q},\\
T_{Q}^{J+}:=\left\{ (\lambda,\sigma)\in T_{Q}\mid\lambda\in\Gamma_{J\cap
Q},~G(\lambda,\sigma)\succ0\right\} ,\\
T_{Q,\operatorname{col}}^{J+}:=\left\{ (\lambda,\sigma)\in
T_{Q,\operatorname{col}}\mid\lambda\in\Gamma_{J\cap Q},~G(\lambda
,\sigma)\succeq0\right\} \supseteq T_{Q}^{J+},\end{gathered}$$ as well as the sets$$T:=T_{\emptyset},\quad T_{\operatorname{col}}:=T_{\emptyset,\operatorname{col}},\quad T^{+}:=T_{\emptyset}^{\emptyset+},\quad T_{\operatorname{col}}^{+}:=T_{\emptyset,\operatorname{col}}^{\emptyset+};$$ in general $T_{Q}^{J+}$ and $T_{Q,\operatorname{col}}^{J+}$ are not convex, unlike their corresponding sets $Y^{+}$, $Y_{\operatorname{col}}^{+}$ and $S^{+}$, $S_{\operatorname{col}}^{+}$ from [@Zal:18b] and [@Zal:18c], respectively. However, taking into account Remark \[rem-afin\], $T_{\operatorname{col}}$, $T_{Q}^{J+}$ and $T_{Q,\operatorname{col}}^{J+}$ are convex whenever $Q_{0}=\overline{1,m}$. In the present context it is natural (in fact necessary) to take $\lambda\in\Gamma_{Q_{0}}$. As in [@Zal:18b] and [@Zal:18c], we consider the (dual objective) function $$D:T_{\operatorname{col}}\rightarrow\mathbb{R},\quad D(\lambda,\sigma
):=\Xi(x,\lambda,\sigma)\text{ with }G(\lambda,\sigma)x=F(\lambda,\sigma);$$ $D$ is well defined by [@Zal:18b Lem. 1 (ii)]. Consider $$\xi:T\rightarrow\mathbb{R}^{n},\quad\xi(\lambda,\sigma):=G(\lambda
,\sigma)^{-1}F(\lambda,\sigma). \label{r-xls}$$ For $(\lambda,\sigma)\in T$ we obtain that$$\begin{aligned}
D(\lambda,\sigma) & =\Xi(G(\lambda,\sigma)^{-1}F(\lambda,\sigma
),\lambda,\sigma)=\Xi(\xi(\lambda,\sigma),\lambda,\sigma)\nonumber\\
& =-\tfrac{1}{2}\left\langle F(\lambda,\sigma),G(\lambda,\sigma
)^{-1}F(\lambda,\sigma)\right\rangle +E(\lambda,\sigma)-\sum\nolimits_{k=0}^{m}\lambda_{k}V_{k}^{\ast}(\sigma_{k}). \label{r-p2d}$$ Taking into account the second formula in (\[r-d1xx2\]), we have that $\Xi(\cdot,\lambda,\sigma)$ is \[strictly\] convex for $(\lambda,\sigma)\in
T_{\operatorname{col}}^{+}$ $[(\lambda,\sigma)\in T^{+}]$, and so $$D(\lambda,\sigma)=\min_{x\in\mathbb{R}^{n}}\Xi(x,\lambda,\sigma)\quad
\forall(\lambda,\sigma)\in T_{\operatorname{col}}\text{ such that }G(\lambda,\sigma)\succeq0, \label{r-pd2}$$ the minimum being attained uniquely at $\xi(\lambda,\sigma)$ when, moreover, $G(\lambda,\sigma)\succ0$.
\[p-perfdual\]Let $(\overline{x},\overline{\lambda},\overline{\sigma})\in\mathbb{R}^{n}\times\mathbb{R}^{m}\times I^{\ast}$ be such that $\nabla_{x}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})=0$, $\frac{\partial\Xi}{\partial\sigma_{0}}(\overline{x},\overline{\lambda
},\overline{\sigma})=0$, and $\left\langle \overline{\lambda},\nabla_{\lambda
}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})\right\rangle =0$. Then $(\overline{\lambda},\overline{\sigma})\in T_{\operatorname{col}}$ and$$f(\overline{x})=\Xi(\overline{x},\overline{\lambda},\overline{\sigma
})=D(\overline{\lambda},\overline{\sigma}). \label{r-px2pd}$$
Proof. Because $\nabla_{x}\Xi(\overline{x},\overline{\lambda},\overline
{\sigma})=0$, $(\overline{\lambda},\overline{\sigma})\in T_{\operatorname{col}}$ and the second equality in (\[r-px2pd\]) holds by the definition of $D$. Since $\Lambda_{0}(\overline{x})-V_{0}^{\ast\prime}(\overline{\sigma}_{0})=\frac{\partial\Xi}{\partial\sigma_{0}}(\overline{x},\overline{\lambda
},\overline{\sigma})=0$, we have that $V_{0}\left( \Lambda_{0}(\overline
{x})\right) =\overline{\sigma}_{0}\Lambda_{0}(\overline{x})-V_{0}^{\ast
}(\overline{\sigma}_{0})$ by Lemma \[lem-gj\] for $k:=0$. Therefore, $$f(\overline{x})=q_{0}(\overline{x})+V_{0}\left( \Lambda_{0}(\overline
{x})\right) =q_{0}(\overline{x})+\overline{\sigma}_{0}\Lambda_{0}(\overline{x})-V^{\ast}(\overline{\sigma}_{0}),$$ whence $$\Xi(\overline{x},\overline{\lambda},\overline{\sigma})=\overline{\lambda}_{0}\left[ q_{0}(x)+\sigma_{0}\Lambda_{0}(x)-V_{0}^{\ast}(\sigma_{0})\right]
+\left\langle \overline{\lambda},\nabla_{\lambda}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})\right\rangle =f(\overline{x}).$$ Hence the first equality in (\[r-px2pd\]) holds, too. $\square$
Formula (\[r-px2pd\]) is related to the so-called complimentary-dual principle (see [GaoRuaLat:16]{}, [@GaoRuaLat:17 p. 13]) and sometimes is called the perfect duality formula.
Observe that $T\cap(\mathbb{R}^{m}\times\operatorname*{int}I^{\ast})\subset\operatorname*{int}T$, and for any $\overline{\sigma}\in I^{\ast}$ we have that the set $\{\lambda\in\mathbb{R}^{m}\mid(\lambda,\overline{\sigma
})\in T\}$ is open. Similarly to the computation of $\frac{\partial
D(\lambda)}{\partial\lambda_{j}}$ in [@Zal:18b p. 5], using the expression of $D(\lambda,\sigma)$ in (\[r-p2d\]), we get $$\begin{aligned}
\frac{\partial D(\lambda,\sigma)}{\partial\lambda_{j}} & =\tfrac{1}{2}\left\langle \xi(\lambda,\sigma),(A_{j}+\sigma_{j}C_{j})\xi(\lambda
,\sigma)\right\rangle -\left\langle b_{j}+\sigma_{j}d_{j},\xi(\lambda
,\sigma)\right\rangle +c_{j}+\sigma_{j}e_{j}-V_{j}^{\ast}(\sigma
_{j})\nonumber\\
& =q_{j}\left( \xi(\lambda,\sigma)\right) +\sigma_{j}\Lambda_{j}\left(
\xi(\lambda,\sigma)\right) -V_{j}^{\ast}(\sigma_{j})\quad\forall
j\in\overline{1,m},~~\forall(\lambda,\sigma)\in T, \label{r-d1p2m}$$ and $$\begin{aligned}
\frac{\partial D(\lambda,\sigma)}{\partial\sigma_{k}} & =\lambda_{k}\left[
\tfrac{1}{2}\left\langle \xi(\lambda,\sigma),C_{k}\xi(\lambda,\sigma
)\right\rangle -\left\langle d_{k},\xi(\lambda,\sigma)\right\rangle
+e_{k}-V_{k}^{\ast\prime}(\sigma_{k})\right] \nonumber\\
& =\lambda_{k}\left[ \Lambda_{k}\left( \xi(\lambda,\sigma)\right)
-V_{k}^{\ast\prime}(\sigma_{k})\right] \quad\forall k\in\overline
{0,m},~~\forall(\lambda,\sigma)\in T\cap(\mathbb{R}^{m}\times
\operatorname*{int}I^{\ast}). \label{r-d1p2s}$$
\[lem1\]Let $(\overline{\lambda},\overline{\sigma})\in\left(
\mathbb{R}^{m}\times\operatorname*{int}I^{\ast}\right) \cap T$ and set $\overline{x}:=\xi(\overline{\lambda},\overline{\sigma})$. Then $$\nabla_{x}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})=0~~\wedge
~~\nabla_{\lambda}\Xi(\overline{x},\overline{\lambda},\overline{\sigma
})=\nabla_{\lambda}D(\overline{\lambda},\overline{\sigma})~~\wedge
~~\nabla_{\sigma}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})=\nabla_{\sigma}D(\overline{\lambda},\overline{\sigma}).$$ In particular $(\overline{x},\overline{\lambda},\overline{\sigma})$ is a critical point of $\Xi$ if and only if $(\overline{\lambda},\overline{\sigma
})$ is a critical point of $D$.
Proof. Using (\[r-d1xx2\]) we get $\nabla_{x}\Xi(\overline{x},\overline
{\lambda},\overline{\sigma})=0$. From (\[r-d1p2m\]) and (\[r-d1mx2\]) for $j\in\overline{1,m}$ we get $$\frac{\partial D}{\partial\lambda_{j}}(\overline{\lambda},\overline{\sigma
})=q_{j}(\overline{x})+\overline{\sigma}_{j}\Lambda_{j}(\overline{x})-V_{j}^{\ast}(\overline{\sigma}_{j})=\frac{\partial\Xi}{\partial\lambda_{j}}(\overline{x},\overline{\lambda},\overline{\sigma}),$$ while from (\[r-d1p2s\]) and (\[r-d1sx2\]) for $k\in\overline{0,m}$ we get$$\frac{\partial D}{\partial\sigma_{k}}(\overline{\lambda},\overline{\sigma
})=\overline{\lambda}_{k}\left[ \Lambda_{k}(\overline{x})-V_{k}^{\ast\prime
}(\overline{\sigma}_{k})\right] =\frac{\partial\Xi}{\partial\sigma_{k}}(\overline{x},\overline{\lambda},\overline{\sigma}).$$ The conclusion follows. $\square$
Similarly to [@Zal:18b], we say that $(\overline{x},\overline{\lambda})\in
X_{0}\times\mathbb{R}^{m}$ is a $J$-LKKT point of $L$ if $\nabla
_{x}L(\overline{x},\overline{\lambda})=0$ and $$\big[\forall j\in J^{c}:\overline{\lambda}_{j}\geq0~\wedge~\frac{\partial
L}{\partial\lambda_{j}}(\overline{x},\overline{\lambda})\leq0~\wedge
~\overline{\lambda}_{j}\frac{\partial L}{\partial\lambda_{j}}(\overline
{x},\overline{\lambda})=0\big]~\wedge~\big[\forall j\in J:\frac{\partial
L}{\partial\lambda_{j}}(\overline{x},\overline{\lambda})=0\big],$$ or, equivalently, $$\overline{x}\in X_{J}~~\wedge~~\overline{\lambda}\in\Gamma_{J}~~\wedge
~~\left[ \forall j\in J^{c}:\overline{\lambda}_{j}g_{j}(\overline
{x})=0\right] ;$$ moreover, we say that $\overline{x}\in X_{0}$ is a $J$-LKKT point of $(P_{J})$ if there exists $\overline{\lambda}\in\mathbb{R}^{m}$ such that $(\overline
{x},\overline{\lambda})$ is a $J$-LKKT point of $L$. Inspired by these notions, we say that $(\overline{x},\overline{\lambda},\overline{\sigma})\in\mathbb{R}^{n}\times\mathbb{R}^{m}\times\operatorname*{int}I^{\ast}$ is a $J$-LKKT point of $\Xi$ if $\nabla_{x}\Xi(\overline{x},\overline{\lambda
},\overline{\sigma})=0$, $\nabla_{\sigma}\Xi(\overline{x},\overline{\lambda
},\overline{\sigma})=0$ and $$\big[\forall j\in J^{c}:\overline{\lambda}_{j}\geq0~\wedge~\frac{\partial\Xi
}{\partial\lambda_{j}}(\overline{x},\overline{\lambda},\overline{\sigma})\leq0~\wedge~\overline{\lambda}_{j}\frac{\partial\Xi}{\partial\lambda_{j}}(\overline{x},\overline{\lambda},\overline{\sigma})=0\big]~\wedge
~\big[\forall j\in J:\frac{\partial\Xi}{\partial\lambda_{j}}(\overline
{x},\overline{\lambda},\overline{\sigma})=0\big], \label{r-kkt-x2}$$ and $(\overline{\lambda},\overline{\sigma})\in\left( \mathbb{R}^{m}\times\operatorname*{int}I^{\ast}\right) \cap T$ is a $J$-LKKT point of $D$ if $\nabla_{\sigma}D(\overline{\lambda},\overline{\sigma})=0$ and $$\big[\forall j\in J^{c}:\overline{\lambda}_{j}\geq0~\wedge~\frac{\partial
D}{\partial\lambda_{j}}(\overline{\lambda},\overline{\sigma})\leq
0~\wedge~\overline{\lambda}_{j}\frac{\partial D}{\partial\lambda_{j}}(\overline{\lambda},\overline{\sigma})=0\big]~\wedge~\big[\forall j\in
J:\frac{\partial D}{\partial\lambda_{j}}(\overline{\lambda},\overline{\sigma
})=0\big].$$
In the case in which $J=\emptyset$ we obtain the notions of KKT points for $\Xi$ and $D$. So, $(\overline{x},\overline{\lambda},\overline{\sigma})\in\mathbb{R}^{n}\times\mathbb{R}^{m}\times\operatorname*{int}I^{\ast}$ is a KKT point of $\Xi$ if $\nabla_{x}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})=0$, $\nabla_{\sigma}\Xi(\overline{x},\overline{\lambda
},\overline{\sigma})=0$ and$$\overline{\lambda}\in\mathbb{R}_{+}^{m}~~\wedge~~\nabla_{\lambda}\Xi
(\overline{x},\overline{\lambda},\overline{\sigma})\in\mathbb{R}_{-}^{m}~~\wedge~~\,\left\langle \overline{\lambda},\nabla_{\lambda}\Xi
(\overline{x},\overline{\lambda},\overline{\sigma})\right\rangle =0,
\label{r-k2t-x2i}$$ and $(\overline{\lambda},\overline{\sigma})\in\mathbb{R}^{m}\times
\operatorname*{int}I^{\ast}$ is a KKT point of $D$ if $\nabla_{\sigma
}D(\overline{\lambda},\overline{\sigma})=0$ and$$\overline{\lambda}\in\mathbb{R}_{+}^{m}~~\wedge~~\nabla_{\lambda}D(\overline{\lambda},\overline{\sigma})\in\mathbb{R}_{-}^{m}~~\wedge
~~\,\big\langle\overline{\lambda},\nabla_{\lambda}D(\overline{\lambda
},\overline{\sigma})\big\rangle=0. \label{r-k2t-pdi}$$
\[rem-kkt\]The definition of a KKT point for $\Xi$ is suggested in the proof of [@RuaGao:17 Th. 3] (the same as that of [RuaGao:16]{}). Observe that $(\overline{x},\overline{\lambda},\overline{\sigma
})$ verifying the conditions in (\[r-k2t-x2i\]) is called critical point of $\Xi$ in [@GaoRuaShe:09 p. 477].
\[c-lkkt\]Let $(\overline{\lambda},\overline{\sigma})\in\left(
\mathbb{R}^{m}\times\operatorname*{int}I^{\ast}\right) \cap T$.
*(i)* If $\overline{x}:=\xi(\overline{\lambda},\overline{\sigma})$, then $(\overline{x},\overline{\lambda},\overline{\sigma})$ is a $J$-LKKT point of $\Xi$ if and only if $(\overline{\lambda},\overline{\sigma})\ $is a $J$-LKKT point of $D$.
*(ii)* If $M_{\neq}(\overline{\lambda})=\overline{1,m}$, then $(\overline{x},\overline{\lambda},\overline{\sigma})$ is a $J$-LKKT point of $\Xi$ if and only if $(\overline{x},\overline{\lambda},\overline{\sigma})$ is a critical point of $\Xi$, if and only if $\overline{x}=\xi(\overline{\lambda
},\overline{\sigma})$ and $(\overline{\lambda},\overline{\sigma})\ $is a critical point of $D$.
Proof. (i) is immediate from Lemma \[lem1\], while (ii) is an obvious consequence of (i) and the definitions of the corresponding notions. $\square$
\[rem-skQ\]Taking into account Remark \[rem-afin\], as well as (\[r-d1xx2\]), (\[r-xls\]) and Lemma \[lem1\], the functions $\nabla
_{x}\Xi$, $\xi$, $\nabla_{\sigma}D$ do not depend on $\sigma_{k}$ for $k\in
Q$. Consequently, if $(\overline{x},\overline{\lambda},\overline{\sigma})$ is a $J$-LKKT point of $\Xi$ then $\overline{\sigma}_{k}=0$ for $k\in Q\cap
M_{\neq}(\overline{\lambda})$, and $(\overline{x},\overline{\lambda},\tilde{\sigma})$ is also a $J$-LKKT point of $\Xi$, where $\tilde{\sigma}_{k}:=0$ for $k\in Q$ and $\tilde{\sigma}_{k}:=\overline{\sigma}_{k}$ for $k\in\overline{0,m}\setminus Q$. Conversely, taking into account that $\nabla_{\sigma}D$ does not depend on $\sigma_{k}$ for $k\in Q$, if $(\overline{\lambda},\overline{\sigma})\in T$ is a $J$-LKKT point of $D$ then $(\overline{\lambda},\tilde{\sigma})$ is also a $J$-LKKT point of $D$, where $\tilde{\sigma}_{k}:=0$ for $k\in Q$ and $\tilde{\sigma}_{k}:=\overline
{\sigma}_{k}$ for $k\in\overline{0,m}\setminus Q$.
Having in view the previous remark, without loss of generality, in the sequel we shall assume that $\overline{\sigma}_{k}=0$ for $k\in Q$ when $(\overline{x},\overline{\lambda},\overline{\sigma})\in\mathbb{R}^{n}\times\mathbb{R}^{m}\times\operatorname*{int}I^{\ast}$ is a $J$-LKKT point of $\Xi$, or $(\overline{\lambda},\overline{\sigma})\in T$ is a $J$-LKKT point of $D$.
The main result
================
\[pei\]Let $(\overline{x},\overline{\lambda},\overline{\sigma})\in\mathbb{R}^{n}\times\mathbb{R}^{m}\times\operatorname*{int}I^{\ast}$ be a $J$-LKKT point of $\Xi$ such that $\overline{\sigma}_{k}=0$ for $k\in Q$.
*(i)* Then $\overline{\lambda}\in\Gamma_{J}$, $(\overline{\lambda
},\overline{\sigma})\in T_{Q,\operatorname{col}}$, $\left\langle
\overline{\lambda},\nabla_{\lambda}\Xi(\overline{x},\overline{\lambda
},\overline{\sigma})\right\rangle =0$, $L(\overline{x},\overline{\lambda})=\Xi(\overline{x},\overline{\lambda},\overline{\sigma})$, $\nabla
_{x}L(\overline{x},\overline{\lambda})=0$, and (\[r-px2pd\]) holds.
*(ii)* Moreover, assume that $Q_{0}^{c}\subset M_{\neq}(\overline
{\lambda})$. Then $\nabla_{\lambda}L(\overline{x},\overline{\lambda})=\nabla_{\lambda}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})$, $(\overline{x},\overline{\lambda})$ is a $J$-LKKT point of $L$ and $\overline{x}\in X_{J\cup Q_{0}^{c}}$.
*(iii)* Furthermore, assume that $\overline{\lambda}_{j}>0$ for all $j\in
Q_{0}^{c}$ and $G(\overline{\lambda},\overline{\sigma})\succeq0$. Then $\overline{x}\in X_{J\cup Q_{0}^{c}}\subset X_{J}\subset X_{J\cap Q}$, $(\overline{\lambda},\overline{\sigma})\in T_{Q,\operatorname{col}}^{J+}$, and $$f(\overline{x})=\inf_{x\in X_{J\cap Q}}f(x)=\Xi(\overline{x},\overline
{\lambda},\overline{\sigma})=L(\overline{x},\overline{\lambda})=\sup
_{(\lambda,\sigma)\in T_{Q,\operatorname{col}}^{J+}}D(\lambda,\sigma
)=D(\overline{\lambda},\overline{\sigma}); \label{r-minmaxei}$$ moreover, if $G(\overline{\lambda},\overline{\sigma})\succ0$ then $\overline{x}$ is the unique global solution of problem $(P_{J\cap Q})$.
Proof. (i) Because $(\overline{x},\overline{\lambda},\overline{\sigma})$ is a $J$-LKKT point, from its very definition we have that $\overline{\lambda}\in\Gamma_{J}$, $\left\langle \overline{\lambda},\nabla_{\lambda}\Xi
(\overline{x},\overline{\lambda},\overline{\sigma})\right\rangle =0$, $\nabla_{x}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})=0$ and $\nabla_{\sigma}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})=0$. Using Lemma \[lem-nXiL\] and we obtain that $\nabla_{x}L(\overline
{x},\overline{\lambda})=\nabla_{x}\Xi(\overline{x},\overline{\lambda
},\overline{\sigma})=0$ and $L(\overline{x},\overline{\lambda})=\Xi
(\overline{x},\overline{\lambda},\overline{\sigma})$, while using Proposition \[p-perfdual\] we get $(\overline{\lambda},\overline{\sigma})\in
T_{Q,\operatorname{col}}$ and that (\[r-px2pd\]) holds.
\(ii) Because $Q_{0}^{c}\subset M_{\neq}(\overline{\lambda})$ we get $\nabla_{\lambda}L(\overline{x},\overline{\lambda})=\nabla_{\lambda}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})$ by Lemma \[lem-nXiL\], and so $(\overline{x},\overline{\lambda})$ is a $J$-LKKT point of $L$ because $(\overline{x},\overline{\lambda},\overline{\sigma})$ is a $J$-LKKT point of $\Xi$. Hence $g_{j}(\overline{x})=0$ for $j\in J$, and $\overline{\lambda}_{j}g_{j}(\overline{x})=0$, $g_{j}(\overline{x})\leq0$ for $j\in J^{c}$. Taking into account that $Q_{0}^{c}\subset M_{\neq}(\overline{\lambda})$, the preceding condition shows that $g_{j}(\overline
{x})=0$ for $j\in Q_{0}^{c,}$ and so $\overline{x}\in X_{J\cup Q_{0}^{c}}$.
\(iii) Our hypothesis shows that $Q_{0}^{c}\subset M_{\neq}(\overline{\lambda
})$. From (i) and (ii) we have that $\overline{\lambda}\in\Gamma_{J}$, $(\overline{\lambda},\overline{\sigma})\in T_{Q,\operatorname{col}}$, $\overline{x}\in X_{J\cup Q_{0}^{c}}\subset X_{J}\subset X_{J\cap Q}$; moreover, $\overline{\lambda}\in\Gamma_{J\cap Q}$ because $\overline{\lambda
}_{j}\geq0$ for $j\in J^{c}\cup Q_{0}^{c}=(J\cap Q)^{c}$, and so $(\overline{\lambda},\overline{\sigma})\in T_{Q,\operatorname{col}}^{J+}$. Using now Lemma \[lem-xiL\], obvious inequalities, (\[r-pd2\]), and (i), as well as the obvious inclusion $T_{J,Q\operatorname{col}}^{+}\subset
\Gamma_{J\cap Q}\times I_{J,Q}$ with $I_{J,Q}$ defined in (\[r-ikss\]), we get $$\begin{aligned}
f(\overline{x}) & \geq\inf_{x\in X_{J\cap Q}}f(x)=\inf_{x\in X_{J\cap Q}}\sup_{\lambda\in\Gamma_{J\cap Q}}L(x,\lambda)=\inf_{x\in X_{J\cap Q}}\sup_{(\lambda,\sigma)\in\Gamma_{J\cap Q}\times I_{J,Q}}\Xi(x,\lambda
,\sigma)\\
& \geq\inf_{x\in X_{J\cap Q}}\sup_{(\lambda,\sigma)\in
T_{J,Q\operatorname{col}}^{+}}\Xi(x,\lambda,\sigma)\geq\sup_{(\lambda
,\sigma)\in T_{J,Q\operatorname{col}}^{+}}\inf_{x\in X_{J\cap Q}}\Xi
(x,\lambda,\sigma)\\
& \geq\sup_{(\lambda,\sigma)\in T_{Q\operatorname{col}}^{J+}}\inf
_{x\in\mathbb{R}^{n}}\Xi(x,\lambda,\sigma)=\sup_{(\lambda,\sigma)\in
T_{Q\operatorname{col}}^{J+}}D(\lambda,\sigma)\geq D(\overline{\lambda
},\overline{\sigma}),\end{aligned}$$ which implies (\[r-minmaxei\]) by (i).
Assume, moreover, that $G(\overline{\lambda},\overline{\sigma})\succ0$; hence $(\overline{\lambda},\overline{\sigma})\in T_{Q}^{J+}$. Consider $x\in
X_{J\cap Q}\setminus\{\overline{x}\}$. Using the strict convexity of $\Xi(\cdot,\overline{\lambda},\overline{\sigma})$ and Lemma \[lem-xiL\] we get $f(\overline{x})=\Xi(\overline{x},\overline{\lambda},\overline{\sigma
})<\Xi(x,\overline{\lambda},\overline{\sigma})\leq L(x,\overline{\lambda})\leq
f(x)$. It follows that $\overline{x}$ is the unique global solution of $(P_{J\cap Q})$ \[and $(P_{J})$, too\]. $\square$
The variant of Proposition \[pei\] in which $Q$ is not taken into consideration, that is the case when one does not observe that $V_{k}\circ\Lambda_{k}=0$ for some $k$, is much weaker; however, the conclusions coincide for $Q=\{0\}$.
\[pei-i\]Let $(\overline{x},\overline{\lambda},\overline{\sigma})\in\mathbb{R}^{n}\times\mathbb{R}^{m}\times\operatorname*{int}I$ be a $J$-LKKT point of $\Xi$.
*(i)* Then $\overline{\lambda}\in\Gamma_{J}$, $(\overline{\lambda
},\overline{\sigma})\in T_{\operatorname{col}}$, $\left\langle \overline
{\lambda},\nabla_{\lambda}\Xi(\overline{x},\overline{\lambda},\overline
{\sigma})\right\rangle =0$, $L(\overline{x},\overline{\lambda})=\Xi
(\overline{x},\overline{\lambda},\overline{\sigma})$, $\nabla_{x}L(\overline{x},\overline{\lambda})=0$, and (\[r-px2pd\]) holds.
*(ii)* Assume that $M_{\neq}(\overline{\lambda})=\overline{1,m}$. Then $\nabla_{\lambda}L(\overline{x},\overline{\lambda})=\nabla_{\lambda}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})=0$, whence $(\overline{x},\overline{\lambda},\overline{\sigma})$ is a critical point of $\Xi$, $(\overline{x},\overline{\lambda})$ is a critical point of $L$, and $\overline{x}\in X_{e}\subset X_{J}\subset X_{i}$.
*(iii)* Assume that $\overline{\lambda}\in\mathbb{R}_{++}^{m}$ and $G(\overline{\lambda},\overline{\sigma})\succeq0$. Then $\overline{x}\in
X_{e}$, $(\overline{\lambda},\overline{\sigma})\in T_{\operatorname{col}}^{+}$ and $$f(\overline{x})=\inf_{x\in X_{i}}f(x)=\Xi(\overline{x},\overline{\lambda
},\overline{\sigma})=L(\overline{x},\overline{\lambda})=\sup_{(\lambda
,\sigma)\in T_{\operatorname{col}}^{+}}D(\lambda,\sigma)=D(\overline{\lambda
},\overline{\sigma});$$ moreover, if $G(\overline{\lambda},\overline{\sigma})\succ0$ then $(\overline{\lambda},\overline{\sigma})\in T^{+}$ and $\overline{x}$ is the unique global solution of problem $(P_{i})$.
The remark below refers to the case $Q=\emptyset$. A similar remark (but a bit less dramatic) is valid for $Q_{0}\neq\emptyset$.
\[rem-cdt\]It is worth observing that given the functions $f$, $g_{1}$, ..., $g_{m}$ of type $q+V\circ\Lambda$ with $q,\Lambda$ quadratic functions and $V\in\Gamma_{sc}$, for any choice of $J\subset\overline{1,m}$ one finds the same $\overline{x}$ using Proposition \[pei-i\] (iii). So, in practice, if one wishes to solve one of the problems $(P_{e})$, $(P_{i})$ or $(P_{J})$ using CDT, it is sufficient to find those critical points $(\overline
{x},\overline{\lambda},\overline{\sigma})$ of $\Xi$ such that $\overline
{\lambda}\in\mathbb{R}_{++}^{m}$ and $G(\overline{\lambda},\overline{\sigma
})\succ0$; if we are successful, $\overline{x}\in X_{e}$ and $\overline{x}$ is the unique solution of $(P_{i})$, and so $\overline{x}$ is also solution for all problems $(P_{J})$ with $J\subset\overline{1,m}$; moreover, $(\overline
{\lambda},\overline{\sigma})$ is a global maximizer of $D$ on $T_{\operatorname{col}}^{+}$.
The next example shows that the condition $Q_{0}^{c}\subset M_{\neq}(\overline{\lambda})$ is essential for $\overline{x}$ to be a feasible solution of problem $(P_{J})$; moreover, it shows that, unlike the quadratic case (see [@Zal:18b Prop. 9]), it is not possible to replace $T_{Q\operatorname{col}}^{J+}$ by $\{(\lambda,\sigma)\in T_{\operatorname{col}}\mid\lambda\in\Gamma_{J},~G(\lambda,\sigma)\succeq0\}$ in (\[r-minmaxei\]). The problem is a particular case of the one considered in [LatGao:16]{}, which is very simple, but important in both theoretical study and real-world applications since the constraint is a so-called double-well function, the most commonly used nonconvex potential in physics and engineering sciences \[7\];[^1] more precisely, $q:=1$, $c:=6$, $d:=4$, $e:=2$.
\[ex1\]Let us take $n=m=1$, $J\subset\{1\}$, $q_{0}(x):=\frac{1}{2}x^{2}-6x$, $\Lambda_{1}(x):=\frac{1}{2}x^{2}-4$, $q_{1}(x):=\Lambda_{0}(x):=0$, $V_{0}(t):=V_{1}(t)+2:=\tfrac{1}{2}t^{2}$ for $x,t\in\mathbb{R}$. Then $f(x)=\frac{1}{2}x^{2}-6x$ and $g_{1}(x)=\tfrac{1}{2}\left( \frac{1}{2}x^{2}-4\right) ^{2}-2$. Hence $Q=\{0\}$ (whence $Q_{0}=\emptyset$) and $X_{e}=\{-2\sqrt{3},2\sqrt{3},-2,2\}\subset\lbrack-2\sqrt{3},-2]\cup
\lbrack2,2\sqrt{3}]=X_{i}$. $$\Xi(x;\lambda;\sigma_{0},\sigma)=\tfrac{1}{2}x^{2}-6x-\tfrac{1}{2}\sigma
_{0}^{2}+\lambda\left[ \sigma_{1}\left( \tfrac{1}{2}x^{2}-4\right)
-\tfrac{1}{2}\sigma_{1}^{2}-2\right] .$$ We have that $G(\lambda,\sigma)=1+\lambda\sigma_{1}$, $T_{\operatorname{col}}=T=\{(\lambda,\sigma)\in\mathbb{R}\times\mathbb{R}^{2}\mid1+\lambda\sigma
_{1}\neq0\}$ and $$D(\lambda;\sigma_{0},\sigma_{1})=-\frac{18}{1+\lambda\sigma_{1}}-\tfrac{1}{2}\sigma_{0}^{2}-\lambda\left( \tfrac{1}{2}\sigma_{1}^{2}+4\sigma
_{1}+2\right) .$$ The critical points of $\Xi$ are $\left( 2;-1;(0,-2)\right) $, $\left(
-2;2;(0,-2)\right) $, $\left( 6;0;(0,14+8\sqrt{3})\right) $, $\big(6;0;(0,14+8\sqrt{3})\big)$, $\left( -2\sqrt{3};-\frac{1}{2}\sqrt
{3}-\frac{1}{2};(0,2)\right) $, $\left( 2\sqrt{3};\frac{1}{2}\sqrt{3}-\frac{1}{2};(0,2)\right) $, and so $1+\overline{\lambda}\overline{\sigma
}_{1}\in\{3,-3,1,-\sqrt{3},\sqrt{3}\}$ for $(\overline{x},\overline{\lambda
},\overline{\sigma})$ critical point of $\Xi$, whence $(\overline{\lambda
},\overline{\sigma})$ $(\in T)$ is critical point of $D$ by Lemma \[lem1\]. For $\overline{\lambda}=0$ the corresponding $\overline{x}$ $(=6)$ is not in $X_{i}\supset X_{e}$; in particular, $(\overline{x},\overline{\lambda})$ is not a critical point of $L$. For $\overline{\lambda}\neq0$, Proposition \[pei\] says that $(\overline{x},\overline{\lambda})$ is a critical point of $L$; in particular $\overline{x}\in X_{e}$. For $\overline{\lambda}\in\{2,-\frac{1}{2}\sqrt{3}-\frac{1}{2}\}$, $1+\overline{\lambda}\overline{\sigma}_{1}<0$, and so Proposition \[pei\] says nothing about the optimality of $\overline{x}$ or $(\overline{\lambda},\overline{\sigma})$; in fact, for $\overline{\lambda}=-\frac{1}{2}\sqrt{3}-\frac{1}{2}$, the corresponding $\overline{x}$ $(=-2\sqrt{3})$ is the global maximizer of $f$ on $X_{e}$. For $\overline{\lambda}:=\frac{1}{2}\sqrt{3}-\frac{1}{2}>0$, $1+\overline{\lambda}\overline{\sigma}_{1}=\sqrt{3}>0$, and so Proposition \[pei\] says that $\overline{x}=2\sqrt{3}$ $(\in X_{e})$ is the global solution of $(P_{i})$, and $(\overline{\lambda},\overline{\sigma})=\left(
\frac{1}{2}\sqrt{3}-\frac{1}{2};(0,2)\right) $ is the global maximizer of $D$ on $T_{\operatorname{col}}^{+}=T^{+}=\{(\lambda,\sigma)\in\mathbb{R}_{+}\times\mathbb{R}^{2}\mid1+\lambda\sigma_{1}>0\}$. For $\overline{\lambda}=-1$, $1+\overline{\lambda}\overline{\sigma}_{1}=3>0$, but $(\overline{\lambda
},\overline{\sigma})$ is not a local extremum of $D$, as easily seen taking $\sigma_{0}:=0$, $(\lambda,\sigma_{1}):=(t-1,t-2)$ with $\left\vert
t\right\vert $ sufficiently small.
When $Q=\overline{0,m}$ problem $(P_{J})$ reduces to the quadratic problem with equality and inequality quadratic constraints considered in [@Zal:18b $(P_{J})$], which is denoted here by $(P_{J}^{q})$. Of course, in this case $X=X_{0}=\mathbb{R}^{n}$, and so $$\Xi(x,\lambda,\sigma)=L(x,\lambda)-\tfrac{1}{2}\sum\nolimits_{k=0}^{m}\lambda_{k}\sigma_{k}^{2}\quad(x\in\mathbb{R}^{n},~\lambda\in\mathbb{R}^{m},~\sigma\in\mathbb{R}\times\mathbb{R}^{m})$$ with $\lambda_{0}:=1$. It follows that $$\begin{gathered}
\nabla_{x}\Xi(x,\lambda,\sigma)=\nabla_{x}L(x,\lambda),\quad\nabla_{\sigma}\Xi(x,\lambda,\sigma)=-\left( \lambda_{k}\sigma_{k}\right) _{k\in
\overline{0,m}},\\
\nabla_{\lambda}\Xi(x,\lambda,\sigma)=\nabla_{\lambda}L(x,\lambda)-\tfrac
{1}{2}\left( \sigma_{j}^{2}\right) _{j\in\overline{1,m}}=\left(
q_{j}(x)-\tfrac{1}{2}\sigma_{j}^{2}\right) _{j\in\overline{1,m}}.\end{gathered}$$
Moreover, $G(\lambda,\sigma)=A(\lambda)$, $F(\lambda,\sigma)=b(\lambda)$, $E(\lambda,\sigma)=c(\lambda)$, and so $T=Y\times\mathbb{R}^{1+m}$, $T_{\operatorname{col}}=Y_{\operatorname{col}}\times\mathbb{R}^{1+m}$, $D(\lambda,\sigma)=D(\lambda)-\tfrac{1}{2}\sum_{k=0}^{m}\lambda_{k}\sigma
_{k}^{2}$, where $A(\lambda)$, $b(\lambda)$, $c(\lambda)$, $Y$, $Y_{\operatorname{col}}$, $D$ are introduced in [@Zal:18b]; we set $D_{L}:=D$ in the present case. Applying Proposition \[pei-i\] for this case we get the next result.
\[c-pei-i\]Let $(\overline{x},\overline{\lambda})\in\mathbb{R}^{n}\times\mathbb{R}^{m}$ be a $J$-LKKT point of $L$.
*(i)* Then $\overline{\lambda}\in Y_{\operatorname{col}}^{J}:=Y_{\operatorname{col}}\cap\Gamma_{J}$, $\left\langle \overline{\lambda
},\nabla_{\lambda}L(\overline{x},\overline{\lambda})\right\rangle =0$, and $q_{0}(\overline{x})=L(\overline{x},\overline{\lambda})=D_{L}(\overline
{\lambda})$.
*(ii)* Assume that $M_{\neq}(\overline{\lambda})=\overline{1,m}$. Then $\nabla_{\lambda}L(\overline{x},\overline{\lambda})=0$, and so $(\overline
{x},\overline{\lambda})$ is a critical point of $L$, and $\overline{x}\in
X_{e}\subset X_{J}\subset X_{i}$.
*(iii)* Assume that $\overline{\lambda}\in\mathbb{R}_{++}^{m}$ and $A(\overline{\lambda})\succeq0$. Then $\overline{x}\in X_{e}$, $\overline
{\lambda}\in Y_{\operatorname{col}}^{+}$ and$$q_{0}(\overline{x})=\inf_{x\in X_{i}}q_{0}(x)=L(\overline{x},\overline
{\lambda})=\sup_{\lambda\in Y_{\operatorname{col}}^{i+}}D_{L}(\lambda
)=D_{L}(\overline{\lambda});$$ moreover, if $A(\overline{\lambda})\succ0$ then $\overline{\lambda}\in Y^{i+}$ and $\overline{x}$ is the unique global solution of problem $(P_{i})$.
However, applying Proposition \[pei\] we get assertion (i) and last part of assertion (ii) of [@Zal:18b Prop. 9].
As seen in [@Zal:18b Prop. 9] the most part of the results obtained by DY Gao and his collaborators for quadratic minimization problems are very far from those obtained studying directly those quadratic problems. In this sense it is worth quoting the following remark from the very recent Ruan and Gao’s paper [@RuaGao:17b]:
> “*Remark 1*. As we have demonstrated that by the generalized canonical duality (32), all KKT conditions can be recovered for both equality and inequality constraints. Generally speaking, the nonzero Lagrange multiplier condition for the linear equality constraint is usually ignored in optimization textbooks. But it can not be ignored for nonlinear constraints. It is proved recently \[26\] that the popular augmented Lagrange multiplier method can be used mainly for linear constrained problems. Since the inequality constraint $\mu\not =0$ produces a nonconvex feasible set $\mathcal{E}_{a}^{\ast}$, this constraint can be replaced by either $\mu<0$ or $\mu>0$. But the condition $\mu<0$ is corresponding to $y\circ(y-e_{K})\geq0$, this leads to a nonconvex open feasible set for the primal problem. By the fact that the integer constraints $y_{i}(y_{i}-1)=0$ are actually a special case (boundary) of the boxed constraints $0\leq y_{i}\leq1$, which is corresponding to $y\circ(y-e_{K})\geq0$, we should have $\mu>0$ (see \[8\] and \[12, 16\]). In this case, the KKT condition (43) should be replaced by
>
> $\mu>0,~~y\circ(y-e_{K})\leq0,~~\mu^{T}[y\circ(y-e_{K})]=0.\quad$ (47)
>
> Therefore, as long as $\mu\neq0$ is satisfied, the complementarity condition in (47) leads to the integer condition $y\circ(y-e_{K})=0$. Similarly, the inequality $\tau\neq0$ can be replaced by $\tau>0$."[^2]
In fact the positivity of the Lagrange multipliers $\lambda_{j}$ is needed for recovering the Lagrangian $L$ from $\Xi$ \[see (\[r-lgk\])\], while the non vanishing condition on $\overline{\lambda}_{j}$ is needed to get $L(\overline{x},\overline{\lambda})=\Xi(\overline{x},\overline{\lambda
},\overline{\sigma})$ and $\nabla_{x}L(\overline{x},\overline{\lambda})=\nabla_{x}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})$ when $\nabla_{\sigma}\Xi(\overline{x},\overline{\lambda},\overline{\sigma})=0$, as seen in Lemma \[lem-nXiL\]. Of course, such conditions are not needed in quadratic minimization problems, as observed after Corollary \[c-pei-i\].
Relations with previous results
===============================
In this section we analyze results obtained by DY Gao and his collaborators in papers dedicated to constrained optimization problems. Because the quadratic problems (with quadratic constraints) are discussed in [@Zal:18b], we discuss only those constrained optimization problems with non quadratic objective function or with at least one non quadratic constraint. In the survey paper [@GaoRuaLat:17] (the same as [@GaoRuaLat:16]) there are mentioned the following papers: [@GaoYan:08], [@GaoRuaShe:09], [@LatGao:16], [@MorGao:17] (with its preprint version [@MorGao:12]); besides these papers we add the retracted version [@MorGao:16] of [@MorGao:17], and [@RuaGao:17].
A detailed discussion of [@GaoYan:08] was done in [@VoiZal:11]; we discuss the corrected versions [@MorGao:12]–[@MorGao:17] of [@GaoYan:08] at the end of this section.
The problem considered by Gao, Ruan and Sherali in [@GaoRuaShe:09] is of type $(P_{i})$, that is $J=\emptyset$ with our notation, with $f$ a quadratic function. Taking $q_{j}:=0$ for $j\in
\overline{1,m}$, our problem $(P_{i})$ is a particular case of the problem $(\mathcal{P})$ from [@GaoRuaShe:09]. In this framework, that is $V_{0}(y)=\tfrac{1}{2}y^{2}$, $V_{j}\in\Gamma_{sc}$, $\Lambda_{0}:=q_{j}:=0$ for $j\in\overline{1,m}$, with our notations, we mention only the following result of [@GaoRuaShe:09].
Theorem 2 (Global Optimality Condition)". *Let $(\overline{x},\overline{\lambda},\overline{\sigma})\in
X\times\mathbb{R}_{+}^{m}\times I^{\ast}$ be a critical point of $\Xi$. If $G(\overline{\lambda},\overline{\sigma})\succeq0$, then $(\overline{\lambda
},\overline{\sigma})$ is a global maximizer of $D$ on $T_{\operatorname{col}}^{+}$, $\overline{x}$ is a global minimizer of $f$ on $X_{i}$ and $f(\overline{x})=\min_{x\in X_{i}}f(x)=\max_{(\lambda,\sigma)\in
T_{\operatorname{col}}^{+}}D(\lambda,\sigma)=D(\overline{\lambda},\overline{\sigma}).$*
This theorem is false because in the mentioned conditions $\overline{x}$ is not necessarily in $X_{i}$, as Example \[ex1\] shows. Indeed, $\left( 6;0;(0,14+8\sqrt{3})\right) $ is a critical point of $\Xi$, but $6\notin X_{i}$. It follows that also Theorem 1 (Complementary-Dual Principle)“ and Theorem 3 (Triality Theory)” of [@GaoRuaShe:09] are false because $(\overline{\lambda
},\overline{\sigma})=\left( 0;(0,14+8\sqrt{3})\right) $ is a critical point of $D$ (by Lemma \[lem1\]), but the assertion $\overline
{x}$ is a KKT point of $(\mathcal{P})$" is not true.
It is shown in [@VoiZal:11b Ex. 6] that the double-min or double-max" duality of [@GaoRuaShe:09 Theorem 3 (Triality Theory)], that is its assertion in the case $G(\overline{\lambda},\overline{\sigma
})\prec0$, is also false.
The problem considered by Latorre and Gao in [@LatGao:16] is of type $(P_{J})$ in which $\Lambda_{k}$ are quadratic and $V_{k}$ are differentiable canonical functions“. In our framework (which, apparently, is more restrictive) and with our notations, the following set is used in [@LatGao:16]:$$\mathcal{S}_{0}:=\{\lambda\in\mathbb{R}^{m}\mid\left[ \forall j\in
J:\lambda_{j}\neq0\right] ~\wedge~\left[ \forall j\in J^{c}:\lambda_{j}\geq0\right] \}\subset\Gamma_{J}.$$ The motivation for defining $\mathcal{S}_{0}$ like this is given in the following text from [@LatGao:16 p. 1767]: From the second and third equation in the (10), it is clear that in order to enforce the constrain $h(x)=0$, the dual variables $\mu_{i}$ must be not zero for $i=1,...,p$. *This is a special complementarity condition for equality constrains, generally not mentioned in many textbooks.* However, the implicit constraint $\mu\neq0$ is important in nonconvex optimization. Let $\sigma
_{0}=(\lambda,\mu)$. The dual feasible spaces should be defined as $\mathcal{S}_{0}$ ...”.[^3]
Besides the set $\mathcal{S}_{0}$ mentioned above, the following sets are also considered in [@LatGao:16]: $$\mathcal{S}_{1}:={\textstyle\prod\nolimits_{k=0}^{m}}\operatorname*{dom}V_{k}=I^{\ast},\quad\mathcal{S}_{a}:=T_{\operatorname{col}}\cap\left(
\mathcal{S}_{0}\times\mathcal{S}_{1}\right) ,\quad\mathcal{S}_{a}^{+}:=\left\{ (\lambda,\sigma)\in T^{+}\mid J\subset M_{\neq}(\lambda
)\right\} .$$
In this context the main results of [@LatGao:16] are the following.
Theorem 1 (Complementarity Dual Principle)". *Let $(\overline{x},\overline{\lambda},\overline{\sigma})\in
X\times\mathbb{R}_{+}^{m}\times I^{\ast}$ be a critical point of $\Xi$. Then $\overline{x}$ is a $J$-KKT of $(P_{J})$, $(\overline{\lambda},\overline
{\sigma})$ is a $J$-LKKT point of $D$ and $f(\overline{x})=\Xi(\overline
{x},\overline{\lambda},\overline{\sigma})=D(\overline{\lambda},\overline
{\sigma})$.*
Theorem 2 (Global Optimality Conditions)". *Let $(\overline{x},\overline{\lambda},\overline{\sigma})\in
X\times\mathbb{R}_{+}^{m}\times I^{\ast}$ be a critical point of $\Xi$ with $(\overline{\lambda},\overline{\sigma})\in\mathcal{S}_{a}^{+}$. If $\mathcal{S}_{a}^{+}$ is convex then $(\overline{\lambda},\overline{\sigma})$ is the global maximizer of $D$ on $\mathcal{S}_{a}^{+}$ and $\overline{x}$ is the global minimizer of $f$ on $X_{J}$, that is $f(\overline{x})=\min_{x\in
X_{J}}f(x)=\max_{(\lambda,\sigma)\in\mathcal{S}_{a}^{+}}D(\lambda
,\sigma)=D(\overline{\lambda},\overline{\sigma})$.*
Note first that it is not clear what is meant by $J$-LKKT point of $D$ (called KKT point) in [@LatGao:16 Th. 1] when $(\overline{\lambda
},\overline{\sigma})\notin T$. As in the case of [@GaoRuaShe:09 Th. 1], Example \[ex1\] shows that [@LatGao:16 Th. 1] is false because $\big(6;0;(0,14+8\sqrt{3})\big)$ is a critical point of $\Xi$, but $6\notin
X_{J}$ $(=X_{i})$; even without assuming that $\mathcal{S}_{a}^{+}$ is convex in [@LatGao:16 Th. 2], for the same reason, this theorem is false.
Having in view that there are not nonempty open convex subsets $C\subset
\mathbb{R}^{2}$ such that the mapping $C\ni(u,v)\mapsto uv\in\mathbb{R}$ is convex, the hypothesis that $\mathcal{S}_{a}^{+}$ is convex in the statement of [@LatGao:16 Th. 2] is very strong. Moreover, it is not clear how this hypothesis is used in the proof of [@LatGao:16 Th. 2].[^4]
The results established by Ruan and Gao in Sections 3 of [@RuaGao:16] and [@RuaGao:17] (which are practically the same) refer to $(P_{i})$ in which $q_{k}=0$, $\Lambda_{k}$ are Gâteaux differentiable on their domains and $V_{k}$ are canonical functions" for $k\in\overline{0,m}$. In our framework (which is more restrictive) and with our notations, the following sets are used in [@RuaGao:17]:$$\mathcal{S}_{a}:=T_{\operatorname{col}}\cap\left( \mathbb{R}_{+}^{m}\times
I^{\ast}\right) ,\quad\mathcal{S}_{a}^{+}:=\left\{ (\lambda,\sigma)\in
T^{+}\mid M_{\neq}(\lambda)=\overline{1,m}\right\} .$$
In this context the results of [@RuaGao:16] and [@RuaGao:17] we are interested in are the following.
Theorem 3. *Let $(\overline{x},\overline{\lambda},\overline{\sigma})\in
X\times\mathbb{R}_{+}^{m}\times I^{\ast}$ be a KKT point of $\Xi$. Then $\overline{x}$ is a KKT of $(P_{i})$, $(\overline{\lambda},\overline{\sigma})$ is a KKT point of $D$ and $f(\overline{x})=\Xi(\overline{x},\overline{\lambda
},\overline{\sigma})=D(\overline{\lambda},\overline{\sigma})$.*
Theorem 4. *Let $(\overline{x},\overline{\lambda},\overline{\sigma})\in X\times\mathbb{R}_{+}^{m}\times I^{\ast}$ be a KKT point of $\Xi$ with $(\overline{\lambda},\overline{\sigma})\in\mathcal{S}_{a}^{+}$. If $\mathcal{S}_{a}^{+}$ is convex then $(\overline{\lambda
},\overline{\sigma})$ is a global maximizer of $D$ on $\mathcal{S}_{a}^{+}$ and $\overline{x}$ is a global minimizer of $f$ on $X_{i}$, that is $f(\overline{x})=\min_{x\in X_{i}}f(x)=\max_{(\lambda,\sigma)\in
\mathcal{S}_{a}^{+}}D(\lambda,\sigma)=D(\overline{\lambda},\overline{\sigma})$.*
As in [@LatGao:16 Th. 1], it is not clear what is meant by KKT point of $D$ in [@RuaGao:17 Th. 3] when $(\overline{\lambda},\overline{\sigma})\notin T$. As in the case of [@GaoRuaShe:09 Th. 1], Example \[ex1\] shows that [@RuaGao:17 Th. 3] is false because $\big(6;0;(0,14+8\sqrt{3})\big)$ is a critical point of $\Xi$, hence a KKT point of $\Xi$, but $6\notin X_{i}$. In what concerns [@RuaGao:17 Th. 4], because $M_{\neq}(\lambda)=\overline{1,m}$, $(\overline{x},\overline{\lambda
},\overline{\sigma})$ is a critical point of $\Xi$ and $\overline{x}\in X_{e}$; moreover, in our framework (that is $V_{k}\in\Gamma_{sc}$ for $k\in\overline{0,m}$), this theorem is true without assuming that $\mathcal{S}_{a}^{+}$ is convex. Notice that the proof of [RuaGao:17]{} is not convincing.
Morales-Silva and Gao in [@MorGao:12]–[@MorGao:17] consider the problem $(\mathcal{P})$ of minimizing $\tfrac{1}{2}\left\Vert y-z\right\Vert
^{2}$ for $x:=(y,z)\in\mathcal{Y}_{c}\times\mathcal{Z}_{c}$ with $\mathcal{Y}_{c}:=\left\{ y\in\mathbb{R}^{n}\mid h(y)=0\right\} $ and $\mathcal{Z}_{c}:=\left\{ z\in\mathbb{R}^{n}\mid h(z)=0\right\} $, where $h(y):=\tfrac{1}{2}\left( \left\langle y,Ay\right\rangle -r^{2}\right) $ and $h(z):=\tfrac{1}{2}\alpha\big(\tfrac{1}{2}\left\Vert z-c\right\Vert ^{2}-\eta\big)^{2}-\left\langle f,z-c\right\rangle $; here $A\in\mathfrak{S}_{n}$ is positive definite, $c,f\in\mathbb{R}^{n}$ and $\alpha,\eta,r\in(0,\infty)$ are taken such that $h(z)>0$ for every $z\in\mathcal{Z}_{c}$. Of course, this problem is of type $(P_{e})$ for which Proposition \[pei-i\] applies. Because [@MorGao:12] is the preprint version of [@MorGao:17], we refer mostly to [@MorGao:17] and [@MorGao:16].[^5] In [@MorGao:17] one considers the sets
$\mathcal{S}_{a}=\{(\lambda,\mu,\varsigma)\in\mathbb{R}\times\mathbb{R}\times\mathcal{V}_{a}^{\ast}:(1+\mu\varsigma)(I+\lambda A)-I$ is invertible$\}.\quad\mathcal{(}10\mathcal{)}$
$\mathcal{S}_{a}^{+}=\{(\lambda,\mu,\varsigma)\in\mathcal{S}_{a}:I+\lambda
A\succ0$ and $(1+\mu\varsigma)(I+\lambda A)-I\succ0\}.\quad\mathcal{(}19\mathcal{)}$"
where $\mathcal{V}_{a}^{\ast}:=[-\alpha\eta,\infty)$, and one states the following results:
Theorem 1 (Complementary-dual principle). *If $(\overline{x},\overline{\lambda},\overline{\mu},\overline{\varsigma})$ is a stationary point of $\Xi$ such that $(\overline{\lambda},\overline{\mu
},\overline{\varsigma})\in\mathcal{S}_{a}$ then $\overline{x}$ is a critical point of $(\mathcal{P})$ with $\overline{\lambda}$ and $\overline{\mu}$ its Lagrange multipliers, $(\overline{\lambda},\overline{\mu},\overline{\varsigma
})$ is a stationary point of $\Pi^{d}$ and $\Pi(\overline{x})=L(\overline
{x},\overline{\lambda},\overline{\mu})=\Xi(\overline{x},\overline{\lambda
},\overline{\mu},\overline{\varsigma})=\Pi^{d}(\overline{\lambda},\overline{\mu},\overline{\varsigma}).\quad(17)$*"
Theorem 2. *Suppose that $(\overline{\lambda
},\overline{\mu},\overline{\varsigma})\in\mathcal{S}_{a}^{+}$ is a stationary point of $\Pi^{d}$ with $\overline{\mu}\geq0$. Then $\overline{x}$ defined by (11) is the only global minimizer of $\Pi$ on $\mathcal{X}_{c}$, and $\Pi(\overline{x})=\min_{x\in\mathcal{X}_{c}}\Pi(x)=\max_{(\lambda
,\mu,\varsigma)\in\mathcal{S}_{a}^{+}}\Pi^{d}(\lambda,\mu,\varsigma)=\Pi
^{d}(\overline{\lambda},\overline{\mu},\overline{\varsigma}).\quad
(20)$*
Theorem 2.2 of [@MorGao:12] coincides with [MorGao:17]{}, while in Theorem 2.3 of [@MorGao:12] $\overline{\mu}\geq0$" and Eq. (20) from the statement of [MorGao:17]{} are missing.
In [@MorGao:16], in the context of the problem $(\mathcal{P})$ above, one considers the sets
$\mathcal{S}_{a}=\{(\lambda,\mu,\varsigma)\in\mathbb{R}\times\mathbb{R}\times\mathcal{V}_{a}^{\ast}:\lambda\neq0,~\mu\neq
0,~\det\left[ (1+\mu\varsigma)(I+\lambda A)-I\right] \neq0\}.\quad
\mathcal{(}27\mathcal{)}$
$\mathcal{S}_{c}^{+}=\{(\lambda,\mu,\varsigma)\in\mathcal{S}_{a}:\lambda
>0,\mu>0,~I+\lambda A\succ0$ and $(1+\mu\varsigma)(I+\lambda A)-I\succ
0\}.\quad\mathcal{(}30\mathcal{)}$".
With this new $\mathcal{S}_{a}$, [@MorGao:16 Th. 2] has the same statement as [@MorGao:17 Th. 1]; moreover, replacing $\mathcal{S}_{a}^{+}$ with $\mathcal{S}_{c}^{+}$ in the statement of [@MorGao:17 Th. 2] one gets the statement of [@MorGao:16 Th. 3].
Notice that there is not a proof of the equality $\max
_{(\lambda,\mu,\varsigma)\in\mathcal{S}_{a}^{+}}\Pi^{d}(\lambda,\mu
,\varsigma)=\Pi^{d}(\overline{\lambda},\overline{\mu},\overline{\varsigma})$ in [@MorGao:17], and there is not a proof of the equality $\max
_{(\lambda,\mu,\varsigma)\in\mathcal{S}_{c}^{+}}\Pi^{d}(\lambda,\mu
,\varsigma)=\Pi^{d}(\overline{\lambda},\overline{\mu},\overline{\varsigma})$ in [@MorGao:16]. However, there is a proof" of the equality $\max_{(\lambda,\mu,\varsigma)\in\mathcal{S}_{c}}P^{d}(\lambda
,\mu,\varsigma)=P^{d}(\overline{\lambda},\overline{\mu},\overline{\varsigma})$ from Theorem 2 of [@GaoYan:08] even if $\mathcal{S}_{c}$ defined in [@GaoYan:08 Eq. (16)] includes $\mathcal{S}_{a}^{+}$ defined in [@MorGao:17 Eq. (19)]; see the discussion form [VoiZal:11]{}.[^6]
Setting $g_{1}:=h$ and $g_{2}:=g$, we have that $Q=\{0,1\}$ and $J=\{1,2\}$ in problem $(\mathcal{P})$ of [@MorGao:12]–[@MorGao:17]. Because $\mathcal{Y}_{c}\cap\mathcal{Z}_{c}=\emptyset$ and taking into account [@VoiZal:11 Assertion II, p. 596],[^7] under the hypothesis of [@MorGao:17 Th. 1] one has $\overline{\lambda}\neq0\neq\overline{\mu}$, and so $M_{\neq}(\overline
{\lambda},\overline{\mu})=\{1,2\}$. Using Proposition \[pei-i\] (ii) we obtain that $(\overline{x},\overline{\lambda},\overline{\mu})$ is a critical point of $L$ and [@MorGao:17 Eq. (17)] holds. The conclusion of [@MorGao:16 Th. 3] is obtained using Proposition \[pei-i\] (iii) \[taking into account Corollary \[c-lkkt\] (ii)\]. In what concerns [@MorGao:17 Th. 2], its conclusion follows using Proposition \[pei\] (iii) because the condition \[$\mathcal{Y}_{c}\cap\mathcal{Z}_{c}=\emptyset$ $\wedge$ $\overline{\mu}\geq0$\] imply $\overline{\mu}>0$, and so $Q_{0}^{c}=\{2\}\subset M_{\neq}(\overline{\lambda},\overline{\mu})$.
Below we show that the equality $\max_{(\lambda,\mu,\varsigma
)\in\mathcal{S}_{a}^{+}}\Pi^{d}(\lambda,\mu,\varsigma)=\Pi^{d}(\overline
{\lambda},\overline{\mu},\overline{\varsigma})$ from [MorGao:17]{} is not true. For this consider $n:=1$, $A:=1$, $r:=\alpha
:=\eta:=c:=1$ and $f:=\frac{9}{8}\sqrt{2}$; this is a particular case $(\gamma:=\frac{9}{8}\sqrt{2})$ of the problem $(\mathcal{P})$ considered in [@VoiZal:11]. In this situation (with the calculations and notations from [@VoiZal:11]), the equation $\varsigma^{4}=8\gamma^{2}(\varsigma+1)$ has the solutions $\overline{\varsigma}:=\varsigma_{1}\in(-1,0)$ (and so $\overline{\varsigma}+\gamma>0$), and $\varsigma_{2}=3$. Taking $\overline
{\lambda}:=\frac{\overline{\varsigma}^{2}}{2\gamma}>0$ and $\overline{\mu
}:=\frac{\overline{\varsigma}^{2}}{2\gamma^{2}-\overline{\varsigma}^{3}}>0$, we have that $(\overline{\lambda},\overline{\mu},\overline{\varsigma})$ is a critical point of $D$ $(=\Pi^{d})$; moreover, $1+\overline{\lambda}>0$ and $(1+\overline{\lambda})(1+\overline{\mu}\overline{\varsigma})-1=\overline{\mu
}(\gamma+\overline{\varsigma})>0$, and so $(\overline{\lambda},\overline{\mu
},\overline{\varsigma})\in\mathcal{S}_{a}^{+}$, where $\mathcal{S}_{a}^{+}$ is defined in [@MorGao:17 Eq. (19)]. In fact, in the present case, $$D(\lambda,\mu,\varsigma)=\Pi^{d}(\lambda,\mu,\varsigma)=-\frac{\mu^{2}\left(
\lambda+1\right) \left( \varsigma^{3}+2\varsigma^{2}+\gamma^{2}\right)
+\mu\lambda\left( \varsigma^{2}+\varsigma\lambda+2\varsigma-2\gamma\right)
+\lambda^{2}}{2\left( \lambda+\varsigma\mu+\varsigma\lambda\mu\right) }$$ for all $(\lambda,\mu,\varsigma)\in\mathcal{S}_{a}$. Applying [MorGao:17]{} we must have that $\max_{(\lambda,\mu,\varsigma)\in\mathcal{S}_{a}^{+}}\Pi^{d}(\lambda,\mu,\varsigma)=\Pi^{d}(\overline{\lambda},\overline{\mu},\overline{\varsigma})$. However, this is not possible because $\sup_{(\lambda,\mu,\varsigma)\in\mathcal{S}_{a}^{+}}\Pi^{d}(\lambda
,\mu,\varsigma)=\infty$. Indeed, there exists $\tilde{\varsigma}<0$ such that $\nu:=\tilde{\varsigma}^{3}+2\tilde{\varsigma}^{2}+\gamma^{2}<0$. Then $(\overline{\lambda},\mu,\tilde{\varsigma})\in\mathcal{S}_{a}^{+}$ for every $\mu<0$ because $1+\overline{\lambda}>0$ and $(1+\overline{\lambda})(1+\tilde{\varsigma}\mu)-1\geq(1+\overline{\lambda})-1=\overline{\lambda}>0$. It follows that $$\begin{aligned}
D(\overline{\lambda},\mu,\tilde{\varsigma}) & =-\frac{\mu^{2}\left(
\overline{\lambda}+1\right) \left( \tilde{\varsigma}^{3}+2\tilde{\varsigma
}^{2}+\gamma^{2}\right) +\mu\overline{\lambda}\left( \tilde{\varsigma}^{2}+\tilde{\varsigma}\overline{\lambda}+2\tilde{\varsigma}-2\gamma\right)
+\overline{\lambda}^{2}}{2\left( \overline{\lambda}+\tilde{\varsigma}\mu+\tilde{\varsigma}\overline{\lambda}\mu\right) }\\
& =-\frac{\mu^{2}\left( \overline{\lambda}+1\right) \nu+\mu\overline
{\lambda}\left( \tilde{\varsigma}^{2}+\tilde{\varsigma}\overline{\lambda
}+2\tilde{\varsigma}-2\gamma\right) +\overline{\lambda}^{2}}{2\left(
\overline{\lambda}+\tilde{\varsigma}\mu+\tilde{\varsigma}\overline{\lambda}\mu\right) }\quad\forall\mu<0,\end{aligned}$$ and so $$\begin{aligned}
\sup_{(\lambda,\mu,\varsigma)\in\mathcal{S}_{a}^{+}}\Pi^{d}(\lambda
,\mu,\varsigma) & \geq-\lim_{\mu\rightarrow-\infty}\frac{\mu^{2}\left(
\overline{\lambda}+1\right) \nu+\mu\overline{\lambda}\left( \tilde
{\varsigma}^{2}+\tilde{\varsigma}\overline{\lambda}+2\tilde{\varsigma}-2\gamma\right) +\overline{\lambda}^{2}}{2\left( \left[ \overline{\lambda
}+\left( \overline{\lambda}+1\right) \tilde{\varsigma}\mu\right] \right)
}\\
& =-\lim_{\mu\rightarrow-\infty}\frac{\mu^{2}\left( \overline{\lambda
}+1\right) \nu}{\left( \overline{\lambda}+1\right) \tilde{\varsigma}\mu
}=-\lim_{\mu\rightarrow-\infty}\frac{\nu}{\tilde{\varsigma}}\mu=\infty.\end{aligned}$$
In [@VoiZal:11] we provided an example with $n=2$ for which the solution(s) of problem $(\mathcal{P})$ from [@GaoYan:08] (which clearly always exists) can not be obtained (found) using [@GaoYan:08 Th. 2]; we concluded that the consideration of the function $\Xi$ is useless, at least for the problem studied in \[3\]“.[^8] In [@MorGao:12]–[@MorGao:17] the authors sustain that this is caused by the non uniqueness of the solution of problem $(\mathcal{P})$ from our example, but a solution can be obtained, even in such a case, by perturbation: The combination of the perturbation and the canonical duality theory is an important method for solving nonconvex optimization problems which have more than one global optimal solution (see also \[15\]).”[^9]
In fact, the same example given in [@VoiZal:11] but for $n=1$ shows that even the results from [@MorGao:12]–[@MorGao:17] do not provide the global solution of problem $(\mathcal{P})$. Indeed, as in [VoiZal:11]{}, take $\gamma:=\sqrt{6}/96$; because $n=1$, we have that $c=1\in\mathbb{R}$. Then the critical points of $\Xi$ with $(\lambda
,\mu,\varsigma)\in\mathcal{S}_{a}$ are, as indicated in [VoiZal:11]{}, the following:$$\begin{gathered}
(\overline{x}_{1},\overline{y}_{1},\overline{\lambda}_{1},\overline{\mu}_{1},\overline{\varsigma}_{1}):=\big(1,1+\tfrac{1}{2}\sqrt{6},\tfrac{1}{2}\sqrt{6},\tfrac{48}{13},-\tfrac{1}{4}\big),\\
(\overline{x}_{2},\overline{y}_{2},\overline{\lambda}_{2},\overline{\mu}_{2},\overline{\varsigma}_{2}):=\big(-1,1+\tfrac{1}{2}\sqrt{6},-2-\tfrac{1}{2}\sqrt{6},\tfrac{16}{13}(3+2\sqrt{6}),-\tfrac{1}{4}\big)\\
(\overline{x}_{3},\overline{y}_{3},\overline{\lambda}_{3},\overline{\mu}_{3},\overline{\varsigma}_{3}):=\left(
1,2.603797322,1.603797322,-3.701\,325488,0.2860829239\right) ,\\
(\overline{x}_{4},\overline{y}_{4},\overline{\lambda}_{4},\overline{\mu}_{4},\overline{\varsigma}_{4}):=\left(
-1,2.603797322,-3.603797322,-8.317027781,0.2860829239\right) .\end{gathered}$$
Using Corollary \[c-lkkt\], we have that $(\overline{\lambda}_{i},\overline{\mu}_{i},\overline{\varsigma}_{i})$ with $i\in\overline{1,4}$ are the only critical points of $D$ $(=\Pi^{d})$. For $i\in\{1,3\}$ we have that $(1+\overline{\lambda}_{i})(1+\overline{\mu}_{i}\overline{\varsigma}_{i})-1<0$, while for $i\in\{2,4\}$ we have that $1+\overline{\lambda}_{i}<0$ and so $(\overline{\lambda},\overline{\mu},\overline{\varsigma})\notin
\mathcal{S}_{a}^{+}$ ($\mathcal{S}_{a}^{+}$ defined in [MorGao:12]{} and [@MorGao:17 Eq. (19)]) and $(\overline{\lambda},\overline{\mu},\overline{\varsigma})\notin\mathcal{S}_{c}^{+}$ ($\mathcal{S}_{c}^{+}$ defined in [@MorGao:16 Eq. (30)]). Therefore, the unique solution $(1,1+\tfrac{1}{2}\sqrt{6})$ of problem $(\mathcal{P})$ is not provided by either [@MorGao:12 Th. 2.3], or [@MorGao:17 Th. 2], or [@MorGao:16 Th. 3]. The use of the perturbation method suggested in these papers is useless for this example.
Conclusions
===========
– We provided a rigorous treatment (study) for constrained minimization problems using the Canonical duality theory developed by DY Gao.
– Proposition \[p-perfdual\] shows that the so-called perfect duality holds under quite mild assumptions on the data of the problem; however, in our opinion this formula is not very useful because for the found element $(\overline{x},\overline{\lambda},\overline{\sigma})$, $\overline{x}$ could not be feasible for the primal problem and/or $(\overline{\lambda},\overline{\sigma})$ could not be feasible for the dual problem.
– Proposition \[pei-i\] and Remark \[rem-cdt\] show that even if CDT can be used for equality and/or inequality constrained optimization problems, it is more appropriate for problems with inequality constraints.
– The most important drawback of CDT is that it could find at most those solutions of the primal problem for which all non quadratic constraints are active; even more, the Lagrange multipliers corresponding to non quadratic constraints must be strictly positive.
– Moreover, the solutions found using CDT are among those found using the usual Lagrange multipliers method. Using the extended Lagrangian" $\Xi$ could be useful to decide if the found $\overline{x}$ is a global minimizer of the primal problem.
– The consideration of the dual function $D$ does not seem to be useful for constrained minimization problems with at least one non quadratic constraint because $D$ is not concave, unlike the case of quadratic constraints.
**Acknowledgement** We thank Prof. Marius Durea for reading a previous version of the paper and for his useful remarks.
[99]{}
DY Gao, *On unified modeling, canonical duality-triality theory, challenges and breakthrough in optimization*, arXiv:1605.05534v3 (2016).
DY Gao, V Latorre, N Ruan (eds), *Canonical Duality Theory. Unified Methodology for Multidisciplinary Study*, Advances in Mechanics and Mathematics 37. Cham: Springer (2017).
DY Gao, N Ruan, V Latorre, RETRACTED: *Canonical duality-triality theory: bridge between nonconvex analysis/mechanics and global optimization in complex system*, Mathematics and Mechanics of Solids, 21(3) (2016), NP5–NP36 (see also arXiv:1410.2665).
DY Gao, N Ruan, V Latorre, *Canonical duality-triality theory: bridge between nonconvex analysis/mechanics and global optimization in complex system*, in: DY Gao, V Latorre, N Ruan (eds) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol 37. Springer, Cham, pp. 1–47 (2017).
DY Gao, N Ruan, H Sherali, *Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality*. J Global Optim 45 (2009), 473–497.
DY Gao, WC Yang, *Complete solutions to minimal distance problem between two nonconvex surfaces*, Optimization 57 (2008), 705–714.
V Latorre, DY Gao, *Canonical duality for solving general nonconvex constrained problems*. Optim. Lett. 10 (2016), 1763–1779 (see also arXiv:1310.2014).
D Morales-Silva, DY Gao, *On the minimal distance between two surfaces*, arXiv:1210.1618 \[math.OC\] (2012) (compare with [@MorGao:16] and [@MorGao:17]).
D Morales-Silva, DY Gao, *RETRACTED: On the minimal distance between two non-convex surfaces*, Mathematics and Mechanics of Solids, 21(3) (2016), NP225–NP237 (compare with [@MorGao:12] and [@MorGao:17]).
D Morales-Silva, DY Gao, *On minimal distance between two surfaces*, in: DY Gao, V Latorre, N Ruan (eds) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol 37. Springer, Cham, pp. 359–371 (2017) (compare with [@MorGao:12] and [@MorGao:16]).
RT Rockafellar, *Convex Analysis*, Princeton University Press, N.J., 1972.
N Ruan, DY Gao, *RETRACTED: Canonical duality theory for solving nonconvex/discrete constrained global optimization problems*, Mathematics and Mechanics of Solids, 21(3) (2016), NP194–NP205.
N Ruan, DY Gao, *Canonical Duality theory for solving nonconvex/discrete constrained global optimization problems*, in: DY Gao, V Latorre, N Ruan (eds) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol 37. Springer, Cham, pp. 187–201 (2017).
N Ruan, DY Gao, *Global optimal solution to quadratic discrete programming problem with inequality constraints*, in: DY Gao, V Latorre, N Ruan (eds) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol. 37, pp. 315–338. Springer, Cham (2017).
MD Voisei, C Zalinescu, *A counter-example to ‘minimal distance between two non-convex surfaces’*, Optimization 60 (2011), 593–602.
MD Voisei, C Zalinescu, *Counterexamples to some triality and tri-duality results*, J. Global Optim. 49 (2011), 173–183.
C Zalinescu, *On V. Latorre and D.Y. Gao’s paper Canonical duality for solving general nonconvex constrained problems*, Optim. Lett. 10 (2016), 1781–1787.
C Zalinescu, *On quadratic optimization problems and canonical duality theory,* arXiv:1809.09032v1 (2018).
C Zalinescu, *On unconstrained optimization problems solved using CDT and triality theory,* arXiv:1809.09032v1 (2018).
[^1]: The reference 7\] is Gao, D.Y.: Nonconvex semi-linear problems and canonical duality solutions, in Advances in Mechanics and Mathematics II. In: Gao, D.Y., Ogden R.W. (eds.), pp. 261–311. Kluwer Academic Publishers (2003)".
[^2]: The reference 26\]“ mentioned in [@RuaGao:17b Rem. 1] is the item [@LatGao:16] from our bibliography, the others being the following: 8. Fang, S.C., Gao, D.Y., Sheu, R.L., Wu, S.Y.: Canonical dual approach to solving 0–1 quadratic programming problems. J. Ind. Manag. Optim. 4(4), 125–142 (2008)”, 12. Gao, D.Y.: Solutions and optimality criteria to box constrained nonconvex minimization problem. J. Ind. Manag. Optim. 3(2), 293–304 (2007)“, and 16. Gao, D.Y., Ruan, N.: Solutions to quadratic minimization problems with box and integer constraints. J. Glob. Optim. 47, 463–484 (2010)”, respectively.
[^3]: The emphasized text can be found also in [@GaoRuaLat:16 p. NP26] and [@GaoRuaLat:17 p. 33]. One must also observe that for Latorre and Gao $\mu\neq0$ is equivalent to $\mu_{j}\neq0$ $\forall j=1,...,p$", and $(\lambda,\mu)\in\mathbb{R}^{m\times
p}$ if $\lambda\in\mathbb{R}^{m}$ and $\mu\in\mathbb{R}^{p}.$
[^4]: It is worth quoting DY Gao’s comment from [@Gao:16 p. 19] on our remark from [@Zal:16 p. 1783] that the proof of [@LatGao:16 Th. 2] is not convincing: Regarding the so-called not convincing proof, serious researcher should provide either a convincing proof or a disproof, rather than a complaint. Note that the canonical dual variables $\sigma_{0}$ and $\sigma_{1}$ are in two different levers (scales) with totally different physical units$^{14}$, it is completely wrong to consider $(\sigma_{0},\sigma_{1})$ as one vector and to discuss the concavity of $\Xi_{1}\left( x,(\cdot,\cdot)\right) $ on $\mathcal{S}_{a}^{+}$. The condition $\mathcal{S}_{a}^{+}$ is convex in Theorem 2 \[5\] should be understood in the way that $\mathcal{S}_{a}^{+}$ is convex in $\sigma_{0}$ and $\sigma_{1}$, respectively, as emphasized in Remark 1 \[5\]. Thus, the proof of Theorem 2 given in \[5\] is indeed convincing by simply using the classical saddle min-max duality for $(x,\sigma_{0})$ and $(x,\sigma_{1})$, respectively."
Note 14 from the text above is Let us consider Example 1 in \[5\]. If the unit for $x$ is the meter $(m)$ and for $q$ is $Kg/m$, then the units for the Lagrange multiplier $\mu$ (dual to the constraint $g(x)=\tfrac
{1}{2}(\tfrac{1}{2}x^{2}-d)^{2}-e$) should be $Kg/m^{3}$ and for $\sigma$ (canonical dual to $\Lambda(x)=\tfrac{1}{2}x^{2}$) should be $Kg/m$, respectively, so that each terms in $\Xi_{1}(x,\mu,\sigma)$ make physical sense“; 5\]” is our reference [@LatGao:16].
[^5]: Excepting [@MorGao:17], there are very few differences between the papers published in [@GaoLatRua:17] and those having the same title from the retracted issue of the journal Mathematics and Mechanics of Solids dedicated to CDT.
[^6]: In fact, we did not find a correct proof of the min-max" duality (like Eq. (20) from [@MorGao:17]) in DY Gao’s papers in the case $Q_{0}\neq\emptyset.$
[^7]: Referring to [MorGao:16]{}, which is a reformulation of [@VoiZal:11 Assertion II], the authors say: The following Lemma is well known in mathematical programming (cf. Latorre and Gao \[12\] and Voisei and Zalinescu \[13\])“; the references 12\] and 13\]” are our items [@LatGao:16] and [@VoiZal:11], respectively. Of course, [@MorGao:16 Lem. 1] is not well known in mathematical programming", being very specific to the problem considered in [@GaoYan:08]. The reference [@LatGao:16] is not mentioned in [@MorGao:17] with respect to [MorGao:16]{}.
[^8]: The reference 3\]" is the item [@GaoYan:08] from our bibliography.
[^9]: The text is quoted from [@MorGao:17 p. 370]; here the reference 15\]“ is Wu, C., Gao, D.Y.: Canonical primal-dual method for solving nonconvex minimization problems. In: Gao, D.Y., Latorre, V., Ruan, N. (eds.) Advances in Canonical Duality Theory. Springer, Berlin”. Note that the same text can be found in [@MorGao:12 p. 9] without any reference, as well as in [MorGao:16]{}, where the indicated reference is Wu, C, Li, C, and Gao, DY. Canonical primal-dual method for solving nonconvex minimization problems. arXiv:1212.6492, 2012.“ Observe that the main difference between arXiv:1212.6492 and reference 15\]” of [@MorGao:17] consists in the list of the authors, the content being practically the same.
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